Mathematical Programming Approach to Hybrid Systems ...cas.ensmp.fr/~levine/MorariTransp.pdf · • Car suspension system • Gas supply system • Hydroelectric power plant... •

Mathematical Programming Approach toMathematical Programming Approach toHybrid SystemsHybrid Systems

Analysis and ControlAnalysis and Control

Automatic Control LaboratoryAutomatic Control LaboratorySwiss Federal Institute of TechnologySwiss Federal Institute of Technology

ETHETH Zürich Zürich

Manfred MorariAlberto Bemporad Giancarlo Ferrari Trecate

Mato Baotic, Francesco Borrelli, Francesco Cuzzola,Tobias Geyer, Domenico Mignone, Fabio Torrisi

Drivers for Control Research

Novel Applicationsenabled by

• new computer power• new actuators• new sensors

Novel Theorymotivated by

• system integration• system failures

Hybrid systemsHybrid systems

Hybrid SystemsHybrid Systems

dtdx(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)

úS , (X;U; ');X = f1; 2; 3; 4; 5g;U = fa; b; cg;' : X â U ! X

ComputerScience

ControlTheory

Finitestate

machines

Continuous dynamical

systemssystemu(t) y(t)

x 2 Rn; u 2 Rm

y 2 Rp

Hybrid Systems in Control - MotivationHybrid Systems in Control - Motivation• Switches occuring naturally

because plant operates in different modes

• Switches introduced by controller

to accommodate constraints: anti-windup, MPC to implement sequence: PLC

•• Switches introduced by controller: Switches introduced by controller: Model Predictive Control (MPC) Model Predictive Control (MPC)

Theorem: The solution of the MPC problem yields a

piecewise affine state feedback law.

(Bemporad, Morari, Dua, Pistikopoulos, 2000)

Example: y =s 2 + s + 2

s + 1u T s = 0:2

à 16 u6 1 x1 ; x2> à 0 :5

• Switches introduced by controller: MPC• Explicit MPC = PWA controller

x2

x1

#5

#1#2

#3

#4

u=

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

¡ 1:0000 if

" 0:2425 0:00000:0000 0:2425¡2:5336 ¡1:3548¡2:4411 0:55700:0000 ¡2:0000

#x ·

"1:00001:0000¡1:00001:00001:0000

#(Region #1)

[¡4:3828 1:0000]x¡ 2:7954 if

" 0:0000 0:2425¡2:0000 0:00000:6615 ¡0:8424¡1:1548 0:26352:4411 ¡0:5570

#x ·

" 1:00001:0000¡1:00001:0000¡1:0000

#(Region #2)

[¡2:5336 ¡1:3548]x if

"¡0:6615 0:8424¡2:5336 ¡1:35482:5336 1:3548¡2:0000 0:00000:0000 ¡2:0000

#x ·

"1:00001:00001:00001:00001:0000

#(Region #3)

1:0000 if

· 0:0000 ¡2:00002:5336 1:3548¡0:6659 ¡1:7922¡2:0000 0:0000

¸x·

·1:0000¡1:00001:00001:0000

¸(Region #4)

• Closed-loop MPC





• Switches introduced by model simplification

to realize model hierarchy: approximatelower level dynamics by switches





• Switches introduced by model simplification

to realize model hierarchy: approximatelower level dynamics by switches

Modeling FrameworkModeling Framework

• Complex enough to be practically important

• Simple enough to allow analysis and synthesis

"Things should be made as simple as possible but not simpler" ….Einstein

x(t + 1) = Aix(t) +Biu(t) + fi

y(t) = Cix(t) + gi

Lix(t) +Miu(t) ô Ni

8>>><>>>:

Discrete Time Piecewise Affine Systems

Modeling FrameworkModeling FrameworkToo restrictive?

Piecewise Affine Systems, equivalent to:

•Mixed Logic Dynamical Systems•(Extended) Linear Complementarity Systems•Max-Min-Plus -Scaling Systems

(Heemels, De Schutter, Bemporad, 2000)

(Each framework has its advantages)

May be too general, …..

Hybrid SystemsHybrid SystemsHybrid Control SystemsHybrid Control Systems Switched (PWA) SystemsSwitched (PWA) Systems

Sastry, Lygeros, Tomlin, Godbole, PappasAlur, Pnueli, Maler, Henzinger, Krogh, ...

