Introduction to Model Predictive Control (MPC) Oscar Mauricio - - PowerPoint PPT Presentation

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
• f performance.
• It is easy to explain to operators and

engineers

Principle of predictive control

1 k  2 k  3 k 

Prediction horizon

k N  k 1 k  2 k 

( ) u k ( ) y k

Prediction of

( ) y k

Reference

Future Past

measurement

ref

y 

 

2 ref ( ), , ( 1) 1

min ( )

N u k u k N i

y y k i

  

 



subject to

• model of the process
• input constraints
• output / state constraints

Why to use MPC ?

time

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

3

Some applications of MPC

Control of synthesis section of a urea plant

MPC strategies have been used for stabilizing and maximizing the throughput of the synthesis 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

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

4

Some applications of MPC

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

Ammonia Carbon dioxide Ammonium carbamate

Reaction 1: Fast and Exothermic

NH2COONH4  NH2CONH2 + H2O

Ammonium carbamate Urea Water

Reaction 2: Slow and Endothermic

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

5

Some applications of MPC

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 Limburg Control Strategy: PLC logic (e.g., three-pos controller) A Nonlinear MPC control strategy has been implemented (2016) for avoiding future floodings of the Demer river in Belgium. Partners: STADIUS, Dept. Civil Engineering of KU Leuven, IPCOS, IMDC, Antea Group, and Cofely Fabricom.

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

6

Some applications of MPC

Flood Control: The Demer

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

DIEST HASSELT

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

7

Some applications of MPC

Flood Control: The Demer

Upstream part of the Demer that is modelled and controlled in a preliminary study carried

• ut 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 rain predictions into account!

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

8

Some applications of MPC

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 re ref ref 1 f ,

( ) ( ) min ( ) ( ) ( ( ( ) ) ( ) )

N N

N N T i i T

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

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 u

with:

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC Algorithm

10

Typical MPC control Loop MPC Algorithm

At every sampling time:

MPC Plant Observer

( ) k u ( ) k y ˆ( ) k x

ref ( )

k x

ref ( )

k u

ZOH

( ) t u ( ) t y

s

T

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 u

Solution 

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

LQR and Classical MPC

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

LQR CLassical Linear MPC

, 1

( ) min ( ) ( ( ) )

T T k

k k k k

 

 





x u

x Q R u x 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 , 1

( ) ( min ( ) ( ) )

N N

N T i k N T

k k k k

  





x u

u R u x Q x

k = 0

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

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 re ref ref 1 f ,

( ) ( ) min ( ) ( ) ( ( ( ) ) ( ) )

N N

N N T i i T

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

with:

 

( 1); ( 2); ; ( ) ,

x

n N N

k k k N



     x x x x

 

( ); ( 1); ; ( 1)

u

n N N

k k k N



     u u u u

x

1 min 2

T T



x

x Hx f x

subject to

e e

 A x b

i i

 A x b

with:

 

 

;

x u

N n n N N  

  x x u

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC optimization problem – Implementation Details

13

where

 

 

 

 

2

x u x u

N n n N n n     

                    Q Q H R R

 

 

ref ref ref ref

(1) ( ) (0) ( 1)

x u

N n n

N N

 

                        x x f H u u

 

 

 

 

2 i

x x x u x u u u

n N n N N n n N n n n N n N         

                  I I A I I

 

max 2 max i max max

x u

N n n  

                      x x b u u

N times N times 2N times 2N times

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

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC optimization problem – Implementation Details

14

   

 

e

x x x x u x

n n N n N n n n    

                     I B A I B A A I B

e

( )

x

N n

k



              Ax b

N 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)

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

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

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC with terminal cost

16

       

1 ref ref ref ref , 1 1

min ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

N N

N T 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 N  x

Effect of the stabilizing control law u(k) = -Kx(k)

1( )

x k

2( )

x k

ref ( )

k  x ( ) k x ( ) k   x

steady-state

Introduction to Model Predictive Control Course: Computergestuurde regeltechniek

MPC with terminal cost

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

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

 

 

 

 

2

x u x u

N 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 bert.pluymers@esat.kuleuven.be

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

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

2

Overview

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Lesson 1 : Introduction

• Motivating example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

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H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

3

Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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

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4

Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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

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Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

LQR controller

50 100 150 200 250 300

• 0.5

0.5 1 k xk,1 50 100 150 200 250 300

• 30
• 20
• 10

10 k uk

6

Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

LQR controller with clipped inputs

50 100 150 200 250 300

• 1
• 0.5

0.5 1 k xk,1 50 100 150 200 250 300

• 0.1
• 0.05

0.05 0.1 0.15 k uk

7

Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

LQR controller with R=100

50 100 150 200 250 300

• 0.5

0.5 1 k xk,1 50 100 150 200 250 300

• 0.1
• 0.05

0.05 0.1 k uk

8

Motivating Example

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be 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
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Process industry in ’70s : how to control a process ??? and… easy to understand (i.e. teach) and implement !

