Control Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

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1 Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft

2 Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few Examples Duality Lack of Controllability and of Observability SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 1

3 Controllability Consider d x(t) = Ax(t) + Bu(t), dt x(0) = x 0 R n with solution x(t) = e At x 0 + t 0 e A(t τ) Bu(τ)dτ Definition: System is controllable if for any x 0 R n, x t R n there exists input u(τ) in τ [0, t], 0 < t <, such that x(t) = x t SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 2

4 Controllability and Reachability Notice distinction in nomenclature, as in AM book 168 CHAPTER 6. STAT Pick point x 0, time T 0. Define time-dependent set R(x 0, T ) x(t ) x 0 R(x 0, T ) (a) Reachable set Discussion on computation of reachability sets (b) Reachability through cont Figure 6.1: The reachable set for a control system. The set R(x 0, T ) shown in (a of points reachable from x 0 in time less than T. The phase portrait in (b) shows the for a double integrator, with the natural dynamics drawn as horizontal arrows and t inputs drawn as vertical arrows. The set of achievable equilibrium points is the x setting the control inputs as a function of the state, it is possible to steer the syst origin, as shown on the sample path. Connections with other topics in systems and control theory SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 3 The definition of reachability addresses whether it is possible to reac

5 Algebraic Conditions for Controllability Controllability depends on form e At B, i.e. on matrices A, B: e At B = [I + At + A 2 t2 2! +...]B = [B AB A2 B A 3 B...] 1 t t 2 2!. Theorem: (A, B) controllable if and only if rank [B AB A 2 B... A n 1 B] = n (= number of rows) Proof: consider characteristic polynomial of A det[si A] = s n +a 1 s n a n 1 s+a n = 0, for an n n matrix A SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 4

6 Lemma (Cayley-Hamilton): Any square matrix satisfies its own characteristic equation Using above Lemma, A n = a 1 A n 1 a 2 A n 2... a n 1 A a n I Col(A n B) Col[B AB A 2 B A 3 B... A n 1 B] Col[B AB...] Col [B AB... A n 1 B] rank[b AB A 2 B A 3 B...] = rank[b AB A 2 B A 3 B... A n 1 B] SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 5

7 Extensions to time-varying, non-linear case through notion of state-transition matrix (also mentioned in Lec. 2) see bibliography for mode details The test is known as Kalman rank condition rank [B AB A 2 B... A n 1 B] = n In MATLAB, use command ctrb over ss structure to obtain controllability matrix SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 6

8 Consider: A = Controllability: An Example B = Compute: [B AB A 2 B... A n 1 B] = Perform elementary column operations (2 o + 1 o, 3 o 1 o ): rank (A, B) not controllable = rank = 2 < n = 3 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 7

9 Controllability: a Second Example Consider the following simple model (double integrator) { ẋ1 = x 2 = u ẋ 2 CHAPTER 6. STATE FEEDBACK x(t ) E (a) Reachable set R(x 0, T ) (b) Reachability through control SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 8 he reachable set for a control system. The set R(x 0, T ) shown in (a) is the set chable from x 0 in time less than T. The phase portrait in (b) shows the dynamics integrator, with the natural dynamics drawn as horizontal arrows and the control

10 Controllability: a Third Example Inverted pendulum on cart (Segway) d dt p θ ṗ θ = mg/m (M + m)g/ml 0 0 p θ ṗ θ /M 1/Ml u Here n = 4. Controllability matrix is [B AB A 2 B A 3 B]: 0 1/M 0 mg/m 2 l 0 1/Ml 0 (M + m)g/(ml) 2 1/M 0 mg/m 2 l 0 1/Ml 0 (M + m)g/(ml) 2 0 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 9

11 Controllability matrix has full rank (i.e., equal to 4) (A, B) is controllable. (perhaps this is one good reason to buy a Segway... ) 1 1 Notice though that we have only shown that we can control the linearized model... SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 10

12 Observability Consider autonomous system: d dt x(t) = Ax(t), x(0) = x 0 R n ; y(t) = Cx(t) with solution y(t) = Ce At x 0 Definition: System is observable if any x 0 R n can be derived from observation y(τ) in interval τ [0, t], t > 0 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 11

13 Observability follows from Ce At, hence it depends on matrices A, C: Ce At = C ] [I + At + A 2t2 2! +... = ] [1 t t2 2!... C CA CA 2. CA n 1 Theorem: (A, C) observable if and only if rank C CA CA 2. CA n 1 = n (= number of columns) SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 12

14 In MATLAB, use command obsv over ss structure to obtain observability matrix SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 13