Sontag, Branicky, Johansson, Rantzer, Morse, Hespanha, van der Schaft, Tsitsiklis, Blondel, ...

xç =úfi(t; x; u) if x 2 Ri

i = 1; . . .; N

xç =úAix + Biu + fi if Hix ô Ki

i = 1; . . .; N

automaton / logicautomaton / logicsymbols δ0symbols δi

continuous dynamical system

continuous dynamical systemcontinuous

statesinputs

A/DD/A

Truth value operator:

From Algebraic Equalities to From Algebraic Equalities to MixedMixed--Integer Linear InequalitiesInteger Linear Inequalities

Xb c 2 f0;1gP(X1; . . .; Xn)b c = 1 Aîöô B

îö= î1; . . .; în[ ]0 2 f0; 1gn

z = îxî 2 f0; 1gx 2 [m;M]

ú z ô Mîz õ mîz ô x àm(1 à î)z õ x àM(1 à î)

8><>:

x ô 0b c = î x ôM(1 à î)x õ ï+ (m + ï)î

ú

Mixed product

Propositionallogic

Threshold condition

Algebraic equalities MI linear inequalities

(Williams, 1977)

(Glover, 1975)(Witsenhausen, 1966)

MLD Hybrid ModelsMLD Hybrid Models

x(t + 1) = Ax(t) +B1u(t)y(t) = Cx(t) +D1u(t)

Mixed Logical Dynamical (MLD) form

+B2î(t) +B3z(t)+D2î(t) +D3z(t)

E2î(t) + E3z(t) ô E4x(t) + E1u(t) + E5

(Bemporad, Morari, Automatica, March 1999)

• Well-Posedness Assumption :

are single valuedWell posedness allows defining trajectories in x- and y-space

î(t) = F(x(t); u(t))z(t) = G(x(t); u(t))

fx(t); u(t)g ! fx(t + 1)gfx(t); u(t)g ! fy(t)g

x; y; u = ?c?`

ô õ; ?c 2 Rnc; ?` 2 f0; 1gn`; z 2 Rrc; î 2 f0; 1gr`

Major Advantage of PWA/MLD Framework

All problems of analysis:• Stability• Verification• Controllability / Reachability• Observability

All problems of synthesis:• Controller Design• Filter Design / Fault Detection & Estimation

can be expressed as (mixed integer) mathematical programmingproblems for which many algorithms and software tools exist.However, all these problems are NP-hard.

MLD/PWAMLD/PWAHybrid SystemsHybrid Systems

• Reachability / Verification• Stability• Observability

• Control (MPC)• Explicit PWA MPC controllers• State estimation (MHE)/fault detection

• Car suspension system• Gas supply system • Hydroelectric power plant ...

• HYSDEL

• Identification

Research TopicsResearch Topics(Bemporad, Borrelli, Ferrari-Trecate, Mignone, Torrisi, Morari)

AnalysisSynthesis

ApplicationsModeling

HYbridHYbrid System Description System Description LAnguageLAnguage (HYSDEL) (HYSDEL)

HYSDEL Model

MLD + PWA Model

Process

Controller Design Reachability Analysis

Filter Design Stability Analysis

• Planned integration with CHECKMATE (CMU)

Identification of Hybrid systemsyk uk[ ]2C1yk uk[ ]2C2yk uk[ ]2C3

yk+1 =

0:9 0:2 0[ ] yk uk 1[ ]0

0:5 0:4 2[ ] yk uk 1[ ]0

0:3 à0:3 à5[ ] yk uk 1[ ]0

8>>>><>>>>:

Model and datapoints

Problem:Identify a piecewise ARX model from a finite set of noisy measurements.

Useful when the switches between different submodels cannot be measured

The estimation of the submodels cannot be separated from the problem of estimating the regions

Identification Algorithm Identification Algorithm Exploit the combined use of

• clustering ⇒ “K-means” like procedure• linear identification ⇒ weighted least squares• classification ⇒ linear support vector machines

Model and datapoints Estimated model and classified datapoints

G. Ferrari-Trecate, M. Muselli, D. Liberati, M. Morari,A Clustering Technique for the Identification of Piecewise Affine Systems, HS2001, Section FA

Dialysis Therapy

• Blood urea concentration is measured

• Bi-exponential dynamic (Liberati et. al., 1993)

- First part (30-40 minutes)Fast decrease

- Second part (3-4 hours) Slow decrease

An early estimation of both the time constants and the switching timeallows the assessment of the total duration of the therapy

