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

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.

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. 𝜆_max(𝐴) and 𝜆_min(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, 𝑅^𝑛 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,∞). 𝐷_𝑥𝑓=𝜕𝑓(𝑥,𝑦)/𝜕𝑥 and 𝐷_𝑦𝑓=𝜕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 𝑥̇₁=𝑓_1,𝜎(𝑡)𝑥₁,𝑡+𝐵_1,𝜎(𝑡)(𝑡)𝑥₂+𝐻_1,𝜎(𝑡)𝑥₁,𝑡𝜔₁(𝑡),𝑥̇₂=𝑓_2,𝜎(𝑡)(𝑥,𝑡)+𝐵_2,𝜎(𝑡)(𝑡)𝑢_𝜎(𝑡)+Δ𝐵_2,𝜎(𝑡)(𝑥,𝑡)+𝐻_2,𝜎(𝑡)(𝑥,𝑡)𝜔₂(𝑡),𝑦=𝑥₁, (1) where 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂]^𝑇 is a physically measurable state vector, 𝑥₁∈𝑅^𝑛 is the null space dynamics, 𝑥₂∈𝑅^𝑚 is the range space dynamics, and 𝜔₁∈𝑅^ℎ1 and 𝜔₂∈𝑅^ℎ2 are the external disturbance. 𝜎∶[0,∞)→𝑀={1,2,…,𝑚} is the right continuous piecewise constant switching signal to be designed; 𝑢_𝑖∈𝑅^𝑚 stands for the control input of the 𝑖th subsystem, the mappings 𝑓_1,𝑖(𝑥₁,𝑡)∈𝑅^𝑛,  𝑓_2,𝑖(𝑥,𝑡)∈𝑅^𝑚,  𝐵_1,𝑖(𝑡)∈𝑅^𝑛×𝑚,  𝐵_2,𝑖(𝑡)∈𝑅^𝑚×𝑚,  𝐻_1,𝑖(𝑥₁,𝑡)∈𝑅^𝑛×ℎ1, and 𝐻_2,𝑖(𝑥,𝑡)∈𝑅^𝑚×ℎ2 are known and smooth with respect to 𝑥 and continuous with respect to time 𝑡, and Δ𝐵_2,𝑖∈𝑅^𝑚,  𝑖∈𝑀, denote the uncertainties in the control input. The relation 𝑚≤𝑛 holds for the system (1).

Corresponding to the switching signal 𝜎(𝑡), we have the switching sequence =𝑥^𝑇₁𝑡₀,𝑥^𝑇₂𝑡₀^𝑇;𝑖₀,𝑡₀,𝑖₁,𝑡₁,…,𝑖_𝑘,𝑡_𝑘,…,∣𝑖_𝑘∈𝑀,𝑘=0,1,…, (2) which means that the 𝑖_𝑘th subsystems are active when 𝑡∈[𝑡_𝑘,𝑡_𝑘+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 constants 𝑏₁ and 𝑏₂ such that for all 𝑥₁∈𝑅^𝑛,  𝑡≥0, 0<𝑏²₁𝐼_𝑚≤𝐵^𝑇_1,𝑖(𝑡)𝐵_1,𝑖(𝑡)≤𝑏²₂𝐼_𝑚,𝑖∈𝑀, (3) where 𝐼_𝑚 is the identity matrix. Moreover, 𝐵_2,𝑖(𝑡),  𝑖∈𝑀 are assumed to be invertible.

Assumption 2. The uncertainties 𝜔₁(𝑡),  𝜔₂(𝑡) and 𝜂_𝑖(𝑥,𝑡),𝑖∈𝑀 in (1) are bounded as 𝜔₁(𝑡)≤𝑙₁,𝜔₂(𝑡)≤𝑙₂,Δ𝐵_2,𝑖(𝑥,𝑡)≤𝑙_𝑏,𝑖∈𝑀, (4) where 𝑙₁,  𝑙₂, and 𝑙_𝑏 are known positive constants.

In this paper, the output of the system (1) is required to track a given reference model: 𝑦⇒𝑦_𝑑=𝑥_1𝑟; that is, the 𝑥₁ subpart is required to track the desired reference model 𝑥̇_1𝑟=𝑓_𝑟𝑥_1𝑟,𝑟(𝑡),𝑡, (5) where 𝑟(𝑡) is a smooth reference input. Define the tracking error as 𝑧₁=𝑥₁−𝑥_1𝑟. Then, the error dynamics of the 𝑥₁-subpart can be transformed into 𝑧̇₁=𝑓_1,𝑖𝑥₁,𝑡+𝐵_1,𝑖(𝑡)𝑥₂+𝐻_1,𝑖𝑥₁,𝑡𝜔₁(𝑡)−𝑓_𝑟𝑥_1𝑟,𝑟(𝑡),𝑡,𝑖∈𝑀. (6)

Assumption 3. There exists a smooth function 𝜓(⋅) such that the following matching condition holds: 𝑓_1,𝑖𝑥₁,𝑡−𝑓_𝑟𝑥_1𝑟,𝑟(𝑡),𝑡=𝑔_1,𝑖𝑧₁,𝑡+𝐵_1,𝑖(𝑡)𝜓𝑥₁,𝑥_1𝑟,𝑟(𝑡),𝑡,𝑖∈𝑀, (7) where 𝜉̇=𝑔_1,𝑖(𝜉,𝑡) is asymptotically stable.

