# Disturbance Attenuation and Rejection for a Class of Switched Nonlinear Systems Subject to Input and Sensor Saturations

Authors: Yunliang Wei; Liping Sun; Shengsen Jia; Kunming Liu; Fanwei Meng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper investigates the problem of disturbance attenuation and rejection for a class of switched nonlinear systems subject to input and sensor saturations, in which exosystem generated disturbances andH2-norm bounded disturbances are considered. The full-order and reduced-order observers are designed according to whether the system states are available or not. Based on the estimating values of the system states and exosystem generated disturbances, the design schemes for the composite controllers are put forward based on the full-order and reduced-order observers, respectively. For a switched system, the input and sensor saturations would influence the effective synthesis of observer and controller. By sector nonlinearity technology, the stability of the augmented closed-loop systems under the proposed composite controllers are analyzed, and the conditions of synthesis of the observers and controllers are further presented to ensure the augmented systems to be robustly asymptotically stable with a weighted H∞ performance level. An example is given to guarantee the effectiveness of the proposed control schemes.


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## Body

## 1. Introduction

In past decades, the research on switched systems has attracted more and more attentions [1, 2]. Switched systems often are encountered in practical control fields, such as mechanical control, aircraft and air traffic control, and switching power converter, which consist of a finite number of subsystems and a switching law that orchestrates the switching between these subsystems [3]. The most important research direction of switched systems is to investigate the stability of switched systems. There are two aspects of this problem: one is how to obtain stable condition of switched systems under the arbitrary switching law [4–6]; the other is how to construct a switching law to guarantee the switched systems to be stable [7, 8]. Based on the discussion about the stability, some schemes of controller synthesis have been further put forward [9, 10]. Recently, many excellent results on the stability analysis and synthesis of discrete-time switched systems have been established [11, 12].The disturbances are mainly caused by measurement noise, load changing, external surroundings, sensor and actuator errors, modeling errors, and structural changes in many actual processes. In recent years, the attenuation and rejection of disturbances have always been hot research topics. Various ways have been tried to deal with disturbances, so that the performance of the system could be less degraded and the system could be more stable [13–23]. In [13], a robust adaptive full-order observer is proposed with adaptation laws for the unknown disturbances and parameters, and a reduced-order observer-based response system is developed under the same assumptions, which can synchronize part of states of drive system. In [14], the problem of composite antidisturbance control for time-varying delay Markovian jump nonlinear systems with multiple disturbances is investigated; by combining disturbance observer and L2-L∞control method, the disturbances are attenuated and rejected. Recently, a composite fault-tolerant control with disturbance observer scheme is considered for a class of stochastic systems with faults and multiple disturbances. A stochastic disturbance observer is designed to estimate exogenous disturbances and a composite fault-tolerant controller is proposed to compensate disturbances and faults [15]. A hierarchical antidisturbance control methodology is given for Markovian jump systems with nonlinearity and multiple disturbances [16]. It is noteworthy that how to restrain disturbance more effectively for switched systems is still an open and attractive research direction.It is well known that actuator saturation, a quite common and inherent phenomenon for practice controlled systems, will lead to poor performance of the closed-loop system and may even cause the system to lose its stability. Therefore, in the past few decades, the study of saturated system has attracted considerable attentions [24–32]. In [25], an antiwindup design is proposed to compensate the stabilizing controller for a linear system subject to nested input saturation, such that the negative effects from saturation are eliminated. In [26], the robust finite-horizon output feedback control problem for a class of uncertain discrete stochastic nonlinear time-varying systems with both sensor and actuator saturations is investigated. An output feedback controller is proposed such that the H∞ disturbance attenuation level is guaranteed for the nonlinear stochastic polytopic system in the presence of saturated sensor and actuator outputs. Recently, the stabilization problem of a class of nonlinear systems with actuator saturation is investigated via active disturbance rejection control (ADRC) and a LMI-based algorithm is correspondingly established to design the linear ADRC controller [27]. Further, in [28], a new compensation scheme is proposed to attenuate the effects from both the randomly occurring faults and the sensor saturations onto the estimation performance. Recently, the optimal control, antiwindup control, and model predictive control have become the most basic methods to deal with saturation problems [33, 34].Motivated by the above-mentioned literatures, this paper investigates the problem about analysis and synthesis of a class of switched nonlinear systems with input and sensor saturations. In the past few decades, the researchers only considered switching and input saturation [29, 30, 35] or only considered input and sensor saturation [26]. When a switched system is subject to both input and sensor saturation, the analysis and synthesis would be very difficult. The main contribution and innovation in this paper are included as follows: firstly, the full-order and reduced-order observers are designed to the values of the system states and matched disturbances, respectively; secondly, by sector nonlinearities technology, the design conditions of the controller ensuring the augmented system is robustly asymptotically stable in the absence of the mismatched disturbances for all initial conditions under arbitrarily switching law; thirdly, a H∞ control scheme is presented to attenuate the mismatched disturbances for the augmented closed-loop system.


