# Observer-Based Robust Tracking Control for a Class of Switched Nonlinear Cascade Systems

Authors: Ben Niu; Deqing Zhao; Xudong Zhao; Hongyi Li; Xiangyong Chen; Xiaoxiao Dong
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is devoted to robust output feedback tracking control design for a class of switched nonlinear cascade systems. The main goal is to ensure the global input-to-state stable (ISS) property of the tracking error nonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances. First, a nonlinear observer is constructed through state transformation to reconstruct the unavailable states, where only one parameter should be determined. Then, by virtue of the nonlinear sliding mode control (SMC), a discontinuous nonlinear output feedback controller is designed using a backstepping like design procedure to ensure the ISS property. Finally, an example is provided to show the effectiveness of the proposed approach.


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## Body

## 1. Introduction

Switched systems are a special class of hybrid systems in engineering applications and have attracted much attention from many researchers [1–8]. A switched system consists of a family of distinct active subsystems subject to a certain switching rule which chooses one of them being active during a certain time. The research on switched systems is motivated by two important practical considerations: (i) many real-world systems exhibit a fundamental characteristic of switching between different system structures; (ii) multicontroller switching provides an effective mechanism to handle highly complex systems and/or systems with large uncertainties. Therefore, switched systems have been a very active area of research in the past twenty years and have motivated a large and growing body of research work on a diverse array of issues, including modeling [9, 10], optimization [11, 12], stability analysis [13–17], and H∞ control [18–20].Output feedback tracking control is a fundamentally important issue in control field and has been extensively studied over the last several decades. In the literature, several approaches have been developed to handle the output feedback control in the presence of structured or unstructured uncertainties: variable structure control approach [21], adaptive control approach [22], output dynamics controller with almost disturbance decoupling [23], and so forth. Inspired by these facts, for switched systems, output feedback tracking control is also a challenging issue for both theoretical investigation as well as practical applications [24–26]. Such a problem usually involves observer design [27], controller design [28], and switching law design [29]. However, to the best of the authors’ knowledge, the output feedback tracking control of switched nonlinear cascade systems by designing nonlinear state observer has not been investigated yet.In sliding mode control (SMC), sliding mode surface design and discontinuous reaching control law are two of the basic control issues. A common practice in SMC is to design a sliding mode surface according to the null space dynamics, which must ensure a stable sliding manifold when the system is in the sliding mode [30]. However, if there exist uncertainties in the null space nonlinear dynamics, sliding mode surface design becomes extremely difficult. Traditionally, the reaching control law is to force the system to reach and stay on the sliding mode surface. Nevertheless, this feature alone is no longer sufficient in the presence of unmatched uncertainties. Due to the effect of the unmatched uncertainties, the nonlinear dynamics may become divergent in a period shorter than the reaching time, if the input-to-state stable (ISS) property does not hold during the reaching phase. Hence, ISS property should be guaranteed either in the sliding phase or in the reaching phase.In this paper, a class of switched nonlinear cascade systems with null space dynamics and range space dynamics are addressed for the tracking control task. Assuming that the full states are not available for measurement, the main objective of the paper is to ensure the global ISS property of the tracking error nonlinear dynamics while achieving a small tracking error bound. The features of the proposed approach are the following: (i) a nonlinear observer is designed for the switched system in which only one parameter needs to be determined; (ii) the resulting sliding manifold in the sliding phase possesses the desired ISS property and to certain extent the optimality through solving a Hamilton-Jacoby inequality; (iii) associated with the sliding mode surface, SMC is applied to the second subsystem that achieves the desired tracking.Notations. We use standard notations throughout this paper. λmax(A) and λmin(A) stand for the maximum and minimum eigenvalues of a symmetric matrix A, respectively. {A}n-×n- denotes the first n rows and n columns in A, and {A}m_×n_ denotes the last m rows and n columns in A. R+ denotes the set of nonnegative real numbers, Rn denotes an n-dimension real vector space, ∥·∥ is the Euclidean norm and induced matrix norm, and L∞[0,∞) is the space of uniformly bounded functions on [0,∞). Dxf=∂f(x,y)/∂x and Dyf=∂f(x,y)/∂y are row vectors, and σ(·) denotes the largest singular value of a matrix.


