High-order fully actuated system approaches: Part X. Basics of discrete-time systems

Abstract

A basic introduction to high-order fully actuated (HOFA) approaches for discrete-time systems is given. Firstly, it is shown that, different from the continuous-time systems, general dynamical discrete-time systems can be represented by two types of discrete-time HOFA models, namely, the step forward HOFA models and the step backward HOFA models. Secondly, controllers for the step backward HOFA models are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructures. The related problem of feasibility is also discussed. The well-known discrete-time feedback linearisable systems and strict-feedback systems are shown to be equivalent to both the step forward and step backward HOFA models. Finally, a generalised step backward HOFA model containing control vectors of different time instants is proposed and investigated and demonstrated with a type of proposed pseudo feed-forward systems. The contribution in this paper has laid a fundamental basis for discrete-time HOFA approaches. Further analysis and design problems can be naturally established parallel to the continuous-time system case.

Keywords:

• Discrete-time systems
• fully actuated systems
• step forward models
• step backward models
• feasibility

1. Introduction

With the advent and rapid updating of computers and microprocessors in the 1980s, the implementations of control systems are more and more dependent on the digital hardware. This has greatly promoted the development of the digital control theory. In practice, there are mainly two ways to obtain a digital controller. One way is to first synthesise the controller in the continuous-time domain, and then to discretise the controller yielding a discrete-time approximation version of the controller. The other one is to first discretise the system to yield a discrete-time system, and then to design the discrete-time controller for the discrete-time system. The study of the latter one has produced the complete discrete-time control system theory.

1.1. Discrete-time systems

Parallel to the continuous-time system case, researches on the linear and the nonlinear discrete-time systems have received considerable attention and fruitful results have been obtained during the past few decades.

For the general linear discrete-time systems, relatively complete and systematical analysis and design methods have been developed. These include response analysis, internal and input–output stability, controllability and observability, observer design (Rugh, 1996), pole assignment and eigenstructure assignment, linear quadratic regulation, robust control (Amato, 2006; K. Zhou et al., 1996), identification and adaptive control (Landau & Zito, 2007), and input constrained control (B. Zhou et al., 2009).

As for the nonlinear discrete-time systems, due to their complexity and diversity, it is hard to establish a universal control approach for all systems, instead, different control methods are proposed for different kinds of systems. In general, these control methods can be roughly classified into several categories:

• Optimisation-based approaches, such as the discrete-time nonlinear optimal and inverse optimal control approach (Haddad & Chellaboina,2011; Lewis et al., 2012), the discrete-time state-dependent Riccati equation approach (Cimen,2012), the model predictive approach (Pannek & Grüne, 2011), and the adaptive dynamic programming approach (H. Zhang et al., 2012).

• Lyapunov-based constructive approaches, of which the core idea is to find an energy-like (Lyapunov) function and construct a control law such that the ‘energy’ of the closed-loop system is dissipative. This type of approaches mainly include the discrete-time sliding mode approach (Argha et al., 2018), the discrete-time backstepping approach (Y. Zhang et al., 2000), the discrete-time dynamic surface approach (Yoshimura, 2018), and the feedback passivation approach (Zhao & Gupta, 2016).

• Linearisation-based approaches, including the feedback linearisation approach (Grizzle & Kokotovic, 1988), the fuzzy T-S model approach (Kruszewski et al., 2010), the gain scheduling approach, etc.

• Intelligence-based approaches, for example, the iterative learning approach (J. X. Xu, 2009), and the fuzzy logic approach (Abidi & Xu, 2015).

Some other important and interesting topics on the control of discrete-time nonlinear systems are also discussed in the literature. An interesting one is that how much uncertainty can be dealt with by the feedback mechanism. It is deeply discussed by Guo and his coauthors, see for instance, Guo (1997) and Xie and Guo (2000).

1.2. Fully actuated system approaches

Nonlinear control system theories based on state-space approaches have appeared more than half a century, and yet systematic and effective methods are still lacking for the various analysis and design problems. To a certain extent, discrete-time system theories are parallel to the continuous-time system theories. Therefore, certain obstacles encountered in continuous-time systems with state-space approaches generally exist in discrete-time systems.

In the recent two series of papers, Duan (2021h,2021f, 2021g, 2021i, 2020a, 2020b, 2020c), and Duan (2021a, 2021b, 2021c, 2021d, 2021e), a different approach other than the state-space one is proposed for continuous-time dynamical systems, which is termed as the high-order fully actuated (HOFA) system approach. It has been sufficiently shown to be extremely effective and simple in dealing with the control problems of general dynamical nonlinear systems.

As the last one in this HOFA approach series, this paper gives a basic introduction to the HOFA approaches for discrete-time systems. Two main topics are covered, namely, HOFA models for general discrete-time dynamical systems, and control of discrete-time HOFA systems.

For HOFA models of general dynamical discrete-time systems, the linear system case is firstly discussed. Different from the continuous-time case, two types of discrete-time HOFA models are proposed, one is the step forward HOFA model, and the other is the step backward HOFA model. An important fact for continuous-time systems is extended to the discrete-time system case, that is, any controllable linear discrete-time system is equivalent to both a step forward and a step backward HOFA model. Such a fact clearly reflects two aspects, one is the generality of HOFA models, and the other is that HOFA model representation reveals in fact the nature of controllability. Secondly, HOFA models for nonlinear systems are also discussed, both affine and non-affine discrete-time HOFA models are proposed. As in the linear case, the models are composed of step forward and step backward HOFA models. The two types of models are essentially different in that the former is, in nature, a time-delay system with a delay in the control input, while the latter is only a discrete-time with state delays only. A common aspect of the two models is that they coincide with each other in the special case that the orders of the systems are both of 1, which corresponds to a normal state-space model.

For control of discrete-time step backward HOFA models, simple controllers in state feedback form are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructures. However, since a step forward discrete-time HOFA model is in nature a system with a delay in the control input, the control of a step forward HOFA model inevitably involves a prediction procedure and will be addressed in a separate paper.

As examples of the proposed HOFA models, the well-known discrete-time feedback linearisable systems and strict-feedback systems are considered. As a result, these two types of discrete-time systems are shown to possess both a step forward and a step backward HOFA model.

Besides the above-mentioned discrete-time HOFA models, a generalised discrete-time step backward HOFA model is also proposed, which takes into consideration of the effect of the control vector at more time instants. Control of this type of models can be also very easily realised and the closed-loop system resulted in is also a constant and linear one with desired eigenstructures. As an example, a type of pseudo feed-forward systems are proposed and are shown to be representable by this generalised HOFA model.

State-space models concentrate on the state variables and integrate all the independent state variables together. They are a kind of models with the intention and possibility of solving out all the state variables and are therefore convenient for the problems of state solution (response analysis) and estimation (observation, filtering and prediction). Although control problems are also tackled with the state-space approaches, the results are not as satisfactory as desired in the nonlinear system case.

On the contrary, HOFA models, proposed in Duan (2021h, 2020a, 2021c) and this paper, concentrate on the control variables, and have the property of being able to solve out the control vectors. Therefore, they are consequently convenient for control problems. The full-actuation property of a HOFA model allows one to cancel the nonlinearities in the system and eventually obtain a constant linear system with arbitrarily assignable eigenstructure. Technically, this control feature of discrete-time HOFA models proposed in this paper has adequately laid a solid basis for the development of a general HOFA system framework for discrete-time systems. Several aspects can be foreseen as follows:

Firstly, since the closed-loop system is already constant and linear, response analysis and stability analysis no longer stand as a problem. Secondly, arbitrary assignability of the closed-loop eigenstructure reflects exactly the meaning of controllability, hence a general theory on controllability and stabilisability of general dynamical discrete-time systems can be invented parallel to those of the continuous-time systems, as in Duan (2020b, 2021c). Finally, this important control feature also allows one to convert many control design problems into corresponding ones for linear systems. Hence problems of robust control, adaptive control, disturbance rejection,asymptotical signal tracking, and optimal controlcan all be effectively solved as the continuous-timecase (Duan, 2021g, 2021i, 2021a, 2021b,2021d, 2021e).

This paper is organised into eight sections. The next section gives certain symbols and notions used in the paper. In Sections 3 and 4, the discrete-time HOFA models for linear systems and nonlinear systems are proposed, respectively. Section 5 presents the designs of controllers for the proposed discrete-time HOFA systems. As examples, the discrete-time feedback linearisable systems and strict-feedback systems are treated in Section 6, and in Section 7 the proposed HOFA models are further generalised and a type of pseudo feed-forward systems are proposed and analysed, followed by a brief concluding remark in Section 8.

2. Notations

In this section, certain notations used in the paper are explained.

2.1. General notations

In the sequential sections, $I_n$ denotes the identity matrix, $\varnothing$ denotes the null set, $\mathbb{N}$ is the set of natural numbers, $\mathrm{\Omega }\mathrm{\setminus }\mathrm{\Theta }$ represents the complement of the set Θ in set Ω, and ${\mathbb{R}}^n$ and ${\mathbb{R}}^{m\times n}$ denote the spaces of n-dimensional vectors and $m\times n$ dimensional matrices, respectively. ${\mathbb{B}}_{\mathbb{N}}^n$ represents the set of bounded discrete vector functions defined on $\mathbb{N}$. Furthermore, $det(A)$ and $A^{-1}$ denote the determinant and the inverse of a matrix A, respectively.

For $x_i\in {\mathbb{R}}^m,i=1,2,\dots ,n,$ we denote $\begin{array}{r}{x}_{i\sim j}\left(k\right)=\left[\begin{array}{c}{x}_i\left(k\right)\\ x_{i+1}\left(k\right)\\ ⋮\\ x_j\left(k\right)\end{array}\right],i\le j.\end{array}$For $A_i\in {\mathbb{R}}^{m\times m},$ $i=1,2,\dots ,n,$ as in Duan (2021g), the following symbols are used: $\begin{array}{rl}{A}_{0\sim n}& =[A_0{A}_1\cdots A_n],\\ \mathrm{\Psi }(A_{0\sim n})& =\left[\begin{array}{cccc}{A}_0& \cdots & A_{n-1}& A_n\\ I& & & \\ & \ddots & & \\ & & I& 0\end{array}\right].\end{array}$

2.2. Step backward operations

For $x(k)\in {\mathbb{R}}^m,$ it is well-known that the one-step backward operator is denoted by $q^{-1},$ which operates in the following way: $\begin{array}{rl}{q}^{-1}x\left(k\right)& =x(k-1),\\ q^{-i}x\left(k\right)& =x(k-i).\end{array}$For convenience, in this paper, we denote the above operation by the following notation: $\begin{array}{r}{x}^{\lceil i\rceil }\left(k\right)=x(k-i).\end{array}$For $x\in {\mathbb{R}}^m,$ $n,n_i\in \mathbb{N},i=1,2,\dots ,n,$ parallel to those in Duan (2021g), the following symbol is used in the paper: $\begin{array}{r}{x}^{\lceil n_1\sim n_2\rceil }\left(k\right)=\left[\begin{array}{c}{x}^{\lceil n_1\rceil }\left(k\right)\\ x^{\lceil n_1+1\rceil }\left(k\right)\\ ⋮\\ x^{\lceil n_2\rceil }\left(k\right)\end{array}\right],n_1\le n_2.\end{array}$hence $\begin{array}{r}{x}^{\lceil 0\sim n\rceil }\left(k\right)=\left[\begin{array}{c}x\left(k\right)\\ x(k-1)\\ ⋮\\ x(k-n)\end{array}\right].\end{array}$In addition, for $x_i\in {\mathbb{R}}^m,$ $n_0\in \mathbb{N},n_0<n_i,i=1,2,\dots ,n,$ the following symbols are further defined: $\begin{array}{rl}{x}_{i\sim j}^{\lceil n_1\sim n_2\rceil }\left(k\right)& =\left[\begin{array}{c}{x}_i^{\lceil n_1\sim n_2\rceil }\left(k\right)\\ x_{i+1}^{\lceil n_1\sim n_2\rceil }\left(k\right)\\ ⋮\\ x_j^{\lceil n_1\sim n_2\rceil }\left(k\right)\end{array}\right],i\le j,n_1\le n_2,\\ x_p^{\lceil n_p\rceil }\left(k\right){|}_{p=i\sim j}& =\left[\begin{array}{c}{x}_i^{\lceil n_i\rceil }\left(k\right)\\ x_{i+1}^{\lceil n_{i+1}\rceil }\left(k\right)\\ ⋮\\ x_j^{\lceil n_j\rceil }\left(k\right)\end{array}\right],i\le j,\\ x_p^{\lceil n_0\sim n_p\rceil }\left(k\right){|}_{p=i\sim j}& =\left[\begin{array}{c}{x}_i^{\lceil n_0\sim n_i\rceil }\left(k\right)\\ x_{i+1}^{\lceil n_0\sim n_{i+1}\rceil }\left(k\right)\\ ⋮\\ x_j^{\lceil n_0\sim n_j\rceil }\left(k\right)\end{array}\right],i\le j.\end{array}$

