Analysis and Control of Linear Systems - Chapter 12 pptx

Predictive Control The developments presented in this chapter aim to cover the main ideas of predictive control and then to indicate the details of the analytical minimization of the cri

Predictive Control

The developments presented in this chapter aim to cover the main ideas of predictive control and then to indicate the details of the analytical minimization of the criterion for two individual structures enabling the elaboration of the equivalent polynomial regulator The choice of adjustment parameters will also be analyzed, providing some simple rules that guarantee the corrected system good stability and robustness

12.1 General principles of predictive control

Predictive control is based on some relatively old and intuitive ideas [RIC 78], but it has been developed as an advanced control technique mainly since the 1980s This development was done mainly according to two privileged main lines:

– generalized predictive control (GPC) by Clarke (1985);

– functional predictive control (FPC) by Richalet (1987)

The philosophy of predictive control lies on the definition of five great ideas, common to all the methods

12.1.1 Anticipative aspect

This anticipative effect is obtained by using explicit knowledge on the evolution

of the trajectory to be followed in the future (necessary knowledge required at least

Chapter written by Patrick BOUCHER and Didier DUMUR

on the horizon of some points beyond the present moment) This constraint which makes it possible to make good use of all the resources of the method, necessarily restricts the application field to the control of the systems for which the trajectory to follow is perfectly known and stored pixel by pixel in the computer It is the case of the numerical control of machine-tools (cutting the pieces), of the control of robots arms, of monitoring the temperature profile of the applications in home automation, etc

12.1.2 Explicit prediction of future behavior

The method requires the definition of a numerical model of the system, which makes it possible to predict the future behavior of the system This discrete model results mainly from a preliminary offline identification This feature makes it possible to classify predictive control in the big family of Model Based Control (MBC)

12.1.3 Optimization by minimization of a quadratic criterion

The optimization which makes it possible to obtain the control law is done by minimizing a quadratic criterion with finite horizon referring to the errors of future prediction, the variance between the predicted output of the system and the future setting or the reference trajectory inferred from this setting

12.1.4 Principle of the sliding horizon

The elaboration of a sequence of future controls results from the preceding minimization, which is optimal in what the quadratic criterion is concerned, out of which only the first value is applied to the system and the model

The preceding steps are then repeated during the following sampling period according to the principle of sliding horizon, as seen in Figure 12.1

Figure 12.1 Principle of sliding horizon

The objective of the polynomial predictive regulator obtained by minimizing the criterion is that the predicted output joins the setting or the reference trajectory on a given prediction horizon The principles that we have just mentioned make it possible to establish the operation diagram in Figure 12.2

Figure 12.2 Operation principle of a predictive algorithm

Hence, the principle of the sliding horizon means that only the control at the

present moment u(t) is applied on the system Therefore, it is possible to limit the

number of estimated values of the sequence

12.2 Generalized predictive control (GPC)

12.2.1 Formulation of the control law

The objective of this section is to indicate the fundamental points of the

predictive structure considered [CLA 87a, CLA 87b], in the monovariable case,

from the mathematical translation of the preceding general concepts up to the

obtaining the equivalent polynomial regulator

12.2.1.1 Definition of the numerical model

All the forms are allowable for the model but the input/output polynomial

approach by transfer functions is preferred

Traditionally the model is represented as CARIMA (Controlled AutoRegressive

Integrated Moving Average):

)(

)()1()()(

)

∆+

−

=

q

t t

u q B t y

q

where ∆(q−1)=1−q−1, u (t) and y (t) are the input and output of the model, ξ (t)

is a centered white noise, q−1 is the delay operator and A(q−1) and B(q−1) are

polynomials defined by:

=

+++

a a n n

n n

q b q

b b q

B

q a q

a q

A

)

(

1)

(

1 1 0 1

1 1 1

This model, which is also called incremental model, introduces an integral action

and makes it possible to undo all the static errors with respect to the input or step

function interference

12.2.1.2 Optimal predictor

The predicted output y(t+ j/t) is traditionally decomposed into a free and

forced response [FAV 88], including a polynomial form meant to properly conclude

the final polynomial synthesis:

forced response free response

y t+j t =F q− y t +H q− ∆u t− +G q− ∆u t+ − +j J q− ξ t+j

The unknown polynomials F j,G j,H j, J j are single solutions of Diophantus

equations, which are obtained by equality of the inputs and output of transfer

functions of equations [12.1] and [12.3] and they are solved recursively:

)()()()