## 2. System Description and Problem Statement

This paper is concerned with the following switched nonlinear cascade system described by(1)x˙1=f1,σ(t)(x1,t)+B1,σ(t)(t)x2+H1,σ(t)(x1,t)ω1(t),x˙2=f2,σ(t)(x,t)+B2,σ(t)(t)[uσ(t)+ΔB2,σ(t)(x,t)]+H2,σ(t)(x,t)ω2(t),y=x1,
where x=[x1T,x2T]T is a physically measurable state vector, x1∈Rn is the null space dynamics, x2∈Rm is the range space dynamics, and ω1∈Rh1 and ω2∈Rh2 are the external disturbance. σ:[0,∞)→M={1,2,…,m} is the right continuous piecewise constant switching signal to be designed; ui∈Rm stands for the control input of the ith subsystem, the mappings f1,i(x1,t)∈Rn,  f2,i(x,t)∈Rm,  B1,i(t)∈Rn×m,  B2,i(t)∈Rm×m,  H1,i(x1,t)∈Rn×h1, and H2,i(x,t)∈Rm×h2 are known and smooth with respect to x and continuous with respect to time t, and ΔB2,i∈Rm,  i∈M, denote the uncertainties in the control input. The relation m≤n holds for the system (1).Corresponding to the switching signalσ(t), we have the switching sequence
(2)∑={(x1T(t0),x2T(t0))T;(i0,t0),(i1,t1),…,(ik,tk),…,iiiiiiiii∣ik∈M,k=0,1,…(x1T(t0),x2T(t0))T},
which means that the ikth subsystems are active when t∈[tk,tk+1). In addition, we assume that the state of the system (1) does not jump at the switching instants; that is, the trajectory x(t) is everywhere continuous.In this paper, the following assumptions are adopted to develop the main results.Assumption 1.
There exist two positive constantsb1 and b2 such that for all x1∈Rn,  t≥0,
(3)0<b12Im≤B1,iT(t)B1,i(t)≤b22Im,i∈M,
where Im is the identity matrix. Moreover, B2,i(t),  i∈M are assumed to be invertible.Assumption 2.
The uncertaintiesω1(t),  ω2(t) and ηi(x,t),i∈M in (1) are bounded as
(4)|ω1(t)|≤l1,|ω2(t)|≤l2,|ΔB2,i(x,t)|≤lb,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∈M,
where l1,  l2, and lb are known positive constants.In this paper, the output of the system (1) is required to track a given reference model: y⇒yd=x1r; that is, the x1 subpart is required to track the desired reference model
(5)x˙1r=fr(x1r,r(t),t),
where r(t) is a smooth reference input. Define the tracking error as z1=x1-x1r. Then, the error dynamics of the x1-subpart can be transformed into
(6)z˙1=f1,i(x1,t)+B1,i(t)x2+H1,i(x1,t)ω1(t)-fr(x1r,r(t),t),i∈M.Assumption 3.
There exists a smooth functionψ(·) such that the following matching condition holds:
(7)f1,i(x1,t)-fr(x1r,r(t),t)f1,i=g1,i(z1,t)+B1,i(t)ψ(x1,x1r,r(t),t),i∈M,
where ξ˙=g1,i(ξ,t) is asymptotically stable.According to Assumption3, the error dynamics (6) and system (1) with the tracking objective (5) can be rewritten as
(8)z˙1=g1,i(z1,t)+B1,i(t)[x2+ψ(x1,x1r,r(t),t)]+H1,i(x1,t)ω1(t),x˙2=f2,i(x,t)+B2,i(t)[ui+ΔB2,i(x,t)]+H2,i(x,t)ω2(t),i∈M,y=x1.Definition 4 (input-to-state stable (ISS) [31, 32]).
Consider a nonlinear dynamical system of the form(9)x˙=f(x,u),
where x and u are the states and the inputs of (9), respectively. The system (9) is said to be locally input-to-state stable if there exist a class KL function β, a class K function γ, and constants k1,k2∈R+ such that
(10)∥x(t)∥≤β(∥x0∥,t)+γ(∥uT(·)∥L∞),≤β(∥x0∥,t)iiii+∀t≥0,0≤T≤t,
for all x0∈D and u∈Du satisfying ∥x0∥<k1 and supt>0∥uT(t)∥=∥uT(·)∥L∞<k2,  0≤T≤t. It is said to be input-to-state stable or globally ISS if D=Rn,  Du=Rm, and (10) is satisfied for any initial state and any bounded input u.Control Objective. Under Assumptions 1–3, design a nonlinear observer for the system (1). Based on the observer, design a controller uσ(t) and a switching law σ(t) such that(i)
the tracking error norm∥z1(t)∥ in (8) tends to a ball Bs in finite time, where the ball Bs is defined as
(11)Bs={z1(t):∥z1(t)∥≤s},wheres is a positive constant;(ii)
the closed-loop system (8) possesses ISS property with respect to the disturbances ωi=[ω1,ω2,ΔB2,i]T,  i∈M.


