Composite anti-disturbance model reference adaptive control for switched systems

Information Sciences 485 (2019) 71–86

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Composite anti-disturbance model reference adaptive control for switched systems Jing Xie a,∗, Dong Yang b, Jun Zhao c a

School of Information Science and Engineering, Shenyang University of Technology, Shenyang, China Shool of Engineering, Qufu Normal University, Rizhao, Shandong, China c College of Information Science and Engineering, Northeastern University, and the State Key Laboratory of Synthetical Automation for Process Industries (Northeastern University), Shenyang, China b

a r t i c l e

i n f o

Article history: Received 23 September 2018 Revised 10 December 2018 Accepted 7 February 2019 Available online 8 February 2019 Keywords: Switched systems Model reference adaptive control Composite anti-disturbance Adaptive state-disturbance observer

a b s t r a c t This paper investigates the problem of the composite anti-disturbance model reference adaptive control for switched systems with parametric uncertainties and multiple types of disturbances. It is worth emphasizing that the measurability of the system state and the disturbance generated by an exosystem is unnecessary. A key point is to design a composite anti-disturbance model reference adaptive control strategy to achieve the state tracking and the anti-disturbance performance. First, a switched adaptive state-disturbance observer is designed to estimate the system state and the disturbance generated by the exosystem simultaneously. Secondly, based on the switched adaptive state-disturbance observer, a composite switched adaptive controller and a state-dependent switching law are designed to solve the composite anti-disturbance model reference adaptive control problem for switched systems. Thirdly, a sufficient condition ensuring the solvability of the composite anti-disturbance model reference adaptive control problem for switched systems is given, even if the problem is unsolvable for individual subsystems. Finally, an example of the electrohydraulic system is used to verify the availability of the acquired approach. © 2019 Elsevier Inc. All rights reserved.

1. Introduction As an important subclass of hybrid systems, switched systems have received much attention, in part due to the challenges they present to the control theorists and in part due to the powerful ability to approximate many practical processes with complex nonlinearity. Recently, the study of switched systems has been one of the most active research fields [13,19,24,32,38,41]. Meanwhile, some analysis and design methods have been exploited [2,13,40]. However, if switched systems suffer uncertainties, which is an ubiquitous problem for a practical system, the conventional stability analysis and design methods are not applied directly, mainly because the co-existence of the switching signals and the uncertainties makes the analysis design of the switched systems with uncertainties more complex. At present, a few of results have been made to study this topic [14,16,25]. To handle parametric uncertainties, the adaptive control plays a significant role owing to its superior adapting ability to achieve a desired level of performance by combining parameter estimation with control. Among the various types of ∗

Corresponding author. E-mail address: [email protected] (J. Xie).

https://doi.org/10.1016/j.ins.2019.02.016 0020-0255/© 2019 Elsevier Inc. All rights reserved.

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adaptive control strategies [1,18], model reference adaptive control (MRAC) is one of the most active subjects [7,10]. The objective of the MRAC is to design a controller with an adaptive law forcing the state or the output of the uncertain system to track the state or the output of a reference model with desirable performance. In recent years, MRAC has been extended to switched systems [3,21,28,30,31]. For conventional state tracking MRAC problem, the controller is designed relying on the system state, and the adaptive law is constructed relying on the state tracking error. While if the system sate is unmeasurable, obviously, the controller and the adaptive law can not be designed by the conventional design method. To deal with the state unmeasurable problem, for a certain system, usually, a state observer is constructed. Then, based on the estimation state, the controller is designed. And yet, if the system with parametric uncertainties, the problem is difficult. First of all, in MRAC scheme, constructing an observer to estimate the state of the system is not easy due to the existence of the parametric uncertainties. Then, it is very hard to design a controller and an adaptive law dependent on the estimation state to achieve the control objective. Moreover, for the state tracking MRAC problem of switched systems, the problem becomes rather complicated and difficulty. Beside the above difficulties, another difficulty has to be faced— the design of the switching signals, which also can not rely on the system state or the state tracking error. The co-design of the controllers, adaptive laws and switching laws leads to the difficulty increasing greatly. Thus, if the system state is infeasible, we must explore the other design methods to achieve the state tracking for switched systems. In our previous paper [31], under unmeasurable system state, the H∞ state tracking MRAC problem was investigated, in which the output of the system and the state of the reference model were used in the design of the controllers and adaptive laws. Besides, a new state was constructed to design the switching law. On the other hand, external disturbances extensively exist in most practical controlled processes, which may result in poor system performance, oscillation, or even instability. Thus, how to attenuate and reject the disturbances becomes a crucial problem. Several effective disturbance rejection methods have been proposed, such as output regulation theory [8,15], H∞ control [22,39], disturbance observer based control (DOBC) strategy [17] and so on. But, when the systems encounter multiple types of disturbances, the control effects of the methods mentioned above are all unsatisfactory. Motivated by this, a composite anti-disturbance control strategy consisting of DOBC and other control such as robust control [26], adaptive control [5], H∞ control [35] and so on, has been proposed to achieve the anti-disturbance performance. During the past two decades, the composite anti-disturbance control strategy has been successfully developed for linear and non-linear systems [27,35]. Inevitably, multiple types of disturbances also exist in switched systems, and the composite anti-disturbance control strategy has further been extended to switched systems. By the average dwell time method, Sun and Hou [23] solved the composite anti-disturbance control problem for a discrete-time time-varying delay system with actuator failures. Yang and Zhao [36] investigated the disturbance attenuation properties of switched liner systems via mixed state-dependent and time-driven switching. It is worth mentioning that the results above all assume that the state of the systems is measurable. However, in practice, the state of the system is hard to be measured. For the unavailable state, only a few results have been reported. In Han et al. [9], the composite anti-disturbance control problem was solved by designing a state and disturbance observers-based polynomial fuzzy controller. It is note that, in Han et al. [9], the bounded disturbances and the non-switched systems were considered. Then, for switched systems, if the state is not measurable, for the composite anti-disturbance control strategy, the disturbance observer can not be constructed depending on the state of the systems. Meanwhile, the design of the controllers and the switching laws can not depend on the system state also. In this situation, how to design the disturbance observer, the controllers and the switching laws to achieve the control objective? Obviously, this increases the difficulty of the control design. Moreover, apart from multiple types of disturbances, if switched systems suffer from parametric uncertainties simultaneously, under the unmeasurable state, how to design the composite anti-disturbance control to attenuate and reject the disturbances? Apparently, this is a challenge in the switched systems field. So far, to our knowledge, no relevant results have been reported. Motivated by the aforementioned discussion, in this paper, we study the problem of composite anti-disturbance MRAC (CADMRAC) for the switched systems via the multiple Lyapunov functions technique, where the switched systems are subjected to parametric uncertainties and multiple types of disturbances at the same time. The measurability of the system state and the disturbance generated by the exosystem and the solvability of the CADMRAC of each subsystem are not required. The contributions of this note are summarized as follows:

