OFF-LINE ROBUSTIFICATION OF EXPLICIT MPC LAWS

The Case of Polynomial Model Representation

Pedro Rodríguez-Ayerbe and Sorin Olaru

Department of Automatic Control, Supélec, 3 rue Joliot Curie, F91192 Gif-sur-Yvette, France

Keywords: Piecewise affine controller, Robustification, Youla-Kucera parameter, Model Predictive Control.

Abstract: The paper deals with the predictive control for linear systems subject to constraints, technique which leads

to nonlinear (piecewise affine) control laws. The main goal is to reduce the sensitivity of these schemes with

respect to the model uncertainties and avoid in the same time a fastidious on-line optimisation which may

reduce the range of application. In this idea a two stage predictive strategy is proposed, which synthesizes in

a first instant an analytical (continuous and piecewise linear) control law based on the nominal model and

secondly robustify the central controller (the controller obtained when no constraint is active). This

robustification is then expanded to all the space of the piecewise structure by means of its corresponding

noise model.

1 INTRODUCTION

The model predictive control (MPC) laws are

optimization based techniques which allow

constraints handling from the design stage. The

analytical formulation of the optimum and its on-line

evaluation avoids a challenging optimization from

the point of view of the real-time control

environment. Solutions in this direction exist at least

for two important classes of problems (linear and

quadratic) subject to linear constraints due to the

Abadie constraint qualification (Goodwin et al.,

2004). It must be said that these are in fact a part of

a larger class of multiparametric convex programs

(Bemporad et al., 2002b) for which exact or

approximate algorithms exist (TØndel et al., 2003,

Seron et al, 2003, Olaru and Dumur, 2004;

Bemporad and Filippi, 2006).

In the case of robust predictive control laws, the

model uncertainties and the disturbances can be

taken into account at the design stage. A popular

technique in this sense is the use of a min-max

criterium (in the case when the extreme combination

of disturbances or uncertainties are known)

(Kerrigan and Maciejowski, 2004; Bemporad et al.,

2002a) which comes finally to the resolution of a

single multiparametric linear program. The structure

of this ultimate optimization is however quite

complex and large prediction horizons cannot be

handled due to the exponential growth of

disturbances realization to be taken into account. In

a slightly different manner, by constructing an

estimation mechanism (Goodwin et al., 2004) for the

constrained variables, one can obtain alternatively a

robust control structure, but the multiparametric

optimization remains intricate.

A first study on the robustness improvements for

the explicit affine feedback policy constructed upon

constrained predictive control strategies was

presented in (Olaru and Rodriguez-Ayerbe, 2006).

The simplest way to proceed is to consider an

observer of the state variables (Goodwin et al.,

2004), the dimension of the state space being

preserved and the piece-wise structure of controller

unchanged. The same observer can be used for all

feasible regions and can be viewed as noise

characterisation of the model. Nevertheless, the

observer does not describe the entire class of

stabilizing controllers. The present paper presents an

improved result based on the Youla-Kučera

parametrization which spans the space of stabilizing

controllers. For a two-degree of freedom controller,

one has access to all the stabilizing controllers that

preserve the same input/output behavior, so the

Youla-Kučera parameter offers more degrees of

freedom than the use of an observer.

The robustification is made such that the state

space dimension of the controller is augmented. The

main contribution here is the reconstruction of the

noise model induced by the central Youla-Kučera

Rodríguez-Ayerbe P. and Olaru S. (2008).

OFF-LINE ROBUSTIFICATION OF EXPLICIT MPC LAWS - The Case of Polynomial Model Representation.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 96-101

DOI: 10.5220/0001481300960101

 SciTePress

parameter, in order to use it to generate the

corresponding robust piece-wise controller.

In the following, section 2 briefly recalls the

predictive control and the Youla-Kučera

parametrization. Section 3 details the explicit

formulation of the control laws obtained in the

constrained case. Section 4 contains the main

contribution: the noise model of the Youla-Kučera

parameter and the numerical examples are presented

in section 5 and the final conclusions in section 6.

2 PREDICTIVE CONTROL

The Generalized Predictive Control (GPC) strategy,

introduced in (Clarke et al., 1987), uses for the

prediction a CARIMA plant model:

)(

)()(

−

−−

uqByqA

(1)

with u, y the input and output,

a white noise, A

and B polynomials in the backward shift operator of

degrees

n and

n respectively, and

1)(

−−

−=Δ qq the difference operator. The C

polynomial is the model argument taking into

account the noise influence on the system. In the

GPC case the cost function to be minimized over a

receding horizon is quadratic:

[][]

∑∑

−+

Δ+−=

jtj

jtjt

uywJ

(2)

where

, NN are the costing horizons,

N the

control horizon,

the control weighting factor and

w the set-point.

