Dissipative dynamic output feedback control for switched systems via multistep Lyapunov function approach

Introduction

The basic problem in the control theory is to explore control laws such that desired control performance can be guaranteed (Riggs, 1971). Among various control strategies, state feedback and output feedback are most well-known, where the former one has critical requirement on measured states (Geromel and Deaecto, 2009) and the latter one is relatively easy in application due to the accessibility of outputs (Li et al., 2019). Therefore, burgeoning results on output feedback have been reported in existing literatures and such a control approach can be roughly divided into static and dynamic one. The structure of static output feedback (SOF) is simple and its implementation is convenient (Boukas, 2006; Lee et al., 2006). However, it is usually difficult to formulate the stabilization problem of SOF into a standard linear matrix inequality (LMI), especially for complex systems. Even though bilinear matrix inequality (BMI) formulation (Dinh et al., 2012) is developed to deal with such a problem but as mentioned in Blondel and Tsitsiklis (1997), BMI is non-convex and NP-hard. Unlike SOF that still remains difficult solvability problems (Bara and Boutayeb, 2005), dynamic output feedback (DOF) control is relatively mature in theoretical design.

In terms of engineering applications, DOF drives extensive investigations. For instance, in networked control systems (NCSs), the plant and controller are spatially separated, where the control-loop is physically linked by a communication network, and it may lead to network-induced delays and data packet dropouts (Bara and Boutayeb, 2005). Generally, obtaining an accurate knowledge of the plant state to achieve a routine state feedback control is quite challenging since some inner state variables cannot be directly measured by sensor nodes. Then, a natural consideration is to design observer-based controllers. However, the estimated error is difficult to be obtained due to the time delay between the control input at the plant side and observer side. Therefore, DOF can be applied to NCS with transmission time delay and data packet dropouts (Chae and Nguang, 2014). Another typical example of the DOF application is wireless control system (WCS) with control network. In a WCS, it is very complicated to share local control of each node when constructing an observer. That is to say, observer-based controllers need a unified control input for the purpose of observing error computing. Therefore, it is reasonable to design DOF controllers when each node can design and perform local control without a unified control input knowledge.

On the other hand, dissipativity theory provides a framework for the analysis and synthesis of control systems (Tan et al., 1999). Specifically, it provides a new method for the construction of Lyapunov function (LF). LF is quite often used in modern control theory while there are few practical construction methods. The storage function of dissipative systems can be constructed as LF under certain conditions. Furthermore, it can be easily extended to many applicable problems, such as system stabilization, adjustment, robust control, adaptive control, optimal control, H ∞ control and other important topics. In terms of engineering application, the dissipative theory can be applied to many practical systems, such as robot systems (Bowyer and Rodriguez, 2015), traffic signal control (Motawej et al., 2011), internal combustion engine systems (Storm, 2003), chemical process (Alonso et al., 2010), and so forth. To this end, dissipativity theory plays a very important role, sometimes even indispensable. General theory of dissipative dynamical systems was first outlined in Willems (1972a) and the dissipativity was characterized by storage functions and supply rates. A state space approach for linear dissipative dynamical systems was further addressed in Willems (1972b). Extension results to the case of affine nonlinear systems were carried out in Hill and Moylan (1976, 1977). Some important advances have been reported for the dissipativity property of switched systems. In Zhao and Hill (2008), novel cross-supply rates among subsystems were introduced to describe the changes of storage function with inactive subsystems. The concept of decomposable dissipativity was firstly proposed in Liu and Hill (2011) and the non-positive condition of the supply rate was relaxed by the decomposition of supply rate. By imposing decreasing condition on storage functions at switching instants, the dissipativity and induced Lyapunov stability of switched systems was established (Xiang et al., 2015). In general cases, the dissipativity of switched systems is established by constructing LF. Such a decreasing requirement on LFs at switching instants was replaced by an increasing one in Park et al. (2014). However, the LFs in existing literature are required to be monotonicly decreasing within subsystems, which motivates us to explore a non-monotonic LF approach to further reduce such a conservatism.

For the average dwell time (ADT) switched systems (Yin, 2017a, 2017b), the LFs are allowed to increase at the switching instant rather than within the subsystem (Zhao et al., 2012). Some pioneering works on the non-monotonic LF have been reported. For instance, non-monotonic LFs are firstly developed in Ahmadi and Parrilo (2008) to stabilize nonlinear discrete-time switched systems such that less conservatism results can be obtained. Based on this fundamental work, a one-step ahead LF approach that can ensure globally uniformly asymptotically stable (GUAS) property for arbitrarily switched systems with further less conservativeness was formulated in Wen et al. (2018). Similar method was adopted to reduce conservatism in a class of non-homogenous Markovian jump systems by allowing LF to increase during the period of several sampling time step ahead of the current time within each jump mode (Wen et al., 2017). However, to our best knowledge, the exploration of non-monotonic results for ADT switched systems still remains open, especially when the N-step ahead scenario is taken into account (Xie et al., 2017). The essential difficulty is to construct an exponential damping law of the LF decreasing points, that is, to find the joint-point between the switching interval and the predictive horizon.

