Introduction to Model Predictive Control (MPC)maapc/static/files/CACSD/... · 2017-05-15 · Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 3 Some

Introduction to Model Predictive Control (MPC)

Oscar Mauricio Agudelo Mañozca

Computergestuurde regeltechniek

Course :

Bart De Moor

ESAT - KU Leuven

May 11th, 2017

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 2

Basic Concepts

Control method for handling input and state constraints within an optimal control setting.

It handles multivariable interactions

It handles input and state constraints

It can push the plants to their limits of performance.

It is easy to explain to operators and engineers

Principle of predictive control

1k 2k 3k

Prediction horizon

k Nk1k 2k

( )u k

( )y k

Prediction of ( )y k

Reference

FuturePast

measurement

ref y

2

ref( ), , ( 1)

1

min ( )N

u k u k Ni

y y k i

subject to

model of the process

input constraints

output / state constraints

Why to use MPC ?

time


Some applications of MPC

Control of synthesis section of a urea plant

MPC strategies have been used for stabilizing and maximizing the throughput of thesynthesis section of a urea plant, while satisfying all the process constraints.

Urea plant of Yara at Brunsbüttel (Germany), where a MPC control system has been set by IPCOS



Control of synthesis section of a urea plant

Results of a preliminary study done by

Throughput increment of 11.81 t/h thanks to the MPC controller

2NH3 + CO2 NH2COONH4

AmmoniaCarbon dioxide

Ammonium carbamate

Reaction 1: Fast and Exothermic

NH2COONH4 NH2CONH2 + H2O

Ammonium carbamate

Urea Water

Reaction 2: Slow and Endothermic



Flood Control: The Demer

The Demer in Hasselt

Flooding events due to heavy rainfall:

1905, 1926, 1965, 1966, 1993-1994,

1995, 1998, 2002 and 2010.

The Demer and its tributaries in the south of the province of LimburgControl Strategy: PLC logic

(e.g., three-pos controller)

A Nonlinear MPC control strategy has been implemented (2016) for avoiding futurefloodings of the Demer river in Belgium. Partners: STADIUS, Dept. Civil Engineering of KULeuven, IPCOS, IMDC, Antea Group, and Cofely Fabricom.




Flooded area during the flood event of 1998. Control Strategy: PLC logic (e.g., three-position controller)

DIEST

HASSELT




Upstream part of the Demer that is modelledand controlled in a preliminary study carriedout by STADIUS

Maximal water levels for the

five reaches for the current

three-pos. controller and the

MPC controller together with

their flood levels (Flood event

2002) .

Notice: The MPC controller takes rainpredictions into account!



In addition MPC, has been used

in all sort of petrochemical and chemical plants,

in food processing,

in automotive industry,

in the control of tubular chemical reactors,

in the normalization of the blood glucose level of critical ill patients,

in power converters,

for the control of power generating kites under changing wind conditions,

in mechatronic systems (e.g., mobile robots),

in power generation,

in aerospace,

in HVAC systems (building control)

…

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

Basic Concepts

9

Kinds of MPC

Linear MPC : it uses a linear model of the plant

Convex optimization problem

Nonlinear MPC: it uses a nonlinear model of the plant

Non-convex optimization problem

( 1) ( ) ( )k k k x Ax Bu

( 1) ( ), ( )k k k x f x u

Linear MPC formulation (Classical MPC)

Remark: Since linear MPC includes constraints, it is a non-linear control strategy !!!

1

ref reref ref

1

f,

0

( ) ( )min ( ) ( )( (( ) ) ( ))N N

NNT

ii

Tk i k i k i k k i k i k i k ii

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1,k i k i k i i N x Ax Bu

max( ) , 0,1, , 1,k i i N u u

max( ) , 1,2, , ,k i i N x x

Model of the plant

Input constraints

State constraints

( 1); ( 2); ; ( ) ,N k k k N x x x x ( ); ( 1); ; ( 1)N k k k N u u u uwith:


MPC Algorithm

10

Typical MPC control Loop

MPC Algorithm

At every sampling time:

MPC Plant

Observer

( )ku

( )ky

ˆ( )kx

ref ( )kx

ref ( )ku ZOH( )tu

( )ty

sT

Digital system

Read the current state of the process, x(k)

Compute an optimal control sequence by solving the MPC optimization problem

Apply to the plant ONLY the first element of such a sequence u(k)

( ), ( 1), ( 2), , ( 1)k k k k N u u u uSolution


LQR and Classical MPC

11

For simplicity, Let’s assume that the references are set to zero.

LQR CLassical Linear MPC

,1

( )min ( ) (( ) )TT

k

kk kk

x u

x Q R ux u

subject to

( 1) ( ) ( ), 0,1, ,k k k k x Ax Bu

subject to

( 1) ( ) ( ), 0,1, , 1,k k k k N x Ax Bu

max( ) , 0,1, , 1,k k N u u

max( ) , 1,2, , ,k k N x x

Constraints are not taken into account

No predictive capacity

The optimal solution has the form: ( ) ( )k k u Kx

PRO

Explicit, Linear solution

Low online computational burden

CON

PRO

Takes constraints into account

Proactive behavior

CON

High online computational burden

No explicit solution

feasibility? Stability?