Using the model (1) and the solution of some

Diophantine equations (Clarke et al., 1987), this

control strategy leads to two-degrees of freedom

RST controller, implemented through a difference

equation (Figure 1):

)(

1−

)(tu

)(tw

)(

1−

Δ qS

)(

1−

)(ty

)(td

)(

11 −−

qBq

)(qT

Figure 1: Two-degrees of freedom GPC controller.

In (Yoon and Clarke, 1995) the relation between

the RST controller obtained with C=1 and

1≠C is

studied. Considering

TSR

′′′

,, the controller

obtained with C=1 and

TSR ,, whose obtained with

1≠C , the following relations are obtained:

CTTBMqCSSMACRR

′

=−

′

=Δ+

′

−1

(3)

with:

∑

−

′

)()(

EECqqM

(4)

[]

21 NN

K=m

(5)

m being the first row of

T1T

)( GΛGG

−

+ .

The set of all stabilizing controllers for the system

shown in Figure 1 is given by the Youla-Kučera

parametrization as follows (Maciejowski, 1989):

⎪

⎩

⎪

⎨

⎧

′

−

′

Δ+

′

−

den

numden

den

num

QTT

BQqQSS

AQQRR

(6)

where )(

1−

qQ is a stable transfer function.

The choice of the Q parameter is a complex

problem on its own but it is not the subject of the

current paper. The methods presented in (Rodriguez

and Dumur, 2005; Rossiter 2003; Ansay et al., 1998;

Yoon and Clarke, 1995; Kouvaritakis et al., 1992)

can be used for the choice of this parameter.

Comparing (3) and (6) it turns out that the

controller for

1≠C is obtained for Q=M/C. As M

depends of C as shown by (4), the robustification by

the C polynomial has less degrees of freedom than

the robustification by Youla-Kučera parameter

(Yoon and Clarke, 1995).

3 EXPLICIT CONSTRAINED

GPC LAWS

In the case when the GPC law is subject to

constraints, the optimization has to be solved with

respect to a feasible domain. If the considered

constraints are stated on the control action, on the

control increment, on the plant outputs or any other

signal related by a CARIMA model to the control

signal, then one can restate them in a form

depending only on the control increment, leading to

a set of linear constraints (Ehrlinger et al., 1996):

OFF-LINE ROBUSTIFICATION OF EXPLICIT MPC LAWS - The Case of Polynomial Model Representation

⎩

⎨

⎧

+≤

eqtequeq

intinuin

bpBkA

(7)

involving the optimization argument )(t

k and the

vector of context parameters:

{}

[]

TTT

wΔuγyp

pastpastpastt

(8)

which regroups a finite sequence of past inputs

Δu

future setpoints

, past outputs y and present and

future values of the signals under constraints (noted

for short

past

It is interesting to observe that this set of

constraints characterizes in fact a parameterized

polyhedron (Olaru and Dumur, 2004) in the

optimization argument space. The optimum will lie

on a combination of its parameterized vertices and

thus one can construct the explicit solution for the

multiparametric optimization:

The use of a dual representation of the feasible

domain and projection mechanisms (see Olaru and

Dumur, 2004 and 2005) can be advantageous in

order to express the optimality conditions if there

exist unbounded feasible directions.

⎩

⎨

⎧

+≤

eqtequeq

intinuin

bpBkA

)(

:tosubject

min

)(

(9)

Subsequently, the predictive control law can be

described explicitly upon the solution of (9) as a

piecewise affine function of the vector of parameters

(Seron et al, 2003; Bemporad et al., 2002; Olaru and

Dumur, 2004).

itliLinu

CRKK

∈+= pppk for)(

(10)

with CR

, critical regions in the space of context

parameters, The GPC subjet to constraints has a

piecewise RST polynomial form:

ipastitititi

tlVwqTyqRuqS +++−=Δ

−−

)()()(

(11)

for

{}

[]

ipastpastpast

CR∈

TTT

wΔuγy

The structure of such a piecewise controller is

shown in Figure 2. Once the look-up table of local

polynomial RST laws is available, an efficient

positioning mechanism (based on a search tree) can

be constructed such that the on-line evaluation

routine can find the optimal control action according

to the GPC philosophy (TØndel et al., 2003).

4 ROBUSTIFICATION

At this stage, it is assumed that the design of initial

controller has been performed with

N ,

adjusted to satisfy the required input/output

behavior. The resulting picewise two-degrees of

freedom RST controller will be denoted

iii

TSR

′′′

,, , in

the following sections.

