Design of hybrid control systems for continuous-time plants: from the Clegg integrator to the hybrid H controller

Transcription

1 Design of hybrid control systems for continuous-time plants: from the Clegg integrator to the hybrid H controller Luca Zaccarian LAAS-CNRS, Toulouse and University of Trento University of Oxford November 12, 213

2 Outline 1 Clegg integrators and First Order Reset Elements (FORE) and an overview of hybrid dynamical systems 2 Exponential stability of FORE control systems 3 Stability/Performance analysis for a larger class of hybrid systems 4 Synthesis of higher order hybrid controllers 5 Conclusions and perspectives

3 An analog integrator and its Clegg extension (1956) Integrators: core components of dynamical control systems v ẋ B c c A c Example: PI controller ẋ c = A c + B c v v k p k i C v R ẋ c = 1 RC v v C In an analog integrator, the state information is stored in a capacitor:

4 An analog integrator and its Clegg extension (1956) Integrators: core components of dynamical control systems v ẋ B c c A c Example: PI controller C ẋ c = A c + B c v v k p k i v R d R R v C1 - - v C2 Clegg's integrator (1956): feedback diodes: the positive part of is all and only coming from the upper capacitor (and viceversa) input diodes: when v the upper capacitor is reset and the lower one integrates (and viceversa) [R d 1] As a consequence v and never have opposite signs C

5 Hybrid dynamics may ow or jump Hybrid Clegg integrator: ẋ c = 1 v, allowed when xc v, RC x + c =, allowed when v, Flow set C: where may ow (1st eq'n) Jump set D: where may jump (2nd eq'n) v C D C C v R d R R v C1 - - v C2 Clegg's integrator (1956): feedback diodes: the positive part of is all and only coming from the upper capacitor (and viceversa) input diodes: when v the upper capacitor is reset and the lower one integrates (and viceversa) [R d 1] As a consequence v and never have opposite signs C

6 Hybrid dynamical systems review: dynamics H = (C, D, F, G) ẋ F (x) x + G(x) n N (state dimension) C R n (ow set) D R n (jump set) F : C R n (ow map) G : D R n (jump map) C { ẋ F (x), x C H : x + G(x), x D D

7 Hybrid dynamical systems review: continuous dynamics H = (C, D, F, G) n N (state dimension) C R n (ow set) D R n (jump set) F : C R n (ow map) G : D R n (jump map) { ẋ F (x), x C H : x + G(x), x D x 2 { ẋ1 = x 2 ẋ 2 = x 1 + x 2 (1 x1 2) 2 2 Van der Pol x 1

8 Hybrid dynamical systems review: discrete dynamics H = (C, D, F, G) n N (state dimension) C R n (ow set) D R n (jump set) F : C R n ow map) G : D R n (jump map) { ẋ F (x), x C H : x + G(x), x D {, 1} if x = x + {, 2} if x = 1 {1, 2} if x = 2 A possible sequence of states from x = is: ( 1 2 1) i i N

9 Hybrid dynamical systems review: trajectories x 3 C x 2 x x 1 x 4 x 5 x 6 x 7 { ẋ F (x), x C H : x + G(x), x D D

10 Hybrid dynamical systems review: hybrid time The motion of the state is parameterized by two parameters: t R, takes into account the elapse of time during the continuous motion of the state; j Z, takes into account the number of jumps during the discrete motion of the state. ξ(, ) ξ(5, 2) ξ(5, ) ξ(5, 1) ξ(8, 2) ξ(8, 3) τ [, 5], ξ(τ, ) τ [5, 8], ξ(τ, 2) τ 8, ξ(τ, 3)

11 Hybrid dynamical systems review: hybrid time E R Z is a compact hybrid time domain if J 1 E = ([t j, t j+1 ] {j}) j= where = t t 1 t J. j (t 3,3) (t 4,3) (t 5,5) (t 4,4) (t 5,4) E is a hybrid time domain if for all (T, J) R Z E ([, T ] {, 1,..., J}) is a compact hybrid time domain. (t,) (t 1,) (t 2,2)=(t 3,2) (t 1,1) (t 2,1) t

12 Hybrid dynamical systems review: solution Formally, a solution satises the ow dynamics when owing and satises the jump dynamics when jumping ξ 1 t 1 t 2 t 3 t 4 t 2 j 4 3

