Observer-based event-triggered control and application in active suspension vehicle systems

Abstract

This paper focuses on event-trigger control of automotive suspension systems. Firstly, fuzzy T-S systems which suitable for automobile suspension systems are studied. Parameter uncertainties and disturbances are included in the fuzzy T-S systems. Secondly, based on linear matrix inequalities (LMIs) and Lyapunov function, the conditions for the stability of fuzzy T-S systems are given. In the meantime, the controller and observer of the fuzzy T-S systems are designed. Finally, the theory is applied to the automotive suspension systems in the simulation.

Keywords:

• Event-trigger control
• suspension vehicle systems
• output-feedback
• disturbances
• parameter uncertainties

1. Introduction

More and more attention has been paid to the quality of automobile performance. Comfort is an important factor to measure the automobile performance. How to improve the driving comfort is a hot issue, the automobile suspension systems are closely related to the comfort. Because of the change of vehicle load and passenger number, the vehicle suspension systems become parameter uncertain systems. It is more suitable to use T-S systems to study automobile suspension systems (Jamal et al., 2019; Li et al., 2011; Wang et al., 2019).

Event-trigger is a core concept in modern industrial information technology. In recent years, the research of event-triggered technology has made rapid progress in the control field (Han & Antsaklis, 2013; Hu & Yue, 2012; Wang et al., 2017). For linear networked systems, Hu and Yue (2012) designed event-triggered control design. Han and Antsaklis (2013) gave event-triggered observer and observer-based controller for networked systems. However, there is no vehicle suspension system involved in the above event-trigger control.

In this paper, for the automotive suspension systems, we study event-trigger control. Firstly, fuzzy T-S systems which parameter uncertainties and disturbances are considered. Using LMIs and Lyapunov function, the conditions of the stability are proposed. In numerical simulation, the fuzzy model is applied to the vehicle suspension systems.

The main contributions of this paper can be summarized as follows:

1. The paper studies observer-based event-triggered control and application in active suspension vehicle systems. Note that (Han & Antsaklis, 2013; Hu & Yue, 2012) also studied event-trigger control, however, control methods are not applied to the vehicle suspension system.

2. Event-trigger control methods are studied in active suspension vehicle systems. Note that (Jamal et al., 2019; Wang et al., 2021) also studied the vehicle suspension system, however, Jamal et al. (2019) and Wang et al. (2021) investigated control methods are not event-triggered control methods.

2. Problem statement

The dynamic equation of the quarter-car suspensionsystem is established as Wang et al. (2019) (1) $\begin{array}{r}\{\begin{array}{l}{m}_s{\ddot{z}}_s(t)+c_s[{\dot{z}}_s(t)-{\dot{z}}_u(t)]+k_s[z_s(t)-z_u(t)]=u(t)\\ m_u{\ddot{z}}_u(t)+c_s[{\dot{z}}_u(t)-{\dot{z}}_s(t)]+k_s[z_u(t)-z_s(t)]\\ +k_t[z_u(t)-z_r(t)]+c_t[{\dot{z}}_u(t)-{\dot{z}}_r(t)]=-u(t)\end{array}\end{array}$(1) where $z_s(t)-z_u(t)$ is relative displacement of vehicle body, $z_u(t)-z_r(t)$ is relative displacement of vehicle wheel, ${\dot{z}}_s(t)$ is velocity of vehicle wheel vibration.Denote $x(t)=[z_s(t)-z_u(t)z_u(t)-z_r(t){\dot{z}}_s(t){\dot{z}}_u(t)]{}^T\in R^4$, $z(t)=[{\dot{z}}_s(t)z_s(t)-z_u(t)z_u(t)-z_r(t)]{}^T\in R^3$, $y(t)=[{\dot{z}}_s(t)z_s(t)-z_u(t)]{}^T\in R^2$. The quarter-car suspension system is shown in Figure 1.

Figure 1. Quarter-car suspension system.

In order to study the automobile suspension system, we first study the following fuzzy system. In the simulation, the theoretical part is applied to the automobile suspension system. The considered T-S fuzzy systems with the following state space equation.

If ${\sigma }_1(t)$ is $F_{i,1}$ and ···  and ${\sigma }_{p(t)}$ is $F_{ip}$, then (2) $\begin{array}{r}\{\begin{array}{l}\dot{x}(t)=(A_i+\mathrm{\Delta }{A}_i)x(t)+(B_i+\mathrm{\Delta }{B}_i)u(t)+W_{1i}{w}_1(t)\\ z(t)=C_{1i}x(t)\\ y(t)=C_{2i}x(t)+W_{2i}{w}_2(t)\end{array}\end{array}$(2) where $F_{ij}$ denotes fuzzy set; ${\sigma }_j(t)$ denotes premise variable vector; $x(t)$ denotes state vector; $u(t)$ denotes input vector; $y(t)$ denotes output vector; $w_1(t)$ denotes input disturbance; $w_2(t)$ denotes output disturbance; $A_i$, $B_i$, $C_{1i}$, $C_{2i}$, $W_{1i}$, $W_{2i}$ denote given matrices; $\mathrm{\Delta }{A}_i$, $\mathrm{\Delta }{B}_i$ denote the parametric uncertainties.

