solving multi-parameteric optimization problems and/or gridding techniques. There- fore, in this thesis, we aim at tackling the stability problem for switched nonlinear using a different method that is not as computationally complex as the conditions in (6.20). This method is extensively discussed in Chapter 9.
Moreover, the disturbance attenuation problem for switched systems has also at- tracted attention of researchers in recent years. L2-gain analysis andH∞control have been developed for switched linear systems based on the extension of algebraic Ric- cati inequalities [158]. For the particular cases of switched nonlinear systems, theH∞
control problem is proposed based on the Hamilton-Jacobi inequalities for nonlinear systems [111, 225, 245].
In [245], L2-gain analysis and H∞ control for switched nonlinear systems is ad- dressed. The approach is basically a generalization of the well-known min-switching strategy [150]. For the general model
˙
x(t)=fσ(x(t))+gσ(x(t))u(t)+pσ(x(t))ω(t), (6.22)
y(t)=hσ(x(t)), (6.23)
withfi,gi,pi,i∈{1,... ,N}, nonlinear vector functions of states, [245] proposes the fol-
lowing results.
Theorem 6.8. [245] Suppose there exist positive definite and smooth functions Vi(x), with Vi(0)=0, continuous functionsµi(x)≤0, smooth functionsβi j(x)withβi j(0)=0and βii(x)=0, such that ∂Vi ∂x fi+ 1 2 ∂Vi ∂x ³1 γ2pip T i −gig T i ´∂TVi ∂x + 1 2h T ihi+ m X j=1 µi j(x) ¡ Vi(x)−Vj(x)+βi j(x) ¢ ≤0, ∀i∈{1,... ,N}, (6.24) ∂βi j ∂x ³ fi(x)−gi(x)giT(x)∂TVi ∂x (x) ´ ≤0, ∀i,j∈{1,... ,N}, (6.25) βi j(x)+βj k(x)≤min©0,βik(x)ª, ∀i,j,k∈{1,... ,N}, (6.26) ∂βi j ∂x pi=0, ∀i,j∈{1,... ,N}. (6.27)
Then, the feedback controllers
ui(x)= −giT(x)∂TVi
∂x (6.28)
along with the switching law
σ(t)=i ifσ(t−)=iandx(t)∈int(Ωi), (6.29) σ(t)=j ifσ(t−)=iandx(t)∈Ω˜i j, (6.30) whereΩi andΩ˜i jare defined as
Ωi=©x ¯
¯Vi(x)−Vj(x)+βi j(x)≤0, j=1,2,... ,mª, (6.31)
˜
6
make the closed-loop system globally asymptotically stable whenω≡0. Also the overall L2-gain fromωto y on any finite time interval[0,T]will be less than or equal toγ.
Compared to the conventional min-switching scheme as in Theorem 6.7, conditions of Theorem 6.8 and the consequently the switching law (6.29) allow the Lyapunov func- tionsVi to grow during the periods in which their corresponding subsystems are active.
6.4.SUMMARY
In this chapter we have presented the general definition of switched systems as a class of hybrid systems. Several categories of these systems based on the dynamics of the sub- systems and the nature of the switching signals have been introduced. Moreover, from Section 6.2 to the end of this chapter, we have focused on the stability analysis and con- trol synthesis for continuous-time switched linear systems and their nonlinear counter- parts. First, we have discussed the concept of common Lyapunov function to conclude stability of switched linear systems under arbitrary switching patterns. Next, we have introduced the multiple Lyapunov functions approach and the notions of the minimum and the average dwell times in order to conclude that with arbitrary but slow switch- ing between stable subsystems we can maintain global stability. Furthermore, we have utilized the multiple Lyapunov functions approach for the design of state-based stabi- lizing switching laws. Multiple methods from the literature have been briefly explained. Among them, we have presented the Lyapunov-Metzler approach [35, 73] in more de- tail as we will use the main concept of this method in the next chapters. Moreover, we have defined theL2-gain for switched systems and further, have discussed some robust H∞switching control schemes from the literature. Finally, we have presented stability analysis and control of switched nonlinear systems. As mentioned before, most of the literature deals with particular cases of these systems. We have briefly discussed some of them and further, we have presented the main results from [35, 245] for stabilization andH∞control of more general cases of switched nonlinear systems. Methods from these two papers along with the ones from [4, 110] will be more elaborately addressed in the next chapters.
7
STABILIZATION AND
ROBUST
H∞
CONTROL FOR
SWITCHED
NONLINEAR
SYSTEMS
This chapter presents robust switching control strategies for switched nonlinear sys- tems with constraints on the control inputs. First, a model transformation is proposed such that the constraint on the continuous control inputs is relaxed. Next, the effect of disturbances is taken into account and theL2-gain analysis and theH∞control design problem for switched nonlinear systems are formulated. Furthermore, in the case study section, the robust switching control approach is utilized for urban network control us- ing the MFD-based modeling framework discussed in Chapter 4.
7
7.1.INTRODUCTION
T
HEdisturbance attenuation problem for switched systems has attracted attention of researchers in recent years [124, 147, 158].L2-gain analysis andH∞control have been developed for switched linear systems based on the extension of algebraic Ric- cati inequalities [158]. For the particular cases of switched nonlinear systems, theH∞control problem has been proposed based on the Hamilton-Jacobi inequalities for non- linear systems [111, 225, 245]. As an example, in [245] a nonlinear switched system is considered that is affine both in the control input and the disturbance input. The model contains a set of nonlinear subsystems each controlled with an unconstrained contin- uous control input. Further, a switching signal determines the active subsystem. How- ever, the design procedure for the switching rule and the continuous feedback control is based on the fact that the control input is not constrained. In this chapter, we study the stabilization problem for switched nonlinear systems that are affine in the control and disturbance inputs. The aim is to extend the current results on stabilization and
H∞control to the constrained control case.
The chapter is organized as follows. First, we present the problem formulation along with a model transformation in Section 7.2. Next, we discuss stability analysis and sta- bilization in the absence of disturbances Section 7.3. In Section 7.4 the effect of dis- turbances is taken into account and theL2-gain is defined for the switched nonlinear
system. Further, we presentH∞control via switching between modes. Next, we eval- uate the performance of theH∞switching controller for an urban network case study. Finally, Section 7.6 contains the concluding remarks.
7.2.PROBLEMSTATEMENT
Consider the following switched nonlinear system ˙
x(t)=fσ(t)¡x(t)¢+gσ(t)¡x(t)¢·u(t)+pσ(t)¡x(t)¢ω(t), x(0)=x0, (7.1)
wherex∈Rnx is the state,u∈Rnu is the control input, andω∈Rnωis the disturbance
input. The switching signal is denoted byσ(t) and is assumed to be piecewise constant. The variableσtakes values from a pre-defined index set {1,... ,N}, and for each value thatσ(t) assumes, the state space model (7.1) is governed by a different set of vector functionsfi,gi, andpifrom the following sets:
fσ(t)∈{f1,... ,fN}, (7.2) gσ(t)∈{g1,... ,gN}, (7.3) pσ(t)∈{p1,... ,pN}. (7.4)
The vector functionsfi,gi, andpi are continuous functions of states such thatfi(0)=0,
gi(0)=0, andpi(0)=0. Moreover, the control inputuis constrained as follows:
u(t)∈[0,1]nu. (7.5)
This constraint on the control input is common in particular applications such as urban traffic control. As presented in Chapter 4, the perimeter control inputu is in fact the ratio of green and red phases of a traffic signal.