Mikles J., Fikar M. Process Modelling, Identification, and Control
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442 9 Predictive Control
down the closed-loop system substantially, gave good results. This is because
a neutralisation reactor control is known to be not very succesful with linear
controllers.
9.9 References
There are several books dealing with predictive control:
D. M. Prett and C. E. Garc´ıa. Fundamental Process Control. Butterworths,
1988.
R. R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control. Prentice
Hall, Englewood Cliffs, New Jersey, 1990.
E. F. Camacho and C. Bordons. Model Predictive Control. Springer-Verlag,
London, 1999.
B. Kouvaritakis and M. Cannon, editors. Non-linear Predictive Control:
Theory and Practice. The Institution of Engineering and Technology,
London, 2001.
J. M. Maciejowski. Predictive Control with Constraints. Prentice Hall, Lon-
don, 2002.
J. A. Rossiter. Model-Based Predictive Control: A Practical Approach.CRC
Press Inc., 2003.
Some selected journal papers dealing with predictive control and stability:
D. W. Clarke, C. Mohtadi, and P. S. Tuffs. Generalized predictive control -
part I. The basic algorithm. Automatica, 23(2):137 – 148, 1987.
D. W. Clarke and R. Scattolini. Constrained receding-horizon predictive con-
trol. IEE Proc. D, 138(4):347 – 354, 1991.
E. Mosca and J. Zhang. Stable redesign of predictive control. Automatica, 28:
1229 – 1233, 1992.
B. Kouvaritakis, J. A. Rossiter, and A. O. T. Chang. Stable generalised
predictive control: an algorithm with guaranteed stability. IEE Proc. D,
139(4):349 – 362, 1992.
M. Fikar and S. Engell. Receding horizon predictive control based upon Youla-
Kuˇcera parametrization. European Journal of Control, 3(4):304 – 316,
1997.
M. Fikar, S. Engell, and P. Dost´al. Design of predictive LQ controller. Ky-
bernetika, 35(4):459–472, 1999.
H. Chen and F. Allg
¨
ower. A quasi-infinite horizon nonlinear model predictive
control scheme with guaranteed stability. Automatica, 34(10):1205 – 1217,
1998.
W. H. Kwon and A. E. Pearson. On feedback stabilization of time-varying
discrete linear systems. IEEE Tr. Aut. Control, 23(3):479 – 481, 1978.
9.9 References 443
J.-W. Lee, W. H. Kwon, and J. Choi. On stability of constrained receding
horizon control with finite terminal weighting matrix. In CD-ROM Pro-
ceedings of ECC’97, Bruxelles, Paper No. 93, 1997.
J. A. Rossiter, J. R. Gossner, and B. Kouvaritakis. Infinite horizon stable
predictive control. IEEE Trans. Automatic Control, 41(10):1522 – 1527,
1996.
J. B. Rawlings and K. R. Muske. The stability of constrained receding horizon
control. IEEE Trans. Automatic Control, 38(10):1512 – 1516, 1993.
A. Zheng and M. Morari. Stability of model predictive control with mixed
constraints. IEEE Trans. Automatic Control, 40(10):1818 – 1823, 1995.
P. Zelinka, B. Rohal’-Ilkiv, and A.G. Kuznetsov. Experimental verification of
stabilising predictive control. Control Engineering Practice, 7(5):601–610,
1999.
Tuning of the MBPC parameters can be found for example in these papers:
A. R. McIntosh, S. L. Shah, and D. G. Fisher. Analysis and tuning of adaptive
generalized predictive control. Can. J. Chem. Eng., 69:97 – 110, 1991.
R. Shridhar and D. J. Cooper. A tuning strategy for unconstrained SISO
model predictive control. Ind. Eng. Chem. Res., 36(3):729 – 746, 1997.
R. Shridhar and D. J. Cooper. A tuning strategy for unconstrained multivari-
able model predictive control. Ind. Eng. Chem. Res., 37(10):4003 – 4016,
1998.
References to explicit predictive control and Multi-Parametric Toolbox
that can efficiently solve a broad range of explicit and on-line formulations:
A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit
linear quadratic regulator for constrained systems. Automatica, 38(1):
3–20, 2002.
P. Grieder, M. Kvasnica, M. Baoti´c, and M. Morari. Stabilizing low complexity
feedback control of constrained piecewise affine systems. Automatica,41
(10):1683–1694, 2005.
M. Kvasnica, P. Grieder, and M. Baoti´c. Multi-Parametric Toolbox (MPT),
2004. Available from http://control.ee.ethz.ch/~mpt/.
V. Sartori, E. Zanderigo, M. Kvasnica, and M. Morari. Optimal Infu-
sion of Intravenous Morphine and Ketamine – A Mixed-Integer Linear
Programming Application. In IEEE Engineering in Medicine and Biology
Society, Shanghai, China, 2005.
