Controllability & Observability

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Thomasine Little
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1 Cotrollability & Observability 可制御性可観測性 ()= t A()+ t Bu(), t ()= y()= t C() t Cotrollable ( 可制御 ): A system is said to be cotrollable if there eists a eteral iput u to move the iteral state from ay iitial state to ay other fial state * i a fiite time iterval. 適当な制御入力 u を用いて状態を任意の目標状態 * に一致させることができる Observable ( 可観測 ): A system is said to be observable if the curret state ca be determied i fiite time usig oly the output y. 出力 y を測定 ( 観測 ) することによって状態を決定できる

2 Cotrollability ad Observability Cotrollable ( 可制御 ): 適当な制御入力 u を用いて状態を任意の目標状態 * に一致させることができる Observable ( 可観測 ): 出力 y を測定 ( 観測 ) することによって状態を決定することができる b = or cut u = a ( + u) u does NO affect b b a a c c + + y c = or cut y = c ( + ) is NO trasmitted to y caot be cotrolled Ucotrollable caot be kow Uobservable

3 Judge Cotrollability/Observability ()= t A()+ t bu(), t ()=, yt ()= c () t () t R, A R, b R, c R {A, b}: Cotrollable Cotrollability matri U C : osigular {c, A}: Observable Observablity matri U O : osigular U C = b Ab A b A b U O = c A c A c A c

4 System dyamics Judgig Cotrollability u () t = 6 () t + (), t () = * * () = 5, t = Desirable velocity at t= *() () 5 t () = = = 4 Cotrol iput to achieve desirable velocity at t= * * u () = ( () 6 ()) = ( 4 6 ) = () t = 6 () t, () = Cotrolled system ( t) () t = 6 (), () = () = 7 () = 7 = 5

5 Cotrollability judgig equatio real ideal ideal real ()= t A()+ t bu(), t ()= ()= t A()+ t Bu(), t, u R *()= A()+ Bu* () If B R is o-sigular u*()= B *() A ( ) impractical u*()= b ( *() A ) ()= t A ()+ t bu() t = A( A+ bu) + bu = A+ Abu+ bu = A+ Ab b u u *() = *() *() [ ] ( Ab b A ) { [ ] if o-sigular *()=( * )/ t u u

6 Cotrollability judgig equatio ()= t A()+ t bu(), t ()= b ( A ) = + u+ u = + u*()= *() u u A Ab b A Ab b *() = *() *() [ ] ( Ab b A ) { [ ] if o-sigular u u cotiued 3 u 3 = A + A b + Ab+ b= A + A b Ab b u u u { u u u u ( ) colum vectors () = A + A b A b Ab b { ( ) u u

7 y y y y Judgig Observability = c ( + ) = c = c A bu = c A + c bu = c = c A + c bu = c A + c Abu+ c bu = c A + c A bu + c A bu + + c bu ( ) ( ) Vector-Matri epressio y c u y c A c b u y = + u c A c Ab c b ( ) 3 ( ) y c A c A b c A b c b u ŷ Measured output Kow Uo Ukow State to determie Γ û Geerated cotrol Kow

8 Judgig Observability ()= t A()+ t bu(), t ()=, yt ()= c () t cotiued ˆy = U + o Γuˆ If U o is osigular ( Γu ) = Uo ˆ ˆy

9 Eample Judge Cotrollability/Observability ()= t ()+ (), ()=[ 3 t u t y t ] () t 3 c A Cotrollability U 3 [ ] C = b Ab Observability c UO = c A b = 3 Ab = A b [ 4 ] detu C =-5 U C : osigular {A, b}: Cotrollable 3 = 4 (= ) (= ) detu O =-4 U O : osigular {c, A}: Observable c A = c A = Determiat Nosigular

10 Cotrollable Caoical Form 可制御正準形 State equatio for trasformed state vector z(t) z () t = Az() t + bu(), t z() = yt () = z() t c a a a a z() t = z() t + u() t, z() = z y () t = c c c c () t z o special form rasfer fuctio Gs () = c s + + c s+ c s s s a a a

11 Cotrollable Caoical Form Cotrollability matri U C l l UC = b Ab A b UC = l rasformatio matri l l A = l A Caoical system A = b= a a a a c= [ c c c c ] No special structure

