Composite Disturbance Observer-Based Control and H ∞ Output Tracking Control for Discrete-Time Switched Systems with Time-Varying Delay

This paper considers the problem of H ∞ output tracking control for discrete-time switched systems with time-varying delay and external disturbances.The control scheme combining disturbance observer-based control (DOBC) andH ∞ control is proposed.The disturbances are assumed to include two parts. One part is generated by an exogenous system,which imposes on systemwith control inputs in the same channel. The other part is supposed to have the bounded H 2 norm. A new disturbance observer is developed to estimate and reject the first case disturbances for switched system with time-varying delay, and the second case disturbances are attenuated byH ∞ control scheme.The stability analysis of the closed-loop system is developed by switched Lyapunov function, and a solvable delay-dependent sufficient condition is presented in terms of linearmatrix inequalities (LMIs) and cone complement linearization (CCL) methods. A numerical example is given to demonstrate the effectiveness of the proposed composite control scheme.


Introduction
Switched systems are a special class of hybrid control systems, which are composed of a family of subsystems described by continuous or discrete time dynamics and a switched rule among them [1,2].Switched systems can be employed to represent many practical systems, for example, power electronics, embedded systems, chemical processes, computer-controlled systems, automotive industries, and so on.Hence, switched systems have attracted considerable attention during the last decade, and many meaningful results have been reported on stability analysis and controller design for switched systems (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14]).By far, there are a number of methodologies on dealing with the stability analysis and control synthesis of switched systems, such as common Lyapunov function [1,11], multiple Lyapunov function [3,12], dwell time and average dwell time method [7,13], and switched Lyapunov function [4,14].
On the one hand, time delay always encounters in various engineering systems, such as long transmission lines in pneumatic, hydraulic, and chemical processes, economic and rolling mill systems, to name a few, which usually leads to instability and poor performance of the closedloop system.Recently, many researches have paid more and more attention to stabilization and performance analysis of systems or switched systems with delays [15][16][17][18][19][20].On the other hand, disturbances widely exist in systems, which may degrade the closed-loop system performance.Usually, some performance indexes, that is,  ∞ and  2 −  ∞ ( 2 −  ∞ ), are employed to deal with external disturbances.Many results have been developed with these performance indexes [10,13,18,19,[21][22][23][24][25].However, disturbances need to satisfy the bounded  2 norm.In reality, many disturbances do not satisfy this condition, for example, constant disturbances or harmonics disturbances.To improve the ability of disturbance attenuation and rejection, disturbance observer-based control (DOBC) scheme has been studied from 1980s and applied in many control fields [26][27][28][29][30][31][32][33][34][35][36][37], which can be regarded as an active disturbance rejection method.In this method, disturbances are estimated by disturbance observer (DOB), and the disturbance estimation is employed to feedforward compensation.
The problem of output tracking control is important and numerous results have been developed for kinds of systems in the literature [38][39][40].Along with the development of switched system theory, increased attention has been paid to studying output tracking controller design for switched systems.In [41][42][43], the tracking control problems have been considered for kinds of continuous switched systems.Exponential  2 −  ∞ output tracking control problem has been solved in [13] for a class of discrete-time switched systems.However, in [13], disturbances are assumed to be the bounded  2 norm.
In this paper, a composite control scheme is developed to solve the problem of  ∞ output tracking control for discrete-time switched systems with time-varying delay and external disturbances.Here, external disturbances can be divided into two parts.One part is generated by exogenous system, which imposes on system in the same channel with control inputs.The other part satisfies bounded  2 norm.A new disturbance observer is designed to estimate and reject the first case disturbances for switched system with time-varying delay. ∞ control is employed to analyze attenuation performance with respect to the second case disturbances.Hence, a composite control method, consisting of DOBC and  ∞ control, is proposed.By resorting to the switched Lyapunov function approach and inspired by [44,45], some delay-dependent conditions for the problem of  ∞ output tracking control are presented.In order to obtain the desired controller and observer gains, a cone complement linearization (CCL) method is used to transform the nonconvex feasibility problem to some sequential optimization problem subject to linear matrix inequalities (LMIs) constraints.Finally, a numerical example is provided to demonstrate the effectiveness of the main result.

Problem Formulation and Preliminaries
Consider the following discrete-time switched systems with time-varying delays described by where () ∈ R  is the states, () ∈ R  is the signal to be estimated, and () is the control input.() : Z + → N = {1, 2, . . ., } is the switching signal, which specifies which subsystem will be activated at a certain discrete time instant and   ,   ,   ,  1 ,   ,   ,   ,  ∈ N, are constant matrices with appropriate dimensions.() is a time-varying delay of the system and satisfies where , , () are nonnegative integer numbers.() is the initial condition. 2 () ∈ R  is the external disturbances, which is assumed to belong to  2 [0, ∞). 1 () is also the external disturbances, which is generated by the exogenous system where (,   ) is observable.
Remark 1.It is clear that () is an interval-like time-varying delay.When  = 0, it is reduced to 0 ≤ () ≤ , for all  ∈ Z + , which was recently discussed in [23].Therefore, this note can be viewed as some extensions of their results.
Remark 2. System Σ 0 contains two kinds of external disturbances: matched external disturbances and mismatched external disturbances.Two different methods are employed to deal with these disturbances.First, a disturbance observer is introduced to estimate matched disturbances; then the estimation values are used to feedforward compensation.Then, a performance index, that is,  ∞ performance index, is presented to reject and attenuate mismatched disturbances.
Assume that the reference signal   () is generated by the following system: where   () is reference states, () ∈  2 [0, ∞) is reference input, and   is Hurwitz matrix with an appropriate dimension.Here we are interested in designing a state feedback controller by the following formula: where d1 () is the estimation of the disturbances  1 (), which is obtained by the following disturbance observer: ] ( + 1) =  (] () −  ()) +  ( ()  () +  ()  ( −  ()) Remark 3. In this paper, we assume that the switching signal () is not known a priori but its instantaneous value is available in real time [4].Here we only consider the case of synchronous switching; that is, the controller switches just as the system Σ 0 does.
Applying controller (5) to system Σ 0 and combining (3) and (7), we obtain where Let then we have the following augmented switched system: where By defining then we have and the controller can be rewritten as In order to prepare for a precise formulation, we introduce the following definition.

