Observer-based non-fragile robust control for incremental quadratic nonlinear systems with uncertain perturbations

Abstract

This article focuses on the issue of non-fragile robust control for a class of nonlinear systems with uncertain structural perturbations based on state observer. The nonlinearities are described by incremental quadratic constraints, which unify numbers of nonlinearities encompassed in recent literature in a framework. A non-fragile observer is first proposed to estimate the state of this system. As a consequence, non-fragile robust control design is utilized to construct an observer-based controller to render asymptotical stabilization. Moreover, a sufficient condition guaranteeing the asymptotical stability of the closed-loop system is formulated in terms of linear matrix inequalities without employing any equality constraints. Numerical simulations on a Lorenz chaotic system demonstrate the validity of the observer-based non-fragile control scheme.

Introduction

During the past two decades, non-fragile control has been attracting lots of attention since inaccuracies or uncertainties may occur in the implementation of a constructed controller. The fragile problem of a designed controller can be dated back to the pioneering work of Keel and Bhattacharyya, ¹ where they demonstrated that the optimal control system may lead to unstable even for a tiny change in the controller’s parameters. Since then, various intriguing results on non-fragile controllers were reported by researchers in lots of literature. For instance, Balandin and Kogan ² investigated the synthesis of non-fragile controllers utilizing linear matrix inequalities (LMIs). Ma et al. ³ addressed the issue of non-fragile guaranteed cost control for microbial fuel cells. Non-fragile consensus control for nonlinear multi-agent systems under periodic denial of service attacks was studied. ⁴

It is worth mentioning that most of the aforementioned non-fragile controllers or filters were designed via state feedback. However, in many physical systems the whole or partial states, the external disturbances may not all be measurable. As a consequence, state estimation is of great significance in control design. Over the past decades, numerous effective approaches of observer design have been developed in existing literature (see, just to mention a few,^5–8,10). For example, Liu et al. ⁸ introduced an intermediate observer to estimate the relative motion information and the disturbance for delayed electromagnetic docking spacecraft in elliptical orbits, which was further developed by Lyu et al. ⁹ A predefined-time disturbance observer was designed in Zhang et al. ¹⁰ to estimate and compensate the external disturbances for attitude tracking control of quadrotors. More recently, Shen et al. ¹¹ developed a novel iterative proportional-integral observer and extended it to get tighter interval estimation for system states and disturbances of discrete-time linear systems.^12,13

Consequently, on the basis of the estimated states, how to construct a non-fragile controller or filter has became a fundamental issue for non-fragile control synthesis. Till now, this issue has been well-studied for linear systems by means of observer-based feedback. For instance, Lyu et al. ¹⁴ designed a hybrid non-fragile observer to estimate state and disturbance simultaneously for flexible spacecraft. However, for general nonlinear systems, the problem of observer-based non-fragile control remains unsolved (see, e.g. Liu et al., ¹⁵ Yang and He, ¹⁶ Lu and Ren ¹⁷ ). For uncertain time-delay Lipschitz nonlinear systems, Liu et al. ¹⁵ proposed a sliding-mode control approach based on non-fragile observers. For a class of Lipschitz nonlinear systems with structural uncertainties, Yang and He ¹⁶ presented a non-fragile observer-based robust control scheme. Very recently, Lu and Ren ¹⁷ constructed a novel non-fragile observer-based controller for discrete fuzzy descriptor systems with nonlinear disturbances.

Note that in order to tackle the state estimation problem for different kinds of nonlinear systems, many helpful observer design methods have been reported in literature. For example, for Lipschitz nonlinear systems, observers were developed in Yang and He ¹⁶ and Zemouche and Boutayeb ¹⁸ Observers for systems satisfying one-sided Lipschitz (OSL) constraints were investigated in Zhang et al.,^19,20 which showed that the OSL constants could be significantly smaller than the usual Lipschitz constants. Recently, Açıkmeşe and Corless ²¹ introduced the so-called incremental quadratic constraint (IQC). This condition unifies numerous types of common nonlinearities by introducing incremental multiplier matrices. It is shown that both Lipschitz and OSL conditions could be included in the framework of IQC.^22,23 As a consequence, observers for nonlinear systems satisfying IQC have been developed in,^24–28 which were applied to secure communication in Zhao et al. ²⁹ for chaotic systems. More recently, Xu et al. ³⁰ has developed observer-based controllers for systems with IQC and external disturbances. The above-mentioned observations inspired us to investigate the issue of observer-based non-fragile control for incremental quadratic nonlinear systems with uncertain perturbations.

On the other hand, there still exist many unsolved difficulties on solving the problem of bilinear matrix inequalities (BMIs) for observer-based control (see, e.g. Zhang et al., ³¹ Lien, ³² Kheloufi et al., ³³ Wu et al., ³⁴ Li et al., ³⁵ and the reference therein). To tackle the problem, Lien ³² utilized an equality constraint, which may not be satisfied with many physical systems. And then Kheloufi et al. ³³ proposed a new approach by employing the Young relation. However, Wang and Jiang ³⁶ compared these two methods and showed that Kheloufi’s method was more conservative than Lien’s. Therefore, how to cope with the BMI problem in observer-based control without utilizing any equality constraints also motives the current study.

