Control Group in Siena. HYSDEL Models and Controller Synthesis for Hybrid Systems. Summary of my talk

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HYSDEL Models and Controller Synthesis for Hybrid Systems Dept. of Information Engineering University of Siena, Italy bemporad@dii.unisi.

1 HYSDEL Models and Controller Synthesis for Hybrid Systems Dept. of Information Engineering University of Siena, Italy Alberto Bemporad Automatic Control Laboratory Swiss Federal Institute of Technology Control Group in Siena 4 professors 4 postdocs 4 PhD students (+2 to come) Research activities: robust control identification mobile robotics & dynamic vision sequencing and scheduling telelaboratory hybrid systems model predictive control Summary of my talk 1. Models of hybrid systems 2. The HYSDEL language 3. Controller synthesis for hybrid systems 4. Safety analysis of hybrid systems Models of Hybrid Systems (that can be handled by HYSDEL) 5. Examples 6. Ongoing research 1

2 Hybrid Dynamics - Continuous Part Hybrid Dynamics - Discrete Part unit delay Switched continuous dynamics Automata continuous states continuous inputs switching index derivatives discrete-time discrete-events logic states logic inputs internal events discrete-time discrete-events Hybrid Dynamics Hybrid Dynamics unit delay Continuous dynamics change according to: 1. Exogenous logic inputs 2. Threshold conditions Discrete dynamics change according to a Boolean fnc of: 1. Previous logic states 2. Exogenous logic inputs 3. Time conditions 3. Threshold conditions 4. Any logic combination of the former Reset conditions: possible (continuous time case more tricky) 2

Mixed Logical Dynamical Systems Computational Models of Hybrid Systems Mixed Logical Dynamical (MLD) form x(t +1)=Ax(t) +B 1 u(t) + B 2 î(t) +B 3 z(t) y(t) =Cx(t) +D 1 u(t) + D 2 î(t) +D 3 z(t) E 2

` (Bemporad, Morari, Automatica, 1999) HYSDEL (HYbrid Systems DEscription Language) Describehybrid systems: HYSDEL Automata Logic Lin.

3 Mixed Logical Dynamical Systems Computational Models of Hybrid Systems Mixed Logical Dynamical (MLD) form x(t +1)=Ax(t) +B 1 u(t) + B 2 î(t) +B 3 z(t) y(t) =Cx(t) +D 1 u(t) + D 2 î(t) +D 3 z(t) E 2 î(t) +E 3 z(t) ô E 4 x(t) +E 1 u(t) +E 5 ô õ x, y, u =?c,? c R n c,?` {, 1 n `, z R rc, î {, 1 r`?` (Bemporad, Morari, Automatica, 1999) HYSDEL (HYbrid Systems DEscription Language) Describehybrid systems: HYSDEL Automata Logic Lin. Dynamics Interfaces Constraints symbols δ i A/D continuous states inputs automaton / logic continuous dynamical system symbols δ D/A inputs Automatically generate MLD models in Matlab MLD model is not unique in terms of the number of auxiliary variables optimize model (minimize # binary variables!) (Torrisi, Bemporad, Mignone, 2) 3

SYSTEM motor { INTERFACE { STATE { REAL ucomp; INPUT { REAL u [,1]; PARAMETER { REAL ut = 1; REAL e = 1e-6; /* end interface */ AD and DA u comp IMPLEMENTATION { AUX { REAL unl;

3*ut ELSE u ; CONTINUOUS { ucomp = unl; /* end implementation */ /* end system */ Nonlinear amplification unit u = u ( s s ) s brake alarm tunnel fire fire u alarm LOGIC SYSTEM

AUTOMATA { brake = decision; MUST { fire -> alarm; /* end implementation */ /* end system */ CONTINUOUS AUTOMATA d i = C u() t dt SYSTEM capacitord { INTERFACE { STATE { REAL u;

uopen, ustop; /* end of interface */ IMPLEMENTATION { Apply forward difference rule: T uk ( + 1) = uk ( ) + ik ( ) C CONTINUOUS { u = u - T/C/R*i; /* end implementation */ /* end

