Analysis and control of linear time–varying 2 systems Before considering the actual subject matter, i.e., analysis and control design of nonlinear systems, at first linear time-varying systems or LTV systems of the form x˙ A(t)x B(t)u, t t , x(t ) x (2.1a) Æ Å È 0 0 Æ 0 y C(t)x D(t)u, t t (2.1b) Æ Å ¸ 0 with x(t) Rn, u(t) Rm, y(t) Rp and 2 2 2 n n n m p n p m A(t): RÅ R £ , B(t): RÅ R £ , C(t): RÅ R £ , D(t): RÅ R £ t 0 ! t 0 ! t 0 ! t 0 ! are considered. Here RÅ : {t R t t 0} denotes the set of real numbers larger or equal to t0. t 0 Æ 2 j ¸ The motivation behind this sequential approach is that LTV systems on the one hand show certain similarities to linear time–invariant systems but are distinguished by significant differences in their dynamic analysis. In this sense, they — at least to some extend — resemble nonlinear systems. On the other hand, the linearization of a nonlinear system around a solution trajectory t (x¤(t),u¤(t)) 7! yields an LTV system. Hence the subsequently introduced tools can be used for control and observer design at least in a neighborhood of the trajectory (x¤(t),u¤(t)). 2.1 Transition matrix and solution of the state differential equations At first free or autonomous systems x˙ A(t)x, t t , x(t ) x (2.2) Æ È 0 0 Æ 0 are addressed. Under the assumption that any element of A(t) is bounded in the considered time interval t [t ,t ] the (local) existence and uniqueness theorem for autonomous nonlinear systems, 2 0 1 see, e.g., Khalil, 2002 or Meurer, 2019, Theorem 2.1, implies solution existence and uniqueness of (2.2). For this it is necessary to verify that A(t)x fulfills a Lipschitz condition. Let A(t) denote the induced k k matrix norm1, then A(t)x A(t)x A(t)¡x x ¢ A(t) x x k 1 ¡ 2k · k 1 ¡ 2 k · k kk 1 ¡ 2k implies the Lipschitz constant A(t) . Hence, the solution of (2.2) can be determined using successive k k approximation or Picard iteration, which starting at x (t) x 0 Æ 0 1In finite–dimensional spaces such as Rn all norms are equivalent so that any norm can be bounded from below and from above by means of any other norm. There exist numerous matrix norms. Some commonly used norms are the Frobenius norm, the 1–norm (maximal column sum) or the –norm (maximal row sum). The notions 1–norm and 1 –norm should not be confused with the L1–norm and the max– or sup–norms, respectively, that are sometimes also 1 labeled by these terms. 23 implies the sequence Z t ³ Z t ´ x1(t) x0 A(¿0)x0(¿0)d¿0 I A(¿0)d¿0 x0 Æ Å t 0 Æ Å t 0 Z t ³ Z ¿0 ´ x2(t) x0 A(¿0) I A(¿1)d¿1 x0d¿0 Æ Å t 0 Å t 0 ³ Z t Z t ³ Z ¿0 ´ ´ I A(¿0)d¿0 A(¿0) A(¿1)d¿1 d¿0 x0 Æ Å t 0 Å t 0 t 0 . ³ Z t Z t Z ¿0 xk (t) I A(¿0)d¿0 A(¿0) A(¿1)d¿1d¿0 Æ Å t 0 Å t 0 t 0 Z t Z ¿0 Z ¿1 A(¿0) A(¿1) A(¿2)d¿2d¿1d¿0 ... Å t 0 t 0 t 0 Å Z t Z ¿0 Z ¿k 2 ¡ ´ A(¿0) A(¿1) A(¿k 1)d¿k 1 d¿1d¿0 x0. Å t 0 t 0 ¢¢¢ t 0 ¡ ¡ ¢¢¢ Here, I is the (n n) identity matrix. The solution of (2.2) is obtained as £ x(t) ©(t,t )x (2.3) Æ 0 0 in terms of the so–called Peano–Baker series Z t Z t Z ¿0 ©(t,t 0) I A(¿0)d¿0 A(¿0) A(¿1)d¿1d¿0 Æ Å t 0 Å t 0 t 0 Z t Z ¿0 Z ¿1 A(¿0) A(¿1) A(¿2)d¿2d¿1d¿0 ... Å t 0 t 0 t 0 Å Z t Z ¿0 Z ¿k 2 ¡ A(¿0) A(¿1) A(¿k 1)d¿k 1 d¿1d¿0 . (2.4) Å t 0 t 0 ¢¢¢ t 0 ¡ ¡ ¢¢¢ Å ¢¢¢ In direct analogy to linear time–invariant systems ©(t,t 0) is called transition matrix. Exercise 2.2. Prove (2.4) using induction. It is an easy exercise to show that ©(t ,t ) I. Thus, x(t) ©(t,t )x fulfills both the differential 0 0 Æ Æ 0 0 equation (2.2) and the initial condition x(t ) ©(t ,t )x x . Further