outline multiparametric linear programming with

Automatic Control Laboratory, ETH Zürichwww.control.ethz.ch

Multiparametric Linear Programmingwith

Applications to Control

Manfred Morari

Colin Jones, Miroslav Baric, Melanie Zeilinger

Outline

1. Motivation

2. Geometry of Parametric Programming

3. Approximate Controllers

4. Industrial Applications

Motivation

ControllerComplexity

ControlPerformance

Motivation

Why is complexity important?

• Design / maintenance effort– Process industries

• Implementation cost (hardware)– Automotive– Electronics– Electrical

DC/DC Converterfor Mobile Phones

ComplexityPe

rfor

man

ce

Current

Target Performance

MPC

Goal: Minimize complexity for given performance

Complexity=

Grams silicon=

Torque Control of anInduction Machine

Complexity

Perf

orm

ance

CurrentAvailableProcessor

Goal: Maximise performance for specific processor

Specific Example:Optimal Control Synthesis

Given a performance index

u = f(x)

Plantcontrolaction u

plantstate x

Compute control action

Obtain U*(x)

Plant output y

plant state xcontrol u0*

Receding Horizon ControlOn-Line Optimization

Optimization Problem

Model Predictive Control (MPC)

ParametricOptimization

Explicit Solution

Plant output y

plant state xcontrol u*

(=Look-Up Table)

Receding Horizon ControlOff-Line Optimization

off-line

Seron, De Doná and Goodwin, 2000Johansen, Peterson and Slupphaug, 2000Bemporad, Morari, Dua and Pistokopoulos, 2000

Explicit MPC

Solution ofBellman equation

Explicit MPC

Explicit MPC Controller Properties

Explicit MPC control law = MPC control law

Guarantees using MPC design techniques• Stability• Feasibility• Invariance

Pros– Easy to implement– Fast on-line evaluation– Analysis of closed-loop system possible

Challenges– Number of controller regions can become large– Computation time may become prohibitive– Numerics

Multi-parametric controllers Online vs. Offline Complexity

Com

puta

tion

Tim

e

Complexity ofonline optimization

log(Problem Complexity)Complexity ofexplicit controller

Optimality Trade-off

Complexity

Perf

orm

ance

Explicit

MPC

Goal: Trade-off performance and complexity

Research Directions1. Reduce complexity of online optimization

– Rao, Wright, Rawlings, 1998– Diehl, Björnberg, 2004– Wang, Boyd, 2007

2. Reduce complexity of explicit solution(i.e., number of regions)– Jones, Baric, Morari, 2007– Lincoln, Rantzer, 2006– Bemporad, Filippi, 2004– Johansen, Grancharova, 2003

3. Combination of 1. and 2.– Panocchia, Rawlings, Wright, 2006– This presentation

Outlook

Complexity

Perf

orm

ance

Explicit

Goal: Trade-off performance and complexity

Outline

1. Motivation




MPC → Parametric Programming

Primal

p = 1,1

MPC → Parametric Programming

Primal

p = 1,1

Parameter = state

Geometry of mpLP : PrimalPrimal

u

x

P


u

x

P

Feasible inputs for givenstate x0

x0

P


u

x

J

c0u

x0

Cost of applying inputs for givenstate x0


x

J

Optimal cost and input forstate x0

x0

P

u

Goal : Minimize J

P


u

x

J

Cost/control affine in each regionCompute optimal input for each state

Goal: Compute facetsof cost function

Parametric = Facet/Vertex EnumerationPrimal Dual

Facet Enumeration ofImplicit polyhedron

Vertex Enumeration ofImplicit polyhedron

Current pLP AlgorithmsGoal: Compute the vertices (facets) of implicitly

defined polyhedronBased on classic vertex/facet enumeration

Primal : Gift Wrapping Dual : Pivoting













mpLP in ControlNumber of facets/regions…

– online computation time– storage– offline processing time

… can grow very quickly with problem size

Current algorithms are non-incremental– output meaningless until complete– cannot stop early

Outline

1. Motivation




Approximation as Separation

J*(x)

Goal: Compute ‘simpler’ approximation of J*Without computing J¤!

Approximation as SeparationFind function such that :• Stability :• Invariance :• Performance :

J*(x)

Related work : Lincoln and Rantzer 2006

Approximation as SeparationFind function such that :• Stability :• Invariance :• Performance :

All separatingfunctions are

stabilizing

Level setsare invariant

Polyhedral SeparationGiven polyhedra S µ P find minimum complexitypolyhedron Q such that S µ Q µ P

P

Q

S

Polyhedral SeparationApplications:1. Geometric surface approximation2. Multi-resolution transmission3. Polyhedral function approximation4. … P

Q

S

Classic method for vertex enumeration

Rediscovered many (many) times

Goal: Compute the convex hull of a set of points

Idea: Build convex hull iteratively by addingpoints one at a time

Beneath/Beyond

Fourier1824

Motzkin1953

Kallay1981

Jones, Baric, Morari, 2007

Parametric Linear Optimisation

How to recover the approximate control law?

