Intro. Computer Control Systems: F9


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1 Intro. Computer Control Systems: F9 Statefeedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21
2 F8: Quiz! 2 / 21
3 F8: Quiz! 1) For an observable system a the effect of all x(t) can be observed in y(t) b we have det O = 0 c we have stability 2 / 21
4 F8: Quiz! 1) For an observable system a the effect of all x(t) can be observed in y(t) b we have det O = 0 c we have stability 2) If a statespace form of G(s) is a minimal realization, a A:s eigenvalues < G(s):s poles b A:s eigenvalues = G(s):s poles c there exist more compact statespace forms 2 / 21
5 F8: Quiz! 1) For an observable system a the effect of all x(t) can be observed in y(t) b we have det O = 0 c we have stability 2) If a statespace form of G(s) is a minimal realization, a A:s eigenvalues < G(s):s poles b A:s eigenvalues = G(s):s poles c there exist more compact statespace forms 3) For a controllable system with statefeedback control a the closedloop system is stable b the poles of the closedloop system can be designed arbitrarily c no information about the system is required 2 / 21
6 Statefeedback control 3 / 21
7 Statefeedback control Statespace form of linear timeinvariant system ẋ = Ax + Bu y = Cx G(s) = C(sI A) 1 B u x y (si A) 1 B C 4 / 21
8 Statefeedback control Controller using state feedback u = Lx + l 0 r gives closedloop system ẋ = (A BL)x + Bl 0 r y = Cx where r is the reference signal. 4 / 21
9 Statefeedback control Controler using state feedback gives closedloop system u = Lx + l 0 r ẋ = (A BL)x + Bl 0 r y = Cx where r is the reference signal. rl 0 u x y (si A) 1 B C + L G c (s) = C(sI A + BL) 1 Bl 0 4 / 21
10 Pole placement Rules of thumb for designing L Eigenvalues/poles given by det(si A + BL) = 0, which we can design Im{s} Re{s} 5 / 21
11 Pole placement Rules of thumb for designing L Eigenvalues/poles given by det(si A + BL) = 0, which we can design Im{s} Re{s} Distance to the origin: Quick system but also sensitive to disturbances 5 / 21
12 Estimating the states via simulation 6 / 21
13 Estimating the states Via simulation Controller u = Lx + l 0 r requires states x which are often unknown. 7 / 21
14 Estimating the states Via simulation Controller u = Lx + l 0 r requires states x which are often unknown. In practice, feedback using u = Lˆx + l 0 r where ˆx is an estimate of x. 7 / 21
15 Estimating the states Via simulation Controller u = Lx + l 0 r requires states x which are often unknown. In practice, feedback using u = Lˆx + l 0 r where ˆx is an estimate of x. Naive idea: Estimate x by simulating the states where ˆx 0 is an initial guess. ˆx = Aˆx + Bu, ˆx(0) = ˆx 0 7 / 21
16 Build intuition from simple systems State estimation via simulation Ex.: Damper y Statespace form: [ ] [ ] ẋ(t) = x(t) + u(t), x(0) = x k/m 0 1/m 0 y(t) = [ 1 0 ] x(t) u 8 / 21
17 Build intuition from simple systems State estimation via simulation Example using impulse u input u(t), output y(t) u(t) y(t) t [s] System with unknown initial state x 0 8 / 21
18 Build intuition from simple systems State estimation via simulation Naive estimate using perfect initial guess: ˆx = Aˆx + Bu, ˆx 0 = x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
19 Build intuition from simple systems State estimation via simulation Naive estimate using perfect initial guess: ˆx = Aˆx + Bu, ˆx 0 = x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = 20 8 / 21
20 Build intuition from simple systems State estimation via simulation Naive estimate using perfect initial guess: ˆx = Aˆx + Bu, ˆx 0 = x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
21 Build intuition from simple systems State estimation via simulation Naive estimate using wrong initial guess: ˆx = Aˆx + Bu, ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
22 Build intuition from simple systems State estimation via simulation Naive estimate using wrong initial guess: ˆx = Aˆx + Bu, ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = 20 9 / 21
23 Build intuition from simple systems State estimation via simulation Naive estimate using wrong initial guess: ˆx = Aˆx + Bu, ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
24 Build intuition from simple systems State estimation via simulation x och ˆx correspond to different outputs: y = Cx versus ŷ = C ˆx output y(t) y(t) C ˆx(t) t [s] 9 / 21
25 Estimating the states via observer 10 / 21
26 Estimating the states Correcting the state estimates Idea: Feedback the prediction error y C ˆx to correct ˆx Observer: an estimator with a correction term ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx(0) = ˆx 0 }{{} correction 11 / 21
