1 Introduction

Switched systems, an important class of hybrid systems, are often encountered in many practical situations. Normally, the switched system is defined as dynamical system which is comprised by series of continuous-time or discrete-time subsystems and a logical rule that decides the switching mechanism among the individual subsystems^[1]. In the past two decades, more and more researchers and experts have been drawn by switched systems because of a number of applications and the theoretical importance. It is well known that the stability problem is one of key issues of studying switched systems. In order to study the stability of switched systems, some approaches have been proposed; for example, the common Lyapunov function method, dwell time or average dwell time method, multiple Lyapunov function method^[2-15]. Among them, the common Lyapunov function method is a particular method which only finds a common Lyapunov function that ensures stability of the whole switched systems and switching signals is arbitrary^[2-4]. This method is usually used when we cannot know the switching signal or it is too complex to apply directly in studying the stability problem.

In the real world, many control systems often company with uncertainties including parametric, structural and environmental uncertainties^{[9, 12]}. Such uncertainties often appear in aircraft engine, mechanical systems, chemical engineering and so on. The existence of uncertainties makes the study of system stability more complicated. As is well known, adaptive control is a powerful tool to deal with uncertainties, which provides adaptive control laws to real-time adjust controllers or system parameters for obtaining desired system performance^[16]. From a perspective of practicality, the adaptive control can be classified into three types: model reference adaptive control (MRAC), self-tuning control and other adaptive control^[17]. Among them, MRAC is an extensively used method and has been matured theoretically^[18-22]. The purpose of MRAC is to make the state (or output) of the controlled system track the state (or output) of a preset reference model by designing the controller and adaptive law. In the last decades, there are many results in this field for non-switched systems^[18-22].

However, few results on MRAC for switched systems with parametric uncertainties have been reported so far.Chiang and Fu^[23] studied a class of signal-input-signal-output switched systems by means of a variable structure MRAC method, while the requirement limits the design flexibility which the reference model is common for all subsystems. When the state of the system is fully measurable, Liu and Xiang^[24] solved the exponential H_∞ output tracking control problem for a class of uncertain switched neutral systems. They presented a concurrent learning model reference adaptive control architecture for this switched systems and the problem of closed loop parameter estimation was solved in the paper^[25].Based on the minimal control synthesis algorithm, Bernardo, Montanaro and Santini^[26] presented a generic MRAC scheme so that the state of a multimodal piecewise affine system can track asymptotically a given multimodal piecewise affine (or smooth) reference model. But, what needs to point out is that all the above results require the state of controlled system being measurable while it is not reasonable for many practical systems. Therefore, how to deal with the MRAC problem for switched systems which have parametric uncertainties is a practical problem when the state of the system is immeasurable, which is a motivation for this paper.

On the other hand, as we all know, what restricts the application of MRAC is the poor transient performance. So it must do something to improve MRAC. Recently, several researchers proposed a new class of closed-loop reference model in which the reference model has an observer-like feedback term containing the state error and a feedback gain^[18-22]. Because of the introduction of the feedback term, the transient performance can be improved by adjusting the feedback gain. In the paper, Gibson, Annaswamy and Lavertsky^[18] analyzed how the closed-loop reference model improved the transient performance of MRAC and compared the transient performance of the closed-loop reference model adaptive control system with the classical open-loop reference model control system. Lately, Zheng^[22] studied the MRAC state tracking problem with the closed-loop reference model when the state is not available for measurement. However, only non-switched systems are studied. A natural problem arises: how to improve the transient performance of switched systems. Clearly, this is a challenge problem for the switched systems. This is another motivation for this paper.

This paper studies the MRAC problem for switched systems with parametric uncertainties under arbitrary switching while it is not required the state of the controlled system should be fully measurable. In this paper, the contributions are as follows: first of all, we only use the output and the state of the given closed-loop reference model to design controllers and adaptive laws because the requirement that the controlled system state is fully measurable for is hard to achieve in many practical systems. Secondly, all reference models and controllers for subsystems are not common which remarkably add the flexibility of designing. Finally, the introduction of the closed-loop reference model can improve the transient performance of the switched system.