In the article, we focus on the observer-based non-fragile robust control for uncertain systems with nonlinearities satisfying IQC. Motivated by Badreddine et al., ³⁷ an LMI-based method is put forward to address the BMIs issue for incremental quadratic nonlinear systems, which can convert BMIs into LMIs without utilizing any equality constraints. Firstly, a non-fragile observer is constructed for incremental quadratic nonlinear systems with uncertain perturbations. A sufficient condition in terms of LMIs is put forward to render the asymptotical stability of the whole closed-loop system. Furthermore, the proposed non-fragile observer and controller gains are figured out via solving LMIs. Finally, an example on the controlled Lorenz chaotic system is given to verify the validity of the observer-based non-fragile control strategy.

Unlike the existing results in related work, the key contributions of this article can be listed as follows. Firstly, a non-fragile control design is proposed to deal with uncertain perturbations in both the observer and controller gains. Meanwhile, a non-fragile observer-based robust controller for incremental quadratic nonlinear systems is provided. Secondly, a novel LMI approach is presented to tackle the BMI problem in designing non-fragile observer-based feedback without employing any equality constraints.

Notations: ${\mathbb{R}}^s$ and ${\mathbb{R}}^{s\times t}$ stands for the sets of $s$ -dimensional real vectors and $s\times t$ real matrices, respectively. $〈x,y〉$ denotes the inner product of two vectors $x$ and $y$ . $||x||$ represents the Euclidean norm of $x$ . The symbol * denotes a term induced by symmetry. $I_s$ is a $s$ -dimensional identity matrix. For a matrix $M$ , its transposition is denoted by $M^T$ , and $\mathrm{He}\left(M\right)=M+M^T$ . Moreover, $M>(\ge )0$ implies that $M$ is positive (semi-) defined.

Preliminaries and problem statement

Consider the following nonlinear system with uncertain structural perturbations

$$\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=(A+\mathrm{\Delta }A\left(t\right))x\left(t\right)+\mathrm{Bu}\left(t\right)+\mathrm{Gf}\left(\mathrm{Hx}\left(t\right)\right)\hfill \\ \multicolumn{1}{c}{y\left(t\right)=\mathrm{Cx}\left(t\right)}\end{array}$$

(1)

where $x\in {\mathbb{R}}^n$ , $u\in {\mathbb{R}}^m$ and $y\in {\mathbb{R}}^p$ are the state vector, the control input and the measured output, respectively. $A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m},C\in {\mathbb{R}}^{p\times n}$ and $G\in {\mathbb{R}}^{n\times s}$ are known matrices, $f\left(\psi \right):{\mathbb{R}}^s\to {\mathbb{R}}^s$ is a vector-valued nonlinear function with $\psi =\mathrm{Hx},H\in {\mathbb{R}}^{s\times n},f\left(0\right)=0$ , and $\mathrm{\Delta }A\left(t\right)$ is a perturbed time-varying matrix. Additionally, assume that $f\left(\psi \right)$ satisfies incremental quadratic constraints (IQC) as follows.

• Definition 1: (Açıkmeşe and Corless ²¹ ) The nonlinear vector-valued function $f\left(\psi \right)$ verifies the incremental quadratic constraints (IQC) if there is a symmetric matrix $\mathcal{M}$ satisfying the following inequality

$${\left[\begin{array}{c}{\psi }_2-{\psi }_1\hfill \\ \multicolumn{1}{c}{f\left({\psi }_2\right)-f\left({\psi }_1\right)}\end{array}\right]}^T\mathcal{M}\left[\begin{array}{c}{\psi }_2-{\psi }_1\hfill \\ \multicolumn{1}{c}{f\left({\psi }_2\right)-f\left({\psi }_1\right)}\end{array}\right]\ge 0,$$

(2)

where $\mathcal{M}$ is partitioned as

$$\mathcal{M}=\left[\begin{array}{cc}{\mathcal{M}}_{11}\hfill & {\mathcal{M}}_{12}\hfill \\ \multicolumn{1}{c}{{\mathcal{M}}_{12}^T}& {\mathcal{M}}_{22}\hfill \end{array}\right],$$

${\mathcal{M}}_{11}\in {\mathbb{R}}^{s\times s}$ , ${\mathcal{M}}_{22}\in {\mathbb{R}}^{s\times s}$ , and $\mathcal{M}$ is called the incremental multiplier matrix (IMM).

It can be seen from Definition 1 that the IMM provides a characterization of the nonlinear function $f\left(\psi \right)$ , while different nonlinearities can be described by choosing appropriate IMMs. Consequently, various types of encountered nonlinearities in the existing literature could be encompassed in the IQC framework with suitable IMMs, such as the following examples.

• Remark 1: Let us recall the OSL nonlinearity. If $f\left(\psi \right)$ satisfies the OSL conditions, then there is a constant $\rho \in \mathbb{R}$ such that

$$〈f\left({\psi }_2\right)-f\left({\psi }_1\right),{\psi }_2-{\psi }_1〉\le \rho ||{\psi }_2-{\psi }_1|{|}^2$$

holds, which is equivalent to

$$\rho {({\psi }_2-{\psi }_1)}^T({\psi }_2-{\psi }_1)-{(f\left({\psi }_2\right)-f\left({\psi }_1\right))}^T({\psi }_2-{\psi }_1)\ge 0.$$

In this case, the corresponding IMM is

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{M}=\gamma \left[\begin{array}{cc}\rho I\hfill & -\frac{1}{2}I\hfill \\ \multicolumn{1}{c}{-\frac{1}{2}I}& 0\hfill \end{array}\right]}\end{array}$$

where $\gamma >0$ .

