A new model reference adaptive control structure for uncertain switched systems with unmodeled input dynamics

Abstract

In this paper, a new model reference adaptive controller (MRAC) for uncertain switched linear systems is developed. A class of uncertain switched linear systems with parametric mismatched and input matched uncertainties, control input effectiveness and unmodeled dynamics is studied in this paper. The difference in input matrix for different switching modes which means the input channel degradation is investigated through this paper. By using common Lyapunov function method and developing new linear matrix inequality based sufficient conditions, uniform ultimate boundedness of the reference tracking error is guaranteed by switched MRAC and by using a novel nonlinear controller term, the size of the uniform ultimate bounded region can be made arbitrarily small which could enhance the performance of the overall control structure. Modified versions of MRAC systems using low-frequency learning adaptation MRAC is also used for better transient performance at switching instants. An illustrative numerical example demonstrates the performance of the proposed approach. Results also show that for the cases without parametric uncertainty in the linear part, asymptotic stability of the reference tracking error is guaranteed.

Keywords

Uncertain switched systems model reference adaptive control linear matrix inequality modified structure adaptive control

Introduction

Hybrid systems as an attractive area for control engineers tend to continuous-time systems with switching which we refer as switched systems. Specifically, a switched system is a hybrid system that consists of a family of subsystems and a switching law that orchestrates switching between these subsystems. It is well known that many physical models such as power systems, aircraft control systems, chemical control processes, manufacturing systems and communication network systems can be transformed into this kind of systems (Liberzon, 2003; Liberzon and Morse, 1999).

Therefore, due not only to their success in applications but also to their importance in theory, the last decade has witnessed burgeoning research activities on their stability (Liberzon and Morse, 1999; Michel, 1999), controllability, observability (Sun et al., 2002), etc., that aim at designing switched systems with guaranteed stability and performance. Among these topics, designing controllers to obtain stability of switched systems with various kinds of model uncertainty have attracted the most attention (He et al., 2012; Lian and Zhao, 2009a,b; Liu et al., 2013; Sun and Ge, 2005). These efforts have been resulted in the development of various stability criteria and analysis tools for switched systems, such as common Lyapunov function (CLF), multiple Lyapunov functions, and the average dwell-time method.

Adaptive control of linear uncertain dynamical systems has been widely studied for systems with significant uncertainties and unmodeled dynamics. The widely studied model reference adaptive control (MRAC) framework has in particular many theoretical results, proving asymptotic tracking error convergence or ultimate boundedness of tracking error in the presence of significant modeling uncertainty, and several experimental results (Tao, 2003).

In addition to removing the restrictive conditions of normal adaptive control, switching is also used to enhance the control performance. More precisely, the switching logic becomes a selecting algorithm that chooses the best controller among the candidate controllers and connects it to the plant. For example, the multiple models adaptive control method improves the transient response of adaptive systems in a stable fashion (Hespanha et al., 2003), and the switching between a robust controller and an adaptive controller by Quang (2006) enhances both the transient and the steady performance dramatically. For switched systems with external switching signal, the importance of synchronously switching the controller with the plant switching mode is necessary in which all switching modes, the controller must have good performance (El Rifai et al., 2005). El Rifai et al. (2005) proposed a controller design algorithm for switching systems with an external periodic switching law.

Among several controller design problems for switched linear systems with uncertainty, the sliding mode method and robust H_∞ controller are the most popular. Switching model reference adaptive controller is another solution for this problem that has been used recently and various kinds of switched MRAC have been proposed in papers published in the last decade. Wang et al. (2012a) and references therein provide a good survey. MRAC for switched systems has been shown to have good performance on various kinds of uncertainty resources and unmodeled dynamics, and is easy to implement in comparison with other switching uncertainty dealing controllers. Standard MRAC systems in the face of uncertainty and dynamic modeling seem to be nonfragile but practically they are limited to some predetermined structure for uncertainty. Recent papers tend to use robust MRAC (Wang et al., 2012a) which is a developed version of robust adaptive control for switched systems or neural-network (NN)-based MRAC (Singh and Sukavanam, 2012) and concurrent learning adaptive structured (De La Torre et al., 2013) to deal with the above-mentioned problems. Practically, much attention has also been focused on new versions of MRAC systems which are similar to the standard structure but with some modification terms and are more easy to understand and implement than robust MRAC or NN-based MRAC. The idea behind these adaptive structures is to attenuate the effect of uncertainty in the learning phase of adaptation law (Schatz et al., 2013a,b; Yucelen and Haddad, 2013; Yucelen et al., 2009). Surprisingly, the learning phase of switched adaptive controller in each active mode is when the spotlight problems occur in switched systems. For switched systems with external fast switching laws, a fast response adaptive reference tracking control system is necessary.

In this paper, we have developed a novel structure model reference adaptive controller for uncertain switched systems. Various kinds of uncertainty and unmodeled dynamics for switched system are considered. The effect of the input channel which could be a cause of actuator deterioration is also considered. The structure of MRAC in this paper is based on the modified versions of standard adaptive controller for uncertain switched systems. It will be proven by CLFs that the proposed structure for switched systems will guarantee the stability of closed-loop systems and by using modified versions of adaptive laws such as low-frequency learning, the quality of responses at switching events will be improved.