The observer based robustification corresponds

to the consideration of an observer of the state. In

the case of GPC, this corresponds to a choice of a C

polynomial. The roots of this polynomial correspond

to the poles of the observer. The obtained piecewise

controller can be implemented as in Figure 2, see

also (Olaru and Rodriguez-Ayerbe, 2006, Camacho

and Bordons, 2004 ; Bitmead et al.,1990).

obustified piecew is

ST controller

Positioning

mechanism

)

(

−

)(

−

ipasti

tlV +

)(

−

Δu

)(

−

(q)

)(

−

)(

1−

)(

1−

− qCS

Figure 2: Piecewise RST formulation for the GPC law

under constraints and robustification using the C

polynomial.

The Youla-Kučera parameter based robusti-

fication has more degrees of freedom than the use of

an observer. It allows to access the entire space of

stabilizing controllers. The idea is thus to robustify

the central RST law of the piecewise controller, that

is, the law where the constraints are not activated,

and to expand this robustification to the others RST

laws of the initial piecewise controller. The choice

of this parameter can be done using stability

robustness and nominal performance specifications,

see (Rodriguez and Dumur, 2005; Rossiter 2003;

Ansay et al., 1998; Yoon and Clarke, 1995;

Kouvaritakis et al., 1992). In the following the

selection will be done according to these principles.

As the use of an observer corresponds to the

consideration of a C polynomial for the noise

influence on the CARIMA model (1), the idea is to

find the corresponding noise model of the Youla-

Kučera parameter. To do this, an extra polynomial is

added in the model of the system. A

)(

1−

polynomial will appear as following:

)()(

)(

)()(

−−

−

−−

qqD

uqByqA

(12)

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

With this new model, Diophantine equations are:

)()()()()()(

111111

−−−−−−−

=+Δ

qDqEqBqHqqCqG

qCqFqqEqDqAq

(13)

Finding the relation between the controller

obtained for C=D=1 and the one obtained for

1≠C

and

1≠D , we obtain something similar to (3).

Considering

TSR

′′′

,, the controller obtained with

C=D=1 and

TSR

whose obtained with 1≠C and

1≠D

, the following relations are obtained:

CTTMBqCSSMACRR

′

=−

′

=Δ+

′

−

(14)

With, (see Appendix for structural details) :

∑

−

′

)

()(

DEECqqM

(15)

So, the D polynomial corresponding to the

considered Youla-Kučera parameter must verify:

∑

−−

−

′

)

()(

)(

inum

DEECqqMqQ

(16)

Once the corresponding noise model has been

obtained, it can be used to regenerate the piecewise

affine controller. The same input/output behaviour

as for the initial one is assured, in the ideal case of

no model errors. A modified close loop behaviour

will be observed with respect to disturbance

rejection, robustness, etc.

The resolution of (16) is a non linear problem

that can be undertaken with standard optimization

methods. Nevertheless, is not always possible to

guarantee a real solution. The resolution of (16) and

its limitations are raising interesting questions,

research being currently conducted on this subject.

From a practical point of view, any such limit case

can be avoided by retuning the initial predictive

control parameters or the robustification

specification.

5 EXAMPLE

Consider the position control of an induction motor,

with 1.0724 ms as sampling period

)998.01)(1(

)8206.0821.0(10

)(

214

ref

−−

−−−

−

−−

(17)

Constraints in control amplitude are considered:

],[

maxmaxref

−∈ and 8.1

max

. An initial

GPC controller is designed with

1== DC with the

following tuning parameters:

=N ,16

0001.0=

and 2=

N . The position of the motor

is obtained through an encoder of 14400 points per

rotation, and the highs dynamics of the system

(current loop, inverter dynamic, mechanic dynamics

in high frequency) have been not identified.

This initial controller is obtained with (9). A

piecewise linear controller with 9 regions is

obtained. The central region corresponds to the case

where no constraint is active. This controller will be

noted

000

,, TSR .

To robustify off-line this piecewise controller,

the idea is to robustify the central one (

000

,, TSR )

and expand this robustification to other regions. In

this way a Youla-Kučera parameter has been

obtained by method described in (Rodriguez and

Dumur, 2005). The following parameter is

considered.

4321

321

7.0973.2838.4565.31

93.254117.890299.104992.4196

−−−−

−−−

+−+−

+++−

qqqq

qqq

(18)

With (6), we obtain the controller

QQQ

TSR

000

,, .

Solving (16) with

den

QC = , the following D

polynomial is obtained:

4321

426.0018.0472.0873.01

−−−−

+−+−= qqqqD

(19)

This value has been obtained by available

optimization methods (classical Matlab routines in

occurrence) as long as (16) represents a set of non

linear equations difficult to solve analytically.