The T-S fuzzy systems can be described as follows: (3) $\begin{array}{r}\{\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{(A_i+\mathrm{\Delta }{A}_i)x(t)+(B_i+\mathrm{\Delta }{B}_i)u(t)}\\ +W_{1i}{w}_1(t)\}\\ z(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))C_{1i}{x}_i(t)}\\ y(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{C_{2i}{x}_i(t)+W_{2i}{w}_2(t)}\end{array}\end{array}$(3) where ${\alpha }_i(\sigma (t))=\prod _{j=1}^p{F}_{ij}(\sigma (t))$, ${\alpha }_i(\sigma (t))\ge 0$, ${\mu }_i(\sigma (t))={\alpha }_i(\sigma (t))/\phantom{{\alpha }_i(\sigma (t)){\sum }_{i=1}^r{\alpha }_i(\sigma (t))}-{\sum }_{i=1}^r{\alpha }_i(\sigma (t))$, $\sum _{i=1}^r{\mu }_i(\sigma (t))=1$.

The fuzzy state observer is expressed by (4) $\begin{array}{r}\{\begin{array}{l}\dot{\hat{x}}(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))[A_i\hat{x}(t)+B_{i}u(t)+L_i(y(t)-\hat{y}(t))]}\\ \hat{y}(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))C_{2i}\hat{x}(t)}\end{array}\end{array}$(4) The event-trigger condition is designed as follows (Wang et al., 2017) (5) $f(t)=e^T(t){\mathrm{\Gamma }}_{l}e(t)-\rho {\hat{x}}^T(t){\mathrm{\Gamma }}_l\hat{x}(t)<0$(5) where ρ is a positive number, ${\mathrm{\Gamma }}_l$ is a designed matrix, $e(t)$ is estimation error, $e(t)=\hat{x}(t_{r}h)-\hat{x}(t)$, $t\in [t_r,t_{r+1})$.

3. Controller design and main results

The controller is designed based on the sampled observer If ${\sigma }_1(t_{r}h)$ is $F_{i1}$, and ${\sigma }_2(t_{r}h)$ is $F_{i2}$ and ···  and ${\sigma }_p(t_{r}h)$ is $F_{ip}$, then (6) $u(t)=K_i\hat{x}(t_{r}h),i=1,\dots ,N$(6) where $K_i$ are control gain matrices with appropriate dimension. Fuzzy controller is designed as follows: (7) $u(t)=\sum _{i=1}^r{\mu }_i(\sigma (t_{r}h))K_i\hat{x}(t_{r}h)$(7) The difference between ${\mu }_i(\sigma (t))$ and ${\mu }_i(\sigma (t_{r}h))$ is considered, the following inequalities hold (Zhao et al., 2015) $\begin{array}{rl}{\overline{\mu }}_i& ={\epsilon }_i{\mu }_i,\left|{\overline{\mu }}_i-{\mu }_i\right|\le {\mathrm{\Delta }}_i\\ {\mu }_i& ={\mu }_i(\sigma (t)),{\overline{\mu }}_i={\mu }_i(\sigma (t_{r}h))\end{array}$where ${\epsilon }_i$ and ${\mathrm{\Delta }}_i$ are two positive scalars.

By inserting (7(7) $u(t)=\sum _{i=1}^r{\mu }_i(\sigma (t_{r}h))K_i\hat{x}(t_{r}h)$(7) ) into (3(3) $\begin{array}{r}\{\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{(A_i+\mathrm{\Delta }{A}_i)x(t)+(B_i+\mathrm{\Delta }{B}_i)u(t)}\\ +W_{1i}{w}_1(t)\}\\ z(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))C_{1i}{x}_i(t)}\\ y(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{C_{2i}{x}_i(t)+W_{2i}{w}_2(t)}\end{array}\end{array}$(3) ), observer equation is expressed as follows: (8) $\begin{array}{rl}\dot{\hat{x}}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{(A_i+B_i{K}_j)\hat{x}(t)+B_i{K}_{j}e(t)\\ & +L_i{C}_{2j}\tilde{x}(t)+L_i{W}_{2i}{w}_2(t)\end{array}$(8) Let $\tilde{x}(t)=x(t)-\hat{x}(t)$, we can obtain the fuzzy closed-loop system (9) $\begin{array}{r}\{\begin{array}{rl}\dot{x}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{((A_i+\mathrm{\Delta }{A}_i)+(B_i+\mathrm{\Delta }{B}_i)K_j)x(t)\\ & -(B_i+\mathrm{\Delta }{B}_i)K_j\tilde{x}(t)+(B_i+\mathrm{\Delta }{B}_i)K_{j}e(t)\\ & +W_{1i}{w}_1(t)\}\\ \dot{\tilde{x}}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{(A_i-\mathrm{\Delta }{B}_i{K}_j)\tilde{x}(t)\\ & +(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j)x(t)+\mathrm{\Delta }{B}_i{K}_{j}e(t)+W_{1i}{w}_1(t)\\ & -L_i{C}_{2j}\tilde{x}(t)-L_i{W}_{2i}{w}_2(t)\end{array}\end{array}$(9)

