Control Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft

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1 Control Systems Design, SC426 SC426 Fall 2, dr A Abate, DCSC, TU Delft

2 Lecture 5 Controllable Canonical and Observable Canonical Forms Stabilization by State Feedback State Estimation, Observer Design Stabilization by Output Feedback: Separation Principle SC426 Fall 2, dr A Abate, DCSC, TU Delft

3 Controllable Canonical Form Transform SISO model coordinates as z = T x to obtain the following A = a a 2 a 3 a n, B =, any C, D Its block diagram contains the very constants appearing in the model 6 REACHABILITY 73 y d b b 2 b n b n u z z 2 z n z n a a 2 a n a n Figure 64: Block diagram for a system in reachable canonical form The individual states of the system are represented by a chain of integrators whose input depends on the weighted values of the states The output is given by an appropriate combination of the system input and other states SC426 Fall 2, dr A Abate, DCSC, TU Delft 2 by λ(s) = s n + a s n + + a n s + a n (67) The reachability matrix also has a relatively simple structure:

4 The characteristic polynomial (same as that of original model) is: λ(s) = s n + a s n + + a n s + a n This canonical form also yields the following controllability matrix: [B AB A 2 B A 3 B A n B] = a a 2 a 2 a a a (A, B) in controllable canonical form is controllable (as expected) SC426 Fall 2, dr A Abate, DCSC, TU Delft 3

5 Recall (exercise set 4) that coordinate transformation does not affect controllability How to find a T that gets the canonical controllable form? From (A, B) (with W c ) to (T AT, T B) (with T W c ) T In conclusion, Fact: IF (A, B) is controllable, THEN there exists a transformation T that yields the controllable canonical form SC426 Fall 2, dr A Abate, DCSC, TU Delft 4

6 Observable Canonical Form Transform SISO model coordinates as z = T x to obtain A = a a 2 a n a n, any B, D, C = [ ] Its block diagram contains the very constants appearing in the model 7 OBSERVABILITY 25 u b n b n b 2 b d z n z n z 2 z y a n a n a 2 a Figure 73: Block diagram of a system in observable canonical form The states of the system are represented by individual integrators whose inputs are a weighted combination of the next integrator in the chain, the first state (rightmost integrator) and the system input The output is a combination of the first state and the input SC426 Fall 2, dr A Abate, DCSC, TU Delft 5 canonical form if its dynamics are given by a a b b

7 Can be verified that its characteristic polynomial is: λ(s) = s n + a s n + + a n s + a n This canonical form also yields the following observability matrix: C CA CA 2 CA n = a a a a (A, C) in observable canonical form is observable SC426 Fall 2, dr A Abate, DCSC, TU Delft 6

8 Again, recall that similarity transformations do not affect observability Fact: IF (A, C) is observable, THEN there exists a transformation T that yields the observable canonical form SC426 Fall 2, dr A Abate, DCSC, TU Delft 7

9 Stabilization by State Feedback Want to build feedback that depends on state, here available SC426 Fall 2, dr A Abate, DCSC, TU Delft 8

10 More generally, can set u = Kx + k r r, where r is a (constant) reference signal Obtain closed-loop system ẋ = (A BK)x + Bk r r Objective: select K so that closed loop has assigned, desired characteristic polynomial eigenvalue assignment - pole placement Controllability canonical form can be used for the purpose SC426 Fall 2, dr A Abate, DCSC, TU Delft 9

11 Stabilization by State Feedback: 2-d Example Consider 2-d control-dependent model, where A = [ ], B = [ γ ], C = [ ], D = It is controllable Introduce state feedback law u = Kx + k r r = k x k 2 x 2 + k r r Obtain (A BK) = [ ] γk γk 2, B k k k r = 2 [ ] γkr k r Characteristic polynomial is: p(s) = s 2 + (γk + k 2 )s + k SC426 Fall 2, dr A Abate, DCSC, TU Delft

12 Consider desired characteristic polynomial: p(s) = s 2 + 2ξ c ω c s + ω 2 c s,2 = ω c ( ξ c ± ξ 2 c ) Set k = ω 2 c, k 2 = 2ξ c ω c γω 2 c 78 CHAPTER 6 STATE FEEDBACK 78 CHAPTER 6 STATE FEEDBACK Lateral position y/b Steering angle δ [rad] Lateral position y/b Steering angle δ [rad] 5 4 ω2 c ω c ω c ω c Normalized Normalized time v time t v t Lateral position y/b Steering angle δ [rad] Lateral position y/b Steering angle δ [rad] 5 5 ζ c ζ c ζ c ζ c!5! Normalized Normalized time vtime t v t (a) Step(a) response Step response for varying for varying ω c ω c (b) Step (b) response Step response for varying for varying ζ c ζ c Figure Figure 66: State 66: feedback State feedback control control of a steering of a steering system system Step responses Step responses obtained obtained with with controllertrollers designed designed with ζ c with = 7 ζ c = and7 ω c and = ω5, c = 5, and2 and [rad/s] 2 [rad/s] are shown are shown in (a) in Notice (a) Notice that that con- response response speed increases speed increases with increasing with increasing ω c, but ω c that, butlarge that large ω c also ω c give also large give large initial initial control control actions actions Step responses Step responses obtained obtained with awith controller a controller designed designed with ωwith c = ω c and = ζand c = ζ5, c = 7 5, 7 and are andshown are shown in (b) in (b) SC426 Fall 2, dr A Abate, DCSC, TU Delft