P. Tøndel, T.A. Johansen, and A. Bemporad. An algorithm for multipara-
metric quadratic programming and explicit MPC solutions. Automatica,
39(3):489–497, 2003.
Simulation and experimental studies from section 9.8 have been published
in more detail in:
444 9 Predictive Control
A. Rusn´ak, K. Fikar, M. Najim, and A. M´eszaros. Generalized predictive con-
trol based on neural networks. Neural Processing Letters, 4(2):107 – 112,
1996.
M. Fikar and A. Draeger. Adaptive predictive control of a neutralization
reactor. In J. Mikleˇs, editor, Preprints of 10th Conf. Process Control’95,
June 4 –7, Tatransk´e Matliare, Slovakia, volume 1, pages 153–156, 1995.
9.10 Exercises
Exercise 9.1:
Derive the control law from Example 9.1 on page 413 for the state-space model
and show that it is equivalent to the input-output model.
Exercise 9.2:
One possible way of including an integrator to a state-space model has been
demonstrated in section 9.3.4. Another possibility is to consider the new state
vector of the form
¯x(k)=
Δx(k)
y(k)
(9.157)
Derive the modified state-space description in this case.
Exercise 9.3:
Unstable processes can pose problems when the predictor is derived in sec-
tion 9.3.4. Calculation of the terms A
i
for unstable matrix A can in this case
be ill-conditioned as higher powers of A
i
do not converge to zero. One possible
remedy is to stabilise the controlled process using the state feedback control
law of the form
u(k + i)=−Kx(k + i)+c
i
(9.158)
where c
i
is a new optimised parameter and it is supposed that it is zero after
k + N
u
. Derive prediction equations and find the solution minimising the cost
function without constraints.
Exercise 9.4:
Consider an unstable process with the transfer function of the form
G(z
−1
)=
1
1 −2z
−1
+ z
−2
(9.159)
• Calculate the closed-loop poles for N
1
=1,λ =0,N
u
=1aN
2
=
1, 2, 3,...,10. Find the smallest N
2
that guarantees stable closed-loop
system.
• Choose suitable parameters of the predictive controller to realise the dead-
beat control.
10
Adaptive Control
Dynamic and static properties of controlled processes often change in time.
Process control is always based on a process model. If process characteristics
are changing in time, it seems to be appropriate to use a time-variant process
model – model that adapts to changes in the process.
There are many approaches and definitions of adaptive control.
Adaptive control is such a control that adapts to unknown and changing
process behaviour.
Adaptive control can be in principle realised in several ways. One of them
is the self-tuning control. Here, one starts with a control design for a known
process. However, as process parameters are unknown, they are obtained us-
ing a recursive parameter identification algorithm assuming that the process
model is linear.
There are several reasons for use of self-tuning controllers. The first are
changes of operating points of technologic processes that are nonlinear. Lin-
ear models for each operating point are different and the controller should
change for each of them. Another reason are changes in physical process prop-
erties caused by material ageing, wearing etc. The third reason is the presence
and elimination of unmeasurable disturbances, perturbations, and unmodelled
process dynamics.
The self-tuning controller structure consists of two loops. The inner loop
contains a linear feedback controller with time-varying parameters. The outer
loop contains identification algorithm and a box that adjusts the controller
parameters based on the identification results. The linear controller design is
separated from the identification part. It can be based on LQ, LQG or other
control design. Parameter estimation is usually implemented using a least
squares method. A self-tuning control system is a complex nonlinear system
that converges under some conditions.
To start adaptive control, it is generally advisable to use a nominal sta-
bilising controller until the identified process parameters converge. As it was
shown in Section 6.3.1, if parameter estimation algorithm starts without any
a priori knowledge about the process (covariance matrix contains large values
446 10 Adaptive Control
on the main diagonal), convergence can be attained in at minimum so many
steps as there is the number of parameters. After that it is possible to switch
the controllers usually using some kind of bumpless transfer methods.
Self-tuning control can be used based on both discrete-time and continuous-
time models.
10.1 Discrete-Time Self-Tuning Control
This section deals with an explicit self-tuning control based on discrete-time
process models. The explicit self-tuning algorithm is implemented in three
steps.
The first step is to estimate process model parameters of a given structure.
In the second step controller parameters are calculated given the process
model parameters from the first step and a suitable design method.
Manipulated variable is calculated and realised in the third step based on
the control law.
This procedure is repeated in each sampling instant.
The self-tuning control system with a discrete-time controller calculates
identified parameters and a new controller in each sampling instant. The ma-
nipulated variable and identified parameters then remain constant until the
new sampling time.