According to Assumption 3, the error dynamics (6) and system (1) with the tracking objective (5) can be rewritten as 𝑧̇₁=𝑔_1,𝑖𝑧₁,𝑡+𝐵_1,𝑖(𝑡)𝑥₂+𝜓𝑥₁,𝑥_1𝑟,𝑟(𝑡),𝑡+𝐻_1,𝑖𝑥₁,𝑡𝜔₁(𝑡),𝑥̇₂=𝑓_2,𝑖(𝑥,𝑡)+𝐵_2,𝑖(𝑡)𝑢_𝑖+Δ𝐵_2,𝑖(𝑥,𝑡)+𝐻_2,𝑖(𝑥,𝑡)𝜔₂(𝑡),𝑖∈𝑀,𝑦=𝑥₁. (8)

Definition 4 (input-to-state stable (ISS) [31, 32]). Consider a nonlinear dynamical system of the form 𝑥̇=𝑓(𝑥,𝑢), (9) 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 ‖𝑥(𝑡)‖≤𝛽𝑥₀,𝑡+𝛾𝑢_𝑇(⋅)_𝐿∞,∀𝑡≥0,0≤𝑇≤𝑡, (10) for all 𝑥₀∈𝐷 and 𝑢∈𝐷_𝑢 satisfying ‖𝑥₀‖<𝑘₁ and sup_𝑡>0‖𝑢_𝑇(𝑡)‖=‖𝑢_𝑇(⋅)‖_𝐿∞<𝑘₂,  0≤𝑇≤𝑡. It is said to be input-to-state stable or globally ISS if 𝐷=𝑅^𝑛,  𝐷_𝑢=𝑅^𝑚, 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 𝑢_𝜎(𝑡) and a switching law 𝜎(𝑡) such that(i) the tracking error norm ‖𝑧₁(𝑡)‖ in (8) tends to a ball 𝐵_𝑠 in finite time, where the ball 𝐵_𝑠 is defined as 𝐵_𝑠=𝑧₁(𝑡)∶𝑧₁(𝑡)≤𝑠, (11)  where 𝑠 is a positive constant;(ii) the closed-loop system (8) possesses ISS property with respect to the disturbances 𝜔_𝑖=[𝜔₁,𝜔₂,Δ𝐵_2,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: 𝑥̇=𝑓_𝑖(𝑥,𝑡)+𝑘_𝑖𝑢_𝑖,𝑡+𝐵_𝑖(𝑡)𝑥+𝐻_𝑖(𝑥,𝑡)𝜔_𝑖(𝑥,𝑡),𝑦=𝐶𝑥,𝑖∈𝑀, (12) where 𝑓_𝑖(𝑥,𝑡)=[𝑓_1,𝑖(𝑥₁,𝑡),𝑓_2,𝑖(𝑥,𝑡)]^𝑇 and 𝑘_𝑖𝑢_𝑖=0𝐵,𝑡_2,𝑖(𝑡)𝑢,𝐵_𝑖(𝑡)=0𝐵_1,𝑖(𝑡),𝜔00_𝑖=𝜔₁(𝑡)𝜔₂(𝑡)Δ𝐵_2,𝑖(𝑡),𝐻_𝑖()=𝐻𝑥,𝑡^1,𝑖𝑥₁,𝑡000𝐻_2,𝑖(𝑥,𝑡)𝐵_2,𝑖(𝑡),𝐶^𝑇=𝐼_𝑛0_𝑚. (13)

Define the transformation matrices 𝑇_𝑖(𝑡),  𝑖=1,…,𝑚 and the matrices Ξ_𝜃,  𝐴, and 𝐶 as 𝑇_𝑖(𝑡)_{2𝑛×(𝑛+𝑚)}=𝐼_𝑛00𝐵_1,𝑖(𝑡),Ξ_𝜃=𝐼_𝑛00𝐼_𝑛𝜃,𝐴=0𝐼_𝑛0_𝑛0,𝐶^𝑇=𝐼_𝑛0_𝑛. (14)