## 3. Nonlinear Observer Design

This section is devoted to the design of a nonlinear observer for the system (1). Motivated by the work in [32, 33], a nonlinear observer is constructed through a state transformation which converts the system (1) into a new form such that the observer gain can be designed in a straightforward manner.First, the system in (1) can be rewritten as the following:
(12)x˙=fi(x,t)+ki(ui,t)+Bi(t)x+Hi(x,t)ωi(x,t),y=Cx,i∈M,
where fi(x,t)=[f1,i(x1,t),f2,i(x,t)]T and
(13)ki(ui,t)=[0B2,i(t)u],Bi(t)=[0B1,i(t)00],ωi=[ω1(t)ω2(t)ΔB2,i(t)],Hi(x,t)=[H1,i(x1,t)000H2,i(x,t)B2,i(t)],CT=[In0m].Define the transformation matricesTi(t),  i=1,…,m and the matrices Ξθ,  A, and C- as
(14)Ti(t)2n×(n+m)=[In00B1,i(t)],Ξθ=[In00Inθ],A=[0In0n0],C-T=[In0n].Therefore, we obtainΞθAΞθ-1=θA,  C-TC-Ξθ=C-TC-, and
(15)ζ(t)=Ti(t)x=[x1B1,i(t)x2],Ti(t)Bi(t)=ATi(t).
Denote Ti+(t) as the left inverse of the matrix Ti(t). Then, the ζ system can be written as
(16)ζ˙=Ti(t)x˙+T˙i(t)x=Ti(t)[fi(x,t)+ki(ui,t)Tii(t)+Bi(t)x+Hi(x,t)ωi(x,t)]+T˙i(t)x=Aζ+Ti[fi(x,t)+ki(ui,t)Aζi+Ti+Hi(x,t)ωi(x,t)]+T˙ix=Aζ+Ti[fi(Ti+ζ,t)+ki(ui,t)Aζ+iTi+Hi(Ti+ζ,t)ωi(Ti+ζ,t)]+T˙i(t)Ti+ζ,y=C-ζ.Thus, the observer for the transformedζ system in (16) can be constructed as
(17)ζ^˙=Aζ^+Ti[fi(Ti+ζ^,t)+ki(ui,t)]+T˙iTi+ζ^+θΞθ-1P-1C-T(y-C-ζ^),
where P is the symmetric positive definite solution of the following algebraic Lyapunov equation:
(18)P+ATP+PA-C-TC-=0.Theorem 5.
Assume that the system in (12) satisfies Assumptions 1-2. Then, under arbitrary switchings, the estimation error of the states has the following property:
(19)∥ex(t)∥=∥x-x^∥≤qθ∥ex(0)∥+β0ε,
where qθ=π1θλmax(P)/λmin(P)e-((θ-c1)/2)tπ2,  β0=π1(c2θ/(θ-c1)λmin(P)),  ε=l12+l22+lb2, and c1<θ,  c2,  π1, and π2 are positive constants.Proof.
Defineeζ(t)=ζ(t)-ζ^(t). From (16) and (17), the estimation error dynamics of eζ(t) becomes
(20)e˙ζ=ζ˙(t)-ζ^˙(t)=Aζ+Ti[fi(Ti+ζ,t)+ki(ui,t)Aζ+iTi+Hi(Ti+ζ,t)ωi(Ti+ζ,t)]+T˙iTi+ζ-Aζ^+Ti[fi(Ti+ζ^,t)+ki(ui,t)]+T˙iTi+ζ^+θΞθ-1P-1C-T(y-C-ζ^)=θΞθ-1(A-P-1C-TC-)Ξθeζ+Ti[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]+T˙iTi+eζ+TiHi(Ti+ζ,t)ωi(Ti+ζ,t).