27 Estimating the states Correcting the state estimates Idea: Feedback the prediction error y C ˆx to correct ˆx Observer: an estimator with a correction term Using matrix ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx(0) = ˆx 0 }{{} correction we can design the estimator. k 1 k 2 K =. k n 11 / 21
28 Estimating the states Correcting the state estimates Idea: Feedback the prediction error y C ˆx to correct ˆx Observer: an estimator with a correction term ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx(0) = ˆx 0 }{{} correction u x y (si A) 1 B C ˆx Obs. 11 / 21
29 Build intuition using simple systems State estimation using observer Estimation using observer: ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
30 Build intuition using simple systems State estimation using observer Estimation using observer: ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
31 Build intuition using simple systems State estimation using observer Estimation using observer: ˆx = Aˆx + Bu + K ( y C ˆx ), ˆx 0 x x 2 (velocity) [m/s] x 1 (position) [m] x versus ˆx at t = / 21
32 State estimation Estimation error and observability Estimation error: x x ˆx [Board: derive evolution of estimation errors] 13 / 21
33 State estimation Estimation error and observability Estimation error: x x ˆx [Board: derive evolution of estimation errors] Result Errors of observer described as system x(t) = e (A KC)t x(0) and therefore x(t) decays at a rate given by maximum Re{ s i } where s i are observer poles/eigenvalues of (A KC). 13 / 21
34 State estimation Estimation error and observability Estimation error: x x ˆx Result 9.2 Statespace form is observable (cf. det O 0) matrix K can be chosen such that x vanish arbitrarily quick 14 / 21
35 State estimation Estimation error and observability Estimation error: x x ˆx Result 9.2 Statespace form is observable (cf. det O 0) matrix K can be chosen such that x vanish arbitrarily quick K is solved by polynomial det(si A + KC) = 0 with desired roots in left halfplane 14 / 21
36 State estimation Estimation error and observability Estimation error: x x ˆx Result 9.2 Statespace form is observable (cf. det O 0) matrix K can be chosen such that x vanish arbitrarily quick K is solved by polynomial det(si A + KC) = 0 with desired roots in left halfplane Quick observer ˆx is however sensitive to measurement noise! 14 / 21
37 Combining feedback with estimated states 15 / 21
38 Feedback using estimated states Controller the Laplace domain rl 0 + u x y (si A) 1 B C L ˆx Obs. 16 / 21
39 Feedback using estimated states Controller the Laplace domain rl 0 + u x y (si A) 1 B C L ˆx Obs. System and controller with observer: { { ẋ = Ax + Bu u and y = Cx ˆx = Lˆx + l 0 r = Aˆx + Bu + K(y C ˆx) 16 / 21
40 Feedback using estimated states Controller the Laplace domain rl 0 + u x y (si A) 1 B C L ˆx Obs. Controller with observer: { U(s) = L L : X(s) + l 0 R(s) s X(s) = A X(s) + BU(s) + K ( Y (s) C X(s) ) [Board: solve for controller] 16 / 21
41 Feedback using estimated states General linear feedback control General linear feedback form, ch.9.5 G&L Controller with observer can be written as where U(s) = F r (s)r(s) F y (s)y (s), F r (s) = (1 L(sI A + KC + BL) 1 B)l 0 F y (s) = L(sI A + KC + BL) 1 K r F r + u G y F y 17 / 21
42 Resulting closedloop system 18 / 21
43 Feedback using estimated states Effect of estimation error rl 0 + u x y (si A) 1 B C L ˆx Obs. Study system and controller with observer: { ẋ = Ax + Bu and u = Lˆx + l 0 r y = Cx by substituting ˆx = x x [Board: derive the closedloop system with estimation error x] 19 / 21
44 Feedback using estimated states Effect of estimation error rl 0 + u x y (si A) 1 B C L ˆx Obs. Yields closedloop system: ẋ = (A BL)x + y = Cx with additional error states effect of estimation error {}}{ BL x +Bl 0 r x = (A KC) x 0 [Board: write the closedloop system in statespace form] 19 / 21
45 Feedback using estimated states The closedloop system with observer The closedloop system with estimation error can be written as [ẋ ] [ ] [ ] [ ] A BL BL x B = + l x 0 A KC x 0 0 r }{{}}{{} Ã B y = [ C 0 ] }{{} C with extended state vector. [ x x This yields transfer function from r to y: ] G c (s) = C(sI Ã) 1 B 20 / 21
46 Feedback using estimated states The closedloop system with observer Closedloop system transfer function, ch.9.5 G&L Insert matrices Ã, B and C yields G c (s) = C(sI Ã) 1 B = C(sI A + BL) 1 Bl 0 with same poles as if states were known and K is gone! r F r + u G y F y 20 / 21
47 Summary and recap Rules of thumb for pole placement Estimation using observer Feedback using estimated states Closedloop system with observer 21 / 21
Intro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of statespace descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
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