Barycentric Interpolation

Weight vertices based onintegral of the surface area ofthe polar dual (Stokes Law).

Goal: Given set of vertices V andfunction f, compute a smoothfunction g such that

1. g(v) = f(v) for all v in V2. Domain of g = conv(V)3. (v,g(v)) 2 conv({vi,f(vi)})

Schaefer, 2007

Beneath/Beyond in ControlBenefits1. Constructive procedure2. Stop at a given complexity (very nice)3. Stabilizing and invariant4. Tradeoff complexity with region of attraction

and/or performance5. Optimal solution in finite time

Jones, Baric and Morari, 2007

Approximate Solutions

• Random system• 808 regions

-8 -6 -4 -2 0 2 4 6 8 10-2

0

2

4

6

8

10

Approximate Solutions

• 808 40 regions• Stabilizing• Small performance loss

-8 -6 -4 -2 0 2 4 6 8 10-2

0

2

4

6

8

10

Approximate Solutions: Example

12128 357 regions

Beneath/Beyond in ControlLimitations1. Approximation >> optimal (exponential)2. Compute vertices and halfspaces (exponential)3. B/B dislikes degeneracy (exponential)4. Greedy approximation (not optimal)

Great for some, rather less so for others!

Jones, Baric and Morari, 2007

Extensions of theory and computationControl :• System type

– Switched linear / piecewise affine– Uncertainty in inputs and/or parameters

• Performance Objective– Quadratic– Min / max– Infinite horizon

• Mixed online/offline computation– Warm start with online complexity guarantees– Continuous trade-off between online/offline effort

Optimal tradeoff

Perf

orm

ance

Arithmetic Operations

Combination of online and offline methods to trade-offperformance and complexity

Onlineoptimization

(Approximate)Explicit solution

Online optimizationwith warmstart

Online/Offline Tradeoff: ExampleOnline Optimization:

0 0.5 1 1.5 2 2.5

x 106

0

1

2

3

4

# arithmetic operations

Stability Bounderro

r

Phase 1

Simplex

Worst case assessed by sampling

0 5 10 15

x 105

0

1

2

3

4

erro

r


Stability Bound

Simplex

Beneath/Beyond

extrapolated

practically solvable

Online/Offline Tradeoff: ExampleApproximate Explicit Solution:


0 5 10 15

x 105

0

1

2

3

4

erro

r


Stability Bound

Simplex

Beneath/Beyond

Online/Offline Tradeoff: ExampleOnline optimization starting from approximate explicitsolution:

Active Set

optimal

Combination between online and offline solutionoutperforms explicit solution


0 5 10 15

x 105

0

1

2

3

4

erro

r


Stability Bound

Simplex

Beneath/Beyond

Online/Offline Tradeoff: ExampleOnline optimization starting from approximate explicitsolution:

Active Set

0 1 2 3 4 5 6 7

x 105

0

0.2

0.4

0.6

0.8

1

erro

r


Stabilizing region

offline

online/

offline

offline online/offline

0 0.2 0.4 0.6 0.8 110

3

104

105

106

107

θ=online/offline tradeoff

optimal

# ar

ithm

etic

ope

ratio

ns

Online Offline

Tradeoff map: Example

Look for the minimum

Get best tradeoff

0 0.2 0.4 0.6 0.8 110

3

104

105

106

107

optimal

# ar

ithm

etic

ope

ratio

ns

Online Offlineθ=online/offline tradeoff

Tradeoff map: Example

errorlevel curves

Best tradeoff

Parametric Programming and ControlParametric linear programming (many, many more)1950s : Orchard-Hays, …1970s : Klatte, Kummer, …1990s- : Baotić, Barić, Bemporad, Borelli, Filippi, Gal,

Johansen, Jones, Murty, Spøtvold, Terlaky, Tøndel, …

Explicit MPC1990s- : Baotić, Barić, Bemporad, De Dona, Dua, Dumur,

Filippi, Goodwin, Grieder, Johansen, Jones, Kerrigan, Kvasnica,Lincoln, Löfberg, Maciejowski, Mayne, Morari, Olaru, Parrilo,Pistikopulous, Raković, Rantzer, Sakizlis, Schroeder, Seron,Tøndel, …

• All results were obtained with the MPT toolbox

http://control.ethz.ch/~mpt

• MPT is a MATLAB toolbox that provides efficientcode for– (Non)-Convex Polytopic Manipulation– Multi-Parametric Programming– Control of PWA and LTI systems

Conclusions

MPT in the World

10000+ downloads in 5 years

Outline

1. Motivation

2. Geometry of Parametric Programming– Focus on linear



Switch-mode DC-DC Converter

d

unregulated DC voltage low-pass filter load

dually operated switches

Switched circuit: supplies power to load with constant DC voltage

Illustrating example: synchronous step-down DC-DC converter

regulated DC voltage

S1

S2

Operation Principle

k-th switching period

duty cycle

• Length of switching period Ts constant (fixed switching frequency)