13 Hybrid dynamical systems review: Lyapunov theorem Theorem Given the Euclidean norm x = x T x and a hybrid system { ẋ = f (x), x C H : x + = g(x), x D, aassume that function V : R n R satises for some scalars c 1, c 2 positive and c 3 positive: c 1 x 2 V (x) c 2 x 2, x C D G(D) V (x), f (x) c 3 x 2, x C, V (g(x)) V (x) c 3 x 2, x D, then the origin is uniformly globally exponentially stable (UGES) for H, namely there exist K, λ > such that all solutions satisfy ξ(t, j) Ke λ(t+j) ξ(, ), (t, j) dom ξ Note: Lyapunov conditions comprise ow and jump conditions. Note: UGAS is characterized in terms of hybrid time (t, j)

14 Hybrid dynamical systems review: Lyapunov theorem Theorem Given a closed set A R n and a hybrid system { ẋ F (x), x C H : x + G(x), x D, aassume that function V : R n R satises for some α 1, α 2 K and ρ positive denite: α 1 ( x A ) V (x) α 2 ( x A ), x C D G(D) V (x), f ρ( x A ), V (g) V (x) ρ( x A ), x C, f F (x), x D, g G(x) then A is uniformly globally asymptotically stable (UGAS) for H, namely there exists β KL such that all solutions satisfy ξ(t, j) A β( ξ(, ) A, t + j), (t, j) dom ξ Note: Lyapunov conditions comprise ow and jump conditions. Note: UGAS is characterized in terms of hybrid time (t, j)

15 Hybrid dynamics and the Clegg integrator (recall) Hybrid Clegg integrator: ẋ c = 1 v, allowed when xc v, RC x + c =, allowed when v, Flow set C: where may ow (1st eq'n) Jump set D: where may jump (2nd eq'n) v C D C C v R d R R v C1 - - v C2 Clegg's integrator (1956): feedback diodes: the positive part of is all and only coming from the upper capacitor (and viceversa) input diodes: when v the upper capacitor is reset and the lower one integrates (and viceversa) [R d 1] As a consequence v and never have opposite signs C

16 Hybrid dynamics of the Clegg integrator (revisited) Hybrid Clegg integrator: ẋ c(t, j) = (RC) 1 v(t, j), (t, j)v(t, j), (t, j + 1) =, (t, j)v(t, j), C D Flow set C := {(, v) : v } is closed Jump set D := {(, v) : v } is closed Stability is robust! (Teel 26212) v C Previous models (Clegg '56, Horowitz '73, Hollot '4): ẋ c = (RC) 1 v, if v, x + c =, if v =, Imprecise: solutions s.t. v <, but Clegg's and v always have same sign! Unrobust: C is almost all R 2 (arbitrary small noise disastrous) Unsuitable: Adds extra solutions Lyapunov results too conservative! v D C C C D

17 Stabilization using hybrid jumps to zero First Order Reset Element (Horowitz '74): ẋ c = a c + b c v, v, x + c =, v, FORE Theorem If P is linear, minimum phase and relative degree one, then a c, b c or (a c, b c ) large enough uniform global exponential stability v u d P y Theorem In the planar case, γ dy shrinks to zero as parameters grow Simulation uses: P = 1 s b c = 1 Plant output Linear (a = 1) c a c = 3 a c = 1 a =1 c a =3 c Time Interpretation: Resets remove overshoots, instability improves transient

18 Piecewise quadratic Lyapunov function construction Given N 2 (number of sectors) Patching angles: θ ɛ = θ < θ 1 < < θ N = π 2 + θ ɛ Patching hyperplanes (C p = [ 1]) Θ i = [ 1 (n 2) sin(θ i ) cos(θ i ) ] T Sector matrices: S := Θ Θ T N + Θ N Θ T S i := (Θ i Θ T i 1 + Θ i 1 Θ T i ), i = 1,..., N, (n 2) (n 2) S ɛ1 := sin(θ ɛ ) sin(θ ɛ ) 2 cos(θ ɛ ) (n 2) (n 2) S ɛ2 := 2 cos(θ ɛ ) sin(θ ɛ ) sin(θ ɛ ) θ 1 θ 2 θ S 2 S 1 S P N S ɛ2 θ N 2 θ N 1 θ N S N 1 SN P N 1 P P 1 P 2 S axis Hybrid closed-loop: ẋ = A F x + B w w, x C x + = A J x, x D S ɛ1 y axis

19 Example 1: Clegg (ac = ) connected to an integrator Block diagram: Clegg a c = d 1 s Output response (overcomes linear systems limitations) 1 y Gain γ dy estimates (N = # of sectors) N gain γ dy A lower bound: π Lyapunov func'n level sets for N = x r 1 x r 1.5 Clegg (a c =) Linear (a c =) Quadratic Lyapunov functions are unsuitable y y P 1,..., P 4 cover 2nd/4th quadrants P covers 1st/3rd quadrants

