Anti-windup solutions for input saturated control systems

Transcription

1 Anti-windup solutions for input saturated control systems Luca Zaccarian LAAS-CNRS and University of Trento thanks to A.R. Teel F. Dabbene, S. Galeani, A. Garulli, T. Hu, S. Tarbouriech, R. Tempo, M. Turner, S. Gayadeen, J. Marcinkovski, S. Podda, V. Vitale, E. Weyer, D. Dai, S. Formentin, F. Forni, G. Grimm, F. Morabito, S. Onori, F. Todeschini, G. Valmorbida Genova, October 1, / 58

2 Outline 1 Saturation and its effects on the performance metric 2 Direct Linear approaches and static anti-windup 3 Incorporating robustness in Direct Linear Anti-Windup 4 Model recovery anti-windup solution 5 Some applications of Model Recovery Anti-Windup 6 Fully nonlinear Anti-windup for nonlinear plants 2 / 58

3 Active control provides extreme vibration isolation Newport Corporation s Elite 3 TM vibration isolation table Useful, for example, in high-precision microscopy semiconductor manufacturing Actuators: piezoelectric stack Sensors: geophones Passive Isolation Sensors Geophones Actuators Piezoelectric A/D Active Isolation D/A DSP Control Algorithm PLANT CONTROLLER 3 / 58

4 Input saturation confuses the base control algorithm Extreme vibration suppression (4 db) up to t = 23 s Controller Output 1 1 SAT SAT Plant Output At t = 23 s someone walks close to the table 4 / 58

5 Performance with saturation depends on size of disturbance Saturation: an abrupt nonlinearity: Small signals: sat(u) = u no effect Large signals: sat(u) bounded severe effect ( Signal size (L 2 norm): z 2 := z(t) 2 dt z L 2 (square integrable) if z 2 < ) 1 2 u w sat(u) K P z y Closed-loop performance measures: Finite L 2 gain (linear H norm): γ wz R : z 2 γ wz w 2 for all w L 2 Nonlinear L 2 gain: a function s γ wz (s): Megretski [1996] z 2 γ wz (s) w 2 for all w satisfying w 2 s 5 / 58

6 Example demonstrates relevance of nonlinear gains Controller K cancels the plant dynamics and stabilizes (before saturation) w sat(u) P z y P : ż = az sat(u) w K : u = az 1z u K Three representative cases Sontag [1984], Lasserre [1992] Nonlinear Gains γ wz (i.e., H norm of P wz ) a=-1 a= a= L norm (size) of w 2 6 / 58

7 Nonlinear L 2 gains are estimated using Lyapunov functions Hu et al. [26], Dai et al. [29b], Garulli et al. [213] Quadratic functions (LMIs Boyd et al. [1994]) V 1 (x) = x T Px Max of quadratics (BMIs) V 2 (x) = max x T P j x j {1,...,J} Convex Hull of quadratics (BMIs) V 3 (x) = ( ) 1 min γ j : x T j γ j γ jq j x j =1 Piecewise quadratic (LMI-BMI) [ ] T [ ] x x V 4 (x) = P dz(u(x)) dz(u(x)) Piecewise Polynomial (LMI-BMI) [ ] {m}t [ x x V 5 (x) = ˆP dz(u(x)) dz(u(x)) ] {m} V 1 γ dz (s) z 2 γ dz (s) w 2 < A possible level set 7 / 58

8 Nonlinear L 2 gains are estimated using Lyapunov functions Hu et al. [26], Dai et al. [29b], Garulli et al. [213] Quadratic functions (LMIs Boyd et al. [1994]) V 1 (x) = x T Px Max of quadratics (BMIs) V 2 (x) = max x T P j x j {1,...,J} Convex Hull of quadratics (BMIs) V 3 (x) = ( ) 1 min γ j : x T j γ j γ jq j x j =1 Piecewise quadratic (LMI-BMI) [ ] T [ ] x x V 4 (x) = P dz(u(x)) dz(u(x)) Piecewise Polynomial (LMI-BMI) [ ] {m}t [ x x V 5 (x) = ˆP dz(u(x)) dz(u(x)) ] {m} V 1 γ dz (s) z 2 γ dz (s) w 2 < A possible level set 8 / 58

