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

This paper is devoted to robust output feedback tracking control design for a class of switched nonlinear cascade systems.Themain goal is to ensure the global input-to-state stable (ISS) property of the tracking error nonlinear dynamicswith 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.


Introduction
Switched systems are a special class of hybrid systems in engineering applications and have attracted much attention from many researchers [1][2][3][4][5][6][7][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 realworld 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][14][15][16][17], and  ∞ control [18][19][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][25][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-tostate 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. max () and  min () stand for the maximum and minimum eigenvalues of a symmetric matrix , respectively.{} × denotes the first  rows and  columns in , and {} × denotes the last  rows and  columns in . + denotes the set of nonnegative real numbers,   denotes an dimension real vector space, ‖ ⋅ ‖ is the Euclidean norm and induced matrix norm, and  ∞ [0, ∞) is the space of uniformly bounded functions on [0, ∞).   = (, )/ and    = (, )/ are row vectors, and (⋅) denotes the largest singular value of a matrix.
where  1 ,  2 , and   are known positive constants.
In this paper, the output of the system (1) is required to track a given reference model:  ⇒   =  1 ; that is, the  1 subpart is required to track the desired reference model where () is a smooth reference input.Define the tracking error as  1 =  1 −  1 .Then, the error dynamics of the  1subpart can be transformed into
According to Assumption 3, the error dynamics (6) and system (1) with the tracking objective (5) can be rewritten as Definition 4 (input-to-state stable (ISS) [31,32]).Consider a nonlinear dynamical system of the form where  and  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  function , a class  function , and constants for all  0 ∈  and  ∈   satisfying ‖ 0 ‖ <  1 and sup It is said to be input-to-state stable or globally ISS if  =   ,   =   , and ( 10) is satisfied for any initial state and any bounded input .
Control Objective.Under Assumptions 1-3, design a nonlinear observer for the system (1).Based on the observer, design a controller  () and a switching law () such that (i) the tracking error norm ‖ 1 ()‖ in (8) tends to a ball   in finite time, where the ball   is defined as where  is a positive constant; (ii) the closed-loop system (8) possesses ISS property with respect to the disturbances

Controller Design and Stability Analysis
Before the controller design, we would like to rewrite the observer dynamics in (27) as where Define ẑ1 = x1 −  1 .In terms of the observer dynamics (22) and the desired trajectory (4), we have the following error dynamics: In what follows, we first choose a sliding mode surface for the error dynamics of the null space dynamics ẑ1 .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  2, (ẑ 1 , ) and functions   (ẑ 1 , ) ≤ 0, ,  = 1, . . .,  such that then, under the nonlinear sliding mode and the switching law the tracking error norm ‖ 1 ()‖ tends to a ball   in finite time, where the ball   is defined as where  3 and  are positive constants.
Proof.First, we now define the following piecewise Lyapunov function candidate: where  2, (ẑ 1 , ) is switched among the solution  2, (ẑ 1 , )'s of (34) in accordance with the piecewise constant switching signal .
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.
Then, we have From ( 44) and (54), we have which implies that the system ( 8) is globally ISS with respect to the external disturbance input, and the tracking error norm ‖ 1 ‖ is bounded in   in finite time.

Illustrative Example
In this section, we present a simulation example to illustrate the applicability and effectiveness of the proposed approach.
Let the initial states be (−0.6,1.9, −0.56)  .Figures 1 and  2 show the responses of the states  11 and  12 , respectively.The tracking errors  11 and  12 are shown in Figures 3 and  4, respectively, which demonstrate the tracking errors of the states  11 and  12 that are bounded with fast convergence.All the figures indicate the feasibility of our results.

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.