## 2. Problem Formulation

Consider the following discrete-time switched system in presence of nonlinearities and saturation constraints(1)xk+1=Aσkxk+Bσksatuk+d1k+Gσk,1fσk,1xk,k+Eσk,1d2kyk=satCσkxk+Gσk,2fσk,2xk,k+Dσkd1k+Eσk,2d2k,where x(k)∈Rn, u(k)∈Rm, and y(k)∈Rp are the state, input, and output vectors of the system, respectively; d2(k)∈Rw is the disturbance which belongs to L2[0,∞). σ(k):Z→I[1,q] is the switching law. When σ(k)=i∈I[1,q], the ith subsystem is activated. For the sake of brevity, in the following, we assume that σ(k)=i. The matrices Ai, Bi, Ci, Di, Gi1, Gi2, Ei1, and Ei2 are known, real matrices of appropriate dimensions. d1(k)∈Rm is the unknown disturbance satisfying the following assumption.Assumption 1.
The disturbanced1(k) in the control input path can be formulated by the following exogenous system:(2)ωk+1=Wσkωk+Mσkd3kd1k=Vσkωk,where Wi∈Rr×r, Mi∈Rr×l, and Vi∈Rm×r are known matrices for σ(k)=i∈I[1,q]. d3(k)∈Rl is supposed to belong to L2[0,∞).Remark 2.
In practical engineering, many kinds of disturbances can be described as the dynamic system (2), such as the constant and harmonics noises with unknown phase and magnitude. The systems with disturbances generated by the exogenous model have been widely researched. In (2), d3(t) is an additional disturbance which results from the perturbations and uncertainties of the exogenous model [16].For further discussion, we need to make an assumption as follows.Assumption 3.
For anyi∈I[1,q], (Ai,Bi) is controllable and (Wi,BiVi) is observable.Definition 4.
A nonlinearityΨ:Rj→Rj is said to satisfy a sector condition if Ψ(0)=0 and(3)Ψx-Ψy-H1x-yTΨx-Ψy-H2x-y≤0,∀x,y∈Rj,where H1 and H2 are known constant matrices, and H=H1-H2 is positive definite matrix. It is customary that Ψ is said to belong to the sector [H1,H2].Remark 5.
The nonlinearitiesfi1(x(k),k) and fi2(x(k),k), without loss of generality, are assumed to belong to the sectors [Ui11,Ui12] and [Hi21,Hi22], respectively.Remark 6.
In the most exiting results, the system nonlinearities are assumed to satisfy the Lipschitz condition. The sector nonlinearity can contain the Lipschitz nonlinearity by choosingH1=H2 to be positive definite matrix, which means that the sector nonlinearity is quite general.s a t ( · ) : R i → R i denotes the vector valued standard saturation function, which is described as sat(v)=[sat1(v1),sat2(v2),⋯,sati(vi)]T with satj(vj)=signvjmax⁡{1,|vj|,j∈I[1,i]}. Without loss of generality, we assume that the saturation level is unity.Assuming that there exist the diagonal matricesRi11, Ri12 and Ri21, Ri22 such that 0≤Ri11<I≤Ri12 and 0≤Ri21<I≤Ri22, then the saturation functions sat(u¯(t)) and sat(y(k)) can be rewritten as(4)satu¯k=Ri11u¯k+Ψi1u¯ksatCixk=Ri21Cixk+Ψi2Cixk,where u¯(k)=u(k)+d1(k), Ψi1(u¯(k)) and Ψi2(y(k)) are nonlinear functions and belong to [0,Ri1] and [0,Ri2], respectively, with Ri1=Ri12-Ri11 and Ri2=Ri22-Ri21, i.e., Ψ1(u(t)+d1(t)) and Ψ2(y(t)) satisfies the following inequalities:(5)Ψi1u¯kΨi1u¯k-Ri1u¯k≤0Ψi2CixkΨi2Cixk-Ri2Cixk≤0,Define a polyhedral set(6)LRi11u¯k=Ri11u¯kl≤1,l∈I1,m.Remark 7.
For saturation nonlinearity, the equivalent form (4) comprising linear term and sector nonlinear term is frequently used and valid. Supposing Ki, i=I[1,q] to be the gains of controller, the matrix Ri11 needs to guarantee Ai+Ri11Ki is stable. Therefore, we need to consider (4) for the input saturation in the region (6). However, for the sensor saturation, Ri21 is chosen to be nonzero and based on the maximum value of Cix in the engineering practice [24, 26].Lemma 8 (S-procedure).
LetY0(η),Y1(η),…,Yp(η) be quadratic functions of η∈Rn, Yi(η)=ηTTiη, i=0,1,…,p with Ti=TiT. Then, the implication Y1(η)≤0,…,Yp(η)≤0⇒Y0(η)≤0 holds if there exist τ1,…,τp>0 such that (7)T0-∑i=1pτiTi≤0.Our aim in this paper is to solve the problem of composite control for the nonlinear saturating system (1) with multiple saturations and disturbances. Based on whether the system states are available, the full-order and reduced-order observers will be designed respectively. Here, the disturbance observer control approach will be applied for rejecting and attenuating the input disturbance, while the external disturbance will be attenuated by a H∞ control scheme.


## 3. Main Result

In this section, based on whether the system states are available, the composite full-order and reduced-order observer-based controllers are designed respectively. The input disturbance is rejected while external disturbance is attenuated.