The organization of this paper is as follows. Section 2 gives the problem statement and preliminaries.The design of controllers and adaptive laws and the stability theorem of switched error system are in Section 3.A numerical Simulink example is established to prove the correctness of this theory in Section 4. Section 5 is the conclusion.

Notation:   The standard Euclidean or the induced matrix two-norm is denoted as ‖·‖. The positive-definite matrix is P >0. A^-1 represents the inverse of a matrix A and A^T represents the transpose of a matrix A. The smallest eigenvalue of matrix A is defined as λ_min(A) and the trace of square matrix A is defined as tr(A). We use I and 0 to denote identity matrix and zero matrix in a block matrix respectively.

2 System Statement and Preliminaries

The switched system with parametric uncertainties in this paper can be described as:

$ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot x}} = {\mathit{\boldsymbol{A}}_\sigma }\mathit{\boldsymbol{x}} + {\mathit{\boldsymbol{B}}_\sigma }{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_\sigma }\mathit{\boldsymbol{u}}}\\ {\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{C}}_\sigma ^{\rm{T}}\mathit{\boldsymbol{x}}} \end{array}} \right. $

(1)

where x∈Rⁿ is the plant vector, the control input is denoted as u∈R^m, y∈R^m denotes the plant output. σ(t):[0, +∞)→I={1, 2, …, N} denotes switching signal of this switched systems and the quantity of the subsystems is N.B_i∈R^n×m and C_i∈R^n×m are known constant matrices while A_i∈R^n×n and Λ_i∈R^m×m are unknown, and only y is measurable.

The closed-loop switched reference model is:

$ \left\{\begin{array}{l}{\dot{\boldsymbol{x}}_{m}=\boldsymbol{A}_{m \sigma} \boldsymbol{x}_{m}+\boldsymbol{B}_{\sigma} \boldsymbol{r}-\boldsymbol{L}_{\sigma}\left(\boldsymbol{y}-\boldsymbol{y}_{m}\right)} \\ {\boldsymbol{y}_{m}=\boldsymbol{C}_{\sigma}^{\mathrm{T}} \boldsymbol{x}_{m}}\end{array}\right. $

(2)

where r∈R^m is the input signal of reference model and L_i∈R^n×m are feedback gains that will be designed suitably. e=x-x_m denotes the state error. There is one-to-one correspondence between the subsystems of the reference model and the controlled system. In this paper, the control objective is to design a switched controller and switched adaptive laws so that the state error converges to 0 i.e., the state of controlled system x tracks the state of given closed-loop reference model x_m.

Remark 1   In the classical switched MRAC control problem^[23-27], the switched reference model is open-loop switched reference model and the form is given by:

$ \left\{\begin{array}{l}{\dot{\overline{\boldsymbol{x}}}_{m}=\boldsymbol{A}_{m \sigma} \overline{\boldsymbol{x}}_{m}+\boldsymbol{B}_{\sigma} \boldsymbol{r}} \\ {\overline{\boldsymbol{y}}_{m}=\boldsymbol{C}_{\sigma}^{\mathrm{T}} \overline{\boldsymbol{x}}_{m}}\end{array}\right. $

Compared with the open-loop reference model, the closed-loop reference model has an additional feedback term. When we choose L_i=0, i∈M or the state error e→0, x_m→x_m, thus x→x_m, so the closed-loop switched reference model in Eq.(2) degenerates into the open reference switched reference model. Besides, the introduction of the feedback L_i can improve the transient performance due to the additional degree of freedom for each subsystem^[18-22].

The following assumptions are requisite to achieve the control objective.

Assumption 1   The product C_i^T B_i is full rank, i∈I.