Consider a transformation on the error ase-ζ(t)=Ξθeζ(t). Then, we have
(21)e-˙ζ=θ(A-P-1C-TC-)e-ζ+ΞθTi[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]+ΞθT˙iTi+Ξθ-1e-ζ+ΞθTiHi(Ti+ζ,t)ωi(Ti+ζ,t).
ChoosingV1=(1/2)e-ζPe-ζ(t), where P is the solution of (18), we obtain
(22)V˙1=-θV1-θ2e-ζTC-TC-e-ζ+e-ζTPΞθTi[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]+e-ζTPΞθTiHi(Ti+ζ,t)ωi(Ti+ζ,t)+e-ζTPΞθT˙iTi+Ξθ-1e-ζ=-θV1+e-ζTPΞθTi[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]+e-ζTPΞθTiHi(Ti+ζ,t)ωi(Ti+ζ,t)+e-ζTPΞθT˙iTi+Ξθ-1e-ζ.
For anyθ>1, we can infer that ∥ΞθTi[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]∥≤bf∥e-ζ∥,  ∥ΞθTiHi(Ti+ζ,t)∥≤bh,  ∥ωi(Ti+ζ,t)∥≤ε, and ∥ΞθT˙iTi+Ξθ-1∥≤bt, where bf,  bh, and bt do not depend on θ. Then, (16) is transformed into
(23)V˙1≤-θV1+λmax(P)(bf+bt)∥e-ζ∥2+λmax(P)bhε∥e-ζ∥≤-(θ-c1)V1+c2εV1,
where c1=2(λmax(P)/λmin(P))(bf+bt) and c2=λmax(P)bh(2/λmin(P)). If θ>max{1+c1} is selected, then (23) becomes
(24)dV1dt≤-(θ-c1)2V1+c2ε2,⟹V1≤e-((θ-c1)/2)tV1(0)⟹V1≤+c2εθ-c1[1-e-((θ-c1)/2)t],⟹∥e-ζ∥≤λmax(P)λmin(P)e-((θ-c1)/2)t∥e-ζ(0)∥⟹∥e-ζ∥≤+c2ε(θ-c1)λmin(P).
Using∥e-ζ(t)∥≤∥eζ(t)∥≤θ∥e-ζ(t)∥, (24) becomes
(25)∥eζ∥≤θλmax(P)λmin(P)e-((θ-c1)/2)t∥e-ζ(0)∥+θc2ε(θ-c1)λmin(P)≤qθ1∥eζ(0)∥+β01ε,
where qθ1=θλmax(P)/λmin(P)e-((θ-c1)/2)t∥e-ζ(0)∥ and β01=θc2/(θ-c1)λmin(P). Thus, ∥eζ(t)∥=∥ζ(t)-ζ^(t)∥≤qθ1∥eζ(0)∥+β01ε. Furthermore, from Assumption 1, we have ∥Ti+(t)∥≤π1 and ∥Ti(t)∥≤π2, where π1 and π2 are constants.
Based onζ(t)=T(t)x(t) and ζ^(t)=T(t)x^(t), we get
(26)∥ex(t)∥=∥x(t)-x^(t)∥=∥Ti+eζ(t)∥≤π1qθ1π2∥ex(0)∥+π1β01χ=qθ∥ex(0)∥+β0χ
with qθ=π1qθ1π2,  β0=π1β01. This completes the proof.From (16) and ζ^˙(t)=Ti(t)x^˙(t)+T˙i(t)x^, the observer to the original coordinate is
(27)x^˙(t)=Ti+(t)[ζ^˙(t)-T˙i(t)x^]=Ti+(t){Aζ^+Ti[fi(Ti+ζ^,t)+ki(ui,t)]Ti+ii(t)+T˙iTi+ζ^+θΞθ-1P-1C-T(y-C-ζ^)Ti+ii(t)-T˙i(t)x^[fi(Ti+ζ^,t)+ki(ui,t)]}=Ti+(t)ATix^+fi(x^,t)+ki(ui,t)+θTi+(t)Ξθ-1P-1C-T(y-C-x^)=Bi(t)x^+fi(x^,t)+ki(ui,t)+θTi+(t)Ξθ-1P-1C-T(y-C-x^),(28)y=x1.Hence, the estimation error dynamics in thex-coordinate with ex(t)=x(t)-x^(t) becomes
(29)e˙x=Bi(t)ex+fi(x,t)-fi(x^,t)+Hi(x,t)ωi(x,t)-θTi+(t)Ξθ-1P-1C-TC-ex.