### 3.1. Full-Order Observer-Based Controller

First of all, we suppose that the system states are not available and the nonlinearitiesfi1(x),fi2(x) are known.Defining the extended vectorx¯(t)=[xT(k),ωT(k)]T, together with (2) and (4), for σ(k)=i, given diagonal matrices Ri11≥0, Ri12≥0, Ri21≥0, Ri22≥0 satisfying Ri12≥I>Ri11≥0 and Ri22≥I>Ri21≥0, and any u¯(t)∈L(Ri11u¯(k)), the system (1) can rewritten as follows:(8)x¯k+1=A¯ix¯k+G¯i1fx¯k,k+B¯iRi11uk+B¯iΨi1k+E¯i1dkyk=Ri21C¯i+D¯ix¯k+G¯i2fx¯k,k+Ψi2k+E¯2dk,wherefi1(x¯)=fi1(x),fi2(x¯)=fi2(x) and (9)fix¯k,k=fi1x¯fi2x¯,dk=d2kd3k,A¯i=AiBiRi11Vi0Wi,B¯i=Bi0,G¯i1=Gi1000,E¯i1=Ei100Mi,C¯i=Ci0,D¯i=0DiVi,G¯i2=Gi20,E¯i2=Ei20.The full-order observer-based controller for the extended system (8) is designed as(10)x¯^k+1=A¯ix¯^k+B¯iRi11uk+G¯i1fix¯^k,k+Liy^k-yky^k=Ri21C¯ix¯^k+G¯i2fix¯^k,k+D¯ikx¯^kuk=Kix^k-d^1k,wherefi1(x¯^)=fi1(x^),fi2(x¯^)=f2(x^) and(11)x¯^k=x^kω^k,Li=Li1Li2,fix¯^k,k=fi1x¯^fi2x¯^.The estimation error is denoted as (12)ek=x¯k-x¯^k=exkeωk=xk-x^kωk-ω^k,and then the error dynamic can be obtained(13)ek+1=A¯i-LiRi21C¯i-LiD¯iek+S¯iΨik+E¯i1-LiE¯i2dk+G¯i1-LiG¯i2fix¯-fix¯^,where (14)S¯i=B¯i-Li,Ψik=Ψi1kΨi2k.Based on (10), (6) can be rewritten as(15)LRi11u¯k=Ri11K~ilx~≤1,l∈I1,m,where (16)x~k=xkek,K~i=Ki-KiVi.For any x~(k)∈L(Ri11u¯(k)) with a given Ri11, by (4), the closed-loop system can be described as(17)x~k+1=A~ix~k+G~if~ik+E~idk+S~iΨik,where (18)A~i=Ai+BiRi11KiBiRi11-KiVi0A¯i-LiC¯i-LiD¯i,f~ik=fi1xfix¯-fix¯^,S~i=Bi0S¯i,G~i=Gi100G¯i1-LiG¯i2,E~=Ei10E¯i1-LiE¯i2.The reference output is chosen to be(19)zk=x~k.In the following theorem, we present some sufficient conditions under which a full-order observer-based controller can be designed to stabilize the closed-loop system (17) without external disturbances d(k) and satisfy the H∞ performance requirement when the system is subject to external disturbances.Theorem 9.
For given diagonal matricesRi11≥0,Ri12≥0 satisfying Ri12≥I>Ri11≥0, diagonal matrices Ri21≥0,Ri22≥0 satisfying Ri22≥I>Ri21≥0 and γ>0, under arbitrarily switching law, the closed-loop system (17) under the full-order state observer (10) for the augmented system (8) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1} and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices P1>0, P2>0, P3>0, S1>0, Yi1, Yi2, Yi3 and parameters τi>0, i∈I[1,5] such that(20)P1S1=I.(21)1Ri11Yi1l-Ri11Yi1lRi11Vil∗P100∗∗P20∗∗∗P3>0,l∈I1,m,(22)Φi11Φi120Φi14∗Φi220Φi24∗∗-γ2IΦi34∗∗∗Φi44<0,where (23)U˘i1=Ui11TUi12+Ui12TUi112,U˘i2=-Ui11T+Ui12T2H˘i1=Hi11THi12+Hi12THi112,H˘i2=-Hi11T+Hi12T2,Θi1=-P1-τi1U˘i1+I,Θi2=-P2-τi2U˘i1-τi3H˘i1+I,Φi11=Θi100-τi1U˘i2000Θi200-τi2U˘i2-τi3H˘i200-P3+I000-τi1U˘i2T00-τi1I000-τi2U˘i2T00-τi2I00-τi3H˘i2T000-τi3I,Φi12T=Ri1Yi1-Ri1Yi1Ri1Vi000τi5Ri2Ci00000,Φi14=AiT+Yi1TRi11BiT00-Yi1TRi11BiTAiTP2-CiTYi2TCiTYi3TViTRi11BiTViTRi11BiTP2T-ViDiTYi2TWiP3+ViDiTYi3TGi1T000Gi1TP2-Gi2TYi2T-Gi2TYi3T000,Φi22=-2τi4I00-2τi5I,Φi24=BiTBiTP200-Yi2T-Yi3T,Φi34T=Ei10P2Ei1-Yi1Ei200P3Mi-Yi3Ei2,Φi44=-S1000-P2000-P3.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(24)Ki=Yi1,Li1=P2-1Yi2,Li2=P3-1Yi3.Proof.
Choose the common Lyapunov candidate as follows:(25)Vx~k=x~TkPx~k,where P=diag{P1,P2,P3} with Pj>0, j=1,2,3.
For anyx~(k)∈L(Ri11u¯(t)), along the closed-loop augmented system (17) with d(k)≡0, the difference of V(k) can be calculated as(26)∆Vx~k=x~Tk+1Px~k+1-x~TkPx~k=ηkTAi0TPAi0-Pηk,where (27)ηTk=x~Tkf~iTk,ΨiTk,A0i=A~iG~iS~i,P=diagP,0,0.From the assumption that the nonlinearities f1(k) and f2(k) belong to sectors [Ui11,Ui12] and [Hi21,Hi22], it can be verified that(28)xkfi1xTU˘i1U˘i2∗Ixkfi1x≤0,exkfi1x¯-fi1x¯^TU˘i1U˘i2∗Iexkfi2x¯-fi2x¯^≤0,exkfi2x¯-fi2x¯^TH˘i1H˘i2∗Iexkfi2x¯-fi2x¯^≤0,where U˘i1, U˘i2, H˘i1, and H˘i2 are defined in (1).
From (22), the following inequality can be obtained:(29)Φ~i11Φi12Φi14∗Φi22Φi24∗∗Φi44<0,where Φ~i11=Φi11-diag{II,I,0,0,0}.
Noting the facts ofYi1=Ki, Yi2=P2Li1, and Yi3=P3Li2, before and after multiplying (29) by diag{Iτi4-1,I,P1,I} and by the condition (20), it yields that(30)Φ~i11Φi12Φ~i14∗Φi22Φ~i24∗∗P<0,where Φ~i14=Φi14P~1 and Φ~i24=Φi24P~1 with P~1=diag{P1,I,I}.
Together (5), (28) with (30), using Lemma 8 and Schur complement, we can get(31)∆Vx~k<0,x~k∈LRi11u¯t.Using Schur complement, (21) is equivalent to(32)Ri11K~lP-1Ri11K~lT≤1,which means that(33)ΩP⊂LRi11K~ix~k.From (30) and (33), we know that the augmented closed system (17) without external disturbance d(k) is asymptotically stable with an initial set Ω(P).
Next, we suppose that system (17) is subject to external disturbances d(k)≠0.
For givenγ>0, consider the index as follows:(34)JN=∑k=1NzTkzk-γ2dTkdk.Under the zero initial condition, from (25), we have(35)JN≤∑k=1NzTkzk-γ2dTkdk+∆Vk.From the above discussion and conditions (20), (22), we can readily derive(36)JN≤0,and then, for nonzero d(k)∈L(0,∞), one has(37)limN→∞JN≤0,which means that the H∞ performance z(k)2≤γd(k)2 is satisfied.