If N∞, and constraints are not considered the MPC and LQR give the same solution

1

,01

( ) (min ( ) () )N N

NT

ik

NTk k k k

x u

u R ux Q xk = 0


MPC optimization problem – Implementation details

12

The following optimization problem,

can be rewritten as a LCQP (Linearly Constrained Quadratic Program) problem in as follows:

1

ref reref ref

1

f,

0

( ) ( )min ( ) ( )( (( ) ) ( ))N N

NNT

ii

Tk i k i k i k k i k i k i k ii

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1,k i k i k i i N x Ax Bu

max( ) , 0,1, , 1,k i i N u u

max( ) , 1,2, , ,k i i N x x

with: ( 1); ( 2); ; ( ) ,xn N

N k k k N

x x x x ( ); ( 1); ; ( 1)un N

N k k k N

u u u u

x

1min

2

T Tx

x Hx f x

subject to

e eA x b

i iA x b

with:

;x uN n n

N N

x x u


MPC optimization problem – Implementation Details

13

where

2x u x uN n n N n n

Q

QH

R

R

ref

ref

ref

ref

(1)

( )

(0)

( 1)

x uN n n N

N

x

xf H

u

u

2

i

x

xx u x u

u

u

n N

n NN n n N n n

n N

n N

I

IA

I

I

max

2 max

i

max

max

x uN n n

x

xb

u

u

N times

N times

2N times

2N times

nx = number of states, nu = number of inputs, N = prediction horizon


MPC optimization problem – Implementation Details

14

e

x

xx x u

x

n

nN n N n n

n

I B

A I BA

A I B

e

( )

xN n

k

Ax

0b

0N times

N times

nx = number of states, nu = number of inputs, N = prediction horizon

Remarks:

The problem is convex if Q and R are positive semi-definite

The hessian matrix H, and the matrices Ae and Ai are sparse.

Number of optimization variables: N(nx + nu). This number can be reduced to N·nu (Condensed

form of the MPC) through elimination of the states x(k) but at the cost of sparsity !!!

Matlab function for solving the LCQP optimization problem

x_tilde = quadprog (H, f, Ai, bi , Ae, be)


MPC with terminal cost

15

Short Prediction Horizon

Large Prediction Horizon

N

What happens?

We have a winner !!!

N

What happens?

We have an accident !!!

N should be large enough in order to keep the process under control



16

1

ref ref ref ref,

1 0

1

min ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )N N

NT T

i

N

i

k i k i k i k i k i k i k i k i

x u

x x Q x x u u R u u

subject to

( 1 ) ( ) ( ), 0,1, , 1,k i k i k i i N x Ax Bu

max( ) , 0,1, , 1,k i i N u u

max( ) , 1,2, , ,k i i N x x

ref ref( ) ( ) ( ) ( )T

k N k N k N k N x x S x x

Main goal of the terminal cost:

What do we gain?

To include the terms for which i ≥ N in the cost

function (To extend the prediction horizon to infinity)

“Stability”

( 1)k x( 2)k x

( )k Nx

Effect of the stabilizing

control law u(k) = -Kx(k)

1( )x k

2 ( )x k

ref ( ) 0k x

( )kx

( )k x

steady-state



17

How to calculate S ?

By solving the discrete-time Riccati equation,

How to carry out this calculation in Matlab?

1( )( ) ( )T T T T A SA S A SB B SB R B SA Q 0

where u(k) = -Kx(k) is the stabilizing control law. 1

.T T

K B SB R B SA

[K,S] = dlqr (A, B, Q, R )

Implementation details of the MPC with terminal constraint

The only change in the LCQP is the hessian matrix

2x u x uN n n N n n

Q

Q

H S

R

R

N - 1 times

N times

Bert Pluymers

Prof. Bart De Moor

Katholieke Universiteit Leuven, Belgium

Faculty of Engineering Sciences

Department of Electrical Engineering (ESAT)

Research Group SCD-SISTA

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction [email protected]

H0K03a : Advanced Process Control Model-based Predictive Control 1 : Introduction

1

Overview

• MPC 1 : Introduction

• MPC 2 : Dynamic Optimization

• MPC 3 : Stability

• MPC 4 : Robustness

• Industry Speaker : Christiaan Moons (IPCOS)

(november 3rd)

Signal processing

Identification

System Theory Automation


• Overview

• Motivating Example

• MPC Paradigm

• History

• Mathematical Formulation

• MPC Basics

2

Overview

• Overview


• MPC Paradigm

• History


• MPC Basics

Lesson 1 : Introduction

• Motivating example

• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



3

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Consider a linear discrete-time state-space model

called a ‘double integrator’.