• Unavailable state. A switched adaptive state-disturbance observer is designed to estimate the system state and the disturbance simultaneously. In the most of existing literatures [6,24,35,42], for the measurable state, the disturbance observer is designed depending on the state. Thus, from practical point of view, our result is more applicable for physical systems. Moreover, the systems we studied suffer uncertainties, which are not considered in the most of existing literatures. • New design technique. In term of design technique, almost all existing design techniques depend on exact information state. As mentioned earlier, the state often unavailable in practice. Then, the existing design techniques cannot be applied directly to handle the CADMRAC problem for switched uncertain systems. What we do is to design a composite switched adaptive controller and a switching law depending on the state estimation and the output of the system. • No solvability of the CADMRAC problem for any subsystem is required, which greatly relaxes the conditions in theory. To the best of our knowledge, for switched uncertain systems with unstable subsystems, the CADMRAC problem has not been explored yet in the existing literatures.

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Fig. 1. Electrohydraulic system. LVDT, linear variable differential transformer.

The remainder of this paper is organized as follows. A motivating example is described in Section 2. Section 3 gives the problem formulation and preliminaries. The main results are shown in Section 4. An example is presented to illustrate the effectiveness of the proposed strategy in Section 5. 2. Motivating example As described in Fig. 1. the electrohydraulic system is made up of a hydraulic pump, a hydraulic arm, two accumulators, a MOOG760 torque motor/flapper-operated four-way double-acting servovalve (East Aurora, NY, USA) and a single rod actuator. In this paper, we select two operating conditions with respect to two different supply pressures, 11.0 MPa and 1.4 MPa. Different supply pressures will make plant parameters variation [20]. According to Yuan et al. [37] and Wu et al. [28], the electrohydraulic system with two different supply pressures 11.0 MPa and 1.4 MPa is described by the following switched system

x˙ (t ) = Aσ x(t ) + Bσ uσ (t ), y(t ) = Cσ x(t ),

(1)

where x = [x1 , x2 ]T , x1 represents the displacement of arm, x2 represents the velocity of the arm and u is the control voltage imposing on the torque motor current amplifier, y is the sum of the arm displacement and the arm velocity. The MRAC of the system (1) has been widely studied [28] and [37]. But the system (1) does not consider the effect of disturbance signals. Practically, due to decreased/increased demands from other services, servo-hydraulic systems may encounter an increase/decrease in supply pressure. Then, the electro-hydraulic actuator plant will be subjected to different types disturbances relating to perturbed supply pressure. These different types of disturbance signals greatly reduce system performance. In these cases, the approaches in Wu et al. [28] and Yuan et al. [37] become infeasible. Here, we consider a realistic circumstance that the electrohydraulic system suffers with parameter uncertainties and multiple types of disturbance signals simultaneously, which can be described by

x˙ (t ) = Aσ x(t ) + Bσ [uσ (t ) + d1 (t ) + θσ fσ (x )] + Eσ d2 (t ), y(t ) = Cσ x(t ) + Dσ d1 (t ),

(2)

and the disturbance control voltage d1 (t) is generated from the following exosystem:

w˙ (t ) = Sσ w(t ) + Fσ θσ fσ (x ) + Hσ d3 (t ), d1 (t ) = Gσ w(t ),

(3)

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d2 (t), d3 (t ) ∈ Ln2 [0, ∞] are the additional disturbance signals from the perturbations and uncertainties. It is worth pointing out that no results about CADMRAC problem of the system (2) have been reported. There are two main issues to be addressed: 1) when the system state is unavailable, how to design the disturbance observers to estimate the unknown disturbance signals and construct the composite anti-disturbance controllers to achieve the tracking and the anti-disturbance performance? 2) When the system state is unavailable and the CADMRAC problem is unsolvable for individual subsystems, how to find a sufficient condition for the solvability of the problem above for switched systems by the design of suitably switching signal? The study of the CADMRAC for switched systems under state unavailable is of great significance and remains an open area. In the following, we first present a systematic design procedure for the CADMRAC for more general class of systems. 3. Problem formulation and preliminaries Consider the following switched system with parametric uncertainties

x˙ (t ) = Aσ x(t ) + Bσ [uσ (t ) + d1 (t ) + θσ fσ (x )] + Eσ d2 (t ), y(t ) = Cσ x(t ) + Dσ d1 (t ),

(4)

where x(t) is the system state, d1 (t) is the unknown external disturbance which can represent the constant and harmonic noises and d1 (t) is generated from an exosystem

w˙ (t ) = Sσ w(t ) + Fσ θσ fσ (x ) + Hσ d3 (t ), d1 (t ) = Gσ w(t ),

(5)

where w(t) is the state of the exosystem, d3 (t ) ∈ is the additional disturbance from the perturbations and uncertainties in the exosystem. d2 (t) is another disturbance which is assumed to be an arbitrary signal in Ln2 [0, ∞]. M is the number of subsystems, σ (t ) : [0, +∞ ) → I = {1, 2, · · · , M} is a switching signal, uk (t), k ∈ I are control inputs, the matrices Ak , Bk , Ek , Ck , Dk , Sk , Fk , Hk , Gk are known constant matrices with appropriate dimensions. fk (x) are known continuous vector functions, while θ k are unknown constant matrices. We assume that only the output y(t) is to be available for measurement, and the eigenvalues of matrices Sk are distinct with zero real parts, which implies that the persistent disturbances are imposed on the system (4). Ln2 [0, ∞]

Remark 1. The arbitrary disturbance signal has been widely studied [43–45]. It is note that they are supposed to be Ln2 [0, ∞] usually. However, many types of disturbances are not to be Ln2 [0, ∞], such as unknown constant and harmonics with unknown phase and magnitude [6], which can be described by the system (5). In fact, the system (5) can represent a class of disturbances with partially-known information in practical engineering [6,24,35]. In this paper, the exosystem with parametric uncertainties is considered, which is more appropriate for more practical situations. For this kind of disturbance, it is hard to achieve the disturbance rejection performance by the method in [43–45]. The system (4) with parametric uncertainties is desired to track the switched reference model given by

x˙ m (t ) = Amσ xm (t ) + Bmσ r (t ), ym (t ) = Cσ xm (t ),

(6)

where r(t) is a bounded reference input signal. Denote e(t ) = x(t ) − xm (t ) as the state tracking error. Definite 1. For the system (4) with unknown external disturbance d1 (t) and unmeasurable system state x(t), the CADMRAC problem is to design a switched adaptive control signal uσ (t) and a switching law σ (t) such that (i) When d (t ) = [d2T (t ), d3T (t )]T = 0, the state x(t) asymptotically tracks xm (x) and keep all the signals in the resulting closed-loop system bounded. (ii) When d (t ) = [d2T (t ), d3T (t )]T = 0, under the zero-initial condition,



