6pre-determined As another example of restricted switching, it might be desired to have

a certain limit on the time interval between two consecutive switching time instants. The reason might be related to the fact that the state trajectories have to stay for some time interval in a certain set before traveling to another sets. With this a priori informa- tion about the switching signal and the restrictions, we may be able to obtain stronger stability results for a given switched system rather than in the arbitrary switching case where we in fact, consider the worst case scenarios [150].

This section will present stability analysis of the switched systems under the re- stricted switching signals. Having this problem solved, we will be able to find out ap- propriate restrictions that must be imposed on the switching signals in order to ensure the stability of switched systems. The restrictions on switching signals may be in the time domain (e.g. dwell time and average dwell time between switching signals) or in the state space (e.g. abstractions from partitions of the state space).

In case of stable subsystems, fast enough switching may lead to instability. This might be explained by failing to absorb the energy increase caused by the switching [51]. On the other hand, when there is an unstable subsystem, if the system stays too long or switches too frequently to this subsystem, the stability may be destroyed. There- fore, if the system dynamics are governed by the stable subsystems long enough and it switches less frequently, then the system may be able to attenuate the energy increase resulted from switching or from staying in unstable modes and preserve the stability. This idea is mathematically formulated in the concepts of dwell time and average dwell time switching proposed in [110, 173].

Definition 6.1. A positive constant TDis called the dwell time of a switching signal if the time interval between any two consecutive switchings is not smaller than TD.

It can be proved that it is always possible to preserve stability when all subsystems are stable and the switching is slow enough, meaning thatTDis sufficiently large [173].

On the other hand, if occasionally the time interval between two successive switching becomes smaller than the dwell timeTD, provided this does not occur too frequently,

overall stability may be preserved. This idea is captured by the concept ofaverage dwell timein [110].

Definition 6.2. A positive constant TADis called the average dwell time for a switching signalσif

Nσ(t0,t)≤N0+t−t0

TAD (6.7)

holds for all t ≥t0and a constant parameter N0≥0. The value Nσ(t0,t)denotes the number of switchings that occur over the interval(t0,t).

It can be inferred from (6.7) that on average the dwell time between any two consec- utive switching instants is not smaller thanTAD. It is proved in [110] that if all subsystems

are exponentially stable, then the switched system is exponentially stable provided that the average dwell time is sufficiently large. Moreover, it can be shown that using the av- erage dwell time notion, we will be able to characterize a larger class of stable switching signals than by using the fixed dwell time concept as in Definition 6.1. Interested readers may refer to [33, 35, 53, 112, 130, 242] for further details and recent applications of the concept of average dwell time for stability and stabilization of switched systems.

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6.2.3.R

S

S

L

S

In the previous two sections, we have discussed stability properties of switched systems under given switching signals, which may be restricted or arbitrary. The problem stud- ied was under what conditions on the dynamics of the subsystems and/or on the switch- ing signals the switched system is stable. Another interesting problem for switched sys- tems is the synthesis of stabilizing switching signals for a given set of dynamical subsys- tems, called the switching stabilization problem.

The stability analysis and design of stabilizing switching laws have been usually per- formed in the framework of multiple Lyapunov functions (MLF). The main idea is that multiple Lyapunov-like functions each corresponding to a single subsystem or a certain region in the state space, are concatenated to make a global Lyapunov function. The MLF might not monotonically decrease along the state trajectories and may have dis- continuities and therefore, be piecewise differentiable. However, often the only require- ment is that the MLF must have nonpositive Lie-derivatives for particular subsystems in particular regions of the state space, instead of having globally negative derivative. There are several results regarding the MLF concept in the literature [56, 73, 208]. The MLF approach in [51] corresponds to the case in which the Lyapunov-like function is decreasing whenever the corresponding subsystem becomes active and its value does not increase at each switching instant. However, one may be able to obtain less con- servative results. For instance, the switching signals may be constrained such that at every time when the system switches away from a subsystem, the value of the corre- sponding Lyapunov function must be smaller than its value at the previous switching time instant. Hence, the switched system would be asymptotically stable [23]. In other words, for each subsystem the values of the corresponding Lyapunov-like function at switching time instants in which the subsystems is inactivated, construct a monotoni- cally decreasing sequence. Moreover, as a different approach, the Lyapunov-like func- tion may increase its value during a time interval, only if the increment is bounded by a certain type of continuous-time functions [119].

In the switching stabilization literature, most of the papers focus on the quadratic stabilization. A system is quadratically stable if there exists a quadratic Lyapunov func- tionV(x)=xTP xwith a quadratic bound on the derivative of the Lyapunov function with respect to time of the form ˙V(x)≤ −ǫkxk2for someǫ>0. A necessary and sufficient condition for a switched system composed of two linear subsystems to be quadratically stabilizable is the existence of a stable convex combination of the twoAimatrices [233]. A generalization to more than two linear subsystems is proposed in [192] by using a min-projection strategy, as presented in the following theorem.

Theorem 6.3. [192] If there exist a positive definite matrix P >0and constantsαi ∈

[0,1],Pi_∈N αi=1such thatPi_∈N αiAiis stable, i.e.: X

i_∈N αi

¡

AT_iP+P Ai¢<0 (6.8)

then the min projection scheme

σ(t)=arg min

i∈Nx(t) T

P Aix(t) (6.9)

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