9 Nonlinear L 2 gains are estimated using Lyapunov functions Hu et al. [26], Dai et al. [29b], Garulli et al. [213] Quadratic functions (LMIs Boyd et al. [1994]) V 1 (x) = x T Px Max of quadratics (BMIs) V 2 (x) = max x T P j x j {1,...,J} Convex Hull of quadratics (BMIs) V 3 (x) = ( ) 1 min γ j : x T j γ j γ jq j x j =1 Piecewise quadratic (LMI-BMI) [ ] T [ ] x x V 4 (x) = P dz(u(x)) dz(u(x)) Piecewise Polynomial (LMI-BMI) [ ] {m}t [ x x V 5 (x) = ˆP dz(u(x)) dz(u(x)) ] {m} V 1 γ dz (s) z 2 γ dz (s) w 2 < A possible level set 9 / 58

10 Nonlinear L 2 gains are estimated using Lyapunov functions Hu et al. [26], Dai et al. [29b], Garulli et al. [213] Quadratic functions (LMIs Boyd et al. [1994]) V 1 (x) = x T Px Max of quadratics (BMIs) V 2 (x) = max x T P j x j {1,...,J} Convex Hull of quadratics (BMIs) V 3 (x) = ( ) 1 min γ j : x T j γ j γ jq j x j =1 Piecewise quadratic (LMI-BMI) [ ] T [ ] x x V 4 (x) = P dz(u(x)) dz(u(x)) Piecewise Polynomial (LMI-BMI) [ ] {m}t [ x x V 5 (x) = ˆP dz(u(x)) dz(u(x)) ] {m} V 1 γ dz (s) z 2 γ dz (s) w 2 < A possible level set 1 / 58

11 Nonlinear L 2 gains are estimated using Lyapunov functions Hu et al. [26], Dai et al. [29b], Garulli et al. [213] Quadratic functions (LMIs Boyd et al. [1994]) V 1 (x) = x T Px Max of quadratics (BMIs) V 2 (x) = max x T P j x j {1,...,J} Convex Hull of quadratics (BMIs) V 3 (x) = ( ) 1 min γ j : x T j γ j γ jq j x j =1 Piecewise quadratic (LMI-BMI) [ ] T [ ] x x V 4 (x) = P dz(u(x)) dz(u(x)) Piecewise Polynomial (LMI-BMI) [ ] {m}t [ x x V 5 (x) = ˆP dz(u(x)) dz(u(x)) ] {m} V 1 γ dz (s) z 2 γ dz (s) w 2 < A possible level set 11 / 58

12 H design generalizes to linear input-saturated plants Dai et al. [29a] d sat(u) P z y Given P linear, design K, namely C linear plant-order AW static: linear gain K u AW static C Performance objective: given s, minimize γ dz (s ) Linear controller K equations ẋ c = Ax c By E 1 (sat(y c ) y c ) y c = Cx c Dy E 2 (sat(y c ) y c ) Synthesis is a convex problem (generalizes LMI-H Gahinet and Apkarian [1994], Iwasaki and Skelton [1994]) Synthesis without AW is nonconvex Generalized hybrid approaches for sampled-data design Dai et al. [21] 12 / 58

13 Pure linear anti-windup design is also convex Grimm et al. [23a], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] d sat(u) P AW static or dynamic z y Given P linear, C linear, design only AW linear static or plant-order Performance objective: given s, minimize γ dz (s ) Necessary conditions: linear feedback (P, C) exp stable K u C [ ( V ([ xp x c ]) = [ xp x c ] T Q p ] Q pc 1[ x p Q T pc Q c x c ]) K s.t. A p B p,u K exp stable (V K (x p ) = x T p Q p 1 x p, K s ) Static anti-windup construction (convex, LMIs) feasible if Q p = Q p : quasi-common quadratic Lyapnuov function Plant-order anti-windup construction (convex, LMIs) always feasible as long as V K and V above exist 13 / 58