Assumption 2   The pair (A_mi, C_i^T) is observable, i∈I.

Assumption 3   The subsystems in controlled system (1) are minimum phase.

Assumption 4   There exists Θ_i^*∈R^n×m and K_i^*∈R^m×m such that A_i+B_iΛ_iΘ_i^*T=A_mi and Λ_iK_i^*T=I, i∈I, respectively.

Assumption 5   Λ_i is diagonal with positive elements, i∈I.

Assumption 6   The unknown control input matrices Λ_i and the uncertain matching parameters Θ_i^* have priori known upper bounds and the following hold:

$ \bar \lambda = \mathop {\max }\limits_{i \in I} \left( {\sup \left\| {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}} \right\|} \right), {\bar \theta ^*} = \mathop {\max }\limits_{i \in I} \left( {\sup \left\| {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}} \right\|} \right) $

(3)

Remark 2   Assumption 1 actually is the requirement that M_i is nonsingular. Assumption 2 is needed since the feedback gains of the reference model L_i are required to use in the following result. Assumption 3 makes the condition of the KYP lemma hold. Assumption 4 is the matching conditions and is commonly used in MRAC^[17]. Assumption 5 is about the uncertainties and it is routinely satisfied in the area of aerospace industry in which control directions are generally known but we do not know their magnitudes^[27]. Assumption 6 is for a suitable choice of L_i(i∈M).

Lemma 1^{[21, 28]}   For each subsystem of the switched system (1) satisfying Assumptions 1-3, and M_i choose as in Eq.(8). There always exists L_si such that M_i^TC_i^T(sI-A_mi-L_siC_i^T)^-1B_i are strict positive realness (SPR).

Lemma 2^{[21, 29]}   If L_i=L_si-ρ_iB_iM_i^T are chosen and ρ_i>0 are arbitrary, then the transfer functions M_i^TC_i^T(sI-A_mi-L_sC_i^T)^-1B_i are SPR and the following hold:

$ \left(\boldsymbol{A}_{m i}+\boldsymbol{L}_{i} \boldsymbol{C}_{i}^{\mathrm{T}}\right)^{\mathrm{T}} \boldsymbol{P}_{i}+\boldsymbol{P}_{i}\left(\boldsymbol{A}_{m i}+\boldsymbol{L}_{i} \boldsymbol{C}_{i}^{\mathrm{T}}\right)=-\boldsymbol{Q}_{i} $

(4a)

$ \boldsymbol{Q}_{i} \triangleq \boldsymbol{Q}_{s i}+2 \rho_{i} \boldsymbol{C}_{i} \boldsymbol{M}_{i} \boldsymbol{M}_{i}^{\mathrm{T}} \boldsymbol{C}_{i}^{\mathrm{T}} $

(4b)

$ \boldsymbol{P}_{i} \boldsymbol{B}_{i}=\boldsymbol{C}_{i} \boldsymbol{M}_{i}, i \in \boldsymbol{I} $

(4c)

where L_si are in Lemma 1, P_i=P_i^T>0 and Q_si=Q_si^T>0 are ρ_i and M_i.

3 Stability Analysis

For the switched system in Eqs. (1) and (2) satisfying the all above assumptions, we design the following switched adaptive controller and adaptive laws:

$ \boldsymbol{u}_{i}(t)=\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{i}^{\mathrm{T}}(t) \boldsymbol{x}_{m}+\boldsymbol{K}_{i}^{\mathrm{T}}(t) \boldsymbol{r}, i \in \boldsymbol{I} $

(5)

$ {\mathit{\boldsymbol{ \boldsymbol{\dot \varTheta} }}_i}(t) = \left\{ {\begin{array}{*{20}{l}} { - {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\Theta }_i}}}{\mathit{\boldsymbol{x}}_\mathit{\boldsymbol{m}}}(\mathit{\boldsymbol{t}})\mathit{\boldsymbol{e}}_y^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}, }&{i = \sigma }\\ {0, }&{i \ne \sigma } \end{array}} \right. $