4 SYSTEM motor { INTERFACE { STATE { REAL ucomp; INPUT { REAL u [,1]; PARAMETER { REAL ut = 1; REAL e = 1e-6; /* end interface */ AD and DA u comp IMPLEMENTATION { AUX { REAL unl; BOOL th; AD { th = ut - u <= ; u ( u < ut ) = 2.3u 1.3 ut ( u ut) DA { unl = { IF th THEN 2.3*u - 1.3*ut ELSE u ; CONTINUOUS { ucomp = unl; /* end implementation */ /* end system */ Nonlinear amplification unit u = u ( s s ) s brake alarm tunnel fire fire u alarm LOGIC SYSTEM train { INTERFACE { STATE { BOOL brake; INPUT { BOOL alarm, tunnel, fire; /* end interface */ IMPLEMENTATION { AUX { BOOL decision; LOGIC { decision = alarm & (~tunnel ~fire); AUTOMATA { brake = decision; MUST { fire -> alarm; /* end implementation */ /* end system */ CONTINUOUS AUTOMATA d i = C u() t dt SYSTEM capacitord { INTERFACE { STATE { REAL u; PARAMETER { REAL R = 1e4; REAL C = 1e-4; REAL T = 1e-1; /* end interface */ IMPLEMENTATION { SYSTEM outflow { INTERFACE { STATE { BOOL closing, stop, opening; INPUT { BOOL uclose, uopen, ustop; /* end of interface */ IMPLEMENTATION { Apply forward difference rule: T uk ( + 1) = uk ( ) + ik ( ) C CONTINUOUS { u = u - T/C/R*i; /* end implementation */ /* end system */ AUTOMATA { closing = (uclose & closing) (uclose & stop); stop = ustop (uopen & closing) (uclose & opening); opening = (uopen & stop) (uopen & opening); MUST { ~(uclose & uopen); ~(uclose & ustop); ~(uopen & ustop); /* end implementation */ /* end system */ 4

5 MUST SYSTEM watertank { INTERFACE { STATE { REAL h; INPUT { REAL Q; PARAMETER { REAL hmax =.3; REAL k = 1; /* end interface */ IMPLEMENTATION { CONTINUOUS { h = h + k*q; Realization and Transformation (other state-space hybrid models) h h max MUST { h - hmax <= ; -h <= ; /* end implementation */ /* end system */ Existing Hybrid Models Piecewise affine (PWA) systems (Sontag, 1981, 1996) Existing Hybrid Models Linear complementarity (LC) systems (Heemels, 1999) state+input space Polyhedral partition of state+input space ô X i = { x õ ô õ : H x i ô K i, i =1,...,s u u x(t +1)=Ax(t) +B 1 u(t) +B 2 w(t) y(t) =Cx(t) +D 1 u(t) +D 2 w(t) v(t) =E 1 x(t) +E 2 u(t) +E 3 w(t) +e 4 ô v(t) w(t) õ Ex: mechanical systems circuits with diodes etc. Affine dynamics in each region x(t +1)=A i(t) x(t) +B i(t) u(t) +f i(t) ô õ x(t) if X u(t) i(t) Can approximate nonlinear dynamics arbitrarily well Extended linear complementarity (ELC) systems Generalization of LC systems Min-max-plus-scaling (MMPS) systems x(t +1)=M x (x(t),u(t),d(t)) y(t) =M y (x(t),u(t),d(t)) õ M c (x(t),u(t),d(t)) Example: x(t + 1) = 2 max (x(t), ) + min (à 2 1 u(t), 1) (De Schutter, Van Den Boom, 2) MMPS function: defined by the grammar M := x i ë max (M 1,M 2 ) min (M 1,M 2 ) M 1 + M 2 ìm 1 Used for modeling discrete-event systems (t=event counter) (De Schutter, De Moor, 2) 5

Equivalence Results Example: Bouncing Ball x ẍ =à g x ô xç(t + )=à (1 à ë)xç(t à ) F= m g N ë [, 1] (Heemels, De Schutter, Bemporad, Automatica, 21 + CDC21) Theoretical properties and