• Remark 2: Let us revisit the incremental sector bounded (ISB) nonlinearity. ²² Note that if $f\left(\psi \right)$ satisfies the ISB condition, then there exist matrices ${\mathcal{K}}_1,{\mathcal{K}}_2\in {\mathbb{R}}^{s\times s}$ such that for all $\mathrm{\Theta }\in \mathcal{X}$

$$\begin{array}{c}{[(f\left({\psi }_2\right)-f\left({\psi }_1\right))-{\mathcal{K}}_1({\psi }_2-{\psi }_1)]}^T\mathrm{\Theta }\hfill \\ [{\mathcal{K}}_2({\psi }_2-{\psi }_1)-(f\left({\psi }_2\right)-f\left({\psi }_1\right))]\ge 0,\hfill \end{array}$$

holds, where $\mathcal{X}$ is an invariant set under multiplication by a positive number. Hence, we can select the IMM as

$$\mathcal{M}=\gamma \left[\begin{array}{cc}-{{\mathcal{K}}_1}^T\mathrm{\Theta }{\mathcal{K}}_2-{{\mathcal{K}}_2}^T\mathrm{\Theta }{\mathcal{K}}_1\hfill & {({\mathcal{K}}_1+{\mathcal{K}}_2)}^T\mathrm{\Theta }\hfill \\ \multicolumn{1}{c}{\mathrm{\Theta }({\mathcal{K}}_1+{\mathcal{K}}_2)}& -2\mathrm{\Theta }\hfill \end{array}\right]$$

where $\gamma >0$ .

For system (1), we construct an observer-based non-fragile robust controller

$$\{\begin{array}{c}\stackrel{.}{\hat{x}}\left(t\right)=A\hat{x}\left(t\right)+\mathrm{Bu}\left(t\right)+\mathrm{Gf}\left(H\hat{x}\left(t\right)\right)\hfill \\ +(L+\mathrm{\Delta }L\left(t\right))[y\left(t\right)-C\hat{x}\left(t\right)]\hfill \\ u\left(t\right)=-(K+\mathrm{\Delta }K\left(t\right))\hat{x}\left(t\right)\hfill \end{array}$$

(3)

where $\hat{x}\left(t\right)\in {\mathbb{R}}^n$ , $K\in {\mathbb{R}}^{m\times n}$ , and $L\in {\mathbb{R}}^{n\times p}$ are respectively the estimation of state, the control gain and the observer gain, $\mathrm{\Delta }K\left(t\right)$ and $\mathrm{\Delta }L\left(t\right)$ are perturbed real matrices.

• Remark 3: It should be noted that the adopted traditional observer (3) could be further improved by the iterative proportional-integral observer developed in Shen et al. ¹¹ In order to address the BMI problem, for simplicity we only consider the classical Luenberger observer.

Assume that $\mathrm{\Delta }A\left(t\right)$ , $\mathrm{\Delta }K\left(t\right)$ , and $\mathrm{\Delta }L\left(t\right)$ are time-varying uncertain perturbations defined as follows (see, e.g. Liu et al. ⁸ and Lien ³² )

$$\left[\begin{array}{c}\mathrm{\Delta }A\left(t\right)\hfill \\ \multicolumn{1}{c}{\mathrm{\Delta }K\left(t\right)}\\ \multicolumn{1}{c}{\mathrm{\Delta }L\left(t\right)}\end{array}\right]=\left[\begin{array}{ccc}{M}_1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & M_3\hfill \end{array}\right]\left[\begin{array}{ccc}{F}_1\left(t\right)\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& F_2\left(t\right)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & F_3\left(t\right)\hfill \end{array}\right]\left[\begin{array}{c}{N}_1\hfill \\ \multicolumn{1}{c}{N_2}\\ \multicolumn{1}{c}{N_3}\end{array}\right]$$

(4)

and the unknown matrices $F_j\left(t\right)$ , $j=1,2,3$ , are assumed to satisfy the following condition

$$F_j^T\left(t\right)F_j\left(t\right)\le I,$$

(5)

where $M_j$ , $N_j$ , $j=1,2,3,$ are known real matrices.

Based on (1) and (3), one can rewrite the corresponding closed-loop system as

$$\begin{array}{c}\{\begin{array}{c}\stackrel{·}{x}\left(t\right)=(A+\mathrm{\Delta }A\left(t\right)-\mathrm{BK}-B\mathrm{\Delta }K\left(t\right))x\left(t\right)+\mathrm{Gf}\left(\psi \right)\hfill \\ +(\mathrm{BK}+B\mathrm{\Delta }K\left(t\right))e\left(t\right)\hfill \\ \stackrel{·}{e}\left(t\right)=\mathrm{\Delta }A\left(t\right)x\left(t\right)+(A-\mathrm{LC}-\mathrm{\Delta }\mathrm{LC})e\left(t\right)+G\tilde{f}\hfill \end{array}\hfill \end{array}$$

(6)

where $e\left(t\right)=x\left(t\right)-\hat{x}\left(t\right)$ is the estimation error, and $\tilde{f}=f\left(\psi \right)-f\left(\hat{\psi }\right)$ .

In what following, we attempt to select two appropriate gains $K$ and $L$ such that system (6) is asymptotically stable. To begin with, we need the following lemmas.