2.3. Step forward operations

For $x(k)\in {\mathbb{R}}^m,$ it is well-known that the one-step forward operator is denoted by q, which operates in the following way: $\begin{array}{rl}qx\left(k\right)& =x(k+1),\\ q^{i}x\left(k\right)& =x(k+i).\end{array}$For convenience, in this paper, we denote the above operation by the following notation: $\begin{array}{r}{x}^{\lfloor i\rfloor }\left(k\right)=x(k+i).\end{array}$For $x\in {\mathbb{R}}^m,$ $n,n_i\in \mathbb{N},i=1,2,\dots ,n,$ the following symbol is used in the paper: $\begin{array}{r}{x}^{\lfloor n_1\sim n_2\rfloor }\left(k\right)=\left[\begin{array}{c}{x}^{\lfloor n_2\rfloor }\left(k\right)\\ x^{\lfloor n_2-1\rfloor }\left(k\right)\\ ⋮\\ x^{\lfloor n_1\rfloor }\left(k\right)\end{array}\right],n_1\le n_2,\end{array}$hence $\begin{array}{r}{x}^{\lfloor 0\sim n\rfloor }\left(k\right)=\left[\begin{array}{c}x(k+n)\\ ⋮\\ x(k+1)\\ x\left(k\right)\end{array}\right].\end{array}$In addition, for $x_i\in {\mathbb{R}}^m,$ $n_0\in \mathbb{N},n_0<n_i,i=1,2,\dots ,n,$ the following symbols are also used in the paper: $\begin{array}{rl}{x}_{i\sim j}^{\lfloor n_1\sim n_2\rfloor }\left(k\right)& =\left[\begin{array}{c}{x}_i^{\lfloor n_1\sim n_2\rfloor }\left(k\right)\\ x_{i+1}^{\lfloor n_1\sim n_2\rfloor }\left(k\right)\\ ⋮\\ x_j^{\lfloor n_1\sim n_2\rfloor }\left(k\right)\end{array}\right],i\le j,n_1\le n_2,\\ x_p^{\lfloor n_p\rfloor }\left(k\right){|}_{p=i\sim j}& =\left[\begin{array}{c}{x}_i^{\lfloor n_i\rfloor }\left(k\right)\\ x_{i+1}^{\lfloor n_{i+1}\rfloor }\left(k\right)\\ ⋮\\ x_j^{\lfloor n_j\rfloor }\left(k\right)\end{array}\right],i\le j,\\ x_p^{\lfloor n_0\sim n_p\rfloor }\left(k\right){|}_{p=i\sim j}& =\left[\begin{array}{c}{x}_i^{\lfloor n_0\sim n_i\rfloor }\left(k\right)\\ x_{i+1}^{\lfloor n_0\sim n_{i+1}\rfloor }\left(k\right)\\ ⋮\\ x_j^{\lfloor n_0\sim n_j\rfloor }\left(k\right)\end{array}\right],i\le j.\end{array}$By the above notations, the following basic relations clearly hold: $\begin{array}{r}{x}^{\lfloor i\rfloor }(k-i)=x^{\lceil i\rceil }(k+i)=x\left(k\right),\end{array}$and $\begin{array}{rl}{x}^{\lceil 0\sim n\rceil }(k+n)& =x^{\lfloor 0\sim n\rfloor }\left(k\right),\\ x^{\lfloor 0\sim n\rfloor }(k-n)& =x^{\lceil 0\sim n\rceil }\left(k\right).\end{array}$

3. Linear HOFA models

Corresponding to the following continuous-time linear state-space model (1) $\begin{array}{r}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),\end{array}$(1) a discrete-time linear state-space model is in the form of (2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) where $x\in {\mathbb{R}}^n$ and $u\in {\mathbb{R}}^r$ are the state vector and input vector, respectively, and $A\in {\mathbb{R}}^{n\times n},$ and $B\in {\mathbb{R}}^{n\times r}$ are the coefficient matrices.

For the above first-order state-space systems (1(1) $\begin{array}{r}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),\end{array}$(1) ) and (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ), it is generally not realistic to require $\mathrm{rank}B=r=n$. However, the following proposition shows that, for continuous-time high-order systems, such a condition is not a restriction at all.

Proposition 3.1

The linear system (1(1) $\begin{array}{r}\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right),\end{array}$(1) ) is controllable if and only if it can be converted equivalently into a high-order system in the form of (3) $\begin{array}{r}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]=\left[\begin{array}{c}{L}_1\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ L_2\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ ⋮\\ L_{\eta }\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\end{array}\right]+\hat{B}u,\end{array}$(3) where ${\mu }_p,p=1,2,\dots ,\eta ,$ are a set of integers, $x_p,p=1,2,\dots ,\eta ,$ are a set of state vectors of proper dimensions, $L_p(\cdot ),p=1,2,\dots ,\eta ,$ are a set of linear functions, and $\hat{B}$ is a square upper triangular matrix with diagonal elements all being 1.

The above result is in fact a modified form of the Theorem 2 in Duan (2020b). It can be proven simply following the proof therein.

A system in the form of (3(3) $\begin{array}{r}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]=\left[\begin{array}{c}{L}_1\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ L_2\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ ⋮\\ L_{\eta }\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\end{array}\right]+\hat{B}u,\end{array}$(3) ) is called a linear HOFA system with multiple orders if (4) $\begin{array}{r}\mathrm{rank}\hat{B}=r=n.\end{array}$(4) A typical special continuous-time linear HOFA system is the following single-order one: (5) $\begin{array}{r}{x}^{\left(n\right)}=\sum _{i=0}^{n-1}{A}_i{x}^{\left(i\right)}+Bu,\end{array}$(5) where $A_i\in {\mathbb{R}}^{r\times r},i=1,2,\dots ,n,$ are a set of matrices, and the matrix $B\in {\mathbb{R}}^{r\times r}$ is nonsingular.

3.1. Definitions

3.1.1. HOFA models with single order

Corresponding to the continuous-time linear HOFA model (5(5) $\begin{array}{r}{x}^{\left(n\right)}=\sum _{i=0}^{n-1}{A}_i{x}^{\left(i\right)}+Bu,\end{array}$(5) ), we have the following definition of discrete-time linear HOFA models.

Definition 3.2

Let $A_i\in {\mathbb{R}}^{r\times r},i=0,1,\dots ,n-1,$ be a set of matrices, and the matrix $B\in {\mathbb{R}}^{r\times r}$ be nonsingular. Then, a system in the form of (6) $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{n-i-1}x(k+i)+Bu\left(k\right),\end{array}$(6) is called a step forward HOFA model of order n, while (7) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+Bu\left(k\right),\end{array}$(7) is called a step backward HOFA model of order n.

By our notations, the above step forward HOFA model (6(6) $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{n-i-1}x(k+i)+Bu\left(k\right),\end{array}$(6) ) can be also written in the form of (8) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=\sum _{i=0}^{n-1}{A}_{n-i-1}{x}^{\lfloor i\rfloor }\left(k\right)+Bu\left(k\right),\end{array}$(8) or (9) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=A_{0\sim n-1}{x}^{\lfloor 0\sim n-1\rfloor }\left(k\right)+Bu\left(k\right).\end{array}$(9) Parallelly, the step backward HOFA model (7(7) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+Bu\left(k\right),\end{array}$(7) ) can be also written as (10) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu\left(k\right),\end{array}$(10) or (11) $\begin{array}{r}x(k+1)=A_{0\sim n-1}{x}^{\lceil 0\sim n-1\rceil }\left(k\right)+Bu\left(k\right).\end{array}$(11) Furthermore, if we denote $\begin{array}{r}{\mathrm{\Gamma }}_{\mathrm{c}}=\left[\begin{array}{c}{I}_r\\ 0\\ ⋮\\ 0\end{array}\right],\end{array}$then the step forward HOFA system (9(9) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=A_{0\sim n-1}{x}^{\lfloor 0\sim n-1\rfloor }\left(k\right)+Bu\left(k\right).\end{array}$(9) ), hence (8(8) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=\sum _{i=0}^{n-1}{A}_{n-i-1}{x}^{\lfloor i\rfloor }\left(k\right)+Bu\left(k\right),\end{array}$(8) ), can be written in the following state-space form: (12) $\begin{array}{rl}& x^{\lfloor 0\sim n-1\rfloor }(k+1)\\ & =\mathrm{\Psi }\left(A_{0\sim n-1}\right)x^{\lfloor 0\sim n-1\rfloor }\left(k\right)+{\mathrm{\Gamma }}_{\mathrm{c}}Bu\left(k\right),\end{array}$(12) and the backward HOFA system (11(11) $\begin{array}{r}x(k+1)=A_{0\sim n-1}{x}^{\lceil 0\sim n-1\rceil }\left(k\right)+Bu\left(k\right).\end{array}$(11) ), hence (10(10) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu\left(k\right),\end{array}$(10) ), has a state-space model as (13) $\begin{array}{rl}& x^{\lceil 0\sim n-1\rceil }(k+1)\\ & =\mathrm{\Psi }\left(A_{0\sim n-1}\right)x^{\lceil 0\sim n-1\rceil }\left(k\right)+{\mathrm{\Gamma }}_{\mathrm{c}}Bu\left(k\right).\end{array}$(13) The two types of HOFA systems (8(8) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=\sum _{i=0}^{n-1}{A}_{n-i-1}{x}^{\lfloor i\rfloor }\left(k\right)+Bu\left(k\right),\end{array}$(8) ) and (10(10) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu\left(k\right),\end{array}$(10) ) are essentially different. As a matter of fact, shifting the time index by n−1 steps in the step forward HOFA system (8(8) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=\sum _{i=0}^{n-1}{A}_{n-i-1}{x}^{\lfloor i\rfloor }\left(k\right)+Bu\left(k\right),\end{array}$(8) ), gives (14) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu(k-(n-1)).\end{array}$(14) This differs with the step backward HOFA system (10(10) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu\left(k\right),\end{array}$(10) ) by n−1 steps delay in the control vector. So, step forward HOFA systems are in fact a type of systems with both state delays and an input delay.

3.1.2. HOFA models with multiple orders

In general, corresponding to the continuous-time linear HOFA model (3(3) $\begin{array}{r}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]=\left[\begin{array}{c}{L}_1\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ L_2\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\\ ⋮\\ L_{\eta }\left(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\right)\end{array}\right]+\hat{B}u,\end{array}$(3) ), we can also introduce discrete-time linear HOFA models with multiple orders.

Definition 3.3

The following systems (15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) and (16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) where ${\mu }_p$, $x_p$, $L_p(\cdot ),p=1,2,\dots ,\eta$, and $\hat{B}$ are as stated in Proposition 3.1, are called a step forward HOFA model with multiple orders and a step backward HOFA model with multiple orders, respectively.

Correspondingly, the two types of HOFA models (8(8) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=\sum _{i=0}^{n-1}{A}_{n-i-1}{x}^{\lfloor i\rfloor }\left(k\right)+Bu\left(k\right),\end{array}$(8) ) and (10(10) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_i{x}^{\lceil i\rceil }\left(k\right)+Bu\left(k\right),\end{array}$(10) ) are called HOFA models with single orders.

If we denote $\begin{array}{r}L(\cdot ,k)=\left[\begin{array}{c}{L}_1(\cdot ,k)\\ L_2(\cdot ,k)\\ ⋮\\ L_{\eta }(\cdot ,k)\end{array}\right],\end{array}$then the above step forward and step backward HOFA models (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) and (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ) can be compactly written as (17) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =L(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)+\hat{B}u\left(k\right),\end{array}$(17) and (18) $\begin{array}{r}{x}_{1\sim \eta }(k+1)=L(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+\hat{B}u\left(k\right),\end{array}$(18) respectively.

In contrast to the above HOFA systems with multiple orders, the systems (9(9) $\begin{array}{r}{x}^{\lfloor n-1\rfloor }(k+1)=A_{0\sim n-1}{x}^{\lfloor 0\sim n-1\rfloor }\left(k\right)+Bu\left(k\right).\end{array}$(9) ) and (11(11) $\begin{array}{r}x(k+1)=A_{0\sim n-1}{x}^{\lceil 0\sim n-1\rceil }\left(k\right)+Bu\left(k\right).\end{array}$(11) ) are called HOFA systems with single orders.

3.2. Controllability canonical forms

Corresponding to the above Proposition 3.1, it can also be proven (see the appendix) that the discrete-time controllable state-space system (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ) can be also represented by a HOFA model.

Theorem 3.4

The linear system (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ) is controllable if and only if it can be converted equivalently into both a step forward HOFA system in the form of (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ), and a step backward HOFA system in the form of (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ).

The above theorem is a very important and insightful discovery, which indicates that the step forward and the step backward HOFA models are in fact the controllability canonical forms of the state-space system (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ).

Remark 3.1

The above Theorem 3.4 obviously indicates that a step forward HOFA system in the form of (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) is equivalent to a step backward HOFA system in the form of (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ). While this is certainly not true since the step forward HOFA system in the form of (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) is in essence a time-delay system as revealed above. Such an equivalence would clearly contradict with the Law of Cause and Effect since $z_p^{\lfloor 1\sim {\mu }_p-1\rfloor }(k){|}_{p=1\sim \eta }$ is generally considered to be unknown. The Theorem is somehow misleading due to an implicit reason behind this fact, that is, both the step backward and the step forward systems are converted under the assumption that the state vector $x(k)$ of the controllable linear system (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ) is available. When $x(k)$ is available, so is $z=T(x),$ hence $z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }(k){|}_{p=1\sim \eta }$ is also available. It is this hidden hypothesis that has made this fact holds true. Without this hypothesis, the step forward HOFA system (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) and the step backward HOFA system (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ) can be by no ways equivalent.

Remark 3.2

It is clearly seen that the step forward HOFA system (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) and the step backward HOFA system (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ) are both time-delay systems. Particularly, the step forward HOFA system in the form of (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) is a discrete-time system with an input delay. Therefore, this paper has actually included contents in the scope of discrete-time delay systems. Since the problem of controller design for the step forward HOFA system in the form of (15(15) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(15) ) is essentially a controller design problem for a system with delay in the control vector, it turns out to be much more complicated, and will be investigated in a separatepaper.

To end this section, we finally mention the following two high-order discrete-time systems (self-recursive model) frequently encountered in linear system theory: (19) $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{n-i-1}x(k+i)+\sum _{i=0}^m{B}_{m-i}u(k+i),\end{array}$(19) and (20) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+\sum _{i=0}^m{B}_{i}u(k-i),\end{array}$(20) It can be easily reasoned that, when $B=B_0$ is a square nonsingular matrix, the system (19(19) $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{n-i-1}x(k+i)+\sum _{i=0}^m{B}_{m-i}u(k+i),\end{array}$(19) ) can be converted equivalently into the following step forward HOFA model: $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{i}x(k+i)+B{u}^{\mathrm{\prime }}(k+m),\end{array}$where (21) $\begin{array}{rl}{u}^{\mathrm{\prime }}(k+m)& =B^{-1}\sum _{i=0}^m{B}_{m-i}u(k+i)\\ & =u(k+m)+\sum _{i=0}^{m-1}{B}^{-1}{B}_{m-i}u(k+i),\end{array}$(21) while the system (20(20) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+\sum _{i=0}^m{B}_{i}u(k-i),\end{array}$(20) ) can be converted equivalently into the following step backward HOFA model: $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+B{u}^{\mathrm{\prime }\mathrm{\prime }}\left(k\right),\end{array}$where (22) $\begin{array}{rl}{u}^{\mathrm{\prime }\mathrm{\prime }}\left(k\right)& =B^{-1}\sum _{i=0}^m{B}_{i}u(k-i)\\ & =u\left(k\right)+\sum _{i=1}^m{B}^{-1}{B}_{i}u(k-i).\end{array}$(22) In both cases, the original control vectors in the two systems can be uniquely solved through (21(21) $\begin{array}{rl}{u}^{\mathrm{\prime }}(k+m)& =B^{-1}\sum _{i=0}^m{B}_{m-i}u(k+i)\\ & =u(k+m)+\sum _{i=0}^{m-1}{B}^{-1}{B}_{m-i}u(k+i),\end{array}$(21) ) and (22(22) $\begin{array}{rl}{u}^{\mathrm{\prime }\mathrm{\prime }}\left(k\right)& =B^{-1}\sum _{i=0}^m{B}_{i}u(k-i)\\ & =u\left(k\right)+\sum _{i=1}^m{B}^{-1}{B}_{i}u(k-i).\end{array}$(22) ), respectively.