0

∞

eT ( τ )e ( τ )d τ ≤ γ 2



∞ 0

d T ( τ )d ( τ )d τ

(7)

holds. In this paper, we mainly investigate the CADMRAC problem for the system (4) with various types of disturbances and the unmeasurable system state. Now, we present several lemmas that are essential in the main results. Lemma 1. [4] Let M, N be real matrices of appropriate dimensions. Then, for any matrix Q > 0 of appropriate dimension and any scalar γ > 0, it holds that

MN + N T MT ≤ γ −1 MQ −1 MT + γ N T QN. Lemma 2. [29] Let T, M, F(t) and N be real matrices of appropriate dimensions with holds that

T + MF (t )N + N T F T (t )MT ≤ T + ε MMT + ε −1 N T N.

(8) FT (t)F(t) ≤ I,

then, for any a scalar ε > 0, it

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4. Main results In this section, a composite switched adaptive controller and a state-dependent switching law are designed to solve the CADMRAC problem of the system (4). Firstly, a switched adaptive state-disturbance observer is constructed. Then, based on the switched adaptive state-disturbance observer, a composite switched adaptive controller and a state-dependent switching law are designed. Finally, we give a sufficient condition ensuring that the problem of CADMRAC of the system (4) is solvable. 4.1. Switched adaptive state-disturbance observer Since the system state and the disturbance d1 (t) are not measurable, we have to construct a switched adaptive statedisturbance observer to estimate the state x(t) and the disturbance d1 (t) simultaneously. The switched adaptive statedisturbance observer is designed by:

xˆ˙ (t ) = Aσ xˆ(t ) + Bσ [uσ (t ) + dˆ1 (t ) + θˆ1σ (t ) fσ (x )] + L1σ (y(t ) − yˆ(t )),

υ˙ (t ) = (Sσ + LBσ Gσ − L2σ Dσ Gσ )(υ (t ) − Lxˆ(t )) + L[Aσ xˆ(t ) + Bσ (uσ (t ) + θˆ1σ (t ) fσ (x ))] + L2σ (y(t ) − ym (t )) + Fσ θˆ1σ (t ) fσ (x ), yˆ(t ) = Cσ xˆ(t ) + Dσ dˆ1 (t ), ˆ (t ), dˆ1 (t ) = Gσ w ˆ (t ) = υ (t ) − Lxˆ(t ), w

θˆ˙ 1k (t ) =



−1k BTk p11k em (t ) fkT (x ) + 1k MkT ey (t ) fkT (t ), k = σ , 0 k = σ ,

(9)

ˆ (t ), xˆ(t ), dˆ1 (t ), yˆ(t ) and θˆ1k (t ) are the estimation of the w(t), x(t), d1 (t), y(t) and θ k , respectively, em (t ) = xˆ(t ) − where w xm (t ), ey (t ) = y(t ) − yˆ(t ), emy (t ) = y(t ) − ym (t ), Mk = Ck Bk , k ∈ I, and L, L1k , L2k are the gains of the switched adaptive statedisturbance observer. Remark 2. In the most of the existing literatures [23,36], since the system state is measurable, the disturbance observer is designed depending on the system state. In this paper, the system state is not measurable, moreover, the system (4) and the exosystem (5) all suffer parametric uncertainties. Thus, the conventional design method of the disturbance observer in [23,36] can not be applied directly. To solve this problem, we design the switched adaptive state-disturbance observer to estimates the state and the disturbance simultaneously. 4.2. Composite switched adaptive controller Based on the switched adaptive disturbance-state observer (9), a composite switched adaptive controller is designed as follows:

uσ (t ) = K1σ xˆ(t ) + K2σ r (t ) − θˆ2σ (t ) fσ (x ) − dˆ1 (t ),

θˆ˙ 2k (t ) =



2k BTk p11k em (t ) fkT (x ), 0

k = σ, k = σ ,

(10)

where θˆ2k (t ) are the estimation of θ k also, K1k , K2k satisfy

Ak + Bk K1k = Amk , Bk K2k = Bmk , k ∈ I.

(11)

A plot for the signal input is given in Fig. 2. Remark 3. For the conventional MRAC, the controller depends the state x(t) and the reference signal r(t). In this paper, due to the unmeasurability of the system state and the disturbance generated by the exosystem, the composite switched adaptive controller (10) depends on the state estimation xˆ(t ), the disturbance estimation dˆ1 (t ) and the reference signal r(t). Remark 4. The composite switched adaptive controller (10) combines the normal adaptive state feedback control law and the disturbance estimation. The normal adaptive state feedback control law is to guarantee the state x(t) tracking xm (t) asymptotically. The disturbance estimation is to compensate the disturbance d1 (t) generated by the exosystem. Remark 5. θˆ1k (t ) and θˆ2k (t ) are both used to estimate the unknown parameters θ k . However, the design goal of the update laws of them is substantially different. The update laws of θˆ1k (t ) in (9) are designed such that the switched adaptive statedisturbance observer can estimate the state x(t) and disturbance d1 (t) at the same time. Then, the update laws of θˆ2k (t ) in (10) are designed to render the state x(t) asymptotically tracking xm (x). ˜ (t ) = w(t ) − w ˆ (t ) as the disturbance estimation We denote e˜(t ) = x(t ) − xˆ(t ) as the state estimation error, and denote w error. By (4), (9) and (10), we have

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Fig. 2. A plot for the signal input.

e˜˙ (t ) = Aσ e˜(t ) + Bσ (d1 (t ) − dˆ1 (t ) + (θσ − θˆ1σ (t )) fσ (x )) + Eσ d2 (t ) − L1σ (y(t ) − yˆ(t )) ˜ (t ) = (Aσ − L1σ Cσ )e˜(t ) + (Bσ Gσ − L1σ Dσ Gσ )w + Bσ θ˜1σ (t ) fσ (x ) + Eσ d2 (t ),

(12)

where θ˜1k (t ) = θk − θˆ1k (t ) are estimation errors. Combining (5) and (9) yields