Theorem 3.1

Given scalars $\gamma >0$, I is identify matrix, if there exist positive definite matrices $P_1$, $P_2$, such that the inequality holds (10) $\mathrm{\Theta }=\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -P_1{B}_i{K}_j& P_1{W}_{1i}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -P_2{L}_i{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0$(10) $\begin{array}{rl}{\tilde{\mathrm{\Theta }}}_{11}& =P_1{A}_i+A_i^T{P}_1+P_1{B}_i{K}_j+K_j^T{B}_i^T{P}_1+4P_1{D}_i{D}_i^T{P}_1\\ & +2E_{i1}^T{E}_{i1}+\rho K_j^T{K}_j+(2+2\rho )K_j^T{E}_{i2}^T{E}_{i2}{K}_j\\ & +P_1{B}_i{B}_i^T{P}_1+C_{1i}^T{C}_{1i}\\ {\mathrm{\Theta }}_{22}& =P_2{A}_i+A_i^T{P}_2+4P_2{D}_i{D}_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j\\ & -P_2{L}_i{C}_{2j}-C_{2j}^T{L}_j^T{P}_2\end{array}$then the fuzzy system (9(9) $\begin{array}{r}\{\begin{array}{rl}\dot{x}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{((A_i+\mathrm{\Delta }{A}_i)+(B_i+\mathrm{\Delta }{B}_i)K_j)x(t)\\ & -(B_i+\mathrm{\Delta }{B}_i)K_j\tilde{x}(t)+(B_i+\mathrm{\Delta }{B}_i)K_{j}e(t)\\ & +W_{1i}{w}_1(t)\}\\ \dot{\tilde{x}}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{(A_i-\mathrm{\Delta }{B}_i{K}_j)\tilde{x}(t)\\ & +(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j)x(t)+\mathrm{\Delta }{B}_i{K}_{j}e(t)+W_{1i}{w}_1(t)\\ & -L_i{C}_{2j}\tilde{x}(t)-L_i{W}_{2i}{w}_2(t)\end{array}\end{array}$(9) ) is asymptotically stable. The control and observe gain matrices are designed as follows $K_j=R_j{P}_1,L_j=P_2^{-1}{N}_j.$

Proof.

Consider Lyapunov function as follows (11) $\begin{array}{rl}V(x(t),\tilde{x}(t))& =\left[\begin{array}{cc}{x}^T(t)& {\tilde{x}}^T(t)\end{array}\right]\left[\begin{array}{cc}{P}_1& 0\\ 0& P_2\end{array}\right]\left[\begin{array}{c}x(t)\\ \tilde{x}(t)\end{array}\right]\end{array}$(11) (12) $\begin{array}{rl}\dot{V}(t)& ={\dot{x}}^T(t)P_{1}x(t)+x^T(t)P_1\dot{x}(t)+{\dot{\tilde{x}}}^T(t)P_2\tilde{x}(t)\\ & +{\tilde{x}}^T(t)P_2\dot{\tilde{x}}(t)\end{array}$(12) By inserting the closed-loop system (9(9) $\begin{array}{r}\{\begin{array}{rl}\dot{x}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{((A_i+\mathrm{\Delta }{A}_i)+(B_i+\mathrm{\Delta }{B}_i)K_j)x(t)\\ & -(B_i+\mathrm{\Delta }{B}_i)K_j\tilde{x}(t)+(B_i+\mathrm{\Delta }{B}_i)K_{j}e(t)\\ & +W_{1i}{w}_1(t)\}\\ \dot{\tilde{x}}(t)& =\sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j\{(A_i-\mathrm{\Delta }{B}_i{K}_j)\tilde{x}(t)\\ & +(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j)x(t)+\mathrm{\Delta }{B}_i{K}_{j}e(t)+W_{1i}{w}_1(t)\\ & -L_i{C}_{2j}\tilde{x}(t)-L_i{W}_{2i}{w}_2(t)\end{array}\end{array}$(9) ) into (12(12) $\begin{array}{rl}\dot{V}(t)& ={\dot{x}}^T(t)P_{1}x(t)+x^T(t)P_1\dot{x}(t)+{\dot{\tilde{x}}}^T(t)P_2\tilde{x}(t)\\ & +{\tilde{x}}^T(t)P_2\dot{\tilde{x}}(t)\end{array}$(12) ) (13) $\begin{array}{rl}\dot{V}(t)& \le \sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j{x}^T(t)[P_1(A_i+\mathrm{\Delta }{A}_i)+P_1(B_i+\mathrm{\Delta }{B}_i)K_j\\ & +(A_i+\mathrm{\Delta }{A}_i{)}^T{P}_1+K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_1)x(t)\\ & -x^T(t)P_1(B_i+\mathrm{\Delta }{B}_i)K_j\tilde{x}(t)-{\tilde{x}}^T(t)K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_{1}x(t)\\ & +x^T(t)P_1(B_i+\mathrm{\Delta }{B}_i)K_{j}e(t)+e^T(t)K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_{1}x(t)\\ & +2x^T(t)P_1{W}_{1i}{w}_1(t)]+[{\tilde{x}}^T(t)P_2(A_i-\mathrm{\Delta }{B}_i{K}_j)\tilde{x}(t)\\ & +{\tilde{x}}^T(t)(A_i-\mathrm{\Delta }{B}_i{K}_j{)}^T{P}_2\tilde{x}(t)+{\tilde{x}}^T(t)P_2(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j)x(t)\\ & +x^T(t)(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j{)}^T{P}_2\tilde{x}(t)+{\tilde{x}}^T(t)P_2\mathrm{\Delta }{B}_i{K}_{j}e(t)\\ & +e^T(t)K_j^T\mathrm{\Delta }{B}_i^T{P}_2\tilde{x}(t)+2{\tilde{x}}^T(t)P_2{W}_{1i}{w}_1(t)\\ & -{\tilde{x}}^T(t)(P_2{L}_i{C}_{2j}+C_{2j}^T{L}_j^T{P}_2)\tilde{x}(t)-2{\tilde{x}}^T(t)P_2{L}_i{W}_{2i}{w}_2(t)]\end{array}$(13) The uncertainties of system can be represented as $\left[\begin{array}{cc}\mathrm{\Delta }{A}_i& \mathrm{\Delta }{B}_i\end{array}\right]=D_i{F}_i(t)\left[\begin{array}{cc}{E}_{i1}& E_{i2}\end{array}\right]$where $F_i(t)$ are unknown uncertainties satisfying$F_i^T(t)F_i(t)\le I$.