The uniformly ultimate boundedness of error signals is proven in this paper in a similar manner to the existing works but due to the novel nonlinear controller block considered in this paper, a degree of freedom to trade off between control signal domain and radius of the uniform ultimate bounded (UUB) region is available in the proposed method which could be useful in practical implementations. The quality of the error signals in learning phase (transient) is also enhanced by using a modified version of the MRAC structure. Using such easy adaptation laws with inclusive performance in comparison with the existing robust MRAC or NN-based controllers is the key advantage of the method proposed in this paper.

This paper is organized as follows. The next section provides some preliminary definitions and gives a brief description of the problem formulation. Followed by introducing a new structure for MRAC of uncertain switched systems, developing the closed-loop error dynamics and presenting sufficient conditions for stability and convergence. An illustrative numerical example is given to demonstrate the efficacy of the proposed structure and finally the conclusions are given.

Notations

The superscript A^T stands for matrix transposition, A⁻¹ for inverse and A^# for pseudo-inverse of a matrix. The subscript A_m is for any matrix is related to reference model and subscript A_i stands as representative for system’s index. Here $R^{n}$ denotes the n-dimensional Euclidean space; the notation P > 0 means that P is real symmetric and positive definite; I and 0 represent the identity matrix and a zero matrix, respectively; ‖·‖ denotes the Euclidean norm of a vector or the spectral norm of a matrix. Matrices, if their dimensions are not stated explicitly, are assumed to be compatible for algebraic operations.

Problem formulation and preliminaries

Consider a class of uncertain switched systems with the following state-space representation:

\begin{matrix} \overset{\cdot}{x} (t) = & (A_{σ (t)} + Δ A_{σ (t)}) x (t) + B [Λ_{σ (t)} u (t) + f_{σ (t)} (x, t)] . \end{matrix}

(1)

where $x (t) \in R^{n}$ is the accessible state vector, $u \in R^{m}$ is the control input, σ(t): [0,+∞) ⇒M = {1,2,…, m} is the switching law, f_σ_(t)(x, t) represents matched bounded disturbance or time-varying matched uncertainty that is state dependent, A_σ_(t), B are known constant matrices of appropriate dimensions and matrix B is of the full column rank and assumed to be identical for all subsystems, with a priori assumption for model reference control which states that the pairs (A_σ_(t), B) are controllable, $Λ_{σ (t)} \in R^{m \times m} \cap D^{m \times m}$ is an unknown control effectiveness matrix and ΔA_σ_(t) represents the parameter uncertainty.

The following assumptions are given for the switched system (1).

Assumption 1. The switching law $σ (t) : R \Rightarrow M$ is the constant piecewise function which could be dependent on either time (event) t or state x. In this study we assume that the switching rule is not known a priori but σ(t) is available at each t. This relaxation corresponds to practical implementations where the switched system is supervised by a discrete-event system or operator allowing for σ(t) to be known in real time (Hou et al., 2008; El Rifai et al., 2005). On the other hand, studying the stability under arbitrary switching signal is more general than regulated switching signals, once the stability of a switched system under arbitrary switching is guaranteed, engineers have more freedom to design a switching signal for better performance, unaffected by stability considerations. For each switching signal σ(t) = i, i ∈ M, we will denote the system associated with the ith subsystem as

\begin{matrix} A_{σ (t)} ≜ A_{i}, & Δ A_{σ (t)} ≜ A_{i}, \\ Λ_{σ (t)} ≜ Λ_{i}, & f_{σ (t)} (x, t) ≜ f_{i} (x, t) . \end{matrix}

Assumption 2. The parametric uncertainty is a norm bounded real matrix function satisfying

Δ A_{i} (t) = D_{i} Σ_{i} (t) E_{i}, i \in M,

(2)

where D_i and E_i are known constant matrices andΣ_i(t) is an unknown time-varying matrix function satisfying

Σ_{i}^{T} (t) Σ_{i} (t) \leq I .

(3)

Assumption 3. Matched uncertainty in (1) can be parametrized as

f_{i} (x, t) = W_{i} {(t)}^{T} β_{i} (x),

(4)

where $W_{i} (t) \in R^{s_{i} \times m}$ is an unknown time-varying weight matrix and $β_{i} (x) : R^{n} \to R^{s}$ is a known basis function of the form β_i(x) = [b_1i(x),…,b_si(x)]^T.

Remark 1. In Assumption 3 it should be noted that for each subsystem a different matched uncertainty is considered which could be different in either structure or parameters.

Remark 2. Assumption 3 is based on the fact that uncertainty could be exactly parametrized and the basis function is known. By using NN approximation or any other parametric identification method, the parametrization in Assumption 3 could be relaxed by considering

f_{i} (x, t) = W_{i}^{T} β_{i} (x) + ε_{i} (x),

(5)

where ε _i (x) is unknown residual error and W_i is constant unknown weighting matrices (Schatz et al., 2013a; Singh and Sukavanam, 2012). As each subsystem has an independent dynamic model we define $E_{m} ≜ max_{1 \leq i \leq n} \sup_{t \geq 0} ε_{i} (x (t))$ as the maximum parameterization error for all switched system modes defined by equation (1).