• Lemma 1: ³⁵ Let $M,N$ be real matrices and $ϵ>0$ . Assume that $F\left(t\right)$ is a time-varying matrix satisfying $F^T\left(t\right)F\left(t\right)\le I$ , then

$$\mathrm{MF}\left(t\right)N+N^T{F}^T\left(t\right)M^T\le {ϵ}^{-1}M{M}^T+ϵN^{T}N.$$

(7)

• Lemma 2: ³⁸ Let $\Sigma ,X,Y$ and $Q$ be real matrices and $\kappa >0$ be a constant. Then

$$\Sigma +\mathrm{He}\left(\mathrm{XY}\right)<0$$

(8)

holds if

$$\left[\begin{array}{cc}\Sigma \hfill & \kappa X+Y^T{Q}^T\hfill \\ \multicolumn{1}{c}{*}& -\kappa Q-\kappa Q^T\hfill \end{array}\right]<0.$$

(9)

Main results

This section proposes an LMI-based condition that guaranteeing the stability of system (6), from which one can figure out the gains of (3). We have the following theorem.

• Theorem 1: Assume that system (1) satisfies (2) and (4). Then under controller (3), the closed-loop system (6) is asymptotically stable if there exist some positive constants $\kappa ,\lambda ,\mu ,{ϵ}_i,i=1,2,\dots ,5$ , matrices $P>0$ , $R>0$ , and $Q\in {\mathbb{R}}^{m\times m}$ , $\hat{K}\in {\mathbb{R}}^{m\times n}$ , $\hat{L}\in {\mathbb{R}}^{n\times p}$ , such that

$$\left[\begin{array}{cc}{\mathrm{\Pi }}_{11}\hfill & {\mathrm{\Pi }}_{12}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Pi }}_{12}^T}& {\mathrm{\Pi }}_{22}\hfill \end{array}\right]<0$$

(10)

where

$${\mathrm{\Pi }}_{11}=\left[\begin{array}{ccccc}{\mathrm{\Lambda }}_1\hfill & B\hat{K}\hfill & {\mathrm{\Lambda }}_3\hfill & 0\hfill & {\mathrm{\Lambda }}_4\hfill \\ \multicolumn{1}{c}{*}& {\mathrm{\Lambda }}_2\hfill & 0\hfill & \mathrm{RG}+\lambda H^T{\mathcal{M}}_{12}\hfill & {\hat{K}}^T\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\xi \mathrm{He}\left(Q\right)\hfill \end{array}\right],$$

$${\mathrm{\Pi }}_{12}=\left[\begin{array}{ccccc}P{M}_1\hfill & \mathrm{PB}{M}_2\hfill & 0\hfill & \mathrm{PB}{M}_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & R{M}_3\hfill & 0\hfill & R{M}_1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right],$$

$${\mathrm{\Pi }}_{22}=\left[\begin{array}{ccccc}-{ϵ}_{1}I\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -{ϵ}_{2}I\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{ϵ}_{3}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -{ϵ}_{4}I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & -{ϵ}_{5}I\hfill \end{array}\right],$$

$$\begin{array}{c}{\mathrm{\Lambda }}_1=A^{T}P+\mathrm{PA}+{ϵ}_1{N}_1^T{N}_1+{ϵ}_2{N}_2^T{N}_2+{ϵ}_5{N}_1^T{N}_1\hfill \\ -{\hat{K}}^T{B}^T-B\hat{K}+\mu H^T{\mathcal{M}}_{11}H,\hfill \\ {\mathrm{\Lambda }}_2=A^{T}R+\mathrm{RA}-C^T{\hat{L}}^T-\hat{L}C+{ϵ}_3{C}^T{N}_3^T{N}_{3}C\hfill \\ +{ϵ}_4{N}_2^T{N}_2+\lambda H^T{\mathcal{M}}_{11}H,\hfill \\ {\mathrm{\Lambda }}_3=\mathrm{PG}+\mu H^T{\mathcal{M}}_{12},{\mathrm{\Lambda }}_4=\kappa (\mathrm{PB}-\mathrm{BQ})-{\hat{K}}^T.\hfill \end{array}$$

Furthermore, the gain matrices $K$ and $L$ are respectively given by

$$K=Q^{-1}\hat{K},L=R^{-1}\hat{L}.$$

(11)

• Proof: Choose the following Lyapunov candidate function for system (6)

$$V(x,e)=x^T\mathrm{Px}+e^T\mathrm{Re},$$

(12)

where $P>0$ and $R>0$ .