### 3.2. Reduced-Order Observer-Based Controller

Assuming that the states of the system (1) are available, a disturbance observer can be formulated as(38)d^1k=Vσkω^kω^k=vk-Lσkxk,where v(k) is generated by the following dynamic:(39)vk+1=Wσk+LσkBσkRσk,11Vσkvk-Lσkxk+LσkAσkxt+BσkRσk,11uk+Gσk,1fσk,1xk,k.Then, the dynamic of the estimation of disturbance error eω(k) is(40)eωk+1=Wσk+LσkBσkRσk,11Veωk+LσkBσkΨσk,1k+Mσkd3k+LσkEσk,1d2k.A disturbance observer-based controller is designed as(41)uk=Kσkxk-d^1k.Combining (1), (38), and (40), for σ(k)=i∈I[1,q], the closed-loop system for the extended state x(k)=[xT(k)eωT(k)]T can be described as(42)xk+1=Aixk+Gifik+SiΨ1k+Eidk,where (43)Ai=Ai+BiRi11KiBiRi11Vi0Wi+LiBiRi11V,Si=BiLiBi,Gi=Gi10,Ei=Ei10LiEi1Mi.For this case, (6) can be rewritten as (44)LRi11u¯k=Ri11Kixl≤1,l∈I1,m,where Ki=[Ki,Vi]. The reference output is set as(45)zk=xk,Theorem 10.
Given diagonal matricesRi11≥0, Ri12≥0 satisfying Ri12≥I>Ri11≥0 and γ>0, under arbitrarily switching law, the augmented system (42) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1},P=diag{P1,P2}, and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices P1>0, P2>0, S1>0, Yi1, Yi2 and parameters τi>0, i∈I[1,5] such that(46)P1S1=I,(47)1Ri11Yi1lRi11Vil∗P10∗∗P2>00,l∈I1,m,(48)Ψi110Ψi13∗-γ2IΨi23∗∗Ψi33<0,where U˘i1 and U˘i2 were defined in (1), and(49)Ψi11=-P1-τi1U˘i1+I0-τi1U˘i2Yi1TRi10-P2+I0ViTRi1-τi1U˘i2T0-τi1I0Ri1Yi1Ri1Vi0-2τi2I,Ψi13T=Ai+BiRi11Yi1BiRi11ViGi1Bi0P2Wi+Yi2BiRi11Vi0Yi2Bi,Ψi23=Ei1TEi1TYi2T0MiTP2,Ψi33=-S100-P2.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(50)Ki=Yi1,Li=P2-1Yi2.Proof.
A Lyapunov candidate function is chosen as(51)Vxt=xTtPxt,where P=diag{P1,P2} with Pi>0, i=1,2.
For anyx(k)∈L(Ri11Kix(k)), the difference of V(k) along system (42) in the absence of disturbance d(k) is(52)∆Vk=xTk+1Pxk+1-xkTPxk=ΘTkA0iTPA0i-PΘk,where(53)Θk=xkfikΨik,A0i=AiGiSi,P=diagP,0,0.Defining Yi1≜qKi and Yi2≜qP2Li and performing a congruence transformation to (48) by diag{I,I,P1,I}, (48) is equivalent to(54)Ψi110Ψ¯i13∗-γ2IΨ¯i23∗∗Ψ¯i33<0,where (55)Ψ¯i13T=P1Ai+P1BiRi11Yi1P1BiRi11ViP1Gi1P1Bi0P2Wi+P2LiBiRi11Vi0P2LiBi,Ψ¯i23T=P1Ei10P2LiEi1P2Mi,Ψ¯ii33=-P100-P2.Equation (56) yields that(56)∆Vk-τi1fi1k-Ui11xfi1k-Ui12x-2τi2Φi1TkΦi1-Ri1kxk≤0Based on the assumption that fi1 belongs to the sector [Ui11,Ui12] and (28), by Lemma 8, we have(57)∆Vk≤0.Under the conditions of zero initial state and nonzero disturbance d(k), defining the H∞ performance index for given γ>0 as(58)JN=∑k=1NzTkzk-γ2dTkdk.By Schur complement to (56) and (47), along the same line in the proof of Theorem 9, we can represent that(59)z2≤γdk,fordk∈L0,∞,and(60)ΩP⊂LRi11Kxk.From (57), (59), and (60), the desired results are obtained and the proof is completed.Remark 11.
The composite antidisturbance control problem has been solved by designing disturbance observer-based controller for the switched system (1) subject to input saturation and multiple disturbances. The disturbance observer-based control method is used to reject the input path disturbances generated by system (2), and for the external disturbances which belong to L[0,∞), the attenuation performance are achieved by H∞ control method. The most existing results about composite antidisturbance control do not consider the switched systems subject to saturation, such as actuator saturation and sensor saturation. We solved the problem from comprehensive perspective and gave the results from both full-order and reduced-order observer-based composite control methods.Remark 12.
Although, due to (20) and (46), the conditions in Theorems 9 and 10 are not rigid constraints in the framework of linear matrix inequalities (LMIs), here, the following algorithms are proposed to this nonconvex feasibility problem.Algorithm 13.
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Step  1 (initialization). Choose the appropriate initial positive definite diagonal matrices Ri11, Ri12, Ri21, Ri22 satisfying Ri12≥I>Ri11≥0 and Ri22≥I>Ri21≥0, scalar λ>0, and find a feasible point Mk, Nk. If there are none, exit. Set k=0.
Step  2. Solve matrices P1,S1 by the following problem: (61)MinimizeTRMkP1+NkS1subjecttoaP1IIS1≥0,binequalities21and22hold.
Step  3. Set Mk+1=S1 and Nk+1=P1. If TR{Mk+1Nk+1}=n, stop the iteration; otherwise, go back to Step 2.Algorithm 14.
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Step  1 (initialization). Choose the appropriate initial positive definite diagonal matrices Ri11,Ri12 satisfying Ri12≥I>Ri11≥0 and scalar λ>0, and find a feasible point Mk, Nk. If there are none, exit. Set k=0.
Step  2. Solve matrices P1,S1 by the following problem: (62)MinimizeTRMkP1+NkS1subjecttoaP1IIS1≥0,binequalities47and48hold.
Step  3. Set Mk+1=S1 and Nk+1=P1. If TR{Mk+1Nk+1}=n, stop the iteration; otherwise, go back to Step 2.Remark 15.
With the help of the Algorithms13 and 14, we will propose further the following algorithms to optimize the design schemes of the controllers for estimation of the domain of initial states as large as possible.Algorithm 16.
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Step  1. Choose the initial scalar γ>0, the appropriate matrix R>0, the initial matrices P1>0 and S1>0 which are obtained from Algorithm 13.
Step  2. Use the initialization from Step 1 to solve the following problem for α: (63)InfP2>0,P3>0,τi>0,i=1,2,3,4,5,Yij,j=1,2,3αs.t.aP-αR≤0,whereP=diagP1,P2,P3;binequalities21and22hold.Algorithm 17.
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Step  1. Choose the initial scalar γ>0, the appropriate matrix R>0, the initial matrices P1>0 and S1>0 which are obtained from Algorithm 13.
Step  2. Use the initialization from Step 1 to solve the following problem for α: (64)InfP2>0,Yij,j=1,2αs.t.aP-αR≤0,whereP=diagP1,P2;binequalities47and48hold.Remark 18.
The results of both Theorems9 and 10 are established under the arbitrary switching rules due to the common Lyapunov function being chosen, and how to choose less conservative Lyapunov function is one of our future works. When the switching rules σ(k)≡i, that means the switched system always stays in the ith subsystem, the result can be used for each subsystem as a special and independent case.Assumption 19.
The nonlinearityfi1(x(k),k) satisfies the Lipschitz condition, namely, fi1(0,0)=0 and (65)fi1xk,k-fi1yk,k-Ui11xk-yk≤0.Corollary 20.
Given diagonal matricesRi11≥0,Ri12≥0 satisfying Ri12≥I>Ri11≥0 and γ>0, under Assumption 19 and arbitrarily switching law, the augmented system (42) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1}, P=diag{P1,P2}, and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices X1>0, P2>0, Yi1, Yi2 and parameters τi>0, i∈I[1,5] such that(66)1Ri11Yi1lRi11Vil∗-X10∗∗P2<0,l∈I1,m,(67)Λi110Λi13∗-γ2IΛi23∗∗Λi33<0,where U˘i1 and U˘i2 were defined in (1), and(68)Λi11=-X10-τi1X1U˘i11Yi1TRi10-P2+I0ViTRi1-τi1U˘i11TX10-τi1I0Ri1Yi1Ri1Vi0-2τi2I,Λi13T=AiX1+BiRi11Yi1BiRi11ViGi1Bi0Wi+LiBiRi11Vi0LiBiX1000,Λi23=Ei1TEi1TLiT00MiT0,Λi33=-X1000-P2000-I.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(69)P1=X1-1,Ki=Yi1P1,Li=Yi2.Proof.
Setting the notation (69), performing a congruence transformation to (66) by diag{IP1,I} and to (67) by diag{P1,I,I,I,I,I,I,I,I} and using Schur complement, from the fact Ui11TUi11≥0, one has (21) and (56), under Assumption 19, and the proof follows Theorem 10.Remark 21.
The nonlinearitiesfi1 under Assumption 19 are investigated widely in the most of results of the composite antidisturbance control [36]. The result of Corollary 20 is convex due to being in the framework of linear matrix inequalities (LMIs); therefore, it is feasible.Remark 22.
In the above discussion, the nonlinearitiesfi1 and fi2 are assumed to be known. The results about the case of the unknown nonlinearities are easy to be obtained by removing the information of fi1 and fi2 in the observers (10) and (40); the details are omitted here for space saving.


