H∞ model reference adaptive control for switched systems based on the switched closed-loop reference model

Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

H∞ model reference adaptive control for switched systems based on the switched closed-loop reference model Jing Xie a,b , Jun Zhao a, * a

College of Information Science and Engineering, Northeastern University; State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, China b Department of Mathematics and Physics, Shenyang University of Chemical Technology, Shenyang, 110142, China

article

info

Article history: Received 15 October 2016 Accepted 21 July 2017

Keywords: Switched systems Model reference adaptive control Multiple Lyapunov functions H∞ control The switched closed-loop reference model

a b s t r a c t This paper is concerned with the problem of H∞ state tracking model reference adaptive control for switched systems by the multiple Lyapunov functions method. Neither the measurability of the system state nor the solvability of the H∞ state tracking model reference adaptive control for each individual subsystem is required. First, to improve the transient performance of switched systems, the closed-loop reference model is introduced to switched systems. Second, the H∞ state tracking model reference adaptive control problem for switched systems is solved by designing adaptive controllers for subsystems and a switching law. Then, a solvability condition of the H∞ state tracking model reference adaptive control problem for switched systems is developed. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction A switched system is a dynamical system which comprises a family of subsystems and a logical rule. The subsystems are described by continuous or discrete-time differential equations and the logical rule decides which subsystem is being activated [1]. In the past decades, switched systems have drawn extensive attention mainly because they allow one to describe the behavior of many practical or man-made systems which own inherently some jumping characteristics. In the study of switched systems, stability is a key issue and several approaches have been developed so far, such as the common Lyapunov function method, the single Lyapunov function method, the multiple Lyapunov functions method and so on [2–10]. However, most of the current works are focused on switched systems without uncertainties. Dealing with uncertainties, especially for parametric uncertainties, the adaptive control is an effective method. Meanwhile, the adaptive control for switched systems is a difficult topic. One of the main reasons is that the existence of the parametric uncertainties makes the conventional stability analysis methods mentioned above to be not applicable directly. Therefore, the study of the adaptive control for switched systems is more complicated [11–20]. For pre-given switching signals, by using passivity, in [14,16], the problem of controlling multi-modal piecewise linear affine (PWA) systems was studied by designing an MRAC control law, in which, the switch of the reference model is independent from the plant. The results have been extended to discrete-time PWA plants in [18]. The problem of controlling bimodal piecewise linear affine (PWA) systems was investigated in [17] and the adaptive method presented in [17] was experimentally validated in [19]. Besides, in [20], an output based MRAC approach for piecewise linear systems was proposed. In recent years, model reference control (MRC) has been widely studied. The objective of MRC is to design a controller such that the state or the output of a plant tracks the state or the output of a given reference model as closely as possible.

*

Corresponding author. E-mail addresses: [email protected] (J. Xie), [email protected] (J. Zhao).

http://dx.doi.org/10.1016/j.nahs.2017.07.003 1751-570X/© 2017 Elsevier Ltd. All rights reserved.

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

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For switched systems, the H∞ tracking MRC problem has been addressed [21–25]. In particular, when the state is fully measurable, [21] investigated the problem of exponential H∞ output tracking control for a class of switched neutral systems based on the average dwell time approach. When the state is not available for measurement, observer-based tracking control was proposed for switched linear systems in [24]. However, it is worth noting that in [21–24], the reference input is seen as a part of external disturbances, which is not always reasonable. Therefore, it is necessary to search a more effective method to solve this problem, which partly makes motivation for this study. Although the H∞ tracking MRC problem has been extensively studied, only a few results focus on the H∞ tracking MRC problem for switched systems with parametric uncertainties. [26] considered the H∞ state tracking model reference adaptive control (MRAC) for switched linear systems under the average dwell time method. However, in [26], the system state is assumed to be fully measurable and the H∞ tracking MRAC problem for each individual subsystem is required to be solvable. Obviously, it put forward high demand for each individual subsystem and is usually hard to be satisfied in practice. Thus, a natural question arises: can we still solve the H∞ tracking MRAC problem for the switched systems by designing adaptive controllers and a switching law, when the system state is not fully measurable and the problem for each individual subsystem is not solvable? Obviously, this is a challenge problem which has never been solved in the MRAC. This makes another motivation for this study. On the other hand, it is well known that the restriction of the adaptive control is slow adaptation and poor transient response, especially when there exist large initial estimation errors. For this reason, many researchers have devoted their efforts to improve the transient performance of the adaptive control systems [27–29]. Recently, a class of closed-loop reference model have been proposed to guarantee transient performance, where an observer-like feedback term containing the state error between the reference model and the controlled system is added into the reference model [30–34]. In fact, the transient performance is improved by the introduction of the feedback gain which is designed to shift the eigenvalue of the closed-loop matrix to shape the convergence speed. Therefore, the feedback gain provides one more degree-of-freedom to the adaptive control system and thus has been received growing attention [30–34]. With the closed-loop reference model, [30] and [31] studied the state tracking of MRAC under state measurable and state unmeasurable, respectively. Then, the stability, robustness, transient, as well as oscillations and peaking were analyzed in [32–34]. However, the transient performance of switched adaptive control systems has been rarely addressed, although this is also an important problem. Allowing for this point, in this paper, we construct a switched closed-loop reference model to improve the transient performance of switched systems. In this paper, H∞ tracking MRAC problem for switched systems is investigated. We construct a switched closed-loop reference model and the transient performance can be improved by designing the feedback gain for each individual subsystem. The contributions of this note are as follows: first, for switched systems, the closed-loop reference model is introduced to improve the transient performance. Second, the state is not required to be fully measurable. We only use the output and the state of the closed-loop reference model in the design of controllers and adaptive laws. Finally, we do not assume the problem of H∞ tracking MRAC of any subsystem is to be solvable. We construct a new state by which the switching law is designed and the H∞ tracking MRAC problem for switched systems is still solved, at the same time, the reference input is not required to be regarded as external disturbance. Notation: the notation used in this paper is standard. We use P > 0 (P < 0) to denote a positive-definite (negative definite) matrix P. A−1 represents the inverse of matrix A and AT represents the transpose of matrix A. I and 0 represent identity matrix and zero matrix in a block matrix, respectively. In a matrix, the symmetric terms are denoted by ∗. Rn stands for the n-dimensional Euclidean space and Rn×m is the set of all n × m real matrices. L2n [0, +) is the space of n-dimensional square integrable function vector over [0, +∞]. λmax (B) (λmin (B)) √ is the maximum (minimum) eigenvalue of matrix B. diag