14 Direct Linear static linear anti-windup design (LMI) Mulder et al. [21], Grimm et al. [23a], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] u K w sat(u) dz(u) u P D aw C v z y Given P linear, C linear, design only [ ] Daw,1 linear anti-windup gain D aw = D aw,2 Performance objective: given s, minimize γ dz (s ) Linear controller K equations ẋ c = Ax c By D aw,1 (u sat(u)) y c = Cx c Dy D aw,2 (u sat(u)) LMI-based design Mulder et al. [21], Grimm et al. [23a], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] Preserve of small signal response (D aw multiplies dz(u) = u sat(u)) Asymptotically recover large signal response (global not always possible) Results generalize nontrivially to the plant-order case 14 / 58

15 Compact representation of the closed-loop system u w sat(u) P z y K dz(u) D aw u C v D aw w v dz(u) H x cl u sat(u) z u H : x cl = A cl x cl B cl,d (u sat(u)) B cl,v v B cl,w w u = C cl,u x p D cl,ud (u sat(u)) D cl,uv v D cl,uw w z = C cl,z x p D cl,zd (u sat(u)) D cl,zv v D cl,zw w, }{{} dz(u) 15 / 58

16 Quadratic analysis conditions are convex Mulder et al. [21], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] Proposition: Given the NOMINAL system and s >, if the LMI problem ˆγ 2 (s) = min {γ 2,Q,Y,U} γ2 subject to Q = Q T >, U > diagonal, A cl Q B cl,d U B cl,v D aw U Y T B cl,w [ ] He C cl,u Q D cl,ud U D cl,uv D aw U U D cl,uw Q T Y[k] I /2, Y [k] ū 2, k /s2 C cl,z Q D cl,zd U D cl,zv D aw U D cl,zw γ2 2 I k = 1,..., n u is feasible, then the following holds for the saturated closed-loop: 1 [Stab] the origin is locally exponentially stable with region of attraction containing the set E(Q, s) := {x : x T Q 1 x s 2 }; 2 [Reach] the reachable set from x() = with w 2 s is contained in E(Q, s); 3 [L 2 Perf] for each w such that w 2 s, the zero state solution satisfies the L 2 gain bound: z 2 ˆγ(s) w 2 16 / 58

17 Quadratic analysis conditions easily lead to synthesis Mulder et al. [21], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] Proposition: Given the NOMINAL system and s >. If the LMI problem ˆγ 2 (s) = min {γ 2,Q,Y,U} γ2 subject to Q = Q T >, U > diagonal, A cl Q B cl,d U B cl,v D aw U Y T B cl,w [ ] He C cl,u Q D cl,ud U D cl,uv D aw U U D cl,uw Q T Y[k] I /2, Y [k] ū 2, k /s2 C cl,z Q D cl,zd U D cl,zv D aw U D cl,zw γ2 2 I k = 1,..., n u is feasible, then the following holds for the saturated closed-loop: 1 [Stab] the origin is locally exponentially stable with region of attraction containing the set E(Q, s) := {x : x T Q 1 x s 2 }; 2 [Reach] the reachable set from x() = with w 2 s is contained in E(Q, s); 3 [L 2 Perf] for each w such that w 2 s, the zero state solution satisfies the L 2 gain bound: z 2 ˆγ(s) w 2 17 / 58

18 Quadratic synthesis conditions are convex Mulder et al. [21], Gomes da Silva Jr and Tarbouriech [25], Hu et al. [28] Proposition: Given the NOMINAL system and s >. If the LMI problem ˆγ 2 (s) = min {γ 2,Q,Y,U,X } γ2 subject to Q = Q T >, U > diagonal, A cl Q B cl,d U B cl,v X Y T B cl,w [ ] He C cl,u Q D cl,ud U D cl,uv X U D cl,uw Q T Y[k] I /2, Y [k] ū 2, k /s2 C cl,z Q D cl,zd U D cl,zv X D cl,zw γ2 2 I k = 1,..., n u is feasible, then, selecting the static AW gain as D aw = XU 1 1 [Stab] the origin is locally exponentially stable with region of attraction containing the set E(Q, s) := {x : x T Q 1 x s 2 }; 2 [Reach] the reachable set from x() = with w 2 s is contained in E(Q, s); 3 [L 2 Perf] for each w such that w 2 s, the zero state solution satisfies the L 2 gain bound: z 2 ˆγ(s) w 2 18 / 58

