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 = + =