{A, B} represents the block diagonal matrix of A and B. ∥x∥ = √ ∥A∥ = λmax (AAT ): the norm of matrix A.

∑n

i=1

|xi |2 : the norm of a vector x = (x1 , x2 , . . . , xn )T .

2. Problem formulation and preliminaries Consider the switched system x˙ (t) = Aσ x(t) + Bσ Λσ uσ (t) + Eσ ω(t), y(t) = CσT x(t),

(1)

where the plant state x(t) ∈ Rn , the control input u(t) ∈ Rm , an arbitrary external disturbance w (t) ∈ L2n [0, ∞) and the measurement output y ∈ Rp . σ (t) : [0, +∞) → I = {1, 2, . . . , M } is a switching signal which is a piecewise continuous function depending on time or on state or both, and M is the number of subsystems. The matrices Ai , Λi are unknown constant matrices, while Bi , Ci are known constant matrices, and only y is assumed to be available for measurement. The switching signal σ (t) can be characterized by the switching sequence

∑

= {x0 ; (i0 , t0 ), (i1 , t1 ), . . . , (in , tn ), . . . |in ∈ I , n ∈ N },

in which t0 is the initial time, x0 is the initial state, and N is the set of nonnegative integers. When t ∈ [tk , tk+1 ), σ (t) = ik , that is the ik th subsystem is active. Thus, when t ∈ [tk , tk+1 ), the trajectory x(t) of the system (1) is defined as the trajectory xik (t) of the ik th subsystem. It is assumed that only limited number of switches happen in any finite time interval which has been widely used in the study of switched systems [2,3].

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The dynamics of the switched closed-loop reference model are described as follows: x˙ m (t) = Amσ xm (t) + Bσ r(t) + Lσ (ym − y), ym (t) = CσT xm (t),

(2)

where r(t) ∈ Rm is a bounded reference input signal, Li , i ∈ I are feedback gains that will be designed suitably. e(t) = x(t) − xm (t) is the state tracking error. We assume that the output matrices of the plant are known and they can be chosen the same as those of reference model, which have been used in [31]. Remark 1. In the switched MRAC control, the open-loop switched reference model is given by x˙ m0 (t) = Amσ xm0 (t) + Bσ r(t), ym0 (t) = CσT xm0 (t).

(3)

Compared to the switched MRAC [11–14], we introduce an additional observer-like feedback terms Li (y − ym ), i ∈ I in the reference model. We refer to Li as feedback gains. Note that when Li = 0, the open-loop reference model (3) is recovered. It is well known that, in the classical adaptive control, the gain adaption is the only adjustable parameter to shape a better transient performance. In this note, the introduction of the feedback gain Li provides the additional degree of freedom for each subsystem to improve the transient performance. The control objective is to design a switched controller with adaptive laws and a state-dependent switching law when the state of system (1) is not completely measurable to enforce. (i) When w (t) = 0, all the signals in the resulting closed-loop system are bounded and the state tracking error e(t) = x(t) − xm (t) converges to 0. (ii) When w (t) ̸ = 0, under the zero-initial condition, the following inequality holds: ∞

∫

e (τ )e(τ )dτ ≤ γ T

0

2

∞

∫

wT (τ )w(τ )dτ .

(4)

0

To achieve the control objective, the following assumptions are necessary. Assumption 1. The pairs {Ami , CiT } are observable, i ∈ I. Assumption 2. There exist Θi∗ ∈ Rn×m and Ki∗ ∈ Rm×m , such that Ai + Bi Λi Θi∗T = Ami , Λi Ki∗T = I, i ∈ I. Assumption 3. Λi are diagonal with positive elements, i ∈ I. Assumption 4. The uncertain matching parameters Θi∗ , and the control uncertainty matrices Λi have priori known upper bounds, that is,

θ¯i∗ ≜ sup ∥Θi∗ ∥,

λ¯ i ≜ sup ∥Λi ∥, i ∈ I .

(5)

Remark 2. Assumption 1 is necessary, since we use observer like-feedback gains Li in the reference model. Assumption 2 is the well known model matching conditions which have been widely used in MRAC for nonswitched systems [35]. Assumption 3 is about the control uncertainties which is routinely satisfied in the area of aerospace applications, where the control directions are usually known but their magnitudes are not known [36]. Assumption 4 is need for us to appropriate choice of Li , i ∈ I . Before concluding this section, the following lemmas are given which will be used in the later development. Lemma 1 ([37]). 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 M T ≤ γ −1 MQ −1 M T + γ N T QN .

(6)

Lemma 2 ([38]). Let T , M , F (t) and N be real matrices of appropriate dimensions with F T (t)F (t) ≤ I, then for any a scalar ε > 0, it holds that T + MF (t)N + N T F T (t)M T ≤ T + ε MM T + ε −1 N T N .