Therefore, we obtain Ξ_𝜃𝐴Ξ_𝜃⁻¹=𝜃𝐴,  𝐶^𝑇𝐶Ξ_𝜃=𝐶^𝑇𝐶, and 𝜁(𝑡)=𝑇_𝑖(𝑡)𝑥𝑥=¹𝐵_1,𝑖(𝑡)𝑥₂,𝑇_𝑖(𝑡)𝐵_𝑖(𝑡)=𝐴𝑇_𝑖(𝑡). (15) Denote 𝑇⁺_𝑖(𝑡) as the left inverse of the matrix 𝑇_𝑖(𝑡). Then, the 𝜁 system can be written as 𝜁̇=𝑇_𝑖(𝑡)𝑥̇+𝑇̇_𝑖(𝑡)𝑥=𝑇_𝑖(𝑡)𝑓_𝑖(𝑥,𝑡)+𝑘_𝑖𝑢_𝑖,𝑡+𝐵_𝑖(𝑡)𝑥+𝐻_𝑖(𝑥,𝑡)𝜔_𝑖(𝑥,𝑡)+𝑇̇_𝑖(𝑡)𝑥=𝐴𝜁+𝑇_𝑖𝑓_𝑖(𝑥,𝑡)+𝑘_𝑖𝑢_𝑖,𝑡+𝐻_𝑖(𝑥,𝑡)𝜔_𝑖(𝑥,𝑡)+𝑇̇_𝑖𝑥=𝐴𝜁+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡+𝑘_𝑖𝑢_𝑖,𝑡+𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡+𝑇̇_𝑖(𝑡)𝑇⁺_𝑖𝜁,𝑦=𝐶𝜁. (16)

Thus, the observer for the transformed 𝜁 system in (16) can be constructed as ̇̂̂𝜁=𝐴𝜁+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁̂,𝑡+𝑘_𝑖𝑢_𝑖+𝑇̇,𝑡_𝑖𝑇⁺_𝑖𝜁̂+𝜃Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝑦−𝐶𝜁̂, (17) where 𝑃 is the symmetric positive definite solution of the following algebraic Lyapunov equation: 𝑃+𝐴^𝑇𝑃+𝑃𝐴−𝐶^𝑇𝐶=0. (18)

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: 𝑒_𝑥(𝑡)=‖𝑥−𝑥‖≤𝑞_𝜃𝑒_𝑥(0)+𝛽₀𝜀, (19) where 𝑞_𝜃=𝜋₁𝜃𝜆_max(𝑃)/𝜆_min(𝑃)𝑒^{−((𝜃−𝑐1)/2)𝑡}𝜋₂,  𝛽₀=𝜋₁(𝑐₂𝜃/(𝜃−𝑐₁)𝜆_min(𝑃)),  𝜀=𝑙²₁+𝑙²₂+𝑙²_𝑏, and 𝑐₁<𝜃,  𝑐₂,  𝜋₁, and 𝜋₂ are positive constants.