19 Cross-directional dynamics application: Synchrotron Queinnec et al. [215] Model is based on a suitable Singular Value Decomposition (SVD) P (z) d p(z) sat - u u p ζ y ΣΨ T Φ dz y c ΨΣ 1 C(z) ν... p(z) c(z)... c(z) Equivalent dynamics requires generalized nonlinearity dz ΣΨ T ΨΣ 1 - u p ν p(z)... p(z) c(z)... c(z) u c ζ u c Φ T y Φ d Φ Φ Φ T y d Collaboration with Diamond Light Source synchrotron (Oxforshire, UK) 19 / 58

20 Closed loop now depends on uncertain parameter q Q Turner et al. [27], Grimm et al. [24], Formentin et al. [213, 214] u H(q) : K w sat(u) dz(u) u P(q) D aw C v z y D aw w v dz(u) H(q) x cl u sat(u) x cl = A cl (q)x cl B cl,d (q)(u sat(u)) B cl,v (q)v B cl,w (q)w u = C cl,u (q)x p D cl,ud (q)(u sat(u)) D cl,uv (q)v D cl,uw (q)w z = C cl,z (q)x p D cl,zd (q) (u sat(u)) D cl,zv (q)v D cl,zw (q)w, }{{} dz(u) z u Robust designs follow a deterministic worst case paradigm, imposing strong convexity conditions Turner et al. [27], Grimm et al. [24] 2 / 58

21 Static AW synthesis based on scenario with certificates Formentin et al. [213, 214] Theorem (Robust static AW using scenario with certificates) Fix a positive value s w 2, ε (, 1), β (, 1), and select N satisfying B(N, ε, n θ ) β, with n θ = 1 n u n u(n u n c) Extract N samples of the uncertain matrices according to the probability distribution Solve γ 2 sc (s) = min {γ 2,U,X},{Q i,y i } γ 2, A (i) cl Q i B (i) cl,d U B(i) cl,v X Y i T B (i) cl,w He C (i) cl,u Q i D (i) cl,ud U D(i) cl,uv X U D(i) cl,uw I /2 C (i) cl,z Q i D (i) cl,zd U D(i) cl,zv X subject to Q i = Q i T >, U > diagonal, D(i) cl,zw γ2 2 I <, [ ] Qi Yi[k] T Y i[k] ūk 2/s2 k = 1,..., n u i = 1,..., N If the above LMIs are feasible, select the static anti-windup gain D aw = XU 1 Then, for each w 2 < s, the zero initial state solution of the closed loop satisfies Pr( z 2 > γ sc(s) w 2 ) ε, with probability no smaller than 1 β. Analysis conditions can also be easily formulated, 21 / 58

22 Illustrative example: A passive networkgrimm et al. [23b] Uncertain parameters with (known) Gaussian distribution parameter mean std dev parameter mean std dev R Ω ± 1 R 5 1 F ± 1 R 2 2 Ω ± 1 C 1.1 F ± 1 R Ω ± 1 C 2.1 F ± 1 R 4 17 Ω ± 1 c 3.1 F ± 1 Input generator voltage constrained: u(t) = V i (t) [ ū, ū] = [ 1Volt, 1Volt] Design parameters are ε =.1, β = 1 6, s =.3, n θ = 35 N = 227 (not 7565) for design based on sequential algorithm 22 / 58

23 Deterministic and Randomized nonlinear L 2 gains Robust compensator shows better robust performance (red curves) The nominal behavior slightly deteriorated (thin curves) γ no AW (r) no AW (n) naw (r) naw (n) raw (r) raw (n) s Without anti-windup (black dashed), with nominal anti-windup (blue dashed-dotted) and with robust anti-windup (red solid) 23 / 58

24 Time responses confirm nonlinear L 2 gain trends uncon no AW naw raw Time [ ] γ no AW (r) no AW (n) naw (r) naw (n) raw (r) raw (n) unconstrained nominal AW Time [ ] s no AW randomized AW Time [ ] 24 / 58

25 Anti-windup extends to fully nonlinear compensation Teel and Kapoor [1997], Zaccarian and Teel [22, 211] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 25 / 58