Assumption 4. The control input matrix B is considered identical for all subsystems. Since the actual control input matrix for the class of uncertain switched systems (1) is F = BΛ_iand to address cases such as nominal controller failure or degradation in the control effectiveness, one can consider the differences in control input matrices of subsystems by defining Λ_i = Λ_0i+ΔΛ_i where $Λ_{0 i} \in R^{m \times m}$ is a known positive-definite matrix and $Δ Λ_{i} \in R^{m \times m}$ is an unknown parametric uncertainty matrix satisfying‖ΔΛ_i‖≤ 1 such thatΛ_i is positive definite.

Remark 3. For the sake of simplicityΛ_0iis considered to be the identity and $Λ_{i} \in R^{m \times m}$ is considered an uncertain diagonal matrix with diagonal elements λ_j ≥ 0 ( Yucelen et al., 2009 ).

For each nominal subsystem of (1),

\overset{\cdot}{x} (t) = A_{σ (t)} x (t) + Bu (t),

(6)

we consider a reference model as follows:

{\overset{\cdot}{x}}_{m} (t) = A_{m_{σ (t)}} x_{m} (t) + B_{m_{σ (t)}} r (t),

(7)

where $x_{m} \in R^{n}$ is the reference state vector, $r (t) \in R^{m}$ is a bounded piecewise continuous reference input, (A_mi, B_mi) is a reference model set for switched system and {A_mi, for all i ∈ M} are Hurwitz stable matrices.

Our aim of having a model reference system is to construct a feedback law u(t) such that the states of the uncertain system (1), in the presence of uncertainty, asymptotically track the states of the reference model (7) for each active subsystem and error signals stay bounded for all time especially at switching instants.

In order to achieve these purposes, we consider the control signal as follows:

u (t) = u_{n_{i}} (t) + u_{{ad}_{i}} (t) + u_{v_{i}} (t), i \in M,

(8)

where $u_{n_{i}} (t)$ is the nominal controller for each nominal subsystem considering (6), (7) which is designed in order to achieve perfect model following, $u_{{ad}_{i}} (t)$ is the adaptive controller in order to attenuate uncertainty of the system and $u_{v_{i}} (t)$ is nonlinear control term which is proposed to enhance the performance of closed-loop system and will be introduced in next sections.

The nominal control law for each subsystem is considered as

u_{n_{i}} (t) = K_{1 i} x (t) + K_{2 i} r (t), i \in M,

(9)

where $K_{1 i} \in R^{m \times n}$ , $K_{2 i} \in R^{m \times m}$ are respectively nominal feedback and feedforward constant gains. These gains are designed such that each nominal subsystem tracks related reference subsystem perfectly (Wang et al., 2012a) by completely attenuating error in state signals due to perfect model following controller gains:

A_{m_{i}} = A_{i} + {BK}_{1 i}, B_{m_{i}} = {BK}_{2 i}, \det (K_{2 i}) \neq 0, i \in M .

Now suppose that the nominal controller is implemented on the model equation (1) according to (8), (9) along with an approximate parameterization in Assumption 3 for matched uncertainty. We will have

\begin{matrix} \overset{\cdot}{x} (t) & = (A_{i} + Δ A_{i}) x (t) + B Λ_{i} u_{i} (t) + {BW}_{i}^{T} β_{i} (x) + B ε_{i} (x) \\ = (A_{i} + {BK}_{1 i}) x (t) + {BK}_{2 i} r (t) + Δ A_{i} x (t) \\ + B (Λ_{i} K_{1 i} - K_{1 i}) x (t) + B (Λ_{i} K_{2 i} - K_{2 i}) r (t) \\ + B Λ_{i} u_{{ad}_{i}} (t) + B Λ_{i} u_{v_{i}} (t) + {BW}_{i}^{T} β_{i} (x) + B ε_{i} (x) \\ = A_{m_{i}} x (t) + B_{m_{i}} r (t) + Δ A_{i} x (t) \\ + B Λ_{i} (I - Λ_{i}^{- 1}) K_{1 i} x (t) + B Λ_{i} (I - Λ_{i}^{- 1}) K_{2 i} r (t) \\ + B Λ_{i} u_{{ad}_{i}} (t) + B Λ_{i} u_{v_{i}} (t) + B Λ_{i} Λ_{i}^{- 1} W_{i}^{T} β_{i} (x) + B ε_{i} (x) \\ = A_{m_{i}} x (t) + B_{m_{i}} r (t) + Δ A_{i} x (t) \\ + B Λ_{i} [W_{ai}^{T} β_{ai} (x, t) + u_{{ad}_{i}} (t)] + B Λ_{i} u_{v_{i}} (t) + B ε_{i} (x), \end{matrix}

