Figure 2.19: Lyapunov function V for system (2.89).
-0.8-0.6 -0.4-0.2 0.2 0.4 0.6 0.8 0
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
y
x
Figure 2.20: Level curves ofV for values 0.213, 0.426 and 0.64.
2.3 Concluding remarks and open questions
In this chapter, a novel technique for the computation of CPA Lyapunov functions has been proposed given the system isKL-stable. For a suitable triangulation of a compact subset of state space, the Yoshizawa functions values at vertices of each simplex are computed. Then we construct a CPA function based on these vertex values. Furthermore, we check if such a CPA function is a Lyapunov function by verifying the inequalities (2.7) and (2.62). If the CPA function is not a Lyapunov function, then we could re-calculate the Yoshizawa functions values at vertices and construct a CPA function after refining the suitable triangulation. The obtained CPA Lyapunov function is a true Lyapunov function rather a numerical approxi-mation of a Lyapunov function, since the interpolation errors are incorporated in the linear inequalities (2.7) and (2.62).
For continuous time dynamical systems, we assume that the Yoshizawa function (2.35) is C2. Then there exists a suitable triangulation such that the constructed CPA function is a Lyapunov function (see Theorem 2.1.7). However, the Yoshizawa function is only Lipschitz continuous. Thus, there may not exist a suitable triangulation such that the computed CPA function is a Lyapunov function.
For discrete time dynamical systems, if the Yoshizawa function (2.79) is differentiable with bounded derivative, then there exists a suitable triangulation such that the constructed CPA function is a Lyapunov function (see Theorem 2.2.8). However, it is only proved that the Yoshizawa function is Lipschitz continuous. If we cannot prove that the Yoshizawa function is differentiable, then there exists a possibility that our approach for computation of Lyapunov function cannot succeed. However, it is worth to point out that the cost of computation
60 2. Computation of Lyapunov functions using the Yoshizawa constructions
of CPA Lyapunov functions by our proposed method is less expensive than by solving a linear optimization problem which also delivers a true Lyapunov function. For Example 1 from Section 2.2.3, it took 12 hours to get the solution by our proposed method using using a Laptop computer with a 32 bit, AMD Athlom (tm) II 360 Dual-core Processor 2.30GHZ.
However, it took more than 36 hours to solving the corresponding linear optimization problem for the same suitable triangulation.
The triangulation of the state space described in Section 2.1.3 was proposed in [27, Section 2]. If system is exponentially stable, then we could construct a CPA Lyapunov function on a subset of state space without excluding the small neighbourhood of the equilibrium with a suitable triangulation, which is discussed in [28, 40]. This is why we use this type of suitable triangulation.
In Section 2.1.2 and Section 2.2.2, we present two types of stability estimate. For each type of stability, the formulation ofα1, α2 are given. However, for general cases, there are some difficult problems of finding a stability estimateβ ∈ KL, and constructing functionsα1, α2 ∈ K∞ such that (1.51) and (1.62) hold. Besides, the cost of computation becomes more expensive as the dimension of considered systems increases. Because of these difficulties, the proposed methods can not be widely applied in computing Lyapunov functions for general dynamical systems.
Based on the above summary, for large scale systems, it is not easy to construct a Lyapunov function by the proposed method. In order to analyse stability of a large scale system, we consider it as interconnected systems. We will investigate stability of interconnected systems in Chapter 3. We will first study stability of each subsystem by Zubov’s method for perturbed systems, since Zubov’s method delivers a maximal Lyapunov function. Then stability of interconnected systems will be analysed by small gain theorems which play a central part in stability analysis of interconnected systems.
3 Stability of two interconnected sys-tems and estimate of the domain of attraction
After the concepts of ISS and iISS were introduced, many small gain theorems have been proposed. The stability of interconnected systems has been investigated using small gain theorems and ISS, iISS Lyapunov functions for the subsystems. We are interested in the small gain theorem in comparison form (Theorem 1.6.3) since the small gain theorem can be used to study stability of two interconnected iISS systems. In this chapter, our aim is to investigate stability of two interconnected systems which are locally iISS. To this end, we first study how to construct local iISS Lyapunov functions for dynamical systems with perturbations. We then restrict our attention to analysing stability of the coupling of two nonlinear systems in a feedback interconnection by small gain theorems.
