An extended-observer approach to robust stabilisation of linear differential-algebraic systems

ABSTRACT

This paper addresses the problem of robustly stabilising a class of linear differential-algebraic systems characterised by autonomous and asymptotically stable zero dynamics, in spite of parameter uncertainties ranging over a priori fixed bounded sets. We exploit recent results related to the structural properties and normal forms of this class of systems and propose a robust control that asymptotically recovers, in practical terms, the performance of a nominal, though non-implementable, stabilising control. The proposed control combines a partial output feedback control, aimed at letting the system behave as a regular system, and a robust control, based on an extended observer, using which the dynamic of the closed loop system is rendered arbitrarily close to the one of a properly selected stable system. The extended observer, originally conceived in the context of standard differential systems, is here shown to be the key ingredient for robustly stabilising the targeted class of differential-algebraic systems.

KEYWORDS:

• Linear differential-algebraic systems
• robust stabilisation
• extended observer

ORCID

Alessandro DiGiorgio http://orcid.org/0000-0002-4171-526X

Antonio Pietrabissa http://orcid.org/0000-0003-0188-3346

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Throughout most of the paper, we follow the notation used in Berger (2016). Specifically, here by $\mathbb{R}\left[s\right]$ we denote the ring of polynomials with coefficients in $\mathbb{R}$.

2 See Berger (2016) for a precise definition of the concept of an autonomous zero dynamics. Roughly speaking, the zero dynamics are said to be autonomous if the choice of the input $u\left(t\right)$ cannot have an influence on the (forced) internal motions that are consistent with the constraint $y\left(t\right)\equiv 0$. In the case of a system modelled by ordinary differential equations, the zero dynamics are autonomous whenever ${\mathcal{R}}^{\ast }$, the largest controllability subspace contained in $\ker \left(C\right)$, is $\left\{0\right\}$.

3 Note, in this respect, that assumption (2(2) ${\mathrm{rank}}_{\mathbb{R}\left[s\right]}\left[\begin{array}{cc}sE-A& -B\\ -C& 0\end{array}\right]=n+m.$(2) ) implies $\ell +p\ge n+m$

4 Nota also that the internal dynamics consistent with the constraint $y\left(t\right)\equiv 0$ are those of ${\dot{x}}_1=A_{11}{x}_1$.

5 Here $\mathbb{R}\left(s\right)$ denotes the quotient field of $\mathbb{R}\left[s\right]$. The matrix $L\left(s\right)$ is any matrix with entries in $\mathbb{R}\left(s\right)$ satisfying $L\left(s\right)\left(\begin{array}{cc}sE-A& -B\\ -C& 0\end{array}\right)=I_{n+m}.$

6 See Berger (2016), Lemma A.1.

7 The procedure described below can be extended, without difficulties, to the case m>p.

8 It is worth observing that, in case m=1, (13(13) $\parallel [{\overline{E}}_{11}^{-1}-B_0]\mathrm{\Lambda }{B}_0^{-1}\parallel \le {\delta }_0$(13) ) reduces to $|\frac{E_{11}^{-1}-B_0}{B_0}|\le {\delta }_0<1$ which, in turn, holds if there exists two numbers $b_{\min },b_{\max }$ such that $0<b_{\min }\le |E_{11}^{-1}|\le b_{\max }$. The condition in (13(13) $\parallel [{\overline{E}}_{11}^{-1}-B_0]\mathrm{\Lambda }{B}_0^{-1}\parallel \le {\delta }_0$(13) ) can be regarded as a multivariable version of such assumption.

9 Knowing that the states $(\hat{\xi },\sigma )$ of the observer can be uniquely expressed as functions of $\overline{x}$ and e, we denote by $(\overline{x},e)$ the arguments of the functions defined below.

10 Here and in the remaining portion of the paper, we denote by $B_R$ the closed ball of radius R, with the tacit understanding that the space in which the ball is considered is specified by the context.

11 Equation (35(35) $\begin{array}{rl}\dot{z}& =Fz+G_1\xi +G_2{x}_2,\\ \dot{\xi }& =A_{11}\xi +A_{12}{x}_2+H_{1}z+B{G}_L\left(\psi \right(\hat{\xi },\sigma \left)\right),\\ \dot{e}& =\kappa [\mathbf{\text{A}}-{\mathbf{\text{B}}}_2{\mathrm{\Delta }}_0(\overline{x},e\left)\mathbf{\text{C}}\right]e+{\mathbf{\text{B}}}_1{\mathrm{\Delta }}_1(\overline{x},e)\\ & +{\mathbf{\text{B}}}_2{\mathrm{\Delta }}_2(\overline{x},e)+{\kappa }^2{\mathbf{\text{B}}}_3(\overline{x},e)x_2,\\ 0& =A_{21}\xi +[A_{22}+hI]x_2+H_{2}z+S_0{G}_L\left(\psi \right(\hat{\xi },\sigma \left)\right)\end{array}$(35) ) is a DAE of the form (49) $E\dot{\mathcal{X}}\left(t\right)=\mathcal{F}\left(\mathcal{X}\right(t\left)\right)$(49)

in which $\mathcal{F}:{\mathbb{R}}^{\nu }\to {\mathbb{R}}^{\nu }$ is locally Lypschitz and $\mathcal{X}\left(t\right)=\mathrm{col}\left(z\right(t),\xi (t),e(t),x_2(t\left)\right)$. Let I be an open interval of $\mathbb{R}$, containing the origin. Given ${\mathcal{X}}_0\in {\mathbb{R}}^{\nu }$, a continuously differentiable function $\mathcal{X}:I\to {\mathbb{R}}^{\nu }$ is a solution of the initial value problem $\mathcal{X}\left(0\right)={\mathcal{X}}_0$ if $\mathcal{X}\left(t\right)$ satisfies (49(49) $E\dot{\mathcal{X}}\left(t\right)=\mathcal{F}\left(\mathcal{X}\right(t\left)\right)$(49) ) for all $t\in I$ and $\mathcal{X}\left(0\right)={\mathcal{X}}_0$. The initial value $\mathcal{X}\left(0\right)$ is consistent if at least one solution exists to the initial value problem $\mathcal{X}\left(0\right)={\mathcal{X}}_0$. It is seen from all of the above that, if h is large, $x_2\left(t\right)=X_{2h}\left(z\right(t),\xi (t),e(t\left)\right)$. Thus the initial value ${\mathcal{X}}_0=\mathrm{col}(z_0,{\xi }_0,e_0,x_{2,0})$ is consistent if and only if $x_{2,0}=X_{2h}(z_0,{\xi }_0,e_0)$. For all such consistent initial values, a solution of Equation (34) can be determined inserting $x_2=X_{2h}(z,\xi ,e)$ in the top three equations and then solving the resulting ODE. In what follows, we will show that – if the design parameters are appropriately chosen – a unique solution of such ODE exists, defined over an interval $I\supset [0,\mathrm{\infty })$.

12 Use the comparison Lemma to get $U\left(t\right)\le {\mathrm{e}}^{-2{\alpha }_{\kappa }t}U\left(0\right)+\frac{{\epsilon }^2}{2{\alpha }_{\kappa }}$

from which, using the estimates (37(37) $a_1\parallel e{\parallel }^2\le U\left(e\right)\le a_2\parallel e{\parallel }^2,$(37) ) and the fact that $2{\alpha }_{\kappa }{a}_1>1$, the claimed inequality follows.

13 Note that μ depends on ϵ and, actually, increases as ϵ decreases. This fact, however, does not affect the prior conclusion.

14 For the notion of input-to-state stability and the associated small-gain theorem, see e.g. Khalil (2002).