26 Anti-windup extends to fully nonlinear compensation Teel and Kapoor [1997], Zaccarian and Teel [22, 211] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 26 / 58

27 Anti-windup extends to fully nonlinear compensation Pagnotta et al. [27] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 27 / 58

28 Anti-windup extends to fully nonlinear compensation Zaccarian and Teel [25] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 28 / 58

29 Anti-windup extends to fully nonlinear compensation Forni et al. [212, 21] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 29 / 58

30 Anti-windup extends to fully nonlinear compensation Zaccarian et al. [25] K v d sat(u) y cl P AW ˆP C x aw y l z y y aw Model Recovery Anti-Windup (MRAW) Framework for nonlinear AW: AW is a model ˆP of P v = k(x aw ) is a (nonlinear) stabilizer whose construction depends on P AW is controller-independent: any (nonlinear) C allowed Useful feature of MRAW: C receives linear plant output y l C delivers linear plant input y cl Reduced order ˆP possible (tested on adaptive noise suppression) MRAW allows for bumpless transfer among controllers MRAW generalizes to rate and curvature saturation MRAW generalizes to dead time plants (Smith predictor) 3 / 58

31 Ad hoc gain adaptation induces very slow isolation recovery Effect of a footstep at the side of the table (recovery > 1 minute) Controller Output SAT SAT Plant Output / 58

32 MRAW dramatically reduces isolation recovery time Teel et al. [26], Zaccarian et al. [2] Effect of a footstep at the side of the table (recovery 4 s) Controller Output 1 1 SAT SAT Plant Output Controller input / 58

33 Even a bat strike does not confuse the MRAW controller Teel et al. [26], Zaccarian et al. [2] Hitting with a baseball bat the table leg (recovery 5 s) Controller Output 1 1 SAT SAT Time [sec].4.2 Plant Output Controller input / 58

34 Bumpless transfer enables smooth controller activation Teel et al. [26], Zaccarian et al. [2] Controller is gradually activated in bumpless transfer scheme System power on Controller Output Time [sec].4 Plant Output Time [sec].4 Controller Input Time [sec] 34 / 58

35 Rate Saturated McDonnell Douglas TAFA dynamics Barbu et al. [25] Linearized longitudinal dynamics (α=angle of attack; q=pitch rate) [ ] ż := α q = [ ] [ Zα Z q z M α M q Mδ =: A z B u δ ] δ Saturation: M = 2 deg, R = 4 deg/s. [ ( u ) ] δ = R sgn M sat δ, M Study a flight trim condition with one exp unstable mode [ ] [ ] [ ] ẋs 4 bs ẋ := = x δ 1 ẋ u b u 35 / 58

36 Problems due to magnitude saturation Unconstrained trajectory may exit the null-controllability region Pitch rate [deg/sec] Angle of atack [deg] Unconstrained ( ), possible desired trajectories ( and ) 36 / 58

37 Problems due to magnituderate saturation Unconstrained trajectory may exit the null-controllability region α q δ [deg] Unconstrained ( ), possible desired trajectories ( and ) 37 / 58

38 Close the position loop using a pilot model Use a simple crossover model d - Pilot Model Antiwindup Closed-loop System q d q R Study the maneuverability of the aircraft with anti-windup Study the possible occurrence of PIOs (Pilot Induced Oscillations) Compare the response to the optimal response using static command limiting Use a step reference θ d = 4 deg 38 / 58

39 Piloted flight simulation 5 Pitch Angle [deg] Control [deg] Time[s] (unconstrained, anti-windup, optimal trajectory with static limiting ) 39 / 58

40 Unconstrained response information (linear case) Plant P ẋ = A x B d d B u u z = C z x D dz d D uz u y = C y x D dy d D uy u Anti-windup filter ˆP { ẋaw = A x aw B u (u y c) y aw = C y x aw D uy (u y c) Unconstrained controller C { ẋc = A c x c B cu u c B cr r y c = C c x c D cu u c D cr r Interconnections { u = sat(yc v), u c = y y aw v 1 : to be selected! Coordinate transformation: (x l, x c, x aw ) = (x x aw, x c, x aw ) { ẋ Unconstrained dynamics P ˆP: l = A x l B d d B u y c y y aw = C y x l D dy d D uy y c Information about the unconstrained response embedded within the scheme! 4 / 58