𝐻 ∞ Output Tracking Performance Analysis.
In this subsection, we focus on developing delay-dependent condition to solve the  ∞ output tracking control problem formulated in previous section.
Without loss of generality, we assume that ( + 1) = , () = , for all ,  ∈ N. Then taking the forward difference yields where Direct computation gives ()  3  () . ( Note that Hence we obtain After some manipulations, the following inequality is satisfied: Observe that the following equalities hold naturally: Mathematical Problems in Engineering 7 where From ( 22)-( 29), and by some manipulations, we obtain where Now, we develop the conclusion from two aspects.We first establish the asymptotic stability of system Σ  under the condition of zero disturbances.In fact, when V() = 0, it is verified that Δ () ≤ λ () Φ1 λ () +   () Φ2  () where Applying Schur complement formula, we obtain Δ() < 0 if ( 18) is true.Therefore, it is easy to see that the error system Σ  is asymptotically stable by the Lyapunov-Krasovskii stability theorem.
Next, we will present that under the zero-initial condition, the time-delay system Σ  satisfies ( 17) for all nonzero V() ∈  2 [0, ∞).To this end, we introduce Then, by the Schur complement formula, it easily follows from ( 18) and ( 31) that (36) which implies that ‖()‖ 2 < ‖V()‖ 2 holds under the zeroinitial condition.This completes the proof.Remark 6.It is obvious that the condition in (18) is not an LMI with respect to the parameters  1 , . . .,   ,  1 ,  2 , .In order to solve the controller in the form of (5), we will cast the  ∞ output tracking control problem into an LMI framework.

𝐻 ∞ Output Tracking Controller Design.
In this subsection, we try to obtain a solvable condition for the problem of  ∞ output tracking controller design using CCL method. Define Pre-and postmultiplying Γ and Γ  on (18), and using Schur complement formula, we obtain where Theorem 7 (consider the system Σ  ).For scalars  > 0,  > 0,  > 0, the error system Σ  is asymptotically stable with an  ∞ performance index, if there exist matrices It is clear that (38) is a nonlinear matrix inequality due to the existence of terms  1 ,  2 ,    1   and    2   .In the sequel, the CCL method is resorted to solve the desired controller gains and observer gain.
Introduce two new variables  1 and  2 such that    1   ≥  1 and    2   ≥  2 , then we obtain the following results.
From ( 41) and ( 42), we know that the conditions in Theorem 8 are not strict linear matrix inequalities.By the assistance of the CCL method [46], the nonconvex feasibility problem formulated by ( 40)-( 42) can be transformed into the following nonlinear minimization problem: Subject to (40) and If the solution of the above minimization problem is (6 + 4), then system Σ  is asymptotically stable with an  ∞ performance index via controller (5) and observer (6) with gains   =    −1  and  =  −1 , respectively.

A Numerical Example
Now, we provide an example to show the effectiveness of the main result in this paper.Consider discrete-time switched system with parameters as follows: The reference model is given by the following parameters: Curves of   and   are depicted in Figure 1 under input signal (48).From Figure 1, we can see that the system output can effectively track the reference model output in presence of matched disturbances and mismatched disturbances, which demonstrates the effectiveness of the proposed method.In order to evaluate the effectiveness of the DOB, curves of disturbances and disturbances estimation are shown in Figure 2. It illustrates that DOB can effectively estimate disturbances.
Figure 3 presents the curves of   and   under input signal (49), which demonstrates that the proposed method obtains good tracking performance in spite of matched disturbances and mismatched disturbances.Figure 4 shows the disturbances estimation results, which depicts that the DOB can effectively estimate disturbances.Remark 9.In this paper, a composite tracking controller is designed for a class of discrete-time switched systems with time-varying delay.In reality, many physical systems can be modelled as switched system, for example, flight control systems [47,48] and inverted-pendulum system [49].Take flight control system; for example, the flight control system can be modelled as switched system corresponding to finite operating points within the flight envelope, where  2 () is denoted as wind gust and  1 () is regarded as unknown harmonic disturbances.In order to better serve engineering, we will pay attention to studying a tracking controller design for a practical system based on our method in future.

Conclusions
The problem of  ∞ output tracking control for discretetime switched systems subject to time-varying delay and disturbances has been studied. ∞ control has achieved the attenuation performance with respect to norm bounded disturbances.DOBC has been employed to reject the disturbances with some known information.In this paper, a composite control scheme, that is, consisting of  ∞ control method and DOBC technique, has been proposed, which can effectively attenuate and reject the external disturbances.A numerical example has been provided to show the effectiveness of the proposed algorithm.
A     A     A   Φ 2,10 Φ 2,11 Φ 2,12 Suppose the switching sequence as 121212 ⋅ ⋅ ⋅ .The initial value of the states is chosen as() = [−0.2−0.1 − ]  ,and the reference model of initial condition is selected as  = [0.1 − 0.2]  .In the sequel, two kinds of reference inputs, that is, step reference input and sinusoidal reference input, are considered to demonstrate the effectiveness of the proposed method.