## 3.1. Full-Order Observer-Based Controller

First of all, we suppose that the system states are not available and the nonlinearitiesfi1(x),fi2(x) are known.Defining the extended vectorx¯(t)=[xT(k),ωT(k)]T, together with (2) and (4), for σ(k)=i, given diagonal matrices Ri11≥0, Ri12≥0, Ri21≥0, Ri22≥0 satisfying Ri12≥I>Ri11≥0 and Ri22≥I>Ri21≥0, and any u¯(t)∈L(Ri11u¯(k)), the system (1) can rewritten as follows:(8)x¯k+1=A¯ix¯k+G¯i1fx¯k,k+B¯iRi11uk+B¯iΨi1k+E¯i1dkyk=Ri21C¯i+D¯ix¯k+G¯i2fx¯k,k+Ψi2k+E¯2dk,wherefi1(x¯)=fi1(x),fi2(x¯)=fi2(x) and (9)fix¯k,k=fi1x¯fi2x¯,dk=d2kd3k,A¯i=AiBiRi11Vi0Wi,B¯i=Bi0,G¯i1=Gi1000,E¯i1=Ei100Mi,C¯i=Ci0,D¯i=0DiVi,G¯i2=Gi20,E¯i2=Ei20.The full-order observer-based controller for the extended system (8) is designed as(10)x¯^k+1=A¯ix¯^k+B¯iRi11uk+G¯i1fix¯^k,k+Liy^k-yky^k=Ri21C¯ix¯^k+G¯i2fix¯^k,k+D¯ikx¯^kuk=Kix^k-d^1k,wherefi1(x¯^)=fi1(x^),fi2(x¯^)=f2(x^) and(11)x¯^k=x^kω^k,Li=Li1Li2,fix¯^k,k=fi1x¯^fi2x¯^.The estimation error is denoted as (12)ek=x¯k-x¯^k=exkeωk=xk-x^kωk-ω^k,and then the error dynamic can be obtained(13)ek+1=A¯i-LiRi21C¯i-LiD¯iek+S¯iΨik+E¯i1-LiE¯i2dk+G¯i1-LiG¯i2fix¯-fix¯^,where (14)S¯i=B¯i-Li,Ψik=Ψi1kΨi2k.Based on (10), (6) can be rewritten as(15)LRi11u¯k=Ri11K~ilx~≤1,l∈I1,m,where (16)x~k=xkek,K~i=Ki-KiVi.For any x~(k)∈L(Ri11u¯(k)) with a given Ri11, by (4), the closed-loop system can be described as(17)x~k+1=A~ix~k+G~if~ik+E~idk+S~iΨik,where (18)A~i=Ai+BiRi11KiBiRi11-KiVi0A¯i-LiC¯i-LiD¯i,f~ik=fi1xfix¯-fix¯^,S~i=Bi0S¯i,G~i=Gi100G¯i1-LiG¯i2,E~=Ei10E¯i1-LiE¯i2.The reference output is chosen to be(19)zk=x~k.In the following theorem, we present some sufficient conditions under which a full-order observer-based controller can be designed to stabilize the closed-loop system (17) without external disturbances d(k) and satisfy the H∞ performance requirement when the system is subject to external disturbances.Theorem 9.
For given diagonal matricesRi11≥0,Ri12≥0 satisfying Ri12≥I>Ri11≥0, diagonal matrices Ri21≥0,Ri22≥0 satisfying Ri22≥I>Ri21≥0 and γ>0, under arbitrarily switching law, the closed-loop system (17) under the full-order state observer (10) for the augmented system (8) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1} and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices P1>0, P2>0, P3>0, S1>0, Yi1, Yi2, Yi3 and parameters τi>0, i∈I[1,5] such that(20)P1S1=I.(21)1Ri11Yi1l-Ri11Yi1lRi11Vil∗P100∗∗P20∗∗∗P3>0,l∈I1,m,(22)Φi11Φi120Φi14∗Φi220Φi24∗∗-γ2IΦi34∗∗∗Φi44<0,where (23)U˘i1=Ui11TUi12+Ui12TUi112,U˘i2=-Ui11T+Ui12T2H˘i1=Hi11THi12+Hi12THi112,H˘i2=-Hi11T+Hi12T2,Θi1=-P1-τi1U˘i1+I,Θi2=-P2-τi2U˘i1-τi3H˘i1+I,Φi11=Θi100-τi1U˘i2000Θi200-τi2U˘i2-τi3H˘i200-P3+I000-τi1U˘i2T00-τi1I000-τi2U˘i2T00-τi2I00-τi3H˘i2T000-τi3I,Φi12T=Ri1Yi1-Ri1Yi1Ri1Vi000τi5Ri2Ci00000,Φi14=AiT+Yi1TRi11BiT00-Yi1TRi11BiTAiTP2-CiTYi2TCiTYi3TViTRi11BiTViTRi11BiTP2T-ViDiTYi2TWiP3+ViDiTYi3TGi1T000Gi1TP2-Gi2TYi2T-Gi2TYi3T000,Φi22=-2τi4I00-2τi5I,Φi24=BiTBiTP200-Yi2T-Yi3T,Φi34T=Ei10P2Ei1-Yi1Ei200P3Mi-Yi3Ei2,Φi44=-S1000-P2000-P3.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(24)Ki=Yi1,Li1=P2-1Yi2,Li2=P3-1Yi3.Proof.
Choose the common Lyapunov candidate as follows:(25)Vx~k=x~TkPx~k,where P=diag{P1,P2,P3} with Pj>0, j=1,2,3.
For anyx~(k)∈L(Ri11u¯(t)), along the closed-loop augmented system (17) with d(k)≡0, the difference of V(k) can be calculated as(26)∆Vx~k=x~Tk+1Px~k+1-x~TkPx~k=ηkTAi0TPAi0-Pηk,where (27)ηTk=x~Tkf~iTk,ΨiTk,A0i=A~iG~iS~i,P=diagP,0,0.From the assumption that the nonlinearities f1(k) and f2(k) belong to sectors [Ui11,Ui12] and [Hi21,Hi22], it can be verified that(28)xkfi1xTU˘i1U˘i2∗Ixkfi1x≤0,exkfi1x¯-fi1x¯^TU˘i1U˘i2∗Iexkfi2x¯-fi2x¯^≤0,exkfi2x¯-fi2x¯^TH˘i1H˘i2∗Iexkfi2x¯-fi2x¯^≤0,where U˘i1, U˘i2, H˘i1, and H˘i2 are defined in (1).
From (22), the following inequality can be obtained:(29)Φ~i11Φi12Φi14∗Φi22Φi24∗∗Φi44<0,where Φ~i11=Φi11-diag{II,I,0,0,0}.
Noting the facts ofYi1=Ki, Yi2=P2Li1, and Yi3=P3Li2, before and after multiplying (29) by diag{Iτi4-1,I,P1,I} and by the condition (20), it yields that(30)Φ~i11Φi12Φ~i14∗Φi22Φ~i24∗∗P<0,where Φ~i14=Φi14P~1 and Φ~i24=Φi24P~1 with P~1=diag{P1,I,I}.
Together (5), (28) with (30), using Lemma 8 and Schur complement, we can get(31)∆Vx~k<0,x~k∈LRi11u¯t.Using Schur complement, (21) is equivalent to(32)Ri11K~lP-1Ri11K~lT≤1,which means that(33)ΩP⊂LRi11K~ix~k.From (30) and (33), we know that the augmented closed system (17) without external disturbance d(k) is asymptotically stable with an initial set Ω(P).
Next, we suppose that system (17) is subject to external disturbances d(k)≠0.
For givenγ>0, consider the index as follows:(34)JN=∑k=1NzTkzk-γ2dTkdk.Under the zero initial condition, from (25), we have(35)JN≤∑k=1NzTkzk-γ2dTkdk+∆Vk.From the above discussion and conditions (20), (22), we can readily derive(36)JN≤0,and then, for nonzero d(k)∈L(0,∞), one has(37)limN→∞JN≤0,which means that the H∞ performance z(k)2≤γd(k)2 is satisfied.