41 The unconstrained response information (nonlinear case) Plant P { ẋ = f (x, u) z = h(x, sat(u)) Anti-windup filter ˆP { ẋaw = f (x, u) f (x x aw, y c) y aw = x aw Unconstrained controller C { ẋc = g(x c, u c, r) y c = k(x c, u c, r) Interconnections { u = sat(yc v), u c = x x aw v 1 : to be selected! Coordinate transformation: (x l, x c, x aw ) = (x x aw, x c, x aw ) { ẋl = f (x Unconstrained dynamics P ˆP: l, y c) u c = x x aw = x l Information about the unconstrained response embedded within the scheme! 41 / 58

42 Anti-windup for nonlinear systems: resulting scheme r u c Unconstrained Controller y c α Saturation Nonlinearity Nonlinear plant u p x x aw x y aw Need extra plant state measurements Recall that x aw = x x l : very useful information - worry about stability looking at x worry about performance looking at x aw A few application examples: Anti-windup Compensator Anti-windup for robot manipulators Morabito et al. [24] Anti-windup for Brake-by-Wire systems Todeschini et al. [215] 42 / 58

43 A SCARA robot manipulator example Morabito et al. [24] SCARA robot with limited torque/force inputs Link m i 55 Nm 45 Nm 7 N 25 Nm Ö ÈÁ ÓÒØÖÓÐÐ Ö Ù ÊÓ ÓØ ÝÒ Ñ ÒÚ Ö ÓÒ Õ Õ P I D Feedback linearizing controllerpid action (computed torque) induces decoupled linear performance (for small signals) 43 / 58

44 A slight saturation can be disastrous The reference is r = [6 deg, 4 deg, 4 cm, 8 deg] Position 3 Position 1 Position 2 Position Torque 1 Torque Force 3 Torque Without anti windup With anti windup Without saturation Stability is recovered, performance is almost fully preserved 44 / 58

45 Anti-windup injects signals and then fades out The reference is a sequence of little step followed by a large step Position 3 Position 4 Position 1 Position Without anti windup With anti windup Without saturation Torque 1 Torque 2 AW action (v1) 1 AW action (v1) The anti-windup action dies away to recover the unconstrained closed-loop 45 / 58

46 SCARA: large signals (nonlinear v 1 ) The reference is r = [15 deg, 1 deg, 1 m, 2 deg] Position 1 Position Position 3 Position Torque 1 Torque Force 3 Torque Without anti windup With anti windup Without saturation Performance is dramatically improved (input authority is almost fully explotied) 46 / 58

47 MRAW applies to nonlinear fully actuated robots Example: a SCARA robot (planar robot) following a circular motion Saturated computed torque controller goes postal (unstable) Nonlinear MRAW provides slight performance degradation Position 1 [deg] Torque 1 [Nm] Position 2 [deg] Without anti windup Without saturation 1 With anti windup 15 Y axis [m] With anti windup Without saturation Without anti windup Torque 2 [Nm] X axis [m] 47 / 58

48 Outline Saturated performance Direct Linear AW Robust designs Model Recovery AW MRAW Applications Nonlinear AW References Nonlinear anti-windup for a Brake By Wire System Todeschini et al. [215] Brake-by-wire system in motorcycles corresponds to a nonlinear plant The main nonlinear effect can be easily isolated in the model: 2 ku 1 s2 k2 sk1 x p(x) y y [bar] u 1 kp Experimental Data p(x) pl (x) x [mm] / 58

49 BBW solution uses nonlinear MRAW Deadzone compensation scheme provides nonlinear baseline controller Fully Nonlinear anti-windup addresses saturation with nonlinear plant and nonlinear controller k nc r r ul u y LC BBW y nc u nc u σ(u) - - K nc BBW x nc udc u - aw x DC x aw ydc y aw AW - yl - y Pressure [bar] Current [A] r PID SAT 1 PID DC SAT PID DC SAT AW PID DC SAT IMC Step response reveals successful anti-windup action Driver would get confused by large overshoots Alternative existing solutions (nonlinear IMC-based anti-windup) are unacceptably slow (black) 49 / 58