(10)

where $W_{ai} = {[(I - Λ_{i}^{- 1}) K_{1 i}, (1 - Λ_{i}^{- 1}) K_{2 i}, Λ_{i}^{- 1} W_{i}^{T}]}^{T}$ is an unknown (aggregated) weighting matrix, β_ai(x, t) = [x(t)^T, r(t), β_i(x)^T]^T is a known (aggregated) basis function. The adaptive switching control will be considered as

u_{{ad}_{i}} (t) = - {\hat{W}}_{ai}^{T} β_{ai} (x, t), i \in M,

(11)

where ${\hat{W}}_{ai} (t)$ is obtained by the standard update law. Since the controller u(t) switches synchronously with the subsystems in (1), the state error only reflects the control effect of the active adaptive controller with its own estimation vector. It is improper that other estimation vectors update their values during their controller’s inactive period (Wang et al., 2012b), so the adaptive controller’s update law are considered as

{\overset{\cdot}{\hat{W}}}_{ai} (t) = {\begin{matrix} Γ_{i} β_{ai} (x, t) e^{T} (t) P_{M} B & for & σ (t) = i, \\ 0 & for & σ (t) \neq i, \end{matrix}

(12)

where Γ_i > 0 is learning rate and P_M is positive-definite matrix satisfying the linear matrix inequality (LMI) condition which will be stated in next section.

Using the standard adaptation law (12) requires a high learning rate for better performance such as low transient error distance (learning phase for adaptive controllers). However, update laws with high learning rates in the face of large system uncertainties and abrupt changes which is frequent in switched systems dynamics, may yield to signals with high-frequency oscillations, which can violate actuator rate saturation constraints and/or excite unmodeled system dynamics resulting in system instability for practical applications (Rohrs et al., 1985).

Remark 4. Yucelen and Haddad (2013) proposed a new adaptive control architecture for nonlinear uncertain dynamical systems to address the problem of achieving fast adaptation using high-gain learning rates. The proposed framework involves a new and novel controller architecture involving a modification term in the update law. Specifically, this modification term filters out the high-frequency content contained in the update law while it does not violate the stability of the closed-loop dynamics. Switched system with an external switching law may be in face of fast switching signal which needs a fast reference tracking controller. This quality is not established by standard adaptive control laws. Two other modification terms such as σ-modification and e-modification is also proposed and a comprehensive comparison is given by Yucelen et al. (2009). A summary of these results will be given in the illustrative examples section. The adaptation law using low-frequency learning is as follows:

{\overset{\cdot}{\hat{W}}}_{ai} (t) = {\begin{matrix} Γ_{i} [β_{ai} (x, t) e^{T} (t) P_{M} B - σ ({\hat{W}}_{ai} (t) - {\hat{W}}_{ai}^{f} (t))] & for & σ (t) = i, \\ 0 & for & σ (t) \neq i, \end{matrix}

(13)

where σ > 0 is the modification gain.

To address the high-frequency oscillation prevalent due to adaptation and switching instant sudden changes, ${\hat{W}}_{ai}^{f} (t)$ , a low-pass filtered signal of ${\hat{W}}_{ai} (t)$ is introduced as

{\overset{\cdot}{\hat{W}}}_{ai}^{f} (t) = Γ_{i}^{f} [{\hat{W}}_{ai} (t) - {\hat{W}}_{ai}^{f} (t)], i \in M,

(14)

where $Γ_{i}^{f}$ is the positive-definite filter gain (for $W_{ai} (t) \in R$ the case $Γ_{i}^{f}$ is related to the low-pass filter time constant).

Remark 5. Owing to the parametric uncertainty and external disturbance in switched linear system (1), it will be shown that we cannot conclude the convergence of the error signals to zero. Therefore, it is only reasonable to expect that all error signals defined on the MRAC structure converge into a neighborhood with reachable radius and remains within it thereafter, which is the so-called uniformly ultimate boundedness (UUB).

The novel nonlinear term in comparison with Wang et al. (2012a) and Hou et al. (2008) is chosen as a signal smoothing term, since the matched uncertainty could be attenuated by the switching adaptive controller; this term is just for adjusting the size of the UUB region due to parametric uncertainty for reference tracking error. This novel term is considered as

u_{vi} (t) = - K_{n_{i}} B^{T} P_{M} e (t), i \in M,

(15)

where $K_{n_{i}} \geq 0$ is an arbitrary chosen constant parameter which determines the size of the ultimate bound in presence of parametric uncertainty or approximation error in parametrization. The size of the ultimate bound for reference tracking error is in indirect relation to the oscillating control signal bound, so a trade off between these two signals for better performance is intransitive.