6 Equivalence Results Example: Bouncing Ball x ẍ =à g x ô xç(t + )=à (1 à ë)xç(t à ) F= m g N ë [, 1] (Heemels, De Schutter, Bemporad, Automatica, 21 + CDC21) Theoretical properties and analysis/synthesis tools can be transferred from one class to another! How to model this system in MLD form? HYSDEL - Bouncing Ball SYSTEM ball { INTERFACE { /* Description of variables and constants */ STATE { REAL height [-1,1]; REAL velocity [-1,1]; PARAMETER { REAL g=9.8; REAL dissipation=.4; /* =elastic, 1=completely anelastic */ REAL e=1e-6; REAL Ts=.5; IMPLEMENTATION { AUX { REAL hnext; REAL vnext; BOOL negative; AD { negative = height <= [hmax,hmin,e]; DA { hnext = { IF negative THEN height-ts*velocity ELSE height+ts*velocity-ts*ts*g ; logic Modeling Flow g23x(t) +h23u(t) <k23 x(t +1)= x(t +1)= x(t +1)= A1x(t)+B1u(t)+f1 A 3x(t)+B 3u(t)+f 3 A 2x(t) + B 2u(t) + f 2 g13x(t) +h13u(t) <k13 g12x(t) +h12u(t) <k12 g32x(t) +h32u(t) <k32 Designer Hysdel Hysdel Source HYbrid Systems Description Language vnext = { IF negative THEN -(1-dissipation)*velocity ELSE velocity-ts*g ; CONTINUOUS { height = hnext; velocity = vnext; A/D continuous D/A MLD Mixed Logical Dynamical form PWA Piecewise Affine form 6

7 System Theory for Hybrid Systems Analysis Realization & Transformation Well-posedness Stability Reachability (=Verification) Observability Controller Synthesis Synthesis Control State estimation Identification Modeling language Optimal Control of Hybrid Systems Finite-time optimal control problem: Tà1 min J(ø, x(),r), P kq (y(k) à r)k + kr(u(k) à u r )k ø k= + û ( kî(k) à î rk + kz(k) à z rk + kx(k) à x rk) MLD model subj. to x(t t) =x(t) x(t + T t) =x r ø =[u(),...,u(t à 1),î(),...,î(T à 1),z(),...,z(T à 1)] Solution Squared 2-norm: Mixed-Integer Quadratic Program (MIQP) or 1-norm: Mixed Integer Linear Program (MILP) At time t : Model Predictive Control Solve an optimal control problem over a finite future horizon p : minimize performance subject to constraints Only apply the first optimal move r(t) Predicted outputs u(t+k) Manipulated Inputs t t+1 t+p u ã (t) t+1 t+2 t+p+1 Get new measurements, and repeat the optimization at time t +1 Advantage of on-line optimization: FEEDBACK! 7

8 Closed-Loop Stability Mixed-Integer Program Solvers Mixed-Integer Programming is NP-hard BUT General purpose Branch & Bound/Branch & Cut solvers available for MILP (CPLEX) and MIQP (Fletcher-Leyffer, Sahinidis, XPRESS-MP) Free Matlab MILP/MIQP solver (Bemporad, Mignone, 1999) More solvers and benchmarks: (Bemporad, Morari, Automatica, 1999) No need to reach global optimum (see proof of the theorem), although performance deteriorates Proof: use optimal value function as a Lyapunov function On-Line vs. Off-Line Optimization min J(U, x(t)), P U k= MLD model subj. to x(t t) =x(t) x(t + T t) = Tà1 kqy(t + k +1 t)k + kru(t + k)k On-line optimization: given x(t), solve the problem at each time step t Mixed-Integer Linear Program (MILP) Good for large sampling times (e.g., 1 h) / expensive hardware but not for fast sampling (e.g. 1 ms) / cheap hardware! Off-line optimization: get the explicit solution of the MPC controller by solving the MILP for all x(t) min J(ø, x(t)), f ø ø s.t.gø ô W + Fx(t) multi-parametric Mixed Integer Linear Program (mp-milp) Explicit Form of Model Predictive Control via Multiparametric Programming 8

9 Example of Multiparametric Solution Multiparametric LP ( ø R) x CR{1,4 CR{1,2,3 CR{1,3 CR{2, x 1 Multiparametric MILP min f ø c + d ø d ø={ øc,ø d s.t. Gø c + Eø d ô W + Fx ø c R n ø d {, 1 m mp-milp can be solved (by alternating MILPs and mp-lps) (Dua, Pistikopoulos, 1999) Theorem: The multiparametric solution ø ã (x is ) piecewise affine Corollary: The MPC controller is piecewise affine in x Remarks on explicit MPC law: Automatic partitioning of state-space (no gridding!) Stability guarantee (value function=pwl Lyapunov function) Reachability Analysis/Verification (Bemporad, Torrisi, Morari, HSCC 2) X() X Z1 () Z 1 Reachability Analysis (Verification) X Z2 () X ZL () Z 2 Z L Efficient algorithms for reachability analysis of MLD/PWA systems were developed during the VHS project Software available in Matlab (requires fabio.dll) 9