50 Anti-windup designs apply to additional applications Zaccarian et al. [27], Vitelli et al. [21] Control of irrigation channels Gate saturation problems: bumpless transfer from manual control with small flows in the pools with large disturbances (rain, etc) Challenge: plant is ANCBC (poles in ) y1 h1 p1 yd y2 h2 p2 Gate 1 Gate 2 Small signal nonlinearity compensation in high-power circulating current amps Thyristors have a min current threshold: below the treshold: circulating current this generates a undesired nonlinearity possibly destabilizing outer feedback Challenge: reverse anti-windup problem 5 / 58

51 Summary of the presented works with references Summary of the proposed solutions in Galeani et al. [29], Zaccarian and Teel [211] Lyapunov certificates of performance in Dai et al. [29b], Garulli et al. [213], Hu et al. [26] LMI-based (Direct Linear) anti-windup proposed in Mulder et al. [21], Tarbouriech et al. [211], Grimm et al. [23a,b], Hu et al. [28] Generalization to saturated H in Dai et al. [29a, 21] Robust extensions of the approaches Formentin et al. [213, 214], Grimm et al. [24], Turner et al. [27] Application to synchrotron control Queinnec et al. [215] Model-Recovery anti-windup proposed in Teel and Kapoor [1997], Zaccarian and Teel [22] Bumpless transfer extensions Zaccarian and Teel [25] Generalizations to rate and curvature saturations Forni et al. [21, 212] Applications: flight Barbu et al. [25], vibration isolation Teel et al. [26] Nonlinear MRAW for NL plants in Morabito et al. [24], Todeschini et al. [215] 51 / 58

52 Bibliography I C. Barbu, S. Galeani, A.R. Teel, and L. Zaccarian. Nonlinear anti-windup for manual flight control. Int. J. of Control, 78(14): , September 25. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Society for Industrial an Applied Mathematics, D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Output feedback design for saturated linear plants using deadzone loops. Automatica, 45(12): , 29a. D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations. Systems and Control Letters, 58(5): , 29b. D. Dai, T. Hu, A.R. Teel, and L. Zaccarian. Output feedback synthesis for sampled-data system with input saturation. In American Control Conference, pages , Baltimore (MD), USA, June 21. S. Formentin, S.M. Savaresi, L. Zaccarian, and F. Dabbene. Randomized analysis and synthesis of robust linear static anti-windup. In Conference on Decision and Control, pages , Florence (Italy), December / 58

53 Bibliography II S. Formentin, F. Dabbene, R. Tempo, L. Zaccarian, and S.M. Savaresi. Scenario optimization with certificates and applications to anti-windup design. In IEEE Conference on Decision and Control, pages , Los Angeles (CA), USA, December 214. F. Forni, S. Galeani, and L. Zaccarian. An almost anti-windup scheme for plants with magnitude, rate and curvature saturation. In American Control Conference, pages , Baltimore (MD), USA, June 21. F. Forni, S. Galeani, and L. Zaccarian. Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants. Automatica, 48 (8): , 212. P. Gahinet and P. Apkarian. A linear matrix inequality approach to H control. Int. J. Robust and Nonlinear Control, 4: , S. Galeani, S. Tarbouriech, M.C. Turner, and L. Zaccarian. A tutorial on modern anti-windup design. European Journal of Control, 15(3-4):418 44, 29. A. Garulli, A. Masi, G. Valmorbida, and L. Zaccarian. Global stability and finite L 2m-gain of saturated uncertain systems via piecewise polynomial Lyapunov functions. IEEE Transactions on Automatic Control, 58(1): , / 58

54 Bibliography III J.M. Gomes da Silva Jr and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an LMI-based approach. IEEE Trans. Aut. Cont., 5(1):16 111, 25. G. Grimm, J. Hatfield, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Antiwindup for stable linear systems with input saturation: an LMI-based synthesis. IEEE Transactions on Automatic Control, 48(9): , September 23a. G. Grimm, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Case studies using linear matrix inequalities for optimal anti-windup synthesis. European Journal of Control, 9(5): , 23b. G. Grimm, A.R. Teel, and L. Zaccarian. Robust linear anti-windup synthesis for recovery of unconstrained performance. Int. J. Robust and Nonlinear Control, 14(13-15): , 24. T. Hu, A.R. Teel, and L. Zaccarian. Stability and performance for saturated systems via quadratic and non-quadratic Lyapunov functions. IEEE Transactions on Automatic Control, 51(11): , November / 58

