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Lecture Signals, systems & inference – Lecture 7: Full modal solution, asymptotic stability, reachability and observability

The following will be discussed in this chapter: Modal solution of CT system ZIR, asymptotic stability of CT system, the DT case: linearization at an equilibrium, modal solution of driven DT system, underlying structure of LTI DT statespace system with L distinct modes, reachability and Observability,. | Lecture Signals, systems & inference – Lecture 7: Full modal solution, asymptotic stability, reachability and observability Full modal solution, asymptotic stability, reachability and observability 6.011, Spring 2018 Lec 7 1 Modal solution of CT system ZIR L X q(t) = ↵ i v i e>i t 1 with the weights {↵i }L 1 determined by the initial condition: L X q(0) = ↵i vi 1 2 Asymptotic stability of CT system In order to have q(t) ! 0 for all q(0) , we require {Re( i ) < 0}L 1 i.e., all eigenvalues (natural frequencies) in open left half plane 3 The DT case: linearization at an equilibrium e , x[n] = x¯ + x[n] ¯ + q[n] DT case: q[n] = q e , q[n + 1] = f (q[n], x[n]) # h @f i h @f i e + 1] ⇡ q[n e + q[n] e x[n] @q ¯ ¯ q,x @x ¯ ¯ q,x e for small perturbations q[n] e and x[n] from equilibrium 4 Modal solution of DT system ZIR Could parallel CT development, but let’s proceed di↵erently: 2 3 A1 0 0 ··· 0 6 0 A2 0 ··· 0 7 6 7 6 7 A[ v1 v2 · · · vL ] = [ v1 v2 · · · vL ] 6 7 6 . . 7 4 . . . 5 0 0 0 ··· AL or AV = V⇤ or A = V⇤V-1 or An = (V⇤V-1 ) · · · (V⇤V-1 ) = V⇤n V-1 5 q[n] = An q[0] = V⇤n V-1 q[0] | {z } 2 3 ↵1 6 7 6 ↵2 7 6 7 6 7 6 . 7 6 7 4 5 ↵L 2 3 n 1 0 0 ··· 0 2 3 6 ↵1 6 0 n 2 0 ··· 0 77 6 ↵2 7 so q[n] = [ v1 v2 · · · vL ] 6 6 76 7 6 7 7 6 7 4 . 5 4 . . . . . 5 n ↵L 0 0 0 ··· L L X n 6 = ↵i v i i 1 Asymptotic stability of DT system In order to have q[n] ! 0 for all q[0] , we require {|�i | < 1}L 1 i.e., all eigenvalues (natural frequencies) inside unit circle 7 An for increasing n 0.6 0.6 101 100 A1 = , A2 = 0.6 0.6 101 100 100.5 100 0.6 100 A3 = , A4 = . 100.5 100 0 0.5 8 An for increasing n n n A1 = 0.6 0.6 n 0.5 0.5 = (1.2) 0.6 0.6 0.5 0.5 n An2 = 101 100 = A2 101 100 9 An for increasing n n n 100.5 100 n 201 200 A3 =