˜˙ (t ) = Sσ w(t ) + Fσ θσ fσ (x ) + Hσ d3 (t ) + LBσ dˆ1 (t ) w ˆ (t ) + LL1σ Cσ e˜(t ) − L2σ Cσ (em (t ) − (Sσ + LBσ Gσ − L2σ Dσ Gσ )w + e˜(t )) − L2σ Dσ d1 (t ) + LL1σ Dσ (d1 (t ) − dˆ1 (t )) − Fσ θˆ1σ fσ (x ) ˜ (t ) + (LL1σ Cσ − L2σ Cσ )e˜(t ) = (Sσ − L2σ Dσ Gσ + LL1σ Dσ Gσ )w − L2σ Cσ em (t ) + Fσ θ˜1σ fσ (x ) + Hσ d3 (t ),

(13)

where em (t ) = xˆ(t ) − xm (t ). By (6), (9)–(11), we have the dynamics of em (t ) = xˆ(t ) − xm (t )

e˙ m (t ) = Aσ xˆ(t ) + Bσ [uσ (t ) + dˆ1 (t ) + θˆ1σ (t ) fσ (x )] + L1σ (y(t ) − yˆ(t )) − Amσ xm (t ) − Bmσ r (t ) = Amσ em (t ) + L1σ Cσ e˜(t ) − Bσ θ˜1σ (t ) fσ (x ) ˜ (t ), + Bσ θ˜2σ (t ) fσ (x ) + L1σ Dσ Gσ w

(14)

where θ˜2k (t ) = θk − θˆ2k (t ) are estimation errors. Augmenting (12), (13) and (14) gives

X˙ (t ) = A¯ σ X (t ) + B¯ 1σ θ˜1σ (t ) fσ (x ) + B¯ 2σ θ˜2σ (t ) fσ (x ) + E¯σ d (t ), where

˜ T (t )]T , d (t ) = [d2T (t ), d3T (t )]T , X = [eTm (t ), e˜T (t ), w



A¯ k =

Amk 0 −L2kCk

 B¯ 1k =



L1kCk Ak − L1kCk LL1kCk − L2kCk

−Bk Bk , B¯ 2k = Fk

 

Bk 0 , E¯k = 0



L1k Dk G k Bk G k − L1k Dk G k , Sk − L2k Dk Gk + LL1k Dk Gk



0 Ek 0



0 0 . Hk

4.3. Composite anti-disturbance performance In the following, a sufficient condition is presented, which solves the CADMRAC problem for the system (4).

(15)

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Theorem 1. Consider the augmented system (15) and the switching law

σ (t ) = argminX T (t )P¯k X (t ),

(16)

k∈I

if under the switching law (16), the state xm (t) in (6) is bounded, and there exist positive definite matrices p11k , p22 , p33 , constants γ k > 0, β kj ≥ 0, k, j ∈ I, such that

⎡

11k ⎢∗ ⎢∗ ⎣ ∗ ∗

p11k L1kCkT + I

22k ∗ ∗ ∗

13k

23k

33k

∗ ∗

0 p22 Ek 0 −γk2 I ∗

⎤

0 0 ⎥ p33 Hk ⎥ < 0, ⎦ 0 −γk2 I

(17)

p22 Bk = CkT Mk ,

(18)

p33 Fk = GTk DTk Mk

(19)

hold, then, the CADMRAC problem of the system (1) is solved, where

11k = ATmk p11k + p11k Amk + I +

M

βk j ( p11 j − p11k ),

j=1

13k = − CkT LT2k p33 + p11k L1k Dk Gk ,

22k = (Ak − L1kCk )T p22 + p22 (Ak − L1kCk ) + I,

23k = p22 (Bk Gk − L1k Dk Gk ) + (CkT LT1k LT − CkT LT2k ) p33 ,

33k = (Sk − L2k Dk Gk + LL1k Dk Gk )T p33 + P33 (Sk − L2k Dk Gk + LL1k Dk Gk ),   p11k P¯k = 0 0

0 p22 0

0 0 , p33

Mk = Ck Bk .

(20)

Proof. We construct Lyapunov functions for each subsystem of the augment system (15) as follows:

Vk (Z (t )) = X T (t )P¯k X (t ) +

M M tr[θ˜1Tl (t )1−1 θ˜ (t )] + tr[θ˜2Tl (t )2−1l θ˜2l (t )], l 1l l=1

(21)

l=1

T (t ), · · · , θ˜ T (t ), θ˜ T (t ), · · · , θ˜ T (t )]T . Due to θ˜˙ (t ) = −θˆ˙ (t ) and θ˜˙ (t ) = −θˆ˙ (t ), then along with where Z (t ) = [X T (t ), θ˜11 1k 1k 2k 2k 1M 21 2M the kth subsystem of the augmented system (15), we have

V˙ k (Z (t )) = X T (t )[A¯ Tk P¯k + P¯k A¯ k ]X (t ) + 2X T (t )P¯k B¯ 1k θ˜1k (t ) fk (x ) + 2X T (t )P¯k B¯ 2k θ˜2k (t ) fk (x ) + 2X T (t )P¯k E¯k d (t ) − 2t r[θ˜1Tk (t )1−1 θˆ˙ (t )] − 2t r[θ˜2Tk (t )2−1k θˆ˙ 2k (t )]. k 1k

(22)

According to the properties of the trace of a matrix, it follows that

2X T (t )P¯k B¯ 1k θ˜1k (t ) fk (x ) = 2tr[ fk (x )X T (t )P¯k B¯ 1k θ˜1k (t )] = 2t r[θ˜1Tk (t )B¯ T1k P¯k X (t ) fkT (x )], 2X T (t )P¯k B¯ 2k θ˜2k (t ) fk (x ) = 2tr[ fk (x )X T (t )P¯k B¯ 2k θ˜2k (t )] = 2t r[θ˜2Tk (t )B¯ T2k P¯k X (t ) fkT (x )]. By the fact that p22 Bk = CkT Mk , p33 Fk = GTk DTk Mk , then

θˆ˙ 1k = − 1k BTk p11k em (t ) fkT (x ) + 1k MkT ey (t ) fkT (x ) ˜ (t )) fkT (x ) = − 1k BTk p11k em (t ) fkT (x ) + 1k MkT (Ck e˜(t ) + Dk Gk w T T T T ˜ (t ) fkT (x ) = − 1k Bk p11k em (t ) fk (x ) + 1k Bk p22 e˜(t ) fk (x ) + 1k FkT p33 w = 1k B¯ T1k P¯k X (t ) fkT (x ). Thus,

2t r[θ˜1Tk (t )1−1 θˆ˙ (t )] = 2t r[θ˜1Tk (t )B¯ T1k P¯k X (t ) fkT (x )]. k 1k

(23)

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Similarly,

2t r[θ˜2Tk (t )2−1 θˆ˙ (t )] = 2t r[θ˜2Tk (t )B¯ T2k P¯k X (t ) fkT (x )]. k 2k

(24)

Substituting (23) and (24) into (22) leads to

V˙ k (Z (t )) = X T (t )[A¯ Tk P¯k + P¯k A¯ k ]X (t ) + 2X T (t )P¯k E¯k d (t ).