4. Nonlinear HOFA models

Different from the linear case, nonlinear systems are divided into affine ones and non-affine ones.

4.1. Affine HOFA models

The state-space representation of a general nonlinear (affine) continuous-time system is (23) $\begin{array}{r}\dot{x}\left(t\right)=f(x\left(t\right),t)+B(x\left(t\right),t)u\left(t\right),\end{array}$(23) and that of a general nonlinear (affine) discrete-time system is (24) $\begin{array}{r}x(k+1)=f(x\left(k\right),k)+B(x\left(k\right),k)u\left(k\right),\end{array}$(24) where $f(\cdot )\in {\mathbb{R}}^n$ and $B(\cdot )\in {\mathbb{R}}^{n\times r}$ are nonlinear functions. For such general state-space systems, we have $r\le n,$ but with the equality case being seldom valid. However, in the continuous-time case, it has been shown that, under very mild conditions, the first-order system (23(23) $\begin{array}{r}\dot{x}\left(t\right)=f(x\left(t\right),t)+B(x\left(t\right),t)u\left(t\right),\end{array}$(23) ) can be converted equivalently to a HOFA system.

Concretely, in Duan (2021h, 2021c, 2021d, 2021e), the following general continuous-time affine HOFA model is proposed and investigated: (25) $\begin{array}{rl}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]& =\left[\begin{array}{c}{f}_1(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ f_2(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ ⋮\\ f_{\eta }(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\end{array}\right]\\ & +B(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)u,\end{array}$(25) where ${\mu }_p,p=1,2,\dots ,\eta ,$ are a set of distinct integers, $x_p\in {\mathbb{R}}^{r_p},p=1,2,\dots ,\eta ,$ are a set of state vectors, with $r_p,p=1,2,\dots ,\eta ,$ being a set of distinct integers satisfying (26) $\begin{array}{r}{r}_1+r_2+\cdots +r_{\eta }=r.\end{array}$(26) Further, $f_p(\cdot )\in {\mathbb{R}}^{r_p},p=1,2,\dots ,\eta ,$ are a set of nonlinear vector functions, and $B(\cdot )\in {\mathbb{R}}^{r\times r}$ is a nonlinear matrix function which is nonsingular for $t\ge 0$ and $x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta }\in \mathrm{\Omega }$, with Ω being some set with dimension not less than 1.

Clearly, in the case of $\eta =1$, the above HOFA model (25(25) $\begin{array}{rl}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]& =\left[\begin{array}{c}{f}_1(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ f_2(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ ⋮\\ f_{\eta }(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\end{array}\right]\\ & +B(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)u,\end{array}$(25) ) reduces to the form of (27) $\begin{array}{r}{x}^{\left(n\right)}=f(x^{(0\sim n-1)},t)+B(x^{(0\sim n-1)},t)u,\end{array}$(27) which is proposed in Duan (2020a) (see, also Duan,2021h). When nonlinear uncertainties, unknown parameters and disturbances are added to this basic HOFA model, robust and adaptive control as well as disturbance rejection control have been investigated in Duan (2021g, 2021i, 2021a, 2021b). While with the general HOFA model (25(25) $\begin{array}{rl}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]& =\left[\begin{array}{c}{f}_1(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ f_2(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ ⋮\\ f_{\eta }(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\end{array}\right]\\ & +B(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)u,\end{array}$(25) ), the problems of optimal control, generalised PID control and model reference tracking control are investigated in Duan (2021d, 2021e).

Corresponding to the continuous-time HOFA system (27(27) $\begin{array}{r}{x}^{\left(n\right)}=f(x^{(0\sim n-1)},t)+B(x^{(0\sim n-1)},t)u,\end{array}$(27) ), we have the following discrete-time step forward HOFA model with single order (28) $\begin{array}{rl}x(k+n)& =f(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)\\ & +B(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)u\left(k\right),\end{array}$(28) and the following step backward HOFA model with single order (29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) Shifting the time index in the step forward HOFA system (28(28) $\begin{array}{rl}x(k+n)& =f(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)\\ & +B(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)u\left(k\right),\end{array}$(28) ) back by n−1 steps, gives (30) $\begin{array}{rl}& x(k+1)\\ & =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k-(n-1))\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k-\left(n-1\right))u(k-\left(n-1\right)).\end{array}$(30) Comparing this with (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ), we can clearly tell the great difference between the step forward and the step backward models. Like the linear case, the step forward nonlinear HOFA system (28(28) $\begin{array}{rl}x(k+n)& =f(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)\\ & +B(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)u\left(k\right),\end{array}$(28) ) is in nature a system with a control delay, while the step backward HOFA system (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ) is a systems with state delays only.

Corresponding to the general continuous-time HOFA system (25(25) $\begin{array}{rl}\left[\begin{array}{c}{x}_1^{\left({\mu }_1\right)}\\ x_2^{\left({\mu }_2\right)}\\ ⋮\\ x_{\eta }^{\left({\mu }_{\eta }\right)}\end{array}\right]& =\left[\begin{array}{c}{f}_1(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ f_2(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\\ ⋮\\ f_{\eta }(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)\end{array}\right]\\ & +B(x_p^{(0\sim {\mu }_p-1)}{|}_{p=1\sim \eta },t)u,\end{array}$(25) ), we can also propose the following discrete-time step forward HOFA system with multiple orders (31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) and the following step backward HOFA system with multiple orders (32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) where ${\mu }_p,p=1,2,\dots ,\eta ,$ are a set of distinct integers, $x_p\in {\mathbb{R}}^{r_p},p=1,2,\dots ,\eta ,$ are a set of state vectors, with $r_p,p=1,2,\dots ,\eta ,$ being a set of distinct integers satisfying (26(26) $\begin{array}{r}{r}_1+r_2+\cdots +r_{\eta }=r.\end{array}$(26) ). Further, $f_p(\cdot )\in {\mathbb{R}}^{r_p},p=1,2,\dots ,\eta ,$ are a set of nonlinear vector functions, and $B(\cdot )\in {\mathbb{R}}^{r\times r}$ is a nonlinear matrix function of its variables.

Denote (33) $\begin{array}{r}f(\cdot ,k)=\left[\begin{array}{c}{f}_1(\cdot ,k)\\ f_2(\cdot ,k)\\ ⋮\\ f_{\eta }(\cdot ,k)\end{array}\right],\end{array}$(33) then the step forward HOFA system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) and the step backward HOFA system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) can be compactly written as (34) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(34) and (35) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(35) respectively.

Define (36) $\begin{array}{r}\varkappa =\sum _{i=p}^{\eta }{r}_p{\mu }_p,\end{array}$(36) Then it is clearly to see $\begin{array}{r}{x}_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }\in {\mathbb{R}}^{\varkappa }\end{array}$With the above preparation, let us introduce the following definition.

Definition 4.1

A vector function $X(k)\in {\mathbb{B}}_{\mathbb{N}}^{\varkappa }$ is called a feasible trajectory of system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) if it satisfies (37) $\begin{array}{r}detB(X\left(k\right),k)\ne 0\mathrm{or}\mathrm{\infty },\mathrm{\forall }k\ge 0.\end{array}$(37)

Let $\mathbb{F}$ be the set of all feasible singular trajectories of system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ), that is, $\begin{array}{r}\mathbb{F}=\{X\left(k\right)\mid detB(X\left(k\right),k)\ne 0\mathrm{or}\mathrm{\infty },\mathrm{\forall }k\ge 0\}.\end{array}$Then $\begin{array}{r}\mathbb{S}={\mathbb{B}}_{\mathbb{N}}^{\varkappa }\mathrm{\setminus }\mathbb{F}\end{array}$is called the set of singular trajectories of system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ).

Corresponding to the Definition 2.1 in Duan (2021c), the following definition for discrete-time fully actuated systems can be proposed.

Definition 4.2

The system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) is called (globally) fully actuated if it does not have a singular trajectory, that is, $\mathbb{F}\mathbb{=}{\mathbb{B}}_{\mathbb{N}}^{\varkappa };$ and is called sub-fully actuated if it has a feasible trajectory, that is, $\mathbb{F}\ne \varnothing$.

When the system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) is fully actuated, there holds (38) $\begin{array}{r}detB(X,k)\ne 0\mathrm{or}\mathrm{\infty },\mathrm{\forall }X\in {\mathbb{R}}^{\varkappa },k\ge 0.\end{array}$(38) In this case the following control vector transformation can be introduced: $\begin{array}{r}B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right)=\tilde{u}\left(k\right),\end{array}$and now the (globally) step backward fully actuated system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) can be written in the following standard form: (39) $\begin{array}{r}{x}_{1\sim \eta }(k+1)=f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+\tilde{u}\left(k\right).\end{array}$(39)

4.2. Non-affine HOFA models

Parallel to the step forward affine HOFA system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ), the following non-affine one can also be introduced (refer to the Definition 2 in Duan (2020b)): (40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) Parallel to the step backward affine HOFA system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ), we can also define the following non-affine one: (41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) where $f(\cdot ),g(\cdot )\in {\mathbb{R}}^r$ are two proper nonlinear functions. Particularly, $g(\cdot ,k,u)=\tilde{u}$ forms a differential homeomorphism from u to $\tilde{u}$ for some of its variables.

Partition $f(\cdot )$ as in (33(33) $\begin{array}{r}f(\cdot ,k)=\left[\begin{array}{c}{f}_1(\cdot ,k)\\ f_2(\cdot ,k)\\ ⋮\\ f_{\eta }(\cdot ,k)\end{array}\right],\end{array}$(33) ), and $g(\cdot )$ as follows (42) $\begin{array}{r}g(\cdot ,k,u)=\left[\begin{array}{c}{g}_1(\cdot ,k,u)\\ g_2(\cdot ,k,u)\\ ⋮\\ g_{\eta }(\cdot ,k,u)\end{array}\right],\end{array}$(42) with $f_i(\cdot ),g_i(\cdot )\in {\mathbb{R}}^{r_i},i=1,2,\dots ,\eta$, then the step forward HOFA system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) and the step backward HOFA system (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ) can be written as (43) $\begin{array}{rl}& x_i(k+{\mu }_i)\\ & =f_i(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g_i(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\\ & i=1,2,\dots ,\eta ,\end{array}$(43) and (44) $\begin{array}{rl}& x_i(k+1)\\ & =f_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\\ & i=1,2,\dots ,\eta ,\end{array}$(44) respectively.

Parallel to Definition 4.1, the following one can be proposed.

Definition 4.3

Let $\mathbb{F}\subset {\mathbb{B}}_{\mathbb{N}}^{\varkappa }$ be the largest set such that the following mapping (45) $\begin{array}{r}\tilde{u}=g(X,k,u),\mathrm{or}u=g^{-1}(X,k,\tilde{u})\end{array}$(45) forms a differential homeomorphism from u to $\tilde{u}$ for all bounded vector functions $X(k)\in \mathbb{F}$, and $k\ge 0$. Then the set $\mathbb{F}$ is called the set of feasible trajectories of system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ), and any $X(k)\in \mathbb{F}$ is called a feasible trajectory of system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ). Furthermore, the set $\begin{array}{r}\mathbb{S}={\mathbb{B}}_{\mathbb{N}}^{\varkappa }\mathrm{\setminus }\mathbb{F}\end{array}$is called the set of singular trajectories of system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ), and any $X(k)\in \mathbb{S}$ is called a singular trajectory of system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ).

With $\mathbb{F}$ and $\mathbb{S}$ well-defined above, the definitions of full-actuation of system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ) can be immediately given simply by replacing the system (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) or (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) in Definition 4.2 by system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ).

Similarly, when the system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) or (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ) is globally fully actuated, there exists a differential homeomorphism from u to $\tilde{u}$, $\mathrm{\forall }X\in {\mathbb{R}}^{\varkappa }$, and $k\ge 0$, as $\begin{array}{r}\tilde{u}=g(X,k,u).\end{array}$In this case, we have $\begin{array}{r}u=g^{-1}(X,k,\tilde{u}),\end{array}$hence the set of systems in (44(44) $\begin{array}{rl}& x_i(k+1)\\ & =f_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\\ & i=1,2,\dots ,\eta ,\end{array}$(44) ) can be also written as in (39(39) $\begin{array}{r}{x}_{1\sim \eta }(k+1)=f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+\tilde{u}\left(k\right).\end{array}$(39) ).

To conclude this section, let us finally point out that the definition of full-actuation can be slightly modified to define over-actuated systems. In the high-order system representations, over-actuated systems may be occasionally encountered. Such systems can be similarly treated, to an extent, as fully actuated ones in terms of control (refer to the Remark 2.1 in Duan (2021h)).

5. Controller designs

Recall that the step forward HOFA systems are systems with input delays, here we only consider the control of step backward HOFA systems. Control of step forward HOFA systems will be considered in a separate paper.

5.1. Single-order HOFA models

For control of the single-order step backward HOFA model (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ), the following important fact can be easily verified, which reflects the importance of HOFA models.

Theorem 5.1

Let $\varnothing \ne \mathbb{F}\subset {\mathbb{B}}_{\mathbb{N}}^{nr}$ be the set of feasible trajectories of the system (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ). Further, let ϖ be an integer satisfying $1\le \varpi \le n$, and $A_{0\sim \varpi -1}\in {\mathbb{R}}^{\varpi \times r}$ be an arbitrarily given matrix. Then the following controller (46) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}(\cdot )[-f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=A_{0\sim \varpi -1}{x}^{\lceil 0\sim \varpi -1\rceil }\left(k\right)+v\left(k\right),\end{array}\end{array}$(46) for system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) produces the following constant linear closed-loop system $\begin{array}{r}x(k+1)=A_{0\sim \varpi -1}{x}^{\lceil 0\sim \varpi -1\rceil }\left(k\right)+v\left(k\right),\end{array}$provided that the states of the above closed-loop system (49(49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) ) are all feasible trajectories, that is, (47) $\begin{array}{r}{x}^{\lceil 0\sim \varpi -1\rceil }\left(k\right)\in \mathbb{F}.\end{array}$(47)

In the case of $\varpi =1,$ we have $\begin{array}{r}{u}^{\ast }\left(k\right)=A_{0}x\left(k\right)+v\left(k\right),\end{array}$and the closed-loop system is $\begin{array}{r}x(k+1)=A_{0}x\left(k\right)+v\left(k\right).\end{array}$While in the case of $\varpi =n,$ we have $\begin{array}{r}{u}^{\ast }\left(k\right)=A_{0\sim n-1}{x}^{\lceil 0\sim n-1\rceil }\left(k\right)+v\left(k\right),\end{array}$and the closed-loop system is $\begin{array}{r}x(k+1)=A_{0\sim n-1}{x}^{\lceil 0\sim n-1\rceil }\left(k\right)+v\left(k\right).\end{array}$In practical applications, the integer ϖ may be chosen according to the requirement on the closed-loop system. For e.g. if there is a requirement on both $x(k)$ and $x(k-1)$, then ϖ should be chosen at least greater than 2.