Proof. Define 𝑒_𝜁(𝑡)=𝜁(𝑡)−𝜁̂(𝑡). From (16) and (17), the estimation error dynamics of 𝑒_𝜁(𝑡) becomes 𝑒_𝜁̇=𝜁̇(𝑡)−̇̂𝜁(𝑡)=𝐴𝜁+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡+𝑘_𝑖𝑢_𝑖,𝑡+𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡+𝑇̇_𝑖𝑇⁺_𝑖𝜁−𝐴𝜁̂+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁̂,𝑡+𝑘_𝑖𝑢_𝑖+𝑇̇,𝑡_𝑖𝑇⁺_𝑖𝜁̂+𝜃Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝑦−𝐶𝜁̂=𝜃Ξ_𝜃⁻¹𝐴−𝑃⁻¹𝐶^𝑇𝐶Ξ_𝜃𝑒_𝜁+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡−𝑓_𝑖𝑇⁺_𝑖𝜁̂,𝑡+𝑇̇_𝑖𝑇⁺_𝑖𝑒_𝜁+𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡. (20)
Consider a transformation on the error as 𝑒_𝜁(𝑡)=Ξ_𝜃𝑒_𝜁(𝑡). Then, we have 𝑒_𝜁̇‌=𝜃𝐴−𝑃⁻¹𝐶^𝑇𝐶𝑒_𝜁+Ξ_𝜃𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡−𝑓_𝑖𝑇⁺_𝑖𝜁̂,𝑡+Ξ_𝜃𝑇̇_𝑖𝑇⁺_𝑖Ξ_𝜃⁻¹𝑒_𝜁+Ξ_𝜃𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡. (21)
Choosing 𝑉₁=(1/2)𝑒_𝜁𝑃𝑒_𝜁(𝑡), where 𝑃 is the solution of (18), we obtain 𝑉̇₁=−𝜃𝑉₁−𝜃2𝑒^𝑇_𝜁𝐶^𝑇𝐶𝑒_𝜁+𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡−𝑓_𝑖𝑇⁺_𝑖̂+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇̇_𝑖𝑇⁺_𝑖Ξ_𝜃⁻¹𝑒_𝜁=−𝜃𝑉₁+𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡−𝑓_𝑖𝑇⁺_𝑖̂+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇̇_𝑖𝑇⁺_𝑖Ξ_𝜃⁻¹𝑒_𝜁. (22)
For any 𝜃>1, we can infer that ‖Ξ_𝜃𝑇_𝑖[𝑓_𝑖(𝑇⁺_𝑖𝜁,𝑡)−𝑓_𝑖(𝑇⁺_𝑖𝜁̂,𝑡)]‖≤𝑏_𝑓‖𝑒_𝜁‖,  ‖Ξ_𝜃𝑇_𝑖𝐻_𝑖(𝑇⁺_𝑖𝜁,𝑡)‖≤𝑏_ℎ,  ‖𝜔_𝑖(𝑇⁺_𝑖𝜁,𝑡)‖≤𝜀, and ‖Ξ_𝜃𝑇̇_𝑖𝑇⁺_𝑖Ξ_𝜃⁻¹‖≤𝑏_𝑡, where 𝑏_𝑓,  𝑏_ℎ, and 𝑏_𝑡 do not depend on 𝜃. Then, (16) is transformed into 𝑉̇₁≤−𝜃𝑉₁+𝜆_max(𝑃)𝑏_𝑓+𝑏_𝑡𝑒_𝜁²+𝜆_max(𝑃)𝑏_ℎ𝜀𝑒_𝜁≤−𝜃−𝑐₁𝑉₁+𝑐₂𝜀𝑉₁, (23) where 𝑐₁=2(𝜆_max(𝑃)/𝜆_min(𝑃))(𝑏_𝑓+𝑏_𝑡) and 𝑐₂=𝜆_max(𝑃)𝑏_ℎ(√2/𝜆_min(𝑃)). If 𝜃>max{1+𝑐₁} is selected, then (23) becomes 𝑑𝑉₁𝑑𝑡≤−𝜃−𝑐₁2𝑉₁+𝑐₂𝜀2,⟹𝑉₁≤𝑒^{−((𝜃−𝑐1)/2)𝑡}𝑉₁(0)+𝑐₂𝜀𝜃−𝑐₁1−𝑒^{−((𝜃−𝑐1)/2)𝑡},⟹𝑒_𝜁≤𝜆_max(𝑃)𝜆_min(𝑃)𝑒^{−((𝜃−𝑐1)/2)𝑡}𝑒_𝜁(0+𝑐₂)𝜀𝜃−𝑐₁𝜆_min(𝑃). (24)
Using ‖𝑒_𝜁(𝑡)‖≤‖𝑒_𝜁(𝑡)‖≤𝜃‖𝑒_𝜁(𝑡)‖, (24) becomes 𝑒_𝜁≤𝜃𝜆_max(𝑃)𝜆_min(𝑃)𝑒^{−((𝜃−𝑐1)/2)𝑡}𝑒_𝜁(0)+𝜃𝑐₂𝜀𝜃−𝑐₁𝜆_min(𝑃)≤𝑞¹_𝜃𝑒_𝜁(0)+𝛽¹₀𝜀, (25) where 𝑞¹_𝜃=𝜃𝜆_max(𝑃)/𝜆_min(𝑃)𝑒^{−((𝜃−𝑐1)/2)𝑡}‖𝑒_𝜁(0)‖ and 𝛽¹₀=𝜃𝑐₂/(𝜃−𝑐₁)𝜆_min(𝑃). Thus, ‖𝑒_𝜁(𝑡)‖=‖𝜁(𝑡)−𝜁̂(𝑡)‖≤𝑞¹_𝜃‖𝑒_𝜁(0)‖+𝛽¹₀𝜀. Furthermore, from Assumption 1, we have ‖𝑇⁺_𝑖(𝑡)‖≤𝜋₁ and ‖𝑇_𝑖(𝑡)‖≤𝜋₂, where 𝜋₁ and 𝜋₂ are constants.
Based on 𝜁(𝑡)=𝑇(𝑡)𝑥(𝑡) and 𝜁̂(𝑡)=𝑇(𝑡)𝑥(𝑡), we get 𝑒_𝑥(𝑡)=‖𝑥(𝑡)−𝑥(𝑡)‖=𝑇⁺_𝑖𝑒_𝜁(𝑡)≤𝜋₁𝑞¹_𝜃𝜋₂𝑒_𝑥(0)+𝜋₁𝛽¹₀𝜒=𝑞_𝜃𝑒_𝑥(0)+𝛽₀𝜒 (26) with 𝑞_𝜃=𝜋₁𝑞¹_𝜃𝜋₂,  𝛽₀=𝜋₁𝛽¹₀. This completes the proof.