10 Hysdel for Verification x(t +1)= A1x(t)+B1u(t)+f1 g12x(t) +h12u(t) <k12 g23x(t)+h23u(t) <k23 x(t +1)= x(t +1)= A3x(t)+B3u(t)+f3 A2x(t)+B2u(t)+f2 Inputs (e.g.: disturbances, set-points) g13x(t) +h13u(t) <k13 g32x(t)+h32u(t) <k32 logic A/D D/A continuous MLD Designer Hysdel Hysdel Source PWA Finite-time reachability analysis Logic-based, threshold-based, and timebased verification queries Reachability-based optimization Examples Vehicle Traction Control Improve driver's ability to control a vehicle under adverse external conditions (wet or icy roads) Traction Control System (F. Borrelli, A. Bemporad, M. Fodor, D. Hrovat) Model nonlinear, uncertain, constraints Controller suitable for real-time implementation MLD hybrid framework + optimization-based control strategy 1

11 Simple Traction Model Hybrid Model Mechanical system ò ó 1 ü t ωç e = ü c à b e ω e à Je gr Torque Nonlinear tire torque τ t =f( ω, µ) ü t vç v = mv r t Manifold/fueling dynamics ü c = b i ü d(t à ü f) µ Slip PWA Approximation (PWL Toolbox, Julian, 1999) (Ferrari et al., HSCC 1) Tire torque τ t is a function of slip ω and road surface adhesion µ ω e v v ω = à gr rt Wheel slip Torque µ Slip HYSDEL (Hybrid Systems Description Language) Mixed-Logical Dynamical (MLD) Hybrid Model (discrete time) ω EXPERIMENTAL RESULTS Experiment 25 2 Wheel Speeds, [rad/s] Engine Torque Command, [Nm] Controller Region Time, [s] >5 regions 2ms sampling time Pentium 266Mhz + Labview 11

12 Comments from Ford: Performance of the hybrid MPC controller is quite good given the limited development time, oversimplified plant model used, and minimal development iterations. The hybrid MPC controller requires much less supervision by logical constructs than controllers developed with traditional techniques. Hybrid Control Example: Cruise Control System (A. Bemporad, F.D. Torrisi) Hybrid Control Problem Hysdel Model (Bemporad, Torrisi, 2) Renault Clio 1.9 DTI RXE GOAL: command gear ratio, gas pedal, and brakes to track a desired speed and minimize consumption 12

13 Smoother tracking controller Hybrid Controller Max-speed controller Hybrid Controller min J(u t,x(t)), v(t +1 t) à v ut d(t) + ú ω v(t +1 t) à v(t) <T s a max subj. to MLD model x(t t) =x(t) Velocity (km/h) Gear Fraction of Max Torque (Nm) Brakes (Nm) MILP optimization problem Linear constraints Continuous variables Binary variables Parameters Time to solve mp-milp (Sun Ultra 1) Number of regions m 54 v d(t) v(t) Road Slope (deg) Engine speed (rpm) Engine Torque (Nm) Power (kw) Smoother tracking controller Hybrid Controller Velocity (km/h), Desired velocity (km/h) 12 5 Gear Fraction of Max Torque (Nm) 1 Brakes (Nm) Road Slope (deg) Engine speed (rpm) Engine Torque (Nm) Power (kw) 1 5 Verification of a Cruise Control System (F.D. Torrisi, A. Bemporad)

(t) v(t)) + saturation ½ kt (v u t (t) = r (t) v(t)) + i t e(t) if v(t) <v r (t)+1 otherwise ½ kb (v u b (t) = r (t) v(t)) if v(t) v r (t)+1 otherwise (Torrisi, Bemporad, 21) Hysdel Model (HYbrid

$w3, w4, w5, DCe1, DCe2, DCe3, DCe4,zut, zub, ierr, torque, F_brake; BOOL dpwl1, dpwl2, dpwl3, dpwl4, sd, su, verr, sat_torque, sat_f_brake, no_sat; LOGIC { no_sat = ~(sat_torque sat_f_brake) & verr;$