20 Example 2: FORE (any ac) and linear plant (Hollot et al.) x r 1 Block diagram (P = s+1 FORE d s(s+.2) ) P a c = 1: level set with N = x r y Gain γ dy estimates L2 gains a c Time responses 1.5 Linear CLS Reset CLS (Thm 3, ACC 25) Reset CLS (this theorem) Linear (a c = 1) a c = 3 a c = 1 a c =1 a c =3 a c = y y

21 A class of homogeneous hybrid systems ẋ = Ax + Bw ( ) τ τ = 1 dz ρ H x + = Gx τ + = z = C z x + D zw w (x, τ) C (x, τ) D J F C = {(x, τ) : x F or τ [, ρ]} D = {(x, τ) : x J and τ [ρ, 2ρ]} F= { x R n : x Mx } J = { x R n : x Mx } F Ideal behavior J x p

22 Dwell time can force the solution to ow in the set J ẋ = Ax + Bw ( ) τ τ = 1 dz ρ H x + = Gx τ + = z = C z x + D zw w (x, τ) C (x, τ) D J x(t + ρ, ) τ(t + ρ, ) = ρ F x(t, ) τ(t, ) = x p C = {(x, τ) : x F or τ [, ρ]} D = {(x, τ) : x J and τ [ρ, 2ρ]} F x(t 1, 1) τ(t 1, 1) = J F = { x R n : x Mx } J = { x R n : x Mx } The dwell time enables ow in the set J (t 1 = t + ρ) Disadvantage: ow in J

23 x Dwell-time prevents consecutive jumps from being too close ẋ = Ax + Bw ( ) τ τ = 1 dz ρ H x + = Gx τ + = z = C z x + D zw w C = {(x, τ) : x F or τ [, ρ]} (x, τ) C (x, τ) D D = {(x, τ) : x J and τ [ρ, 2ρ]} Dwell-time condition: Each solution ξ to H satises t s ρ for any pair of hybrid times (t, j), (s, i) dom(ξ), (t, j) (s, i) F = { x R n : x Mx } J = { x R n : x Mx } Advantage: persistent ow of all solutions t j

24 From dwell time performance wrt ordinary time t Dwell-time allows us to use classical performance indexes. Denition (t-decay rate) Given a compact set A R n and w =, H has t-decay rate α > if there exists K > such that each solution x satises x(t, j) A K exp( αt) x(, ) A, for all (t, j) dom(x). Denition (t-l 2 gain) Consider a set A R n uniformly globally asymptotically stable for H. H is nite t-l 2 gain stable from w to z with gain (upper bounded by) γ > if any solution x to H starting from A satises x 2t γ w 2t, for all w t-l 2.

25 Performance analysis result: V (x) = x Px Proposition: Consider system H. If there exist matrices P = P >, M = M, non-negative scalars τ F, τ C, τ R R and positive scalars ɛ, γ, such that A P + PA ( M ɛi ) PB C z B P γi D zw C z D zw γi G PG P + τ R M, M τ F M ɛi, G MG + τ C M. Then for any γ satisfying there exists ρ > such that for any ρ (, ρ): <, (1a) (1b) (1c) (1d) γ γ, γ > 2 D zw, (2) 1) the set A = {} [, 2ρ] is uniformly globally exponentially stable for the hybrid system H with w = ; 2) the t-l 2 gain from w to z is less than or equal to γ, for all w t-l 2.

26 An architecture for homogeneous reset controller deisgn May design the reset rules K p, M, ρ only (case 1) or the whole dynamics (case 2) ẋ c = Āc + B c y τ = 1 dz ( ) τ ρ u = C c + D c y u ẋ p = Āpx p + B p u y = C p x p + D p u P y flow:, τ Supervisor x + c xp M xp or τ [,ρ] = K p x p τ + = jump: xp M xp and τ [ρ, 2ρ] H c x p

27 Case 1: Flow dynamics is given, design Jump sets and rules Matrices Ā c, Bc, Cc, Dc are given. Deisgn K p, M and ρ ẋ c = Āc + B c y τ = 1 dz ( ) τ ρ u = C c + D c y u ẋ p = Āpx p + B p u y = C p x p + D p u P y flow:, τ Supervisor x + c xp M xp or τ [,ρ] = K p x p τ + = jump: xp M xp and τ [ρ, 2ρ] H c x p

28 Design paradigm is inherited from Clegg integrator Reset to the minimizer of the hybrid Lyapunov function: x + c = φ(x p ) := argmin V (x p, ) Reset whenever the function V p (x p ) := V (x p, φ(x p )) is nondecreasing: { } J = [ xp ] R n : [ xp ] M [ xp ], [ xp ] M [ xp ] = V p (x p ), A p x p + B p = Vp (x p, ) x p