(

1)()

()(

)

(

1 1 1

1

1 1

1 1

=+

∆

q J q B q H q q

G

q F q q J q A

q

j j

j j

j q J

1 1

1 1

H

q A q

F

j j

The set of calculations may be done in real-time off loop The optimal predictor

is finally defined by considering that the best noise prediction in the future is its

mean (here supposed as zero), let us suppose that:

12.2.1.3 Definition and minimization of the quadratic criterion

The control law is obtained by minimizing a quadratic criterion pertaining to

future errors with a weighting term on the control:

with: ∆u t j( + ≡) 0 for j N≥ u

The criterion requires the definition of four adjustment parameters:

–N1: minimal prediction horizon;

–N2: maximal prediction horizon;

–N u: prediction horizon on the control;

–λ : weighting coefficient on the control

12.2.1.4 Synthesis of the equivalent polynomial RST regulator

The minimization of the criterion is based on writing the prediction equation

[12.5] and the cost function [12.6] in a matrix form, such as:

~

~ w ) 1 ( )

~

G

w ) 1 ( )

−

−

∆ + +

−

−

∆ + +

=

t u t y

t u t y

T 1 1

)1(

)(

~

)()

(

)()

(

2 1

2 1

−+

N N

N t u t

u

q H q

H

q F q

Traditionally, in a predictive control, only the first value of the sequence,

equation [12.8] is applied to the system, according to the principle of the sliding

Based on the above relation, it is finally possible to obtain the polynomial

representation of the equivalent regulator as indicated in Figure 12.3 This traditional

RST structure enables the implementation of the control law by a simple

difference equation:

)()()()()()

Figure 12.3 Structure of the equivalent polynomial regulator

We observe that polynomial T (q) encloses the non-causal structure (positive

power ofq) inherent to the predictive control

The interest resulted from the RST representation (actually very general because

any numerical control law can be modeled this way [LAN 88]) is that, finally, the

real-time loop proves to take little calculation time as the control applied to the

system is calculated through a simple difference equation [12.10] The three

polynomials R, S, T are actually elaborated offline and uniquely defined as soon as

the four adjustment parameters are chosen

Consequently, this type of control favors the selection of short sampling periods

and it proves to be well-adapted to the control of fast electro-mechanical systems

(machine-tool, high-speed machining, etc.)

Another major interest in the RST structure pertains to the study of stability of

the corrected loop and thus the characterization of stability of the elaborated

predictive control, which is from that moment on possible for a set of parameters of

the fixed criterion This study is examined in the following section

12.2.2 Automatic synthesis of adjustment parameters

The definition of the quadratic criterion [12.6] showed that the user must set four

adjustment parameters However, this choice of parameters proves to be difficult for

a person who is not a specialist because there are no empirical relations which make

it possible to relate these parameters to traditional “indicators” in control such as

stability margins or a bandwidth

Based on the study of a great number of single-variable systems, it is however

possible to issue some “rules” based on the traditional criteria of stability and

robustness [BOU 92] that we summarize

12.2.2.1 Criterion of stability and robustness

First of all, the objectives of stability are related to the study in Bode, Black or

Nyquist planes of the transfer function of the open loop corrected by the predictive

regulator:

)()()(

)()()

1 1 1 1

q R q B q q

It is generally agreed that a “good” adjustment is characterized by:

– a phase margin ∆ϕhigher to 45°;

– a minimal gain margin ∆G from 6 to 8 dB (decibels)

The objectives of robustness are linked to the calculation of the delay margin

dB)0

at frequency angular

gaprad,

in

ωϕ

to the study, in the scalar plane, of the direct sensitivity functions σ and d

complementary sensitivity functions σ : c

)()()

()()(

)()()(

1 1 1 1 1 1

1 1 1

∆

∆

=

q R q B q q q S q

A

q q S q A d

)()()

()()(

)()(

1 1 1 1 1 1

1 1 1

∆

=

q R q B q q q S q

A

q R q B q c

It is generally agreed that a “good” adjustment is characterized by:

– a delay margin higher than a sampling period;

– a direct sensitivity function of a module lower than 6 dB;

– a complementary sensitivity function of a module lower than 3 dB

12.2.2.2 Selection procedure of the criterion parameters

From the criteria formulated above with the help of the traditional tools of scalar

Automation, it is possible to choose the sets of satisfactory adjustment parameters:

– N1: prediction horizon lower on the output The product N 1T e (T esampling

period ) is chosen as equal to the pure delay of the system;

