emark 1. Note that Part 2 gives only sufficient conditions for the stability of the equilibrium state. As the following example shows, these conditions are not necessary. le12.2 Consider first the continuous system x=Ox, where O is the zero matrix. Note that all constant functions x(t) = x are solutions and also equilibrium states. Since φ(t,t0) is bounded(being independent of t), all equilibrium states are stable, but o has only one eigenvalue M=0 with zero real part and multiplicity n, where n is the order of the system Consider next the discrete systems x(t 1)=Ix(o), when all constant functions x(o)=x are also solutio and equilibrium states. Furthermore, (,t0)=A I which is obviously bounded. Therefore, all equilibrium states are stable, but the condition of Part 2 of the theorem is violated again, since M= l with unit absolute value having a multiplicity n. Remark 2. The following extension of Theorem 12.5 can be proven. The equilibrium state is stable if and only if for all eigenvalues of A,Reλ:≤0or内≤1), and ifλ M is a repeated eigenvalue of A such that Re= 0 or =1), then the size of each block containing 2 in the Jordan canonical form of A is 1 X1. if the same holds for the equilibrium states of the corres niog homogeneous equation ss Remark 3. The equilibrium states of inhomogeneous equations are stable or asymptotically stable if and only Example 12.3 0 the stability of which was analyzed earlier in Example 12.1 by using the Lyapunov function method. The characteristic polynomial of the coefficient matrix is therefore, the eigenvalues are M=jo and n2=-jo Both eigenvalues have single multiplicities, and Re M1= Re of Part 3 do not hold. Consequenty the system is not asymptotically stable m state is st n2=0. Hence, the conditions of Part 2 are satisfied, and therefore the equilibrium state is stable. The conditions If a time-invariant system is nonlinear, then the Lyapunov method is the most popular choice for stability alysis. If the system is linear, then the direct application of Theorem 12.5 is more attractive, since the igenvalues of the coefficient matrix A can be obtained by standard methods. In addition, several condition are known from the literature that guarantee the asymptotic stability of time-invariant discrete and continuou systems even without computing the eigenvalues For examining asymptotic stability, linearization is an alternative approach to the Lyapunov method as is shown here. Consider the time- invariant continuous and discrete syste x(t)=f(x(t)) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Remark 1. Note that Part 2 gives only sufficient conditions for the stability of the equilibrium state. As the following example shows, these conditions are not necessary. Example 12.2 Consider first the continuous system x· = Ox, where O is the zero matrix. Note that all constant functions x(t) º x are solutions and also equilibrium states. Since is bounded (being independent of t), all equilibrium states are stable, but O has only one eigenvalue l1 = 0 with zero real part and multiplicity n, where n is the order of the system. Consider next the discrete systems x(t + 1) = Ix(t), when all constant functions x(t) º x are also solutions and equilibrium states. Furthermore, which is obviously bounded. Therefore, all equilibrium states are stable, but the condition of Part 2 of the theorem is violated again, since l1= 1 with unit absolute value having a multiplicity n. Remark 2. The following extension of Theorem 12.5 can be proven. The equilibrium state is stable if and only if for all eigenvalues of A, Re li £ 0 (or *li * £ 1), and if li is a repeated eigenvalue of A such that Re li = 0 (or *li * = 1), then the size of each block containing li in the Jordan canonical form of A is 1 3 1. Remark 3. The equilibrium states of inhomogeneous equations are stable or asymptotically stable if and only if the same holds for the equilibrium states of the corresponding homogeneous equations. Example 12.3 Consider again the continuous system the stability of which was analyzed earlier in Example 12.1 by using the Lyapunov function method. The characteristic polynomial of the coefficient matrix is therefore, the eigenvalues are l1 = jw and l2 = –jw. Both eigenvalues have single multiplicities, and Re l1 = Re l2 = 0. Hence, the conditions of Part 2 are satisfied, and