Now by defining the error terms e(t) = x(t) − x_m(t), ${\tilde{W}}_{ai} (t) = {\hat{W}}_{ai} (t) - W_{ai}$ the closed-loop error dynamic will be considered by

\begin{matrix} \overset{\cdot}{e} (t) = A_{mi} e (t) + B Λ_{i} [{\tilde{W}}_{ai}^{T} β_{ai} (x, t)] + Δ A_{i} x (t) + B ε_{i} (x) \\ - B Λ_{i} K_{n} B^{T} P_{M} e (t) \\ = [A_{mi} - B Λ_{i} K_{n} B^{T} P_{M} + Δ A_{i}] e (t) + B Λ_{i} [{\tilde{W}}_{ai}^{T} β_{ai} (x, t)] \\ + Δ A_{i} x_{m} (t) + B ε_{i} (x), \end{matrix}

(16)

along with (12) (due to Assumption 3 and related remarks, the parametrization vector W_ai will be considered constant). The proposed control MRAC architecture consists of three different control blocks satisfying the control objectives as shown in Figure 1. As it is mentioned before, the controller and reference model are switched synchronously with the uncertain plant (1).

Figure 1.

Structure of the model reference adaptive controller of an uncertain switched system.

Stability analysis and main results

In this section, the following lemmas and theorems are given to develop a sufficient condition which guarantees the uniform ultimate boundedness of the error signals.

Definition 1 (Singh and Sukavanam, 2012). The solution of a dynamical system is UUB if there exists a compact set $S \subset R^{n}$ such that for all x(t₀) = x₀ ∈ S, there exists a λ > 0 and a number T(λ, x₀) such that‖x(t)‖ < λ for all t ≥ t₀ + T.

Lemma 1 (Xu and Chen, 2002). Given vectors $x \in R^{p}$ , $y \in R^{q}$ and real matrices M₁, M₂of appropriate dimensions and an unknown time-varying matrixΣ(t) withΣ^T (t)Σ(t) ≤ I, we have

2 x^{T} M_{1} Σ (t) M_{2} y \leq γ x^{T} M_{1} M_{1}^{T} x + γ^{- 1} y^{T} M_{2}^{T} M_{2} y,

(17)

where γ > 0 is any constant scalar.

Theorem 1. If there exist positive numbers ε_i, δ_i and a positive-definite matrix P_M > 0 which is named a CLF such that the following LMI holds for all considered subsystems (for all i ∈ M):

\begin{matrix} (\begin{matrix} A_{m_{i}}^{T} P_{M} + P_{M} A_{m_{i}} + ε_{i}^{- 1} E_{i}^{T} E_{i} + δ_{i} P_{M} & \sqrt{2 ε_{i}} P_{M} D_{i} \\ \sqrt{2 ε_{i}} D_{i}^{T} P_{M} & - I \end{matrix}) & < 0, \\ P_{M} = P_{M}^{T} & > 0, \end{matrix}

(18)

then under an arbitrary switching law, the system and estimation error dynamics e(t) and ${\tilde{W}}_{ai}$ are UUB, and‖e(t)‖can be made small by a suitable choice of controller gain $K_{n_{i}}$ .

We now give a comprehensive proof on the proposed theorems based on the CLF which will result to uniform ultimate boundedness of states. The radius of the UUB region could be adjusted (minimized) by controller gain which will be a good improvement on the design of switched MRAC for uncertain switched system with unmodeled dynamics.

Proof. Consider the following CLF candidate:

V (t) = \frac{1}{2} [e {(t)}^{T} P_{M} e (t) + tr (Ψ {(t)}^{T} ϒ^{- 1} Ψ (t))],

(19)

where e(t) = x(t) − x_m(t), ${\tilde{W}}_{ai} (t) = W_{ai} - {\hat{W}}_{ai} (t)$ and P_M is a positive-definite matrix obtained from Theorem 1. Since the weighting factor for matched uncertainty W_ai is considered constant, we will have ${\overset{\cdot}{\tilde{W}}}_{ai} (t) = - {\overset{\cdot}{\hat{W}}}_{ai} (t)$ . The augmented estimation error matrix is defined by $Ψ (t) = {[{\tilde{W}}_{a 1} Λ_{1}^{1 / 2}, \dots, {\tilde{W}}_{am} Λ_{m}^{1 / 2}]}^{T}$ , $ϒ^{- 1} = diag (Γ_{1}^{- 1}, \dots, Γ_{s}^{- 1})$ . Recall that during each active period only one element of Ψ(t) is changing its value.

Taking the time derivative of V(t) along the trajectories defined on state error dynamic (16) and estimation error dynamics (12) or (13), we obtain

\begin{matrix} \overset{\cdot}{V} (t) = \frac{1}{2} [\overset{\cdot}{e} {(t)}^{T} P_{M} e (t) + e {(t)}^{T} P_{M} \overset{\cdot}{e} (t) \\ + tr ({\overset{\cdot}{\tilde{W}}}_{ai}^{T} Γ_{i}^{- 1} {\tilde{W}}_{ai}) + tr ({\tilde{W}}_{ai}^{T} Γ_{i}^{- 1} {\overset{\cdot}{\tilde{W}}}_{ai})] \\ = \frac{1}{2} e {(t)}^{T} (A_{mi}^{T} P_{M} + P_{M} A_{mi}) e (t) + e {(t)}^{T} P_{M} Δ A_{i} x (t) \\ + e {(t)}^{T} P_{M} B ε_{i} (x) - e {(t)}^{T} P_{M} B Λ_{i} K_{n_{i}} B^{T} P_{M} e (t), \end{matrix}

