Stability of Switched Systems

Ahmet Cetinkaya

March 2009

A switched system is a dynamic system composed of a number of subsystems and a rule that manages the switching between these subsystems. Switched systems are capable of describing systems with multiple dynamics where one of them governs the system at a time [1]. In addition, they can be used to model systems of a single process being controlled by multiple switching controllers. In practice, switched systems can be applied to various modeling and control problems present in robotics, automotive systems, process control, power systems, and air traffic control [2].

In switched systems, a switching signal determines the subsystem that will govern the dynamics of the overall system [1]. Each value of the switching signal corresponds to an index of the subsystems. This value may change depending on the time, the states of the system or the previous values of the switching signal. The switching signal may also be probabilistic [1].

There has been research on stability of switched systems under arbitrary switching. Research in this field revealed that if there exists a common quadratic Lyapunov function for all subsystems of a switched system, then the overall system will be stable under any switching signal [2]. Some subsystems having some particular properties may share a quadratic Lyapunov function. For instance, it is stated in [3] that for a switched system composed of asymptotically stable linear subsystems, if the subsystems are pairwise commutative, then a common quadratic Lyapunov function exists which guarantees the stability of the overall system. If a common Lyapunov function is not known or does not exist, one may investigate different Lyapunov functions for each subsystem. In such cases a Lyapunov function should have a lower value each time when its corresponding subsystem gets active [4].

In addition to the characteristics of subsystems, switching signal is another factor that has to be taken into account for stability. First, even if all the subsystems of a switched system are stable, there may be some switching sequence that the output of the overall system diverges. Second, switching by itself can make the overall system stable even if the subsystems are unstable. Some switched systems do not have stability under arbitrary switching; however such systems may be stable when the switching signals are restricted [2].

Continuous-time switching control systems may have the problem of arbitrarily fast switching which may result in damage to the actuators and/or the plant [5]. As a result, there has been research on switching with dwell time which introduces a restriction on the switching signal. In systems with dwell time, before a switching occurs, a subsystem must be active at least for a predefined amount of time which is also called the dwell time [4].

The realization of arbitrary switching multiple controllers for stabilization of a single process is addressed in [6]. In multiple switching controller systems, in addition to control laws, switching signal is also a design variable. In this case, appropriate controllers and switching sequences are designed together [1]. Synthesis of switching signals that stabilize switched systems is investigated in several papers [7, 8, 9, 10]. Periodic switching signals are addressed in [8]. In [9] design of switching sequences that stabilize discrete-time switched linear systems is considered. The same issue is also addressed in [11], where directed graphs are used to demonstrate the possible transitions between subsystems. Moreover, [11] explores the sets of switching sequences that can stabilize a discrete- time switched system. The periodic and aperiodic switching sequence design problems are explored in [10]. Furthermore, the design of stabilizing switching sequences with minimum number of switchings are investigated in [12]. Controllability issue of switched systems is addressed in [13]. It is asserted in [13] that controllability can be achieved by repetitively using only one switching sequence.

In addition to the deterministic switched system studies, there has also been research on stochastic switched systems. In literature, researchers have proposed different models to describe stochastic switched systems. The place where randomness arise is the main difference between these models [14]. First type of these models introduce the randomness in the dynamics of each mode of the switched system. In this type, subsystem dynamics are described by stochastic differential equations [14]. There has been research on the stability of systems described by these models. Stability analysis for linear stochastic systems is given in [15]. Moreover, for switched stochastic systems, sufficient conditions for stability are explored in [14], where common and multiple Lyapunov functions are used.

Another model for stochastic switched systems introduces the randomness in the transition between subsystems. Here, the switching signal, which manages the transition between the subsystems of a switched system, is probabilistic. If this probabilistic switching signal is the state of a finite-state Markov chain then the overall system is called a Markov jump system. There has been considerable amount of research on Markov jump linear systems due to their applicability in modeling real life systems that are subject to changes in their structure or parameters [16, 17].

Many researchers studying Markov jump linear systems have addressed the stochastic stabilization issue. The different stochastic stability notions used in literature are mean-square stability, δ-moment stability and almost sure stability [18, 19]. Mean-square stability is a special case of δ-moment stability where δ equals two. It requires the convergence of second moment of the state norm. Almost sure stability is assured when the probability that the state's sample path converges to zero is one [16, 20].