14 Cruise Control System Renault Clio 1.9 DTI RXE Gear selector: Cruise Control System w w u w w u w w u w w u GOAL: Verify if a given switching controller satisfies certain specifications Speed controller: e(t +1)=e(t)+T s (v r (t) v(t)) + saturation ½ kt (v u t (t) = r (t) v(t)) + i t e(t) if v(t) <v r (t)+1 otherwise ½ kb (v u b (t) = r (t) v(t)) if v(t) v r (t)+1 otherwise (Torrisi, Bemporad, 21) Hysdel Model (HYbrid Systems DEscription Language) SYSTEM car { INTERFACE { STATE { REAL speed, err, vr; BOOL gear1, gear2, gear3, gear4, gear5; PARAMETER { IMPLEMENTATION { AUX { REAL Fe1, Fe2, Fe3, Fe4, Fe5, w1, w2, w3, w4, w5, DCe1, DCe2, DCe3, DCe4,zut, zub, ierr, torque, F_brake; BOOL dpwl1, dpwl2, dpwl3, dpwl4, sd, su, verr, sat_torque, sat_f_brake, no_sat; LOGIC { no_sat = ~(sat_torque sat_f_brake) & verr; AD {dpwl1 = wpwl1 - (w1 + w2 + w3 + w4 + w5) <= ; dpwl2 = wpwl2 - (w1 + w2 + w3 + w4 + w5) <= ; dpwl3 = wpwl3 - (w1 + w2 + w3 + w4 + w5) <= ; dpwl4 = wpwl4 - (w1 + w2 + w3 + w4 + w5) <= ; sd = (w1 + w2 + w3 + w4 + w5) - wl <= ; su = wu - (w1 + w2 + w3 + w4 + w5) <= ; verr = speed - vr - 2 <= ; sat_torque = - zut + (DCe1 + DCe2 + DCe3 + DCe4) + 1 <= ; sat_f_brake = - zub + max_brake_force <= ; DA {Fe1 = {IF gear1 THEN torque / speed_factor * Rgear1; Fe2 = {IF gear2 THEN torque / speed_factor * Rgear2; Fe3 = {IF gear3 THEN torque / speed_factor * Rgear3; Fe4 = {IF gear4 THEN torque / speed_factor * Rgear4; Fe5 = {IF gear5 THEN torque / speed_factor * Rgear5; w1 = {IF gear1 THEN speed / speed_factor * Rgear1; w2 = {IF gear2 THEN speed / speed_factor * Rgear2; w3 = {IF gear3 THEN speed / speed_factor * Rgear3; w4 = {IF gear4 THEN speed / speed_factor * Rgear4; w5 = {IF gear5 THEN speed / speed_factor * Rgear5; DCe1 = {IF dpwl1 THEN (apwl2) + (bpwl2) * (w1 + w2 + w3 + w4 + w5) ELSE (apwl1) + (bpwl1) * (w1 + w2 + w3 + w4 + w5); DCe2 = {IF dpwl2 THEN (apwl3 - apwl2) + (bpwl3 - bpwl2) * (w1 + w2 + w3 + w4 + w5); DCe3 = {IF dpwl3 THEN (apwl4 - apwl3) + (bpwl4 - bpwl3) * (w1 + w2 + w3 + w4 + w5); DCe4 = {IF dpwl4 THEN (apwl5 - apwl4) + (bpwl5 - bpwl4) * (w1 + w2 + w3 + w4 + w5); zut = {IF verr THEN kt * (vr - speed) + it * err; zub = {IF ~verr THEN - kb * (vr - speed) - ib * err; torque ={IF sat_torque THEN (DCe1 + DCe2 + DCe3 + DCe4) + 1 ELSE zut; F_brake = {IF sat_f_brake THEN max_brake_force ELSE zub; ierr = {IF no_sat THEN err + Ts * (vr - speed); CONTINUOUS { speed = speed + Ts / mass * (Fe1 + Fe2 + Fe3 + Fe4 + Fe5 - F_brake - beta_friction * speed); err = ierr; vr = vr; AUTOMATA { gear1 = (gear2 & sd) (gear1 & ~su); gear2 = (gear1 & su) (gear3 & sd) (gear2 & ~sd & ~su); gear3 = (gear2 & su) (gear4 & sd) (gear3 & ~sd & ~su); gear4 = (gear3 & su) (gear5 & sd) (gear4 & ~sd & ~su); gear5 = (gear4 & su) (gear5 & ~sd ); MUST { -w1 <= -wemin; w1 <= wemax; -w2 <= -wemin; w2 <= wemax; -w3 <= -wemin; w3 <= wemax; -w4 <= -wemin; w4 <= wemax; -w5 <= -wemin; w5 <= wemax; -F_brake <= ; F_brake <= max_brake_force; torque - (DCe1 + DCe2 + DCe3 + DCe4) - 1 <= ; -((REAL gear1) + (REAL gear2) + (REAL gear3) + (REAL gear4) + (REAL gear5)) <= ; (REAL gear1) + (REAL gear2) + (REAL gear3) + (REAL gear4) + (REAL gear5) <= 1.1; dpwl4 -> dpwl3; dpwl4 -> dpwl2; dpwl4 -> dpwl1; dpwl3 -> dpwl2; dpwl3 -> dpwl1; dpwl2 -> dpwl1; (Torrisi, Bemporad, Mignone, 2) MLD model Hybrid Model x(t +1)=Ax(t) +B 1 u(t) + B 2 î(t) +B 3 z(t) y(t) =Cx(t) +D 1 u(t) + D 2 î(t) +D 3 z(t) E 2 î(t) +E 3 z(t) ô E 4 x(t) +E 1 u(t) +E 5 3 continuous states: 5 binary states: 15 auxiliary binary vars: v, v r,e 19 auxiliary continuous vars: g 1,g 2,g 3,g 4,g mixed-integer inequalities (vehicle speed, reference and tracking error) (gears) (5 traction force, 5 engine speed, 5 reset/saturation, 4 PWL max engine torque) (4 PWL max torque breakpoints, 4 saturations 5 logic updates, 2 gear switching conditions) 14