(6)

$ {\mathit{\boldsymbol{\dot K}}_i}(t) = \left\{ {\begin{array}{*{20}{l}} { - {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{K_i}}}{\mathit{\boldsymbol{x}}_m}(t)\mathit{\boldsymbol{e}}_y^{\rm{T}}(t){\mathit{\boldsymbol{M}}_i}, i}&{ = \sigma }\\ {0, }&{i \ne \sigma } \end{array}} \right. $

(7)

where Θ_i(t) and K_i(t) are the estimated values of Θ_i^* and K_i^* respectively. Γ_Θi and Γ_Ki are both positive diagonal matrices which can be designed freely. e_y=y-y_m is the output tracking error and where

$ \boldsymbol{M}_{i} \triangleq \boldsymbol{C}_{i}^{\mathrm{T}} \boldsymbol{B}_{i} $

(8)

Remark 3   In general, it is hard to measure all the state of the controlled systems in many practical systems. Therefore, we choose the state of the given reference model instead of the controlled system to design the control input Eq.(5).

From Eqs.(1), (2) and (5), we can easily get the following expression of the state error

$ \left\{\begin{array}{l} \mathit{\boldsymbol{\dot e}}=\left(\boldsymbol{A}_{m_{\sigma}}+\boldsymbol{L}_{\sigma} \boldsymbol{C}_{\sigma}^{\mathrm{T}}\right) \boldsymbol{e}+\\ \boldsymbol{B}_{\sigma} \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{\sigma}\left(\tilde{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}}_{\sigma}^{\mathrm{T}} \boldsymbol{x}_{m}+\tilde{\boldsymbol{K}}_{\sigma}^{\mathrm{T}} \boldsymbol{r}-\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{\sigma}^{* \mathrm{T}} \boldsymbol{e}\right) \\ \boldsymbol{e}_{y}=\boldsymbol{C}_{\sigma}^{\mathrm{T}} \boldsymbol{e} \end{array}\right. $

(9)

where ${\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i} = {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i} - \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^* $ and $\tilde{\boldsymbol{K}}_{i}=\boldsymbol{K}_{i}-\boldsymbol{K}_{i}^{*} $ are adjustable parameter errors for each i∈I.

In the following, we give a sufficient condition under which the MRAC problem is solved.

Theorem 1   If there exist P=P^T>0 and Q=Q^T>0, such that the following holds:

$ \left(\boldsymbol{A}_{m i}+\boldsymbol{L}_{i} \mathrm{C}_{i}^{\mathrm{T}}\right)^{\mathrm{T}} \boldsymbol{P}+\boldsymbol{P}\left(\boldsymbol{A}_{m i}+\boldsymbol{L}_{i} \boldsymbol{C}_{i}^{\mathrm{T}}\right) \leqslant-\boldsymbol{Q} $

(10a)

$ \boldsymbol{Q}_{s i}+2 \rho \boldsymbol{C}_{i} \boldsymbol{M}_{i} \boldsymbol{M}_{i}^{\mathrm{T}} \boldsymbol{C}_{i}^{\mathrm{T}} \leqslant \boldsymbol{Q} $

(10b)

$ \boldsymbol{P B}_{i}=\boldsymbol{C}_{i} \boldsymbol{M}_{i} $

(10c)

The MRAC problem of switched adaptive system described by Eqs.(1), (2), (5), (6) satisfying Assumptions 1-6 with L_i defined in Eq.(4), M_i stated in Eq. (8), ρ>ρ^* is solved with $ \mathop {\lim }\limits_{t \to \infty } \mathit{\boldsymbol{e}}(t) = 0$, where λ and θ^* are defined in Eq. (3), Q_si is defined in Lemma 2 and