(7)

3. Main results The objective of this section is to design a switched adaptive controller and a state-dependent switching law such that the problem of H∞ state tracking MRAC for the system (1) is solved with disturbance attenuation level γ > 0. First, a switched

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

95

controller with adaptive laws is designed. Second, a new state is constructed by which a switching law is designed. Then, a solvability condition for H∞ state tracking MRAC problem for the system (1) is given. We propose the following switched adaptive controller uσ (t) = ΘσT (t)xm (t) + KσT (t)r(t),

(8)

{ −ΓΘ i xm (t)eTy (t)Mi , i = σ , ˙ Θi (t) = 0, i ̸= σ , { T −ΓKi r(t)ey (t)Mi , i = σ , K˙ i (t) = 0, i ̸= σ ,

(9) (10)

where Θi (t), Ki (t), i ∈ I are the estimates of Θi∗ , Ki∗ respectively, and ΓΘ i , ΓKi are both positive definite free design matrices, ey (t) = y(t) − ym (t) and Mi ≜ CiT Bi , i ∈ I .

(11)

From (1), (2) and (8), the state error e(t) satisfies the following dynamic.

˜ σT xm (t) + K˜ σT (t)r(t) − Θσ∗T e(t)) e˙ (t) =(Amσ + Lσ CσT )e(t) + Bσ Λσ (Θ + Eσ ω(t),

ye (t) =CσT e(t),

(12)

˜ i (t) = Θi (t) − Θi∗ , K˜ i (t) = Ki (t) − Ki∗ , i ∈ I are parameter estimation errors. where Θ

Using the fact that the measurability of x(t) is not obtained, it follows that e(t) is also not available for measurement. Then, the design of switching law cannot dependent on the state e(t). In order to achieve the control objective, we construct a new state by which a switching law is designed. The new state x¯ m (t) = xm (t) − xm0 (t) satisfies x˙¯ m (t) = Amσ (xm (t) − xm0 (t)) + Lσ (ym (t) − y(t)) = Amσ (xm (t) − xm0 (t)) + Lσ CσT (xm (t) − x(t)) = Amσ x¯ m (t) − Lσ CσT e(t), yx¯ m (t) = CσT x¯ m (t).

(13)

Remark 3. In the H∞ tracking MRC [21–24], the reference input r(t) has to be taken as part of external disturbance. This is often unreasonable since the reference input r(t) is usually the control input for the reference model. Here the introduction of the state x¯ m (t) enables us to excluded the reference input r(t) for the whole disturbance. From (12) and (13), the augmented system is described by

˜¯ T (t) x (t) + K˜¯ T (t)r(t) − Θ ¯ σ (Θ ¯ ∗T X (t)) X˙ (t) = A¯ σ X (t) + B¯ σ Λ σ m σ ¯ + Eσ w(t), Y (t) = C¯ σT X (t),

(14)

where X (t) =

¯i = Λ

[

¯ i∗T = Θ

[

]

I 0

]

x¯ m (t) e(t)

[

0

Λi 0 0

, A¯ i =

[

−Li CiT , Ami + Li CiT

]

Ami 0

[

]

B¯ i =

[

[

]

Bi 0

0 Bi

˜¯ T (t) = T , K˜¯ T (t) = T , E¯ = ,Θ i ˜ (t) i K˜ i (t) Θ i

0

Θi∗T

]

, C¯ iT =

0

[

CiT 0

0 CiT

]

0

, Y (t) =

[

yx¯ m (t) ye (t)

]

]

,

[ ] 0 Ei

,

, i ∈ I.

The following theorem presents a sufficient condition to ensure the solvability of H∞ state tracking MRAC problem for the system (1). Theorem 1. For given constants λij ≥ 0, ρi ≥ ρi∗ , i, j ∈ I, if there exists constant γ > 0, positive definite matrices P11i , P22 , Q11i , Q22 , and matrices Li , i ∈ I, such that

⎡ Σ1i ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗ ∗

−P11i Li CiT Σ2i ∗ ∗ ∗ ∗ P22 Bi = Ci Mi , λ¯ 2i θ¯i∗2 ρi∗ = 2λmin (Q22 )

I 0 −1 −Q11i

∗ ∗ ∗

0 P22 Ei 0 −γ 2 I

∗ ∗

0 I 0 0 −1 −Q22

∗

0 Ci Mi ⎥ ⎥ 0 ⎥ ⎥ ≤ 0, 0 ⎥ ⎦ 0 −1 −(2ρi ) I

⎤

(15)

(16) (17)

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J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

hold, and the state xm0 (t) of open-loop reference model (3) is bounded under the switching law

σ (t) = argmin X T (t)P¯ i X (t),

(18)

i∈I

then, the problem of H∞ state tracking MRAC of the system (1) is solved by the adaptive controller (8)–(10) and the switching law (18), where m ∑

Σ1i = ATmi P11i + P11i Ami +

λij (P11j − P11i ),

j=1

Σ2i = (Ami + Li CiT )T P22 + P22 (Ami + Li CiT ) + I , [ ] P 0 P¯ i = 11i . 0

P22

Proof. For the augmented system (14), we construct the following Lyapunov functions: Vi (Z (t)) =X T (t)P¯ i X (t) +

M ∑

˜¯ (t)] ˜¯ T (t)Γ −1 Θ ¯ lΘ tr [Λ l l Θl

l=1

+

M ∑

¯ l K˜¯ Tl (t)ΓKl−1 K˜¯ l (t)], tr [Λ

(19)

l=1 2

T T T T T T T T ˜ 11 ˜ 1m ˜ 21 ˜ Mm (t)]T ∈ Rn(Mm+1)+Mm . (t), . . . , K˜ Mm (t), K˜ 11 (t), . . . , K˜ 1m (t), K˜ 21 where Z (t) = [X (t)T , Θ (t), . . . , Θ (t), Θ (t), . . . , Θ Differentiating Vi (Z (t)) along the trajectory of the ith subsystem of the augmented system (14) gives