It is stated in [20] that mean-square stabilization problem can be reduced to an optimal control problem and the solution of a set of coupled Riccati equations provides sufficient and necessary conditions for mean-square stability. An algorithm for solving coupled Riccati equations is also given in [20]. Furthermore, [21] explores the mean-square stabilization of Markov jump linear systems where the jumping parameters are not exactly known. A method using the estimates of the jumping parameters is employed to derive necessary and sufficient conditions for stability [21].

Relations between different stability notions and Lyapunov exponent method for stabilization are studied in [18]. Almost sure stabilization of continuous- time Markov jump linear systems is discussed and sufficient conditions for the stability are given in several papers [19, 16, 22]. Moreover, [17] presents sufficient conditions for stability of linear and nonlinear Markov jump systems that are subject to disturbance.

In addition to Markov jump linear systems, some studies explored switched systems with a switching signal governed by a Poisson process. Consensus problem for stochastic switched linear systems with Poisson switching is investigated in [23]. Stability analysis and stabilization of switched systems with probabilistic switching are discussed in [24]. In the same study, sufficient conditions for almost sure stability are given for switched systems that have statistically slow switching. The slow switching condition is given in [24] that the number of switchings occur on any time interval is upper bounded by "the probability mass function of a Poisson process". Furthermore, stability of asynchronous systems with Poisson switchings are discussed in [25, 26].

It is asserted in [14] that more general switching system models can be adopted combining probabilistic switching signals with subsystems that have stochastic dynamics. These systems include diffusive motion (such as Brownian motion) in their dynamics.

Yin states in [27] that in spite of the difficulty in their analysis, switching systems with diffusive motion provide more realistic models than the counterpart which lacks diffusion in the dynamics. Such systems have been investigated under the name "switching diffusion processes". A practical use of switching diffusion models is given by Khasminskii et al. in [28]. Khasminskii et al. state that due to changes in economical conditions, stock markets may have different modes: "up"s and "down"s. Switching diffusion models can be employed for describing "stochastic volatility" in stock markets [28].

There has been research devoted on the stability of switching diffusions [27-31]. In these studies stochastic differential equations with Markovian switching are considered. Exponential stability of switching diffusion processes is investigated by Yuan in [29] where Lyapunov functions are employed to give sufficient conditions ensuring the stability. Yuan also considered the systems with uncertain parameters. In [28], necessary conditions for δ-moment stability are presented. In the same study, an example switching system comprising some stable and unstable one-dimensional subsystems is discussed. It is stated that although some of subsystems are individually unstable, the properties of Markovian transition enabled stability for the overall system [28]. In [31] δ-moment stability is investigated for stochastic differential equations with Markovian switching. In the same research impulsive jumps on switching instants are also taken into account. In [30] some sufficient conditions are given for guaranteeing the stability of Markovian switching diffusions. Authors of the same work also studied linear switching diffusions where each subsystem is described by a linear stochastic differential equation. Necessary and sufficient conditions of stability is given for linear switching diffusions [30]. Another result given in this research is that multiplicative Brownian motion in the dynamics can not destabilize a one-dimensional system if the system without the diffusive motion is stable. It is presented in [32] that under certain conditions an unstable linear system with Markovian switching can be stabilized with noisy perturbation. In [27] Yin introduces the notion of practical stability for switching diffusion processes. It is suggested in [27] that in some practical applications even though the stability conditions are not satisfied, the performance level may still be acceptable if the system is "practically stable". It is also stated in the same work that switching by itself can lead to practical stability even some subsystems are unstable [27].

Switching diffusions has possible practical applications. Yin mentions that these processes can be used to model stock price movements and they can be employed in solving control and optimization problems of wireless communications [27]. In [33], the authors mention about the Black-Scholes options pricing model and state that this model assumes constant rate of return and volatility values. However, volatility may change in time and it may actually be a stochastic process. Stochastic volatility models such as Heston model are explored in [33]. Another example use of switching diffusions is analyzing the dynamics of populations. In [34] the authors state that the dynamic population models developed using switching diffusion processes have desired properties. Such models are investigated in [33].