We want to design a state feedback controller

that stabilizes the system (i.e. steers it to x=[0; 0])

starting from x=[1; 0], without violating the imposed

input constraints Signal processing

Identification



4

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Furthermore, we want the controller to lead to a

minimal control ‘cost’ defined as

with state and input weighting matrices

A straightforward candidate is the LQR controller,

which has the form

Signal processing

Identification



5

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



LQR controller

0 50 100 150 200 250 300-0.5

0

0.5

1

k

xk,1

0 50 100 150 200 250 300-30

-20

-10

0

10

k

uk

6

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



LQR controller with clipped inputs

0 50 100 150 200 250 300-1

-0.5

0

0.5

1

k

xk,1

0 50 100 150 200 250 300

-0.1

-0.05

0

0.05

0.1

0.15

k

uk

7

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



LQR controller with R=100

0 50 100 150 200 250 300-0.5

0

0.5

1

k

xk,1

0 50 100 150 200 250 300-0.1

-0.05

0

0.05

0.1

k

uk

8

Motivating Example

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



P

Fuel gas

Feed

EDC

EDC / VC / HCl

Cracking

Furnace

evaporato

r

superheater

waste gas

T

P

L

T F

H

F

condenser

Systematic way to deal with this issue… ?

9

MPC Paradigm

• Overview


• MPC Paradigm

• History


• MPC Basics

Process industry in ’70s : how to control a process ???

and… easy to understand (i.e. teach) and implement !

Signal processing

Identification



10

MPC Paradigm

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



→ Modelbased Predictive Control (MPC)

• Predictive : use model to optimize future input sequence

• Feedback : incoming measurements used to compensate for

inaccuracies in predictions and unmeasured disturbances

11

MPC has earned its place in the control hierarchy…

• Econ. Opt. : optimize profits using market and plant information (~day)

• MPC : steer process to desired trajectory (~minute)

• PID : control flows, temp., press., … towards MPC setpoints (~second)

MPC Paradigm

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



12

History

• Overview


• MPC Paradigm

• History


• MPC Basics

Before 1960’s :

• only input/output models, i.e. transfer functions, FIR

models

• Controllers :

• heuristic (e.g. on/off controllers)

• PID, lead/lag compensators, …

• mostly SISO

• MIMO case : input/output pairing, then SISO control

Signal processing

Identification



13

History

• Overview


• MPC Paradigm

• History


• MPC Basics

Early 1960’s : Rudolf Kalman

• Introduction of the State Space model :

• notion of states as ‘internal memory’ of the system

• states not always directly measurable : ‘Kalman’ Filter !

• afterwards LQR (as the dual of Kalman filtering)

• LQG : LQR + Kalman filter

• But LQG no real succes in industry :

• constraints not taken into account

• only for linear models

• only quadratic cost objectives

• no model uncertainties Signal processing

Identification



14

History

• Overview


• MPC Paradigm

• History


• MPC Basics

During 1960’s : ‘Receding Horizon’ concept

• Propoi, A. I. (1963). “Use of linear programming methods for synthesizing

sampled-data automatic systems”. Automatic Remote Control, 24(7), 837–844.

• Lee, E. B., & Markus, L. (1967). “Foundations of optimal control theory”. New

York: Wiley. :

“… One technique for obtaining a feedback controller synthesis

from knowledge of open-loop controllers is to measure the current

control process state and then compute very rapidly for the open-

loop control function. The first portion of this function is then used

during a short time interval, after which a new measurement of the

function is computed for this new measurement. The procedure is

then repeated. …”

Signal processing

Identification



15

History

• Overview


• MPC Paradigm

• History


• MPC Basics

During 1960’s : ‘Receding Horizon’ concept

Signal processing

Identification



16

History

• Overview


• MPC Paradigm

• History


• MPC Basics

1970’s : 1st generation MPC

• Extension of the LQR / LQG framework through combination with the

‘receding horizon’ concept

• IDCOM (Richalet et al., 1976) :

• IR models

• quadratic objective

• input / output constraints

• heuristic solution strategy

• DMC (Shell, 1973) :

• SR models


• no constraints

• solved as least-squares problem

Signal processing

Identification



17

History

• Overview


• MPC Paradigm

• History


• MPC Basics

Early 1980’s : 2nd generation MPC

• Improve the rather ad-hoc constraint handling of the 1st generation

MPC algorithms

• QDMC (Shell, 1983) :

• SR models


• linear constraints

• solved as a quadratic program (QP)

Late 1980’s : 3rd generation MPC

• IDCOM-M (Setpoint, 1988), SMOC (Shell, late 80’s), …

• Constraint prioritizing

• Monitoring / Removal of ill-conditioning

• fault-tolerance w.r.t. lost signals

Signal processing

Identification



18

History

• Overview


• MPC Paradigm

• History


• MPC Basics

Mid 1990’s : 4th generation MPC

• DMC-Plus (Honeywell Hi-Spec, ‘95), RMPCT (Aspen Tech, ‘96)

• Graphical user interfaces

• Explicit control objective hierarchy

• Estimation of model uncertainty

Currently (in industry) still …

• … no guarantees for stability

• … often approximate optimization methods

• … not all support state-space models

• … no explicit use of model uncertainty in controller design Signal processing

Identification



19

MPC Formulation

• Overview


• MPC Paradigm

• History

• Mathematical Form.