From (16) and ̇̂𝜁(𝑡)=𝑇_𝑖(𝑡)𝑥̇(𝑡)+𝑇̇_𝑖(𝑡)𝑥, the observer to the original coordinate is 𝑥̇(𝑡)=𝑇⁺_𝑖(𝑡)𝜁̇̂(𝑡)−𝑇̇_𝑖(𝑡)𝑥=𝑇⁺_𝑖(𝑡)𝐴𝜁̂+𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁̂,𝑡+𝑘_𝑖𝑢_𝑖+𝑇̇,𝑡_𝑖𝑇⁺_𝑖𝜁̂+𝜃Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝑦−𝐶𝜁̂−𝑇̇_𝑖(𝑡)𝑥=𝑇⁺_𝑖(𝑡)𝐴𝑇_𝑖𝑥+𝑓_𝑖(𝑥,𝑡)+𝑘_𝑖𝑢_𝑖,𝑡+𝜃𝑇⁺_𝑖(𝑡)Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝑦−𝐶𝑥=𝐵_𝑖(𝑡)𝑥+𝑓_𝑖(𝑥,𝑡)+𝑘_𝑖𝑢_𝑖,𝑡+𝜃𝑇⁺_𝑖(𝑡)Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝑦−𝐶𝑥,𝑦=𝑥₁. (27)(28)

Hence, the estimation error dynamics in the 𝑥-coordinate with 𝑒_𝑥(𝑡)=𝑥(𝑡)−𝑥(𝑡) becomes 𝑒_𝑥̇=𝐵_𝑖(𝑡)𝑒_𝑥+𝑓_𝑖(𝑥,𝑡)−𝑓_𝑖(𝑥,𝑡)+𝐻_𝑖(𝑥,𝑡)𝜔_𝑖(𝑥,𝑡)−𝜃𝑇⁺_𝑖(𝑡)Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇𝐶𝑒_𝑥. (29)

4. Controller Design and Stability Analysis

Before the controller design, we would like to rewrite the observer dynamics in (27) as 𝑥̇₁=𝑓_1,𝑖𝑥₁,𝑡+𝐵_1,𝑖(𝑡)𝑥₂+Φ_𝑛,𝑖,𝑥̇₂=𝑓_2,𝑖(𝑥,𝑡)+𝐵_2,𝑖(𝑡)𝑢_𝑖+Φ_𝑚,𝑖, (30) where Φ_𝑛,𝑖=𝜃𝑇⁺_𝑖(𝑡)Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇_𝑛×𝑛𝑦−𝐶𝑥=𝐹_𝑛,𝑖𝑦−𝐶𝑥=𝐹_𝑛,𝑖𝐶𝑒_𝜁=𝐹_𝑛,𝑖𝐶Ξ_𝜃⁻¹𝑒_𝜁=𝐹_𝑛,𝑖𝑒_𝜁,Φ_𝑚,𝑖=𝜃𝑇⁺_𝑖(𝑡)Ξ_𝜃⁻¹𝑃⁻¹𝐶^𝑇_𝑚×𝑛𝑦−𝐶𝑥=𝐹_𝑚,𝑖𝑦−𝐶𝑥=𝐹_𝑚,𝑖𝐶𝑒_𝜁=𝐹_𝑚,𝑖𝐶Ξ_𝜃⁻¹𝑒_𝜁=𝐹_𝑚,𝑖𝑒_𝜁. (31)

Define 𝑧₁=𝑥₁−𝑥_1𝑟. In terms of the observer dynamics (22) and the desired trajectory (4), we have the following error dynamics: 𝑧̇₁=𝑔_1,𝑖𝑧₁,𝑡+𝐵_1,𝑖(𝑡)𝑥₂+𝜓𝑥₁,𝑥_1𝑟,𝑟(𝑡),𝑡+Φ_𝑛,𝑖,𝑥̇₂=𝑓_2,𝑖(𝑥,𝑡)+𝐵_2,𝑖(𝑡)𝑢_𝑖+Φ_𝑚,𝑖. (32)(33)

In what follows, we first choose a sliding mode surface for the error dynamics of the null space dynamics 𝑧₁. 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 functions 𝑉_2,𝑖(𝑧₁,𝑡) and functions 𝛽_𝑖𝑗(𝑧₁,𝑡)≤0,  𝑖,𝑗=1,…,𝑚 such that 𝐷_𝑡𝑉_2,𝑖+𝐷𝑧₁𝑉_2,𝑖𝑔₁+14𝛾²₁𝐷𝑧₁𝑉_2,𝑖𝐹_𝑛,𝑖𝐹^𝑇_𝑛,𝑖𝐷𝑧₁𝑉_2,𝑖^𝑇+𝑧^𝑇₁𝑧₁+^𝑚_𝑗=1𝛽_𝑖𝑗𝑉_2,𝑗−𝑉_2,𝑖≤0, (34) then, under the nonlinear sliding mode 𝑆=𝑥₂+𝜓=0 (35) and the switching law 𝜎(𝑡)=min𝑖∣𝑖=max_𝑖∈𝑀𝑉_1,𝑖𝑒₁,𝑡, (36) the tracking error norm ‖𝑧₁(𝑡)‖ tends to a ball 𝐵_𝑠 in finite time, where the ball 𝐵_𝑠 is defined as 𝐵_𝑠=𝑧₁(𝑡)∶𝑧₁(𝑡)≤𝛾²₃𝜀²=𝑠, (37) where 𝛾₃ and 𝜀 are positive constants.