• Switch-on transition at kTs, k2N • Switch-off transition at (k+d(k))Ts (variable pulse width)

• Duty cycle d(k) is real variable bounded by 0 and 1

S1 = 1S2 = 0

S1 = 0S2 = 1

kTs (k+1)Ts(k+d(k))Ts

S1 = 1S2 = 0

S1 = 0S2 = 1

kTs (k+1)Ts(k+d(k))Ts

duty cycle d(k)

Mode 1 and 2

mode 1: mode 2: d

S1

S2

Control Objective

unregulated DC input voltagedisturbance

duty cyclemanipulated variable

regulated DC output voltagecontrolled variable

inductor currentstate

capacitor voltagestate

Regulate DC output voltage by appropriate choice of duty cycle

Control Objectives

Minimize (average) output voltage error andchanges in duty cycle

Respect constraint on current limit

Translate in Receding Horizon Control (RHC) problem

State-feedback Controller:Polyhedral Partition

Colors correspond to the 121 polyhedra

PWA state-feedbackcontrol law:

computed in 100s using theMPT toolbox

121 polyhedra after simplificationwith optimal merging

Start up: Experiment (1/2)

Inductor current

Output voltage

330mA

2V

Time1msec

6V

~1A

660mA

Input voltage @10V, Output voltage reference @6V, Current limit @1A

Experiments at U. Sannio: Papafotiou, Frasca, Vasca, Borrelli

Start up: Experiment (2/2)

Time1msec

~1A

660mA

Input voltage @10V, Output voltage reference @6V, Current limit @1A

Duty Cycle1

05V

Inductor current

330mA

Experiments at U. Sannio: Papafotiou, Frasca, Vasca, Borrelli

Direct Torque Control

Physical Setup:• Three-level DC link inverter

driving a three-phase symmetricinduction motor

• Binary control inputs

Control Objectives:• Keep torque, stator flux and neutral point potential

within given bounds• Minimize average switching frequency (losses)

Reduction of switching frequency by up to 45 % (in average 25 %)with respect to ABB’s commercial DTC scheme (ACS 6000)

Physical Setup:• Cascade of hydroelectric power plants• Turbines and weirs (adjustable water flow) • Lock operation disturbances

Control Objectives:• Respect constraints on concession level• Minimize flow changes at weirs and turbines• Maximize income from power generation

River Power Plants

Optimal coordination between plants leads to 1.) Major damping of upstream water flow disturbances; 2.) Strict respect of concessions level constraints

Electronic Throttle ControlPhysical Setup:• Valve (driven by DC motor) regulates air inflow to the car engine• Friction nonlinearity• Limp-Home nonlinearity• Physical constraints

Control Objectives:• Minimize steady-state regulation error• Achieve fast transient behavior without overshoot

Systematic controller synthesis procedure. On average twice asfast transient behavior compared to state-of-the-art PID

controller with ad-hoc precompensation of nonlinearities.

Traction ControlPhysical Setup:• Improve driver's ability to control vehicle under adverse external conditions (wet or icy roads)• Tire torque is nonlinear function of slip• Uncertainties and constraints

Control Objectives:• Maximize tire torque by keeping tire

slip close to the desired value

Experimental results: 2000 Ford Focus on a Polished Ice Surface;Receding Horizon controller with 20 ms sampling time

Tire Slip

Tire

Tor

que

Piecewise affineapproximation

Adaptive Cruise ControlPhysical Setup:

Control Objectives:• Track reference speed• Respect traffic rules• Consider all objects on all lanes

Optimal state-feedback control law successfully implemented andtested on a research car Mercedes E430 with 80ms sampling time

Longitudinal and LateralControl

Sensors: IRScanner,Cameras, Radar

TestVehicle

Traffic Scene

Virtual TrafficScene

Smart Damping Materials

– Device suppresses vibration– External power source for operation is not required– Weight and size of the device have to be kept to a

minimum

• Demands

• Idea– Switched Piezoelectric (PZT) Patches

• Problem– What is the optimal switching law for optimal vibration suppression?

PZTPZT

Niederberger, Moheimani

Control of Anaesthesia

Physical Setup:• Patient undergoing surgery• Analgesic infusion pump

Control Objectives:• Minimize stress reaction to

surgical stimulation(by controlling mean arterial pressure)

• Minimize drug consumption

Excellent performance of administration scheme,mean arterial pressure variations kept within bounds

Control of Thermal Print-HeadsPhysical Setup:• Thermal print-head

with ~ 1400 heat elements• Binary control inputs• Printing on a wide range

of materials

Control Objectives:• Maximize printout quality• Achieve robustness to parameter variations

90% quality gain over traditional controllers [ANSI X3.182-1990];Straight-forward design method for print-head controller

Conclusions

• Control synthesis tools emerging fromconfluence of mathematical programming,computational geometry and control theory

• Tools for constrained switched linear systemsaffecting industrial applications

• Performance/complexity trade-off importantfor cost sensitive application domains arebeing addressed

outline multiparametric linear programming with

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