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Chapter 4 Advanced Stability Theory

Chapter 4 Advanced Stability Theory. In the previous chapter, we studied Lyapunov analysis of autonomous systems. In many practical problems, however, we encounter non-autonomous systems. Therefore, stability analysis techniques for non-autonomous systems must be developed.

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Chapter 4 Advanced Stability Theory

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  1. Chapter 4 Advanced Stability Theory

  2. In the previous chapter, we studied Lyapunov analysis of autonomous systems. In many practical problems, however, we encounter non-autonomous systems. Therefore, stability analysis techniques for non-autonomous systems must be developed.

  3. 4.1 Concepts of Stability for Non-Autonomous Systems The concepts of stability for non-autonomous systems are quite similar to those of autonomous systems. However, due to the dependence of non-autonomous system behavior on initial time , the definitions of these stability concepts include explicitly.

  4. Equilibrium Points and Invariant Sets For non-autonomous systems, of the form (4.1) equilibrium points x* are defined by (4.2) Note that this equation must be satisfied implying that the system should be able to stay at the point x* all the time.

  5. For instance, one easily sees that the linear time-varying system (4.3) has a unique equilibrium point at the origin 0 unless A(t) is always singular.

  6. Definition 4.1：The equilibrium point 0 is stable at if for any R> 0, there exists a positive scalar r(R, ) such that (4.6) Otherwise, the equilibrium point 0is unstable. • Extensions of the Previous Stability Concepts

  7. Definition 4.2：The equilibrium point 0 is asymptotically stable at time if • it is stable Definition 4.3：The equilibrium point 0 is exponentially stable if there exist two positive number, , such that for sufficiently small x(t0 ),

  8. Definition 4.4：The equilibrium point 0 is globally asymptotically stable if Example 4.2：A first-order linear time-varying system Consider the first-order system Its solution is

  9. Thus, the system is stable if It is asymptotically stable if It is exponentially stable if there exists a strictly positive number T such that with  being a positive constant.

  10. For instance, • The system • The system is asymptotically stable. • The system is exponentially stable.

  11. Another interesting example is the system whose solution can be expressed as Since the system is exponentially convergent with rate 1/2.

  12. Definition 4.5：The equilibrium point 0 is locally uniformly stable if the scalar r in Definition 4.1 can be chosen independently of • Uniformity in Stability Concepts

  13. Definition 4.6：The equilibrium point at the origin is locally uniformly asymptotically stable if • it is uniformly stable • There exists a ball of attraction , whose radius is independent of such that any system trajectory with initial states in converges to 0 uniformly in

  14. Definition 4.7：A scalar time-varying function V(x, t) is locally positive definite if V(0, t)=0 and there exists a time-invariant positive definite function such that Thus, a time-variant function is locally positive definite if it dominates a time-invariant locally positive definite function.Globally positive definite functions can be defined similarly.

  15. Other related concepts ca be defined in the same way, in a local or a global sense. A function V(x, t) is negative definite if -V(x, t) is positive definite; V(x, t) is positive semi-definite function; V(x, t) is negative semi-definite if -V(x, t) is positive semi-definite. In Lyapunov analysis of non-autonomous systems, the concept of decrescent functions is also necessary.

  16. Definition 4.8：A scalar function V(x, t) is said to be decrescent if V(0, t)=0, and if there exists a time-invariant positive definite function such that In other words, a scalar function V(x, t) is decrescent if it is dominated by a time-invariant positive definite function.

  17. Example 4.4：A simple example of a time-varying positive definite function is because it dominates the function . This function is also decrescent because it is dominated by the function Given a time-varying scalar function V(x, t), its derivative along a system trajectory is

  18. Lyapunov Theorem for Non-autonomous System Stability • Theorem 4.1 (Lyapunov theorem for non-autonomous systems) • Stability • If, in a ball around the equilibrium point 0, there exists a scalar function V(x, t)with continuous partial derivatives such that 1. V is positive definite2. Is negative semi-definite then the equilibrium point 0 is stable in the sense of Lypunov.