(25)

First, we prove that the state x(t) asymptotically tracks xm (x) and the boundedness of the signals in the closed-loop system when d (t ) = 0. Observe that when d (t ) = 0, (25) becomes

V˙ k (Z (t )) = X T (t )[A¯ Tk P¯k + P¯k A¯ k ]X (t ) =



ATmk p11k + p11k Amk T X (t ) ∗ ∗

p11k L1kCk

22k

∗

 13k 23k X (t ),

33k

(26)

where 13k = −CkT LT2k p33 + p11k L1k Dk Gk , 22k = (Ak − L1kCk )T p22 + p22 (Ak − L1kCk ), 23k = p22 (Bk Gk − L1k Dk Gk ) + (CkT LT1k LT − CkT LT2k ) p33 , 33k are given in (20). From (17) and Schur complement lemma, we obtain



11k ∗ ∗

 13k 23k < 0, 33k

p11k L1kCkT + I

22k

∗

(27)

where 13k = −CkT LT2k p33 + p11k L1k Dk Gk , 22k = (Ak − L1kCk )T p22 + p22 (Ak − L1kCk ) + γ −2 p22 Ek EkT p22 + I, 23k = p22 (Bk Gk − L1k Dk Gk ) + (CkT LT1k LT − CkT LT2k ) p33 33k = (Sk − L2k Dk Gk + LL1k Dk Gk )T p33 + P33 (Sk − L2k Dk Gk + LL1k Dk Gk ) + γ −2 p33 Hk HkT p33 ,

11k are given in (20). It follows from the switching law (16) that

σ (t ) = argmin{X T (t )P¯k X (t )} = argmin{eTm (x ) p11k em (x )}. k∈I

Combing (26) and (28), when kth subsystem is active, we have



−I V˙ k (Z (t )) ≤ X T (t ) ∗ ∗

−I −γ −2 p22 Ek EkT p22 − I ∗

By a simple calculation, it holds that



−I ∗ ∗

(28)

k∈I

−I −γ −2 p22 Ek EkT p22 − I ∗

0 0 −γ 2 p33 Hk HkT p33



0 0 X (t ). −γ 2 p33 Hk HkT p33

(29)

 < 0.

(30)

Thus, V˙ k (Z (t )) < 0 follows, which implies Vk (Z(t)) < Vk (Z(0)). By (21), we have X (t ), θ˜1k (t ), θ˜2k (t ) ∈ L∞ . It then follows ˜ (t ), θ1k (t ), θ2k (t ) ∈ L∞ . By the fact that em (x ) = xˆ(t ) − xm (t ), e˜(x ) = x(t ) − xˆ(t ), w ˜ (x ) = w(t ) − w ˆ (t ), then that em (x ), e˜(x ), w ˆ (t ) ∈ L∞ hold since w(t) is bounded, this further implies that fk (x), dˆ1 (t ) are bounded. Thus, e˜˙ (x ) in (12) and xˆ(t ), x(t ), w e˙ m (x ) in (14) and uσ (t) in (10) are all bounded. From (30), it is easy to know that em (x ), e˜(x ) ∈ L2 . One can conclude that em (x ) ∈ L2 ∩ L∞ , e˙ m (x ) ∈ L∞ , e˜(x ) ∈ L2 ∩ L∞ , e˜˙ m (x ) ∈ L∞ . By Barbalat’s lemma, we get em (x ) → 0, e˜(x ) → 0 as t → ∞. Note that e(t ) = em (x ) + e˜(x ), and thus e(t) → 0 as t → 0 also, that is, the state x(t) asymptotically tracks xm (x). Therefore, when d (t ) = 0, all the signals in the closed-loop system (15) are bounded and the state x(t) asymptotically tracks xm (x) under the switching law (16). Next, we show that for any nonzero d (t ) ∈ L2n [0, +∞ ), under Z (0 ) = 0,



∞

0

eT ( τ )e ( τ )d τ ≤ γ 2



∞ 0

d T ( τ )d ( τ )d τ

(31)

holds. By Lemma 1, we have 2X T (t )P¯k E¯k d (t ) ≤ γk−2 X T (t )P¯k Ek EkT P¯k X (t ) + γk2 dT (t )d (t ) for any scalar γ k > 0. Thus, (25) turns into

V˙ k (Z (t )) ≤ X T (t )[A¯ Tk P¯k + P¯k A¯ k ]X (t ) + γk−2 X T (t )P¯k Ek EkT P¯k X (t ) + γk2 dT (t )d (t )



ATmk p11k + p11k Amk = X (t ) ∗ ∗ T

p11k L1kCk

22k

∗



13k

23k X (t ) + γk2 dT (t )d (t ), 33k

where 13k = −CkT LT2k p33 + p11k L1k Dk Gk , 22k = (Ak − L1kCk )T p22 + p22 (Ak − L1kCk ) + γk−2 p22 Ek EkT p22 , L1k Dk Gk ) + (CkT LT1k LT − CkT LT2k ) p33 , 33k are given in (27). Viewing (27) and (28) yileds

(32)

23k = p22 (Bk Gk −

J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86

79

V˙ k (Z (t )) + eT (t )e(t ) − γk2 dT (t )d (t )

 ϒ11k

≤ X T (t ) ∗ ∗

p11k L1kCk + I

22k ∗

 ϒ13k ϒ23k X (t ) < 0, 33k

(33)

where ϒ11k = ATmk p11k + p11k Amk + I, ϒ13k = −CkT LT2k p33 + p11k L1k Dk Gk , ϒ23k = p22 (Bk Gk − L1k Dk Gk ) + (CkT LT1k LT − CkT LT2k ) p33 , 22k , 33k are given in (27). Integrating both sides of the inequality V˙ k (Z (t )) + eT (t )e(t ) − γ 2 dT (t )d (t ) < 0, under the zero k

initial condition, we conclude that

V (Z (t )) − V (Z (0 )) +



t

0

(eT (τ )e(τ ) − γk2 dT (τ )d (τ ))dτ < 0.