$ \rho^{*}=\frac{\overline{\lambda}^{2} \overline{\theta}^{* 2}}{2 \min\limits_{i \in I} \lambda_{\min }\left(\boldsymbol{Q_{s i}}\right)} $

(11)

Proof   The Lyapunov function we choose is:

$ \begin{array}{l} \mathit{\boldsymbol{V}} = {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{Pe}} + \sum\limits_{i = 1}^M {{\bf{Tr}}} \left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}(t)\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i}}^{ - 1}{{\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}}_i}(t)} \right) + \\ \;\;\;\;\;\;\sum\limits_{i = 1}^M {{\bf{Tr}}} \left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}(t)\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\boldsymbol{K}}_\mathit{\boldsymbol{i}}}}^{ - 1}{{\mathit{\boldsymbol{\widetilde K}}}_i}(t)} \right) \end{array} $

(12)

Differentiating V along the active subsystem trajectories in Eq. (9) gives:

$ \begin{array}{l} \mathit{\boldsymbol{\dot V}} = {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\left( {\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}{\mathit{\boldsymbol{x}}_m} + \mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}\mathit{\boldsymbol{r}} - } \right.} \right.\\ \;\;\;\;\;\;\;\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}}){)^{\rm{T}}}\mathit{\boldsymbol{Pe}} + {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}} + {\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\left( {\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}{\mathit{\boldsymbol{x}}_m} + } \right.} \right.\\ \;\;\;\;\;\;\;\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}\mathit{\boldsymbol{r}} - \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}})) + 2{\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}(t)\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i}}^{ - 1}{{\mathit{\boldsymbol{\dot { \boldsymbol{\tilde \varTheta} }}}}_i}(t)} \right) + \\ \;\;\;\;\;\;\;2{\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}(t)\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{K_i}}^{ - 1}{{\mathit{\boldsymbol{\dot {\tilde K}}}}_i}(t)} \right) = {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {{{\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} + } \right.\\ \;\;\;\;\;\;\;\mathit{\boldsymbol{P}}\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}}} \right))\mathit{\boldsymbol{e}} - 2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}} + \\ \;\;\;\;\;\;\;2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}{\mathit{\boldsymbol{x}}_m} + 2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}\mathit{\boldsymbol{r}} - \\ \;\;\;\;\;\;\;2{\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}(t){\mathit{\boldsymbol{x}}_m}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}} \right) - \\ \;\;\;\;\;\;\;2{\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}(t)\mathit{\boldsymbol{r}}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}} \right) \end{array} $

(13)

According to the property of the matrix trace, we have:

$ {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}{\mathit{\boldsymbol{x}}_m} = {\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i^{\rm{T}}(t){\mathit{\boldsymbol{x}}_m}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}} \right) $

(14a)

$ {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}\mathit{\boldsymbol{r}} = {\bf{Tr}}\left( {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{\widetilde K}}_i^{\rm{T}}(t)\mathit{\boldsymbol{r}}{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}} \right) $

(14b)

By Eqs. (8), (10) and (14), Eq. (13) can be written as:

$ \begin{array}{l} \mathit{\boldsymbol{\dot V}} = {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {{{\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}}} \right)}^{\rm{T}}}\mathit{\boldsymbol{P}} + \mathit{\boldsymbol{P}}\left( {{\mathit{\boldsymbol{A}}_{mi}} + {\mathit{\boldsymbol{L}}_i}\mathit{\boldsymbol{C}}_i^{\rm{T}}} \right)} \right)\mathit{\boldsymbol{e}} - \\ \;\;\;\;\;\;2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}} \le \\ \;\;\;\;\;\; - {\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{Qe}} - 2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}} \le \\ \;\;\;\;\;\; - {\mathit{\boldsymbol{e}}^{\rm{T}}}\left( {{\mathit{\boldsymbol{Q}}_{si}} + 2\rho {\mathit{\boldsymbol{C}}_i}{\mathit{\boldsymbol{M}}_i}\mathit{\boldsymbol{M}}_i^{\rm{T}}\mathit{\boldsymbol{C}}_i^{\rm{T}}} \right) - \\ \;\;\;\;\;\;2{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{B}}_i}{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_i}\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_i^{*{\rm{T}}}\mathit{\boldsymbol{e}} \end{array} $