In order to construct local iISS Lyapunov functions for perturbed systems, we introduce auxiliary systems which are locally uniformly asymptotically stable at the equilibrium in Section 3.2. This idea is inspired by the result of [13] where an ISS Lyapunov function in implication formulation is computed via an auxiliary system. In [13], it is proved that if there exists an ISS Lyapunov function in implication formulation for the considered system, then there exists an auxiliary system such that the ISS Lyapunov function in implication formulation is a robust Lyapunov function for the auxiliary system. A robust Lyapunov function for the auxiliary system is then computed by Zubov’s method. Furthermore, in [13] the robust Lyapunov function is proved to be an ISS Lyapunov function in implication formulation for the original system on a subset of the robust domain of attraction of the auxiliary system. In this chapter, under certain conditions, we prove that a robust Lyapunov function for the introduced auxiliary system is an iISS Lyapunov function for the original system on the robust domain of attraction of the auxiliary system. Therefore, the problem of constructing an iISS Lyapunov function for the original system is reduced to constructing a robust Lyapunov function for the auxiliary system.
We recall Zubov’s method developed in [9, 10, 11] for computing robust Lyapunov func-tions for perturbed systems which are uniformly asymptotically stable at the equilibrium in Section 3.3. A first order partial differential equation (Hamilton-Jacobi-Bellman equation) is formulated which has a unique viscosity solution vanishing at the fixed point. This viscosity solution is a maximal Lyapunov function on the robust domain of attraction for perturbed systems. By Zubov’s method for perturbed systems we obtain a robust Lyapunov function for the introduced auxiliary system and then show that it is an iISS Lyapunov function for the original system.
From Section 3.3.1 on, we put our emphasis on investigating stability of two intercon-nected nonlinear systems which are locally iISS. A local iISS Lyapunov function for each subsystem can be obtained by the introduction of an auxiliary system and Zubov’s method for a perturbed system. In Section 3.4, a local version of the small gain theorem in
com-62
3. Stability of two interconnected systems and estimate of the domain of attraction parison form (Theorem 1.6.3) is introduced. We prove that if each subsystem of the two interconnected systems are locally iISS and the conditions of the local version of small gain theorem in comparison form hold, then the interconnected system is asymptotically stable at the equilibrium. Moreover, an estimate of the domain of attraction of the two interconnected systems at the equilibrium is attained.
Based on the obtained iISS Lyapunov functions, we further state that the iISS Lyapunov function is a local ISS Lyapunov function in dissipative formulation for each subsystem on a compact subset of the domain of attraction of the corresponding auxiliary system in Section 3.5. A local version of the small gain theorem in dissipative form (Theorem 1.6.2) is presented.
In order to compare with the results obtained by the local version small gain theorem in comparison form, stability of two interconnected ISS systems will be investigated by the local version of small gain theorem in dissipative form. If each subsystem is locally ISS and conditions from the local version of the small gain theorem in dissipative form hold, then the interconnected system is asymptotically stable at the equilibrium. The procedure of estimating the domain of attraction of the coupled system at the equilibrium is described based on the local version of the small gain theorem in dissipative form.
Additionally, we illustrate the main results of this chapter by an academic example in Section 3.6. For this example, we compare estimates of the domain of attraction obtained by these two small gain theorems and subsystems’ iISS and ISS Lyapunov functions.
3.1 Problem statement
We consider a system described by
x˙1 =f1(x1, x2),
˙
x2 =f2(x1, x2) (3.1)
wherexi ∈Rni,ni ∈Z>0,fi :Rni →Rni.
Letn=n1+n2,x= (x1, x2)>,f = (f1, f2)>, and x0 denote the initial condition.
Then we may equivalently write the system as x˙ =f(x)
x(0) =x0 (3.2)
We assume
• f is Lipschitz continuous,
• system (3.1) has a fixed point in x= 0 which is locally asymptotically stable.
Each subsystem of system (3.1) is treated as a dynamical system with perturbation by con-sidering the effect of the other state as perturbation. A dynamical system with perturbations is of the type
˙
x(t) =f(x(t), u(t)) (3.3)
wherex∈Rnis the state,u∈Rmis the perturbation input andt∈R+. The set of admissible input values is denoted by UR := B2(0, R) ⊂Rm, the set of admissible input functions are defined byUR:={u:R→Rm measurable| kuk∞≤R},R >0, andf(0,0) = 0.