Calculating $\stackrel{·}{V}(x,e)$ along with the trajectories of (6) yields

$$\begin{array}{c}\stackrel{·}{V}=x^T[A^{T}P+\mathrm{\Delta }{A}^{T}P-K^T{B}^{T}P-\mathrm{\Delta }{K}^T{B}^{T}P\hfill \\ +\mathrm{PA}+P\mathrm{\Delta }A-\mathrm{PBK}-\mathrm{PB}\mathrm{\Delta }K]x\hfill \\ +e^T[A^{T}R-C^T{L}^{T}R-C^T\mathrm{\Delta }{L}^{T}R-R\mathrm{\Delta }\mathrm{LC}]e\hfill \\ +x^T[\mathrm{PBK}+\mathrm{PB}\mathrm{\Delta }K+\mathrm{RA}-\mathrm{RLC}+\mathrm{\Delta }{A}^{T}R]e\hfill \\ +e^T[K^T{B}^{T}P+\mathrm{\Delta }{K}^T{B}^{T}P+R\mathrm{\Delta }{A}^T]x+x^T\mathrm{PGf}\left(\psi \right)\hfill \\ +f^T\left(\psi \right)G^T\mathrm{Px}+e^T\mathrm{RG}\tilde{f}+{\tilde{f}}^T{G}^T\mathrm{Re}\hfill \\ =x^T[A^{T}P+\mathrm{PA}-K^T{B}^{T}P-\mathrm{PBK}+{\mathrm{\Delta }}_1]x\hfill \\ +e^T[A^{T}R+\mathrm{RA}-C^T{L}^{T}R-\mathrm{RLC}+{\mathrm{\Delta }}_2]e\hfill \\ +x^T\mathrm{PBKe}+e^T{K}^T{B}^T\mathrm{Px}+{\mathrm{\Delta }}_3+x^T\mathrm{PGf}\left(\psi \right)\hfill \\ +f^T\left(\psi \right)G^T\mathrm{Px}+e^T\mathrm{RG}\tilde{f}+{\tilde{f}}^T{G}^T\mathrm{Re},\hfill \end{array}$$

(13)

where

$$\begin{array}{c}{\mathrm{\Delta }}_1=\mathrm{\Delta }{A}^{T}P+P\mathrm{\Delta }{A}^T-\mathrm{\Delta }{K}^T{B}^{T}P-\mathrm{PB}\mathrm{\Delta }K\hfill \\ =N_1^T{F}_1^T\left(t\right)M_1^{T}P+P{M}_1{F}_1\left(t\right)N_1-\mathrm{PB}{M}_2{F}_2\left(t\right)N_2\hfill \\ -N_2^T{F}_2^T\left(t\right)M_2^T{B}^{T}P,\hfill \\ {\mathrm{\Delta }}_2=-C^T\mathrm{\Delta }{L}^T\left(t\right)R-R\mathrm{\Delta }L\left(t\right)C=-C^T{N}_3^T{F}_3^T\left(t\right)M_3^{T}R\hfill \\ -R{M}_3{F}_3\left(t\right)N_{3}C,\hfill \\ {\mathrm{\Delta }}_3=x^T[\mathrm{PB}\mathrm{\Delta }K\left(t\right)+\mathrm{\Delta }{A}^T\left(t\right)R]e+e^T[\mathrm{\Delta }{K}^T\left(t\right)B^{T}P+R\mathrm{\Delta }A\left(t\right)]x\hfill \\ =x^T\mathrm{PB}{M}_2{F}_2\left(t\right)N_{2}e+x^T{N}_1^T{F}_1^T\left(t\right)M_1^T\mathrm{Re}\hfill \\ +e^{T}R{M}_1{F}_1\left(t\right)N_{1}x+e^T{N}_2^T{F}_2^T\left(t\right)M_2^T{B}^T\mathrm{Px}.\hfill \end{array}$$

In view of (4) and (5) and applying Lemma 1, one has

$$\begin{array}{c}\begin{array}{c}{\mathrm{\Delta }}_1\le {ϵ}_1^{-1}P{M}_1{M}_1^{T}P+{ϵ}_1{N}_1^T{N}_1+{ϵ}_2{N}_2^T{N}_2+{ϵ}_2^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P,\hfill \\ {\mathrm{\Delta }}_2\le {ϵ}_3^{-1}R{M}_3{M}_3^{T}R+{ϵ}_3{C}^T{N}_3^T{N}_{3}C,\hfill \\ {\mathrm{\Delta }}_3\le {ϵ}_4^{-1}{x}^T\mathrm{PB}{M}_2{M}_2^T{B}^T\mathrm{Px}+{ϵ}_4{e}^T{N}_2^T{N}_{2}e+{ϵ}_5{x}^T{N}_1^T{N}_{1}x\hfill \\ +{ϵ}_5^{-1}{e}^{T}R{M}_1{M}_1^T\mathrm{Re}.\hfill \end{array}\hfill \end{array}$$

Consequently, (13) can be continued by

$$\begin{array}{c}\stackrel{·}{V}\le x^T[A^{T}P+\mathrm{PA}-K^T{B}^{T}P-\mathrm{PBK}+{ϵ}_1{N}_1^T{N}_1+{ϵ}_1^{-1}P{M}_1{M}_1^{T}P\hfill \\ +{ϵ}_2^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P+{ϵ}_2{N}_2^T{N}_2]x+e^T[A^{T}R+\mathrm{RA}-C^T{L}^{T}R\hfill \\ -\mathrm{RLC}+{ϵ}_3^{-1}R{M}_3{M}_3^{T}R+{ϵ}_3{C}^T{N}_3^T{N}_{3}C]e\hfill \\ +x^T\mathrm{PBKe}+e^T{K}^T{B}^T\mathrm{Px}+{ϵ}_4^{-1}{x}^T\mathrm{PB}{M}_2{M}_2^T{B}^T\mathrm{Px}+{ϵ}_4{e}^T{N}_2^T{N}_{2}e\hfill \\ +{ϵ}_5^{-1}{e}^{T}R{M}_1{M}_1^T\mathrm{Re}+{ϵ}_5{x}^T{N}_1^T{N}_{1}x\hfill \\ +x^T\mathrm{PGf}\left(\psi \right)+f^T\left(\psi \right)G^T\mathrm{Px}+e^T\mathrm{RG}\tilde{f}+{\tilde{f}}^T{G}^T\mathrm{Re}\hfill \\ =x^T[A^{T}P+\mathrm{PA}-K^T{B}^{T}P-\mathrm{PBK}+{ϵ}_1{N}_1^T{N}_1+{ϵ}_1^{-1}P{M}_1{M}_1^{T}P\hfill \\ +{ϵ}_2{N}_2^T{N}_2+{ϵ}_2^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P+{ϵ}_4^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P+{ϵ}_5{N}_1^T{N}_1]x\hfill \\ +e^T[A^{T}R+\mathrm{RA}-C^T{L}^{T}R-\mathrm{RLC}+{ϵ}_3^{-1}R{M}_3{M}_3^{T}R+{ϵ}_3{C}^T{N}_3^T{N}_{3}C\hfill \\ +{ϵ}_4{N}_2^T{N}_2+{ϵ}_5^{-1}R{M}_1{M}_1^{T}R]e+x^T\mathrm{PBKe}+e^T{K}^T{B}^T\mathrm{Px}\hfill \\ +x^T\mathrm{PGf}\left(\psi \right)+f^T\left(\psi \right)G^T\mathrm{Px}+e^T\mathrm{RG}\tilde{f}+{\tilde{f}}^T{G}^T\mathrm{Re}\hfill \\ =z^T\left(t\right)\mathrm{\Xi }z\left(t\right),\hfill \end{array}$$