• MPC Basics

Basic ingredients :

• prediction model to predict plant response to future input

sequence

• (finite,) sliding window (receding horizon control)

• parameterization of future input sequence into finite number of

parameters

• discrete-time : inputs at discrete time steps

• continuous-time : weighted sum of basis functions

• optimization of future input sequence

• reference trajectory

• constraints

Signal processing

Identification



20

MPC Formulation

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



21

MPC Formulation

• Overview


• MPC Paradigm

• History


• MPC Basics

Assumptions / simplifications :

• no plant-model mismatch :

• no disturbance inputs

• all states are measured

(or estimation errors are negligible)

• no sensor noise

. . . seem trivial issues but form essential difficulties

in applications . . .

Signal processing

Identification



22

Theoretical Formulation (cfr. CACSD)

• Overview


• MPC Paradigm

• History


• MPC Basics

• future window of length ∞

• impossible to solve, because . . .

• infinite number of optimization variables

• infinite number of inequality constraints

• infinite number of equality constraints

Signal processing

Identification



23

Formulation 1

• Overview


• MPC Paradigm

• History


• MPC Basics

• still future window of length ∞, BUT

• quadratic cost function

• no input or state constraints

• linear model

• Optimal solution has the form uk = -Kxk

• Find K by solving Ricatti equation → LQR controller

Signal processing

Identification



24

Formulation 1 : LQR

• Overview


• MPC Paradigm

• History


• MPC Basics

PRO

• explicit, linear solution

• low online computational complexity

CON

• constraints not taken into account

• linear model assumption

• only quadratic cost functions

• no predictive capacity

Signal processing

Identification



25

Formulation 2

• Overview


• MPC Paradigm

• History


• MPC Basics

• changes :

• keep all constraints etc.

• reduce horizon to length N

• solution obtainable through dynamic optimization

• only u0 is applied, in order to obtain feedback at each k about xk

Signal processing

Identification



26

Formulation 2 : Classic MPC

• Overview


• MPC Paradigm

• History


• MPC Basics

PRO

• takes constraints into account

• proactive behaviour

• wide range of (convex) cost functions possible

• also for nonlinear models (but convex ?)

CON

• high online computational complexity

• no explicit solution

• feasibility ?

• stability ?

• robustness ?

• In what follows, we will concentrate on this formulation.

Signal processing

Identification



27

Formulation 3

• Overview


• MPC Paradigm

• History


• MPC Basics

• linear form of feedback law is enforced

• problem can be recast as a convex (LMI-based) optimization problem

more on this later . . .

Signal processing

Identification



28

Summary

• Overview


• MPC Paradigm

• History


• MPC Basics

Formulation 1

• infinite horizon

• no constraints

• explicit solution

• → LQR

Formulation 2

• finite horizon

• constraints

• no explicit solution

• → Classic MPC

Formulation 3

• infinite horizon

• constraints

• explicit solution enforced

• → has elements of LQR ánd MPC

Signal processing

Identification



29

Open vs. closed loop control

• Overview


• MPC Paradigm

• History


• MPC Basics

• Open loop : no state/output feedback : feedforward control

• Closed loop : state/output feedback : e.g. LQR

MPC is a mix of both :

• internally optimizing an open loop finite horizon control problem

• but at each k there is state feedback to compensate unmodelled

dynamics and disturbance inputs → closed loop control paradigm.

• has implications on e.g stability analysis

• Is of essential importance in Robust MPC, more on this later…

Signal processing

Identification



30

Standard MPC Algorithm

• Overview


• MPC Paradigm

• History


• MPC Basics

1. Assume current time = 0

2. Measure or estimate x0 and solve for uN and xN :

3. Apply uo0 and go to step 1

Remarks :

• : Terminal state cost and constraint

• : some kind of norm function

• Sliding Window

Signal processing

Identification



31

MPC design choices

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



1.prediction model

2. cost function

• norm

• horizon N

• terminal state cost

3. constraints

• typical input/state constraints

• terminal constraint

32

Prediction Model

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



• Input/Output or State-Space ?

• I/O restricted to stable, linear plants

• Hence SS-models

• Type of model determines class of MPC algorithm

• Linear model : Linear MPC

• Non-linear model : Non-linear MPC (or NMPC)

• Linear model with uncertainties : Robust MPC

• BUT : MPC is always a non-linear feedback law due to the

constraints

• Type of model determines class of involved optimization problem

• Linear models lead to most efficiently solvable opt.-problems

• Choose simplest model that fits the real plant ‘sufficiently well’

33

Cost Function

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



General Cost Function:

Design Functions and Parameters:

1.

2. Horizon N

3. F(x)

4. Reference Trajectory

34

Cost Function

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



35

Cost Function

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



36

Cost Function

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



37

Constraints

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



38

Constraints

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



39

Constraints

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



40

Constraints

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



41

Terminal State Constraints

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



42

Reference Insertion

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



43

Reference Insertion

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



44

Reference Insertion

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



45

Reference Insertion

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



46

Exercise Sessions

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



• Ex. 1 : Optimization oriented

• Ex. 2 : MPC oriented

• Ex. 3 : real-life MPC/optimization problem

Evaluation

• (brief !) report (groups of 2)

• oral examination → insight !