## 3.2. Reduced-Order Observer-Based Controller

Assuming that the states of the system (1) are available, a disturbance observer can be formulated as(38)d^1k=Vσkω^kω^k=vk-Lσkxk,where v(k) is generated by the following dynamic:(39)vk+1=Wσk+LσkBσkRσk,11Vσkvk-Lσkxk+LσkAσkxt+BσkRσk,11uk+Gσk,1fσk,1xk,k.Then, the dynamic of the estimation of disturbance error eω(k) is(40)eωk+1=Wσk+LσkBσkRσk,11Veωk+LσkBσkΨσk,1k+Mσkd3k+LσkEσk,1d2k.A disturbance observer-based controller is designed as(41)uk=Kσkxk-d^1k.Combining (1), (38), and (40), for σ(k)=i∈I[1,q], the closed-loop system for the extended state x(k)=[xT(k)eωT(k)]T can be described as(42)xk+1=Aixk+Gifik+SiΨ1k+Eidk,where (43)Ai=Ai+BiRi11KiBiRi11Vi0Wi+LiBiRi11V,Si=BiLiBi,Gi=Gi10,Ei=Ei10LiEi1Mi.For this case, (6) can be rewritten as (44)LRi11u¯k=Ri11Kixl≤1,l∈I1,m,where Ki=[Ki,Vi]. The reference output is set as(45)zk=xk,Theorem 10.
Given diagonal matricesRi11≥0, Ri12≥0 satisfying Ri12≥I>Ri11≥0 and γ>0, under arbitrarily switching law, the augmented system (42) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1},P=diag{P1,P2}, and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices P1>0, P2>0, S1>0, Yi1, Yi2 and parameters τi>0, i∈I[1,5] such that(46)P1S1=I,(47)1Ri11Yi1lRi11Vil∗P10∗∗P2>00,l∈I1,m,(48)Ψi110Ψi13∗-γ2IΨi23∗∗Ψi33<0,where U˘i1 and U˘i2 were defined in (1), and(49)Ψi11=-P1-τi1U˘i1+I0-τi1U˘i2Yi1TRi10-P2+I0ViTRi1-τi1U˘i2T0-τi1I0Ri1Yi1Ri1Vi0-2τi2I,Ψi13T=Ai+BiRi11Yi1BiRi11ViGi1Bi0P2Wi+Yi2BiRi11Vi0Yi2Bi,Ψi23=Ei1TEi1TYi2T0MiTP2,Ψi33=-S100-P2.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(50)Ki=Yi1,Li=P2-1Yi2.Proof.
A Lyapunov candidate function is chosen as(51)Vxt=xTtPxt,where P=diag{P1,P2} with Pi>0, i=1,2.
For anyx(k)∈L(Ri11Kix(k)), the difference of V(k) along system (42) in the absence of disturbance d(k) is(52)∆Vk=xTk+1Pxk+1-xkTPxk=ΘTkA0iTPA0i-PΘk,where(53)Θk=xkfikΨik,A0i=AiGiSi,P=diagP,0,0.Defining Yi1≜qKi and Yi2≜qP2Li and performing a congruence transformation to (48) by diag{I,I,P1,I}, (48) is equivalent to(54)Ψi110Ψ¯i13∗-γ2IΨ¯i23∗∗Ψ¯i33<0,where (55)Ψ¯i13T=P1Ai+P1BiRi11Yi1P1BiRi11ViP1Gi1P1Bi0P2Wi+P2LiBiRi11Vi0P2LiBi,Ψ¯i23T=P1Ei10P2LiEi1P2Mi,Ψ¯ii33=-P100-P2.Equation (56) yields that(56)∆Vk-τi1fi1k-Ui11xfi1k-Ui12x-2τi2Φi1TkΦi1-Ri1kxk≤0Based on the assumption that fi1 belongs to the sector [Ui11,Ui12] and (28), by Lemma 8, we have(57)∆Vk≤0.Under the conditions of zero initial state and nonzero disturbance d(k), defining the H∞ performance index for given γ>0 as(58)JN=∑k=1NzTkzk-γ2dTkdk.By Schur complement to (56) and (47), along the same line in the proof of Theorem 9, we can represent that(59)z2≤γdk,fordk∈L0,∞,and(60)ΩP⊂LRi11Kxk.From (57), (59), and (60), the desired results are obtained and the proof is completed.Remark 11.
The composite antidisturbance control problem has been solved by designing disturbance observer-based controller for the switched system (1) subject to input saturation and multiple disturbances. The disturbance observer-based control method is used to reject the input path disturbances generated by system (2), and for the external disturbances which belong to L[0,∞), the attenuation performance are achieved by H∞ control method. The most existing results about composite antidisturbance control do not consider the switched systems subject to saturation, such as actuator saturation and sensor saturation. We solved the problem from comprehensive perspective and gave the results from both full-order and reduced-order observer-based composite control methods.Remark 12.
Although, due to (20) and (46), the conditions in Theorems 9 and 10 are not rigid constraints in the framework of linear matrix inequalities (LMIs), here, the following algorithms are proposed to this nonconvex feasibility problem.Algorithm 13.
​
Step  1 (initialization). Choose the appropriate initial positive definite diagonal matrices Ri11, Ri12, Ri21, Ri22 satisfying Ri12≥I>Ri11≥0 and Ri22≥I>Ri21≥0, scalar λ>0, and find a feasible point Mk, Nk. If there are none, exit. Set k=0.