(14)

where $z^T\left(t\right)=[\begin{array}{cccc}{x}^T\hfill & e^T\hfill & f^T\left(\psi \right)\hfill & {\tilde{f}}^T]\hfill \end{array}$ , and

$$\begin{array}{c}\mathrm{\Xi }=\left[\begin{array}{cccc}{\phi }_1-\mathrm{He}\left(\mathrm{PBK}\right)\hfill & \mathrm{PBK}\hfill & \mathrm{PG}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\phi }_2\hfill & 0\hfill & \mathrm{RG}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & 0\hfill \end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{\phi }_1=A^{T}P+\mathrm{PA}+{ϵ}_1^{-1}P{M}_1{M}_1^{T}P+{ϵ}_1{N}_1^T{N}_1\hfill \\ +{ϵ}_2{N}_2^T{N}_2+{ϵ}_2^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P\hfill \\ +{ϵ}_4^{-1}\mathrm{PB}{M}_2{M}_2^T{B}^{T}P+{ϵ}_5{N}_1^T{N}_1,\hfill \end{array}$$

$$\begin{array}{c}{\phi }_2=A^{T}R+\mathrm{RA}-C^T{L}^{T}R-\mathrm{RLC}+{ϵ}_3^{-1}R{M}_3{M}_3^{T}R\hfill \\ +{ϵ}_3{C}^T{N}_3^T{N}_{3}C+{ϵ}_4{N}_2^T{N}_2\hfill \\ +{ϵ}_{5}R{M}_1{M}_1^{T}R.\hfill \end{array}$$

Hence, the inequality $\stackrel{·}{V}<0$ is satisfied if

$$\begin{array}{c}{z}^T\left(t\right)\mathrm{\Xi }z\left(t\right)<0.\hfill \end{array}$$

(15)

Let ${\psi }_2=\psi$ and ${\psi }_1=\hat{\psi }$ . Then by Definition 1 one has

$$\begin{array}{c}{\left[\begin{array}{c}e\hfill \\ \multicolumn{1}{c}{\tilde{f}}\end{array}\right]}^T{\left[\begin{array}{cc}H\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]}^T\mathcal{M}\left[\begin{array}{cc}H\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]\left[\begin{array}{c}e\hfill \\ \multicolumn{1}{c}{\tilde{f}}\end{array}\right]\ge 0.\hfill \end{array}$$

(16)

Notice that $f\left(0\right)=0$ . Let ${\psi }_2=\psi$ and ${\psi }_1=0$ , then

$$\begin{array}{c}{\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{f\left(\psi \right)}\end{array}\right]}^T{\left[\begin{array}{cc}H\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]}^T\mathcal{M}\left[\begin{array}{cc}H\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{f\left(\psi \right)}\end{array}\right]\ge 0.\hfill \end{array}$$

(17)

Let

$$\mathcal{M}=\left[\begin{array}{cc}{\mathcal{M}}_{11}\hfill & {\mathcal{M}}_{12}\hfill \\ \multicolumn{1}{c}{{\mathcal{M}}_{12}^T}& {\mathcal{M}}_{22}\hfill \end{array}\right],\mathrm{\Psi }=\left[\begin{array}{cccc}H\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& H\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & I\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & I\hfill \end{array}\right].$$

Then (16) and (17) are equivalent to

$$\begin{array}{c}{z}^T\left(t\right){\mathrm{\Psi }}^T\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& \lambda {\mathcal{M}}_{11}\hfill & 0\hfill & \lambda {\mathcal{M}}_{12}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill \end{array}\right]\mathrm{\Psi }z\left(t\right)\ge 0,\hfill \end{array}$$

(18)

and

$$\begin{array}{c}{z}^T\left(t\right){\mathrm{\Psi }}^T\left[\begin{array}{cccc}\mu {\mathcal{M}}_{11}\hfill & 0\hfill & \mu {\mathcal{M}}_{12}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & 0\hfill \end{array}\right]\mathrm{\Psi }z\left(t\right)\ge 0,\hfill \end{array}$$

(19)

respectively, where $\lambda >0$ and $\mu >0$ are two constants.

