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Lecture Signals, systems & inference – Lecture 8: Matrix exponential, ZIR+ZSR, transfer function, hidden modes, reaching target states

The following will be discussed in this chapter: Modal solution of driven DT system, underlying structure of LTI DT statespace system with L distinct modes, reachability and observability, hidden modes,. | Lecture Signals, systems & inference – Lecture 8: Matrix exponential, ZIR+ZSR, transfer function, hidden modes, reaching target states Matrix exponential, ZIR+ZSR, transfer function, hidden modes, reaching target states 6.011, Spring 2018 Lec 8 1 Modal solution of driven DT system q[n + 1] = V⇤ V-1 q[n] +bx[n] , y[n] = cT q[n] + dx[n] | {z } r[n] # -1 T r[n + 1] = ⇤r[n] + V | {z b} x[n] , y[n] = c | {zV} r[n] + dx[n] ⇠T Because ⇤ is diagonal, we get the decoupled scalar equations ⇣X L ⌘ ri [n + 1] =λi ri [n] + β i x[n] , y[n] = ⇠i ri [n] + d[n] 1 2 Underlying structure of LTI DT state- space system with L distinct modes d z - n1 x[n] b1 ξ1 y[n] + z - nL bL ξL 3 Reachability and Observability ⇣X L ⌘ ri [n + 1] = i ri [n] + i x[n] , y[n] = ⇠i ri [n] + d[n] 1 for i = 1, 2, . . . , L # j = 0 , the jth mode cannot be excited from the input i.e., the jth mode is unreachable ⇠k = 0 , the kth mode cannot be seen in the output i.e., the kth mode is unobservable 4 Hidden modes ⇣X L ⌘ i ⇠i H(z) = +d i=1 z i Any modes that are unreachable ( i = 0) or/and unobservable (⇠i = 0) are “hidden” from the input-output transfer function. 5 ZIR + ZSR ri [n] = i ri [n 1] + i x[n 1] # n X k 1 ri [n] = ( ni )ri [0] + i i x[n k] , n 1 | {z } k=1 ZIR | {z } ZSR # L X 6 q[n] = vi ri [n] i=1 More directly q[n] = Aq[n 1] + bx[n 1] # n X q[n] = (An ) q[0] + Ak-1 b x[n k] , n 1 | {z } k=1 ZIR | {z } ZSR (linear jointly in initial state and input sequence) 7 Similarly for CT systems r˙i (t) = i ri (t) + i x(t) # Z t ri (t) = (e i t )ri (0) + e i⌧ ix(t ⌧ ) d⌧ , t 0 | {z } ZIR |0 {z } ZSR # L X q(t) = vi ri (t) 8 i=1 Decoupled structure of CT LTI system in modal coordinates d b1l1 s - n1 x(t) y(t) + bLlL s - nL 9 More generally Z t q(t) = (eAt ) q(0) + eA⌧ b x(t ⌧ ) d⌧ , t 0 | {z } 0 ZIR | {z } ZSR where 2 3 At 2