Fast dynamics

Slow dynamics

Dialysis Therapy

Take the log of the data

Piecewise Affineapproximation

Estimation of thetime constants

The switching time cannot be measured directly

Depends on both the patient physiologyand the clearance rate of the dialyzer

EEG Analysis

"letter composing"task

"multiplication"task

Problem: discriminate the presence of different mental tasks from EEGProposition: EEG in a single mental "state" ≈ AR model of low order

The switch between mental states cannot be measured

Hybrid identification

(C. Anderson et al., 1995)

Application of EEG Analysis:

Brain computer interfacing

• High inter-subjects and intra-subjects variability of EEG

Need to update models easily

• Biofeedback: the subject can be forewarned that he is changing mental state

Epileptic patients: Early seizure detection

• Prompt intervention against epilepsy crisis

EEG Analysis for Brain-Computer Interface

The MITs Technology Review magazine recently listed brain-machine interfaces as one of the 10 emerging technologies that will "soon have a profound impact on the economy and on how we live and work."





• HYSDEL

• Identification


AnalysisSynthesis


Analysis Analysis vsvs. Synthesis. Synthesis

Control of stable system with input constraints

• Analysis of closed loop stability conservative / difficult

• Synthesis of feedback controllers with stability guarantee industrial routine

Laptop Computer

• Analysis: 101020 states 10100 atoms in universe (Wm. A. Wulf, President NAE)

Hybrid Systems Control ReviewHybrid Systems Control Review

• Piecewise Quadratic/Linear Lyapunov functions Linear Matrix Inequalities to characterize stability and performance

• Pontryagin Maximum Principle

• Hamilton Jacobi Bellman equation

• Parametric Programming

• …….

Bemporad, Berardi, Boyd, Borrelli, Branicky, Buss, Burlirsch, De Santis, Di Benedetto, Hassibi, Hedlund,Johansson, Kratz, Lygeros, Mitter, Piccoli, Rantzer, Riedinger, Sastry, Styrk, Sussmann, Tomlin, Zann

Receding Horizon ControlReceding Horizon Control

Predicted outputs

Manipulated (t+k)uInputs

t t+1 t+m t+p

futurepast

t+1 t+2 t+1+m t+1+p

• Optimize at time t (new measurements)

• Only apply the first optimal move u(t)

• Repeat the whole optimization at time t +1

• Advantage of on-line optimization Õ FEEDBACK

• Objective: determine the optimal input sequence

driving the system from to ,

compatibly with the limits on

• Apply the first input move according to RHC

• Repeat the optimization at time t +1

Model Predictive ControlModel Predictive Control

VU� � � �� VU�L YU Y�

VU�L� YU�LkU

6 gVU� VU�� VU�/Vh

TVCKFDU UP VNJO � VU�L � V

NBY

YNJO � YU�LkU � Y

NBY

+PQU �NJO

6

+6�YU QL��

/ l2YU�LkU�Y�l� l3VU� Ll

TZTUFN EZOBNJDT

VU

• Linear Model and 2-norm performance index

Õ Quadratic Program

• Linear Model and ∞-norm performance index

Õ Linear Program

• Hybrid Model and ∞-norm performance index

Õ Mixed Integer Linear Program

Model Predictive ControlModel Predictive Control



NBY


NBY

+PQU �NJO

6

+6�YU QL��


TZTUFN EZOBNJDT

MPC - SolutionMPC - Solution



NBY


NBY

+PQU �NJO

6

+6�YU QL��


TZTUFN EZOBNJDT

(Bemporad et. Al., 2000; Bemporad, Borrelli, Morari, CDC 2000 TuM05-1, WeM01-1)

Theorem: The solution of the MPC problem yields apiecewise affine state feedback law

• The solution of the MPC can be computed explicitly

• Off-line optimization: optimize for all x(t)

MPC - Explicit SolutionMPC - Explicit Solution



NBY


NBY

+PQU �NJO

6

+6�YU QL��


TZTUFN EZOBNJDT

Explicit solvers for QP, LP and MILP are available(Bemporad et. Al., 2000; Bemporad, Borrelli, Morari, CDC 2000 TuM05-1, WeM01-1)

Õ Multi-parametric Program

MPC for MLD SystemsExample:Alternate Heating of Two Furnaces

(Hedlund,Rantzer CDC1999)

xç = à 1 00 à 2

ô õx(t) +Biu0

Bi =

10

ô õif first furnace heated

01

ô õif second furnace heated

00

ô õif no heating

8>>>>>><>>>>>>:

• Objective:

• Control the Temperature of Two

Furnaces

• Constraints:

•Switching Control between three

operation modes:1- Heat only the first furnace

2- Heat only the second furnace

3- Do not heat any furnaces

Amount of energy u0 fixed

Can be parametrized !