Proof. First, we now define the following piecewise Lyapunov function candidate: 𝑉₂𝑒_𝜁,𝑧₁,𝑡=𝑉₁𝑒_𝜁+𝑉_2,𝜎𝑧₁=1,𝑡2𝑒_𝜁𝑃𝑒_𝜁(𝑡)+𝑉_2,𝜎𝑧₁,𝑡, (38) where 𝑉_2,𝜎(𝑧₁,𝑡) is switched among the solution 𝑉_2,𝑖(𝑧₁,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 𝑉_2,𝑖(𝑧₁,𝑡) is 𝑉̇_2,𝑖=𝐷_𝑡𝑉_2,𝑖+𝐷𝑧₁𝑉_2,𝑖×𝑔_1,𝑖+𝐵_1,𝑖(𝑡)×𝑥₂+𝜓𝑥₁,𝑥_1𝑟,𝑟(𝑡),𝑡+Φ_𝑛,𝑖=𝐷_𝑡𝑉_2,𝑖+𝐷𝑧₁𝑉_2,𝑖𝑔_1,𝑖+𝐷𝑧₁𝑉_2,𝑖𝐹_𝑛,𝑖𝑒_𝜁=𝐷_𝑡𝑉_𝑖+𝐷_𝑒1𝑉_𝑖𝑔_1,𝑖+𝛾²₁𝑒^𝑇_𝜁𝑒_𝜁+14𝛾²₁𝐷𝑧₁𝑉_2,𝑖𝐹_𝑛,𝑖𝐹^𝑇_𝑛,𝑖𝐷𝑧₁𝑉_2,𝑖^𝑇−12𝛾₁𝐹^𝑇_𝑛,𝑖𝐷𝑧₁𝑉_2,𝑖^𝑇−𝛾₁𝑒_𝜁²≤𝐷_𝑡𝑉_𝑖+𝐷_𝑒1𝑉_𝑖𝑔_1,𝑖+𝛾²₁𝜔²₁𝜔₁+14𝛾²₁𝐷𝑧₁𝑉_2,𝑖𝐹_𝑛,𝑖𝐹^𝑇_𝑛,𝑖𝐷𝑧₁𝑉_2,𝑖^𝑇. (39)
If there exist solutions of 𝑉_2,𝑖(⋅) such that the inequality in (34) is satisfied, (39) becomes 𝑉̇_𝑖≤−𝑧^𝑇₁𝑧₁+𝛾²₁𝑒^𝑇_𝜁𝑒_𝜁. (40)
From (22) and (40), we have 𝑉̇‌₂𝑒_𝜁,𝑧₁=𝑉̇,𝑡₁𝑒_𝜁+𝑉̇_2,𝑖𝑧₁,𝑡≤−𝜃𝑒^𝑇_𝜁𝑃𝑒_𝜁+𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝑓_𝑖𝑇⁺_𝑖𝜁,𝑡−𝑓_𝑖𝑇⁺_𝑖̂+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖+𝜁,𝑡𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇̇_𝑖𝑇⁺_𝑖Ξ_𝜃⁻¹𝑒_𝜁−𝑧^𝑇₁𝑧₁+𝛾²₁𝑒^𝑇_𝜁𝑒_𝜁≤−𝜃𝜆_min(𝑃)𝑒_𝜁²+𝜆_max(𝑃)𝑏_𝑓+𝑏_𝑡𝑒_𝜁²+𝑒^𝑇_𝜁𝑃Ξ_𝜃𝑇_𝑖𝐻_𝑖𝑇⁺_𝑖𝜁,𝑡𝜔_𝑖𝑇⁺_𝑖−𝑧𝜁,𝑡_𝑇1𝑧₁+𝛾²₁𝑒^𝑇_𝜁𝑒_𝜁≤−𝜃𝜆_min(𝑃)𝑒_𝜁²−𝛾²₁−𝜆_max(𝑃)𝑏_𝑓+𝑏_𝑡−14𝛾²₂𝜆_max(𝑃)²𝑏²_ℎ𝑒_𝜁²−𝑧^𝑇₁𝑧₁+𝛾²₂𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡². (41)
When 𝜃 is selected to be 𝜃𝜆_min(𝑃)𝑒_𝜁²−𝛾²₁−𝜆_max(𝑃)𝑏_𝑓+𝑏_𝑡−14𝛾²₂𝜆_max(𝑃)²𝑏²_ℎ≥0, (42) then the inequality (41) becomes 𝑉̇‌₂𝑒_𝜁,𝑧₁,𝑡≤−𝑧^𝑇₁𝑧₁+𝛾²₂𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡². (43)
Using 𝑒_𝑥,1(𝑡)=𝑥₁(𝑡)−𝑥₁(𝑡)=𝑧₁(𝑡)−𝑧₁(𝑡), ‖𝑧₁(𝑡)‖²≥‖𝑧₁(𝑡)‖²−‖𝑒_𝑥,1(𝑡)‖², and 𝑒_𝑥,1(𝑡)=𝑒_𝜁,1(𝑡) and according to 𝑒_𝑥(𝑡)=𝑇⁺_𝑖Ξ_𝜃⁻¹𝑒(𝑡), (43) becomes 𝑉̇‌₂𝑒_𝜁,𝑧₁,𝑡≤−𝑧^𝑇₁𝑧₁+𝑒_𝑥,1(𝑡)²+𝛾²₂𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡²=−𝑧^𝑇₁𝑧₁+𝑒_𝜁,1(𝑡)²+𝛾²₂𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡²≤−𝑧^𝑇₁𝑧₁+𝑒_𝜁(𝑡)²+𝛾²₂𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡². (44)
From (17), 𝑒_𝜁 is bounded as 𝑒_𝜁≤𝜆_max(𝑃)𝑏_ℎ𝜆_min(𝑃)𝜃−𝜆_max(𝑃)𝑏_𝑓+𝑏_𝑡𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡, (45) where 𝜆_min(𝑃)𝜃−𝜆_max(𝑃)(𝑏_𝑓+𝑏_𝑡)>0 according to (42). Hence, (44) becomes 𝑉̇‌₂𝑒_𝜁,𝑧₁,𝑡≤−𝑧^𝑇₁𝑧₁+𝛾²₃𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡²≤−𝑧^𝑇₁𝑧₁+𝛾²₃𝜀², (46) where 𝛾₃=𝛾²₂+𝜒² and 𝜒=𝑏_ℎ/((𝜆_min(𝑃)/𝜆_max(𝑃))−𝜆_max(𝑃)(𝑏_𝑓+𝑏_𝑡)). Note that lim_𝜃→∞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 𝐵_𝑠=𝑧₁(𝑡)∶𝑧₁(𝑡)≤𝛾²₃𝜀²=𝑠, (47) where 𝐵_𝑠={𝑧₁(𝑡)∶‖𝑧₁(𝑡)‖≤𝛾²₃𝜀²=𝑠}, 𝜀=𝑙²₁+𝑙²₂+𝑙²_𝜂.