29 Design paradigm for the quadratic case V (x) = x T Px Reset to the minimizer of the hybrid [ Lyapunov function: ] x + c = φ(x p ) := argmin [ xp ] P p P pc [ xp P T pc x Pc ] = Pc 1 Ppc T x p = K p x p c Reset to ensure nonincrease of V p (x p ) := V (x p, φ(x p )) = xp T (P p P pc Pc 1 P: pc) T x p }{{} J = P p { } [ xp ] R n : [ xp ] M [ xp ], [ xp ] M [ xp ] = V p (x p ), A p x p + B p + 2 αv p (x p ) = Vp (x p, ) + 2 αv p (x p ) [ ] = [ xp ] 2 Pp(A p+ αi ) PpB p [ xp ] }{{} M x p

30 Construction leaves many degrees of freedom Theorem: Consider system H and assume that He ( Pp (A p + B p K p ) + α ) P p <, Pp = Pp >, α >. (3) 2 Then for each α (, α], there exists a small enough ρ > such that controller H c with ] [ Pp (A M = p + αi ) Pp B p 2 guarantees that: the set A = {} [, 2ρ] is globally exponentially stable for H; any solution with (, ) = satises ( α2 ) t x p (t, j) K exp x p (, ), (t, j) dom(ξ). (4)

31 Positioning system: overshoot reduction A DC motor controlled by a rst oder lter u V K e T m 1 ω 1 θ 1 + τ e s F + Js s P y Plant Output PI Hyb Time.3 Plant Input Time optimal synthesis for overshoot reduction x T p P px p y 2 or improvement of the convergence rate (using α)

32 Multi-objective hybrid H controller synthesis Design all blue parameters Ā c, Bc, Cc, Dc, K p, M and ρ ẋ c = Āc + B c y τ = 1 dz ( ) τ ρ u = C c + D c y w u ẋ p = Āpx p + B p u + B w w y = C p x p + D p u + D w w z = C z x p + D z u + D zw w P z y flow:, τ Supervisor x + c xp M xp or τ [,ρ] = K p x p τ + = jump: xp M xp and τ [ρ, 2ρ] H c x p

33 Multi-objective hybrid H : t-l 2 gain and t-decay rate Analysis conditions: t-l 2 gain A P + PA ( M ɛi ) PB Cz B P γi Dzw C z D zw γi G PG P + τ R M M τ F M ɛi G MG + τ C M < Reset controller: t-decay rate He ( Pp (A p + B p K p ) + α ) P p < 2 Nonlinear coupling constraints: [ ] Pp P P = pc Ppc T P c P p = P p P pc Pc 1 Ppc T = Change of coordinates from Scherer, Gahinet, Chilali 1997 leads to [ ] [ ] W W Y Z P := W W + Z 1, P 1 :=, Pp = Y 1 Z Z [ ] [ Y Z I Π =, ΠP = I W W ]

34 Multi-objective synthesis then becomes an LMI problem Theorem: Consider plant P and any solution to LMIs: [ ] Y I > I W Ā p Y + Bp Ĉ Ā p + Bp ˆD Cp Bw + Bp ˆD Dw Y C z + Ĉ D z He  W Ā p + ˆB Cp W Bw + ˆB Dw C z + C p ˆD D z γ 2 I D zw + D w ˆD D z γ 2 I He (Āp Y + Bp Ĉ + α 2 Y ) < < Then there exists a hybrid controller H c such that: the t-decay rate is equal to α/2, with α (, α]; the t-l 2 gain from w to z less than or equal to γ, for all w t-l 2.

35 y z u γ y z u DC motor: optimal t-l 2 gain for a given α = α = linear syn linear b.r.l. hybrid (SF) α=α L 2 linear 4 hybrid Time linear hybrid Time Time Time Time Free response Time Forced response

36 y z u γ z y u DC motor: optimal t-l 2 gain for a given α = α = linear syn linear b.r.l. hybrid (SF) α=α L linear hybrid Time Time Time Free response linear hybrid Time Time Time Forced response

37 Conclusions and perspectives 1 Objective and motivation exploit hybrid tools to push further the initial idea of Clegg in Revisiting Clegg and FORE new modeling paradigm: ow only in half of the state space can now give Lyapunov guarantees of exponential stability exp instability before reset promises high performance experimental tests on EGR valve control (Diesel engines) 3 Generalized reset controllers A Lyapunov framework for stability and performance analysis A hybrid H controller design 4 Perspectives feedback from observed state (not covered here, partially done) overcome performance limitations improve synthesis scheme to allow for unstable continuous dynamics