• MLD system

• mp-MILP optimization problem

•inside the region:

• Computational complexity of mp-MILP

Alternate Heating of Two FurnacesAlternate Heating of Two Furnaces

minv20

n o J(v20; x(t)),P

k=02 kR(v(k + 1) à v(k))k1 + kQ(x(kjt) à xe)k1

u(k) =1 0 0[ ] if first furnace heated0 1 0[ ] if second furnace heated0 0 1[ ] if no heating

8<:

State x(t) 3 variables

Input u(t) 3 variables

Aux. binary vector ±(t) 0 variables

Aux. continuous vector z(t) 9 variables

1 ô x1 ô 11 ô x2 ô 10 ô u0 ô 1 linear constraints 168

continuous variables 33

binary variables 9

parameter variables 3

time to solve the mp-MILP 5 min

Number of regions 105

X1

X2

Heat 1

Heat 2

No Heat

mpmp-MILP Solution-MILP Solutionu0=0.4 u0=0.8

X1

X2

Heat 1

Heat 2

No Heat

X1

X2

X1

X2

Characteristics of the SolutionCharacteristics of the Solution

u(k) = Fkx(k) +Gk , x(k) 2 Xk; Xk = fxjLkx ô Mkg k = 0; . . .; N:

• Piecewise affine control law, polyhedral regions

• Simultaneous and automatic partitioning and control law synthesis

• Stability guarantee (PWL Lyapunov function)

• On-line implementation does not require storage of all Xk





• HYSDEL

• Identification


AnalysisSynthesis


• Traction control (Ford Research Center )

• Gas supply system (Kawasaki Steel )

• Batch evaporator system (Esprit Project 26270 )

• Anesthesia (Hospital Bern )

• Hydroelectric power plant ( )

• Power generation scheduling ( )

ApplicationsApplications

• must have priority

• in range rather than tracked

• aggressive action upon constraint violation

• constraints prioritization:

Analgesia Control during AnesthesiaAnalgesia Control during AnesthesiaClinicalClinical Goals Goals

��

��

�

�

�

�

1. hypotension

2. overdosing

3. hypertension

4. underdosing

��

��

��

��

�

�

• Explicit MPC Control horizon 3 Dimension 8Prediction horizon 10 weight 150Number of regions 127 weight 1

Controller ImplementationController Implementation

��

Case Study ICase Study I

Case Study IIICase Study III

In the Operating RoomIn the Operating Room

1. Induction

2. Artifact detection

3. Intense stimulation

1. 2.

3.

• Traction control (Ford Research Center )

• Gas supply system (Kawasaki Steel )

• Batch evaporator system (Esprit Project 26270 )

• Anesthesia (Hospital Bern )

• Hydroelectric power plant ( )

• Power generation scheduling ( )

ApplicationsApplications

Optimization of Combined Cycle Power Plants

Air

C

Fuel

CC

Condensate

TG HRSG

ProcessPP

Recycled water

Factory

Exhausts

TV

Deregulated energy market- electricity/gas demands and

prices change rapidly

Optimize the plant hourly

Optimization

Maximize the profit while fulfilling the operating constraints

(ETH-ABB joint project)

Topologies of Combined Cycle Power Plants

Simple plant:

• Two turbines and two binary inputs (on/off commands)

• Several gas/steam turbines, firings ...

Other plants:

The complexity of the example is scalable !

Optimization of Combined Cycle Power Plants

• Gas/Steam turbines can be switched on/off

• Different types of startup procedures

• Minimum up/down times

• Normal operation: Dynamicsof the energy/steam production

• Constraints on the productioncapabilities

Hybrid model

LogicContinuous

Economic optimization ⇒ Predictive control for hybrid systems

ConclusionsConclusions

• Hybrid system models are important

• Control Theory ⇔ Computer Science

• Computations must be an inherent part of any new theory

Mathematical Programming Approach to Hybrid Systems ...cas.ensmp.fr/~levine/MorariTransp.pdf · • Car suspension system • Gas supply system • Hydroelectric power plant... •

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