(20)

by substituting e = x − x_m we will have

\begin{matrix} = & \frac{1}{2} e {(t)}^{T} (A_{mi}^{T} P_{M} + P_{M} A_{mi}) e (t) \\ + e {(t)}^{T} P_{M} Δ A_{i} e (t) + e {(t)}^{T} P_{M} Δ A_{i} x_{m} (t) \\ + e {(t)}^{T} P_{M} B ε_{i} (x) - e {(t)}^{T} P_{M} B Λ_{i} K_{n_{i}} B^{T} P_{M} e (t), \end{matrix}

by using Lemma 1 we will have

\begin{matrix} \overset{\cdot}{V} (t) \leq & \frac{1}{2} e {(t)}^{T} (A_{mi}^{T} P_{M} + P_{M} A_{mi}) e (t) \\ + 2 ϵ e {(t)}^{T} P_{M} {DD}^{T} P_{M} e (t) + ε^{- 1} e {(t)}^{T} E^{T} Ee (t) \\ + ϵ^{- 1} x_{m} {(t)}^{T} E^{T} {Ex}_{m} (t) \\ + e {(t)}^{T} P_{M} B ε_{i} (x) - e {(t)}^{T} P_{M} B Λ_{i} K_{n_{i}} B^{T} P_{M} e (t) . \end{matrix}

now by using LMI condition from Theorem 1 we have

\begin{matrix} \overset{\cdot}{V} (t) \leq - \frac{1}{2} λ_{\min} (δ_{i} P_{M}) | | e (t) | |^{2} + ϵ_{i}^{- 1} | | {Ex}_{m} (t) | |^{2} \\ + | | P_{m} | | | | B | | E_{M} | | e (t) | | - λ_{\min}^{2} (P_{M}) K_{n_{i}} | | B | |^{2} ({\underline{Λ}}_{i}) | | e (t) | |^{2} \\ \overset{\cdot}{V} (t) \leq - | | e (t) | | [\frac{1}{2} λ_{\min} (δ_{i} P_{M}) | | e (t) | | \\ + λ_{\min}^{2} (P_{M}) K_{n_{i}} | | B | |^{2} ({\underline{Λ}}_{i}) | | e (t) | | - λ_{\max} (P_{m}) | | B | | E_{M}] \\ + ϵ_{i}^{- 1} | | {Ex}_{m} (t) | |^{2} . \end{matrix}

The reference model and related subsystems are assumed to be Hurwitz, so x_m(t) will be a bounded signal such that for all t > t₀, ‖x_m(t)‖ ≤ X_m; this implies that

\begin{matrix} \overset{\cdot}{V} (t) \leq - | | e (t) | | [\frac{1}{2} λ_{\min} (δ_{i} P_{M}) | | e (t) | | \\ + λ_{\min}^{2} (P_{M}) K_{n_{i}} {| | B | |}^{2} ({\underline{Λ}}_{i}) | | e (t) | | - λ_{\max} (P_{m}) | | B | | E_{M}] \\ + ϵ i^{- 1} {| | E | |}^{2} {| | X_{m} | |}^{2} . \end{matrix}

Let us define

\begin{matrix} Θ_{1} & = inf_{\forall i \in M} {\frac{1}{2} λ_{\min} (δ_{i} P_{M}) + λ_{\min}^{2} (P_{M}) K_{n_{i}} | | B | |^{2} ({\underline{Λ}}_{i})}, \\ Θ_{2} & = λ_{\max} (P_{m}) | | B | | E_{M}, \\ Θ_{3} & = sup_{\forall i \in M, \forall t > 0} {ϵ_{i}^{- 1} {| | E_{i} | |}^{2} {| | X_{m} (t) | |}^{2}}, \end{matrix}

(21)

Then we have

\overset{\cdot}{V} (t) \leq - | | e (t) | | [Θ_{1} | | e (t) | | - Θ_{2}] + Θ_{3} .

(22)

Accordingly the Θ₁ is a linear function of controller gain $K_{n_{i}}$ where the relation of $K_{n_{i}}$ and radius of UUB region is reversed: the greater the controller gain the smaller the radius of the UUB region. The time derivative of the Lyapunov function $\overset{\cdot}{V} (t)$ will be negative if ‖e(t)‖ ≥ Θ_e where

Θ_{e} = \frac{1}{2} \frac{Θ_{2}}{Θ_{1}} + \sqrt{\frac{Θ_{2}^{2}}{4 Θ_{1}^{2}} + Θ_{3}} .

(23)

Therefore, as $\overset{\cdot}{V} (t)$ is negative outside the compact set defined by (23) during the active period for each subsystem, then we conclude that the proposed switched MRAC structure for switched uncertain system share a CLF and the closed-loop error signal and estimation error signal for the adaptation law will be uniformly bounded. Moreover the ultimate size of the ‖e(t)‖ can easily be made smaller by choosing the controller gain $K_{n_{i}}$ large enough.

