ELEC4410 Control Systems Design Lectre 16: Controllability and Observability Canonical Decompositions Jlio H. Braslavsky jlio@ee.newcastle.ed.a School of Electrical Engineering and Compter Science Lectre 15: Observability p.1/14
Otline Canonical Decompositions Kalman Decomposition and Minimal Realisation Discrete-Time Systems Lectre 15: Observability p.2/14
Canonical Decompositions The Canonical Decompositions of state eqations will establish the relationship between Controllability, Observability, and a transfer matrix and its minimal realisations. Consider the state eqation ẋ = Ax + B y = Cx + D where A Rn n, B R n p, C R p n, D R q p. (SE) Let x = Px, where P is nonsinglar, P R n n. Then we know that the state eqation x = Ā x + B y = C x + D where Ā = PAP 1, B = PB, C = CP 1, D = D, is algebraically eqivalent to (SE). Lectre 15: Observability p.3/14
Canonical Decompositions Theorem (Controllable/Uncontrollable Decomposition). Consider the n-dimensional state eqation (SE) and sppose that [ rank C = rank B AB A n 1 B = n 1 < n (i.e., the system is not controllable). Let the n n matrix of change of coordinates P be defined as P 1 = [q 1 q 2 q n1 q n where the first n 1 colmns are any n 1 independent colmns in C, and the remaining are arbitrarily chosen so that P is nonsinglar. Then the eqivalence transformation x = Px transforms (SE) to x C Ā 12 = ĀC x C + x C 0 Ā C x C [ y = C C x + D C C B C 0 Lectre 15: Observability p.4/14
Canonical Decompositions The states in the new coordinates are decomposed into Controllable C y x C : x C : n 1 controllable states n n 1 ncontrollable states C Uncontrollable Lectre 15: Observability p.5/14
Canonical Decompositions The states in the new coordinates are decomposed into Controllable C y x C : x C : n 1 controllable states n n 1 ncontrollable states C Uncontrollable The redced order state eqation of the controllable states x C = Ā C x C + B C ȳ = C C x + D is controllable and has the same transfer fnction as the original state eqation (SE). The MATLAB fnction ctrbf transforms a state eqation into its controllable/ncontrollable canonical form. Lectre 15: Observability p.5/14
Canonical Decompositions Theorem (Observable/Unobservable Decomposition). Consider the n-dimensional state eqation (SE) and sppose that rank O = rank [ C CA CA n 1 = n 2 < n (i.e., the system is not observable). Let the n n matrix of change of coordinates P be defined as P = p 1 p 2 p n2 p n where the first n 2 colmns are any n 2 independent colmns in O, and the remaining are arbitrarily chosen so that P is nonsinglar. Then the eqivalence transformation x = Px transforms (SE) to x O = ĀO 0 x O + B O xõ Ā 21 ĀÕ xõ BÕ [ y = C O 0 x + D Lectre 15: Observability p.6/14
Canonical Decompositions The states in the new coordinates are decomposed into Õ x O : xõ : n 2 observable states n n 2 nobservable states Unobservable O Observable y Lectre 15: Observability p.7/14
Canonical Decompositions The states in the new coordinates are decomposed into Õ x O : xõ : n 2 observable states n n 2 nobservable states Unobservable O Observable y The redced order state eqation of the observable states x O = Ā O x O + B O ȳ = C O x + D is observable and has the same transfer fnction as the original state eqation (SE). The MATLAB fnction obsvf transforms a state eqation into its observable/nobservable canonical form. Lectre 15: Observability p.7/14
Kalman Decomposition The Kalman decomposition combines the Controllable/Uncontrollable and Observable/Unobservable decompositions. Every state-space eqation can be CÕ Unobservable CO y transformed, by eqivalence transformation, into a canonical form that splits the states into Controllable and observable states CO CÕ Uncontrollable Lectre 15: Observability p.8/14
Kalman Decomposition The Kalman decomposition combines the Controllable/Uncontrollable and Observable/Unobservable decompositions. CÕ Every state-space eqation can be transformed, by eqivalence transfor- Unobservable mation, into a canonical form that splits CO y the states into Controllable and observable states Controllable bt nobservable CO states CÕ Uncontrollable Lectre 15: Observability p.8/14
Kalman Decomposition The Kalman decomposition combines the Controllable/Uncontrollable and Observable/Unobservable decompositions. CÕ Every state-space eqation can be transformed, by eqivalence transfor- Unobservable mation, into a canonical form that splits CO y the states into Controllable and observable states Controllable bt nobservable CO states Uncontrollable bt observable CÕ states Uncontrollable Lectre 15: Observability p.8/14