Block diagram of a self-tuning discrete-time control system of a singlevari-
able process is shown in Fig. 10.1.
_
Parameter
estimation
Process
Sampler
Discrete-time
controller
Calculation of
controller
parameters
y(k)
T
s
u(k)
w(k)
Fig. 10.1. Block diagram of a self-tuning discrete-time controller
10.3 Examples of Self-Tuning Control 447
10.2 Continuous-Time Self-Tuning Control
Block diagram of a self-tuning control with a continuous-time controller and
singlevariable controlled process is shown in Fig. 10.2.
_
Filter
Continuous-time
controller
w(t)
u(t)
T
s
y(t)
Filter
Calculation of
controller
parameters
Parameter
estimation
Process
T
s
Fig. 10.2. Block diagram of a self-tuning continuous-time controller
The self-tuning continuous-time control uses a continuous-time controller
but its parameters are changed in discrete time intervals based on the re-
cursive identification of the continuous-time process. Inputs to identification
are filtered inputs, outputs and their derivatives. Outputs of the identifica-
tion block are piece-wise constant (with respect to time) parameters of the
continuous-time process model. See Section 6.3.3 for more details.
The self-tuning continuous-time control system is implemented in three
steps in the same way as its discrete-time counterpart.
10.3 Examples of Self-Tuning Control
The above described adaptive control algorithms will be tested on several
simulation examples. Testing of algorithms is usually based on step changes
of reference and disturbance variables, slowly varying parameters, etc for an
unknown system after some adaptation phase.
In all studies the recursive parameter estimation LDDIF with directional
forgetting is employed. Dead-beat, pole-placement, LQ, and predictive control
algorithms are used. Simulations results show that adaptive control algorithms
converge quickly.
Simulation studies in MATLAB/Simulink environment use the Polyno-
mial Toolbox for MATLAB and Simulink identification blockset IDTOOL.
Although other choices are possible, this corresponds to the authors and their
colleagues experience.
448 10 Adaptive Control
One of the fundamental problems of the feedback linear control in general
and of ST control in particular is the issue of system structure selection. This
in linear case means determination of degrees of numerator and denominator.
If the model order is chosen too low, control performance will suffer. On
the other side, if the system degree is too high, its transfer function may
contain similar factors in numerator and denominator. As we have seen, the
Diophantine equation can be solved only if both polynomials are coprime.
Therefore, numerical problems and ill-conditioning can be observed in this
case.
Also the choice of the sampling period is very important in the ST control.
If not otherwise explicitly stated, all MATLAB simulations used default
values of the library IDTOOL. This includes LDDIF algorithm with increasing
exponential forgetting strategy (6.98) with λ
0
=0.985, minimum trace of
the covariance matrix δ =0.01, zero initial parameter estimates, and initial
covariance matrix with values of 10
6
on the diagonal.
10.3.1 Discrete-Time Adaptive Dead-Beat Control of a Second
Order System
Consider a continuous-time controlled system with the transfer function
www
G(s)=
b
s1
s + a
s0
a
s2
s
2
+ a
s1
s +1
(10.1)
This system will be controlled by an adaptive feedback controller according to
Simulink scheme ad031s.mdl shown in Fig. 10.3. Initialisation of this diagram
is performed with the script ad031init.m that sets the process and discrete
integrator blocks. This file can be loaded automatically when the diagram is
opened using the MATLAB command (the scheme has to be already opened)
set_param(’ad031s’, ’PreLoadFcn’, ’ad031init’);
It is necessary to save the scheme afterwards. The first parameter of this
function determines the name of the Simulink scheme and the third determines
the name of the script to be executed.
Program 10.1 (Initialisation of the scheme in Fig. 10.3 – ad031init.m)
z=2;t1=1;t2=2;ts=1;
G = tf(z, conv([t1 1],[t2 1]));
zinteg = tf([1 0],[1 -1], ts);
The corresponding discrete-time transfer function is in the fractional rep-
resentation given as
G(z
−1
)=
b
1
z
−1
+ b
2
z
−2
1+a
1
z
−1
+ a
2
z
−2
(10.2)
10.3 Examples of Self-Tuning Control 449
param
To Workspace1
data
To Workspace
Step
Scope
stdiscon
ST discrete
controller design
G
Process
Pred. error
D. identification
SISO
Discrete identification
zinteg
Discrete
Integrator
Cov. matrix
1+.4z +.2z
−1 −2
1+.7z
−1
Controller
u
y
B
A
w
w
Fig. 10.3. Scheme ad031s.mdl in Simulink for dead-beat tracking of a continuous-
time second order system with a discrete-time adaptive controller
Parameters a
1
, a
2
, b
1
, b
2
are estimated in each sampling instant. The
controller without integral part is given by polynomials
Y (z
−1
)=y
0
+ y
1
z
−1
+ y
2
z
−2
,X(z
−1
)=x
0
+ x
1
z
−1
(10.3)
that are solution of the Diophantine equation
(1 + a
1
z
−1
+ a
2
z
−2
)(1 −z
−1
)X(z
−1
)+(b
1
z
−1
+ b
2
z
−2
)Y (z
−1
) = 1 (10.4)
The general transfer function of the controller is then of the form
R(z
−1
)=
Y (z
−1
)+(1− z
−1
)A(z
−1
)T (z
−1
)
X(z
−1
) −B(z
−1
)T (z
−1
)
(10.5)
If T (z
−1
) = 0 then the controller defined as
R(z
−1
)=
y
0
+ y
1
z
−1
+ y
2
z
−2
x
0
+ x
1
z
−1
(10.6)
guarantees the finite-time response to the step change of reference and distur-
bance variables in a minimum number of control steps.