Applying Young's inequality and event trigger condition, (13(13) $\begin{array}{rl}\dot{V}(t)& \le \sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j{x}^T(t)[P_1(A_i+\mathrm{\Delta }{A}_i)+P_1(B_i+\mathrm{\Delta }{B}_i)K_j\\ & +(A_i+\mathrm{\Delta }{A}_i{)}^T{P}_1+K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_1)x(t)\\ & -x^T(t)P_1(B_i+\mathrm{\Delta }{B}_i)K_j\tilde{x}(t)-{\tilde{x}}^T(t)K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_{1}x(t)\\ & +x^T(t)P_1(B_i+\mathrm{\Delta }{B}_i)K_{j}e(t)+e^T(t)K_j^T(B_i+\mathrm{\Delta }{B}_i{)}^T{P}_{1}x(t)\\ & +2x^T(t)P_1{W}_{1i}{w}_1(t)]+[{\tilde{x}}^T(t)P_2(A_i-\mathrm{\Delta }{B}_i{K}_j)\tilde{x}(t)\\ & +{\tilde{x}}^T(t)(A_i-\mathrm{\Delta }{B}_i{K}_j{)}^T{P}_2\tilde{x}(t)+{\tilde{x}}^T(t)P_2(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j)x(t)\\ & +x^T(t)(\mathrm{\Delta }{A}_i+\mathrm{\Delta }{B}_i{K}_j{)}^T{P}_2\tilde{x}(t)+{\tilde{x}}^T(t)P_2\mathrm{\Delta }{B}_i{K}_{j}e(t)\\ & +e^T(t)K_j^T\mathrm{\Delta }{B}_i^T{P}_2\tilde{x}(t)+2{\tilde{x}}^T(t)P_2{W}_{1i}{w}_1(t)\\ & -{\tilde{x}}^T(t)(P_2{L}_i{C}_{2j}+C_{2j}^T{L}_j^T{P}_2)\tilde{x}(t)-2{\tilde{x}}^T(t)P_2{L}_i{W}_{2i}{w}_2(t)]\end{array}$(13) ) converted to (14) $\begin{array}{rl}\dot{V}(t)& \le \sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j{x}^T(t)[P_1{A}_i+A_i^T{P}_1+P_1{B}_i{K}_j\\ & +K_j^T{B}_i^T{P}_1+4P_1{D}_i{D}_i^T{P}_1+2E_{i1}^T{E}_{i1}+\rho K_j^T{K}_j\\ & +P_1{B}_i^T{B}_i{P}_1+2(1+\rho )K_j^T{E}_{i2}^T{E}_{i2}{K}_j)x(t)\\ & -x^T(t)P_1{B}_i{K}_j\tilde{x}(t)-{\tilde{x}}^T(t)K_j^T{B}_j^T{P}_{1}x(t)\\ & +{\tilde{x}}^T(t)(P_2{A}_i+A_i^T{P}_2+4P_2{D}_i{D}_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j\\ & -P_2{L}_i{C}_{2j}-C_{2j}^T{L}_j^T{P}_2)\tilde{x}(t)+2{\tilde{x}}^T(t)P_2{W}_{1i}{w}_1(t)\\ & +2x^T(t)P_1{W}_{1i}{w}_1(t)-2{\tilde{x}}^T(t)P_2{L}_i{W}_{2i}{w}_2(t)]\end{array}$(14) Combining (14(14) $\begin{array}{rl}\dot{V}(t)& \le \sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j{x}^T(t)[P_1{A}_i+A_i^T{P}_1+P_1{B}_i{K}_j\\ & +K_j^T{B}_i^T{P}_1+4P_1{D}_i{D}_i^T{P}_1+2E_{i1}^T{E}_{i1}+\rho K_j^T{K}_j\\ & +P_1{B}_i^T{B}_i{P}_1+2(1+\rho )K_j^T{E}_{i2}^T{E}_{i2}{K}_j)x(t)\\ & -x^T(t)P_1{B}_i{K}_j\tilde{x}(t)-{\tilde{x}}^T(t)K_j^T{B}_j^T{P}_{1}x(t)\\ & +{\tilde{x}}^T(t)(P_2{A}_i+A_i^T{P}_2+4P_2{D}_i{D}_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j\\ & -P_2{L}_i{C}_{2j}-C_{2j}^T{L}_j^T{P}_2)\tilde{x}(t)+2{\tilde{x}}^T(t)P_2{W}_{1i}{w}_1(t)\\ & +2x^T(t)P_1{W}_{1i}{w}_1(t)-2{\tilde{x}}^T(t)P_2{L}_i{W}_{2i}{w}_2(t)]\end{array}$(14) ), we can obtain as follows: (15) $\dot{V}(t)\le \sum _{i=1}^r\sum _{j=1}^r{\mu }_i{\epsilon }_j{\mu }_j{\xi }^T(t)\mathrm{\Theta }\xi (t)$(15) where ${\xi }^T(t,s)=[\begin{array}{cccc}{x}^T(t)& {\tilde{x}}^T(t)& w_1^T(t)& w_2^T(t)\end{array}]$. (16) $\mathrm{\Theta }=\left[\begin{array}{cccc}{\mathrm{\Theta }}_{11}& -P_1{B}_i{K}_j& P_1{W}_{1i}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -P_2{L}_i{W}_{2i}\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & 0\end{array}\right]<0$(16) $\begin{array}{rl}{\mathrm{\Theta }}_{11}& =P_1{A}_i+A_i^T{P}_1+P_1{B}_i{K}_j+K_j^T{B}_i^T{P}_1+4P_1{D}_i{D}_i^T{P}_1\\ & +2E_{i1}^T{E}_{i1}+\rho K_j^T{K}_j+(2+2\rho )K_j^T{E}_{i2}^T{E}_{i2}{K}_j+P_1{B}_i^T{B}_i{P}_1\\ {\mathrm{\Theta }}_{22}& =P_2{A}_i+A_i^T{P}_2+4P_2{D}_i{D}_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j-P_2{L}_i{C}_{2j}\\ & -C_{2j}^T{L}_j^T{P}_2\end{array}$$H_{\mathrm{\infty }}$ performance index is taken as follows: (17) $\begin{array}{r}J={\int }_0^T[z^T(t)z(t)-{\gamma }^2{w}_1^T(t)w_1(t)-{\gamma }^2{w}_2^T(t)w_2(t)]\mathrm{d}t\end{array}$(17) According to Equation (10(10) $\mathrm{\Theta }=\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -P_1{B}_i{K}_j& P_1{W}_{1i}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -P_2{L}_i{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0$(10) ) with zero initial conditions, Equation (17(17) $\begin{array}{r}J={\int }_0^T[z^T(t)z(t)-{\gamma }^2{w}_1^T(t)w_1(t)-{\gamma }^2{w}_2^T(t)w_2(t)]\mathrm{d}t\end{array}$(17) ) can be converted to $\begin{array}{rl}J& ={\int }_0^T[z^T(t)z(t)-{\gamma }^2{w}_1^T(t)w_1(t)-{\gamma }^2{w}_2^T(t)w_2(t)\\ & +\dot{V}(t)-\dot{V}(t)]\mathrm{d}t\\ & \le {\int }_0^T[x^T(t)C_{1i}^T{C}_{1i}x(t)-{\gamma }^2{w}_1^T(t)w_1(t)\\ & -{\gamma }^2{w}_2^T(t)w_2(t)+\dot{V}(t)]\mathrm{d}t\\ & {\int }_0^{+\mathrm{\infty }}[z^T(t)z(t)]\mathrm{d}t\\ & <{\gamma }^2{\int }_0^{+\mathrm{\infty }}[w_1^T(t)w_1(t)+w_2^T(t)w_2(t)]\mathrm{d}t\end{array}$Finally, we get inequalities (10(10) $\mathrm{\Theta }=\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -P_1{B}_i{K}_j& P_1{W}_{1i}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -P_2{L}_i{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0$(10) ).