therefore the equilibrium state is stable. The conditions of Part 3 do not hold. Consequently, the system is not asymptotically stable. If a time-invariant system is nonlinear, then the Lyapunov method is the most popular choice for stability analysis. If the system is linear, then the direct application of Theorem 12.5 is more attractive, since the eigenvalues of the coefficient matrix A can be obtained by standard methods. In addition, several conditions are known from the literature that guarantee the asymptotic stability of time-invariant discrete and continuous systems even without computing the eigenvalues. For examining asymptotic stability, linearization is an alternative approach to the Lyapunov method as is shown here. Consider the time-invariant continuous and discrete systems f( , ) ( ) t t e t t 0 0 = = O - I f(t t, ) t t t t 0 0 0 = = = - - A I I ˙ x = - Ê Ë Á ˆ ¯ ˜ + Ê Ë Á ˆ ¯ ˜ 0 0 0 1 w w x j w w (s) w s s = s - - - Ê Ë Á ˆ ¯ ˜ det = + 2 2 x˙( )t = f x( (t))
x(t+1)=f(x(t) Let j(x)denote the Jacobian of f(x), and let x be an equilibrium state of the system. It is known that the method of linearization around the equilibrium state results in the time-invariant linear systems x6()=J(x)x(t) x6(t+1)=J(x)x(t) where x(t)=x(o)-x. It is also known from the theory of ordinary differential equations that the asymptotic tability of the zero vector in the linearized system implies the asymptotic stability of the equilib For continuous systems the following result has a special importance The equilibrium state of a continuous system [Eq.(12.4) is asymptotically stable if and only if equation A Q+QA has positive definite solution Q with some positive definite matrix M We note that in practical applications the identity matrix is almost always selected for M. An initial stability check is provided by the following result Theorem 12 Let o()=A"+P2-1+..+P,+ Po be the characteristic polynomial of matrix A Assume that all eigenvalues of matrix A have negative real parts. Then P: >0(i=0, 1,.,n-1) Corollary. If any of the coefficients p is negative or zero, the equilibrium state of the system with coefficient matrix A cannot be asymptotically stable. However, the conditions of the theorem do not imply that the eigenvalues of A have negative real part the characteristic polynominal is ((s)=s2+0. Since the coefficient of s is zero, the system of Example 12.3 tically stable The Transfer Function Approach The transfer function of the time invariant linear continuous system (12.7) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC and Let J(x) denote the Jacobian of f(x), and let x be an equilibrium state of the system. It is known that the method of linearization around the equilibrium state results in the time-invariant linear systems and where xd(t) = x(t) – x. It is also known from the theory of ordinary differential equations that the asymptotic stability of the zero vector in the linearized system implies the asymptotic stability of the equilibrium state x in the original nonlinear system. For continuous systems the following result has a special importance. Theorem 12.6 The equilibrium state of a continuous system [Eq. (12.4)] is asymptotically stable if and only if equation (12.6) has positive definite solution Q with some positive definite matrix M. We note that in practical applications the identity matrix is almost always selected for M. An initial stability check is provided by the following result. Theorem 12.7 Let j(l) = l n + pn–1 ln–1 + . . . + p1l + p0 be the characteristic polynomial of matrix A.Assume that all eigenvalues of matrix A have negative real parts. Then pi > 0 (i = 0, 1,..., n – 1). Corollary. If any of the coefficients pi is negative or zero, the equilibrium state of the system with coefficient matrix A cannot be asymptotically stable. However, the conditions of the theorem do not imply that the eigenvalues of A have negative real parts. Example 12.4 For matrix the characteristic polynominal is j(s) = s 2 + w2 . Since the coefficient of s1 is zero, the system of Example 12.3 is not asymptotically stable. The Transfer Function Approach The transfer function of the time invariant linear continuous system (12.7) x(t + 1) = f(x(t)) x˙ ( ) Jxx ( ) ( ) d d t = t x Jxx d d (t + 1) = ( ) (t) A Q QA M T + = - A = - Ê Ë Á ˆ ¯ ˜ 0 0 w w x˙ Ax Bu y Cx = + =