Step  2. Solve matrices P1,S1 by the following problem: (61)MinimizeTRMkP1+NkS1subjecttoaP1IIS1≥0,binequalities21and22hold.
Step  3. Set Mk+1=S1 and Nk+1=P1. If TR{Mk+1Nk+1}=n, stop the iteration; otherwise, go back to Step 2.Algorithm 14.
​
Step  1 (initialization). Choose the appropriate initial positive definite diagonal matrices Ri11,Ri12 satisfying Ri12≥I>Ri11≥0 and scalar λ>0, and find a feasible point Mk, Nk. If there are none, exit. Set k=0.
Step  2. Solve matrices P1,S1 by the following problem: (62)MinimizeTRMkP1+NkS1subjecttoaP1IIS1≥0,binequalities47and48hold.
Step  3. Set Mk+1=S1 and Nk+1=P1. If TR{Mk+1Nk+1}=n, stop the iteration; otherwise, go back to Step 2.Remark 15.
With the help of the Algorithms13 and 14, we will propose further the following algorithms to optimize the design schemes of the controllers for estimation of the domain of initial states as large as possible.Algorithm 16.
​
Step  1. Choose the initial scalar γ>0, the appropriate matrix R>0, the initial matrices P1>0 and S1>0 which are obtained from Algorithm 13.
Step  2. Use the initialization from Step 1 to solve the following problem for α: (63)InfP2>0,P3>0,τi>0,i=1,2,3,4,5,Yij,j=1,2,3αs.t.aP-αR≤0,whereP=diagP1,P2,P3;binequalities21and22hold.Algorithm 17.
​
Step  1. Choose the initial scalar γ>0, the appropriate matrix R>0, the initial matrices P1>0 and S1>0 which are obtained from Algorithm 13.
Step  2. Use the initialization from Step 1 to solve the following problem for α: (64)InfP2>0,Yij,j=1,2αs.t.aP-αR≤0,whereP=diagP1,P2;binequalities47and48hold.Remark 18.
The results of both Theorems9 and 10 are established under the arbitrary switching rules due to the common Lyapunov function being chosen, and how to choose less conservative Lyapunov function is one of our future works. When the switching rules σ(k)≡i, that means the switched system always stays in the ith subsystem, the result can be used for each subsystem as a special and independent case.Assumption 19.
The nonlinearityfi1(x(k),k) satisfies the Lipschitz condition, namely, fi1(0,0)=0 and (65)fi1xk,k-fi1yk,k-Ui11xk-yk≤0.Corollary 20.
Given diagonal matricesRi11≥0,Ri12≥0 satisfying Ri12≥I>Ri11≥0 and γ>0, under Assumption 19 and arbitrarily switching law, the augmented system (42) is robustly asymptotically stable in the absence of the disturbance d(k) for all initial conditions Ω(P)={x~TPx~≤1}, P=diag{P1,P2}, and satisfies the H∞ performance z(k)2<γd(k)2, if there exist matrices X1>0, P2>0, Yi1, Yi2 and parameters τi>0, i∈I[1,5] such that(66)1Ri11Yi1lRi11Vil∗-X10∗∗P2<0,l∈I1,m,(67)Λi110Λi13∗-γ2IΛi23∗∗Λi33<0,where U˘i1 and U˘i2 were defined in (1), and(68)Λi11=-X10-τi1X1U˘i11Yi1TRi10-P2+I0ViTRi1-τi1U˘i11TX10-τi1I0Ri1Yi1Ri1Vi0-2τi2I,Λi13T=AiX1+BiRi11Yi1BiRi11ViGi1Bi0Wi+LiBiRi11Vi0LiBiX1000,Λi23=Ei1TEi1TLiT00MiT0,Λi33=-X1000-P2000-I.Moreover, if the above conditions are feasible, the gains of the desired observer and controller are given by(69)P1=X1-1,Ki=Yi1P1,Li=Yi2.Proof.
Setting the notation (69), performing a congruence transformation to (66) by diag{IP1,I} and to (67) by diag{P1,I,I,I,I,I,I,I,I} and using Schur complement, from the fact Ui11TUi11≥0, one has (21) and (56), under Assumption 19, and the proof follows Theorem 10.Remark 21.
The nonlinearitiesfi1 under Assumption 19 are investigated widely in the most of results of the composite antidisturbance control [36]. The result of Corollary 20 is convex due to being in the framework of linear matrix inequalities (LMIs); therefore, it is feasible.Remark 22.
In the above discussion, the nonlinearitiesfi1 and fi2 are assumed to be known. The results about the case of the unknown nonlinearities are easy to be obtained by removing the information of fi1 and fi2 in the observers (10) and (40); the details are omitted here for space saving.