12 Eample Cotrollable caoical form rasform the followig system ito cotrollable caoical form. ( t)= ( t)+ u( t ), ()+, y( t)= [ 3 ] 3 ( t) A b rasformatio matri UC = [ b Ab] = 5 5 UC = = = l l 6 7 A = = = = l A Caoical form A =, =, 56 =, c= c 3 Ab= A b (= ) b [ ] z z z y z () t () t u(), () 56 = + t =, () t = [ 7 7 ] () t

13 Sigle motor cotrol system State equatio d dt v(t) θ (t) τ (t) Rotary ecoder Jθ() t + D θ() t + Kθ() t = kv() t ( = τ() t ) θ () = θ, θ() = ω θ () t θ() t θ() θ vt (), ω() t = K D ω() t + k = ω() ω J J J yt () A b θ () t = [ ] ω() t c Cotrollable Observable u

14 Coupled motors cotrol systems θ (t) τ (t) v(t) θ (t) τ (t) τ(t)=c(θ(t)-θ(t)) θ (t) θ (t) -τ(t) τ(t) v(t)

15 Coupled motors cotrol system θ() t θ() t K D k d ω() t J J ω() t J = + v() t dt θ() t θ() t ω() t K D ω() t k J J J θ() θ ω() ω = θ() θ ω () ω yt () = [ ] θ() t ω() t θ() t ω () t θ (t) τ (t) θ (t) τ (t) v(t)

16 Coupled motors cotrol system τ(t)=c(θ(t)-θ(t)) θ (t) -τ(t) θ (t) τ(t) ( )) ( ) θ () t + θ () t + θ () t = () t θ () t θ ( t J D K kv c θ () t + θ () t + θ ( t) = θ ( t) θ () t J D K c v(t) θ() t θ ( ) () t K c D c ω () t () k + ω t = + dt θ () t θ () t ω ( ) () t c K c D ω () t + J J J J d J J J J J t v ()

17 Coupled motor cotrol system θ() t θ ( ) () t K c D c ω () t () k + ω t = + dt θ () t θ () t ω ( ) () t c K c D ω () t + J J J J θ() θ θ() t ω() ω ω() t = yt () = [ ] θ() θ θ() t ω() ω ω () t d J J J J J t τ(t)=c(θ(t)-θ(t)) v () θ (t) θ (t) -τ(t) τ(t) v(t)

18 Eercise. Check the cotrollability ad observability of (a) sigle, (b) parallelly-coupled, ad (c) seriallycoupled motor cotrol system for parameter values J=, K=, d=, c=, k=. Calculate the free respose of the three motor cotrol systems with iitial state = [ ] for (a), = [ ] for (b), ad = [ ] for (c), respectively. 3. Draw the graph of the free resposes. Report Deadlie: Jue th

19 State Equatio for Coupled Motor System Parallel Couplig Case d dt A O b = + u O A b A O A O A O A O A O A O = =, = O A O A O A O A O A O A A O b Ab A O b Ab Ab = =, A b = =, O A b Ab O A b Ab Cotrollability Matri U C U C = b Ab A b A b 3 3 = = E. = a b e, c d = f b Ab A b A b b Ab A b A b 3 A b Ucotrollable! ie ae+cf Ha +bcle+hac+cdlf HaHa +bcl +bhac+cdll e+hcha +bcl +dhac+cdll fy f be+df Hab+bdLe+ Hbc+d Lf HaHab+bdL +bhbc+d LL e+hchab+bdl +dhbc+d LL f e ae+cf Ha +bcle+hac+cdlf HaHa +bcl +bhac+cdll e+hcha +bcl +dhac+cdll f j kf be+df Hab+bdLe+ Hbc+d Lf HaHab+bdL +bhbc+d LL e+hchab+bdl +dhbc+d z LL f{

20 Cotrollability Parallel Coectio J= ; K = ; d = ; c = ; k = ; kc = ; i c K J d c J J A= c c k K J J d J y { b= i k J kc J y { i = k y= H L Sequetial Coectio y i y + u { k { J= ; K = ; d = ; c = ; k = ; kc = ; i y i y = + u k { k { y= H L i y Uc = k { Ucotrollable i y 3 Uc = k { Cotrollable

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