Matrix exponential, ZIR+ZSR, transfer function, hidden modes, - PowerPoint PPT Presentation

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 Λ V − 1 q [ n ] + b x [ n ] , T y [ n ] = c q [ n ] + d x [ n ] | {z } r [ n ] ↓ T V r [ n ] + d x [ n ] r [ n + 1] = Λ r [ n ] + V − 1 b x [ n ] , y [ n ] = c | {z } |{z} β ξ T β Because Λ is diagonal, we get the decoupled scalar equations ⇣ L ⌘ X r i [ n + 1] = λ i r i [ n ] + β i x [ n ] , y [ n ] = ξ i r i [ n ] + d [ n ] λ β 1 2

Underlying structure of LTI DT state- space system with L distinct modes d x [ n ] y [ n ] b 1 ξ 1 + z - n 1 ξ L b L z - n L 3

Reachability and Observability ⇣ L X ⌘ r i [ n + 1] = λ i r i [ n ] + β i x [ n ] , y [ n ] = ξ i r i [ n ] + d [ n ] λ β 1 for i = 1 , 2 , . . . , L ↓ β β j = 0 , the j th mode cannot be excited from the input i.e., the j th mode is unreachable ξ k = 0 , the k th mode cannot be seen in the output i.e., the k th mode is unobservable 4

Hidden modes ⇣ L β β i ξ i ⌘ X H ( z ) = + d λ z − λ i i =1 β Any modes that are unreachable ( β i = 0) or/and unobservable ( ξ i = 0) are “hidden” from the input-output transfer function. 5

ZIR + ZSR λ β r i [ n ] = λ i r i [ n − 1] + β i x [ n − 1] − − ↓ n X k − 1 β i x [ n − k ] n λ λ β − ≥ r i [ n ] = | ( λ i ) r i [0] + n ≥ 1 λ i , {z } k =1 ZIR | {z } ZSR ↓ L X q [ n ] = v i r i [ n ] 6 i =1

More directly … − − q [ n ] = Aq [ n − 1] + b x [ n − 1] ↓ X n q [ n ] = ( A n ) q [0] + A k − 1 b x [ n k ] , 1 − n ≥ | {z } k =1 | {z ZIR } ZSR (linear jointly in initial state and input sequence) 7

Similarly for CT systems ˙ i ( t ) = λ i r i ( t ) + β i x ( t ) λ β r ↓ Z t λ λ | e λ i t ) e λ i τ β i x ( t − τ ) dτ r i ( t ) = ( r i (0) + β − t ≥ 0 ≥ , {z } 0 | {z } ZIR ZSR ↓ L X v i r i ( t ) q ( t ) = i =1 8

Decoupled structure of CT LTI system in modal coordinates d y ( t ) x(t) b 1 l 1 + s - n 1 b L l L s - n L 9

More generally Z t A τ b x ( t A t ) q (0) + q ( t ) = ( e e τ ) d τ , 0 t − ≥ | {z } | 0 {z ZIR } ZSR where 3 2 2 t + A 3 t A t e = I + A t + · · · + A 2! 3! Λ t V − 1 = V e 10

6 Key properties of matrix exponential A . 0 e = I d A t A t e = A e = e A t A dt A ( t 1 + t 2 ) A t 1 A t 2 e = e e A 1 + A 2 but e A 1 A 2 e = e unless the two matrices commute 11

In the transform domain … The matrix extension of 1 at ↔ e s a − is A t ↔ ( s I e A ) − 1 − Input-output transfer function: T ( s I H ( s ) = c A ) − 1 b + d 12 −

Reaching a target state from the origin (e.g., in a 2 nd -order system) q [ n + 1] = Aq [ n ] + b x [ n ] , q [0] = 0 b = v 1 β 1 + v 2 β 2 β β Reaching a target state in 2 steps: q [2] = v 1 γ 1 + v 2 γ 2 γ γ ⇓  x [1]  1 � − 1  β 1 � − 1  γ 1 � � � � λ β γ � � λ 1 0 = λ β γ x [0] 1 λ 2 0 β 2 γ 2  λ 2 �  γ 1 /β 1 � λ − λ γ β � 1 − λ 1 = . γ β − λ − λ 1 1 γ 2 /β 2 13

MIT OpenCourseWare https://ocw.mit.edu 6.011 Signals, Systems and Inference Spring 201 8 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. 14