Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems
Abstract.
Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DCDC converter.
1. Introduction
Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Although this fact already amply justifies its study, switched systems are also quite intriguing from a theoretical point of view. It is well known that by judiciously switching between stable subsystems one can render the overall system unstable. This motivated several researchers over the years to understand which classes of switching strategies or switching signals preserve stability (see e.g. [Lib03]). Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. These comprise the synthesis of control strategies guiding the switched systems through predetermined operating points while avoiding certain regions in the state space, enforcing limit cycles and oscillatory behavior, reconfiguration upon the occurrence of faults, etc.
This paper describes one particular approach to address these different objectives based on the construction of symbolic models that are abstract description of the switched dynamics and in which each abstract state, or symbol, corresponds to an aggregate of states in the switched system. When the symbolic models are finite, controller synthesis problems can be efficiently solved by resorting to mature techniques developed in the areas of supervisory control of discreteevent systems [RW87] and algorithmic game theory [AVW03]. The crucial step is therefore the construction of symbolic models that are detailed enough to capture all the behavior of the original system, but not so detailed that their use for synthesis is as difficult as the original model. This is accomplished, at the technical level, by using the notion of approximate bisimulation. Approximate bisimulation has been introduced in [GP07], as an approximate version of the usual bisimulation relation [Mil89, Par81], and in [Tab06] by using setvalued observations. It generalizes the notion of bisimulation by requiring the outputs of two systems to be close instead of being strictly equal. This relaxed requirement makes it possible to compute symbolic models for larger classes of systems as shown recently for incrementally stable continuous control systems [PGT07].
In this paper, we first extend the standard theorems on asymptotic stability of switched systems, i.e. results based on the existence a common Lyapunov function, or multiple Lyapunov functions with dwell time [Lib03], to study incremental stability of switched systems. The main contribution of the paper consists in showing that under the assumptions ensuring incremental stability of a switched system, it is possible to construct a symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. The proof is constructive and it is straightforward to derive a procedure for the computation of these symbolic models. Since in problems of practical interest the state space can be assumed to be bounded, the resulting symbolic model is guaranteed to have finitely many states and can thus be used for algorithmic controller synthesis. To support the computational merits of the proposed approach, we show how to use symbolic models to synthesize controllers for two examples of switched systems. First, we consider the boost DCDC converter, and show how to synthesize a switched controller that regulates the output voltage at a desired level. For this example, it is possible to find a common Lyapunov function, therefore, we consider a second example that illustrates the use of multiple Lyapunov functions with dwell time. A preliminary version of these results appeared in [GPT08].
In the following, the symbols , , , and denote the set of natural, integer, real, positive and nonnegative real numbers respectively. Given a vector , we denote by its th coordinate and by its Euclidean norm.
2. Switched systems and incremental stability
2.1. Switched systems
We shall consider the class of switched systems formalized in the following definition.
Definition 2.1.
A switched system is a quadruple where:

is the state space;

is the finite set of modes;

is a subset of which denotes the set of piecewise constant functions from to , continuous from the right and with a finite number of discontinuities on every bounded interval of ;