47

• Overview


• MPC Paradigm

• History


• MPC Basics

Signal processing

Identification



Bert Pluymers

Prof. Bart De Moor





H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization [email protected]

H0K03a : Advanced Process Control Model-based Predictive Control 2 : Dynamic Optimization

1

Overview

• Overview

• Optimization Basics

• Convex Optimization

• Dynamic Optimization

• Optimization Algorithms

Lesson 2 : Dynamic Optimization

• Optimization basics




Signal processing

Identification



2

General form :

Legend :

• : vector of optimization variables

• : objective function / cost function

• : equality constraints

• : inequality constraints

• : solution to optimization problem

• : optimal function value

Notation

• Overview





Signal processing

Identification



3

Gradient :

(points in direction of

steepest ascent)

Hessian :

(gives information about local curvature of )

Gradient & Hessian

• Overview





Signal processing

Identification



4

Gradient & Hessian

• Overview





Signal processing

Identification



Example :

Gradients for different Eigenvectors of hessian

at the origin ( )

5

Unconstrained Optimality Conditions

• Overview





Signal processing

Identification



Necessary condition for optimality of

Sufficient conditions for minimum

positive definite

Classification of optima :

positive definite minimum

indefinite saddle point

negative definite maximum

6

Introduction of Lagrange multipliers leads to Lagrangian :

with

Lagrange multipliers of the ineq. constraints

Lagrange multipliers of the eq. constriants

Constrained optimum can be found as

Minimization over but Maximization over !!!!

Lagrange Multipliers

• Overview





Signal processing

Identification



7

Constrained optimum can be found as

First-order optimality conditions in


• Overview





Signal processing

Identification



Gradient of Gradient of

ineq.

Gradient of

eq.

Interpretation ???

8


• Overview





Signal processing

Identification



9


• Overview





Signal processing

Identification



10


• Overview





Signal processing

Identification



11

Lagrange Duality

• Overview





Signal processing

Identification



12

Karush-Kuhn-Tucker Conditions

• Overview





Signal processing

Identification



From previous considerations we can now state

necessary conditions for constrained optimality :

These are called the KKT conditions.

13

Optimization Tree

• Overview





Signal processing

Identification



Optimization

discrete continuous

unconstrained constrained

convex

optimization

non-convex

optimization

NLP LP QP SOCP SDP

14

Convex Optimization

• Overview





Signal processing

Identification



An optimization problem of the form

is convex iff for any two feasible points :

• is feasible

•

This is satisfied iff

•`the cost function is a convex function

• the equality constraints or linear or absent

• the inequality constraints define a convex region

15

Convex Optimization

• Overview





Signal processing

Identification



Importance of convexity :

• no local minima, one global optimum

• under certain conditions, primal and dual have

same solution

• efficient solvers exist

• polynomial worst-case execution time

• guaranteed precision

16

From LP to SDP

• Overview





Signal processing

Identification



SDP

SOCP

QP

LP

Semi-Definite Programming

Second Order Cone Progr.

Quadratic Programming

Linear Programming

computational

efficiency

generality

17

General form :

Remarks :

• always convex

• optimal solution always at a corner of ineq. constraints

• typically used in finance / economics / management

Linear Programming (LP)

• Overview





Signal processing

Identification



18

Linear Programming (LP)

• Overview





Signal processing

Identification



Eliminating equality constraints :

Reparametrize optimization vector :

Leading to

19

Quadratic Programming (QP)

• Overview





Signal processing

Identification



General form :

Remarks :

• convex iff

• LP is a special case of QP (imagine )

• Used in all domains of engineering

20

Second-Order Cone Programming (SOCP)

• Overview





Signal processing

Identification



General form :

Remarks :

• Always convex

• Second-Order, Ice-Cream, Lorentz cone :

•

• Engineering applications with sum-of-squares,

robust LP, robust QP

SOC constraint

21

Second-Order Cone Programming (SOCP)

• Overview





Signal processing

Identification



QP as special case of SOCP :

Rewrite this as

which is equivalent to

By introducing an additional variable we get the SOCP

22

Semi-Definite Programming (SDP)

• Overview





Signal processing

Identification



General form :

with

Remarks :

• means that should be positive semi-definite

• means that should be pos. semi-def.

• always convex :

• ineq. constraints called LMI’s : Linear Matrix Ineq.

• LMI’s arise in many applications of

Systems & Control Theory

23


• Overview





Signal processing

Identification



convexity :

Easily verified :

hence

and therefore

which means that LMI’s are convex constraints.

24


• Overview





Signal processing

Identification



Schur Complement :

is equivalent with

More general :

Remarks :

• Originally developed in a statistical framework

• Today widely used in S&C in order to reformulate

problems involving eigenvalues as an LMI.

25


• Overview





Signal processing

Identification



SOCP as special case of SDP :

is equivalent with (by using Schur complement) :

(exercise : apply Schur complement to LMI and

reconstruct SOC constraint)

26

Convexity = SDP ?

• Overview





Signal processing

Identification



Formulated as SDP Convex optimization

Convex optimization formulatable as SDP

Example :

27

Convexity ≠ SDP !