˜¯ T (t)x (t) ¯ iΘ V˙ i (Z (t)) =X T (t)[A¯ Ti P¯ i + P¯ i A¯ ]X (t) + 2X T (t)P¯ i B¯ i Λ m i ¯ i K˜¯ Ti (t)r(t) + 2X T (t)P¯ i E¯ i ω(t) + 2X T (t)P¯ i B¯ i Λ ˙˜¯ ¯ ∗i T X (t) + 2tr [Λ ¯ iΘ ¯˜ Ti (t)ΓΘ−i1 Θ ¯ iΘ − 2X T (t)P¯ i B¯ i Λ i (t)] ˙ ˜ − 1 ˜ T ¯ i K¯ i (t)ΓKi K¯ i (t)]. + 2tr [Λ

(20)

Considering the fact that aT b = tr [baT ] holds for any vectors a and b, we obtain

˜¯ T (t)x (t) = 2tr [Λ ˜¯ T (t)x (t)X T (t)P¯ B¯ ], ¯ iΘ ¯ iΘ 2X T (t)P¯ i B¯ i Λ m m i i i i ¯ i K˜¯ Ti (t)r(t) = 2tr [Λ ¯ i K˜¯ Ti (t)r(t)X T (t)P¯ i B¯ i ]. 2X T (t)P¯ i B¯ i Λ

(21)

˙˜¯ ˙˜¯ ¯˙ ¯˙ Then, it follows from Θ i (t) = Θ (t), K i (t) = K (t), i ∈ I that (20) can be rewritten as ˜¯ T (t)x (t)X T (t)P¯ B¯ ] ¯ iΘ V˙ i (Z (t)) =X T (t)[A¯ Ti P¯ i + P¯ i A¯ ]X (t) + 2tr [Λ m i i i T

¯ i K˜¯ i (t)r(t)X T (t)P¯ i B¯ i ] + 2X T (t)P¯ i E¯ i ω(t) + 2tr [Λ ¯ iΘ ¯ i∗T X (t) + 2tr [Λ ¯ iΘ ¯˜ Ti (t)ΓΘ−i1 Θ ¯˙ i (t)] − 2X T (t)P¯ i B¯ i Λ ¯ K˜¯ T (t)Γ −1 K˙¯ (t)]. + 2tr [Λ i

i

Ki

(22)

i

From (9)–(11) and (16), one has

˙¯ (t) = [0 Θ i [ = 0 [ K˙¯ i (t) = 0 [ = 0

] [ −ΓΘ i xm (t)eTy (t)Mi = 0 ] −ΓΘ i xm (t)eT (t)P22 Bi , ] [ −ΓKi r(t)eTy (t)Mi = 0 ] −ΓKi r(t)eT (t)P22 Bi .

−ΓΘ i xm (t)eT (t)Ci Mi − ΓKi r(t)eT (t)Ci Mi

]

] (23)

This leads to

˜¯ T (t)Γ −1 Θ ˙¯ (t)] ¯ iΘ tr [Λ i i Θi

{[

I 0

= tr

[ = tr

0 0

0

Λi

][

]

[ 0 −1 ˜ iT (t) ΓΘ i 0 Θ 0

−ΓΘ i xm (t)e (t)P22 Bi ]

˜ iT (t)xm (t)eT (t)P22 Bi −Λi Θ T ˜ i (t)xm (t)eT (t)P22 Bi ]. = −tr [Λi Θ

T

]

}

(24)

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

97

In addition,

˜¯ T (t)x (t)X T (t)P¯ B¯ ] ¯ iΘ tr [Λ m i i i

{[ = tr

I 0

0

][

Λi

]

[

]T [

0 x¯ m (t) ˜ iT (t) xm (t) eT (t) Θ

[

0 = tr ˜ iT (t)xm (t)x¯ Tm (t)P11i Bi Λi Θ

P11i 0

0 P22

][

Bi 0

0 ˜ iT (t)xm (t)eT (t)P22 Bi Λi Θ

˜ iT (t)xm (t)eT (t)P22 Bi ]. = tr [Λi Θ

0 Bi

]}

] (25)

From (24) and (25), we have T

˙¯ (t)] = −tr [Λ ˜¯ (t)x (t)X T (t)P¯ B¯ ]. ˜¯ T (t)Γ −1 Θ ¯ iΘ ¯ iΘ tr [Λ i m i i i i Θi

(26)

Similarly,

¯ i K˜¯ Ti (t)r(t)X T (t)P¯ i B¯ i ]. ¯ i K˜¯ Ti (t)ΓKi−1 K˙¯ i (t)] = −tr [Λ tr [Λ

(27)

Substituting (26) and (27) into (23) gives V˙ i (Z (t)) =X T (t)[A¯ Ti P¯ i + P¯ i A¯ ]X (t) + 2X T (t)P¯ i E¯ i ω(t)

¯ i∗T X (t). ¯ iΘ − 2X T (t)P¯ i B¯ i Λ

(28)

Next, we prove that when ω(t) = 0, all the signals in the resulting closed-loop system are bounded and the state tracking error e(t) = x(t) − xm (t) converges to 0. When ω(t) = 0, from (28), the time derivative of Vi (Z (t)) along the trajectory of the ith subsystem of the augmented system (14) turns into

¯ i∗T ]X (t) ¯ iΘ V˙ i (Z (t)) = X T (t)[A¯ Ti P¯ i + P¯ i A¯ − 2P¯ i B¯ i Λ = X T (t)

[

ATmi P11i + P11i Ami

∗

] −P11i Li CiT X (t) Ψ1i

(29)

with Ψ1i = (Ami + Li CiT )T P22 + P22 (Ami + Li CiT ) − 2P22 Bi Λi Θi∗T . It follows from (15) and the Schur complement lemma that

Φ11i

[

] −P11i Li CiT ≤0 Φ22i

−Ci LTi P11i

(30)

hold with

Φ11i =ATmi P11i + P11i Ami +

m ∑

λij (P11j − P11i ) + Q11i ,

j=1

Φ22i =(Ami + Li CiT )T P22 + P22 (Ami + Li CiT ) + I

+ γ −2 P22 Ei EiT P22 + 2ρi Ci Mi MiT CiT + Q22 .