(34)

By the fact that V(Z(t)) ≥ 0 and V (Z (0 )) = 0, it follows that



t 0

eT ( τ )e ( τ )d τ < γ 2



t 0

d T ( τ )d ( τ )d τ

(35)

with γ = max γk . Noting d (t ) ∈ L2n [0, ∞ ) gives k∈I



∞ 0

eT ( τ )e ( τ )d τ ≤ γ 2



∞

0

d T ( τ )d ( τ )d τ

(36)

as t → ∞, which means that (31) hold. This completes the proof.



The following theorem gives the gains of the switched adaptive state-disturbance observer by solving the matrix inequalities. Theorem 2. Consider the augmented system (15). If there exist matrices Yk , L1k , Zk , V, positive definite matrices Xk , p22 , p33 , constants β kj satisfying β kj ≥ 0, j, k ∈ I, and positive scalars γ k , ε 1k , ε 2k , ε 3k , ε 4k , ε 5k , ε 6k , ε 7k , ε 8k , ε 9k , ε 10k , such that

⎡ 11k ⎢∗ ⎣∗ ∗

⎡ 22k ⎢∗ ⎣∗ ∗

⎡ 33k ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢ ⎢∗ ⎢ ⎢∗ ⎢∗ ⎢ ⎣∗ ∗

1k

⎤

L1kCk −ε1k I ∗ ∗

L1k Dk G k 0 −ε4k I ∗

0 ⎥ < 0, 0 ⎦ −Rk

p22 Ek −γk2 I ∗ ∗

p22 Bk Gk 0 −ε5k I ∗

Yk Dk Gk 0 ⎥ ⎦ < 0, 0 −ε6k I

p33 Hk −γk2 I ∗ ∗ ∗ ∗ ∗ ∗ ∗

ZkCk 0 −ε3k I ∗ ∗ ∗ ∗ ∗ ∗

(37)

⎤

0 0 0 −ε7k I ∗ ∗ ∗ ∗ ∗

ZkCk 0 0 0 −ε8k I ∗ ∗ ∗ ∗

(38)

V 0 0 0 0 −ε9k I ∗ ∗ ∗

GTk DTk LT1k 0 0 0 0 0 −ε9−1 I k ∗ ∗

V 0 0 0 0 0 0 −ε10k I ∗

⎤

0 ⎥ 0 ⎥ 0 ⎥ CkT LT1k ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎦ 0 −1 −ε10k I

<0

(39)

p22 Bk = CkT Mk ,

(40)

p33 Fk = GTk DTk Mk

(41)

hold, then under the controller (10) and the switching law (16), the problem of CADMRAC for the system (4) is solved, where

11k = Xk ATmk + Amk Xk + (1 + ε2−1 + ε3k )I −

M

βk j Xk ,

j=1

22k = ATk p22 + p22 Ak − CkT YkT − YkCk + (1 + ε1k + ε2k + ε7k + ε8k )I, 33k = SkT p33 + p33 Sk − GTk DTk ZkT − Zk Dk Gk + (ε4k + ε5k + ε6k )I,

80

J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86



1k = [ βk1 Xk , βk2 Xk , · · · , βkk−1 Xk , βkk+1 Xk , · · · , βkM Xk ], Rk = diag{Xk1 , Xk2 , · · · , Xkk−1 , Xkk+1 , · · · , XkM }, Mk = Ck Bk . Moreover, the gains of the switched adaptive state-disturbance observer in (9) are given by L1k = p−1 Y , L2k = p−1 Z , L = p−1 V. 22 k 33 k 33 Proof. From Theorem 1, the problem of CADMRAC for the system (4) is solved if the conditions (17)–(19) are satisfied. Based on Schur complement lemma, we rewrite (17) as (27). Further, (27) can be reformulated as



11k

0

22k

0 0

0



0 + CkT LT1k p11k + I 0





0 0

33k 0 0 0







0 0 0 + 0 0 0

0 + 0 −p33 L2kCk + GTk DTk LT1k p11k



p11k L1kCk + I 0 0

0 + 0 0

0 0

0 0 0





0 0 0 + 0 0 0





0 0 (Bk Gk − L1k Dk Gk )T p22

0 0 0 + 0 0 0

0 + 0 0

0 0 p33 (CkT LT1k LT − CkT LT2k )T

0 0 0

0 0 0

0 0 0



0 0 0

p11k L1kCk 0 0



0 0 0



=

0 0 0



p11k L1kCk 0 0

p11k L1kCk 0 0

0 0 0

Similarly, we obtain

0 0 0

and



I 0 0



≤ε3−1 k

0 0 0 0 0



0 0 0

≤ε4−1 k





0 0 0 + I 0 0



0 0 0



0 0 0 + CkT LT1k p11k 0 0

p11k L1kCkCkT LT1k p11k 0 0

≤ ε1−1 k





0 0

0 0 0





−CkT LT2k p33 0 0 0 0 0





I 0 0

0 0 0

0 0 0

0 0 0





0 + 0 −p33 L2kCk

0 0 0

0 0 0

I 0 0

0 0 0







+ ε3k



0 + 0 GTk DTk LT1k p11k

p11k L1k Dk Gk GTk DTk LT1k p11k 0 0

0 0 0



0 0 0

I 0 0



0 0 . 0

(43)

(44)

0 0 0 + ε2k 0 0 0

≤ε



0 0 . 0

0 0 0

−1 2k





0 I 0

I 0 0

0 0 p33 L2kCkCkT LT2k p33

p11k L1k Dk Gk 0 0



0 0 0

0 0 0 + ε1k 0 0 0



0 0 0

0 (CkT LT1k LT − CkT LT2k ) p33 0



0 0 0



0 0 0



0 p22 (Bk Gk − L1k Dk Gk ) 0

(42)

By Lemma 2, for any ε 1k > 0, we have





< 0,

where 11k , 22k , 33k are given in (27). Note that





−CkT LT2k p33 + p11k L1k Dk Gk 0 0

0 + 0 0



0 0 0

0 0 0

0 I 0



0 0 , 0

(45)

 

0 0 , 0

(46)





0 0 0

0 0 0 + ε4k 0 0 0

0 0 0



0 0 , I

(47)

J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86



≤ε



−1 5k



≤ε

0 0 0

0 0 0



−1 6k

0 0 0 0 0 0

0 0 0

0 0 0

0 0 0

0 −CkT LT2k p33 0

0 0 0

0 0 0



 ε

0 0 GTk BTk p22









0 + 0 0

 

0 0 0

0 0 , I





0 + 0 0



0 + 0 0





0 0 0



0 0 0 0 0 0



0 0 , I

(49)



0 0 0

 + ε8k

0 0 0



+ ε7k

0 0 p33 L2kCk

0 0 p33 L2kCkCkT LT2k p33

0 0 0

0 0 0 + ε6k 0 0 0

0 0 p33 LL1kCk

(48)