• Overview





Signal processing

Identification



SDP

SOCP

QP

LP

convex optimization

structure easily exploitable

(many toolboxes available)

→ significant efficiency gains !

convexity difficult to exploit

(computationally)

28

• Overview





Signal processing

Identification



MPC Paradigm

• At every discrete time instant , given information about

the current system state , calculate an ‘optimal’ input

sequence over a finite time horizon :

• Apply the first input to the real system

• Repeat at the next time instant , using new state

measurements / estimates.

N

N

29

• Overview





Signal processing

Identification



Dynamic Programming

• Finding the optimal input sequence is done by means

of Dynamic Programming

• Definition* :

“DP is a class of solution methods for solving sequential

decision problems with a compositional cost structure”

• Invented by Richard Bellman (1920-1984) in 1953

30

• Overview





Signal processing

Identification



Example 1 : The darts problem* :

“Obtain a score of 301 as fast as possible while

beginning and ending in a double.”

• Decision : next area towards which to throw the dart

• Cost : time

http://plus.maths.org/issue3/dynamic/

* D. Kohler, Journal of the Operational Research Society, 1982.

Dynamic Programming

31

• Overview





Signal processing

Identification



Example 2 : DNA sequence allignment :

G A A T T C A G T T A (sequence #1)

G G A T C G A (sequence #2)

• Decisions : which nucleotides to match

• Cost : e.g. based on substitution / insertion prob.

• Algorithms : Baum/Welch, Waterman/Smith, …

Dynamic Programming

32

• Overview





Signal processing

Identification



“Series of sequential decisions” :

Dynamic Programming in MPC

Typical optimization problem :

measured

Optimization variables

→ standard QP formulation ?

33

• Overview





Signal processing

Identification



Optimization vector :

Cost function :

For convexity hence .

Linear MPC as standard QP

34

• Overview





Signal processing

Identification



Equality constraints

with

Sparsity : many entries in equal to 0


35

• Overview





Signal processing

Identification



Inequality constraints

with

Sparsity : many entries in equal to 0


36

• Overview





Signal processing

Identification



Alternatives

• slightly faster to solve

• ‘chattering’ : optimal solution jumps around

37

• Overview





Signal processing

Identification



Alternatives

Non-convex optimization in general : to be avoided !!!

38

• Overview





Signal processing

Identification



MPC and optimization

39

• Overview





Signal processing

Identification



MPC and optimization

40

• Overview





Signal processing

Identification



Active Set Methods

41

• Overview





Signal processing

Identification



Active Set Methods

42

• Overview





Signal processing

Identification



Interior Point Methods

1. Choose initial point

2. Linearize KKT conditions around current point

3. Solve Linearized KKT system to obtain search

direction

4. Calculate step length such that

5. Repeat from step 2, until convergence

43

• Overview





Signal processing

Identification



Interior Point Methods

44

• Overview





Signal processing

Identification



Comparison

• ASM allow hot start

• reuse of active set, factorizations, …

• ASM has feasible intermediate results

• by construction

• IPM can exploit sparsity

• solution of KKT system by sparse solver

In industry currently mostly ASM due to first two

advantages, but IPM under consideration…

45

• Overview





Signal processing

Identification



Comparison

46

• Overview





Signal processing

Identification



Comparison

47

• Overview





Signal processing

Identification



Comparison

48

• Overview





Signal processing

Identification



Comparison

49

• Overview





Signal processing

Identification



Conclusion

• convex optimization powerful tool !!!

• different optimization algorithm have pro’s and con’s

• try to avoid NLP’s !!!!

50

• Overview





Signal processing

Identification



References

Bert Pluymers

Prof. Bart De Moor





H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability [email protected]

H0K03a : Advanced Process Control Model-based Predictive Control 3 : Stability

1

Overview

• Introduction

• Example

• Stability Theory

• Set Invariance

• Implementations

Lecture 3 : Stability

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



2

MPC Paradigm

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



• At every discrete time instant , given information about

the current system state , calculate an ‘optimal’ input

sequence over a finite time horizon :

• Apply the first input to the real system

• Repeat at the next time instant , using new state

measurements / estimates.

N

N

3

Optimality of input sequence

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



co

mp

ute

d a

t

to be applied at

Optimal input sequences

Input sequence applied to the system

4

Stability Analysis

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



• Classical Way :

Analyse poles/zeros of and associated

transfer functions.

+ -

plant linear controller

u x r

+ -

plant MPC controller

u x r

• Modelbased Predictive Control :

Lyapunov theory for stability.

5

Inverted Pendulum

• Introduction

• Example


• Set Invariance

• Implementations

• 1 input :

• 4 states :

• open loop unstable system

Signal processing

Identification



6

Inverted Pendulum

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



• 4 different horizon lengths

• 3 different MPC variants (to be defined later)

Non-minimum phase

behaviour

7

Stability Theory

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



• Explicit vs. Optimization-based controller

• Transfer functions → Lyapunov theory

Stability is obtained / proven in 2 steps :

1. Recursive feasibility

i.e. controller well-defined for all k

2. Lyapunov function construction

i.e. trajectories converge to equilibrium

8

Stability Theory

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Limited validity of MPC stability framework :

• only for ‘stabilization’ problems :

• initial state

• system steered towards

• no disturbances allowed (but extension possible)

• no general stability framework for ‘tracking’ problems

9

Stability Theory

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Recursive Feasibility

If the optimization problem is feasible for time , then it

is also feasible for time .