5.2. Multiple order HOFA models

The following important fact about the control of the step backward HOFA system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) or (35(35) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(35) ) can be easily verified.

Theorem 5.2

Let $\varnothing \ne \mathbb{F}\subset {\mathbb{B}}_{\mathbb{N}}^{\varkappa }$ be the set of feasible trajectories of the system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ). Further, let ${\varpi }_i,i=1,2,\dots ,\eta ,$ be a set of integers satisfying $\begin{array}{r}1\le {\varpi }_i,i=1,2,\dots ,\eta ,\end{array}$and $[A_i{]}_{0\sim {\varpi }_i-1}\in {\mathbb{R}}^{r_i\times {\varpi }_i{r}_i},i=1,2,\dots ,\eta$, be a set of arbitrarily given matrices. Then the following controller (48) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=\left[\begin{array}{c}{\left[A_1\right]}_{0\sim {\varpi }_1-1}{x}_1^{\lceil 0\sim {\varpi }_1-1\rceil }\left(k\right)\\ {\left[A_2\right]}_{0\sim {\varpi }_2-1}{x}_2^{\lceil 0\sim {\varpi }_2-1\rceil }\left(k\right)\\ ⋮\\ {\left[A_{\eta }\right]}_{0\sim {\varpi }_{\eta }-1}{x}_{\eta }^{\lceil 0\sim {\varpi }_{\eta }-1\rceil }\left(k\right)\end{array}\right]+v\left(k\right),\end{array}\end{array}$(48) for system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) produces the following constant linear closed-loop system (49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) with $v_p,p=1,2,\dots ,\eta ,$ being defined by (50) $\begin{array}{r}v=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_{\eta }\end{array}\right],v_p\in {\mathbb{R}}^{r_p},\end{array}$(50) provided that the closed-loop system (49(49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) ) gives all feasible trajectories, that is, (51) $\begin{array}{r}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta }\left(k\right)\in \mathbb{F}.\end{array}$(51)

Please note that in this multiple order case, the set of integers ${\varpi }_i,i=1,2,\dots ,\eta ,$ may be chosen to be equal to each other, that is, $\begin{array}{r}{\varpi }_1={\varpi }_2=\cdots ={\varpi }_{\eta }=\varpi .\end{array}$In this case, we get a closed-loop system with a single-order ϖ.

Define (52) $\begin{array}{r}{A}_E=\mathrm{blockdiag}({\left[A_p\right]}_{0\sim {\varpi }_p-1},p=1,2,\dots ,\eta ),\end{array}$(52) then, obviously, the controller (48(48) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=\left[\begin{array}{c}{\left[A_1\right]}_{0\sim {\varpi }_1-1}{x}_1^{\lceil 0\sim {\varpi }_1-1\rceil }\left(k\right)\\ {\left[A_2\right]}_{0\sim {\varpi }_2-1}{x}_2^{\lceil 0\sim {\varpi }_2-1\rceil }\left(k\right)\\ ⋮\\ {\left[A_{\eta }\right]}_{0\sim {\varpi }_{\eta }-1}{x}_{\eta }^{\lceil 0\sim {\varpi }_{\eta }-1\rceil }\left(k\right)\end{array}\right]+v\left(k\right),\end{array}\end{array}$(48) ) can be more compactly written as (53) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=A_E{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right).\end{array}\end{array}$(53) Furthermore, if written in state-space form, the closed-loop system (49(49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) ) can be written as (54) $\begin{array}{rl}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }(k+1)& =\mathrm{\Psi }([A_p{]}_{0\sim {\varpi }_p-1})x_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)\\ & +{\mathrm{\Gamma }}_{\mathrm{c}p}{v}_p\left(k\right),p=1,2,\dots ,\eta ,\end{array}$(54) where (55) $\begin{array}{r}{\mathrm{\Gamma }}_{\mathrm{c}p}=\left[\begin{array}{c}{I}_{r_p}\\ 0\\ ⋮\\ 0\end{array}\right],p=1,2,\dots ,\eta .\end{array}$(55)

Remark 5.1

When $f(\cdot )$ is restricted to be linear and $B(\cdot )$ is a constant nonsingular matrix, the above result turns to be the control of the linear multiple-order HOFA system (16(16) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{L}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ L_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ L_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]+\hat{B}u\left(k\right),\end{array}$(16) ) or (18(18) $\begin{array}{r}{x}_{1\sim \eta }(k+1)=L(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+\hat{B}u\left(k\right),\end{array}$(18) ). For the more general non-affine step backward HOFA system (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ), a result similar to Theorem 5.2 still holds. The controller can be obtained using the differential homeomorphism property of the mapping $\tilde{u}=g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }{|}_{p=1\sim \eta },k,u),$ as (56) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=g^{-1}(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,\tilde{u}\left(k\right))\\ \tilde{u}\left(k\right)=-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)\\ u^{\ast }\left(k\right)=\left[\begin{array}{c}{\left[A_1\right]}_{0\sim {\varpi }_1-1}{x}_1^{\lceil 0\sim {\varpi }_1-1\rceil }\left(k\right)\\ {\left[A_2\right]}_{0\sim {\varpi }_2-1}{x}_2^{\lceil 0\sim {\varpi }_2-1\rceil }\left(k\right)\\ ⋮\\ {\left[A_{\eta }\right]}_{0\sim {\varpi }_{\eta }-1}{x}_{\eta }^{\lceil 0\sim {\varpi }_{\eta }-1\rceil }\left(k\right)\end{array}\right]+v\left(k\right),\end{array}\end{array}$(56) and the same closed-loop system as in (49(49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) ) is achieved.

Remark 5.2

The design given in Theorem 5.2 is a decoupled one, and the $u^{\ast }(t)$ given in the controller (48(48) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=\left[\begin{array}{c}{\left[A_1\right]}_{0\sim {\varpi }_1-1}{x}_1^{\lceil 0\sim {\varpi }_1-1\rceil }\left(k\right)\\ {\left[A_2\right]}_{0\sim {\varpi }_2-1}{x}_2^{\lceil 0\sim {\varpi }_2-1\rceil }\left(k\right)\\ ⋮\\ {\left[A_{\eta }\right]}_{0\sim {\varpi }_{\eta }-1}{x}_{\eta }^{\lceil 0\sim {\varpi }_{\eta }-1\rceil }\left(k\right)\end{array}\right]+v\left(k\right),\end{array}\end{array}$(48) ) is in fact a decentralised one. This is adequate to show the advantage of the HOFA approach. However, a coupled design can also be carried out, with the expression of $u^{\ast }(t)$ in (48(48) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)+u^{\ast }\left(k\right)]\\ u^{\ast }\left(k\right)=\left[\begin{array}{c}{\left[A_1\right]}_{0\sim {\varpi }_1-1}{x}_1^{\lceil 0\sim {\varpi }_1-1\rceil }\left(k\right)\\ {\left[A_2\right]}_{0\sim {\varpi }_2-1}{x}_2^{\lceil 0\sim {\varpi }_2-1\rceil }\left(k\right)\\ ⋮\\ {\left[A_{\eta }\right]}_{0\sim {\varpi }_{\eta }-1}{x}_{\eta }^{\lceil 0\sim {\varpi }_{\eta }-1\rceil }\left(k\right)\end{array}\right]+v\left(k\right),\end{array}\end{array}$(48) ) replaced by $\begin{array}{r}{u}^{\ast }\left(k\right)=K{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }-v\left(k\right),\end{array}$where $K\in {\mathbb{R}}^{r\times \varkappa }$ is a feedback gain. For details, one can refer to the coupled design for continuous-time HOFA systems given in Duan (2021c).

Remark 5.3

Note that $\begin{array}{r}\mathrm{\Psi }\left(A_{0\sim n-1}\right)=\mathrm{\Psi }\left(0_{0\sim n-1}\right)+{\mathrm{\Gamma }}_{\mathrm{c}}{A}_{0\sim n-1},\end{array}$and $[\mathrm{\Psi }(0),{\mathrm{\Gamma }}_{\mathrm{c}}]$ is controllable, the solution to the feedback gains in the proposed controllers really reduces to a problem of finding a real matrix K such that A + BK is Schur for some controllable matrix pair $[A,B]$. This can be easily and completely solved by using the Lemma 5.2 in Duan (2021c), where the matrix F needs to be chosen Schur. Particularly, to find an $A_{0\sim n-1}$ to make $\mathrm{\Psi }(A_{0\sim n-1})$ Schur, the dual form of Theorem 3.4 in Duan (2021b) can be readily used, with the matrix F chosen Schur.

Remark 5.4

Response analysis and stability analysis are two very fundamental and important aspects of system analysis. Generally speaking, these two problems are very hard to treat with the state-space approaches for general nonlinear dynamical control systems. However, as it is clearly seen in the above Theorems 5.1 and 5.2, with the proposed HOFA approaches these two problems really vanish since they are both converted into corresponding problems for linear systems. As is well-known that many control systems design approaches are based on, and also limited by, certain stability analysis results. While it is now clearly seen that the provided HOFA system approaches are such a type of approaches which are not dependent on stability analysis results and therefore are not subject to the conservatism brought about by the stability analysis results.

Remark 5.5

It is again clearly observed in the above Theorems 5.1 and 5.2 that the linear closed-loop systems are obtained with desired coefficient matrices, or equivalently, with desired eigenstructures. Such a property of HOFA systems totally reveals the meaning of controllability of dynamical control systems. Therefore, following the general line in Duan (2021c), controllability and stabilisability of general discrete-time systems can be also readily defined. Specifically, a dynamical system is called controllable if it can be represented by either a step forward or a step backward HOFA model. While a stabilisable dynamical system can be represented by either a step forward or a step backward HOFA model, together with a stable autonomous dynamical system.

Remark 5.6

The feasibility requirements (47(47) $\begin{array}{r}{x}^{\lceil 0\sim \varpi -1\rceil }\left(k\right)\in \mathbb{F}.\end{array}$(47) ) and (51(51) $\begin{array}{r}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta }\left(k\right)\in \mathbb{F}.\end{array}$(51) ) are essential for general HOFA systems. They vanish only for global fully actuated systems. For continuous-time HOFA systems, two types of feasibility conditions are provided, one is dependent on a Lyapunov matrix equation (see the Lemma 5.2 in Duan, 2021d), the other is based on the closed-loop eigenstructure (see the Lemma 6.1 in Duan, 2021e). Both types of conditions turn out to be constraints on the system initial values, and can be both parallelly extended to the discrete-time system case.

6. Feedback linearisable systems and strict-feedback systems

As examples of the proposed discrete-time step forward HOFA systems and discrete-time step backward HOFA systems, in this section two types of state-space systems are treated, namely, the feedback linearisable systems and the strict-feedback systems.

6.1. Feedback linearisable systems

Consider the discrete-time control system (57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) where $x(k)\in {\mathbb{R}}^n$ and$u(k)\in {\mathbb{R}}^r$ are the state vector and input vector, respectively,$f(\cdot )\in {\mathbb{R}}^n$ is a sufficiently differentiable function satisfying the initial condition $f(x_0,u_0)=x_0.$ We are interested in the problem when system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is locally equivalent to a controllable linear system (Jakubczyk, 1987).

Definition 6.1

The system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is called locally feedback linearisable at $(x_0,u_0)$ via the differentiable feedback: (58) $\begin{array}{rl}u\left(k\right)& =\xi (x\left(k\right),v\left(k\right)),\\ \xi (x_0,0)& =u_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }v}\xi (x_0,0)=r,\end{array}$(58) if there exists a sufficiently differentiable transformation of coordinates in the state space: (59) $\begin{array}{r}x=T_0\left(y\right),T_0\left(0\right)=x_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }y}{T}_0\left(0\right)=n,\end{array}$(59) such that the system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is equivalently transformed, under the control transformation (58(58) $\begin{array}{rl}u\left(k\right)& =\xi (x\left(k\right),v\left(k\right)),\\ \xi (x_0,0)& =u_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }v}\xi (x_0,0)=r,\end{array}$(58) ) and the state transformation (59(59) $\begin{array}{r}x=T_0\left(y\right),T_0\left(0\right)=x_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }y}{T}_0\left(0\right)=n,\end{array}$(59) ), into the form of (60) $\begin{array}{rl}& y(k+1)\\ & =Ay\left(k\right)+Bv\left(k\right),y\left(k\right)\in {\mathbb{R}}^n,v\left(k\right)\in {\mathbb{R}}^r,\end{array}$(60) with $[A,B]$ controllable.

The following result indicates that feedback linearisable systems can be represented by an HOFA model.

Proposition 6.2

Let system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) be locally feedback linearisable at $(x_0,u_0)$ via the feedback (58(58) $\begin{array}{rl}u\left(k\right)& =\xi (x\left(k\right),v\left(k\right)),\\ \xi (x_0,0)& =u_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }v}\xi (x_0,0)=r,\end{array}$(58) ). Then the system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is equivalent to both a step forward HOFA system in the form of (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ) within a neighbourhood of $(x_0,u_0)$, and a step backward HOFA system in the form of (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ) within a neighbourhood of $(x_0,u_0)$.

For a proof of the above proposition, one can refer to the appendix.

Particularly, when the system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is locally feedback linearisable via the following feedback (61) $\begin{array}{r}u\left(k\right)=\xi (x\left(k\right),v\left(k\right))=\alpha \left(x\left(k\right)\right)+\beta \left(x\left(k\right)\right)v\left(k\right),\end{array}$(61) where $\alpha (x)$ and $\beta (x)$ satisfy (62) $\begin{array}{r}\alpha \left(x_0\right)=u_0,\mathrm{rank}\beta \left(x\right)=r,\end{array}$(62) the above result becomes the following.

