Control Systems Design, SC4026 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft
Lecture 4 Controllability (a.k.a. Reachability) and Observability Algebraic Tests (Kalman rank condition & Hautus test) A few Examples Duality Lack of Controllability and of Observability SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 1
Controllability Consider d x(t) = Ax(t) + Bu(t), dt x(0) = x 0 R n with known 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), with τ [0, t], 0 < t <, such that x(t) = x t SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 2
Controllability and Reachability (Notice distinction in nomenclature, as in AM book, between reachability & controllability) 168 CHAPTER 6. STAT Pick point x 0, time T 0. Define time-dependent set R(x 0, T ) (reachability set) x(t ) x 0 R(x 0, T ) (a) Reachable set Note on the 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. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 3
Algebraic Conditions for Controllability Controllability depends on the 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) Lemma (Cayley-Hamilton): Any square matrix satisfies its own characteristic equation SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 4
Proof (of Theorem): consider characteristic polynomial of A det[si A] = s n +a 1 s n 1 +...+a n 1 s+a n = 0, for an n n matrix A 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 2010, dr. A. Abate, DCSC, TU Delft 5
The test is known as Kalman rank condition rank [B AB A 2 B... A n 1 B] = n Extensions to time-varying, non-linear case through notion of state-transition matrix (also mentioned in Lec. 2) see bibliography for more details In MATLAB, use command ctrb over ss structure to obtain controllability matrix SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 6
Consider: A = Controllability: An Example 5 4 4 1 0 2 1 1 1 B = Compute: [B AB A 2 B... A n 1 B] = 3 1 1 3 7 19 1 1 1 1 3 9 Perform elementary column operations (2 o + 1 o, 3 o 1 o ): rank 3 7 19 1 1 1 1 3 9 (A, B) not controllable = rank 3 4 16 1 0 0 1 2 8 = 2 < n = 3 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 7
Controllability: a Second Example Consider the following simple model (double integrator) { ẋ1 = x 2 = u CHAPTER 6. STATE FEEDBACK ẋ 2 x(t ) E hable set R(x 0, T ) (b) Reachability through control SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 8 able set for a control system. The set R(x 0, T ) shown in (a) is the set m x 0 in time less than T. The phase portrait in (b) shows the dynamics
Controllability: a Third Example Consider the following simple model { ẋ1 = x 1 + δx 2 + u ẋ 2 = x 2 + δu A = [ 1 δ 0 1 ], B = [ 1 δ ] controllable for δ 0 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 9
Controllability: a Fourth Example Inverted pendulum on cart (Segway) d dt p θ ṗ θ = 0 0 1 0 0 0 0 1 0 mg/m 0 0 0 (M + m)g/ml 0 0 p θ ṗ θ + 0 0 1/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 2010, dr. A. Abate, DCSC, TU Delft 10
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 2010, dr. A. Abate, DCSC, TU Delft 11
Observability Consider autonomous (no control) 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) within the interval τ [0, t], t > 0 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 12
Observability follows from Ce At, hence it depends on matrices A, C: ] ] C Ce At = C [I + At + A 2t2 2! +... = [1 t t2 2!... CA CA 2. Theorem: (A, C) observable if and only if rank C CA CA 2. CA n 1 = n (= number of columns) In MATLAB, use obsv over ss structure to obtain obs. matrix SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 13
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 (will elaborate in exercise session) SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 14
Controllability & Observability: Alternative Test by Hautus Rationale: λi A has rank n for all λ not equal to an eigenvalue of A rank check needs only to be evaluated when λ is equal to an eigenvalue of A Finding uncontrollable or unobservable eigenvalues can be done 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 C ] = n, for all λ C SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 15
Controllability: Example of Hautus Test Consider (A, B) used above: A = 5 4 4 1 0 2 1 1 1 Matrix A has eigenvalues in the set { 1, 2, 3} Perform following computations: rank [ (λi A) λ= 1 B ] = rank B = 1 1 2 1 4 4 4 3 1 1 0 1 eigenvalue 1 is controllable 3 1 1 = 3 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 16
rank [ (λi A) λ= 2 B ] = rank 1 2 2 1 3 4 4 3 1 1 1 1 = 2 eigenvalue 2 is uncontrollable rank [ (λi A) λ= 3 B ] = rank 2 4 4 3 1 3 2 1 1 1 2 1 = 3 eigenvalue 3 is controllable SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 17
A few Reasons for the Emergence of uncontrollability and of unobservability UC 1 - physical uncontrollability - uncoupled variables are not affected by input UC 2 - parallel interconnection controlled by single input UO 1 - directly unmeasured variables are not fed back to measured ones UO 2 - single variables cannot be extracted from global observation function This is in particular true for SISO systems SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 18
θ 2 A few Examples of uncontrollable and unobservable 204 CHAPTER 6. STATE FEEDBACK Models m 1. 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 2010, 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
ble, detectable 117 Controllability/observability 9: pole-zero cancellation 118 relocated by feedback. eigenvalues are owing weaker concepts 0 } are controllable 0 } are observable u(s) z(s) y(s) s +3 s +1 s +1 s +2 2. Consider system in series, described via the following block diagram: Simulation diagram (time domain): u(t) ξ 1 z(t) ξ 2 y(t) 2 1 1 2 is are considered as 1 1 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 (check with Hautus test) Unobservable eigenvalue λ 2 = 1 is cancelled by forming transfer function SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft ( ) 20 s +3 C(sI A) 1 B + D = s +2
Controllability/observability 9: zero-pole cancellation 121 Controllability/observa v(s) x(s) w(s) s +1 s +3 s +2 s +1 3. Consider system in series, described via the following block diagram: Simulation diagram (time domain) v(t) ξ 2 x(t) ξ 1 w(t) 1 2 2 1 Diagram uniquely defines (A [ ] 1 2 A B = 0 2 C D 1 1 [ [ ] 2 4 B AB = 1 2 [ C CA ] = [ ] 1 1 1 0 1 1 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 2010, 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
4. Consider model ẋ = ax + bu. Introduce variables y = cx. Then ẏ = ay + bcu. The controllability matrix of the parallel composition of the two models is ( ) b ab, bc abc which has rank 1. Thus, it is not controllable. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft 22