(31)

The switching law (18) is equivalent to

σ (t) = argmin{¯xTm (t)P11i x¯ m (t) + eT (t)P22 e(t)}, i∈I

= argmin{¯xTm (t)P11i x¯ m (t)}. i∈I

(32)

Using (30) and the switching law (32) results in V˙ i (Z (t)) ≤ X T (t)

[

−Q11i ∗

0

Ψ2i

] X (t)

(33)

with Ψ2i = −I − γ −2 P22 Ei EiT P22 − 2ρi Ci Mi MiT CiT − Q22 − 2P22 Bi Λi Θi∗T . Note that X (t) = [¯xm (t)T e(t)T ]T and P22 Bi = Ci Mi , ey (t) = CiT e(t), it follows that

− 2ρi eT (t)Ci Mi MiT CiT e(t) = −2ρi eTy (t)Mi MiT ey (t), − 2eT (t)P22 Bi Λi Θi∗T e(t) = −2eT (t)Ci Mi Λi Θi∗T e(t) = −2ey (t)T Mi Λi Θi∗T e(t).

(34)

98

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

Then, (33) can be rewritten as V˙ i (Z (t)) xTm (t)

[

≤ ¯

< ¯

xTm (t)

[

[ = x¯ Tm (t)

e (t)

⎡ −Q11i ] ⎢ eTy (t) ⎣ ∗ ∗ ⎡ −Q11i ] eTy (t) ⎣ ∗

eT (t)

∗ ∗ [ ] ¯ x (t) m ] eTy (t) ∆i e(t) ,

T

e (t)

T

0

⎤

0

Θi Λi MiT 2 i Mi MiT

Υi

−

∗

− ρ

0 −Q22

∗T

[

⎥ ⎦

0

−

Θi Λi MiT 2 i Mi MiT

x¯ m (t) e(t) ey (t)

⎤[

x¯ m (t) e(t) ey (t)

∗T

− ρ

]

⎦

]

(35)

ey (t) where

Υi = −I − γ −2 P22 Ei EiT P22 − Q22 , ⎡ ⎤ −Q11i 0 0 ∗T T −Q22 −Θi Λi Mi ⎦ . ∆i = ⎣ ∗

∗

−2ρi Mi MiT

∗

Given ρi > ρi∗ > 0, from (17), −2ρi Mi MiT + Θi∗T Λi MiT Q22 Mi Λi Θi < 0 hold. By Schur complement lemma, ∆i are negative definite. Thus V˙ i (Z (t)) < 0. Moreover, the ‘‘min-switching’’ strategy (32) ensues that the adjacent Vi (Z (t)) are connected at

˜¯ (t), K˜¯ (t) ∈ L . switching points. Thus, Vσ (Z (t)) is continuous and decreasing with respect to time t, which implies X (t), Θ i i ∞ ˙ (t) ∈ L∞ . Moreover, Then, we have x¯ m (t), e(t), ey (t), Θi (t), Ki (t) ∈ L∞ . From (12) and (13),⋂ it follows that x¯ m (t) ∈ L∞ and e⋂ L∞ , x˙¯ m (t) ∈ L∞ . By L∞ , e˙ (t) ∈ L∞ , and x¯ m (t) ∈ L2 x¯ m (t), e(t) ∈ L2 , due to ∆i are negative definite in (35). Thus, e(t) ∈ L2 Barbalat lemma, limt →∞ e(t) = 0, limt →∞ x¯ m (t) = 0 holds. By the fact that e(t) = x(t) − xm (t), x¯ m (t) = xm (t) − xm0 (t), the boundedness of x(t) and xm (t) follows since xm0 (t) is bounded. Therefore, we conclude that all the signals in the closed-loop system (2), (9)–(10) and (14) are bounded and the state tracking error e(t) = x(t) − xm (t) converges to 0 when w (t) = 0. Finally, for any nonzero w (t) ∈ L2n [0, ∞), we prove that ∞

∫

eT (τ )e(τ )dτ ≤ γ 2

∞

∫

wT (τ )w(τ )dτ . 0

0

By Lemma 1, 2xT (t)P¯ i E¯ i ω(t) ≤ γ −2 xT (t)P¯ i E¯ i E¯ iT P¯ i X (t) + γ 2 ωT (t)ω(t), then, the time derivative of Vi (Z (t)) in (28) become

¯ i∗T ]X (t) + γ 2 ωT ω ¯ iΘ V˙ i (Z (t)) ≤ X T (t)[A¯ Ti P¯ i + P¯ i A¯ + γ −2 P¯ i E¯ i E¯ iT P¯ i − 2P¯ i B¯ i Λ = X T (t)

[

] −P11i Li CiT X (t) + γ 2 ωT ω, Ψ3i

ATmi P11i + P11i Ami

∗

(36)

where Ψ3i = (Ami + Li CiT )T P22 + P22 (Ami + Li CiT ) + γ −2 Pi Ei EiT Pi − 2P22 Bi Λi Θi∗T . By (30) and the switching law (32), it follows that