(and hence for all )

Feasible Region

The region in state space, defined by all states for

which the MPC optimization problem is feasible.

→ Recursive feasibility proven : all states within feasible

region lead to trajectories for which the MPC-controller is

feasible and hence well-defined.

10

Stability Theory

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



→ Recursive feasibility proven : all states within feasible

region lead to trajectories for which the MPC-controller is

feasible and hence well-defined.

feasible region

11

MPC Stability Measures

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



12

MPC Stability Measures

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



recursive feasibility

(terminal constraint) Lyapunov stability

(terminal cost)

13

Step 1 : Recursive Feasibility

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Problem : given the optimal (and hence feasible) solution

to the optimization at time , construct a feasible solution

for the optimization at time .

14


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Given :

To be found :

Observe / Choose :

?

15


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Plant state at time predicted at time :

Real plant state at time :

Assumption : No plant model mismatch, i.e.

Hence, reusing the overlapping part of the input sequence

will also result in an identical state sequence

16


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



OK OK

OK OK OK ???

??? ???

17


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Condition 1 :

Satisfied if

Condition 2 :

Condition 3 :

How to choose

and ?

18


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Assume we know a locally stabilizing controller :

i.e. such that is locally stable.

How to choose and ?

Then choose

19


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



OK OK

OK OK OK ???

OK ???

Condition 2 :

Condition 3 :

20


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Condition 2 :

Condition 3 :

Since we know that …

Condition 2 is satisfied if

Condition 3 is satisfied if

21


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Summary & Interpretation

Recursive feasibility is guaranteed if

1)

2)

3)

Terminal constraint is

feasible w.r.t state constraints


feasible w.r.t input constraints


a positive invariant set w.r.t

22

Step 2 : Lyapunov stability

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



•Lyapunov stability :

If such that for some region around 0

then all trajectories starting within asymptotically

evolve towards 0.

• In the MPC context :

• is chosen as the feasible region,

• is chosen as the optimal cost value of the MPC

optimization problem for the given

Under which conditions is a Lyapunov function ???

23


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Under which conditions is a Lyapunov function ???

We have to prove that

Or in other words that

This is satisfied if

24


• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Special relationship between the two cost expressions :

should be < 0

Satisfied if

Condition 4 :

25

Lyapunov inequality

i.e. should ‘overbound’ cost of terminal controller

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Recursive feasibility and asymptotic stability is guaranteed if

1)

2)

3)

4)







Summary & Interpretation

Iff the optimization problem is feasible at time !!!!!

• conditions are sufficient, but not necessary

• is only used implicitly !

26

Set Invariance

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



“Given an autonomous dynamical system, then a set is

(positive) invariant if it is guaranteed that if the current state

lies within , all future states will also lie within .”

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

not invariant invariant

27

• Useful tool for analysis of controllers for constrained systems

• Example :

– linear system

– linear controller

Set Invariance

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



– state constraints

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

‘feasible region’ of closed loop system

28

Consider an autonomous time-invariant system as

defined previously

A set is …

Set Invariance

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



… feasible iff

Problem :

Given an autonomous dynamical system subject to state

constraints, find the feasible invariant set of maximal size.

… invariant iff

29

Given an LTI system subject to linear constraints

then the largest size feasible invariant set can be found as

with a finite integer.

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification






• is constructed by simple forward prediction

• can be proven to be the largest feasible invariant set

• is called the Maximal Admissible Set (MAS)

• is constructed by simple forward prediction

• can be proven to be the largest feasible invariant set

• is called the Maximal Admissible Set (MAS)

Invariant sets for LTI systems (Gilbert et al.,1991, IEEE TAC)

30

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

How to choose such that conditions are

satisfied ?

Different possibilities, depending of

• type of system (linear, non-linear)

• stability of the system

• presence of state constraints

• horizon length

• time constraints during design !

31

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

32

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

33

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

2

1

34

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

35

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

36

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

37

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

Terminal constraint set determines feasible region

38

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Implementations

Solid : standard MPC, dashed : terminal cost, constraint, dotted : terminal equality constr.

39

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



Conclusion

• stability of standard MPC not guaranteed

• pole/zero analysis impossible

• recursive feasibility

• Lyapunov stability

• general stability framework for stabilization problems

• different implementations

• stability measures allow the use of shorter horizons

40

• Introduction

• Example


• Set Invariance

• Implementations

Signal processing

Identification



References

Bert Pluymers

Prof. Bart De Moor





H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness [email protected]

H0K03a : Advanced Process Control Model-based Predictive Control 4 : Robustness

1

Overview

• Example

• Robustness

• Robust MPC

• Conclusion

Lecture 4 : Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



2

Example

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Linear state-space system of the form

with bounded parametric uncertainty

Aim : steer this system towards the origin from initial state

without violating the constraint

3

Example

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Results for 4 different parameter settings :

• Recursive feasibility ?

• Monotonicity of the cost ?

4

Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Robust with respect to what ?