Multiplying the sides of (10(10) $\mathrm{\Theta }=\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -P_1{B}_i{K}_j& P_1{W}_{1i}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -P_2{L}_i{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0$(10) ) by matrix$diag\{\begin{array}{cccc}{P}_1^{-1}& 0& 0& 0\end{array}\}$ and applying Schur complement formula, we obtain the inequalities as follows: (18) $\begin{array}{rl}\mathrm{\Theta }& =\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -B_i{R}_j& W_{1i}{P}_1^{-1}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -N_j{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0\end{array}$(18) $\begin{array}{rl}{\tilde{\mathrm{\Theta }}}_{11}& =A_i{P}_1^{-1}+P_1^{-1}{A}_i^T+B_i{R}_j+R_j^T{B}_i^T+4D_i{D}_i^T\\ & +2P_1^{-1}{E}_{i1}^T{E}_{i1}{P}_1^{-1}+\rho R_j^T{R}_j+(2+2\rho )R_j^T{E}_{i2}^T{E}_{i2}{R}_j\\ & +B_i^T{B}_i+P_1^{-1}{C}_{1i}^T{C}_{1i}{P}_1^{-1}\\ {\mathrm{\Theta }}_{22}& =P_2{A}_i+A_i^T{P}_2+4P_2{D}_i{D}_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j\\ & -N_j{C}_{2i}-C_{2i}^T{N}_j^T\end{array}$Applying the Schur complement formula, ${\tilde{\mathrm{\Theta }}}_{11}<0$ isconverted to (19) $\begin{array}{rl}{\hat{\mathrm{\Theta }}}_{11}& =\left[\begin{array}{ccccc}{{\mathrm{\Theta }}^{\prime }}_{11}& P_1^{-1}{E}_{i1}^T& R_j^T& R_j^T{E}_{i2}^T& P_1^{-1}{C}_{1i}^T\\ \ast & -0.5I& 0& 0& 0\\ \ast & \ast & -{\displaystyle \frac{1}{\rho }}I& 0& 0\\ \ast & \ast & \ast & -{\displaystyle \frac{1}{2+2\rho }}I& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right]<0\\ {\mathrm{\Theta }}_{11}^{\prime }& =A_i{P}_1^{-1}+P_1^{-1}{A}_i^T+B_i{R}_j+R_j^T{B}_i^T+4D_i{D}_i^T+B_i^T{B}_i\end{array}$(19) By solving LMIs (19(19) $\begin{array}{rl}{\hat{\mathrm{\Theta }}}_{11}& =\left[\begin{array}{ccccc}{{\mathrm{\Theta }}^{\prime }}_{11}& P_1^{-1}{E}_{i1}^T& R_j^T& R_j^T{E}_{i2}^T& P_1^{-1}{C}_{1i}^T\\ \ast & -0.5I& 0& 0& 0\\ \ast & \ast & -{\displaystyle \frac{1}{\rho }}I& 0& 0\\ \ast & \ast & \ast & -{\displaystyle \frac{1}{2+2\rho }}I& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right]<0\\ {\mathrm{\Theta }}_{11}^{\prime }& =A_i{P}_1^{-1}+P_1^{-1}{A}_i^T+B_i{R}_j+R_j^T{B}_i^T+4D_i{D}_i^T+B_i^T{B}_i\end{array}$(19) ), the matrices $P_1$ and the control gain matrices $K_j$ are obtained, then put $P_1$ and $K_j$ into (18(18) $\begin{array}{rl}\mathrm{\Theta }& =\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -B_i{R}_j& W_{1i}{P}_1^{-1}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -N_j{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0\end{array}$(18) ).