[ ] −Q11i ∗ [ ] −Q11i e(t) ∗

V˙ i (Z (t)) ≤ x¯ m (t)

[

e(t)

[

= x¯ m (t)

0

][

x¯ m (t) + γ 2 ωT (t)ω(t) e(t)

0

][

x¯ m (t) − eT (t)e(t) e(t)

Ψ4i Ψ5i

]

]

+ γ ω (t)ω(t), 2

T

(37)

where Ψ4i = −I − 2ρ − Q22 − 2P22 Bi Λi Θi , Ψ5i = −2ρ Ci Mi , ey (t) = CiT e(t), by (34), we have T T i Ci Mi Mi Ci

∗T

T T i Ci Mi Mi Ci

− Q22 − 2P22 Bi Λi Θi . Note that P22 Bi = ∗T

V˙ i (Z (t)) + e(t)T e(t) − γ 2 ω(t)T ω(t) xTm (t)

[

≤ ¯

[ = x¯ Tm (t)

T

e (t)

eT (t)

⎡ −Q11i 0 T ⎣ ∗ − Q 22 ey (t) ∗ ∗ [ ] x¯ m (t) ] T ¯ ey (t) ∆i e(t) , ]

ey (t)

0

−

Θi Λi MiT 2 i Mi MiT

⎤[

∗T

− ρ

⎦

x¯ m (t) e(t) ey (t)

]

(38)

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

99

¯ i are negative definite by the Schur complement lemma. Therefore, we have when ρi > ρi∗ , ∆ V˙ i (Z (t)) + eT (t)e(t) − γ 2 ωT (t)ω(t) < 0.

(39)

Under the zero initial condition, integrating both sides of the inequality (39), we get t

∫

e(τ )T e(τ )dτ − γ 2

V (Z (t)) − V (0) +

∫

0

t

ωT (τ )ω(τ )dτ < 0.

(40)

0

Since V (Z (t)) > 0 and V (0) = 0, it follows that t

∫

t

∫

e(τ )T e(τ )dτ < γ 2

ωT (τ )ω(τ )dτ . 0

0

Besides, due to external disturbance w (t) ∈ L2n [0, ∞), which implies ∞

∫

e(τ )T e(τ )dτ ≤ γ 2

∞

∫

ωT (τ )ω(τ )dτ 0

0

holds as t → ∞. This completes the proof. It is clearly seen that the condition (15) in Theorem 1 are bilinear matrix inequalities due to the existence of the terms P11i Li CiT and P22 Li CiT . In order to make the computation easier, we give the following theorem. Theorem 2. For given constants λij ≥ 0, ρi ≥ ρi∗ , εi > 0, i, j ∈ I, if there exists constant γ > 0, positive definite matrices Xi , Q11i , P22 , Q22 , and matrices Li , Yi , i ∈ I, such that

⎡

⎤

M ∑

λij Xi Xi Li CiT ⎢Xi ATmi + Ami Xi − ⎢ j=1 ⎢ ⎢ −1 ∗ −Q11i 0 ⎢ ⎣ ∗ ∗ −εi I ∗ ∗ ∗ ⎡ ⎤ Π2i P22 Ei Ci Mi I ⎥ ⎢ ∗ −γ 2 I 0 0 ⎢ ⎥ ≤ 0, −1 ⎣ ∗ ∗ −(2ρi ) I 0 ⎦ −1 ∗ ∗ ∗ −Q22 P22 Bi = Ci Mi , λ¯ 2i θ¯i∗2 ρ∗ = 2λmin (Q22 )

Π1i ⎥ ⎥ ⎥ ⎥ ≤ 0, 0 ⎥ 0 ⎦

(41)

−Ri (42)

(43) (44)

hold, then the problem of H∞ state tracking MRAC for the system (1) is solved by the adaptive controller (8)–(10) and the switching law (18), where

√ √ √ √ √ Π1i = [ λi1 Xi , λi2 Xi , . . . , λii−1 Xi , λii+1 Xi , . . . , λiM Xi ], Π2i = ATmi P22 + P22 Ami + Ci YiT + Yi CiT + I + εi I , Ri = diag {Xi1 , Xi1 , . . . , Xii−1 , Xii+1 , . . . , XiM }. −1 Moreover, the feedback gains Li = P22 Yi .

Proof. If the conditions (15)–(16) are satisfied, the problem of H∞ state tracking MRAC for the system (1) is solved by Theorem 1. Applying Schur complement lemma and the switching law (32) to (15) immediately gives (30). It is not difficult to know that (30) can be rewritten as

[ Φ11i 0

0

Φ22i

]

[ +

−P11i Li CiT

0 0

]

[ +

0

0 −Ci LTi P11i

]

0 ≤0 0

(45)

with Φ11i , Φ22i are given in (31). Due to

[

−P11i Li CiT

0 0

] =

0

[ −P11i Li CiT 0

][

0 0

I 0

][

0 0

0 0

]

I , 0

(46)

then, based on Lemma 2, for any εi > 0, i ∈ I such that

[

0 0

−P11i Li CiT

≤ εi−1

0

[

]

[

0 + −Ci LTi P11i

P11i Li CiT Ci LTi P11i 0

]

]

0 0

(47)

[

0 0 + εi 0 0

]

0 . I

(48)

100

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

Substituting (47) into (45) leads to

[ Ξ11i 0

0

]

Ξ22i

≤ 0,

(49)

where

Ξ11i =ATmi P11i + P11i Ami +

m ∑

λij (P11j − P11i ) + Q11i + εi−1 P11i Li CiT Ci LTi P11i ,

j=1

Ξ22i =(Ami +

+ 2ρ

Li CiT )T P22 T T i Ci Mi Mi Ci

+ P22 (Ami + Li CiT ) + I + γ −2 P22 Ei EiT P22 + Q22 + εi I .