Kalman Decomposition The Kalman decomposition combines the Controllable/Uncontrollable and Observable/Unobservable decompositions. CÕ Every state-space eqation can be transformed, by eqivalence transfor- Unobservable mation, into a canonical form that splits CO y the states into Controllable and observable states Controllable bt nobservable CO states Uncontrollable bt observable CÕ states Uncontrollable and nobservable Uncontrollable states Lectre 15: Observability p.8/14
Kalman Decomposition The Kalman decomposition brings the system to the form x CO Ā CO 0 Ā 13 0 x CO B CO x C Õ Ā 21 Ā = C Õ Ā 23 Ā 24 x C Õ B + C Õ x CO 0 0 Ā CO 0 x CO 0 x CÕ 0 0 Ā 43 Ā CÕ x CÕ 0 }{{} x [ y = C CO 0 C CO 0 x + D Lectre 15: Observability p.9/14
Kalman Decomposition The Kalman decomposition brings the system to the form x CO Ā CO 0 Ā 13 0 x CO B CO x C Õ Ā 21 Ā = C Õ Ā 23 Ā 24 x C Õ B + C Õ x CO 0 0 Ā CO 0 x CO 0 x CÕ 0 0 Ā 43 Ā CÕ x CÕ 0 }{{} x [ y = C CO 0 C CO 0 x + D A minimal realisation of the system is obtained by sing only the controllable and observable states from the Kalman decomposition. x CO = Ā CO x CO + B CO ȳ = C CO x + D Lectre 15: Observability p.9/14
Kalman Decomposition Example. Consider the system in Modal Canonical Form [ 10 ẋ = x + [ λ1 1 0 0 0 λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 y = [ 1 0 1 1 x From the example seen in the Ttorial, Controllability and Observability in Modal Form eqations, we see that the first λ 1 is controllable and observable λ 2 is not controllable, althogh observable λ 3 is controllable and observable 0 2 Ths a minimal realisation of this system is given by [ x = λ1 0 0 λ 3 x + [ 1 2 with transfer fnction G(s) = y = [ 1 1 x 1 s λ 1 + 2 s λ 3 Lectre 15: Observability p.10/14
Canonical Decompositions Example (Controllable/Uncontrollable decomposition). Consider the third order system [ 1 1 0 [ 0 1 ẋ = 0 1 0 x + 1 0 0 1 1 0 1 y = [ 1 1 1 x Compte the rank of the controllability matrix, rank C = rank [ B AB A 2 B = rank [ 0 1 1 1 2 1 1 0 1 0 1 0 0 1 1 1 2 1 = 2 < 3, ths the system is not controllable. Take the change of coordinates formed by the first two colmns of C and an arbitrary third one independent of the first two, P 1 = Q [ 0 1 1 1 0 0 0 1 0. Lectre 15: Observability p.11/14
Canonical Decompositions Example (contination). By doing ^x = Px we obtain the eqivalent eqations 1 0. 0 ^x = 1 1. 0............ x + 0 0. 1 y = [1 2.. 1 x [ 1 0 0 1...... 0 0 and the redced controllable system ^x = [ 1 0 1 1 x + [ 1 0 0 1 y = [ 1 2 x which has the same transfer matrix than the original system [ G(s) =. s+1 s 2 2s+1 2 s 1 Lectre 15: Observability p.12/14
Discrete-Time Systems For controllability and observability of a discrete-time eqation x[k + 1 = Ax[k + B[k y[k = Cx[k + D[k we can se the same Controllability and Observability matrices rank tests that we have for continos-time systems, rank C = rank [ B AB A n 1 B = n Controllability rank O = rank = n Observability [ C CA CA n 1 Canonical decompositions are analogos. Lectre 15: Observability p.13/14
Smmary When a system is not controllable or not observable, there might be a part of the system that still is controllable and observable. Lectre 15: Observability p.14/14
Smmary When a system is not controllable or not observable, there might be a part of the system that still is controllable and observable. The controllability and observability matrices can be sed to split (by a change of coordinates) a state eqation into its controllable/ncontrollable parts and observable/nobservable parts. Lectre 15: Observability p.14/14
Smmary When a system is not controllable or not observable, there might be a part of the system that still is controllable and observable. The controllability and observability matrices can be sed to split (by a change of coordinates) a state eqation into its controllable/ncontrollable parts and observable/nobservable parts. The controllable and observable part of a state eqation yields minimal realisation. Lectre 15: Observability p.14/14
Smmary When a system is not controllable or not observable, there might be a part of the system that still is controllable and observable. The controllability and observability matrices can be sed to split (by a change of coordinates) a state eqation into its controllable/ncontrollable parts and observable/nobservable parts. The controllable and observable part of a state eqation yields minimal realisation. Ths, we conclde that for a state eqation minimal realisation controllable and observable Lectre 15: Observability p.14/14