Speed (Km/h) 12 1 8 6 4 2 Verification Safety Liveness Verification Results For all v r [3, 7] km/hthe controller satisfies both liveness & safety properties CPU time: ~2.5h on Matlab5.3, PC65MHz.

15 Speed (Km/h) Verification Safety Liveness Verification Results For all v r [3, 7] km/hthe controller satisfies both liveness & safety properties CPU time: ~2.5h on Matlab5.3, PC65MHz Verify that the cruise control reaches the desired speed reference ( Z 2 = {v,t : v<v r 2r toll,t >1/T) s and without driving above the limit ( Z 1 = {v : v>v r + r), toll r toll = 5km/h Speed (Km/h) 12 Safety Liveness For v r [3, 12] km/h the verification algorithm finds the first counterexample after ~7m Hybrid Modeling and Control of a Direct Injection Stratified Charge (DISC) Engine A. Bemporad, N. Giorgetti, I. Kolmanovsky, D. Hrovat States/Controlled outputs: 1. Intake manifold pressure 2. Air-to-fuel ratio 3. Engine brake torque Nonlinear Hybrid Model Inputs (continuous): 1. Mass flow rate through throttle 2. Mass flow rate of fuel 3. Spark timing Inputs(binary): 1. ρ = Regime of combustion (stratified/homogeneous) Constraints on: 1. Air-to-fuel ratio (due to engine roughness, misfiring, smoke emiss.) 2. Spark timing (to avoid excessive engine roughness) 3. Mass flow rate of throttle Dynamic equations are nonlinear Dynamics and constraints depend on ρ! 15

Hybrid MPC Control Closed-loop: Nonlinear hybrid model + Hybrid MPC controller Engine Brake Torque Air-To-Fuel Ratio More about this problem Intake Manifold Pressure at the next CC meeting Combustion

16 Hybrid MPC Control Closed-loop: Nonlinear hybrid model + Hybrid MPC controller Engine Brake Torque Air-To-Fuel Ratio More about this problem Intake Manifold Pressure at the next CC meeting Combustion Regime The End Engine Brake Torque + Noise Engine Brake Torque (nominal MLD model) Announcements 1. CC project web site: Please contribute!!! (mailto: hybrid@dii.unisi.it) 2. IEEE Technical Committee on Hybrid Systems web site: Please contribute!!! (mailto: hybrid@dii.unisi.it) 16

Modeling and Control of Hybrid Systems

Modeling and Control of Hybrid Systems Alberto Bemporad Dept. of Information Engineering University of Siena, Italy bemporad@unisi.it MTNS 2004 Leuven, July 8 COHES Group Control and Optimization of Hybrid