## 4. Simulation Example

In this section, system (1) with two subsystems is presented to illustrate the effectiveness of the proposed control design schemes, the parameters of which are shown as(70)Subsystem1:A1=0.10.20.20.3,B1=0.20.1,G1,1=0.10.2,E1,1=0.20.1,C1T=0.10.1,D1=0.1,G1,2=0.1,E1,2=0.1,W1=0.20.10.40.5,M1=0.10.4,V1=0.20.2,(71)Subsystem2:A2=0.20.10.50.1,B2=0.60.4,G2,1=0.10.1,E2,1=0.50.1,C2T=0.10.1,D2=0.1,G2,2=0.2,E2,2=0.1,W2=0.10.20.20.1,M2=0.10.4,V2=0.10.1.Setting Ri11=0.05, Ri12=0.1 for i=1,2, by utilizing the proposed Algorithms 13 and 16, we can get a=0.5919 and(72)P1=1.40120.02300.02301.4012,P2=1.8336-0.5010-0.50101.7315,P3=1.89600.14130.14132.1901,S1=0.7139-0.0117-0.01170.7139,K11T=0.28020.2962,K21T=-1.9447-0.1290,L11=0.37910.5463,L21=1.65541.4216,L12=0.34040.5321,L22=1.16561.0207;Similarly, solving Algorithms 14 and 17 yields a=1.6385,(73)P1=1.58730.26340.26341.5873,P2=1.97650.47290.47291.9576,S1=0.6478-0.1075-0.10750.6478,K21T=-4.01060.4136,L11=-0.0007-0.0003-0.0155-0.0077,L21=10-3∗-0.4665-0.1546-0.4527-0.1840.With fi1(x(k))=sin⁡(0.1∗x1(k)), fi2(x(k))=0, d2(k)=d3(k)=e-0.1∗k, the simulation results are plotted under the switching signal described in Figure 1. For the case of unavailable states, from Figures 3 and 4, we can see that the states x(k) and the disturbance d1(k) can be estimated effectively by observers (10). With the gains in (72), Figure 2 shows that the system can be stabilized with a H∞ performance level by the composite controller (10). When the states are available, the response curves of System (1) and the disturbance errors are described in Figures 5 and 6, respectively. The stability of the closed-loop system under disturbance-observed-based controller (42) can be achieved with a H∞ performance level.Figure 1
Switching signal.Figure 2
The state trajectories of System (1) under full-order observer-based controller (10).Figure 3
Curves of the state errored(k) under full-order observer (10).Figure 4
Curves of the disturbance erroreω(k) under observer (10).Figure 5
The state trajectories of System (1) under disturbance-observer-based controller (42).Figure 6
Curves of the disturbance erroreω(k) under disturbance observer (42).


## 5. Conclusion

The problem of analysis and synthesis of a class of switched nonlinear systems with saturation and multiple disturbances has been investigated in this paper. Based on designing the full-order and reduced-order observers, a composite controller is constructed to ensure the augmented system is robustly asymptotically stable in the absence of the disturbance. Further, aH∞ control scheme has been proposed to attenuate the mismatched disturbances.



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*Source: 5653619-2019-04-16.xml*