– N2: prediction horizon higher on the output The product N 2T e is limited by

the value of the response time The bigger N2 is, the more stable and slower the

corrected system becomes;

– N u : prediction horizon on the control Choosing N u equal to 1 simplifies the

calculation and does not penalize the stability margins (on the contrary, a higher

value tends to decompose the phase margin);

– λ : weighting coefficient on the control This parameter is related to the gain of

the system, through the empirical relation:

)(

tr GTG

=

opt

The choice of parameters is frequently limited to a bi-dimensional search (N2

and λ ) ending with the selection of a “good” adjustment

12.2.3 Extension of the basic version

Based on the preceding easy version, several derived strategies were developed,

which made it possible to recognize:

– closed loop pre-specified dynamics (structure of multiple reference models);

– several variables to control (cascade structure);

– constraints imposed on the input and output signals

12.2.3.1 Structure of multiple reference models

The aim of this predictive structure of multiple reference models is double

Firstly, it makes it possible to impose a reference trajectory through a stable pursuit

of a model determined by the user who tones down the conformity with the setting

This pursuit model imposes the dynamics of the looped system (input/output

behavior) and it may be considered as a pole placement

It is also a matter of weakening the quick control variations that we can

sometimes recognize through the preceding algorithm, by trying to recreate the

reasonable reference control that must be applied to the system in order to obtain, at

the output, the reference trajectory and by creating in the criterion a minimization on

the control error and not only on the control

The digital model of prediction is defined here again as CARIMA:

)(

)()1()()(

)

∆+

−

=

q

t t

u q B t y

q

The pursuit model chosen by the user makes it possible to specify the reference

trajectory y r (t) that the output of the system will have to follow:

)()()

()

(q 1 y t q 1B q 1 w t

where:B r(q−1)=B(q−1)P(q−1)

B

A q

)

(q−1

A r is generally a second degree polynomial making it possible to impose a

desired response time as well as an adapted damping coefficient

Coupled to the reference trajectory y r (t), a reference control u r (t), which is

allowed by the system, is equally defined, the two trajectories being related by the

relation:

)()()

()

(q 1 y t q 1B q 1u t

In order to avoid the reverse of the model and the stability problems related to

polynomial B(q−1) that may result, equation [12.18] can be formulated again based

on relation [12.17] by:

)()()()()

The cost function is henceforth a weighted sum affecting the squares of the

output predicted errors and the squares of the future control error increments:

t J

∆

=+

+

−+

=+

)()()(

)()()(

j t u j t u j t

j t y j t y j t

r u

r y

ε

ε

Based on relations [12.16], [12.17] and [12.18], we notice that, on the one hand,

the increments of control errors and, on the other hand, the output errors are linked

by the relation:

)()1()()()

which corresponds exactly to the CARIMA structure [12.16], parameterized again in

terms of signals pertaining to input/output errors The entire theory previously

developed in the case of the GPC “traditional” algorithm can be preserved by

replacing in the minimization process y by ε , y ∆ u by ε and w by 0 (the system u

must indeed follow a zero error setting) From this moment on, the minimization of

the quadratic criterion [12.20] reaches the optimal sequence:

with: εu opt =⎡⎣εu( )t opt " εu(t N+ u−1)opt⎤⎦T

Here again, only the first value of the sequence, equation [12.22], is applied to

the system, according to the principle of sliding horizon:

We infer from it the equivalent polynomial regulator of this restated problem in

terms of error signals:

)()()()

R and S(q−1) as those obtained through the traditional algorithm; only

polynomial T (q) is modified, becoming a causal rational fraction and explicitly

considering the pursuit model chosen by the user Furthermore, the calculation of the

input/output closed loop makes it possible to verify that the resulting dynamics is

defined by the pursuit model, which is not at all the case of the transfer function

between the output and the interference

12.2.3.2 Cascade structure

The cascade structure suggested makes it possible, in the case of a two-loop

version, to simultaneously control two variables (for instance speed and position, for

the regulation of the electro-mechanical systems) In the internal loop it includes a

predictive structure with multiple reference models developed above, paired to a

GPC traditional algorithm for the external loop, as indicated in Figure 12.5

The synthesis of the regulator of the internal loop is considered according to the

GPC/MRM strategy of the previous section, in such a way that the internal regulator

The predictive model used for the synthesis of the external regulator consists of

two terms: on the one hand the model corresponding to the asymptotical behavior of

the closed internal loop and on the other hand the model issued from the external