(15)

By the fact that e_y=C_i^Te, we can get $\dot{\boldsymbol{V}}=- \boldsymbol{\varepsilon}^{\mathrm{T}}(\boldsymbol{Q}(\boldsymbol{\rho})) \boldsymbol{\varepsilon}$, where

$ \boldsymbol{Q}(\rho)=\left[\begin{array}{cc}{2 \rho \boldsymbol{M}_{i} \boldsymbol{M}_{i}^{\mathrm{T}}} & {\boldsymbol{M}_{i} \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{i} \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{i}^{* \mathrm{T}}} \\ {\mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{i}^{* \mathrm{T}} \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{i} \boldsymbol{M}_{i}^{\mathrm{T}}} & {\boldsymbol{Q}_{s i}}\end{array}\right]\\ ~~~~~~~~~~~\boldsymbol{\varepsilon}=\left[\begin{array}{l}{\boldsymbol{e}_{y}} \\ {\boldsymbol{e}}\end{array}\right] $

Due to ρ>ρ^*>0 and Eq.(11), we have

$ 2 \boldsymbol{M}_{i} \rho \boldsymbol{M}_{i}^{\mathrm{T}}-\boldsymbol{M}_{i} \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{i} \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{i}^{* \mathrm{T}} \boldsymbol{Q}_{s i}^{-1} \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_{i}^{*} \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{i} \boldsymbol{M}_{i}^{\mathrm{T}}>0 $

According to the Schur complement Lemma, Q(ρ) is a positive-definite matrix. Thus $\mathit{\boldsymbol{\dot V}} < 0 $, so e_y, e, ${\mathit{\boldsymbol{ \boldsymbol{\widetilde \varTheta} }}_i} $ and $ \tilde{\boldsymbol{K}}_{i} \in \boldsymbol{L}_{\infty}$. From Eqs.(2) and (9), it can be gotten that x_m∈L_∞ and $ \mathit{\boldsymbol{\dot e}} \in {\mathit{\boldsymbol{L}}_\infty }$. Furthermore, e∈L₂ because of the positive definiteness of matrix Q(ρ). So we have e∈L₂∩L_∞ and $ \mathit{\boldsymbol{\dot e}} \in {\mathit{\boldsymbol{L}}_\infty }$. By Barbalat Lemma, it follows $\mathop {\lim }\limits_{t \to \infty } \mathit{\boldsymbol{e}}(\mathit{\boldsymbol{t}}) = 0 $. By the fact e=x-x_m, we can get the boundedness of x.

4 Numerical Examples

In this section, we provide a numerical example to prove the effectiveness of the theorem in this paper.

Considering the switched system (1) and the switched closed-loop reference model (2), the system matrices and parameters are given by:

$ A_{1}=\left[\begin{array}{rr}{-2} & {0} \\ {0} & {-3}\end{array}\right], \quad \boldsymbol{B}_{1}=\left[\begin{array}{l}{4} \\ {2}\end{array}\right]\\ \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{1}=0.5, \quad \boldsymbol{C}_{1}=\left[\begin{array}{l}{1} \\ {1}\end{array}\right]\\ {\mathit{\boldsymbol{A}}_2} = \left[ {\begin{array}{*{20}{r}} { - 1.8}&0\\ 0&{ - 3.2} \end{array}} \right], \;\;\;{\mathit{\boldsymbol{B}}_2} = \left[ {\begin{array}{*{20}{l}} 4\\ 2 \end{array}} \right]\\ \mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{2}=0.5, \quad \quad \boldsymbol{C}_{2}=\left[\begin{array}{l}{1} \\ {1}\end{array}\right] $