Signal processing

Identification System Theory Automation

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MPC Paradigm

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

→ 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
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

12

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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

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13

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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 System Theory Automation

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14

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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

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History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• MPC Basics

During 1960’s : ‘Receding Horizon’ concept

Signal processing

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16

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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
• quadratic objective
• no constraints
• solved as least-squares problem

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

17

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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
• quadratic objective
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

18

History

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Formulation
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

19

MPC Formulation

• Overview
• Motivating Example
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

20

MPC Formulation

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

21

MPC Formulation

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

22

Theoretical Formulation (cfr. CACSD)

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

23

Formulation 1

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

24

Formulation 1 : LQR

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

25

Formulation 2

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

26

Formulation 2 : Classic MPC

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

27

Formulation 3

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

28

Summary

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

29

Open vs. closed loop control

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

30

Standard MPC Algorithm

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics
• 1. Assume current time = 0
• 2. Measure or estimate x0 and solve for uN and xN :
• 3. Apply uo

0 and go to step 1

Remarks :

• : Terminal state cost and constraint
• : some kind of norm function
• Sliding Window

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

31

MPC design choices

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

1.prediction model

• 2. cost function
• norm
• horizon N
• terminal state cost
• 3. constraints
• typical input/state constraints
• terminal constraint

32

Prediction Model

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

• 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
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

General Cost Function: Design Functions and Parameters:

1. 2. Horizon N 3. F(x) 4. Reference Trajectory

34

Cost Function

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

35

Cost Function

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

36

Cost Function

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

37

Constraints

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

38

Constraints

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

39

Constraints

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

40

Constraints

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

41

Terminal State Constraints

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

42

Reference Insertion

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

43

Reference Insertion

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

44

Reference Insertion

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

45

Reference Insertion

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

46

Exercise Sessions

• Overview
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

• 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
• Motivating Example
• MPC Paradigm
• History
• Mathematical Form.
• MPC Basics

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 1 : Introduction bert.pluymers@esat.kuleuven.be

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 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

3

Gradient : (points in direction of steepest ascent) Hessian : (gives information about local curvature of )

Gradient & Hessian

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

4

Gradient & Hessian

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Example : Gradients for different Eigenvectors of hessian at the origin ( )

5

Unconstrained Optimality Conditions

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

7

Constrained optimum can be found as First-order optimality conditions in

Lagrange Multipliers

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Gradient of Gradient of ineq. Gradient of eq.

Interpretation ???

8

Lagrange Multipliers

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

9

Lagrange Multipliers

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

10

Lagrange Multipliers

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

11

Lagrange Duality

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

12

Karush-Kuhn-Tucker Conditions

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

From previous considerations we can now state necessary conditions for constrained optimality : These are called the KKT conditions.

13

Optimization Tree

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Optimization discrete continuous unconstrained constrained convex

• ptimization

non-convex

• ptimization

NLP LP QP SOCP SDP

14

Convex Optimization

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

18

Linear Programming (LP)

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Eliminating equality constraints : Reparametrize optimization vector : Leading to

19

Quadratic Programming (QP)

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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

Semi-Definite Programming (SDP)

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

convexity : Easily verified : hence and therefore which means that LMI’s are convex constraints.

24

Semi-Definite Programming (SDP)

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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

Semi-Definite Programming (SDP)

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Formulated as SDP Convex optimization Convex optimization formulatable as SDP Example :

27

Convexity ≠ SDP !

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

SDP SOCP QP LP convex optimization

structure easily exploitable (many toolboxes available) → significant efficiency gains ! convexity difficult to exploit (computationally)

28

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Dynamic Programming

• Finding the optimal input sequence is done by means
• f 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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

“Series of sequential decisions” :

Dynamic Programming in MPC

Typical optimization problem :

measured Optimization variables

→ standard QP formulation ?

33

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Optimization vector : Cost function : For convexity hence .

Linear MPC as standard QP

34

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Equality constraints with Sparsity : many entries in equal to 0

Linear MPC as standard QP

35

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Inequality constraints with Sparsity : many entries in equal to 0

Linear MPC as standard QP

36

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Alternatives

• slightly faster to solve
• ‘chattering’ : optimal solution jumps around

37

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Alternatives

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

38

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

MPC and optimization

39

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

MPC and optimization

40

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Active Set Methods

41

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Active Set Methods

42

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Interior Point Methods

44

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

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
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

46

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

47

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

48

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Comparison

49

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

Conclusion

• convex optimization powerful tool !!!
• different optimization algorithm have pro’s and con’s
• try to avoid NLP’s !!!!

50

• Overview
• Optimization Basics
• Convex Optimization
• Dynamic Optimization
• Optimization Algorithms

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be

References

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 3 : Stability bert.pluymers@esat.kuleuven.be

H0K03a : Advanced Process Control

Model-based Predictive Control 3 : Stability

1

Overview

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Lecture 3 : Stability

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

2

MPC Paradigm

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

• 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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

computed at to be applied at

Optimal input sequences Input sequence applied to the system

4

Stability Analysis

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

• 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
• Stability Theory
• Set Invariance
• Implementations
• 1 input :
• 4 states :
• open loop unstable system

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

6

Inverted Pendulum

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

• 4 different horizon lengths
• 3 different MPC variants (to be defined later)

Non-minimum phase behaviour

7

Stability Theory

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

• 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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

→ 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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

12

MPC Stability Measures

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

recursive feasibility (terminal constraint) Lyapunov stability (terminal cost)

13

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Problem : given the optimal (and hence feasible) solution to the optimization at time , construct a feasible solution for the optimization at time .