## 4. Controller Design and Stability Analysis

Before the controller design, we would like to rewrite the observer dynamics in (27) as
(30)x^˙1=f1,i(x^1,t)+B1,i(t)x^2+Φn,i,x^˙2=f2,i(x^,t)+B2,i(t)ui+Φm,i,
where
(31)Φn,i={θTi+(t)Ξθ-1P-1C-T}n-×n-(y-Cx^)=Fn,i(y-Cx^)=Fn,iC-eζ=Fn,iC-Ξθ-1e-ζ=F-n,ie-ζ,Φm,i={θTi+(t)Ξθ-1P-1C-T}m×n(y-Cx^)=Fm,i(y-Cx^)=Fm,iC-eζ=Fm,iC-Ξθ-1e-ζ=F-m,ie-ζ.Definez^1=x^1-x1r. In terms of the observer dynamics (22) and the desired trajectory (4), we have the following error dynamics:
(32)z^˙1=g1,i(z^1,t)+B1,i(t)[x^2+ψ(x^1,x1r,r(t),t)]+Φn,i,(33)x^˙2=f2,i(x^,t)+B2,i(t)ui+Φm,i.In what follows, we first choose a sliding mode surface for the error dynamics of the null space dynamicsz^1. Second, we design a controller for the augmented system in (21) and (32) such that ISS property is achieved.Theorem 6.
If there exist positive definite, radially unbounded, and smooth functionsV2,i(z^1,t) and functions βij(z^1,t)≤0,  i,j=1,…,m such that
(34)DtV2,i+(Dz^1V2,i)g1iii+14γ12(Dz^1V2,i)F-n,iF-n,iT(Dz^1V2,i)Tiii+z^1Tz^1+∑j=1m‍βij(V2,j-V2,i)≤0,
then, under the nonlinear sliding mode
(35)S=x^2+ψ=0
and the switching law
(36)σ(t)=min{i∣i=maxi∈M{V1,i(e1,t)}},
the tracking error norm ∥z1(t)∥ tends to a ball Bs in finite time, where the ball Bs is defined as
(37)Bs={z1(t):∥z1(t)∥≤γ32ε2=s},
where γ3 and ε are positive constants.Proof.
First, we now define the following piecewise Lyapunov function candidate:(38)V-2(e-ζ,z^1,t)=V1(e-ζ)+V2,σ(z^1,t)=12e-ζPe-ζ(t)+V2,σ(z^1,t),
where V2,σ(z^1,t) is switched among the solution V2,i(z^1,t)’s of (34) in accordance with the piecewise constant switching signal σ.
Using the sliding mode surface constructed in (35) and under the switching law (36), the derivative of V2,i(z^1,t) is
(39)V˙2,i=DtV2,i+Dz^1V2,i×{g1,i+B1,i(t)iii×[x^2+ψ(x^1,x1r,r(t),t)]+Φn,i}=DtV2,i+Dz^1V2,ig1,i+Dz^1V2,iF-n,ie-ζ=DtVi+De1Vig1,i+γ12e-ζTe-ζ+14γ12(Dz^1V2,i)F-n,iF-n,iT(Dz^1V2,i)T-∥12γ1F-n,iT(Dz^1V2,i)T-γ1e-ζ∥2≤DtVi+De1Vig1,i+γ12ω12ω1+14γ12(Dz^1V2,i)F-n,iF-n,iT(Dz^1V2,i)T.
If there exist solutions ofV2,i(·) such that the inequality in (34) is satisfied, (39) becomes
(40)V˙i≤-z^1Tz^1+γ12e-ζTe-ζ.
From (22) and (40), we have
(41)V-˙2(e-ζ,z^1,t)iii=V˙1(e-ζ)+V˙2,i(z^1,t)iii≤-θe-ζTPe-ζiiiiii+e-ζTPΞθTi[fi(Ti+ζ,t)-fi(Ti+ζ^,t)]iiiiii+e-ζTPΞθTiHi(Ti+ζ,t)ωi(Ti+ζ,t)iiiiii+e-ζTPΞθT˙iTi+Ξθ-1e-ζ-z^1Tz^1+γ12e-ζTe-ζiii≤-θλmin(P)∥e-ζ∥2iiiiii+λmax(P)(bf+bt)∥e-ζ∥2iiiiii+e-ζTPΞθTiHi(Ti+ζ,t)ωi(Ti+ζ,t)iiiiii-z^1Tz^1+γ12e-ζTe-ζiii≤-[14γ22θλmin(P)∥e-ζ∥2iiiiiiiiiiii-γ12-λmax(P)(bf+bt)iiiiiiiiiiii-14γ22λmax(P)2bh2]∥e-ζ∥2iiiiii-z^1Tz^1+γ22∥ωi(Ti+ζ,t)∥2.
Whenθ is selected to be
(42)θλmin(P)∥e-ζ∥2-γ12-λmax(P)(bf+bt)iii-14γ22λmax(P)2bh2≥0,
then the inequality (41) becomes
(43)V-˙2(e-ζ,z^1,t)≤-z^1Tz^1+γ22∥ωi(Ti+ζ,t)∥2.