The plant parameters of the closed-loop reference model are:

$ \begin{array}{c} {\mathit{\boldsymbol{A}}_{m1}} = \left[ {\begin{array}{*{20}{c}} { - 3}&{ - 2}\\ { - 0.5}&{ - 4} \end{array}} \right]\\ {\mathit{\boldsymbol{B}}_{m1}} = \left[ {\begin{array}{*{20}{c}} 4\\ 2 \end{array}} \right], \;\;\;{\mathit{\boldsymbol{C}}_{m1}} = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right]\\ {\mathit{\boldsymbol{A}}_{m2}} = \left[ {\begin{array}{*{20}{c}} { - 2.8}&{ - 2}\\ { - 0.5}&{ - 4.2} \end{array}} \right]\\ {\mathit{\boldsymbol{B}}_{m2}} = \left[ {\begin{array}{*{20}{c}} 4\\ 2 \end{array}} \right], \;\;\;{\mathit{\boldsymbol{C}}_{m2}} = \left[ {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right] \end{array} $

The reference input is r(t)=4sin(0.2πt). From Assumption 4, we have

$ \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_1^* = \mathit{\boldsymbol{ \boldsymbol{\varTheta} }}_2^* = {\left[ {\begin{array}{*{20}{c}} { - 0.5}\\ { - 1} \end{array}} \right]^{\rm{T}}}, K_1^{*{\rm{T}}} = K_2^{*{\rm{T}}} = 2 $

The parameters of the adaptive laws in Eqs.(6) and (7) are designed as

$ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\Theta }_1}}} = {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{\mathit{\Theta }_2}}} = \left[ {\begin{array}{*{20}{c}} {400}&0\\ 0&{400} \end{array}} \right], {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{K_1}}} = {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_{{K_2}}} = 450 $

A direct calculation gives

$ \boldsymbol{L}_{s 1}=\left[\begin{array}{c}{2} \\ {1.333}\end{array}\right], \boldsymbol{L}_{s 2}=\left[\begin{array}{c}{1.4} \\ {-1.87}\end{array}\right] $

Then, we choose ρ=1. The feedback gains of the reference model are

$ \boldsymbol{L}_{1}=\left[\begin{array}{c}{-22} \\ {-10.6667}\end{array}\right], \boldsymbol{L}_{2}=\left[\begin{array}{c}{-22.6} \\ {-10.13}\end{array}\right] $

so we have

$ \begin{array}{c} \mathit{\boldsymbol{P}} = \left[ {\begin{array}{*{20}{r}} {16.730}&0&{ - 24.331}&2\\ { - 24.331}&2&{51.563}&1 \end{array}} \right]\\ \mathit{\boldsymbol{Q}} = \left[ {\begin{array}{*{20}{c}} { - 350}&0\\ 0&{ - 350} \end{array}} \right] \end{array} $

From Figs. 1-5, it can be concluded that the switched MRAC problem in Eqs.(1), (2) and (9) with the closed-loop reference model is solved under the adaptive controller (5). Fig. 1 is the switching signal. Fig. 2 shows that the trajectories of the closed-loop reference model are bounded. It shows us that the adaptive parameters of the error systems are bounded from Figs. 3-4. From Fig. 5, we can get the fact that the state error converges to 0 which implies the state of controlled system can track the state of given switched reference model.

5 Conclusions

The MRAC problem for the switched systems with parametric uncertainties which has closed-loop reference model for each subsystem under arbitrary switching signals has been solved in this paper. Firstly, the reference model and the controller were not common for all subsystems. Secondly, to get a better transient performance of the whole switched system, the closed-loop reference model has been introduced. Finally, it is hard to measure all of the state of system in many piratical systems, so the design of control input is depended on the state of closed-loop reference model and the measurable output error in this paper. By appropriate design of the adaptive control laws and the gain of the closed-loop reference models for switched systems, the state tracking errors converged to 0.