15 Duality Controllability/Observability Controllability, observability are dual concepts: (A, B) controllable (A T, B T ) observable (A, C) observable (A T, C T ) controllable The above fact relates to the following propositions: For every property that holds for controllability there exists a dual property in terms of observability State feedback and observer design problems are closely related (as we shall see later in class) Both controllability and observability are invariant under similarity transformations SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 14

16 Controllability & Observability: Another Test by Hautus Rationale: λi A has rank n for all λ not equal to an eigenvalue of A. Criterion needs only to be evaluated for values λ equal to an eigenvalue of A. Finding uncontrollable or unobservable eigenvalues is easier using: Hautus controllability condition: (A, B) is controllable iff rank [ λi A B ] = n for all λ C Hautus observability condition: (A, C) is observable iff [ ] λi A rank = n for all λ C C SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 15

17 Controllability: Example of Hautus Test Consider (A, B) used above: A = Matrix A has eigenvalues in the set { 1, 2, 3} Perform following computations: rank [ (λi A) λ= 1 B ] = rank B = eigenvalue 1 is controllable = 3 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 16

18 rank [ (λi A) λ= 2 B ] = rank = 2 eigenvalue 2 is uncontrollable rank [ (λi A) λ= 3 B ] = rank = 3 eigenvalue 3 is controllable SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 17

19 A few Reasons for the Emergence of uncontrollability and of unobservability UC - physical uncontrollability UC - too much symmetry in the model (redundant state variables) UO - single variables cannot be extracted from global observation function UO - directly unmeasured variables are not fed back to measured ones SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 18

20 θ 2 A few Examples of uncontrollable and unobservable 204 CHAPTER 6. STATE FEEDBACK Models m Consider identical systems in parallel: R l S S v 1 S S R The first is not controllable, whereas the second is not observable Figure 7.2: An unobservable system. Two identical subs The cart pendulum system shown on the left has toaform single the overall system output. The individual states o ual length and mass. Since the forces affectingsince the two the contributions of each to the output are not dis mics are SC4026 identical, Fall 2009, it is dr. not A. Abate, possible DCSC, TUtoDelft arbitrarily thecontrol 19 right is an example of such a system. on the right is a block diagram representation of this

21 ble, detectable 117 Controllability/observability 9: pole-zero cancellation 118 relocated by feedback. eigenvalues are owing weaker concepts Consider system in series: u(s) z(s) y(s) s +3 s +1 s +1 s +2 0 } are controllable 0 } are observable is are considered as Simulation diagram (time domain): u(t) ξ 1 z(t) ξ 2 y(t) ro cancellation 119 Controllability/observability 9: pole-zero cancellation 120 rank is 2 rank is 1 Eigenvectors of λ 1 = 1, λ 2 = 2: [ ] [ ] 1 0 m 1 = m 2 = 1 1 [ ] C m 1 =0 CA }{{} unobservable [ ] C m 2 0 CA }{{} observable It is controllable, though not observable Unobservable eigenvalue λ 2 = 1 is cancelled by forming transfer function SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft ( ) 20 s +3 C(sI A) 1 B + D = s +2

22 Consider system in series: Controllability/observability 9: zero-pole cancellation 121 v(s) x(s) w(s) s +1 s +3 s +2 s +1 Simulation diagram (time domain) v(t) ξ 2 x(t) ξ 1 w(t) Controllability/observa Diagram uniquely defines (A [ ] 1 2 A B = 0 2 C D 1 1 [ [ ] 2 4 B AB = 1 2 [ C CA ] = [ ] Observable, not completely Controllability/observability 9: zero-pole cancellation 123 Eigenvectors of λ 1 = 1, λ 2 = 2: [ ] 1 m 1 = 0 [ ] m 1 / range B AB }{{} uncontrollable m 2 = It is observable, though not controllable [ 2 1 ] [ ] m 2 range B AB }{{} controllable Uncontrollable eigenvalue λ = 1 is cancelled by forming transfer SC4026 Fall 2009, dr. A. Abate, function DCSC, TU Delft (A, B, C) 21is of minimal or ( ) s +3 C(sI A) 1 is observable. Then (A, B B + D = s +2 Result Let R be controllable sub to form of (94), then colu Let N be unobservable su to form of (98), then colu In general, T c and T o will Definition

23 Consider model ẋ = ax + bu. Introduce variables y = cx. Then ẏ = ay + bcu. The controllability matrix is ( b ab bc abc ), which has rank 1. Thus, it is not controllable. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft 22