ADAPTIVE DECOUPLING OF MIMO SYSTEMS WITH CONSTRAINED INPUTS.
BY
MOHAMED BENNAMOUN, B. Sc. (Eng. )
A thesis submitted to the Department of Electrical Engineering in conformity with the requirements
for the degree of Master Science.
Queen's University Kingston, Ontario, Canada
August, 1988
copyright 0 M.Bennamoun, 1988
ACKNoWLmGEMENrs
express .y sincere appreciation for his precious guidance.
Thanks .re due to my friend, Mahmoud Katt.n, whose valuable input gave me r.n.w.d confidence during the most critic.l p.riods.
I would like to
therefore, greatly inspiration during the course of research.
Finally, l take thi. opportunity to thank my f .. ily, ""pecially my p.r.nt. who have .anaS.d, de.pite th. di.tanc., to provide .. with
the nece.sary str.ngth to complete this acadeýc soa1.
This thesis is the culmination of a collective effort. Its completion would not have been possible without the help of Dr. K.
K. Bayoumi who was a constant source of encouragement and
NSERC r"""" rch grant were needed and, appreciated.
Gratitude mu.t al.o be expressed to the Algerian Government for its sponsor.hip. The funds made available through Dr. BayoUlli'.
TABLE
OFCýTS
ABST'B.A.CT 11
ACDOVI.EOOEKERTS ..."..."."""""".""".""""""""""""""""....". 111
TABLE OF CONTENTS i.
NOTA.TION """"""."."""""""""""""""""""""""""""""""""""""""""""""" vi
l IBTl.ODOCTIOR ...""".."""""""""""""""""""""""""""""""""""" 1 2 UVIW 01' STABDüD METHODS ABD TECllRIQUES 1'01.
SELF-TURING COHTaOLLEl DESICR...."""..""""""".."."""""..".""." 7
2.1 Introduction 7
2.2 Sy.t.. IIOdels ..."."...".... 10 2.3 KinL.u. variance and detuned miný variance
""lf-tuning controllers 14
2.3.1 De.ilftS for .calar .y.te 14
2.3.1.1 5150 minimum variance .elf-tuning controll.r"... 14 2.3.1.2 5150 generalized minimum varianca controller .... 20 2.3.1.3 5150 detuned minimum variance self-tuninS
controll.rs 26
2.3.2 o..ian- for Sultivariabl"" y.t 27 2.3.2.1 KIKO minimum varianc. s.lf-tuning controll.r "... 27 2.3.2.2 detun.d minimum variance self-tuninS
controll.r. for KIMO .y.te 38
2.4 Pol.-z.ro placem.nt self-tuninS controll.r "... 39 2.5 Stat"" pac. self-tun.r "... 44
2.5.1 5ISO state-space STC 44
2.5.2 KIKO state-space STC 48
2.6 Parameter estimation 50
2.6.1 Th. RLS algorithm 52
2.6.2 The square root algorithm 53
2.6.3 RLS using Householder transformation 54
2.6.4 R.cursive U-D factorization 56
3 AN ALGORITHM FOI. THE CALCULATION OF THE INTDACTOI. KAnIX "".." 59
3.1 Introduction 59
3.2 Concept .nd history of the Inter.ctor matrix 63
3.3 D.finitions and mathematic.l not.tion"... 64
3.4 The algorithm for a nonsquare .yste "... 67
3.5 Nua.ric.l .xample for th. nonaquare .y.t ""..."... 74
3.6 Coaput.tion burden ...""."."... 75
3.7 The algorithm for. square .y.te "..."... 76
3.8 Numeric.l example for. square .y.t 80 3.9 Conclusions 84 4 ADAPTIVE DICOUPLING OF A CUSS OF HULTIVAnABLI DYRAKIC SYSTEMS USIRG OU'l'PUT FDDBACE ...""".""."""""""."""""".".... 86
4.1 Introduction 86 4.2 B.ckground to the proble. and objective of thi. ch.pt.r 86 4.3 The aultiv.ri.bl "" y.te"" nd the decoupling probl 89 4.4 The adaptive decoupling .lgoritha 94 4.5 S1au.lation .xample "...".""""... 100
4.6 Conclusions ..."... 102
5 CO.CLUSIONS ABD RECOHMERDATIO.S ...".."."."""""""""""... 119
5.1 Conclusions ...".".."..""... 119
5.2 Recomm.ndations for future re.earch 120 lt.EFDEN CES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
APPENDIX A: DEllIVATION OF THE MINIMUM VAIllANCE
sn
OF ASTllOH AND VITTENMARlC (1973) 129 APPENDIX B: D!l.IVATION OF THE DATA MATI.IX """""".""""""".."... 132 VITANOTATION
Abbreviation.
a.8.z
ARMAX
DMVR
CKVC
LQC LS LSE
MFD
MIKO
ML
KLE
NV
KVSTR.
ODE
PK
PPR.
PZP
llLP
RLS
S1S0
STC
STR.