With this D polynomial, the optimization

problem (9) can be solved but this time with

matrices obtained from (13) for

den

QC = and D as

in (19). The solution of this new optimization

problem leads to a new piecewise controller with 9

regions, as the initial one. The central controller of

this piecewise controller correspond to

QQQ

TSR

000

,, .

Figures 3 and 4 show the obtained simulations

results for a filtered step reference considering a

second order neglected dynamic in high frequency

of the following characteristics:

srad /1000

3.0=

In these figures we can observe that the obtained

behaviour is stable in the case of robustified

controller and instable in the case of initial

controller. So, the robustified controller has better

OFF-LINE ROBUSTIFICATION OF EXPLICIT MPC LAWS - The Case of Polynomial Model Representation

behaviour towards uncertainties in high frequency

and the continuity between regions is guaranteed.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

Pos. Ref.

(rad)

0 0.5 1 1.5

-0.2

0.2

Pos. Error

(rad)

0 0.5 1 1.5

-2

Control

signal

0 0.5 1 1.5

Active

Region

time (s)

Figure 3: Position reference, position error, control signal

and active region for the initial controller and uncertain

model.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5

Pos. Ref.

(rad)

0 0.5 1 1.5

-0.2

0.2

Pos. Error

(rad)

0 0.5 1 1.5

-2

Control

signal

0 0.5 1 1.5

Active

Region

time (s)

Figure 4: Position reference, position error, control signal

and active region for the robustified controller and

uncertain model.

6 CONCLUSIONS

The paper investigated the robustification methods

for the control laws obtained in a constrained

predictive control framework. The idea is to design

in a first instance a piecewise polynomial controller

which satisfy the basic demands in terms of tracking

performances. In a second stage, the same predictive

control structure (prediction horizon, weightings,

etc.) is robustified using the model arguments

accounting for the noise influence. The idea is

similar to that of using a fixed observer, but

exploring all the class of stabilizing controllers of

the unconstrained system. This increases the number

of degrees of freedom.

The robustification of initial unconstrained

controller is made using the Youla-Kučera

parametrization, and then this robustification is

expanded to all the piecewise structure of the

controller. For this, the noise model corresponding

to the Youla-Kučera parameter is found, and use to

regenerate the robust piecewise controller by

preserving the same input/output behavior but being

more robust.

The limitations of the method are in the existence

of the corresponding noise model of the Youla-

Kučera parameter. This is transparent in the

resolution of a non linear equation system. The

robustification being done off-line, any infeasibility

can be handled by retuning the GPC parameters.

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APPENDIX

By solving the first Diophantine equation of (13) for

C=D=1 and

1≠C , 1≠D , a relationship between

the

′

polynomial obtained for C=D=1 and the R

obtained for

1≠C ,

1≠D

is obtained:

⎪

⎩

⎪

⎨

⎧

=+Δ

′

−−−−−−−

−−−−−

)()(

)(

)()()(

1)()()()(

111111

1111

qCqFqqEqDqAq

qFqqEqAq

)

(

EDECAqFCF −

′

Δ+

′

=⇒

R is obtained as:

∑

(see Yoon and

Clarke 1995).

For

MARC

EDECqARC

EDECAqFCFR

Δ+

′

−

′

Δ+

′

−

′

Δ+

′

∑

∑∑

)

(

))

((

αα

With the same development, the corresponding

expression for

of (14) is obtained. Solving (16)

comes to the identification of a D polynomial

corresponding to

)(

1−

num

. It must be noted that

the first i coefficients of

1+i

E are the same than

those of

E polynomials. (

E is a polynomial of

degree i-1.) With this, the solution of D for the

numerical example has been obtained solving the

following non linear equations:

−−−

−−

−−−−

+++=

′

++++=

qmqmqmmM

qeqeE

qcqcqcqcC

qdqdqdqdD

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

⎪

⎭

⎪

⎬

⎫

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

⎪

⎩

⎪

⎨

⎧

−

⎟

⎠

⎞

⎜

⎝

⎛

′

⎟

⎠

⎞

⎜

⎝

⎛

1000

000

1000

000

151

432

4321

151

432

4321

ddd

dddd

ccc

cccc

(20)

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

′

⎟

⎠

⎞

⎜

⎝

⎛

100

0001

100

0001

1234

123

1234

123

MOL

dddd

ddd

cccc

ccc

(21)

With

)

(

1514321

eedddd L=x , the

following problem has been solved:

0)( =xF

With F defined by the matrix relations (20) and

(21), and using a standard optimization routine.

OFF-LINE ROBUSTIFICATION OF EXPLICIT MPC LAWS - The Case of Polynomial Model Representation

101