is a collection of vector fields indexed by . For all , is a locally Lipschitz continuous map.
For all , we denote by the continuous subsystem of defined by the differential equation:
(2.1) 
We make the assumption that the vector field is such that the solutions of the differential equation (2.1) are defined on an interval of the form with . Necessary and sufficient conditions to be satisfied by can be found in [AS99]. Simpler, but only sufficient, conditions include linear growth or compact support of the vector field .
A switching signal of is a function , the discontinuities of are called switching times. A piecewise function is said to be a trajectory of if it is continuous and there exists a switching signal such that, at each where the function is continuous, is continuously differentiable and satisfies:
We will use to denote the point reached at time from the initial condition under the switching signal . The assumptions on the vector fields ensure for all initial conditions and switching signals, existence and uniqueness of the trajectory of . Furthermore since switching signals have only a finite number of discontinuities on every bounded interval, Zeno behaviors are ruled out. Let us remark that a trajectory of is a trajectory of associated with the constant switching signal , for all . Then, we will use to denote the point reached by at time from the initial condition .
2.2. Incremental stability
The results presented in this paper rely on some stability notions. A continuous function is said to belong to class if it is strictly increasing and . Function is said to belong to class if it is a function and when . A continuous function is said to belong to class if for all fixed , the map belongs to class and for all fixed , the map is strictly decreasing and when .
Definition 2.2.
[Ang02] The subsystem is incrementally globally asymptotically stable (GAS) if there exists a function such that for all , for all , the following condition is satisfied:
Intuitively, incremental stability means that all the trajectories of the subsystem converge to the same reference trajectory independently of their initial condition. This is an incremental version of the notion of global asymptotic stability (GAS) [Kha96]. Let us remark that when satisfies then GAS implies GAS, as all the trajectories of converge to the trajectory . Further, if is linear then GAS and GAS are equivalent. Similarly to GAS, GAS can be characterized by dissipation inequalities.
Definition 2.3.
A smooth function is a GAS Lyapunov function for if there exist functions , and such that:
(2.2)  
(2.3) 
The following result completely characterizes GAS in terms of existence of a GAS Lyapunov function.
Theorem 2.4.
[Ang02] is GAS if and only if it admits a GAS Lyapunov function.
Remark 2.5.
For the purpose of this paper, we extend the notion of incremental stability to switched systems as follows:
Definition 2.6.
A switched system is incrementally globally uniformly asymptotically stable (GUAS) if there exists a function such that for all , for all , for all switching signals , the following condition is satisfied:
(2.4) 
Let us remark that the speed of convergence specified by the function is independent of the switching signal . Thus, the stability property is uniform over the set of switching signals; hence the notion of incremental global uniform asymptotic stability. Incremental stability of a switched system means that all the trajectories associated with the same switching signal converge to the same reference trajectory independently of their initial condition. This is an incremental version of global uniform asymptotic stability (GUAS) for switched systems [Lib03]. If for all , (i.e. all the subsystems share a common equilibrium), then GUAS implies GUAS as all the trajectories of converge to the constant trajectory . Further, if for all , is linear, GUAS and GUAS are equivalent.
It is well known that a switched system whose subsystems are all GAS may exhibit some unstable behaviors under fast switching signals. The same kind of phenomenon can be observed for switched systems with GAS subsystems. Similarly, the results on common or multiple Lyapunov functions for proving GUAS of switched systems (see e.g. [Lib03]) can be extended to prove GUAS. Let the functions , and the real number be given by , and .
Theorem 2.7.
Consider a switched system . Let us assume that there exists which is a common GAS Lyapunov function for subsystems . Then, is GUAS.
Proof.
When a common GAS Lyapunov function fails to exist, GUAS of the switched system can be ensured by using multiple GAS Lyapunov functions and a restrained set of switching signals. Let denote the set of switching signals with dwell time so that has dwell time if the switching times satisfy and , for all .
Theorem 2.8.
Let and consider a switched system with . Let us assume that for all , there exists a GAS Lyapunov function for subsystem and that in addition there exists such that:
(2.5) 
If , then is GUAS.
Proof.
We shall prove the GUAS property only for switching signals with an infinite number of discontinuities but a proof for signals with a finite number of discontinuities can be written in a very similar way. Let , , let and let denote the value of the switching signal on the open interval , for . From equation (2.3), for all and
Then, for all and ,
(2.6) 
Particularly, for and from equation (2.5), it follows that for all ,
Using this inequality, we prove by induction that for all
(2.7) 
Then, from equations (2.6) and (2.7), for all and ,
Since the switching signal has dwell time , it follows that and therefore for all , . Since , then for all and ,
Hence, for all and
Therefore, for all ,
Equation (2.4) holds with the function given by which belongs to class since by assumption . The same inequality can be shown for switching signals with a finite number of discontinuities; thus, is GUAS. ∎
In the following, we show that under the assumptions of Theorems 2.7 or 2.8, ensuring incremental stability, it is possible to compute approximately equivalent symbolic models of switched systems. We will make the following supplementary assumption on the GAS Lyapunov functions: for all , there exists a function such that
(2.8) 
Note that is not a function of the variable . It is convenient, for later use, to define the function by . We will discuss this assumption later in the paper and we will show that it is not restrictive provided we are interested in the dynamics of the switched system on a compact subset of the state space .
3. Approximate bisimulation
In this section, we present a notion of approximate equivalence which will relate a switched system to the symbolic models that we construct. We start by introducing the class of transition systems which allows us to model switched and symbolic systems in a common framework.
Definition 3.1.
A transition system is a sextuple consisting of:

a set of states ;

a set of labels ;

a transition relation ;

an output set ;

an output function ;

a set of initial states .
is said to be metric if the output set is equipped with a metric , countable if and are countable sets, finite, if and are finite sets.
The transition will be denoted and means that the system can evolve from state to state under the action labelled by . Thus, the transition relation captures the dynamics of the transition system.
Transition systems can serve as abstract models for describing switched systems. Given a switched system where , we define the associated transition system where the set of states is ; the set of labels is ; the transition relation is given by
i.e. subsystem goes from state to state in time ; the set of outputs is ; the observation map is the identity map over ; the set of initial states is . The transition system is metric when the set of outputs is equipped with the metric . Note that the state space of is infinite.
Usual equivalence relationships between transition systems rely on the equality of observed behaviors. In this paper, we are mostly interested in bisimulation equivalence [Mil89, Par81]. Intuitively, a bisimulation relation between two transition systems and is a relation between their set of states explaining how a trajectory of can be transformed into a trajectory of with the same associated sequence of outputs, and vice versa. The requirement of equality of output sequences, as in the classical formulation of bisimulation [Mil89, Par81] is quite strong for metric transition systems. We shall relax this, by requiring output sequences to be close where closeness is measured with respect to the metric on the output space. This relaxation leads to the notion of approximate bisimulation relation introduced in [GP07].
Definition 3.2.
Let , be metric transition systems with the same sets of labels and outputs equipped with the metric . Let be a given precision, a relation is said to be an approximate bisimulation relation between and if for all :