Remark 7. In the nonlinear uncertain system (32), if 𝑔_1,𝑖(𝑧₁,𝑡) can be expressed as 𝑊_1,𝑖(𝑧₁,𝑡)𝑧₁, when 𝑊_1,𝑖(𝑧₁,𝑡) is a matrix-valued smooth function, then the HJI inequality (34) can be simplified into the following differential Riccati inequality: 12𝐸̇_𝑖+12𝐸_𝑖𝑊_1,𝑖+𝑊^𝑇_1,𝑖𝐸_𝑖+14𝛾²₁𝐸_𝑖𝐹_𝑛,𝑖𝐹^𝑇_𝑛,𝑖𝐸_𝑖+𝐼_𝑛×𝑛+^𝑚_𝑗=1𝛽_𝑖𝑗𝐸_𝑗−𝐸_𝑖≤0, (48) where 𝐸_𝑖(𝑧₁,𝑡)𝑧₁,  𝑖∈𝑀, are symmetric positive definite smooth matrices.

Remark 8. In the observer design, the parameter 𝜃 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 Theorem 5 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 𝑢_𝑖=𝑢_1,𝑖+𝑢_2,𝑖,𝑢_1,𝑖=−𝐵⁻¹_2,𝑖𝐷_𝑡𝑆+𝐷_𝑥1𝑟𝑆𝑥̇_1𝑟+𝐿𝑓_1,𝑖+𝐵_1,𝑖𝑥₂+Φ_𝑛,𝑖+𝑓_2,𝑖+Φ_𝑚,𝑖,𝑢_2,𝑖=−𝑘₀𝐵^𝑇_2,𝑖𝑆𝐵^𝑇_2,𝑖𝑆, (49)(50)(51) where 𝐿(𝑥₁,𝑥_1𝑟,𝑡)=𝐷𝑥₁𝑆∈𝑅^𝑚×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 𝐵_𝑠 as in Theorem 6.