The controller parameters are determined in the each sampling instant
from (10.4) based on estimated parameters a
1
, a
2
, b
1
, b
2
. From the actual
control law is calculated the control action. This algorithm is repeated in each
sampling instant.
Parameter estimation is for the case shown in Fig. 10.3 implemented using
the discrete-time SISO identification block from the library IDTOOL and the
controller parameters are calculated in the S-function stdiscon.
We can notice in the Simulink scheme that in contrast to Fig. 10.1 it
contains only one feedback loop consisting of the controlled process and the
controller. The second loop is closed using the information about the ac-
tual controller parameter values from the block stdiscon into the block
450 10 Adaptive Control
Controller. To achieve this, MATLAB function set_param is used within
stdiscon. The implementation of the function stdiscon is as follows.
Program 10.2 (S-function stdiscon.m)
function [sys,x0,str,ts] = stdiscon(t,x,u,flag,tsamp)
switch flag,
case 0,
[sys,x0,str,ts]=mdlInitializeSizes(tsamp);
case 2,
sys=mdlUpdate(t,x,u);
case 3,
sys=mdlOutputs(t,x,u);
case 9,
sys=mdlTerminate(t,x,u);
otherwise
error([’Unhandled flag = ’,num2str(flag)]);
end
function [sys,x0,str,ts]=mdlInitializeSizes(tsamp)
sizes = simsizes;
sizes.NumContStates = 0;
sizes.NumDiscStates = 0;
sizes.NumOutputs = 0;
sizes.NumInputs = 5;
sizes.DirFeedthrough = 1;
sizes.NumSampleTimes = 1;
sys = simsizes(sizes);
x0 = [];
str = [];
ts = [tsamp 0];
% Controller initialisation
Denominator = ’[1 .7]’;
Numerator = ’[1 .4 .2]’;
set_param(’ad031s/Controller’, ’Numerator’, Numerator, ...
’Denominator’, Denominator);
function sys=mdlOutputs(t,x,u)
if t>4 % begin adaptive control
b = [1 zi zi^2] * [0; u(1:2)];
a = [1 zi zi^2] * u(3:5);
a = a * (1-zi); % add integrator
[x0,y0]=axbyc(a,b,1);
tt = 0;
xx=x0-b*tt;
yy=y0+a*tt;
10.3 Examples of Self-Tuning Control 451
Denominator = sprintf(’[%f %f]’,xx{0},xx{1});
Numerator = sprintf(’[%f %f %f]’,yy{0},yy{1}, yy{2});
set_param(’ad031s/Controller’, ’Numerator’, Numerator, ...
’Denominator’, Denominator);
end
sys=[];
function sys=mdlUpdate(t,x,u)
sys = [];
function sys=mdlTerminate(t,x,u)
sys = [];
0 5 10 15 20
−3
−2
−1
0
1
2
3
4
5
6
(a)
t
u,y,w
u
y
w
0 5 10 15 20
−1
−0.5
0
0.5
(b)
t
Param
b1
b2
a1
a2
Fig. 10.4. Trajectories of (a) input, output, and setpoint variables, (b) identified
parameters of the discrete-time process model controlled with a dead-beat controller
Fig. 10.4a shows trajectories of input and output variables of the controlled
process and Fig. 10.4b gives the trajectories of the estimated parameters.
The nominal controller was chosen as
R(z
−1
)=
1+0.4z
−1
+0.2z
−2
1+0.7z
−1
(10.7)
The adaptive closed-loop system was controlled in the first four sampling
periods with this nominal controller. Afterwards, the controller calculated in
the block stdiscon was used. The controlled process used in the simulation
was the same as its model.
10.3.2 Continuous-Time Adaptive LQ Control of a Second Order
System
Consider a continuous-time controlled system with the transfer function
www