The radius of the UUB region (Θ_e) is in direct relation with Θ₂, Θ₃ and reverse relation with Θ₁. Actually our goal is to reduce Θ_e to as small as possible to have $\overset{\cdot}{V} (t)$ negative. Our suggested method is based on the adjusting $K_{n_{i}}$ , the gain of nonlinear controller term, which will affect the size of control signal U(t) besides and may cause practical problems such as saturation of actuators. This problem will limit the maneuverability of this controller but the thing that matters is that the other adjustments occur by systems dynamic and parameters could be used to reduce the Θ_e besides $K_{n_{i}}$ . These parameters are put in Θ₂ and Θ₃. lower parametrization error E_m in Assumption 2 or lower parameter uncertainty ΔA_i will result in a reduced radius of the UUB region; hence, for the systems with bigger (larger norm) parameter uncertainty the radius of the UUB region could not be smaller than a specific value.

Illustrative example

In this section, we provided a numerical example to demonstrate the efficacy of the proposed structure.

Consider the following linear uncertain switched system

\begin{matrix} \overset{\cdot}{x} (t) = (A_{σ (t)} + Δ A_{σ (t)}) x (t) + B [Λ_{σ (t)} u (t) + f_{σ (t)} (x, t)], \end{matrix}

(24)

where σ(t) ∈ M = {1, 2} and for nominal plant we have

A_{1} = (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}), A_{2} = (\begin{matrix} 0 & 1 \\ 10 & 12 \end{matrix}), B = (\begin{matrix} 0 \\ 1 \end{matrix}) .

The parametric uncertainty is considered as D = [0, 0; 0.5, 1.5], E = diag(0.5, 0.2) and Σ₁(t) = −1 ∈ [−1, 1], Σ₂(t) = 0.5 sin(t) ∈ [−1,1]. The unknown control effectiveness matrices are considered constant as Λ₁ = 0.5, Λ₂ = 0.25 and the matched uncertainty is considered parametrized by W₁ = [1.514, 0.5504, 0.0215]^T, W₂ = [0.1514, 0.5504, −0.0624, −0.0095,0.0215]^T and $β_{1} = [x_{1}, x_{2}, x_{1}^{3}]$ , $β_{2} = [x_{1}, x_{2} | x_{1} | x_{2}, | x_{2} | x_{2}, e^{x_{1}}]$ and approximation error is considered as ε = sin(x_1(t)) ≤ 1. The reference model is considered by ζ₁ = 0.707, ζ₂ = 0.85 as the damping ratio and ω₁ = 0.4 rad/s, ω₂ = 0.6 rad/s as the natural frequency of the desired reference models corresponding to their related uncertain subsystems.

The design parameters are as follows. The nominal perfect model following controller gains are

\begin{matrix} K_{11} & = B^{#} (A_{1} - A_{m 1}) = [01600, 05656], \\ K_{12} & = B^{#} (A_{2} - A_{m 2}) = [101600, 126800], \\ K_{21} & = B^{#} (B_{m 1}) = [01600], \\ K_{22} & = B^{#} (B_{m 2}) = [01600] . \end{matrix}

the adaptation rates are Γ = 100, Γ_f = 0.5, σ = 0.1 and novel controller gain K_n = 20. The solution of the LMI satisfying Theorem 1 for the considered example with δ = 2.5, ε = 25 will be

P_{M} = (\begin{matrix} 5.6177 & 8.7280 \\ 8.7280 & 17.5479 \end{matrix}) .

(25)

The radius of the UUB region Θ_e is calculated based on equation (23) for K_n→ 0, K_n→∞. According to equation (23) and (21), the size of the UUB region is in direct relation with K_n and will be 0.10 ≤Θ_e ≤ 0.2240 for any K_n ≥ 0 and for given parameters. Hence, the uniformly ultimately boundedness of error signals is proven for illustrated example.

For simulation we took the switching law σ(t) as a external periodic time function with frequency f = 0.25 Hz and equal duty of 2 seconds for each active switched subsystem. The simulation is done for t = 30 seconds. The following figures are the results for the above-considered example for square pulse setpoint with domain r(t) = {−1,+1}. The responses of state variables x₁(t), x₂(t) for uncertain switched plant are shown in Figure 2 with reference model states x_m₁(t), x_m₂(t) and setpoint input. The control signal u(t) is shown in Figure 3. Figure 4 depicts the error signals for closed-loop and uncertainty simulation, respectively.

Figure 2.

Switching signal and states of the uncertain switched plant and reference model.

Figure 3.

Control input.

Figure 4.

Closed-loop state error and estimation error for aggregated uncertainty.

To investigate the effect of the proposed structure more precisely, the novel controller gain is considered to have increasing values K_n = 0, K_n = 5, K_n = 20. Figure 5 shows the result for comparison of the UUB region size. As is seen, the size of the UUB region is smaller for larger K_n and also the performance at switching instants will be better for high controller gains. To show the efficacy of the proposed structure and observe its performance, a comparison with the standard MRAC structure is given is Figure 6. In this figure a composition structure of standard MRAC and nonlinear controller is also considered. As is seen, the proposed method is better in both transient response and reference tracking. Also using modified version of MRAC reduces the oscillations in the learning phase in comparison with the given composition structure.