In view of (15), (18), and (19), we deduce that (15) is satisfied if

$$\begin{array}{c}{z}^T\left(t\right)\mathrm{\Omega }z\left(t\right)<0,\hfill \end{array}$$

(20)

where

$$\begin{array}{c}\mathrm{\Omega }=\left[\begin{array}{cccc}{\mathrm{\Omega }}_{11}\hfill & \mathrm{PBK}\hfill & {\mathrm{\Omega }}_{13}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\phi }_2+\lambda {\hat{\mathcal{M}}}_{11}\hfill & 0\hfill & {\mathrm{\Omega }}_{24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill \end{array}\right],\hfill \end{array}$$

$$\begin{array}{c}{\mathrm{\Omega }}_{11}={\phi }_1+\mu {\hat{\mathcal{M}}}_{11}-\mathrm{He}\left(\mathrm{PBK}\right),{\hat{\mathcal{M}}}_{11}=H^T{\mathcal{M}}_{11}H,\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{13}=\mathrm{PG}+\mu {\hat{\mathcal{M}}}_{12},{\hat{\mathcal{M}}}_{12}=H^T{\mathcal{M}}_{12},{\mathrm{\Omega }}_{24}=\mathrm{RG}+\lambda {\hat{\mathcal{M}}}_{12}.}\end{array}$$

Notice that the inequality (20) holds if $\mathrm{\Omega }<0$ . However, $\mathrm{\Omega }<0$ is not a strict LMI due to the bilinear terms $\mathrm{PBK}$ . To tackle this problem, let $K$ be decomposed with $K=Q^{-1}\hat{K}$ , then $\mathrm{PBK}$ can be converted into linear term with

$$\begin{array}{c}\mathrm{PBK}=(\mathrm{PB}-\mathrm{BQ})Q^{-1}\hat{K}+B\hat{K}.\hfill \end{array}$$

(21)

On the other hand, $\mathrm{\Omega }$ can be rewritten as

$$\begin{array}{c}\left[\begin{array}{cccc}{\phi }_1+\mu {\hat{\mathcal{M}}}_{11}\hfill & 0\hfill & {\mathrm{\Omega }}_{13}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\hat{\phi }}_2\hfill & 0\hfill & {\mathrm{\Omega }}_{24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill \end{array}\right]\hfill \\ +\left[\begin{array}{cccc}-\mathrm{He}\left(\mathrm{PBK}\right)\hfill & \mathrm{PBK}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{K^T{B}^{T}P}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \end{array}\right]\hfill \\ =\left[\begin{array}{cccc}{\hat{\phi }}_1\hfill & B\hat{K}\hfill & {\mathrm{\Omega }}_{13}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\hat{\phi }}_2\hfill & 0\hfill & {\mathrm{\Omega }}_{24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill \end{array}\right]\hfill \\ +\mathrm{He}\left\{\left[\begin{array}{c}\mathrm{PB}-\mathrm{BQ}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]Q^{-1}{\left[\begin{array}{c}-{\hat{K}}^T\hfill \\ \multicolumn{1}{c}{{\hat{K}}^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]}^T\right\}\hfill \end{array}$$

where

$${\hat{\phi }}_1={\phi }_1+\mu {\hat{\mathcal{M}}}_{11}-{\hat{K}}^{T}B-B\hat{K},{\hat{\phi }}_2={\phi }_2+\lambda {\hat{\mathcal{M}}}_{11}.$$

Thus, one can convert $\mathrm{\Omega }<0$ into the form of (8) by letting

$$\Sigma =\left[\begin{array}{cccc}{\hat{\phi }}_1\hfill & B\hat{K}\hfill & {\mathrm{\Omega }}_{13}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\hat{\phi }}_2\hfill & 0\hfill & {\mathrm{\Omega }}_{24}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill \end{array}\right],X=\left[\begin{array}{c}\mathrm{PB}-\mathrm{BQ}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],$$

$$Y=Q^{-1}\left[\begin{array}{cccc}-\hat{K}\hfill & \hat{K}\hfill & 0\hfill & 0\hfill \end{array}\right].$$

Therefore, by applying Lemma 2, $\mathrm{\Omega }<0$ is equivalent to

$$\begin{array}{c}\left[\begin{array}{ccccc}{\hat{\phi }}_1\hfill & B\hat{K}\hfill & {\mathrm{\Omega }}_{13}\hfill & 0\hfill & {\mathrm{\Lambda }}_4\hfill \\ \multicolumn{1}{c}{*}& {\hat{\phi }}_2\hfill & 0\hfill & {\mathrm{\Omega }}_{24}\hfill & {\hat{K}}^T\hfill \\ \multicolumn{1}{c}{*}& *\hfill & \mu {\mathcal{M}}_{22}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & \lambda {\mathcal{M}}_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\kappa \mathrm{He}\left(Q\right)\hfill \end{array}\right]<0.\hfill \end{array}$$

(22)

Notice that (22) can be converted into the following form

$${\mathrm{\Pi }}_{11}-{\mathrm{\Pi }}_{12}{\mathrm{\Pi }}_{22}^{-1}{\mathrm{\Pi }}_{12}^T<0.$$

(23)

By utilizing the Schur complement lemma, ²⁶ (10) is equivalent to (23). Therefore, system (6) is asymptotically stable if (10) is satisfied. Furthermore, the corresponding gains $K$ and $L$ are then obtained by (11). This concludes the proof.▪