Therefore, from the conservatism reduction point of view, a non-monotonic dissipative DOF control approach is developed for a class of switched systems in this paper. The underlying LF is allowed to increase both at the switching instants and during the running time of each subsystem. The main contributions and novelty of this paper are summarized as follows:

Notations: R n represents the n dimensional Euclidean space and Z + is the set of nonnegative integers. The superscript “ T “ stands for matrix transposition, symbol * is an ellipsis for symmetric terms and diag { ⋯ } stands for a block-diagonal matrix. l 2 [ 0 , ∞ ] is the space of square summable infinite sequence for exogenous noise w k ∈ l 2 [ 0 , ∞ ] . The notation P > 0 ( P ≥ 0 ) denotes a positive definite (positive-semi definite) matrix. ℵ 1 , r represents a set takeing values in an integer set { 1 , 2 , 3 , ⋯ , r } , where r ∈ Z + .

System description and preliminaries

Consider the following discrete-time switched systems

where x ( k ) ∈ R n x is the state, y ( k ) ∈ R n y and z ( k ) ∈ R n z are the measured output and controlled output respectively, ω ( k ) ∈ R n ω represents disturbance belonging to l 2 [ 0 , ∞ ] . The switching signal σ ( k ) takes its value in the finite set ℵ 1 , r and σ ( k ) = i means the i th subsystem is activated. Matrices A i , B i , B i ω , C i , D i , L i and D i ω are are constant matrices with appropriate dimensions. Furthermore, Δ A i η represents norm-bounded uncertainties decomposed as E i F η H i . E i and H i are known matrices, which characterize the structure of the uncertainties. F η ( η ∈ [ k , k + N ) ) models the time-varying element that satisfying F η T F η < I .

Definition 1: (Li et al., 2018) For a finite switching sequence σ ( k ) and any K > k > k 0 , N σ ( K , k ) stands for the switching numbers of σ ( k ) over the interval [ k , K ) . If there exists N 0 > 0 (the chatter bounds) such that N σ ( K , k ) ≤ N 0 + ( K − k ) / τ a holds, then we call τ a as the ADT of the system.

In this paper, the DOF controller is formulated in the following form

where x ^ k is the state vector of the controller, A ^ i , B ^ i and C ^ i are controller matrices with appropriate dimensions to be designed. By constructing an augmented state vector x ~ ( k ) = [ x T ( k ) x ^ T ( k ) ] T , it follows from (1) and (2) that

where

Now, introduce quadratic energy supply function F ( · , · ) associated with the above augmented system

where Q = Q T > 0 , S = S T and R ≤ 0 are real matrices of appropriate dimensions.

Definition 2: (Tan et al., 1999) Under zero initial condition x ( 0 ) = 0 , the system (3) with energy supply (4) is strictly ( Q , S , R ) -dissipative, if there exists a sufficiently small scalar β such that the following inequality holds for all T ≥ 0 and ω k ∈ l 2 [ 0 , T ]

Remark 1: A typical example that well addresses the definition of dissipativity is a resistance, inductance, capacitance (RLC) circuit. When we give a direct current voltage source to such a circuit, the electrical energy is then interactive between inductor and capacitor, and gradually consumed by the resistor. If we choose the capacitor voltage as a state variable, then the state behavior is a damped oscillation process, which means the energy consumed within the system does not exceed the energy supplied by the outside.

In the following, the main objective is to derive general sufficient conditions that guarantee the robust GUAS and strict dissipativity of the system (3).

Dissipative analysis

A MLF technique is addressed in this section and dissipativity conditions are formulated by a set of LMIs.

Theorem 1: Let 0 < α < 1 , μ > 1 be given constants, and Q , S , R be given matrices with Q and R symmetric. The switched system (3) is GUAS and strictly ( Q , S , R ) - dissipative when ω k = 0 , if the ADT of switching signal σ ( k ) satisfies τ a > τ a * = − N ln μ / ln ( 1 − α ) and there exists a set of symmetric matrices Z i , f ( f ∈ ℵ 1 , N − 1 ) and P i > 0 such that the following LMIs hold. That is

for N = 2

for N ≥ 3 and f ∈ ℵ 2 , N − 1

and for all σ ( k l ) = i , σ ( k l − ) = j

Proof: Since the derivation for the cases of N = 2 and N ≥ 3 are in a similar routine, we give the proof for N ≥ 3 in details and only some brief notes for N = 2 .