Figure 5.

The effect of the novel controller gain on the closed-loop error and UUB region.

Figure 6.

Comparison of the proposed method with standard MRAC and standard MRAC with a nonlinear term.

Conclusion

In this paper, we have considered the problem of designing a model reference adaptive controller for uncertain switched systems. The key feature of this paper is that we considered various sources of uncertainty and unmodeled dynamics which have not been mentioned before in the switched systems literature. The MRAC structure is designed for an uncertain switched system and the uniform ultimate boundedness of the error signals have been shown. Accordingly the structure for the controller is a pure adaptive control law with perfect model following and novel controller term which is easy to implement to many kinds of uncertain system in comparison with other controller structures dealing with uncertainty such as robust MRAC, etc.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

De La Torre

Chowdhary

Johnson

(2013) Concurrent learning adaptive control for linear switched systems. In American Control Conference (ACC), June 2013. IEEE, pp. 854–859.

El Rifai

Youcef-Toumi

(2005) On robust adaptive switched control. In Proceedings of the 2005 American Control Conference (ACC), June 2005. IEEE, pp. 18–23.

Sun

Gao

(2012) State estimation and sliding mode control of uncertain switched hybrid systems. International Journal of Innovative Computing, Information and Control 8(10): 7143–7156.

Hespanha

Liberzon

Morse

(2003) Overcoming the limitations of adaptive control by means of logic-based switching. Systems Control Letters 49(1): 49–65.

Hou

Dong

Wang

(2008) Adaptive control scheme for linear uncertain switched systems. In AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, pp. 1–12.

Lian

Zhao

(2009a) Robust H_∞ control of uncertain switched systems: a sliding mode control design. Acta Automatica Sinica 35(7): 965–970.

Lian

Zhao

(2009b) Output feedback variable structure control for a class of uncertain switched systems. Asian Journal of Control 11(1): 31–39.

Liberzon

(2003) Switching in systems and control. Berlin: Springer.

Liberzon

Morse

(1999) Basic problems in stability and design of switched systems. IEEE Control Systems 19(5): 59–70.

10.

Liu

Jia

Niu

Zou

(2013) Design of sliding mode control for a class of uncertain switched systems. International Journal of Systems Science (in press).

11.

Michel

(1999) Recent trends in the stability analysis of hybrid dynamical systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 46(1): 120–134.

12.

Narenda

Balakrishnan

(1994) A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Transactions on automatic control 39(12): 2469–2471.

13.

Quang

(2006) Switching Robust Adaptive Control in Nonlinear Mechanical Systems. Doctoral dissertation, The University of New South Wales.

14.

Rohrs

Valavani

Athans

Stein

(1985) Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics. IEEE Transactions on Automatic Control 30(9): 881–889.

15.

Schatz

Yucelen

Johnson

Holzapfel

(2013a) Constraint enforcement methods for command governor based adaptive control of uncertain dynamical systems. In AIAA Guidance, Navigation, and Control (GNC) Conference, August 2013.

16.

Schatz

Yucelen

Johnson

(2013) Constrained adaptive control with transient and steady-state performance guarantees. In CEAS EuroGNC, Delft, the Netherlands.

17.

Singh

Sukavanam

(2012) Simulation and stability analysis of neural network based control scheme for switched linear systems. ISA Transactions 51(1): 105–110.

18.

Sun

(2005) Analysis and synthesis of switched linear control systems. Automatica 41(2): 181–195.

19.

Sun

Lee

(2002) Controllability and reachability criteria for switched linear systems. Automatica 38(5): 775–786.

20.

Tao

(2003) Adaptive control design and analysis (Wiley Series on Adaptive and Learning Systems for Signal Processing, Communication and Control, Vol. 37). New York: John Wiley & Sons Inc.

21.

Wang

Hou

Dong

(2012) Model reference robust adaptive control for a class of uncertain switched linear systems. International Journal of Robust and Nonlinear Control 22(9): 1019–1035.

22.

Wang

Zhao

Tang

(2012) State tracking model reference adaptive control for switched nonlinear systems with linear uncertain parameters. Journal of Control Theory and Applications 10(3): 354–358.

23.

Chen

(2002) Robust H -infinity control for uncertain stochastic systems with state delay. IEEE Transactions on Automatic Control 47(12): 2089–2094.

24.

Yucelen

Calise

Haddad

Volyanskyy

(2009) A Comparison of a new neuroadaptive controller architecture with the -and e-modification architectures. In AIAA Guidance, Navigation, and Control Conference, Chicago, IL, August 2009, pp. 10–13.

25.

Yucelen

Haddad

(2013) Low-frequency learning and fast adaptation in model reference adaptive control. IEEE Transactions on Automatic Control 58(4): 1080–1085.

26.

Zhang

Chen

Mastorakis

(2008) Robust uniformly ultimate boundedness control for uncertain switched linear systems. Computers Mathematics with Applications 56(7): 1709–1714.