Obviously, the condition (49) implies Ξ11i ≤ 0 and Ξ22i ≤ 0. According to the Schur complement lemma, Ξ11i ≤ 0 turns into

⎡

ℶi

⎣∗ ∗

⎤

P11i Li CiT 0 ⎦≤0 −εi I

I −1 −Q11i

∗

with ℶi = ATmi P11i + P11i Ami +

⎡

P22 Ei

⎢∗ ⎢ ⎣∗ ∗

−γ 2 I ∗ ∗

ℶ1i

∑m

(50)

λ

j=1 ij (P11j

Ci Mi 0 (−2ρi )−1 I

∗

− P11i ). Similarly, Ξ22i ≤ 0 turns into

⎤

I 0 ⎥ ⎥≤0 0 ⎦ −1 −Q22

(51)

−1 , I , I } from both sides gives with ℶ1i = (Ami + Li CiT )T P22 + P22 (Ami + Li CiT ) + I + εi I. Multiplying (50) by diag {P11i

⎡

ℶ2i

⎣∗ ∗

⎤

−1 P11i −1 −Q11i

Li CiT 0 ⎦≤0 −εi I

∗

(52)

∑M

−1 T −1 −1 −1 −1 −1 P11j P11i − P11i ). Let Xi = P11i . By Schur complement lemma, we conclude that the with ℶ2i = P11i Ami + Ami P11i + j=1 λij (P11i condition (41) holds. In addition, let Yi = P22 Li . It follows from (51) that the condition (42) holds. This completes the proof.

Remark 4. Note that LMI (41) depends on the solution of Li in the LMI (42). Therefore, to get the solution, we should solve LMI (42) to get Li firstly, then, enter Li in the LMI (41). Remark 5. From Theorem 1, we have the state tracking error e(t) converges to 0, which indicates that the state xm (t) of the closed-loop reference model (2) converges to the state xm0 (t) of the open-loop reference model (3). Thus, the state x(t) of the system (1) converges to the state xm0 (t) of the open-loop reference model (3) eventually. Remark 6. Often for the adaptive control of switched systems, the switching mechanism is given [14,16–19]. While, for a given switching law, sometimes, a switched system cannot achieve the control objective by designing the controllers only. However, by the dual design of the controllers and an appropriate switching law, the switched system can achieve the control objective. The same situation arises for the study of adaptive control of switched systems. Thus, we construct the switching law (18) which is part of the control design. Remark 7. The switching law (32) can be easily rewritten as

σ (t) = argmin i∈I

where ξm =

x¯ m , ∥¯xm ∥

x¯ Tm P¯ i x¯ m

∥¯xm ∥2

= argmin ξmT P¯ i ξm , i∈I

which implies that the switching law only depends on the direction of x¯ m and Pi , i ∈ I .

Remark 8. The feedback gains Li and adaptive gains ΓΘi , ΓKi make it possible to simultaneously decrease the state tracking error when the subsystem is active. For convenience, choose ΓΘi = ΓKi = Γi , Γi = γi In×n , Ami + Li CiT = gi In×n , where γi > 0, gi < 0. Using the method in [32], for the ith subsystem on the active time interval [ti , ti+1 ), it follows from (35) that

∫ ti

ti+1

˜¯ ∥2 (t ) + ∥K¯˜ ∥2 (t )) λmax (P¯ i )X 2 (ti ) + λ¯ i λmax (Γi−1 )(∥Θ i F i i F i |gi | ¯˜ i ∥2F (ti ) + ∥K¯˜ i ∥2F (ti )) λmax (Pi )X 2 (ti ) λ¯ i (∥Θ = + . |gi | |gi |γi

∥e∥2 dτ ≤

(53)

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

101

Fig. 1. The electrohydraulic system. LVDT, linear variable differential transformer.

We now have two independent parameters |gi | and γi to adjust to simultaneously reduce the two terms on the right hand side of (53). Without |gi |, the first term cannot be changed. Note that |gi | = ∥Ami + Li CiT ∥. Therefore, the feedback gain Li provides additional degree of freedom to reduce the L2 norm of the tracking error between two consecutive terms, which means that a quicker tracking is obtained.

4. Example In this section, we apply the proposed MRAC approach to an electro-hydraulic system which has been studied in [39,40]. 4.1. System description As shown in Fig. 1, the electro-hydraulic system consists of a MOOG 760 torque motor/flapper operated four-way doubleacting servovalve, a hydraulic pump, a single rod actuator, a hydraulic arm and two accumulators. The cylinder has a diameter of 32 mm with its stroke of approximately 100 mm. The end of the hydraulic arm is connected to an inertial load representing an aircraft control surface. The plant input u is the control voltage imposing on the torque motor current amplifier, and the state x = [x1 , x2 ], where x1 is a linear variable differential transducer (LVDT) measure of the arm displacement. x2 is the rate of the linear variable differential transducer (LVDT) measure of the arm displacement. In this paper, we select two operating conditions with respect to two different supply pressures, 11.0 MPa (Condition 1) and 1.4 MPa (Condition 2). According to [39,40], the dynamic of the system satisfies Condition 1:

[ x˙ (t) =

0 0

]

[

]

]

[

]

1 0 x(t) + u(t), −4.58 62.4

y(t) = [1, 1]x(t). Condition 2:

[ x˙ (t) =

0 0

1 0 x(t) + u(t), −9.19 47.2

y(t) = [1, 1]x(t).