Applying the Schur complement formula, (18(18) $\begin{array}{rl}\mathrm{\Theta }& =\left[\begin{array}{cccc}{\tilde{\mathrm{\Theta }}}_{11}& -B_i{R}_j& W_{1i}{P}_1^{-1}& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -N_j{W}_{2i}\\ \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I\end{array}\right]<0\end{array}$(18) ) becomes (20) $\begin{array}{rl}\mathrm{\Theta }& =\left[\begin{array}{ccccc}{\tilde{\mathrm{\Theta }}}_{11}& -B_i{R}_j& W_{1i}{P}_1^{-1}& 0& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -N_j{W}_{2i}& P_2{D}_i\\ \ast & \ast & -{\gamma }^{2}I& 0& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & -0.25I\end{array}\right]<0\end{array}$(20) $\begin{array}{rl}{\tilde{\mathrm{\Theta }}}_{11}& =A_i{P}_1^{-1}+P_1^{-1}{A}_i^T+B_i{R}_j+R_j^T{B}_i^T+4D_i{D}_i^T\\ & +2P_1^{-1}{E}_{i1}^T{E}_{i1}{P}_1^{-1}+\rho R_j^T{R}_j+(2+2\rho )R_j^T{E}_{i2}^T{E}_{i2}{R}_j\\ & +B_i^T{B}_i+P_1^{-1}{C}_{1i}^T{C}_{1i}{P}_1^{-1}\\ {\mathrm{\Theta }}_{22}& =P_2{A}_i+A_i^T{P}_2+2K_j^T{E}_{i2}^T{E}_{i2}{K}_j-N_j{C}_{2i}-C_{2i}^T{N}_j^T\end{array}$By solving LMIs (20(20) $\begin{array}{rl}\mathrm{\Theta }& =\left[\begin{array}{ccccc}{\tilde{\mathrm{\Theta }}}_{11}& -B_i{R}_j& W_{1i}{P}_1^{-1}& 0& 0\\ \ast & {\mathrm{\Theta }}_{22}& P_2{W}_{1i}& -N_j{W}_{2i}& P_2{D}_i\\ \ast & \ast & -{\gamma }^{2}I& 0& 0\\ \ast & \ast & \ast & -{\gamma }^{2}I& 0\\ \ast & \ast & \ast & \ast & -0.25I\end{array}\right]<0\end{array}$(20) ), the matrices $P_2$ and the observe gain matrices $L_j$ are obtained.