Corollary 6.3

Let system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) be locally feedback linearisable at $(x_0,u_0)$ via the feedback (61(61) $\begin{array}{r}u\left(k\right)=\xi (x\left(k\right),v\left(k\right))=\alpha \left(x\left(k\right)\right)+\beta \left(x\left(k\right)\right)v\left(k\right),\end{array}$(61) )–(62(62) $\begin{array}{r}\alpha \left(x_0\right)=u_0,\mathrm{rank}\beta \left(x\right)=r,\end{array}$(62) ). Then the system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is equivalent to both a step forward HOFA system in the affine form of (31(31) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+{\mu }_1)\\ x_2(k+{\mu }_2)\\ ⋮\\ x_{\eta }(k+{\mu }_{\eta })\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(31) ) within a neighbourhood of $(x_0,u_0)$, and a step backward HOFA system in the affine form of (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ) within a neighbourhood of $(x_0,u_0)$.

The proof of the above corollary is also provided in the appendix.

6.2. Strict feedback systems

Parallel to the continuous-time strict-feedback systems (see, Duan, 2021f), the discrete-time strict-feedback systems possess the following general form (Ge et al., 2003, 2008, 2009; W. Xu et al., 2021): (63) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)x_3\left(k\right)\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)\\ +g_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)x_n\left(k\right)\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+g_n\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(63) where u and $x_i\in {\mathbb{R}}^r,i=1,2,\dots ,n$, are the system input vector and state vectors, respectively, $f_i(\cdot )\in {\mathbb{R}}^r,$ and $g_i(\cdot )\in {\mathbb{R}}^{r\times r},i=1,2,\dots ,n$, are nonlinear functions, and particularly, $g_i(\cdot )$ satisfies the following assumption:

Assumption A2

$det{g}_i(x_{1\sim i})\ne 0,\mathrm{\forall }{x}_{1\sim i}\in {\mathbb{R}}^{ir},i=1,2,\dots ,n.$

Proposition 6.4

Let $f_i(\cdot )\in {\mathbb{R}}^r,$ and $g_i(\cdot )\in {\mathbb{R}}^{r\times r},i=1,2,\dots ,n$, are sufficiently differentiable nonlinear functions. Then the above system (63(63) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)x_3\left(k\right)\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)\\ +g_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)x_n\left(k\right)\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+g_n\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(63) ), with Assumption A2 satisfied, can be equivalently converted into both a step forward HOFA system in the form of (28(28) $\begin{array}{rl}x(k+n)& =f(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)\\ & +B(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)u\left(k\right),\end{array}$(28) ), and a step backward HOFA system in the form of (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ).

For a proof of the above proposition, refer to the appendix.

The importance of the above result lies in that, once a strict-feedback system is converted into a HOFA model, the system can be easily controlled in view of Theorems 5.1 and 5.2, with the closed-loop system being a constant linear one with an arbitrarily assignable eigenstructure. It should be noted that such an advantage is not generally achievable with the well-known method of backstepping. We point out that the converted HOFA model is a global fully actuated one due to Assumption A2. Therefore, the thorny problem of singularity vanishes.

To finish this section, let us make some further remarks.

Remark 6.1

As treated in Duan (2021f), the strict-feedback system (63(63) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)x_3\left(k\right)\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)\\ +g_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)x_n\left(k\right)\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+g_n\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(63) ) can be also generalised to the second-order case and the mixed-order case. Particularly, a second-order strict-feedback system in step backward form can be represented as follows: (64) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1^{\lceil 0\sim 1\rceil }\left(k\right)\right)+g_1\left(x_1^{\lceil 0\sim 1\rceil }\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}^{\lceil 0\sim 1\rceil }\left(k\right)\right)+g_2\left(x_{1\sim 2}^{\lceil 0\sim 1\rceil }\left(k\right)\right)x_3\left(k\right)\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}\left(x_{1\sim n-1}^{\lceil 0\sim 1\rceil }\left(k\right)\right)\\ +g_{n-1}\left(x_{1\sim n-1}^{\lceil 0\sim 1\rceil }\left(k\right)\right)x_n\left(k\right)\\ x_n(k+1)=f_n\left(x_{1\sim n}^{\lceil 0\sim 1\rceil }\left(k\right)\right)+g_n\left(x_{1\sim n}^{\lceil 0\sim 1\rceil }\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(64) which can be also shown to be representable by an HOFA model. Moreover, as in Duan (2021f), a second-order backstepping design for the above second-order strict-feedback system can be also proposed.

Remark 6.2

It follows from Theorem 3.4 that a discrete-time controllable linear system can be equivalently converted into a HOFA system. Now in this section, it happens that both the feedback linearisable systems and strict-feedback systems can be equivalently converted into HOFA systems. These facts strongly demonstrate, to an extent, the generality of the HOFA system approaches. With the HOFA approaches, it is clearly seen in Section 5 that a controller for a HOFA system can be immediately designed. We point out that the designs can be also easily generalised to systems with disturbances and uncertainties, as done in the continuous-time cases (see Duan, 2021g, 2021i, 2021a, 2021b, 2021e).

7. Further generalisations

In Remark 3.1, it is pointed out that, for controllable linear systems, a general input–output representation (19(19) $\begin{array}{r}x(k+n)=\sum _{i=0}^{n-1}{A}_{n-i-1}x(k+i)+\sum _{i=0}^m{B}_{m-i}u(k+i),\end{array}$(19) ) or (20(20) $\begin{array}{r}x(k+1)=\sum _{i=0}^{n-1}{A}_{i}x(k-i)+\sum _{i=0}^m{B}_{i}u(k-i),\end{array}$(20) ) can be equivalently converted into both a step forward and a step backward HOFA one. However, for nonlinear systems, the problem may be more complicated. Due to this consideration, in this section, our HOFA models are extended to include the effect of the control vector at more time instants. Due to paper length, here only the backward model is considered.

7.1. Generalised HOFA models

As a further generalisation of the discrete-time step backward HOFA system (32(32) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)u\left(k\right),\end{array}$(32) ), the following discrete-time step backward HOFA system is proposed: (65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) where $m_p,p=1,2,\dots ,\eta +1,$ are a set of integers, while the other variables are as stated before. Remember that the matrix function $B(\cdot )\in {\mathbb{R}}^{r\times r}$ is required to be nonsingular for some of its variables.

Denote (66) $\begin{array}{r}f(\cdot ,\cdot ,k)=\left[\begin{array}{c}{f}_1(\cdot ,\cdot ,k)\\ f_2(\cdot ,\cdot ,k)\\ ⋮\\ f_{\eta }(\cdot ,\cdot ,k)\end{array}\right],\end{array}$(66) then the step backward HOFA system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ) can be compactly written as (67) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)u\left(k\right).\end{array}$(67) where $\begin{array}{r}m=max\{m_i,i=1,2,\dots ,\eta +1\}.\end{array}$For this extended HOFA system (67(67) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)u\left(k\right).\end{array}$(67) ), Definition 4.1 becomes the following.

Definition 7.1

If $X(k)\in {\mathbb{B}}_{\mathbb{N}}^{\varkappa }$ and $U(k)\in {\mathbb{B}}_{\mathbb{N}}^{mr}$ satisfy (68) $\begin{array}{r}detB(X,U,k)=0\mathrm{or}\mathrm{\infty },\mathrm{\exists }k\ge 0,\end{array}$(68) then $(X,U)$ is called a pair of singular trajectories of system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ).

Let $\mathbb{S}$ be the set of all pairs of singular trajectories of system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ), that is, $\begin{array}{r}\mathbb{S}=\{(X,U)\mid detB(X,U,k)=0\mathrm{or}\mathrm{\infty },k\ge 0\}.\end{array}$Then $\begin{array}{r}\mathbb{F}={\mathbb{B}}_{\mathbb{N}}^{\varkappa +mr}\mathrm{\setminus }\mathbb{S}\end{array}$is called the set of feasible trajectories of system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ). The system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ) is generally said to be (global) HOFA if it does not have a pair of singular trajectories, and is called (sub-) HOFA if $\mathbb{F}$ is not empty. A strict definition can be given parallel to Definition 4.2.

In the case of $\mathbb{S}\mathbb{=}\varnothing$, we have (69) $\begin{array}{r}detB(X,U,k)\ne 0\mathrm{or}\mathrm{\infty },\mathrm{\forall }X\in {\mathbb{B}}_{\mathbb{N}}^{\varkappa },U\in {\mathbb{B}}_{\mathbb{N}}^{mr},k\ge 0.\end{array}$(69) Therefore, the following control vector transformation can be introduced: $\begin{array}{r}B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)u\left(k\right)=\tilde{u}\left(k\right),\end{array}$under which the (globally) fully actuated system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ) can be written as (70) $\begin{array}{rl}{x}_i(k+1)& =f_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_i\rceil }\left(k\right),k)\\ & +{\tilde{u}}_i\left(k\right),i=1,2,\dots ,\eta ,\end{array}$(70) where ${\tilde{u}}_i,i=1,2,\dots ,\eta ,$ are defined by (71) $\begin{array}{r}\tilde{u}=\left[\begin{array}{c}{\tilde{u}}_1\\ {\tilde{u}}_2\\ ⋮\\ {\tilde{u}}_{\eta }\end{array}\right],{\tilde{u}}_i\in {\mathbb{R}}^{r_i}.\end{array}$(71) For the control of the extended step backward HOFA system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ) or (67(67) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)u\left(k\right).\end{array}$(67) ), a straight forward extension of Theorem 5.2 can be given as follows.

Theorem 7.2

Let $\varnothing \ne \mathbb{F}\subset {\mathbb{B}}_{\mathbb{N}}^{\varkappa +mr}$ be the set of feasible trajectories of the system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ), and ${\varpi }_p,p=1,2,\dots ,\eta ,$ be a set of integers greater than 1. Then the following controller (72) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=B^{-1}\\ (\cdot )[-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ +u^{\ast }\left(k\right)\phantom{(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)}]\\ u^{\ast }\left(k\right)=K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right),\end{array}\end{array}$(72) for system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ) produces the following constrained constant linear closed-loop system (73) $\begin{array}{r}\{\begin{array}{l}{x}_p(k+1){|}_{p=1\sim \eta }\\ =K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right)\\ (x_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right))\in \mathbb{F},\end{array}\end{array}$(73) where $K_{1\sim \eta }\in {\mathbb{R}}^{r\times {\varkappa }^{\mathrm{\prime }}},$ ${\varkappa }^{\mathrm{\prime }}=\sum _{p=1}^{\eta }{\varpi }_p{r}_p,$ is the designed feedback gain matrix.

As a constrained linear system, the closed-loop system (73(73) $\begin{array}{r}\{\begin{array}{l}{x}_p(k+1){|}_{p=1\sim \eta }\\ =K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right)\\ (x_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right))\in \mathbb{F},\end{array}\end{array}$(73) ) can be easily made stable by properly selecting the feedback gain $K_{1\sim \eta }$. For a systematic method, one can refer to Duan (2021c). Particularly, when $K_{1\sim \eta }$ is chosen as $\begin{array}{r}{K}_{1\sim \eta }=\mathrm{blockdiag}({\left[A_p\right]}_{0\sim {\varpi }_p-1},p=1,2,\dots ,\eta ),\end{array}$the closed-loop system (73(73) $\begin{array}{r}\{\begin{array}{l}{x}_p(k+1){|}_{p=1\sim \eta }\\ =K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right)\\ (x_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right))\in \mathbb{F},\end{array}\end{array}$(73) ) reduces to (49(49) $\begin{array}{rl}{x}_p(k+1)& =[A_p{]}_{0\sim {\varpi }_p-1}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)+v_p\left(k\right),\\ p& =1,2,\dots ,\eta ,\end{array}$(49) ) or equivalently the state-space form (54(54) $\begin{array}{rl}{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }(k+1)& =\mathrm{\Psi }([A_p{]}_{0\sim {\varpi }_p-1})x_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right)\\ & +{\mathrm{\Gamma }}_{\mathrm{c}p}{v}_p\left(k\right),p=1,2,\dots ,\eta ,\end{array}$(54) ). Furthermore, the constraints in the closed-loop system (73(73) $\begin{array}{r}\{\begin{array}{l}{x}_p(k+1){|}_{p=1\sim \eta }\\ =K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right)\\ (x_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right))\in \mathbb{F},\end{array}\end{array}$(73) ) vanishes if $\mathbb{S}\mathbb{=}\varnothing$, or equivalently, the relation in (69(69) $\begin{array}{r}detB(X,U,k)\ne 0\mathrm{or}\mathrm{\infty },\mathrm{\forall }X\in {\mathbb{B}}_{\mathbb{N}}^{\varkappa },U\in {\mathbb{B}}_{\mathbb{N}}^{mr},k\ge 0.\end{array}$(69) ) holds.

Parallel to the extended step backward affine HOFA system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ), the following extended non-affine one can also be defined: (74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) where $f(\cdot ),g(\cdot )\in {\mathbb{R}}^r$ are two proper nonlinear functions, and $g(\cdot ,u)$ forms a differential homeomorphism with respect to u.

Parallel to Definition 7.1, the following one can also be proposed.

Definition 7.3

Let $\mathbb{F}\subset {\mathbb{B}}_{\mathbb{N}}^{\varkappa +mr}$ be the largest set such that the following mapping (75) $\begin{array}{r}\tilde{u}=g(X,U,k,u),\end{array}$(75) forms a differential homeomorphism from u to $\tilde{u}$ for all $(X,U)\in \mathbb{F}$, and $k\ge 0$, then the set $\mathbb{F}$ is called the set of feasible trajectories of system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ), and any $(X(k),U(k))\in \mathbb{F}$ is called a feasible trajectory of system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ). Furthermore, the set $\begin{array}{r}\mathbb{S}={\mathbb{B}}_{\mathbb{N}}^{\varkappa +mr}\mathrm{\setminus }\mathbb{F},\end{array}$is called the set of singular trajectories of system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ), and any $(X(k),U(k))\in \mathbb{S}$ is called a singular trajectory of system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ).