13 22 State EstimationCHAPTER 7 OUTPUT FEEDBACK Process n u ẋ = Ax + Bu y = Cx + Du y Observer ˆx Figure 7: Block diagram for an observer The observer uses the process measurement y (possibly corrupted by noise n) and the input u to estimate the current state of the process, denoted ˆx Find ˆx R n and associated linear model (observer) dˆx = F ˆx + Gu + Hy dt Definition 7 (Observability) A linear system is observable if for any T > it is possible to determine the state of the system x(t ) through measurements of y(t) and such u(t) that on the ˆx(t) interval x(t) [, as T ] t Consider The definition case D above =, holds and for observer nonlinear ˆx = systems Aˆx + as Bu well, Calculate and the results estimation discussed error here x = have x ˆx extensions Error dynamics: to the nonlinear x = case A x The problem of observability is one that has many important applications, even outside feedback systems If a system is observable, then there are no hidden dynamics inside it; we can understand everything that is going on through observation (over time) of the inputs and outputs As we shall see, the problem of observability is of significant practical interest because it will determine if a set of sensors is SC426 Fall 2, dr A Abate, DCSC, TU Delft 2

14 (D = ) - consider observer ˆx = Aˆx + Bu + L(y C ˆx) Error dynamics: x = (A LC) x Perform eigenvalue assignment through gain matrix L (can make estimation speed faster than model s dynamics) Observability canonical form can be used for the purpose Observer design problem is dual to state feedback design problem A LC A BK SC426 Fall 2, dr A Abate, DCSC, TU Delft 3

15 Observer Design: 2-d Example Consider 2-d control-dependent model, where A = [ ] [ γ, B = ], C = [ ] It is observable Introduce observer so that [ ] l (A LC) = det(si A + LC) = s 2 + l l 2 s + l 2 desired char polynomial: p(s) = s 2 + 2ξ o ω o s + ω 2 o s,2 = ω o ( ξ o ± ξ 2 o ) SC426 Fall 2, dr A Abate, DCSC, TU Delft 4

16 STIMATED STATE Set l 2 = ωo, 2 2 l = 2ξ o ω o x, ˆx 6 4 Act 2 Est x ˆx x2, ˆx2 2! Normalized time t x2 ˆx Normalized time t on of an observer for a vehicle driving on a curvy road (left) The observer y error The plots on the middle show the lateral deviation x, the lateral lines andsc426 their Fall estimates 2, dr A Abate, ˆx DCSC, and TU ˆx 2 Delft by dashed lines The plots on the right 5 errors

17 Theorem 62 (Reachable canonical form) Let A and B be the dynamics and Stabilization by Output Feedback (Separation Principle) Recall topic of stabilization by state feedback Design: u = Kx + k r r Objective: select K so that closed loop has assigned characteristic polynomial eigenvalue assignment (pole placement) 62 STABILIZATION BY STATE FEEDBACK 75 Controller d Process r k r u ẋ = Ax + Bu y = Cx + Du y K x Figure 65: A feedback control system with state feedback The controller uses the system state x and the reference input r to command the process through its input u We model disturbances via the additive input d SC426 Fall 2, dr A Abate, DCSC, TU Delft 6

18 Now assume that not all the variables are observed: introduce observer Error dynamics: d x dt = (A LC) x Objective: select L so that closed loop has assigned characteristic polynomial 22 CHAPTER 7 OUTPUT FEEDBACK eigenvalue assignment (pole placement) Process n u ẋ = Ax + Bu y = Cx + Du y Observer ˆx Figure 7: Block diagram for an observer The observer uses the process measurement y (possibly Now, corrupted considerbythe noise following n) and the objective: input u tostabilization estimate the current by output state offeedback the process, denoted ˆx Consider model with state (x, x) T Assume there is no noise Definition SC426 Fall 2, 7 dr (Observability) A Abate, DCSC, TU DelftA linear system is observable if for any T > 7 it is possible to determine the state of the system x(t ) through measurements of y(t) and u(t) on the interval [, T ]

19 Closed-loop system: d dt ( x x ) = ( A BK BK A LC ) ( x x ) + ( Bkr ) r 73 CONTROL USING ESTIMATED STATE 23 r k r u B ẋ x C y Process A K L e B ˆx ˆx C ŷ Observer A Controller Figure 77: Block diagram of an observer-based control system The observer uses the measured output y and the input u to construct an estimate of the state This estimate is used SC426 Fall 2, dr A Abate, by a statedcsc, feedback TUcontroller Delft to generate the corrective input The controller consists of the 8 observer and the state feedback; the observer is identical to that in Figure 75 Theorem 73 (Eigenvalue assignment by output feedback) Consider the system dx

20 Design u = K ˆx + k r r (notice difference with: u = Kx + k r r) Objective: (same as above) select K so that closed loop has assigned characteristic polynomial Consider characteristic polynomial of closed-loop system: λ(s) = det(si A + BK) det(si A + LC) Characteristic polynomial can be decoupled: it can be assigned arbitrary roots if (A, B) is controllable 2 (A, C) is observable Separation Principle: Eigenvalue assignment for output feedback can be neatly split into two separate problems: SC426 Fall 2, dr A Abate, DCSC, TU Delft 9

21 eigenvalue assignment for state feedback 2 eigenvalue assignment for observer Internal model principle SC426 Fall 2, dr A Abate, DCSC, TU Delft 2

22 Stabilization by Output Feedback: an Example Consider 2-d control-dependent model, where [ ] [ ] γ A =, B =, C = [ ] Let us combine the observer with the state feedback: ˆx = Aˆx + Bu + L(y C ˆx) u = K ˆx + k r r ˆx = (ACHAPTER BK 7 LC)ˆx OUTPUT + Ly FEEDBACK + Bk r r inputs: y, r; output: u x2, ˆx2 State feedback Output feedback Reference! 5 5 SC426 Fall 2, dr A Abate, DCSC, TU Delft 2 u, usfb!

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