Members: Mato Baotic

Francesco Borrelli

Tobias Geyer

Pascal Grieder

Frank Christophersen

Fabio Torrisi

Affiliates: Alberto Bemporad (Univ. of Siena)

Giancarlo Ferrari-Trecate (INRIA)

Control and ComputationControl and Computation

ETH ActivitiesETH Activities

Manfred Morari

ETH Zürich

Control EngineeringGroup

Walter SchaufelbergerAdolf H. Glattfelder

Franta J. Kraus

System Dynamics & Control Group

Manfred Morari

AdministrationInfrastructure

Teaching Program

FachgruppenFachgruppen

ETH Zürich

People (November 2000)People (November 2000)

FTE

Professors 2 (+1) 2

Researchers 11 10

Graduate Students 14 14

Visitors 3 3

Administrative/Technical Staff 7 4.6

TOTAL 37 33.6

ETH Zürich

System Dynamics & ControlSystem Dynamics & ControlBiomedical Systems

- Rehabilitation Engineering Popovic, Jezernik, Keller, Pappas

- Anesthesia Glattfelder, Gentilini, Stadler

Active Noise and Vibration Control, Kaiser, Seba

Intelligent Materials

Nonlinear Dynamics of Distillation Columns Bonanomi, Dorn, Ulrich

Hybrid Systems Bemporad, Borrelli, Cuzzola, Ferrari, Kojima,

- Analysis: Controllability, Verification, Stability, ... Mignone, Torrisi

- Synthesis: Control, Estimation, Fault Detection

- Applications

Model Development for Control Pearson

- Nonlinear Model Reduction

- Nonlinear System Identification

Role of ETH in CCRole of ETH in CC

• Controller Synthesis via Optimization

• ABB Case Study

• Coordination of Case Studies

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.

Finite Time Optimal ControlFinite Time Optimal Control

Predicted outputs

Manipulated (t+k)u

Inputs

t t+1 t+m t+p

futurepast

Jopt =minUJ(U,x(t)) :=P

k=0

N kQ(x(t+ k|t)àxö)k+ kRu(t+ k)k

U := {u(t), u(t+1), . . ., u(t+Nu)}

subject to umin ô u(t+k) ô umax

xmin ô x(t+k|t) ô xmax

system dynamics

Model Perf. index Method

Linear Quadratic ProgramLinear Linear ProgramHybrid Mixed Integer Quadr. ProgramHybrid Mixed Integer Linear Program

Finite Time Optimal ControlFinite Time Optimal Control

U := {u(t), u(t+1), . . ., u(t+Nu)}

subject to umin ô u(t+k) ô umax

xmin ô x(t+k|t) ô xmax

Jopt =minUJ(U,x(t)) :=P

k=0

N kQ(x(t+ k|t)àxö)k+ kRu(t+ k)k

system dynamics

k á k∞k á k2k á k∞

k á k2

Feedback Control SolutionFeedback Control Solution

U := {u(t), u(t+1), . . ., u(t+Nu)}

subject to umin ô u(t+k) ô umax

xmin ô x(t+k|t) ô xmax

Jopt =minUJ(U,x(t)) :=P

k=0

N kQ(x(t+ k|t)àxö)k+ kRu(t+ k)k

system dynamics

Theorem: The solution of the finite time optimalcontrol problem yields a piecewise affinestate feedback control law

Computation of Feedback Control LawComputation of Feedback Control Law

U := {u(t), u(t+1), . . ., u(t+Nu)}

subject to umin ô u(t+k) ô umax

xmin ô x(t+k|t) ô xmax

Jopt =minUJ(U,x(t)) :=P

k=0

N kQ(x(t+ k|t)àxö)k+ kRu(t+ k)k

system dynamics

• The feedback controller can be computed explicitly

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

Multi-parametric Program

Solvers for LP, QP, MILP and MIQP? are available

• 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

Current and Future ProjectsCurrent and Future Projects

• Algorithm Improvements (mpMIQP)

• Controller Characterization and Storage

• Controller Simplification- Order reduction

- Decentralized structure

• Robustness