With $\mathbb{F}$ and $\mathbb{S}$ well-defined above, the definitions of full-actuation of system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ) can be also immediately given parallel to Definition 4.2. Similarly, when the system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ) is globally fully actuated, there exists a differential homeomorphism, $\begin{array}{r}\tilde{u}=g(X,U,k,u),\end{array}$from u to $\tilde{u}$, $\mathrm{\forall }X\in {\mathbb{R}}^{\varkappa }$, $U\in {\mathbb{R}}^{mr}$, and $k\ge 0$. In this case, the set of systems in (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ) can be also written equivalently as in (70(70) $\begin{array}{rl}{x}_i(k+1)& =f_i(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_i\rceil }\left(k\right),k)\\ & +{\tilde{u}}_i\left(k\right),i=1,2,\dots ,\eta ,\end{array}$(70) ).

Furthermore, for the more general non-affine step backward HOFA system (74(74) $\begin{array}{rl}& x_{1\sim \eta }(k+1)\\ & =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,u\left(k\right)),\end{array}$(74) ), a result similar to Theorem 7.2 still holds. The controller can be obtained using the differential homeomorphism property of the mapping $\tilde{u}=g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }(k),k,u),$ as (76) $\begin{array}{r}\{\begin{array}{l}u\left(k\right)=g^{-1}(x_p^{\lceil 0\sim {\mu }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k,\tilde{u}\left(k\right))\\ \tilde{u}\left(k\right)=K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+u^{\ast }\left(k\right)\\ u^{\ast }\left(k\right)=-f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right),k)\\ +v\left(k\right),\end{array}\end{array}$(76) and the closed-loop system is the same as that in (73(73) $\begin{array}{r}\{\begin{array}{l}{x}_p(k+1){|}_{p=1\sim \eta }\\ =K_{1\sim \eta }{x}_p^{\lceil 0\sim {\varpi }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }+v\left(k\right)\\ (x_p^{\lceil 0\sim {\varpi }_p-1\rceil }{|}_{p=1\sim \eta },u^{\lceil 1\sim m\rceil }\left(k\right))\in \mathbb{F},\end{array}\end{array}$(73) ).

7.2. Pseudo feed-forward systems

Consider the following pseudo feed-forwardsystem (77) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_n\left(k\right),u\left(k\right))\\ x_2(k+1)=f_2(x_1\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_3(k+1)=f_3(x_{1\sim 2}\left(k\right),x_n\left(k\right),u\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{1\sim n-1}\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+B\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(77) where $x_i\in {\mathbb{R}}^{r_i},i=1,2,\dots ,n$, are the system state vectors, $u\in {\mathbb{R}}^r,r=r_n,$ is the system control vector, $f_i(\cdot )\in {\mathbb{R}}^{r_i},i=1,2,\dots ,n$, are a series of vector functions, and $B(\cdot )\in {\mathbb{R}}^{r\times r}$ is a nonlinear matrix function satisfying the following assumption.

Assumption A3

$detB(x_{1\sim n}(k))\ne 0,\mathrm{\forall }{x}_{1\sim n}(k)\in {\mathbb{R}}^{nr}.$

Proposition 7.4

The above system (77(77) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_n\left(k\right),u\left(k\right))\\ x_2(k+1)=f_2(x_1\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_3(k+1)=f_3(x_{1\sim 2}\left(k\right),x_n\left(k\right),u\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{1\sim n-1}\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+B\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(77) ) satisfying Assumption A3 can be equivalently converted into the step backward HOFA system (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ).

For a proof of the above proposition, one can refer to the appendix.

Example 7.5

Consider a specific pseudo feed-forward system, with $n=3,$ (78) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=x_3\left(k\right)\\ x_2(k+1)=[1+x_1^2\left(k\right)][x_3\left(k\right)+\sigma u\left(k\right)]\\ x_3(k+1)=x_1\left(k\right)x_2\left(k\right)+[1+x_3\left(k\right)]u\left(k\right),\end{array}\end{array}$(78) where σ is a nonzero real scalar. Following the procedure given in the proof of Proposition 7.4, we can obtain its following equivalent HOFA model: $\begin{array}{rl}{x}_3(k+1)& =f\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)+[1+x_3\left(k\right)]u\left(k\right)\\ & +b_1\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)u(k-1),\end{array}$where $\begin{array}{rl}f\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)& =x_3^2(k-1)[1+x_3^2(k-2)],\\ b_1\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)& =\sigma x_3(k-1)[1+x_3^2(k-2)].\end{array}$The feasibility condition of this system is clearly (79) $\begin{array}{r}{x}_3\left(k\right)\ne -1.\end{array}$(79)

For this system, we design the following controller: $\begin{array}{r}\{\begin{array}{l}{\displaystyle u\left(k\right)}\\ =\frac{1}{1+x_3\left(k\right)}(\sum _{i=0}^{\varpi -1}{a}_i{x}_3(k-i)-u^{\ast }\left(k\right)+v\left(k\right))\\ u^{\ast }\left(k\right)=f\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)+b_1\left(x_3^{\lceil 1\sim 2\rceil }\left(k\right)\right)u\left(k-1\right).\end{array}\end{array}$In the case of $\varpi =1,$ the closed-loop system is $\begin{array}{r}{x}_3(k+1)=a_0{x}_3\left(k\right)+v\left(k\right),\end{array}$the system is clearly stable if and only if $|a_0|<1.$ In the case of $v(k)\equiv 0$, the feasibility condition (79(79) $\begin{array}{r}{x}_3\left(k\right)\ne -1.\end{array}$(79) ) is clearly met when the initial value satisfies $|x_3(0)|<1.$

In the case of $\varpi =3,$ the following closed-loop system is obtained: $\begin{array}{r}{x}_3(k+1)=\sum _{i=0}^2{a}_i{x}_3(k-i)+v\left(k\right),\end{array}$which has the state-space form of (80) $\begin{array}{r}{x}_3^{\lceil 0\sim 2\rceil }(k+1)=\mathrm{\Psi }\left(a_{0\sim 2}\right)x_3^{\lceil 0\sim 2\rceil }\left(k\right)+{\mathrm{\Gamma }}_{\mathrm{c}}v\left(k\right),\end{array}$(80) where, by our notations, $\begin{array}{r}\mathrm{\Psi }\left(a_{0\sim 2}\right)=\left[\begin{array}{ccc}{a}_0& a_1& a_2\\ 1& 0& 0\\ 0& 1& 0\end{array}\right],{\mathrm{\Gamma }}_{\mathrm{c}}=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right].\end{array}$Without loss of generality, let us assume that $\mathrm{\Psi }(a_{0\sim 2})$ is similar to a real diagonal Schur matrix $\begin{array}{r}F=\mathrm{diag}(s_i,i=1,2,3),\end{array}$that is, $\begin{array}{r}\mathrm{\Psi }\left(a_{0\sim 2}\right)=\mathrm{\Psi }\left(0_{0\sim 2}\right)-{\mathrm{\Gamma }}_{\mathrm{c}}{a}_{0\sim 2}=VF{V}^{-1},\end{array}$for some nonsingular V. Solution to $a_{0\sim 2}$ satisfying the above equation can be parametrically solved according to the Lemma 5.2 in Duan (2021c).

In the case of $v=0,$ we eventually have $\begin{array}{r}{x}_3\left(k\right)\to 0,\mathrm{as}t\to \mathrm{\infty }.\end{array}$When the initial values are properly chosen, the feasibility requirement (79(79) $\begin{array}{r}{x}_3\left(k\right)\ne -1.\end{array}$(79) ) can be satisfied. For a theoretical guarantee of the feasibility requirement (79(79) $\begin{array}{r}{x}_3\left(k\right)\ne -1.\end{array}$(79) ), we can either apply the feasibility condition based on a Lyapunov matrix equation (see the Lemma 5.2 in Duan, 2021d), or the one based on the closed-loop eigenstructure (see the Lemma 6.1 in Duan, 2021e). In certain cases, the signal v can also be properly chosen to make the steady state of $x_3(k)$ to be further away from the singularity set.

Once $x_3(k),u(k),k=0,1,2,\dots$ are obtained, by the first equation in (78(78) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=x_3\left(k\right)\\ x_2(k+1)=[1+x_1^2\left(k\right)][x_3\left(k\right)+\sigma u\left(k\right)]\\ x_3(k+1)=x_1\left(k\right)x_2\left(k\right)+[1+x_3\left(k\right)]u\left(k\right),\end{array}\end{array}$(78) ) we can get $x_1(k),k=0,1,2,\dots ,$ and then $x_2(k),k=0,1,2,\dots$ by the second one in (78(78) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=x_3\left(k\right)\\ x_2(k+1)=[1+x_1^2\left(k\right)][x_3\left(k\right)+\sigma u\left(k\right)]\\ x_3(k+1)=x_1\left(k\right)x_2\left(k\right)+[1+x_3\left(k\right)]u\left(k\right),\end{array}\end{array}$(78) ).

8. Concluding remarks

As the last one in this HOFA approach series, this paper provides a foundation for HOFA approaches to discrete-time systems.

Regarding models of discrete-time HOFA systems, different from the continuous-time case, there are two types of general HOFA models, one is the type of step forward HOFA models, and the other is the type of step backward HOFA models. It is shown that the two types of models are essentially different in the sense that, once a step forward model is represented in a step backward form, it eventually becomes a system with a time-delay in the control vector.

As a generalisation of the proposed step backward HOFA models, an extended form of the step backward HOFA model is also proposed, which takes consideration of the effect of the control vector at more time instants in the functions $f(\cdot )$ and $B(\cdot )$.

It is proven that controllers for step backward HOFA models can be easily designed such that the closed-loop systems are constant linear ones with arbitrarily assignable eigenstructures. However, due to the delay in the control vector, the problem of controller design for a step forward HOFA system will be address in separate papers on time-delay systems.

The results proposed in the paper are very fundamental and are also vitally important since they lay a solid basis for discrete-time HOFA approaches. Specifically, we can give the following brief comments on discrete-time HOFA system approaches.

Control systems analysis Firstly, the problems of response analysis and stability analysis are not as necessary as they are for state-space approaches since the closed-loop systems designed via HOFA approaches are constant and linear, or with a constant and linear main part (when uncertainties are added). Secondly, parallel to the continuous-time case, a general discrete-time dynamical system can be defined to be controllable if it can be equivalently converted into a step backward HOFA model. Furthermore, an uncontrollable discrete-time system is generally composed of a HOFA model and an autonomous subsystem, and the system is stabilisable if the autonomous subsystem does not exist or is stable (Duan, 2020b, 2021c).

Control systems design Thanks to the full-actuation property of the proposed discrete-time HOFA models, important control features are revealed as shown in Theorems 5.1 and 5.2. As in the continuous-time system case, these basic results allow us to extend the results to many other design problems, such as robust control (Duan, 2021g, 2021a), adaptive control (Duan, 2021i, 2021a), disturbance rejection (Duan, 2021b), optimal control (Duan, 2021d), and signal tracking control (Duan, 2021e). All such control problems, as done in the continuous-time case, can be converted into corresponding ones for linear systems, hence can be effectively solved by adopting existing methods for linear systems. Another aspect associated with the discrete-time HOFA system designs is the complete parameterisation. Since constant linear closed-loop systems are obtained, as done in the continuous-time case, complete parametric expressions for the feedback gains in the controllers as well as the closed-loop systems can be established, which provide all the design degrees of freedom to be used for further improving the system performance.

Finally, it is also mentioned that, like the case of continuous-time systems and discrete-time ones, HOFA approaches can be also applied to other types of systems, such as stochastic systems, and time-delay systems.

Acknowledgments

The author is grateful to his Ph.D. students Guangtai Tian, Qin Zhao, Xiubo Wang, Weizhen Liu, Kaixin Cui, Liyao Hu, and Prof. Y. Cui, for helping him with reference selection and proofreading. His particular thanks go to his students Bin Zhou, Tianyi Zhao and Mingzhe Hou for their helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been partially supported by the Major Program of National Natural Science Foundation of China [grant numbers 61690210, 61690212], the National Natural Science Foundation of China [grant number 61333003] and the Self-Planned Task of State Key Laboratory of Robotics and System (HIT) [grant number SKLRS201716A].

Notes on contributors

Guangren Duan

Guangren Duan received his Ph.D. degree in Control Systems Sciences from Harbin Institute of Technology, Harbin, People's Republic of China, in 1989. After a two-year post-doctoral experience at the same university, he became professor of control systems theory at that university in 1991. He is the founder and the Director of the Center for Control Theory and Guidance Technology at Harbin Institute of Technology, and recently he is also in charge of the Center for Control Science and Technology at the Southern University of Science and Technology. He visited the University of Hull, the University of Sheffield, and also the Queen's University of Belfast, UK, from December 1996 to October 2002, and has served as Member of the Science and Technology Committee of the Chinese Ministry of Education, Vice President of the Control Theory and Applications Committee, Chinese Association of Automation (CAA), and Associate Editors of a few international journals. He is currently an Academician of the Chinese Academy of Sciences, and Fellow of CAA, IEEE and IET. His main research interests include parametric control systems design, nonlinear systems, descriptor systems, spacecraft control and magnetic bearing control. He is the author and co-author of 5 books and over 280 SCI indexed publications.

Appendix

Proofs of certain results

A.1. Proof of Theorem 3.4

In order to prove this theorem, the following lemma is needed, which can be easily proven using the controllability canonical form of linear systems (see, also the proof of the Theorem 2 in Duan, 2020b).

Lemma A.1

Let $[A,B]$ be controllable. Then, there exists a state transformation $z=T(x),$ under which the above system (2(2) $\begin{array}{r}x(k+1)=Ax\left(k\right)+Bu\left(k\right),\end{array}$(2) ) can be transformed into (A1) $\begin{array}{r}\{\begin{array}{l}{z}_{i1}(k+1)=z_{i2}\left(k\right)\\ z_{i2}(k+1)=z_{i3}\left(k\right)\\ ⋮\\ z_{i,{\mu }_i-1}(k+1)=z_{i{\mu }_i}\left(k\right)\\ {\displaystyle z_{i{\mu }_i}(k+1)=\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),}\\ i=1,2,\dots ,r,\end{array}\end{array}$(A1) where ${\mu }_i,i=1,2,\dots ,r$ are the controllability indices of $[A,B]$, and $a_{lj}^{(i)}$, $b_{il}$, $j=1,2,\dots ,{\mu }_l$, $i,l=1,2,\dots ,r$, with $b_{ii}=1,i=1,2,\dots ,r,$ are some real numbers.

Based on the above lemma, the result can be proven as follows.