;

for all , there exists , such that ;

for all , there exists , such that .
The transition systems and are said to be approximately bisimilar with precision , denoted , if:

for all , there exists , such that ;

for all , there exists , such that .
4. Approximately bisimilar symbolic models
In the following, we will work with a subtransition system of obtained by selecting the transitions of that describe trajectories of duration for some chosen . This can be seen as a sampling process. Particularly, we suppose that switching instants can only occur at times of the form with . This is a natural constraint when the switching in has to be controlled by a microprocessor with clock period . Given a switched system where , and a time sampling parameter , we define the associated transition system where the set of states is ; the set of labels is ; the transition relation is given by
the set of outputs is ; the observation map is the identity map over ; the set of initial states is . The transition system is metric when the set of outputs is equipped with the metric .
4.1. Common Lyapunov function
We first examine the simpler case when there exists a common GAS Lyapunov function for subsystems . We start by approximating the set of states by the lattice:
where is a state space discretization parameter. By simple geometrical considerations, we can check that for all , there exists such that .
Let us define the approximate transition system where the set of states is ; the set of labels remains the same ; the transition relation is given by
the set of outputs remains the same ; the observation map is the natural inclusion map from to , i.e. ; the set of initial states is . Note that the transition system is countable. Moreover, it is metric when the set of outputs is equipped with the metric . An illustration of the approximation principle is shown on Figure 1.
We now give the result that relates the existence of a common GAS Lyapunov function for the subsystems to the existence of approximately bisimilar symbolic models for the transition system .
Theorem 4.1.
Consider a switched system with , time and state space sampling parameters and a desired precision . Let us assume that there exists which is a common GAS Lyapunov function for subsystems and such that equation (2.8) holds for some function . If
(4.1) 
then, the transition systems and are approximately bisimilar with precision .
Proof.
We start by showing that the relation defined by , if and only if , is an approximate bisimulation relation. Let , then we have that Thus, the first condition of Definition 3.2 holds. Let , then . There exists such that Then, we have . Let us check that . From equation (2.8),
It follows that
(4.2)  
because is a GAS Lyapunov function for subsystem . Then, from equation (4.1) and since is a function,
Hence, . In a similar way, we can prove that, for all , there is such that . Hence is an approximate bisimulation relation between and .
By definition of , for all , there exists such that . Then,
because of equation (4.1) and is a function. Hence, . Conversely, for all , , then and . Therefore, and are approximately bisimilar with precision . ∎
Let us remark that, for a given time sampling parameter and a desired precision , there always exists sufficiently small such that equation (4.1) holds. This means that for switched systems admitting a common GAS Lyapunov function there exists approximately bisimilar symbolic models and any precision can be reached for all sampling rates.
The approach presented in this section for the computation of symbolic abstractions is quite similar to the approach presented in [PGT07] for GAS continuous control systems. Though, instead of defining the approximate bisimulation relation using the infinity norm as in [PGT07], we use sublevel sets of the common GAS Lyapunov function. This makes it possible, unlike in [PGT07], to compute symbolic models for arbitrary small time sampling parameter . Further, this allows us to extend our approach to switched systems with multiple GAS Lyapunov functions.
4.2. Multiple Lyapunov functions
If a common GAS Lyapunov function does not exist, it remains possible to compute approximately bisimilar symbolic models provided we restrict the set of switching signals using a dwell time . In this section, we consider a switched system where . Let be a time sampling parameter; for simplicity and without loss of generality, we will assume that the dwell time is an integer multiple of : there exists such that . Representing using a transition system is a bit less trivial than previously as we need to record inside the state of the transition system the time elapsed since the latest switching occurred. Thus, the transition system associated with is where:

The set of states is , a state means that the current state of is , the current value of the switching signal is and the time elapsed since the latest switching is exactly , if , or at least , if .

The set of labels is .

The transition relation is given by if and only if and one the following holds:

, , and : switching is not allowed because the time elapsed since the latest switch is strictly smaller than the dwell time;

, , and : switching is allowed but no switch occurs;

, , and : switching is allowed and a switch occurs.


The set of outputs is .

The observation map is given by .

The set of initial states is .
One can verify that the output trajectories of are the output trajectories of associated with switching signals with dwell time . The approximation of the set of states of by a symbolic model is done using a lattice, as previously. Let be a state space discretization parameter, we define the transition system where:

The set of states is .

The set of labels remains the same .

The transition relation is given by if and only if and one of the following holds:

, , and ;

, , and ;

, , and .


The set of outputs remains the same .

The observation map is given by .

The set of initial states is .
Note that the transition system is countable. Moreover, and are metric when the set of outputs is equipped with the metric . The following theorem establishes the approximate equivalence of and .
Theorem 4.2.
Consider , a switched system with , time and state space sampling parameters and a desired precision . Let us assume that for all , there exists a GAS Lyapunov function for subsystem and that equations (2.5) and (2.8) hold for some and functions . If and
(4.3) 
then, the transition systems and are approximately bisimilar with precision .