Simulation study

This section gives a detailed illustration of the validity of the developed observer-based non-fragile controller via a numerical example. Consider a controlled Lorenz chaotic system borrowed from Zhao et al. ²⁹ with the following form

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=10(x_2-x_1)+u_1,\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=28x_1-x_2-x_1{x}_3+u_2,}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_3=-(8/3)x_3+x_1{x}_2-u_1,}\\ \multicolumn{1}{c}{y=x_1}\end{array}$$

One can rewrite the above system in the state-space form of (1) with

$$A=\left[\begin{array}{ccc}-10\hfill & 10\hfill & 0\hfill \\ \multicolumn{1}{c}{28}& -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -8/3\hfill \end{array}\right],B=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill \end{array}\right],G=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right],$$

$$H=I_3,C=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right],f\left(\mathrm{Hx}\left(t\right)\right)={\left[\begin{array}{cc}{x}_1{x}_3\hfill & x_1{x}_2\hfill \end{array}\right]}^T.$$

The nonlinearity $f\left(\mathrm{Hx}\left(t\right)\right)$ belongs to the conic class (see, e.g., ²¹ ). Notice that we cannot select a global Lipschitz constant for this kind of nonlinearities, which means this system could not described by the global Lipschitz condition. Therefore, the existing results^{14–18,31,32,35} cannot be applied to this case.

However, this nonlinearity can be characterized by IQC. In fact, from Zhao et al. ²⁹ we know that the corresponding IMM can be derived in the following form

$$\mathcal{M}=\left[\begin{array}{ccccc}10.1989\hfill & 0\hfill & 0\hfill & -0.9520\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 7.9306\hfill & 0\hfill & 0\hfill & -1.4586\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 7.4234\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-0.9520}& 0\hfill & 0\hfill & -4.0588\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1.4586\hfill & 0\hfill & 0\hfill & -4.0836\hfill \end{array}\right]$$

Moreover, assume that the uncertainties in system (1) and the controller (3) can be characterized with following parameters

$$\begin{array}{c}{M}_1=I_3,M_2=I_2,M_3=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right],\hfill \\ F_1\left(t\right)=\left[\begin{array}{ccc}\alpha \left(t\right)/{\alpha }_m\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \beta \left(t\right)/{\beta }_m\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \gamma \left(t\right)/{\gamma }_m\hfill \end{array}\right],\hfill \end{array}$$

$$F_2\left(t\right)=\left[\begin{array}{cc}\delta \left(t\right)/{\delta }_m\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \eta \left(t\right)/{\eta }_m\hfill \end{array}\right],F_3\left(t\right)=\rho \left(t\right)/{\rho }_m,$$

$$N_1=\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 0\hfill \end{array}\right],N_2=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \end{array}\right],N_3=1.$$

where $\alpha \left(t\right)\le {\alpha }_m$ , $\beta \left(t\right)\le {\beta }_m$ , $\gamma \left(t\right)\le {\gamma }_m$ , $\delta \left(t\right)\le {\delta }_m$ , $\eta \left(t\right)\le {\eta }_m$ , $\rho \left(t\right)\le {\rho }_m$ , and ${\alpha }_m$ , ${\beta }_m$ , ${\gamma }_m$ , ${\delta }_m$ , ${\eta }_m$ , ${\rho }_m$ are known upper bounds. ³²

For these uncertainties, by solving (10) the gains of controller and observer are derived

$$K=\left[\begin{array}{ccc}696.00\hfill & -312.93\hfill & -802.69\hfill \\ \multicolumn{1}{c}{615.51}& 1694.10\hfill & 662.84\hfill \end{array}\right],L=\left[\begin{array}{c}9.52\hfill \\ \multicolumn{1}{c}{58.19}\\ \multicolumn{1}{c}{-0.18}\end{array}\right].$$

Then, with the time-varying parameters $\alpha \left(t\right)=\cos \left(3t\right)$ , $\beta \left(t\right)=1-0.3\sin \left(2t\right)$ , $\gamma \left(t\right)=1.3\sin \left(4t\right)$ , $\delta \left(t\right)=-1.2\sin \left(3t\right)$ , $\eta \left(t\right)=-1.3\cos \left(4t\right)$ , $\rho \left(t\right)=-1.2\cos \left(3t\right)$ and the initial conditions $x\left(0\right)={(23-4)}^T$ , $\hat{x}\left(0\right)={(5-13)}^T$ , the state trajectories are displayed in Figures 1–3, while the estimation error is depicted in Figure 4. It can be seen from Figures 1 –4 that the estimate states rapidly converge to the real states.

Figure 1.

The simulation for state $x_1$ and its estimate.

Figure 2.

The simulation for state $x_2$ and its estimate.

Figure 3.

The simulation for state $x_3$ and its estimate.

Figure 4.

The simulation for the estimation errors.

Conclusion

This article has addressed the issue of observer-based non-fragile robust control for incremental quadratic nonlinear systems with time-varying perturbations. Firstly, a non-fragile observer is put forward to estimate the state of this uncertain system. And then a non-fragile robust control scheme is employed to deal with uncertain time-varying perturbations in both the observer and the controller gains. Resorting to some useful matrix inequalities, we convert the BMIs involved in the Lyapunov stability analysis into LMIs without utilizing any equality constraints. Furthermore, an LMI-based sufficient condition is provided to guarantee the asymptotical stability of the whole closed-loop system. The validity of the proposed observer-based non-fragile robust control design is demonstrated through a simulation example.