Using the first r−1 equations in (A1(A1) $\begin{array}{r}\{\begin{array}{l}{z}_{i1}(k+1)=z_{i2}\left(k\right)\\ z_{i2}(k+1)=z_{i3}\left(k\right)\\ ⋮\\ z_{i,{\mu }_i-1}(k+1)=z_{i{\mu }_i}\left(k\right)\\ {\displaystyle z_{i{\mu }_i}(k+1)=\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),}\\ i=1,2,\dots ,r,\end{array}\end{array}$(A1) ), we can obtain (A2) $\begin{array}{r}{z}_{ij}\left(k\right)=z_{i1}(k+j-1),j=1,2,\dots ,{\mu }_i,i=1,2,\dots ,r,\end{array}$(A2) substituting these into the last one in system (A1(A1) $\begin{array}{r}\{\begin{array}{l}{z}_{i1}(k+1)=z_{i2}\left(k\right)\\ z_{i2}(k+1)=z_{i3}\left(k\right)\\ ⋮\\ z_{i,{\mu }_i-1}(k+1)=z_{i{\mu }_i}\left(k\right)\\ {\displaystyle z_{i{\mu }_i}(k+1)=\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),}\\ i=1,2,\dots ,r,\end{array}\end{array}$(A1) ), yields $\begin{array}{rl}{z}_{i1}(k+{\mu }_i)& =\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right)\\ & =\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{l1}(k+j-1)\\ & +\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),i=1,2,\dots ,r,\end{array}$which can be written in the form of $\begin{array}{rl}{z}_{p1}^{\lfloor {\mu }_p-1\rfloor }(k+1)& =L_i(z_{i1}^{\lfloor 0\sim {\mu }_i-1\rfloor }\left(k\right),i=1,2,\dots ,r)\\ & +\sum _{j=p}^r{b}_{pj}{u}_j\left(k\right),\\ & p=1,2,\dots ,r,\end{array}$where each $L_i(\cdot )$ is a linear function. Clearly, the above system can be written into the form of the following step forward HOFA model: (A3) $\begin{array}{r}{z_{p1}^{\lfloor {\mu }_p-1\rfloor }(k+1)|}_{p=1\sim r}=L\left({z_{p1}^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right)|}_{p=1\sim r}\right)+\hat{B}u\left(k\right),\end{array}$(A3) where $L(\cdot )$ is an extended vector of linear functions, and $\hat{B}$ is an upper triangular matrix with diagonal elements all being 1.

On the other side, it can also be obtained from the first r−1 equations in (A1(A1) $\begin{array}{r}\{\begin{array}{l}{z}_{i1}(k+1)=z_{i2}\left(k\right)\\ z_{i2}(k+1)=z_{i3}\left(k\right)\\ ⋮\\ z_{i,{\mu }_i-1}(k+1)=z_{i{\mu }_i}\left(k\right)\\ {\displaystyle z_{i{\mu }_i}(k+1)=\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),}\\ i=1,2,\dots ,r,\end{array}\end{array}$(A1) ) that (A4) $\begin{array}{r}{z}_{ij}\left(k\right)=z_{i{\mu }_i}(k-{\mu }_i+j),j=1,2,\dots ,{\mu }_i,i=1,2,\dots ,r.\end{array}$(A4) Substituting these into the last one in (A1(A1) $\begin{array}{r}\{\begin{array}{l}{z}_{i1}(k+1)=z_{i2}\left(k\right)\\ z_{i2}(k+1)=z_{i3}\left(k\right)\\ ⋮\\ z_{i,{\mu }_i-1}(k+1)=z_{i{\mu }_i}\left(k\right)\\ {\displaystyle z_{i{\mu }_i}(k+1)=\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),}\\ i=1,2,\dots ,r,\end{array}\end{array}$(A1) ), gives $\begin{array}{rl}{z}_{i{\mu }_i}(k+1)& =\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{lj}\left(k\right)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right)\\ & =\sum _{l=1}^r\sum _{j=1}^{{\mu }_l}{a}_{lj}^{\left(i\right)}{z}_{l{\mu }_l}(k-{\mu }_l+j)+\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right)\\ & =\sum _{l=1}^r\sum _{q=0}^{{\mu }_{l-1}}{a}_{l,{\mu }_l-q}^{\left(i\right)}{z}_{l,{\mu }_l}(k-q)\\ & +\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),i=1,2,\dots ,r,\end{array}$which can be written as $\begin{array}{rl}{z}_{i{\mu }_i}(k+1)& =L_i\left({z_{p{\mu }_p}^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right)|}_{p=1\sim r}\right)\\ & +\sum _{j=i}^r{b}_{ij}{u}_j\left(k\right),i=1,2,\dots ,r,\end{array}$where each $L_i(\cdot )$ is a linear function. While this can clearly be written in the form of the following step backward HOFA model: (A5) $\begin{array}{r}{z_{p{\mu }_p}(k+1)|}_{p=1\sim r}=L\left({z_{p{\mu }_p}^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right)|}_{p=1\sim r}\right)+\hat{B}u\left(k\right),\end{array}$(A5) where $L(\cdot )$ is an extended vector of linear function, and $\hat{B}$ is an upper triangular matrix with diagonal elements all being 1.

Finally, note that the above two processes are both invertible, the whole proof is done.

A.2. Proofs of some propositions

A.2.1. Proofs of Proposition 6.2 and its corollary

Firstly, let us prove Proposition 6.2.

Since system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is locally feedback linearizable at $(x_0,u_0)$ via the feedback (58(58) $\begin{array}{rl}u\left(k\right)& =\xi (x\left(k\right),v\left(k\right)),\\ \xi (x_0,0)& =u_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }v}\xi (x_0,0)=r,\end{array}$(58) ), by definition, there exists a neighbourhood of $(x_0,u_0),$ denoted by $\mathrm{\Omega }(x_0,u_0)$, such that the system is equivalent to the linear controllable system (60(60) $\begin{array}{rl}& y(k+1)\\ & =Ay\left(k\right)+Bv\left(k\right),y\left(k\right)\in {\mathbb{R}}^n,v\left(k\right)\in {\mathbb{R}}^r,\end{array}$(60) ) within $\mathrm{\Omega }(x_0,u_0).$

On the other side, it follows from Theorem 3.4 that there exists a one-to-one transformation (A6) $\begin{array}{r}y=T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right),\end{array}$(A6) under which the linear system (60(60) $\begin{array}{rl}& y(k+1)\\ & =Ay\left(k\right)+Bv\left(k\right),y\left(k\right)\in {\mathbb{R}}^n,v\left(k\right)\in {\mathbb{R}}^r,\end{array}$(60) ) is equivalent to the following step forward HOFA model: (A7) $\begin{array}{r}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }=L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)+\hat{B}v\left(k\right),\end{array}$(A7) where $L(\cdot )\in {\mathbb{R}}^r$ is a linear function, and $\hat{B}\in {\mathbb{R}}^{r\times r}$ is a nonsingular matrix.

Next, it follows from (58(58) $\begin{array}{rl}u\left(k\right)& =\xi (x\left(k\right),v\left(k\right)),\\ \xi (x_0,0)& =u_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }v}\xi (x_0,0)=r,\end{array}$(58) ) that the inverse of $\xi (x,v)$ with respect to v locally exists, that is, (A8) $\begin{array}{r}v={\xi }^{-1}(x,u),(x,u)\in \mathrm{\Omega }(x_0,u_0).\end{array}$(A8) Using the transformation (59(59) $\begin{array}{r}x=T_0\left(y\right),T_0\left(0\right)=x_0,\mathrm{rank}\frac{\mathrm{\partial }}{\mathrm{\partial }y}{T}_0\left(0\right)=n,\end{array}$(59) ), turns the above equation into (A9) $\begin{array}{r}v={\xi }^{-1}(T_0\left(y\right),u),(y,u)\in \mathrm{\Omega }(0,u_0).\end{array}$(A9) Further applying the transformation $T_1$ in (A6(A6) $\begin{array}{r}y=T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right),\end{array}$(A6) ), gives (A10) $\begin{array}{rl}v& ={\xi }^{-1}(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right),u),\\ & (z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta },u)\in \mathrm{\Omega }(0,u_0).\end{array}$(A10) Finally, substituting the above equation into (A7(A7) $\begin{array}{r}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }=L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)+\hat{B}v\left(k\right),\end{array}$(A7) ) produces (A11) $\begin{array}{rl}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }& =L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)\\ & +\hat{g}(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },u\left(k\right)),\end{array}$(A11) where $\begin{array}{r}\hat{g}(\cdot )=\hat{B}{\xi }^{-1}(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)\right),u\left(k\right)).\end{array}$It is easily recognized that, on $\mathrm{\Omega }(0,u_0),$ the function $\hat{g}(\cdot )$ is one-to-one with respect to $u(k).$ Therefore, the above system (A11(A11) $\begin{array}{rl}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }& =L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)\\ & +\hat{g}(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },u\left(k\right)),\end{array}$(A11) ) is clearly a step forward HOFA model in the formof (40(40) $\begin{array}{rl}& x_p(k+{\mu }_p){|}_{p=1\sim \eta }\\ & =f(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)).\end{array}$(40) ).

Again, it follows from Theorem 3.4 that there exists a one-to-one transformation (A12) $\begin{array}{r}y={\tilde{T}}_1\left(z_p^{\lceil 0\sim {\mu }_p-1\rceil }{|}_{p=1\sim \eta }\right),\end{array}$(A12) under which the linear system (60(60) $\begin{array}{rl}& y(k+1)\\ & =Ay\left(k\right)+Bv\left(k\right),y\left(k\right)\in {\mathbb{R}}^n,v\left(k\right)\in {\mathbb{R}}^r,\end{array}$(60) ) is equivalent to the following step backward HOFA model (A13) $\begin{array}{r}{z}_{1\sim \eta }(k+1)=\tilde{L}\left(z_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }\right)+\tilde{B}v\left(k\right),\end{array}$(A13) where $\tilde{L}(\cdot )\in {\mathbb{R}}^r$ is a linear function, and $\tilde{B}\in {\mathbb{R}}^{r\times r}$ is a nonsingular matrix. By a similar procedure, it can be shown that the above system (A13(A13) $\begin{array}{r}{z}_{1\sim \eta }(k+1)=\tilde{L}\left(z_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta }\right)+\tilde{B}v\left(k\right),\end{array}$(A13) ) is equivalent to a step backward HOFA model in the form of (41(41) $\begin{array}{rl}{x}_{1\sim \eta }(k+1)& =f(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k)\\ & +g(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },k,u\left(k\right)),\end{array}$(41) ). The whole proof of Proposition 6.2 is then completed.

Next, let us prove Corollary 6.3.

When the system (57(57) $\begin{array}{r}x(k+1)=f(x\left(k\right),u\left(k\right)),\end{array}$(57) ) is locally feedback linearizable at $(x_0,u_0)$ via the feedback (61(61) $\begin{array}{r}u\left(k\right)=\xi (x\left(k\right),v\left(k\right))=\alpha \left(x\left(k\right)\right)+\beta \left(x\left(k\right)\right)v\left(k\right),\end{array}$(61) )–(62(62) $\begin{array}{r}\alpha \left(x_0\right)=u_0,\mathrm{rank}\beta \left(x\right)=r,\end{array}$(62) ), we have, instead of (A8(A8) $\begin{array}{r}v={\xi }^{-1}(x,u),(x,u)\in \mathrm{\Omega }(x_0,u_0).\end{array}$(A8) ), the following (A14) $\begin{array}{r}v={\beta }^{-1}\left(x\right)u-{\beta }^{-1}\left(x\right)\alpha \left(x\right),(x,u)\in \mathrm{\Omega }(x_0,u_0).\end{array}$(A14) Further, using the transformations $T_0$ and $T_1,$ yields, for the step forward case, (A15) $\begin{array}{rl}v& ={\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right)u\\ & -{\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right)\\ & \times \alpha \left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right),\end{array}$(A15) for $(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta },u)\in \mathrm{\Omega }(0,u_0).$ Substituting the above (A15(A15) $\begin{array}{rl}v& ={\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right)u\\ & -{\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right)\\ & \times \alpha \left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right),\end{array}$(A15) ) into (A7(A7) $\begin{array}{r}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }=L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)+\hat{B}v\left(k\right),\end{array}$(A7) ), produces (A16) $\begin{array}{rl}{z}_p^{\lfloor {\mu }_p-1\rfloor }(k+1){|}_{p=1\sim \eta }& =\breve{f}\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)\\ & +\breve{B}\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)u\left(k\right),\end{array}$(A16) where $\begin{array}{rl}\breve{f}(\cdot )& =L\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }\left(k\right){|}_{p=1\sim \eta }\right)\\ & -\hat{B}{\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right)\\ & \times \alpha \left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right),\end{array}$and $\begin{array}{r}\breve{B}(\cdot )=\hat{B}{\beta }^{-1}\left(T_0\left(T_1\left(z_p^{\lfloor 0\sim {\mu }_p-1\rfloor }{|}_{p=1\sim \eta }\right)\right)\right).\end{array}$The case of step backward systems can be carried out similarly.

A.2.2. Proof of Proposition 6.4

Let us first consider the case of $n=2,$ where the system is (A17) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)u\left(k\right).\end{array}\end{array}$(A17) From the first equation, we get $\begin{array}{r}{x}_2\left(k\right)=g_1^{-1}\left(x_1\left(k\right)\right)[x_1(k+1)-f_1\left(x_1\left(k\right)\right)].\end{array}$Substituting this into the second equation in (A17(A17) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)u\left(k\right).\end{array}\end{array}$(A17) ) gives (A18) $\begin{array}{r}{x}_2(k+1)={\tilde{f}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)+{\tilde{g}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)u\left(k\right),\end{array}$(A18) where ${\tilde{f}}_2(\cdot )$ and ${\tilde{g}}_2(\cdot )$ are some nonlinear functions.