0 0 GTk DTk LT1k p22

0 0 p33 LL1kCkCkT LT1k LT p33





0 0 0

0 0 0 + ε5k 0 0 0

0 p22 L1k Dk Gk GTk DTk LT1k p22 0 0 CkT LT1k LT p33 0

0 0 0

≤

0 + 0 0

0 0 0



−1 8k



0 −p22 L1k Dk Gk 0

0 0 0

ε7−1k

≤



0 p22 Bk Gk GTk BTk p22 0

0 0 0



0 0 0

0 p22 Bk Gk 0

81



0 I 0

0 0 , 0

(50)

 (51)



0 I 0

0 0 . 0

(52)

Substituting (44)–(52) into (42) yields

 11k 0 0

0

22k 0

0 0



33k

≤ 0,

(53)

where

11k = ATmk p11k + p11k Amk + ε1−1 p L C C T LT p k 11k 1k k k 1k 11k +

M

βk j ( p11 j − p11k )(1 + ε2−1k + ε3k )I

j=1

ε4−1k p11k L1k Dk Gk GTk DTk LT1k p11k , 22k =(Ak − L1kCk )T p22 + p22 (Ak − L1kCk ) + γ −2 p22 Ek EkT p22 + (1 + ε1k + ε2k + ε7k + ε8k )I + ε5−1 p B G GT BT p + ε6−1 p L D G G T DT LT p , k 22 k k k k 22 k 22 1k k k k k 1k 22 33k = (Sk − L2k Dk Gk + LL1k Dk Gk )T p33 + P33 (Sk − L2k Dk Gk + LL1k Dk Gk ) + γ 2 p33 Hk HkT p33 + (ε4k + ε5k + ε6k )I + ε3−1 p L C C T LT p k 33 2k k k 2k 33 + ε7−1 p LL C C T LT LT p33 + ε8−1 p L C C T LT p . k 33 1k k k 1k k 33 2k k k 2k 33 +

(54)

Obviously, (53) yields  11k < 0,  22k < 0,  33k < 0. By a simple derivation,  11k < 0 is equivalent to

 11k ∗ ∗

p11k L1kCk −ε1k I ∗

p11k L1k Dk Gk 0 −ε4k I



< 0,

where 11k = ATmk p11k + p11k Amk + (1 + ε2−1 + ε3k )I + k

(55) M  j=1

βk j ( p11 j − p11k ).

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J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86

Pre-multiplying and post-multiplying the inequality (55) by diag[ p−1 , I, I] gives 11k



11k ∗ ∗

L1kCk −ε1k I 0

L1k Dk G k 0 −ε4k I



<0

(56)

 −1 T −1 −1 −1 −1 −1 with 11k = P11 A + Amk P11 + (1 + ε2−1 + ε3k )I + M j=1 βk j (P11k P11k P11k − P11k ). Let Xk = P11k . Applying Schur complement k mk k k lemma to (56) gives rise to condition (37). Slimily,  22k < 0 is equivalent to

⎡

22k ⎢∗ ⎣∗ ∗

p22 Ek −γ 2 I ∗ ∗

⎤

p22 Bk Gk 0 −ε5k I ∗

p22 L1k Dk Gk 0 ⎥ ⎦ < 0, 0 −ε6k I

(57)

with 22k = (Ak − L1kCk )T p22 + p22 (Ak − L1kCk ) + (1 + ε1k + ε2k + ε7k + ε8k )I. Let p22 L1k = Yk , then condition (38) hold. Moreover,  33k < 0 gives rise to

⎡ ⎢∗ ⎢∗ ⎣

p33 Hk −γ 2 I ∗ ∗ ∗

33k

∗ ∗

p33 L2kCk 0 −ε3k I 0 0

p33 LL1kCk 0 0 −ε7k I 0

⎤

p33 L2kCk 0 ⎥ ⎥ < 0, 0 ⎦ 0 −ε8k I

(58)

where 33k = (Sk − L2k Dk Gk + LL1k Dk Gk )T p33 + P33 (Sk − L2k Dk Gk + LL1k Dk Gk ) + (ε4k + ε5k + ε6k )I. From Lemma 2, for any ε 9k > 0, ε 10k > 0, one has

⎡P LL D G 33 1k k k ⎢0 ⎢0 ⎣ 0 0

0 0 0 0 0

⎡

p33 LLT p33 ⎢0 ⎢0 ≤ ε9−1 k⎣ 0 0 and

⎡0 ⎢0 ⎢0 ⎣ 0 0

0 0 0 0 0

⎡

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

⎤

0 0 0 0 0

p33 LL1kCk 0 0 0 0

0 0 0 0 0

⎤

0

0

0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

Substituting (59)-(60) into (58) leads to

⎡ ⎢∗ ⎢∗ ⎣ ∗ ∗

44k

p33 Hk γ 2I 0 ∗ ∗

p33 L2kCk 0 ε3k I ∗ ∗

⎤

⎡

0 0 0 55k ∗

0 0 0 0 0

0 0 0 0 0

GTk DTk LT1k L1k Dk Gk 0 0⎥ ⎢0 0⎥ + ε9k ⎢0 ⎦ ⎣ 0 0 0 0

⎡

0 0 0⎥ ⎢0 0⎥ + ⎢0 ⎦ ⎣ T T T 0 Ck L1k L p33 0 0

p33 LLT p33

⎢ ⎢ −1 ⎢0 ≤ ε10 k ⎢0 ⎣

⎡

0 GTk DTk LT1k LT P33 0⎥ ⎢0 0⎥ + ⎢0 ⎦ ⎣ 0 0 0 0

0 0 0 0 0

⎤

⎡0 ⎥ ⎢0 0⎥ ⎥ + ε10k ⎢0 ⎣ 0⎥ 0 ⎦ 0

0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

⎤

0 0⎥ 0⎥ ⎦ 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0⎥ 0⎥. ⎦ 0 0

⎤

0 0⎥ 0⎥, ⎦ 0 0

(59)

⎤

0 0⎥ 0⎥ ⎦ 0 0

0 0 0 CkT LT1k L1kCk 0

⎤

(60)

⎤

p33 L2kCk 0 ⎥ ⎥<0 0 ⎦ 0 ε8k I

(61)

−1 with 44k = (Sk − L2k Dk Gk )Tk p33 + P33 (Sk − L2k Dk Gk ) + (ε4k + ε5k + ε6k )I + ε9−1 p LLT p33 + ε9k GTk DTk LT1k L1k Dk Gk + ε10 p LLT k 33 k 33 T T p33 , 55k = ε7k I + ε10kCk L1k L1kCk . Letting p33 L2k = Zk , p33 L = V and applying Schur complement lemma to (61) get condition (39). Therefore, if conditons (37)–(39) hold, the CADMRAC problem for the system (4) is solved. In addition, the gains of the switched adaptive state-disturbance observer in (9) are given by L1k = p−1 Y , L2k = p−1 Z and L = p−1 V . This completes 22 k 33 k 33 the proof. 