4. Simulation

The related parameters of Equation (3(3) $\begin{array}{r}\{\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{(A_i+\mathrm{\Delta }{A}_i)x(t)+(B_i+\mathrm{\Delta }{B}_i)u(t)}\\ +W_{1i}{w}_1(t)\}\\ z(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))C_{1i}{x}_i(t)}\\ y(t)={\displaystyle \sum _{i=1}^r{\mu }_i(\sigma (t))\{C_{2i}{x}_i(t)+W_{2i}{w}_2(t)}\end{array}\end{array}$(3) ) are listed asfollows: $\begin{array}{rl}{A}_1& =\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ {\displaystyle \frac{-k_s}{m_s+{\hat{m}}_s}}& 0& {\displaystyle \frac{-c_s}{m_s+{\hat{m}}_s}}& {\displaystyle \frac{c_s}{m_s+{\hat{m}}_s}}\\ {\displaystyle \frac{k_s}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{-k_t}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{c_s}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{-c_s-c_t}{m_u+{\hat{m}}_u}}\end{array}\right],\\ A_2& =\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ {\displaystyle \frac{-k_s}{m_s+{\hat{m}}_s}}& 0& {\displaystyle \frac{-c_s}{m_s+{\hat{m}}_s}}& {\displaystyle \frac{c_s}{m_s+{\hat{m}}_s}}\\ {\displaystyle \frac{k_s}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{-k_t}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{c_s}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{-c_s-c_t}{m_u+{\breve{m}}_u}}\end{array}\right]\\ A_3& =\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ {\displaystyle \frac{-k_s}{m_s+{\breve{m}}_s}}& 0& {\displaystyle \frac{-c_s}{m_s+{\breve{m}}_s}}& {\displaystyle \frac{c_s}{m_s+{\breve{m}}_s}}\\ {\displaystyle \frac{k_s}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{-k_t}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{c_s}{m_u+{\hat{m}}_u}}& {\displaystyle \frac{-c_s-c_t}{m_u+{\hat{m}}_u}}\end{array}\right],\\ A_4& =\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ {\displaystyle \frac{-k_s}{m_s+{\breve{m}}_s}}& 0& {\displaystyle \frac{-c_s}{m_s+{\breve{m}}_s}}& {\displaystyle \frac{c_s}{m_s+{\breve{m}}_s}}\\ {\displaystyle \frac{k_s}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{-k_t}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{c_s}{m_u+{\breve{m}}_u}}& {\displaystyle \frac{-c_s-c_t}{m_u+{\breve{m}}_u}}\end{array}\right]\\ B_1& =\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{1}{m_s+{\hat{m}}_s}}\\ {\displaystyle \frac{-1}{m_u+{\hat{m}}_u}}\end{array}\right].B_2=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{1}{m_s+{\hat{m}}_s}}\\ {\displaystyle \frac{-1}{m_u+{\breve{m}}_u}}\end{array}\right],\\ B_3& =\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{1}{m_s+{\breve{m}}_s}}\\ {\displaystyle \frac{-1}{m_u+{\hat{m}}_u}}\end{array}\right],B_4=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{1}{m_s+{\breve{m}}_s}}\\ {\displaystyle \frac{-1}{m_u+{\breve{m}}_u}}\end{array}\right]\end{array}$$W_{11}=W_{13}=\left[\begin{array}{c}0\\ -1\\ 0\\ \frac{c_t}{m_u+{\hat{m}}_u}\end{array}\right]$, $W_{12}=W_{14}=\left[\begin{array}{c}0\\ -1\\ 0\\ \frac{c_t}{m_u+{\breve{m}}_u}\end{array}\right]$, $C_{11}=C_{12}=C_{13}=C_{14}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]$, $W_{21}=W_{23}=\left[\begin{array}{c}1\\ 0.1\end{array}\right]$, $W_{22}=W_{24}=\left[\begin{array}{c}0.1\\ 1\end{array}\right]$, $C_{21}=C_{22}=C_{23}=C_{24}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\end{array}\right]$, $\rho =1$, $\gamma =6$, $\mathrm{\Delta }{A}_i=0.01\times D_i\times diag[\begin{array}{cccc}\sin t& \sin t& \sin t& \sin t\end{array}]\times E_{1i}$,$\mathrm{\Delta }{B}_i=0.01\times D_i\times diag[\begin{array}{cccc}\sin t& \sin t& \sin t& \sin t\end{array}]\times E_{2i}$, $D_i=0.02I_4$, $E_{1i}=0.1I_4$, $E_{2i}=[0.1;0.1;0.1;0.1]$, $i=1,\cdots ,4$, $\begin{array}{rl}{w}_2(t)& =\sin (t)/\phantom{\sin (t)10}-10,\\ z_r(t)& =\{\begin{array}{ll}\frac{H}{2}(1-\cos (\frac{2\pi v}{L})t),& 0\le t\le \frac{L}{v}\\ 0,& t>\frac{L}{v}\end{array},\end{array}$$w_1(t)={\dot{z}}_r(t)$, $m_s=350\mathrm{kg}$, $m_u=30\mathrm{kg}$, $k_s=10\mathrm{kN}/\phantom{\mathrm{kN}m}\mathrm{m}$, $k_t=180\mathrm{kN}/\phantom{\mathrm{kN}m}\mathrm{m}$, $c_s=1.5\mathrm{kNs}/\phantom{\mathrm{kNs}m}\mathrm{m}$, $c_t=17\mathrm{Ns}/\phantom{\mathrm{Ns}m}\mathrm{m}$.