properties of the transition 0 Æ 0 0 0 Æ 0 matrix are summarized below. Lemma 2.1: Properties of the transition matrix The transition matrix ©(t,t 0) of a linear time–varying system fulfills the following identities (i) ©(t ,t ) I (Initial value) 0 0 Æ (ii) ©(t ,t ) ©(t ,t )©(t ,t ) (Product property) 2 0 Æ 2 1 1 0 1 (iii) ©¡ (t,t ) ©(t ,t) (Inversion) 0 Æ 0 d (2.5) (i v) ©(t,t 0) A(t)©(t,t 0) (Differentiation) dt Æ µZ t ¶ ¡ ¢ 2 (i v) det©(t,t 0) exp tr A(¿) d¿ (Determinant ). Æ t 0 2Herein tr(A) is the trace of the matrix A, i.e., the sum of the diagonal entries of A. 24 Chapter 2 Analysis and control of linear time–varying systems Exercise 2.3. Prove the properties (2.5). Contrary to linear time–invariant systems there is no general solution approach for the determination of the transition matrix ©(t,t 0) for linear time–varying systems. In particular the eigenvalues of the matrix A(t) in general do not provide information concerning the solution behavior. Example 2.1. Let n 1 so that (2.1a) with A(t) a(t) reduces to the scalar differential equation Æ Æ x˙ a(t)x, t t , x(t ) x . (2.6) Æ È 0 0 Æ 0 Its solution can be determined straightforwardly using separation of variables, which yields µZ t ¶ x exp a(¿)d¿ x0. (2.7) Æ t 0 Let a(t) 1/t 2. In this case ¸(t) 1/t 2 is the only eigenvalue of the matrix A(t). Moreover ¸(t) 0 Æ Æ È for all finite t and ¸(t) 0 as t . For t 0 the explicit solution of (2.6) follows as Ç 1 Æ ! §1 0 6Æ µ 1 1¶ x exp x0. (2.8) Æ t 0 ¡ t Obviously x(t) is bounded for all t t and x(t) exp(1/t )x as t although the eigenvalue ¸ 0 ! 0 0 ! 1 ¸(t) has positive real part for all t t . This exemplarily confirms the statement that contrary to ¸ 0 linear time–invariant systems the eigenvalues of the system matrix in the time–varying case do not allow to deduce the solution dynamics. Example 2.2. Contrary to linear time–invariant systems but in some analogy to nonlinear systems it is possible that the solution of a linear time–varying systems shows a finite escape time. To illustrate this consider 1 x˙ x, t t 0, x(t 0) x0 (2.9) Æ¡(t t )2 È Æ ¡ 1 for t t . Using separation of variables the solution is obtained as 1 È 0 µ 1 1 ¶ x exp x0. (2.10) Æ ¡ t t Å t t 0 ¡ 1 ¡ 1 Obviously, x(t) as t t , which implies the finite escape time. The solution to the differential ! 1 ! 1 equation is hence only valid for the time interval t [t ,t ). 2 0 1 2.1.1 Systems admitting closed–form solutions For the special case of a scalar linear time–varying differential equation Example 2.1 shows that µZ t ¶ x exp a(¿)d¿ x0 (2.11) Æ t 0 solves x˙ a(t)x, t t , x(t ) x . (2.12) Æ È 0 0 Æ 0 2.1 Transition matrix and solution of the state differential equations 25 Coupled systems with dimx n 1 admit a closed–form solution, if the system matrix A(t) fulfills the Æ ¸ condition stated in Theorem 2.1 below. Theorem 2.1 Let A(t)A(¿) A(¿)A(t) (2.13) Æ for all t, ¿. Then the transition matrix ©(t,¿) of the linear time–varying system x˙ A(t)x, t t , x(t ) x (2.14) Æ È 0 0 Æ 0 is given by3 µZ t ¶ ©(t,¿) exp A(s)ds . (2.15) Æ ¿ Proof. Taking into account the Taylor series expansion of the exponential function one obtains Z t 1 Z t Z t ©(t,¿) I A(s)ds A(s)ds A(p)dp . Æ Å ¿ Å 2! ¿ ¿ Å ¢¢¢ This implies Z t 1 Z t Z t A(t)©(t,¿) A(t) A(t) A(s)ds A(t) A(s)ds A(p)dp . Æ Å ¿ Å 2! ¿ ¿ Å ¢¢¢ Differentiation of ©(t,¿) with respect to t yields the series d 1 µ Z t Z t ¶ ©(t,¿) A(t) A(t) A(p)dp A(s)ds A(t) , dt Æ Å 2! ¿ Å ¿ Å ¢¢¢ which is identical to A(t)©(t,t) if Z t Z t A(t) A(p)dp A(s)ds A(t).