Usingex,1(t)=x1(t)-x^1(t)=z1(t)-z^1(t), ∥z^1(t)∥2≥∥z1(t)∥2-∥ex,1(t)∥2, and ex,1(t)=e-ζ,1(t) and according to ex(t)=Ti+Ξθ-1e-(t), (43) becomes
(44)V-˙2(e-ζ,z^1,t)iiii≤-z1Tz1+∥ex,1(t)∥2+γ22∥ωi(Ti+ζ,t)∥2iiii=-z1Tz1+∥eζ,1(t)∥2+γ22∥ωi(Ti+ζ,t)∥2iiii≤-z1Tz1+∥eζ(t)∥2+γ22∥ωi(Ti+ζ,t)∥2.
From (17), e-ζ is bounded as
(45)∥e-ζ∥≤λmax(P)bh[λmin(P)θ-λmax(P)(bf+bt)]∥ωi(Ti+ζ,t)∥,
where λmin(P)θ-λmax(P)(bf+bt)>0 according to (42). Hence, (44) becomes
(46)V-˙2(e-ζ,z^1,t)≤-z1Tz1+γ32∥ωi(Ti+ζ,t)∥2≤-z1Tz1+γ32ε2,
where γ3=γ22+χ2 and χ=bh/((λmin(P)/λmax(P))-λmax(P)(bf+bt)). Note that limθ→∞χ=0. Equation (47) shows that the tracking error norm ∥z1(t)∥ in (8) tends to a ball in finite time, which is defined by
(47)Bs={z1(t):∥z1(t)∥≤γ32ε2=s},
where Bs={z1(t):∥z1(t)∥≤γ32ε2=s}, ε=l12+l22+lη2.Remark 7.
In the nonlinear uncertain system (32), if g1,i(z^1,t) can be expressed as W1,i(z^1,t)z^1, when W1,i(z^1,t) is a matrix-valued smooth function, then the HJI inequality (34) can be simplified into the following differential Riccati inequality:
(48)12E˙i+12(EiW1,i+W1,iTEi)+14γ12EiF-n,iF-n,iTEiiiiii+In×n+∑j=1m‍βij(Ej-Ei)≤0,
where Ei(z^1,t)z^1,  i∈M, are symmetric positive definite smooth matrices.Remark 8.
In the observer design, the parameterθ is the only key parameter to be determined. It should be designed such that the two conditions are satisfied in (42) and θ=max{1,c1} simultaneously.Remark 9.
Since the estimation error of the states in Theorem5 has the property (19), the tracking error simply converges to a ball showed in (37).We are now in a position to design the controller to ensure the ISS stability.Theorem 10.
With the sliding mode surface (35), the switching law (36), and the following sliding mode controller
(49)ui=u1,i+u2,i,(50)u1,i=-B2,i-1[DtS+(Dx1rS)x˙1r-iiB2,i-1+L(f1,i+B1,ix^2+Φn,i)+f2,i+Φm,i(Dx1rS)],(51)u2,i=-k0B2,iTS∥B2,iTS∥,
where L(x^1,x1r,t)=Dx^1S∈Rm×n and k0 is a positive constant, the system (8) is globally ISS stable with respect to the external disturbance inputs, and the tracking error norm ∥z1∥ is bounded in Bs as in Theorem 6.Proof.
DefineV3=(1/2)STS. Choose the following piecewise Lyapunov function candidate:
(52)V-3(e-ζ,z^1,x^,x1r,t)=V1(e-ζ)+V2,σ(z^1,t)+12STS=12e-ζPe-ζ(t)+V2,σ(z^1,t)12STS,
where V2,σ(z^1,t) is switched among the solution V2,i(z^1,t)’s of (34) in accordance with the piecewise constant switching signal σ.
Then, we have(53)V˙3=ST[DtS+(Dx1rS)x˙1r+L(f1,i+B1,ix^2+Φn,i)STii+f2,i+B2,iui+Φm,i(Dx1rS)]≤-k0∥B2,iTS∥.
From (44) and (54), we have
(54)V-˙3=V-˙2+V˙≤-z1Tz1+γ32∥ωi(Ti+ζ,t)∥2-k0∥B2,iTS∥≤-z1Tz1+γ32ε2,
which implies that the system (8) is globally ISS with respect to the external disturbance input, and the tracking error norm ∥z1∥ is bounded in Bs in finite time.