Next, using the first equation in (A17(A17) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)u\left(k\right).\end{array}\end{array}$(A17) ) again, yields $\begin{array}{r}{x}_1(k+2)=f_1\left(x_1(k+1)\right)+g_1\left(x_1(k+1)\right)x_2(k+1).\end{array}$Substituting (A18(A18) $\begin{array}{r}{x}_2(k+1)={\tilde{f}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)+{\tilde{g}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)u\left(k\right),\end{array}$(A18) ) into the above equation, produces $\begin{array}{rl}{x}_1(k+2)& =f_1\left(x_1(k+1)\right)+g_1\left(x_1(k+1)\right){\tilde{f}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)\\ & +g_1\left(x_1(k+1)\right){\tilde{g}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)u\left(k\right),\end{array}$which can be written as (A19) $\begin{array}{r}{x}^{\lfloor 1\rfloor }(k+1)=f\left(x^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)+B\left(x^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)u\left(k\right),\end{array}$(A19) where $\begin{array}{rl}f\left(x^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)& =f_1\left(x_1(k+1)\right)\\ & +g_1\left(x_1(k+1)\right){\tilde{f}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right),\\ B\left(x^{\lfloor 0\sim 1\rfloor }\left(k\right)\right)& =g_1\left(x_1(k+1)\right){\tilde{g}}_2\left(x_1^{\lfloor 0\sim 1\rfloor }\left(k\right)\right).\end{array}$This is obviously a step forward HOFA system in the form of (28(28) $\begin{array}{rl}x(k+n)& =f(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)\\ & +B(x^{\lfloor 0\sim n-1\rfloor }\left(k\right),k)u\left(k\right),\end{array}$(28) ), with $n=2.$

For the case of $n>2,$ the proof can be also fulfilled with the help of the method of mathematical induction, as done for the continuous-time case in Duan (2021f).

To show that system (63(63) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1\left(x_1\left(k\right)\right)+g_1\left(x_1\left(k\right)\right)x_2\left(k\right)\\ x_2(k+1)=f_2\left(x_{1\sim 2}\left(k\right)\right)+g_2\left(x_{1\sim 2}\left(k\right)\right)x_3\left(k\right)\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)\\ +g_{n-1}\left(x_{1\sim n-1}\left(k\right)\right)x_n\left(k\right)\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+g_n\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(63) ) can be equivalently converted into a step backward HOFA system in the form of (29(29) $\begin{array}{rl}x(k+1)& =f(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)\\ & +B(x^{\lceil 0\sim n-1\rceil }\left(k\right),k)u\left(k\right).\end{array}$(29) ), without loss of generality, let us treat the case of n = 3, in which the system is (A20) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_1(k))+g_1(x_1(k))x_2(k),\\ x_2(k+1)=f_2(x_{1\sim 2}(k))+g_2(x_{1\sim 2}(k))x_3(k),\\ x_3(k+1)=f_3(x_{1\sim 3}(k))+g_3(x_{1\sim 3}(k))u(k).\end{array}\end{array}$(A20) From the first equation in (A20(A20) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_1(k))+g_1(x_1(k))x_2(k),\\ x_2(k+1)=f_2(x_{1\sim 2}(k))+g_2(x_{1\sim 2}(k))x_3(k),\\ x_3(k+1)=f_3(x_{1\sim 3}(k))+g_3(x_{1\sim 3}(k))u(k).\end{array}\end{array}$(A20) ) we have (A21) $\begin{array}{rl}{x}_1(k+1)& =f_1(x_1(k))+g_1(x_1(k))x_2(k)\\ & \triangleq {\overline{x}}_2(k),\end{array}$(A21) by which and the second equation of (A20(A20) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_1(k))+g_1(x_1(k))x_2(k),\\ x_2(k+1)=f_2(x_{1\sim 2}(k))+g_2(x_{1\sim 2}(k))x_3(k),\\ x_3(k+1)=f_3(x_{1\sim 3}(k))+g_3(x_{1\sim 3}(k))u(k).\end{array}\end{array}$(A20) ), we can further obtain (A22) $\begin{array}{rl}{\overline{x}}_2(k+1)& =f_1(x_1(k+1))+g_1(x_1(k+1))x_2(k+1)\\ & =f_1(x_1(k+1))\\ & +g_1(x_1(k+1))(f_2(x_{1\sim 2}(k))\\ & +g_2(x_{1\sim 2}(k))x_3(k))\\ & \triangleq {\overline{f}}_2(x_{1\sim 2}(k))+{\overline{g}}_2(x_{1\sim 2}(k))x_3(k)\\ & \triangleq {\overline{x}}_3(k),\end{array}$(A22) where ${\overline{g}}_2(\cdot )=g_1{g}_2$ is clearly nonsingular.

Similarly, using the third equation in (A20(A20) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_1(k))+g_1(x_1(k))x_2(k),\\ x_2(k+1)=f_2(x_{1\sim 2}(k))+g_2(x_{1\sim 2}(k))x_3(k),\\ x_3(k+1)=f_3(x_{1\sim 3}(k))+g_3(x_{1\sim 3}(k))u(k).\end{array}\end{array}$(A20) ) gives (A23) $\begin{array}{rl}{\overline{x}}_3(k+1)& ={\overline{f}}_2\left(x_{1\sim 2}(k+1)\right)+{\overline{g}}_2(x_{1\sim 2}(k+1))x_3(k+1)\\ & ={\overline{f}}_2\left(x_{1\sim 2}(k+1)\right)\\ & +{\overline{g}}_2(x_{1\sim 2}(k+1))(f_3(x_{1\sim 3}(k))\\ & +g_3(x_{1\sim 3}(k))u(k))\\ & \triangleq {\overline{f}}_3(x_{1\sim 3}(k))+{\overline{g}}_3(x_{1\sim 3}(k))u(k),\end{array}$(A23) where ${\overline{g}}_3(\cdot )={\overline{g}}_2{g}_3$ is also nonsingular.

Notice that $\begin{array}{r}\left[\begin{array}{c}{\overline{x}}_1\left(k\right)\\ {\overline{x}}_2(k)\\ {\overline{x}}_3(k)\end{array}\right]=\left[\begin{array}{c}{x}_1(k)\\ f_1(x_1(k))+g_1(x_1(k))x_2(k)\\ {\overline{f}}_2\left(x_{1\sim 2}\left(k\right)\right)+{\overline{g}}_2(x_{1\sim 2}\left(k\right))x_3(k)\end{array}\right],\end{array}$from which we can solve recursively $\begin{array}{rl}{x}_1(k)& ={\overline{x}}_1\left(k\right),\\ x_2(k)& =g_1^{-1}({\overline{x}}_1(k))({\overline{x}}_2(k)-f_1({\overline{x}}_1(k)))\\ & \triangleq h_2\left({\overline{x}}_{1\sim 2}\left(k\right)\right),\\ x_3(k)& ={\overline{g}}_2^{-1}(x_{1\sim 2}\left(k\right))({\overline{x}}_3(k)-{\overline{f}}_2\left(x_{1\sim 2}\left(k\right)\right))\\ & \triangleq h_3\left({\overline{x}}_{1\sim 3}\left(k\right)\right).\end{array}$Inserting the above into (A23(A23) $\begin{array}{rl}{\overline{x}}_3(k+1)& ={\overline{f}}_2\left(x_{1\sim 2}(k+1)\right)+{\overline{g}}_2(x_{1\sim 2}(k+1))x_3(k+1)\\ & ={\overline{f}}_2\left(x_{1\sim 2}(k+1)\right)\\ & +{\overline{g}}_2(x_{1\sim 2}(k+1))(f_3(x_{1\sim 3}(k))\\ & +g_3(x_{1\sim 3}(k))u(k))\\ & \triangleq {\overline{f}}_3(x_{1\sim 3}(k))+{\overline{g}}_3(x_{1\sim 3}(k))u(k),\end{array}$(A23) ) and copying (A21(A21) $\begin{array}{rl}{x}_1(k+1)& =f_1(x_1(k))+g_1(x_1(k))x_2(k)\\ & \triangleq {\overline{x}}_2(k),\end{array}$(A21) )–(A22(A22) $\begin{array}{rl}{\overline{x}}_2(k+1)& =f_1(x_1(k+1))+g_1(x_1(k+1))x_2(k+1)\\ & =f_1(x_1(k+1))\\ & +g_1(x_1(k+1))(f_2(x_{1\sim 2}(k))\\ & +g_2(x_{1\sim 2}(k))x_3(k))\\ & \triangleq {\overline{f}}_2(x_{1\sim 2}(k))+{\overline{g}}_2(x_{1\sim 2}(k))x_3(k)\\ & \triangleq {\overline{x}}_3(k),\end{array}$(A22) ), give the following system (which is equivalent to (A20(A20) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_1(k))+g_1(x_1(k))x_2(k),\\ x_2(k+1)=f_2(x_{1\sim 2}(k))+g_2(x_{1\sim 2}(k))x_3(k),\\ x_3(k+1)=f_3(x_{1\sim 3}(k))+g_3(x_{1\sim 3}(k))u(k).\end{array}\end{array}$(A20) )) (A24) $\begin{array}{r}\{\begin{array}{l}{\overline{x}}_1(k+1)={\overline{x}}_2(k),\\ {\overline{x}}_2(k+1)={\overline{x}}_3\left(k\right),\\ {\overline{x}}_3(k+1)={\tilde{f}}_3\left({\overline{x}}_{1\sim 3}\left(k\right)\right)+{\tilde{g}}_3({\overline{x}}_{1\sim 3}\left(k\right))u(k),\end{array}\end{array}$(A24) where ${\tilde{g}}_3(\cdot )$ is nonzero.

Finally, inserting the first two equations of (A24(A24) $\begin{array}{r}\{\begin{array}{l}{\overline{x}}_1(k+1)={\overline{x}}_2(k),\\ {\overline{x}}_2(k+1)={\overline{x}}_3\left(k\right),\\ {\overline{x}}_3(k+1)={\tilde{f}}_3\left({\overline{x}}_{1\sim 3}\left(k\right)\right)+{\tilde{g}}_3({\overline{x}}_{1\sim 3}\left(k\right))u(k),\end{array}\end{array}$(A24) ) into the third one, gives the backward HOFA system $\begin{array}{rl}{\overline{x}}_3(k+1)& ={\tilde{f}}_3({\overline{x}}_3(k-2),{\overline{x}}_3(k-1),{\overline{x}}_3\left(k\right))\\ & +{\tilde{g}}_3({\overline{x}}_3(k-2),{\overline{x}}_3(k-1),{\overline{x}}_3\left(k\right))u(k).\end{array}$The proof is then complete.

A.2.3. Proof of Proposition 7.4

Rewriting the first equation in (77(77) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_n\left(k\right),u\left(k\right))\\ x_2(k+1)=f_2(x_1\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_3(k+1)=f_3(x_{1\sim 2}\left(k\right),x_n\left(k\right),u\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{1\sim n-1}\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+B\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(77) ) as $\begin{array}{r}{x}_1\left(k\right)=f_1(x_n(k-1),u(k-1)),\end{array}$and substituting it into the rest ones in (77(77) $\begin{array}{r}\{\begin{array}{l}{x}_1(k+1)=f_1(x_n\left(k\right),u\left(k\right))\\ x_2(k+1)=f_2(x_1\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_3(k+1)=f_3(x_{1\sim 2}\left(k\right),x_n\left(k\right),u\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{1\sim n-1}\left(k\right),x_n\left(k\right),u\left(k\right))\\ x_n(k+1)=f_n\left(x_{1\sim n}\left(k\right)\right)+B\left(x_{1\sim n}\left(k\right)\right)u\left(k\right),\end{array}\end{array}$(77) ), yield (A25) $\begin{array}{r}\{\begin{array}{l}{x}_2(k+1)=f_2(x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{2\sim n-2}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ +B(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A25) Again, rewriting the first equation in (A25(A25) $\begin{array}{r}\{\begin{array}{l}{x}_2(k+1)=f_2(x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{2\sim n-2}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ +B(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A25) ) as $\begin{array}{r}{x}_2\left(k\right)=f_2(x_n^{\lceil 0\sim 1\rceil }(k-1),u^{\lceil 1\rceil }(k-1)),\end{array}$and substituting it into the rest equations in (A25(A25) $\begin{array}{r}\{\begin{array}{l}{x}_2(k+1)=f_2(x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{2\sim n-2}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))\\ +B(x_{2\sim n-1}\left(k\right),x_n^{\lceil 0\sim 1\rceil }\left(k\right),u^{\lceil 1\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A25) ), give (A26) $\begin{array}{r}\{\begin{array}{l}{x}_3(k+1)=f_3(x_n^{\lceil 0\sim 2\rceil }\left(k\right),u^{\lceil 1\sim 2\rceil }\left(k\right))\\ ⋮\\ x_{n-1}(k+1)=f_{n-1}(x_{3\sim n-1}\left(k\right),x_n^{\lceil 0\sim 2\rceil }\left(k\right),u^{\lceil 1\sim 2\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{3\sim n-1}\left(k\right),x_n^{\lceil 0\sim 2\rceil }\left(k\right),u^{\lceil 1\sim 2\rceil }\left(k\right))\\ +B(x_{3\sim n-1}\left(k\right),x_n^{\lceil 0\sim 2\rceil }\left(k\right),u^{\lceil 1\sim 2\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A26) Continuing this process, after n−2 times of operation, we obtain (A27) $\begin{array}{r}\{\begin{array}{l}{x}_{n-1}(k+1)=f_{n-1}(x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{n-1}\left(k\right),x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))\\ +B(x_{n-1}\left(k\right),x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A27) Finally, substituting the first equation in (A27(A27) $\begin{array}{r}\{\begin{array}{l}{x}_{n-1}(k+1)=f_{n-1}(x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))\\ x_n(k+1)=f_n(x_{n-1}\left(k\right),x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))\\ +B(x_{n-1}\left(k\right),x_n^{\lceil 0\sim n-2\rceil }\left(k\right),u^{\lceil 1\sim n-2\rceil }\left(k\right))u\left(k\right).\end{array}\end{array}$(A27) ) into its second one, gives (A28) $\begin{array}{rl}{x}_n(k+1)& =f_n(x_n^{\lceil 0\sim n-1\rceil }\left(k\right),u^{\lceil 1\sim n-1\rceil }\left(k\right))\\ & +B(x_n^{\lceil 0\sim n-1\rceil }\left(k\right),u^{\lceil 1\sim n-1\rceil }\left(k\right))u\left(k\right).\end{array}$(A28) This is clearly a step backward HOFA model in the form of (65(65) $\begin{array}{rl}& \left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\\ ⋮\\ x_{\eta }(k+1)\end{array}\right]\\ & =\left[\begin{array}{c}{f}_1(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_1\rceil }\left(k\right),k)\\ f_2(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_2\rceil }\left(k\right),k)\\ ⋮\\ f_{\eta }(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta }\rceil }\left(k\right),k)\end{array}\right]\\ & +B(x_p^{\lceil 0\sim {\mu }_p-1\rceil }\left(k\right){|}_{p=1\sim \eta },u^{\lceil 1\sim m_{\eta +1}\rceil }\left(k\right),k)u\left(k\right),\end{array}$(65) ). The proof is completed.