14

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Given : To be found : Observe / Choose :

?

15

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

OK OK OK OK OK ??? ??? ???

17

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Condition 1 : Satisfied if Condition 2 : Condition 3 : How to choose and ?

18

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Assume we know a locally stabilizing controller : i.e. such that is locally stable. How to choose and ? Then choose

19

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

OK OK OK OK OK ??? OK ???

Condition 2 : Condition 3 :

20

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Condition 2 : Condition 3 : Since we know that … Condition 2 is satisfied if Condition 3 is satisfied if

21

Step 1 : Recursive Feasibility

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Summary & Interpretation

Recursive feasibility is guaranteed if 1) 2) 3)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

22

Step 2 : Lyapunov stability

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

• 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
• ptimization problem for the given

Under which conditions is a Lyapunov function ???

23

Step 2 : Lyapunov stability

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Under which conditions is a Lyapunov function ??? We have to prove that Or in other words that This is satisfied if

24

Step 2 : Lyapunov stability

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Recursive feasibility and asymptotic stability is guaranteed if 1) 2) 3) 4)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

“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

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

– state constraints

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

… 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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer.

• 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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

32

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

33

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

2 1

34

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

35

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

36

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

37

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

Terminal constraint set determines feasible region

38

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

Implementations

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

39

• Introduction
• Example
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

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
• Stability Theory
• Set Invariance
• Implementations

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 3 : Stability bert.pluymers@esat.kuleuven.be

References

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 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

2

Example

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Results for 4 different parameter settings :

• Recursive feasibility ?
• Monotonicity of the cost ?

4

Robustness

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Linear Parameter-Varying state space models with polytopic uncertainty description

7

Uncertain Models

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Linear Parameter-Varying state space models with norm-bounded uncertainty description

8

Bounded Disturbances

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be model uncertainty disturbances

Uncertain predictions :

N N

11

Uncertain Predictions

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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

Uncertain Predictions

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LTI

(L=1)

LPV

(L>1, e.g. 2)

13

Uncertain Predictions

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Impose state constraints on all nodes

• f state prediction tree

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

14

Worst-Case Cost Objective

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Step 2) Worst-Case cost minimization Observations :

• depends linearly on
• is a convex polytopic set
• 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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

states inputs

16

Worst-Case Cost Objective

(1-norm)

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LP

17

Worst-Case Cost Objective

(2-norm)

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

CVX ?

18

Worst-Case Cost Objective

(2-norm)

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Constraints of the form :

SOC CVX ?

19

Worst-Case Cost Objective

(2-norm)

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

SOCP

20

Robust MPC

(2-norm)

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

SOCP

By rewriting we now get Terminal cost Terminal constraint

21

Robust Terminal Cost

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

“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 :

• r equivalent :

22

Robust Terminal Cost

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Robust stability condition for terminal cost: Observations :

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

Hence, inequality satisfied iff

23

Robust Terminal Cost : Design

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

• 1. Find a robustly stabilizing controller
• 2. Find a terminal cost satisfying

by solving the following optimization problem :

SDP

• ptimization variables

Minimization of eigenvalues of

24

Robust Terminal Constraint

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Recursive feasibility is guaranteed if 1) 2) 3)

Terminal constraint is feasible w.r.t state constraints Terminal constraint is feasible w.r.t input constraints Terminal constraint is a positive invariant set w.r.t

Reminder : nominal case

remain unchanged Has to be modified in order to Model uncertainty into account Robust positive invariance

25

Robust Terminal Constraint

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Consider linear terminal controller , then the resulting closed loop system is : Robust positive invariance : Again : sufficient to satisfy inclusion

26

Robust Terminal Constraint

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Reminder : invariant sets for LTI systems Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer. Given an LTI system subject to linear constraints then the largest size feasible invariant set can be found as with a finite integer. Comes down to making forward predictions using

27

Robust Terminal Constraint

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

LTI

(L=1,n=2)

LPV

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

28

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

• 1
• 0.8
• 0.6
• 0.4
• 0.2

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

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 System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Initialization

33

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 10

34

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 10 + garbage collection

35

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Iteration 20

36

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Final Result

37

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

Final Result

38

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Polyhedral invariant sets for LPV systems

Example

39

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Recursive feasibility, stability guarantee ?

Open loop

• ptimal input sequence

Closed loop

• ptimal input sequence

NO recursive feasibility !!! Recursive feasibility

40

Example revisited…

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Results for 4 different parameter settings :

• Recursive feasibility ?
• Monotonicity of the cost ?

41

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

Example revisited…

42

• Example
• Robustness
• Robust MPC
• Conclusion

Signal processing

Identification System Theory Automation

H0k03a : Advanced Process Control – Model-based Predictive Control 4 : Robustness bert.pluymers@esat.kuleuven.be

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 !