## 5. Illustrative Example

In this section, we present a simulation example to illustrate the applicability and effectiveness of the proposed approach.Example 1.
Consider a switched nonlinear cascade system as in (1), where
(55)f1,1=f1,2=W1x1=[01-1.6-2.1][x11x12],B1,1(t)=[1+0.8sin(t)001],B1,2(t)=[1+0.8cos(t)001],H1,1=[sin(x11)cos(x11)cos(x12)sin(x12)],H1,2=[cos(x12)sin(x12)sin(x11)cos(x11)],ω1=[e-0.3t,e-0.1t]T,ω2=[e-0.2t,e-0.5t]T,f2,1=2x112x12cos(x2),f2,1=x12sin(x12x2),η1=0.6sin(x11)+0.6sin(x12),η2=-0.6cos(x12)-0.6cos(x2).
The nonlinear observer is designed as in (27). Based on (18), we have the symmetric positive definite solution
(56)P=[10-1010-102].
Then, θ=86 is selected based on Remark 8 with bf=4.8416,  bt=1,  bh=1.128, and γ2=1.
The target trajectory isx11r=0.3sin(πt) and x12r=x˙11r=0.3πcos(πt). From (8), the error dynamics of the x1-subpart can be expressed as
(57)z˙1=W1e1+B1.i[x2+ψ]+H1,iω1,i=1,2,
where ψ(t)=-x˙12r-1.6x11r-2.1x12r.
Letγ1=1. In z^1-subpart, according to Remark 7, we first choose Vi(z^1,t)=(1/2)z^1TEiz^1,  i=1,2, where Ei are determined by the differential Riccati inequality (48). When E˙i=0 and β12=β21=-1, from the linear algebraic matrix inequality
(58)12(E1W1+W1TE1)+14γ12E1H1,1H1,1TE1iiii+I+(E2-E1)≤0,12(E2W1+W1TE2)+14γ12E2H1,2H1,2TE2iiii+I+(E1-E2)≤0,
and using the singular values of the matrices H1,1 and H1,2, we can get two symmetric positive definite smooth matrices
(59)E1=[0.1147990.0003300.0003300.463736],E2=[0.039253-1.025477-1.0254770.070260].
Therefore, the switching surface isS=x^2+ψ=x^2-x˙12r-1.6x11r-2.1x12r. Moreover, the switching law is chosen as
(60)σ(t)={1ifz^1TE1z^1>z^1TE2z1,2ifz^1TE2z1>z^1TE1z1,
according to (36) in Theorem 6, and the controller is constructed according to (49) in Theorem 10.
Let the initial states be(-0.6,1.9,-0.56)T. Figures 1 and 2 show the responses of the states x11 and x12, respectively. The tracking errors z11 and z12 are shown in Figures 3 and 4, respectively, which demonstrate the tracking errors of the states x11 and x12 that are bounded with fast convergence. All the figures indicate the feasibility of our results.Figure 1
The response of the statex11.Figure 2
The response of the statex12.Figure 3
The evolution of the tracking errorz11.Figure 4
The evolution of the tracking errorz12.


## 6. Conclusions

In this paper, we have investigated the tracking control problem for a class of switched nonlinear cascade systems with unknown system uncertainties and external disturbances. A new robust output feedback control approach based on a nonlinear observer is proposed for the switched system. Through solving a Hamilton-Jacoby inequality, the nonlinear control law for the first subsystem specifies a nonlinear sliding mode surface. By virtue of nonlinear control for the first subsystem, the resulting sliding manifold in the sliding phase possesses the desired ISS property. Furthermore, sufficient conditions for the solvability of the tracking control problem of the switched systems and design of both switching law and output feedback controller are presented.



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*Source: 136863-2013-09-19.xml*