  19. Uniform stability and uniform asymptotic stability：If , furthermore, • 3. V is decrescent • then the origin is uniformly stable. If condition 2 is strengthened by requiring that be negative definite, then the equilibrium point is uniformly asymptotically stable.

  20. Global uniform asymptotic stability：If the ball is replaced by the whole state space, and condition 1, the strengthened condition 2, condition 3, and the condition • 4. V(x, t) is radially unbounded • are all satisfied, then the equilibrium point at 0 is globally uniformly asymptotically stable. Translate the definitions of positive definite function s and decrescent function in terms of the so-called class-K functions.

  21. Definition 4.9：A continuous function  : • is said to be of class K (or to belong to class K), if • (0) = 0 • (p) > 0 •  is non-decreasing

  22. Lemma 4.1：A function V(x, t) is locally (or globally) positive definite if, and only if, there exists a function  of class K such that V(0, t)=0 and (or in the whole state space). A function V(x, t) is locally (or globally) decrescent if and only if there exists a class K function  such that V(0, t)=0 and (or in the whole state space).

  23. Theorem 4.1：Assume that, in a neighborhood of the equilibrium point 0, there exists a scalar function V(x, t) with continuous first order derivative and a class-K function  such that , x  0 1. V(x, t)  (||x||) >02. then the origin 0 is Lyapunov stable. If, furthermore, there is a scalar class-K function  such that 3.

  24. then 0 is uniformly stable. If conditions 1 and 3 are satisfied and condition 2a is replaced by condition 2bwith  being another class-K function, there 0 is uniformly asymptotically stable. If conditions 1, 2b and 3 are satisfied in the whole state space, and then 0 is globally uniformly asymptotically stable.

  25. Figure 4.1：A positive definite and decrescent function

  26. Example 4.5：Global Asymptotic Stability Consider the system defined by To determine the stability of the equilibrium point at 0, let us choose the following scalar function

  27. This function is positive definite, because it dominates the time-invariant positive function . It is also decrescent, because it is dominated by the time-invariant positive definite function .Furthermore, This shows that Thus, is negative definite, and therefore, the point 0 is globally asymptotically stable.

  28. Example 4.6：Importance of the decrescence condition Let g(t) be a continuously-differentiable function which coincides with the function except around some peaks where it reaches the value 1. Specifically, is shown in Figure 4.2. There is a peak for each integer value of t. The width of the peak corresponding to abscissa n is assumedto be smaller than . The infinite integral of satisfies

  29. and therefore, the scalar function is positive definite

  30. Figure 4.2：The function

  31. Now consider the first-order differential equation (4.15) If we choose V(x, t) in (4.14) as the Lyapunov function candidate, we easily find that i. e., negative definite. Yet, the general solution of (4.15) is and hence the origin is not asymptotically stable.

  32. Example 4.7：Asymptotic stability with time-varying damping Lyapunov stability of the system (although not its asymptotic stability) can be easily established using the mechanical energy of the system as a Lyapunov function. Let us now use a different Laypunov function to determine sufficient conditions for the asymptotic stability of the origin for the system (4.16). Consider the following positive definite function

  33. where  is any positive constant smaller than , and can be easily computed as Thus, if there exist positive numbers  and  such that Then is negative definite. Assuming in addition that c(t) is upper bounded (guaranteeing the decrescence of V), the above conditions imply the asymptotic convergence of the system.

  34. It can be shown [Rouche, et al., 1997] that, actually, the technical assumption that c(t) is upper bounded ins not necessary. Thus, for instance, the system is asymptotically stable.

  35. 4.2.2 Lyapunov Analysis of Linear Time-Varying Systems Consider linear time-varying systems of the form Since LTI systems are asymptotically stable if their eigenvalues all have negative real parts, one might be tempted to conjecture that system (4.17) will be stable if at any time t  0, the eigenvalues of A(t) all have negative real parts.