The dissipativity investigation in this paper is based on the system stability. Therefore, before performing dissipative analysis, we aim at presenting sufficient conditions for the GUAS of system (3), which can be implied by Theorem 1.

The LF in this paper is taken as V i ( x ~ k ) = x ~ k T P i x ~ k and its N samples variation is

Let us further consider an auxiliary quadratic function

In view of conditions (8)–(10), we obtain the following LMIs by performing some basic matrix manipulations

The following inequalities

can be inferred from the above (13)–(15) when we assume zero disturbance input to the system.

It followed from LMIs (16)–(18) that

Assume that the switching behavior is much slower than sampling time step, it follows from (19) that

Denote k l as the switching times and when condition (11) holds, it results in

Combing (20) and (21), one obtains

If the ADT satisfies τ a > τ a * = − N ln μ / ln ( 1 − α ) , we can directly conclude that V σ ( k ) ( x ~ k ) ≤ μ N 0 V σ ( k 0 ) ( x ~ k 0 ) holds. Therefore, conditions (8)–(10) guarantee the GUAS of system (3) without disturbance for N ≥ 3 .

Then, by assuming system (3) is under zero-initial condition, we are in the position to perform dissipative analysis.

Due to the non-monotonic feature of MLF approach, a set of auxiliary matrices Z i , f are introduced to construct connection from k to k + N . Along the trajectory of system (3)

can be directly inferred from (8)–(10).

It follows from conditions (23)–(25) that

Under zero initial conditions x 0 = x − 1 = x − 2 ⋯ = x − N + 1 = 0 , we have

When T → ∞ , it straightforwardly results in

It is obvious that ∑ k = 0 T F ( ω k , z k ) ≥ 0 and there exists a sufficiently small β such that ∑ k = 0 T F ( ω k , z k ) ≥ β ∑ k = 0 T ω k T ω k . The proof is completed.

Remark 2: It should be noted that N = 1 is one of the special cases of MLF approach since it is required to decrease monotonically within subsystems. The dissipative analysis for N = 1 is omitted since the derivation is quite easy. Furthermore, note that N has direct impact on the number of the inequality constraints. For N = 2 , conditions (16) and (18) can be obtained from (6) and (7), respectively. Then corresponding dissipative analysis can be performed by following the same lines of the proof for N = 3 .

Remark 3: ( Q , S , R ) -dissipative covers various forms of dissipativity, such as H ∞ performance and passivity, as special cases.

Robust dissipative DOF control

In this section, robust dissipative DOF controller design is developed based on the dissipative analysis.

Theorem 2: Let 0 < α < 1 , μ > 1 , ε i > 0 be given constants and matrices Q , S , R be given matrices with Q and R symmetric. For ADT of the switching signal, σ ( k ) satisfies τ a > τ a * = − N ln μ / ln ( 1 − α ) , if there exist matrices T i , V i , A icc and a set of symmetric matrices W i 11 > 0 , W i 12 > 0 , W i 22 > 0 , X i , Y i , K i , f 11 , K i , f 12 , K i , f 22 ( f ∈ ℵ 1 , N − 1 ) such that the following LMIs hold. That is

for N = 2

for N ≥ 3

and for σ ( k l ) = i , σ ( k l − ) = j

where

Then the system (3) is GUAS with the controller in the form of (2) and the control matrices can be obtained as

Proof: In this section, we still focus on the derivation for N = 3 . Let us consider the same LF and auxiliary quadratic function as Theorem 1.

For matrix transformation, matrices G i is introduced as follows

where both X i and Y i are symmetric. Pre- and post- multiply diag { G i P i − 1 , I , I , I } and diag { P i − 1 G i T , I , I , I } to (9), respectively. In view of the fact that − G i T − G i + P i ≤ − G i P i − 1 G i T , one obtains

When taking the uncertainties into account, we denote

Subsequently, (32) can be decomposed as

where

Due to the fact that Γ 0 + E ~ i F ~ η H ~ i + ( E ~ i F ~ η H ~ i ) T ≤ ε i E ~ i E ~ i T + ε i − 1 H ~ i T H ~ i , the following LMI

can be obtained by employing Schur’s complement.

Performing a congruence transformation to (33) by diag { J i T , J i T , I , I } on the left and diag { J i , J i , I , I } on the right with J i = [ Y i I Y i 0 ] .

Then, condition (33) can be transformed into

Set J i T P i J i ≜ W i = [ W i 11 W i 12 * W i 22 ] and J i T Z i , f J i ≜ K i = [ K i , f 11 K i , f 12 ∗ K i , f 22 ] , one can readily obtain J i T ( − G i T − G i + P i ) J i =