Chapter 6 Controllability and Obervability

Transcription

1 Chapter 6 Controllability and Obervability Controllability: whether or not the state-space equation can be controlled from input. Observability: whether or not the initial state can be observed from output.

2 6.2 Controllability Consider the n-dimensional p-input equation & = + Bu Definition 6.2 The pair (, B) is said to be controllable if for any initial state () = and any final state 1, there eists an input that transfer to 1 in a finite time. Eample 6.1: Figure 6.2 (a) and (b) is not controllable.

3 Theorem 6.1 The following statement are equivalent. 1. The n-dimensional pair (, B) is controllable 2. The n n matri W (t) = c t e τ τ t (t τ) (t τ) BB e dτ is nonsingular for any t >. 3. The n np controllability matri C =[B B 2 B n-1 B] has rank n (full row rank). = e BB e dτ

4 4. The n (n+p) matri [-λi B] has full row rank at every eigenvalue, λ, of. 5. If all eigenvalues of have negative real parts, then the unique solution of Wc + Wc T = -BB T is positive definite. The solution is called the controllability Gramian and can be epressed as W c = τ e BB e τ dτ

5 6.2.1 Controllability indices Let and B be n n and n p constant matri. If (, B) is controllable, its controllability matri C = [B B 2 B n-1 B] = [b 1 b p b 1 b p n-1 b 1 n-1 b p ] has rank n and consequently, n linearly independent column.

6 If C has rank n, then µ 1 + µ µ p = n. The set {µ 1, µ 2,,µ p } is called the controllability indices and µ = ma(µ 1, µ 2,,µ p ) is called the controllability inde of (, B). Consequently, if (, B) is controllable, the controllability inde µ is the least integer such that ρ(c µ ) =ρ([b B 2 B µ-1 B]) = n.

7 The controllability inde satifies n / p µ min(n, n p + 1) where r(b) = p, and n B {B,B,..., n 1 B}(Linear combination) Corollary 6.1 The dimensional pair (, B) is controllable if and only if the matri C n-p+1 := [B B n-p B] where ρ(b) = p, has rank n or the n n matri C n-p+1 C T n-p+1 is nonsingular.

8 Theorem 6.2 The controllability property is invariant under equivalence transformation. Theorem 6.3 The set of controllability indices (, B) is invariant under any equivalence transformation and any reordering of the columns of B.

9 6.3 Observability The concept of observability is dual to that of controllability. Controllability studies the possibility of steering the state from input. Observability studies the possibility of estimating the state from output.

10 Consider the n-dimensional p-input q- output state equation & = + Bu y = C + Du Definition 6.O1 The above state equation is said to be onservable if for any unknow initial state (), there eists a finite t 1 > such that the knowledge of the input u and the output y over [, t 1 ] suffices t determine uniquely the initial state ().

11 Theorem 6.4 The state equation is observable if and only if the n n matri t τ τ W = e C Ce dτ is nonsingular for any t >. Theorem 6.5 (Theorem of duality) The pair (, B) is controllable if and only if the pair ( T, B T ) is observable.

12 Theorem 6.O1 The following statement are equivalent. 1. The n-dimensional pair (, C) is observable. 2. The n n matri t τ τ W = τ is e C Ce d nonsingular for any t >. 3. The nq n observability has rank n (full column rank). 4. The (n+q) n matri λi has full column rank at C every eigenvalue. O = C C. C n 1

13 5. If all eigenvalues of have negative real parts, then the unique solution of T W + W T = -C T C is positive definite. If (, C) is observable, its observability matri O has rank n. Let v m be the number of the linearly independent rows associated with c m. If O has rank n, then v 1 + v v q = n

14 The set {v 1,v 2,,v q } is called observablity indices and v = ma(v 1,v 2,,v q ) is called the observablility inde of (, C). If (, C) is observable, it is the least integer such that ρ(o C C ) : = C. v C 2 v = Dual to the controllability part, 1 n / q v min(n, n q + 1) where r(c)=q and n is the degree of the minimal polynomial of. n

15 Corollary 6.O1 The dimensional pair (, C) is observable if and only if the matri has full rank or the n n matri O T n-q+1o n-q+1 is nonsingular. = + q n 2 1 q n C. C C C O

16 Theorem 6.O2 The observability property is invariant under any equivalence transformation. Theorem 6.O3 The set of observability indices of (, C) is invariant under any equivalence transformation and any reordering of the rows of C.

17 6.4 Canonical decomposition Theorem 6.6 Consider the n-dimensional state equation with ρ(c) = ρ([b B n-1 B]) = n 1 < n. We form the n n matri P -1 := [q 1 q n 1 q n ] where the first n 1 columns are any linearly independent columns of C, and the remaining columns can arbitrarily be chosen as long as P is nonsingular.

18 Then the equivalent transformation will obtain where is n 1 n 1 and is (n-n 1 ) (n-n 1 ), and the n 1 -dimensional subequation is controllable. u B c c c c 12 c c c + = & & [ ] Du C C y c c c c + = c c

19 Theorem 6.O6 Consider the n-dimensional state equation with We form the nn matri where n 2 rows are any n 2 linearly independent rows of O. n n C. C C O) ( 2 1 n < = = ρ ρ = n n 1 p. p. p P 2

20 Then the equivalence transformation will obtain where is n 2 n 2 and is (n-n 2 ) (n-n 2 ), and the n 2 -dimensional subequation is observable. u B B o o o o o 12 o o o + = & & [ ] Du C y o o o + = o o

21 Theorem 6.7 Every state-space equation can be transformed into the canonical form u B B co co co co co co co 43 co co co co co co co + = & & & & [ ] Du C C y o c co + =

22 6.5 Conditions in Jordan-form equations If a state equation is transformed into Jordan form, then the controllability and observability conditions become very simple. Consider the state equation & = J + Bu y = C ssume that J has only two distinct eigenvalues and can be written as J = diag(j 1, J 2 )

23 gain assume that J 1 has three Jordan blocks and J 2 has two Jordan blocks or J 1 = diag(j 11, J 12, J 13 ) J 2 = diag(j 21, J 22 ) The row of B corresponding to the last row of J ij is denoted by b lij. The column of C corresponding to the first column of J ij is denoted by c fij.