According to the mechanical structure of the suspension, the above factors change the mass of the spring and the mass under the spring, assuming that they change in a fixed range $m_{smin}\le m_s\le m_{smax}$$m_{umin}\le m_u\le m_{umax}$, $M_1(\sigma (t))=\frac{1/\phantom{1m_s}{m}_s-1/\phantom{1m_{smax}}{m}_{smax}}{1/\phantom{1m_{smin}}{m}_{smin}-1/\phantom{1m_{smax}}{m}_{smax}}$,$M_2(\sigma (t))=\frac{1/\phantom{1m_{smin}}{m}_{smin}-1/\phantom{1m_s}{m}_s}{1/\phantom{1m_{smin}}{m}_{smin}-1/\phantom{1m_{smax}}{m}_{smax}}$, $N_1(\sigma (t))=\frac{1/\phantom{1m_u}{m}_u-1/\phantom{1m_{umax}}{m}_{umax}}{1/\phantom{1m_{umin}}{m}_{umin}-1/\phantom{1m_{umax}}{m}_{umax}}$, $N_2(\sigma (t))=\frac{1/\phantom{1m_{umin}}{m}_{umin}-1/\phantom{1m_u}{m}_u}{1/\phantom{1m_{umin}}{m}_{umin}-1/\phantom{1m_{umax}}{m}_{umax}}$, ${\mu }_1(\sigma (t))=M_1({\sigma }_1(t))\cdot N_1({\sigma }_1(t))$, ${\mu }_2(\sigma (t))=M_1({\sigma }_1(t))\cdot N_2({\sigma }_2(t))$, ${\mu }_3(\sigma (t))=M_2({\sigma }_1(t))\cdot N_1({\sigma }_2(t))$, ${\mu }_4(\sigma (t))=M_2({\sigma }_1(t))\cdot N_2({\sigma }_2(t))$,$x(0)=[-3.22.4-0.15-2.3]$, $\hat{x}(0)=[-2.91.0-0.11-0.5]$.

On the basis of LMIs, $K_i$ and $L_i$ ($i=1\cdots 4$) are obtained as follows: $K_1=[-0.0131-0.0052-0.00050]$, $K_2=[-0.0143-0.0080-0.00050]$, $K_3=[0.00540.0243-0.00060.0001]$, $K_4=[0.01470.0398-0.00070.0001]$. Thus, the simualtion results are displayed in Figures 2–7, where Figures 2–5 are the trajectories of states $x_i$ and their estimations ${\hat{x}}_i(i=1,2,3,4)$; Figure 6 is the curve of control input u; Figure 7 are the curves of event-triggered respobse. $\begin{array}{rl}{L}_1& =\left[\begin{array}{cc}-1.1591& 0.1159\\ -4.4804& 0.4480\\ 0.4740& -0.0474\\ -0.0128& 0.0013\end{array}\right],\\ L_2& =\left[\begin{array}{cc}-10.3014& 20.6547\\ -12.2863& -17.0371\\ 2.0951& -1.6102\\ -0.1403& 0.1924\end{array}\right],\\ L_3& =\left[\begin{array}{cc}-11.2801& 19.1041\\ -12.8266& -15.0428\\ 2.8134& -1.6520\\ -0.0195& 0.1224\end{array}\right],\\ L_4& =\left[\begin{array}{cc}-9.6737& 20.3428\\ -8.5910& -18.7309\\ 1.9707& -1.5349\\ -0.0680& 0.1647\end{array}\right].\end{array}$

Figure 2. Trajectories of $x_1$ and ${\hat{x}}_1$.

Figure 3. Trajectories of $x_2$ and ${\hat{x}}_2$.

Figure 4. Trajectories of $x_3$ and ${\hat{x}}_3$.

Figure 5. Trajectories of $x_4$ and ${\hat{x}}_4$.

Figure 6. Trajectory of input u.

Figure 7. Trajectories of event-triggered response.

5. Conclusion

This paper has designed event-trigger control for the fuzzy T-S systems with parameter uncertainties and disturbances. The conditions of the asymptotically stability are proposed by LMIs and Lyapunov function. Simulation are given for the vehicle suspension systems.

In the future, we will pay more attention to on multi-agent nonlinear systems such as a separation-based methodology to consensus tracking of switched high-order nonlinear multi-agent systems, consensus in high-power multi-agent systems with mixed unknown control directions and so on. It will be a challenging job.

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 61822307].