  36. If this were indeed the case, it would make the case, it would make the analysis of linear time-varying systems very easy. However, this conjecture is not true. Consider for instance the system The eigenvalues of the matrix A(t) are both equal to -1 at all times. (4.18)

  37. Yet, solving first for x2 and then substituting in the equation, one sees that the system verifies and therefor is unstable, since x1 can be viewed as the output of a first-order filter whose input tends to infinity.

  38. A simple result, however, is that the time-varying system (4.17) is asymptotically stable if the eigenvalues of the symmetric matrix (all of which are real) remain strictly in the left-half complex plane (4.19) This can be readily shown using the Lyapunov function since so that and therefore x tends to zero exponentially.

  39. Perturbed linear systems Consider a linear time-varying system of the form (4.20) where the matrix A1 is constant and Hurwitz (i. e., has all its eigenvalues strictly in the left-half plane), and the time-varying matrix A2(t) is such that and (i. e., the integral exists and is finite) Then the system (4.20) is globally exponentially stable.

  40. Sufficient smoothness conditions on the A(t) matrix Consider the linear system (4.17), and assume that at any time t 0, the eigenvalues of A(t) all have negative real parts (4.21) If, in addition, the matrix A(t) remains bounded, and (i.e, the integral exists and is finite) Then the system is globally exponentially stable.

  41. 4.2.3. The Linearization Method for Non-Autonomous Systems Lyapunov's linearization method can also be developed for non-autonomous systems. Let a non-autonomous system be described by (4.1) and 0 be an equilibrium point. Assume that f is continuously differentiable with respect to x. Let us denote (4.22)

  42. Then for any fixed time t( i. e., regarding t as a parameter), a Taylor expansion of f leads to If f can be well approximated by A(t)x for any time t, i. e., (4.23) then the system is said to be the linearization (or linear approximation) of the nonlinear non-autonomous system (4.1) around equilibrium point 0.

  43. Theorem 4.2If the linearized system (with condition (4.23) satisfied) is uniformly asymptotically stable, then the equilibrium point 0 of the original non-autonomous system is alson uniformly asymptotically stable. Theorem 4.3 If the Jacobian matrix A(t) us constant, A(t)=A0 , and if (4.23) is satisfied, then the instability of the linearized system implies that of the original non-autonomous nonlinear system, i. e., (4.1) is unstable if one or more of the eigenvalues of A0 has a positive real part.

  44. 4.3 Instability Theorems • Theorem 4.4(First instability theorem) If, in a certain neighborhood  of the origin, there exists a continuously differentiable, decrescent scalar function V(x, t) such that • can assume strictly positive values arbitrarily close to the origin • is positive definite (locally in  ) Then the equilibrium point 0 at time t0 is unstable.

  45. Example 4.9：Consider the system (4.26) (4.27) Linearization of the system yieldsand . The eigenvalues of this system are +2j and -2j, indicating the inability of Lyapunov's linearzation method for this system. However, if we take

  46. Its derivative is Because of the positive definiteness V and, the above theorem indicates the instability of the system.

  47. Theorem 4.5 (Second instability theorem) • If, in a certain neighborhood  of the origin, there exists a continuously differentiable, decrescent scalar function V(x, t) satisfying • V(0, t0) = 0 and V(x, t0) can assume strictly positive values arbitrarily close to the origin with  being a strictly positive constant, then the equilibrium point 0 at time t0 is unstable.

  48. Theorem 4.6(Third instability theorem) • Let  be a neighborhood of the origin. If there exists a scalar function V(x, t) with continuous first partial derivatives, decrescent in , and a region 1in , such that • V(x, t)and are positive definite in 1 • The origin is a boundary point of 1 • At the boundary points of 1 within , V(x, t)= 0 for all Then the equilibrium point 0 at time t0 is unstable.

  49. Figure 4.3：Geometrical interpretation of the third instability theorem

  50. 4.5 Lyapunov-Like Analysis Using Barbalat's Lemma

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