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Practically, servo-hydraulic systems may encounter an increase/decrease in supply pressure, following decreased/ increased demands from other services. Then, the electro-hydraulic actuator plant will be subjected to largely and nonsmoothly varying dynamics relating to perturbed supply pressure. Besides, variable load and switchable supply pressure provided by dual pressure pump will also induce similar dynamical variation. Therefore, it is of benefit to model the system in a switched system. Here, the switched system contains two subsystems which are corresponding to two operating conditions. Thus, we have the following switched system: x˙ (t) = Aσ x(t) + Bσ Λσ uσ (t) + Eσ ω(t), y(t) = Cσ x(t), σ (t) : [0, +∞] → I = [1, 2],

(54)

where

[ A1 =

]

0 0

[ ]

1 , 1

C1 = E2 =

[

1 0 , A2 = −4.58 0

[ ]

1 , 1

C2 =

[ ] −0.02 , 0.01

]

[

]

[

]

1 0 0 , B1 = , B2 = , −9.19 124.8 94.4

E1 =

[ ] −0.1 , 0.02

Λ1 = Λ2 = 0.5.

The external disturbance input ω(t) = sin(t) exp(−0.01t). 4.2. Simulation results Our purpose is to design the switched adaptive controllers in the form of (8)–(10) and the switching law (18) such that the state of the system (54) tracks the state of the reference model: x˙ m (t) = Amσ x(t) + Bσ r(t), ym (t) = Cmσ xm (t), σ (t) : [0, +∞] → I = [1, 2], where

[ Am1 =

0 −15

]

[

1 0 , Am2 = −8 −27

]

[ ]

1 1 , Cm1 = , −12 1

[ ] Cm2 =

1 , 1

where the reference input r(t) = 1/2 sin π t. From Assumption 2, we obtain

[ Θ1∗ = −0.4808,

]T −0.1096 ,

[ Θ2∗ = −1.1441,

]T −0.1191 .

In order to show how the feedback gains Li influence the transient performance, we consider three cases of parameters. Case 1: 0.8878 L1 = , −80.84

[

]

1.7777 L2 = , −57.0545

[

]

λ11 = 0, λ12 = 0.8, λ21 = 0.8, λ22 = 0, ϵ1 = ϵ2 = 7, ρ1 = ρ2 = 0.023, γ = 11.7248. Case 2:

[

]

1 L1 = , −64627

[

]

2 L2 = , −55444

λ11 = 0, λ12 = 0.8, λ21 = 0.8, λ22 = 0, ϵ1 = ϵ2 = 1, ρ1 = ρ2 = 5, γ = 489.1473. Obviously, (41)–(43) are satisfied for both cases and the H∞ state tracking MRAC problem for the system (1) is solved. To compare with the standard MRAC, we also consider Case 3: L1 = L2 = [0, 0]T . In the simulation, the common adaptive gains are chosen for the three cases:

ΓΘ1 = ΓΘ2 =

[

10 0

]

0 , Γk1 = Γk2 = 10. 10

The simulation results are depicted in Figs. 2–5. Fig. 2 shows that the boundedness of the trajectory of the switched reference model. The switching signal, the trajectories of the state tracking error and the parameter estimates are depicted in Figs. 3–5 for the three cases, respectively. It can be seen from Figs. 3–5 that for all the three cases all the state of the system (54) can track the state of the reference model, and the parameter estimates are bounded. However, they present different transient performances. In Fig. 5, for Case 3, as L1 = L2 = [0, 0]T , the closed-loop reference model degenerates into the open-loop reference model. In this case, although the state of the plant tracks the state of the reference model, the system response suffers the high frequency

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

Fig. 2. The state trajectories of the switched reference model.

Fig. 3. The switching signal, the trajectories of the state tracking error and the parameter estimates for Case 1.

103

104

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

Fig. 4. The switching signal, the trajectories of the state tracking error and the parameter estimates for Case 2.

oscillations. Meanwhile, the high frequency oscillations greatly deteriorate the transient performance and even result in system instability. Taking this into account, we introduce the Li gains to suppress the high frequency oscillations. When introducing the feedback gains Li , the system responses are improved as illustrated in Figs. 3 and 4. Figs. 3 and 4indicate that a larger choice of L1 , and L2 produces lower frequency oscillations in the system response, but the system response presents a slower convergence rate and a faster switching between the subsystems. In contrast, a smaller choice of L1 , and L2 gives higher frequency oscillations, a faster convergence rate, and a slower switching between the subsystems. Thus, a trade-off should be needed to have acceptable frequency of oscillations, switching frequency between subsystems and the convergence rate. This can be done by choosing the appropriate values of Li . 5. Conclusions This paper has studied the H∞ state tracking MRAC problem for switched systems. First, in order to improve the transient performance, we have introduced the closed-loop switched reference model to switched systems. Second, when the system state is not fully measurable and the H∞ state tracking MRAC problem for each individual subsystem is not solvable,

J. Xie, J. Zhao / Nonlinear Analysis: Hybrid Systems 27 (2018) 92–106

105

Fig. 5. The switching signal, the trajectories of the state tracking error and the parameter estimates for Case 3.

the problem for switched systems is still solved by designing adaptive controllers for subsystems and a switching law. The designed adaptive controllers only depend on the state of the reference model and the measurable output error. The switching law depends on a new constructed state. Then, a solvability condition of the H∞ state tracking MRAC of switched systems has been given. Finally, an application example is provided to demonstrate the effectiveness of the main results.

Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant 61773098 and the 111 Project (B16009). Moreover, the General Project of Scientific Research of the Education Department of Liaoning Province supported this work under Grant L2016020.

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