Proof. Define 𝑉₃=(1/2)𝑆^𝑇𝑆. Choose the following piecewise Lyapunov function candidate: 𝑉₃𝑒_𝜁,𝑧₁,𝑥,𝑥_1𝑟,𝑡=𝑉₁𝑒_𝜁+𝑉_2,𝜎𝑧₁+1,𝑡2𝑆^𝑇𝑆=12𝑒_𝜁𝑃𝑒_𝜁(𝑡)+𝑉_2,𝜎𝑧₁1,𝑡2𝑆^𝑇𝑆, (52) where 𝑉_2,𝜎(𝑧₁,𝑡) is switched among the solution 𝑉_2,𝑖(𝑧₁,𝑡)’s of (34) in accordance with the piecewise constant switching signal 𝜎.
Then, we have 𝑉̇₃=𝑆^𝑇𝐷_𝑡𝑆+𝐷_𝑥1𝑟𝑆𝑥̇_1𝑟+𝐿𝑓_1,𝑖+𝐵_1,𝑖𝑥₂+Φ_𝑛,𝑖+𝑓_2,𝑖+𝐵_2,𝑖𝑢_𝑖+Φ_𝑚,𝑖≤−𝑘₀𝐵^𝑇_2,𝑖𝑆. (53)
From (44) and (54), we have 𝑉̇‌₃=𝑉̇₂+𝑉̇≤−𝑧^𝑇₁𝑧₁+𝛾²₃𝜔_𝑖𝑇⁺_𝑖𝜁,𝑡²−𝑘₀𝐵^𝑇_2,𝑖𝑆≤−𝑧^𝑇₁𝑧₁+𝛾²₃𝜀², (54) 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 𝐵_𝑠 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 𝑓_1,1=𝑓_1,2=𝑊₁𝑥₁=01𝑥−1.6−2.1𝑥¹¹₁₂,𝐵_1,1(𝑡)=(𝑡)0,𝐵1+0.8sin01_1,2(𝑡)=(𝑡)0,𝐻1+0.8cos01_1,1=sin𝑥₁₁cos𝑥₁₁cos𝑥₁₂sin𝑥₁₂,𝐻_1,2=cos𝑥₁₂sin𝑥₁₂sin𝑥₁₁cos𝑥₁₁,𝜔₁=𝑒^−0.3𝑡,𝑒^{−0.1𝑡𝑇},𝜔₂=𝑒^−0.2𝑡,𝑒^{−0.5𝑡𝑇},𝑓_2,1=2𝑥²₁₁𝑥₁₂cos𝑥₂,𝑓_2,1=𝑥₁₂sin𝑥₁₂𝑥₂,𝜂₁=0.6sin𝑥₁₁+0.6sin𝑥₁₂,𝜂₂=−0.6cos𝑥₁₂−0.6cos𝑥₂. (55)
The nonlinear observer is designed as in (27). Based on (18), we have the symmetric positive definite solution 𝑃=10−1010−102. (56) Then, 𝜃=86 is selected based on Remark 8 with 𝑏_𝑓=4.8416,  𝑏_𝑡=1,  𝑏_ℎ=1.128, and 𝛾₂=1.
The target trajectory is 𝑥_11𝑟=0.3sin(𝜋𝑡) and 𝑥_12𝑟=𝑥̇_11𝑟=0.3𝜋cos(𝜋𝑡). From (8), the error dynamics of the 𝑥₁-subpart can be expressed as 𝑧̇₁=𝑊₁𝑒₁+𝐵_1.𝑖𝑥₂+𝜓+𝐻_1,𝑖𝜔₁,𝑖=1,2, (57) where 𝜓(𝑡)=−𝑥̇_12𝑟−1.6𝑥_11𝑟−2.1𝑥_12𝑟.
Let 𝛾₁=1. In 𝑧₁-subpart, according to Remark 7, we first choose 𝑉_𝑖(𝑧₁,𝑡)=(1/2)𝑧^𝑇₁𝐸_𝑖𝑧₁,  𝑖=1,2, where 𝐸_𝑖 are determined by the differential Riccati inequality (48). When 𝐸̇_𝑖=0 and 𝛽₁₂=𝛽₂₁=−1, from the linear algebraic matrix inequality 12𝐸₁𝑊₁+𝑊^𝑇₁𝐸₁+14𝛾²₁𝐸₁𝐻_1,1𝐻^𝑇_1,1𝐸₁+𝐼+𝐸₂−𝐸₁1≤0,2𝐸₂𝑊₁+𝑊^𝑇₁𝐸₂+14𝛾²₁𝐸₂𝐻_1,2𝐻^𝑇_1,2𝐸₂+𝐼+𝐸₁−𝐸₂≤0, (58) and using the singular values of the matrices 𝐻_1,1 and 𝐻_1,2, we can get two symmetric positive definite smooth matrices 𝐸₁=,𝐸0.1147990.0003300.0003300.463736₂=.0.039253−1.025477−1.0254770.070260 (59)
Therefore, the switching surface is 𝑆=𝑥₂+𝜓=𝑥₂−𝑥̇_12𝑟−1.6𝑥_11𝑟−2.1𝑥_12𝑟. Moreover, the switching law is chosen as 𝜎(𝑡)=1if𝑧^𝑇₁𝐸₁𝑧₁>𝑧^𝑇₁𝐸₂𝑧₁,2if𝑧^𝑇₁𝐸₂𝑧₁>𝑧^𝑇₁𝐸₁𝑧₁, (60) 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 state 𝑥11.

Figure 2 

The response of the state 𝑥12.

Figure 3 

The evolution of the tracking error 𝑧11.

Figure 4 

The evolution of the tracking error 𝑧12.

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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants (nos. 61304054, 61203002, 61304102, and 61203123), the Program for New Century Excellent Talents in University, the Program for Liaoning Excellent Talents in University, and the Project of Shandong Province Higher Educational Science and Technology Program under Grant (no. J13LI11).