• Disturbances

• Model uncertainty

Cause predictions of

‘nominal’ MPC to be inaccurate

5

Robustness

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Main aims :

• Keep recursive feasibility properties, despite model errors,

disturbances

• Keep asymptotic stability (in the case without disturbances)

We need to have an idea about …

• the size of the model uncertainty

• the size of the disturbances

6

Uncertain Models

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Linear Parameter-Varying state space models with

polytopic uncertainty description

7

Uncertain Models

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Linear Parameter-Varying state space models with

norm-bounded uncertainty description

8

Bounded Disturbances

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



• Typically bounded by a polytope :

• Can be described in two ways

•

•

• Trivial condition for well-posedness :

9

Robust MPC

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Main aims :

• Keep recursive feasibility properties, despite model errors,

disturbances

• Keep asymptotic stability (in the case without disturbances)

Necessary modifications :

•Uncertain predictions (e.g predictions with all models within

uncertainty region)

• worst-case constraint satisfaction over all predictions

• worst-case cost over all predictions

•Terminal cost has to satisfy multiple Lyap. Ineq.

•Terminal constraint has to be a robust invariant set

10

Robust MPC

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



model uncertainty

disturbances

Uncertain predictions :

N

N

11

Uncertain Predictions

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Step 1) Robust Constraint Satisfaction

Result : Sufficient to impose constraint only for vert. of :

Observations :

• depends linearly on

• is a convex polytopic set

• is a convex set

12


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LTI (L=1)

LPV (L>1, e.g. 2)

13


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Impose state constraints on all nodes

of state prediction tree

→ number of constraints increases expon. with incr. !!!

14

Worst-Case Cost Objective

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Step 2) Worst-Case cost minimization

Observations :

• depends linearly on


• cost function typically convex function of

→ Also for objective function sufficient to make

predictions only with vertices of uncertainty polytope

15

Worst-Case Cost Objective

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



sta

tes

inp

uts

16

Worst-Case Cost Objective (1-norm)

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LP

17


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



CVX ?

18


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Constraints of the form :

SOC

CVX ?

19


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



SOCP

20

Robust MPC (2-norm)

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



SOCP

By rewriting we now get

Terminal cost

Terminal constraint

21

Robust Terminal Cost

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



“non-robust” stability condition for terminal cost:

In case of…

• LPV system with polytopic uncertainty

• linear feedback controller

• quadratic cost criterion

• quadratic terminal cost

… this becomes :

or equivalent :

22

Robust Terminal Cost

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Robust stability condition for terminal cost:

Observations :

• inequality is convex and linear in and (i.e. LMI in )


Hence, inequality satisfied iff

23

Robust Terminal Cost : Design

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



1. Find a robustly stabilizing controller

2. Find a terminal cost satisfying

by solving the following optimization problem :

SDP

optimization variables

Minimization of

eigenvalues of

24

Robust Terminal Constraint

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Recursive feasibility is guaranteed if

1)

2)

3)







Reminder : nominal case

remain unchanged

Has to be modified in order to

Model uncertainty into account Robust positive invariance

25


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Consider linear terminal controller ,

then the resulting closed loop system is :

Robust positive invariance :

Again : sufficient to satisfy inclusion

26


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Reminder : invariant sets for LTI systems







Comes down to making forward predictions using

27


• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



LTI (L=1,n=2)

LPV (L>1, e.g. 2, n=2)

28

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

S X

• Constructed by solving semi-definite program (SDP)

• Conservative with respect to constraints

Ellipsoidal invariant sets for LPV systems

(Kothare et al.,1996, Automatica)

29

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Polyhedral invariant sets for LPV systems

A set is invariant with respect to a system defined

by iff

with

Reformulate invariance condition :

Sufficient condition :

Also necessary condition

30

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Advantages :

• in step 2 only ‘significant’ constraints are added to :

significant insignificant

• Initialize

• iteratively add constraints from to until

Algorithm :

31

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Algorithm :

Advantages :

• prediction tree never explicitly constructed

• given a polyhedral set , it is straightforward

to calculate :

• Initialize

• iteratively add constraints from to until

32

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Initialization

33

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 10

34

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 10 + garbage collection

35

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Iteration 20

36

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Final Result

37

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

Final Result

38

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification




Example

39

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Recursive feasibility, stability guarantee ?

Open loop

optimal input sequence

Closed loop

optimal input sequence

NO recursive feasibility !!! Recursive feasibility

40

Example revisited…

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Results for 4 different parameter settings :

• Recursive feasibility ?

• Monotonicity of the cost ?

41

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Example revisited…

42

• Example

• Robustness

• Robust MPC

• Conclusion

Signal processing

Identification



Conclusion

• Robustness w.r.t a) bounded model uncertainty

b) bounded disturbances

• necessary modifications :

• worst-case constraints satisfaction

• worst-case objective function

• terminal cost

• terminal constraint

• “open-loop” vs. “closed-loop” predictions

→ currently hot research topic !

• convex optimization but problem size impractical

→ currently hot research topic !

Introduction to Model Predictive Control (MPC)maapc/static/files/CACSD/... · 2017-05-15 · Introduction to Model Predictive Control Course: Computergestuurde regeltechniek 3 Some

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