24 Theorem The state equation in Jordan form is controllable if and only if {b l11, b l12, b l13 } are linearly independent and {b l21, b l22 } are linearly independent. 2. That is observable if and only if {c f11, c f12, c f13 } are linearly independent and {c f21, c f22 } are linearly independent.

25 6.6 Discrete-time state equations Consider the n-dimensional p-input q-output state equation [k+1] = [k] + Bu[k] y[k] = C[k] Definition 6.D1 The discrete-time state equation or the pair (, B) is said to be controllable if for any initial state () = and any final state 1, there eists an input sequence of finite length that transfers to 1.

26 Theorem 6.D1 The following statements are equivalent: 1. The n-dimensional pair (, B) is controllable. 2. The n n matri W = n 1 m m= m dc [n 1] () BB ( ) is nonsingular. 3. n np controllability matri C d = [B B... has rank n (full row rank). 2 B n 1 B]

27 4. The n (n+p) matri [-λi B] has full row rank at every eigenvalue, λ, of. 5. If alleigenvalues of has magnitudes less than 1, then the unique solution of W dc -W dc T = BB T (Controllability Gramian) is positive definite.

28 Definition 6.D2 The discrete-time state equation or the pair (, C) is said to be observable if for any unknown initial state [], there eists a finite integer k 1 >, such that the knowledge of the input sequence u[k] and the output sequence y[k] from k = to k 1 suffices to determine uniquely the initial state [].

29 Theorem 6.DO1 The following statements are equivalent: 1. The n-dimensional pair (, C) is observable. 2. The n n matri W do n m is nonsingular (positive definite). 3. The nq n observability matri O d = has rank n (full column rank). m [n 1] = 1 = ( ) C C C C. C n 1 m

30 4. The (n+q) n matri λi C has full column rank at every eigenvalue, λ, of. 5. If all eigenvalues of have magnitude less than 1, then the unique solution of W do - T W do = C T C (observability Gramian) is positive definite.

31 There are three different controllability definitions: 1. Definition 6.D1. 2. Controllability to the origin: transfer any state to the zero state. 3. Reachability (controllability from the origin): Transfer the zero state to any state. In continuous-time case, because e t is nonsingular, the three definitions are equivalent.

32 In discrete-time case, if is nonsingular, the three definitions are again equivalent. But if is singular, then 1 and 3 are equivalent, but not 2 and 3.

33 6.7 Controllability after sampling Consider a continuous-time state equation &(t) = (t) + If the input is piecewise constant or u[k] := u[kt] = u(t) for kt t<(k+1)t then the equation can be described by [k + 1] = [k] + with = T e B = ( T Bu(t) Bu[k] a t dt)b = : MB

34 Theorem 6.9 suppose (6.65) is controllable. sufficient condition for its discretized equation in (6.66), with sampling period T, to be controllable is that Im[λ i -λ j ] 2πm/T for m=1, 2,, whenever Re[λ i -λ j ] =. Theorem 6.1 If a continuous-time linear timeinvariant state equation is not controllable, then its discretized state equation, with any sampling period, is not controllable.

35 6.8 LTV state equations Consider the n-dimensional p-input q- output state equation & = (t) + y = C(t) B(t)u In the time-varying case, the specification of t and t 1 is crucial.

36 Theorem 6.11 The n-dimensional pair ((t), B(t)) is controllable at time t if and only if there eists a finite t 1 > t such that n n matri W (t c, t 1 t τ τ τ Φ t 1, )B( )B ( ) (t1 ) = 1 Φ(t, τ) dτ where Φ(t, τ): the state transition matri is nonsingular.

37 The controllability condition without involving Φ(t, τ): Define M (t) = B(t), and then define recursively a sequence of n p matrices M m (t) as Mm + 1(t) : = (t)mm(t) + we have Using Φ(t, t) = (t, t)(t), we compute d dt M Φ( t2, t)b(t) = Φ(t2, t)m(t) t 2 Φ 2 m (t) d [ Φ(t2, t)b(t)] = [ Φ(t2, t)]b(t) + Φ(t2, t) B(t) t t dt d = Φ( t2, t)[ (t)m(t) + M(t)] = Φ(t2, t)m 1(t) dt

38 Thus, we have t m m Φ(t 2, t)b(t) = Φ(t 2, t)m m (t) Theorem 6.12 Let (t) and B(t) be n-1 times continuously differentiable. Then the n- dimensional pair ((t), B(t)) is controllable at t if there eists a finite t 1 >t such that [ M (t ) M (t )... M (t )] n rank n 1 1 = Theorem 6.O11 The pair((t), C(t)) is observable at time t if there eists a finite t 1 >t such that the n n matri t t W (t, t ) = 1 o 1 Φ ( τ, τ )C ( τ)c( τ) Φ( τ, t) dτ

39 Theorem 6.O12 Let (t) and C(t) be n-1 times continuously differentiable. Then the n-dimensional pair ((t), C(t)) is observable at t if there eists a finite t 1 >t such that N N rank Nn where 1 (t1) (t ) 1 =. 1(t1) Nm + 1(t) = Nm(t)(t) + with N = C(t) n d dt N m (t)