55 Bibliography IV T. Hu, A.R. Teel, and L. Zaccarian. Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance. Automatica, 44(2): , 28. T. Iwasaki and R.E. Skelton. All controllers for the general H control problem: LMI existence conditions and state space formulas. Automatica, 3 (8):137 17, August J.B. Lasserre. Reachable, controllable sets and stabilizing control of constrained linear systems. Automatica, 29(2): , A. Megretski. L 2 BIBO output feedback stabilization with saturated control. In 13th Triennial IFAC World Congress, pages , San Francisco, USA, F. Morabito, A.R. Teel, and L. Zaccarian. Nonlinear anti-windup applied to Euler-Lagrange systems. IEEE Trans. Rob. Aut., 2(3): , 24. E.F. Mulder, M.V. Kothare, and M. Morari. Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica, 37(9): , September / 58

56 Bibliography V L. Pagnotta, L. Zaccarian, A. Constantinescu, and S. Galeani. Anti-windup applied to adaptive rejection of unknown narrow band disturbances. In European Control Conference, pages , Kos (Greece), July 27. I. Queinnec, S. Tarbouriech, S. Gayadeen, and L. Zaccarian. Static anti-windup design for discrete-time large-scale cross-directional saturated linear control systems. In IEEE Conference on Decision and Control, Osaka, Japan, December 215. E.D. Sontag. An algebraic approach to bounded controllability of linear systems. Int. J. of Control, 39(1): , S. Tarbouriech, G. Garcia, J.M. Gomes da Silva Jr., and I. Queinnec. Stability and stabilization of linear systems with saturating actuators. Springer-Verlag London Ltd., 211. A.R. Teel and N. Kapoor. The L 2 anti-windup problem: Its definition and solution. In European Control Conference, Brussels, Belgium, July A.R. Teel, L. Zaccarian, and J. Marcinkowski. An anti-windup strategy for active vibration isolation systems. Control Engineering Practice, 14(1): 17 27, / 58

57 Bibliography VI F. Todeschini, S. Formentin, G. Panzani, M. Corno, S. Savaresi, and L. Zaccarian. Nonlinear pressure control for bbw systems via dead zone and anti-windup compensation. IEEE Transactions on Control Systems Technology, to appear, 215. M.C. Turner, G. Herrmann, and I. Postlethwaite. Incorporating robustness requirements into antiwindup design. IEEE Transactions on Automatic Control, 52(1): , 27. R. Vitelli, L. Boncagni, F. Mecocci, S. Podda, V. Vitale, and L. Zaccarian. An anti-windup-based solution for the low current nonlinearity compensation on the FTU horizontal position controller. In Conference on Decision and Control, pages , Atlanta (GA), USA, December 21. L. Zaccarian and A.R. Teel. A common framework for anti-windup, bumpless transfer and reliable designs. Automatica, 38(1): , 22. L. Zaccarian and A.R. Teel. The L 2 (l 2) bumpless transfer problem: its definition and solution. Automatica, 41(7): , 25. L. Zaccarian and A.R. Teel. Modern anti-windup synthesis: control augmentation for actuator saturation. Princeton University Press, Princeton (NJ), / 58

58 Bibliography VII L. Zaccarian, A.R. Teel, and J. Marcinkowski. Anti-windup for an active vibration isolation device: theory and experiments. In Proceedings of the American Control Conference, pages , Chicago (IL), USA, June 2. L. Zaccarian, D. Nešić, and A.R. Teel. L 2 anti-windup for linear dead-time systems. Systems and Control Letters, 54(12): , 25. L. Zaccarian, E. Weyer, A.R. Teel, Y. Li, and M. Cantoni. Anti-windup for marginally stable plants and its application to open water channel control systems. Control Engineering Practice, 15(2): , / 58