Remark 6. In Theorem 1, in order to cope with unavailable state, we need the common p22 and p33 existing for all subsystems to design the switching law. Meanwhile, the two equalities (19)-(20) are needed to design the adaptive laws.

J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86

83

Remark 7. Notice that the conditions (41)-(42) are not in the strictly linear matrix inequality form. In order to deal with this problem, from computational point of view, we can replace (41)-(42) by the inequalities [33]:

 



−α1i I ∗

p22 Bk − CkT Mk , I

−α2i I ∗

p33 Fk − CkT DTk Mk . I

(62)



(63)

When α 1k , α 2k are sufficiently small positive scalars, (62)-(63) are closed to (41)-(42). Thus, p22 , p33 , XK , Yk , L1k , Zk and V can be easily obtained by solving conditions (38)–(40) and (62)-(63) using the LMI toolbox. 5. Example In this section, we apply the CADMRAC scheme to an electrohydraulic system described in Motivating example. 5.1. Description of the electrohydraulic system According to Wu et al. [28] and Yuan et al. [37], the parameters in the system (2) and (3) are given by



A1 =

 B2 =

 E2 =



0 0







1 0 , A2 = −4.58 0



 0 , C1 = 1 47.2





1 ,





2.495 F1 = , 2.496



 S1 =



1.888 F2 = , 1.888

G1 = G2 = 0.01





0 −1





C2 = 1



0.01 , D1 = D2 = 20, 2



1 0 , B1 = , −9.19 62.4 1 ,

E1 =









0.1 , 1

1 0 , S2 = 0 −2







−2 2 H1 = , H2 = , −5 −5





cos(x ) 0.01 , f1 (x ) = , sin(2x )





2 , 0



cos(0.2x ) f 2 (x ) = , e−0.1x

θ1 = [0.1, −0.1], θ2 = [−1, 0.5] are unknown constant parameters. d2 (t ) = sin(t ) exp(−0.01t ), d3 (t ) = t exp(−0.1t ) are the external disturbances. The dynamics of the switched reference model are given by

x˙ m (t ) = Amσ xm (t ) + Bmσ r (t ), ym (t ) = Cσ xm (t ), where



Am1

0 = −15

 

Bm1 =

0 , 1



(64)



1 0 , Am2 = −8 −27

 

Bm2 =



1 , −12

0 , 1

the reference input r (t ) = 2 sin(π t ) + 3sin(0.1π t ). 5.2. Simulation results Our purpose is to solve the CADMRAC problem of the system (2). Comparing with existing results [6,24,35], we need to compute more design parameters. Using the LMI toolbox, from (37)–(39), (62)-(63), we get the switched adaptive state0.2168 0.1396 0.0369 0.0419 0.0029 0.0 0 01 disturbance observer (9) with the gains L11 = [ ], L12 = [ ], L21 = [ ], L22 = [ ], L = [ ]. 0.5227 0.2417 0.0369 0.0419 0.0 0 01 0.0029 Solving (11) gets the gains of the controller (10) K11 = [−0.2404, −0.0548], K12 = [−0.5720, −0.0595], K21 = 0.016, K22 = 0.0212. Choose the adaptive gains 11 = 12 = 21 = 22 = 10. We conduct the simulation. Fig. 3 shows the switching signal σ (t). The state trajectories of the switched reference model are given in Fig. 4, which indicates that the state trajectories of the switched reference model are bounded under the switching signal σ (t). Fig. 5 describes the tracking error of e(t) with e(0 ) = [−2.5, 1.5]. Fig. 5 indicates that the state of the system (1) asymptotically tracks the state of the reference model (64). The state x(t) and its estimate xˆ(t ), and the disturbance d1 (t) and its estimate dˆ1 (t ) are presented in Figs. 6–7, respectively. The parameter estimations θˆ11 (t ), θˆ12 (t ), θˆ21 (t ), θˆ22 (t ) are given in Fig. 8. From Figs. 4–8, we can see that all the signals in the closed-loop system (15) are bounded.

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J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86

Fig. 3. The switching signal σ (t).

Fig. 4. The trajectories of the switched reference model xm (t).

Fig. 5. The trajectory of the tracking error e(t).

Fig. 6. The trajectory of the state x(t) and the state estimation xˆ(t ).

J. Xie, D. Yang and J. Zhao / Information Sciences 485 (2019) 71–86

85

Fig. 7. The trajectory of the disturbance d1 (t) and the disturbance estimation dˆ1 (t ).

Fig. 8. The trajectory of the parameter estimations.

Remark 8. The problem of the proposed model in the sensor or actuator fault is not considered in this paper. While, the literatures [11,12,34] may provide us some possible solutions. 6. Conclusions In this paper, we have studied the CADMRAC problem for the switched systems with parametric uncertainties and multiple types of disturbances. Multiple types of disturbances include the norm-bounded disturbance and the disturbance generated by the exosystem. Under the unmeasurable state and the unmeasurable disturbance generated by the exosystem, by design of the controllers and the switching law, achieve the state tracking and the anti-disturbance performance. Firstly, a switched adaptive state-disturbance observer was constructed to estimate the system state and the disturbance simultaneously. Secondly, the composite switched adaptive controller and the switching law were designed to solve the CADMRAC problem for switched systems. Further, in the case that the CADMRAC problem of each subsystem was not solvable, a solvability condition ensuring the CADMRAC of the switched systems was given. Finally, the CADMRAC of an electrohydraulic system has been studied to illustrate the effectiveness of the proposed control design scheme. In conclusion, the proposed method can cope with more complex situation originating from practice, namely, the system with unavailable state and uncertainties. Meanwhile, the proposed design technique can be applied to more general class of systems. In the further, we will attempt to extend the result to the generalized multiple Lyapunov functions method [41]. Acknowledgments This work was supported by the National Natural Science Foundation of China under grants 61803225, 61773098, the 111 Project(B16009), and the General Project of Scientific Research of the Education Department of Liaoning Province under grant L2016020. References [1] L.W. An, G.H. Yang, Improved adaptive resilient control against sensor and actuator attacks, Inf. Sci. 423 (2018) 145–156. [2] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control 43 (4) (1998) 475–482.

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