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Full modal solution, asymptotic stability, reachability and observability 6.011, Spring 2018 Lec 7 1

Modal solution of CT system ZIR L X α i v i e λ i t q ( t ) = 1 with the weights { α i } L determined by the initial condition: 1 L X q (0) = α i v i 1 2

Asymptotic stability of CT system In order to have q ( t ) → 0 for all q (0) , we require { Re ( λ i ) < 0 } L 1 i.e., all eigenvalues (natural frequencies) in open left half plane 3

The DT case: linearization at an equilibrium DT case: q [ n ] = ¯ + q [ n ] , x [ n ] = ¯ e x + x [ n ] , e q q [ n + 1] = f ( q [ n ] , x [ n ]) ↓ � � � � h ∂ f i h ∂ f i � � � � q [ n + 1] ≈ e q ,x e q [ n ] + x [ n ] q ,x e � � � � ∂ q ∂ x ¯ ¯ ¯ ¯ for small perturbations q [ n ] and x [ n ] from equilibrium e e 4

Modal solution of DT system ZIR Could parallel CT development, but let’s proceed di ff erently:  λ 1 0  0 0 · · · 0 0 · · · 0 λ 2     A [ v 1 v 2 · · · v L ] = [ v 1 v 2 · · · v L ]   . . . .   . . . . . .   . . . . .   0 0 0 · · · λ L or AV = V Λ − 1 or A = V Λ V − 1 ) · · · ( V Λ V − 1 ) = V Λ n V or A n = ( V Λ V − 1 5

A n q [0] = V Λ n V − 1 q [0] q [ n ] = | {z }  α 1   α 2      .   .   .     α L 2 λ n λ 3 2 α 1 0 0 0 3 · · · 1 λ λ n 0 0 0 6 7 · · · 2 α 2 6 7 6 7 so q [ n ] = [ v 1 v 2 · · · v L ] 6 7 6 7 . 6 7 6 7 . . . . . ... . 6 7 4 5 . . . . . . . . 4 5 α L λ λ n 0 0 0 · · · L L X n = α i v i λ i 6 1

Asymptotic stability of DT system In order to have q [ n ] → 0 for all q [0] , we require {| λ i | < 1 } L 1 i.e., all eigenvalues (natural frequencies) inside unit circle 7

A n for increasing n  0 . 6 � � �  � 0 6 101 100 A 1 = . A 2 = , 0 . 6 0 6 , − 101 − 100 .  0 . 6 � �  � � 100 . 5 100 100 A 3 = A 4 = , . − 100 . 5 − 100 0 0 5 . 8

A n for increasing n  0 . 6 � � � n  � 0 6 0 . 5 0 5 A n = = (1 . 2) n . . 1 0 . 6 0 6 0 . 5 0 5 . . � � n  101 100 A n 2 = = A 2 − 101 − 100 9

A n for increasing n � n � �   � 100 . 5 100 201 200 A n = = (0 . 5) n 3 − 100 . 5 − 100 − 201 − 200  0 . 6 n � n � �  � 0 . 6 100 1000(0 . 6 n − 0 . 5 n ) A n = = 4 0 0 5 0 0 . 5 n . 10

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 ] ↓ r [ n + 1] = Λ r [ n ] + V − 1 b x [ n ] , T V r [ n ] + d 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 11

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 12

Reachability and Observability ⇣ L ⌘ λ β X r i [ n + 1] = λ i r i [ n ] + β i x [ n ] , [ n ] = ξ i r i [ n ] + d [ n ] y 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 13

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