U-D
WKVC
Almost all z
Autoregressive moving average with exogeneoua variable.
Detuned minimum variance controller Ceneralized .iný variance controller Linear quadratic Caua.ian
Least squares
Least .quare. e.tillate.
Matrix fraction de.cription Multi-input multi-output Maximum likelihood
Kaxý likelihood estimate Minimum variance
Minimum variance self-tuninc recul.tor Ordinary differential equation
Polynoaial matrix
Pole placement regulator Pole-zero placement aelatively left prime aecur.ive least square.
Single-input single-output Self-tuning controller Self-tuning regulator Upper-diagonal
Weighted minimum variance controller
Roman symbols
A(z -1) A(z -1) B(z -1) B(z -1) C(Z -1) C(Z -1) D(z) e(t) E
l
B(z)
h (z)
lj
J
k
n
n "
n
c
N
R(z)
u(t) u(t) uO(t)
output polynomial in the system description
output polynomial matix in the system description input polynomial in the system description
input polynomial matrix in the system description noise polynomial in the system description
noise polynomial matrix in the system description diagonal term of the lnteractor
noise vector
component of D(z) in the interactor .. trix unimodular matrix in the interactor .. trlx polynomial element of B(z)
cost function
system time delay (integer) upper bound for system dynamics
degree of output polynomial in the system description degree of input polynomial in the system description degree of noise polynomial in the system description number of observations
Numerator of the system transfer function matrix output weighting matrix in a minimum cost function input weighting matrix in a minimum coat function inverse information matrix for LSE
integer sampling time
system transfer function matrix input variable at time t
input vector at time t
optimal control at time t
w forgetting factor
x(t) state vector at time t
x(t) memory vector at time t
Mathematical convention
k summation operator det determinant operator
data matrix
data vector component of the data _trlx covariance matrix
variance
parameter vector parameter matrix
filtered partial decoupling error partial decoupling error
input polynomial on the lea.t square. .odel memory vector at time t
output polynomial on the least square" .odel reference vector at tt.. t
output vector at time t
reference value at ti.. t
output variable at time t
optimal prediction of th. output vetor
a2
"
T vector or matrix transposition
q forward shift operator
-1
q backward shift operator
E expectation operator z(t)
y
_.
t)y (t)
r
Creek symbols Q(. -1)
IJ(z -1)
ý(t) ý(t)
6
,
y(t) y(t)
y (t)
Z'
CHAPTER 1 INTRODUCTION
The subject area of designing controllers can be viewed as dealing with the manipulation of the inputs to a system so that the out- puta may achieve certain specified objectivea. Typical applica- tions include:
*
Control of prosthetic devices and robots.*
Control of the aileron and elevators on an aircraft.*
Control of the flow of raw materials in an industrial plantto yield a desired product.
*
Control of intereat and tariff rates to regulate the economy.*
Control of anesthetic dosage to produce a desired level ofunconsciousness in a patient
Thus, one can see that the application of control theory extends over many areas. These include the technological, biological, and socio- economic systems.
In regards to the applications in technological systems, one of the elements that has made a great impact on the state of the art of control systems is the digital computer. The great advances in cOlliputers have made them increasingly important as elements in
control systems. In particular, the development of cheaper and more reliable computers and the dramatic advances in micro- electronics have made it possible to implement more complex regulators which in the past would have been a formidable task if
undertaken manually or with the help of analog controllers.
Controllers become more attractive when they have the ability of
self-modification or self-adjustement in the face of varying parameters and environment, which is often the case in industry.
In that case adaptive controllers may be used advantageously.
Systems may be classified into:
Static or dynamic systems. Static systems are composed of simple linear gains or nonlinear devices and described by algebraic equations, and dynamic systems are described by differential equations .
Continuous-ttae or discrete-time systems. Continuous-tiae dynamic syste.s are described by differential equations, and discrete-time dynamic systems by difference equations.
Linear or nonlinear systems. Linear dynamic systems are described
by differential (or difference) equations having solutions that Equations describing non- are linearly related to their inputs.
linear dynamic systems contain one or more nonlinear terms.
Luaped or distributed parameters. Lumped-parameters, continuous- ti.e, dynamic systems are described by ordinary differential equations. Distributed-parameter, continuous-time, dynamic sys- tems by partial differential equations.
Ttme-Tarying or time-inTariant systems. Time-varying dynamic systems are described by differential (or difference) equations having one or more coefficients as functions of time. Time- in- variant (constant parameter) dynamic systems are described by
differential (or difference) equations having only constant coefficients.
Deterministic or stochastic systems. Deterministic systems have fixed (nonrandom) parameters and inputs, and stochastic systems have randomness in one or more parameters or inputs.
performance, constraints and robustness.
This is concerned with stability of the system,
1. Stability.
4. Constraints. Usually we are limited by physical constraints the system to reproduce desired output values.
3. Tracking performance. This is concerned with the ability of terms of bandwidth, damping, resonance, and so on.
effort, limits in the rate of change of control signals and so
percent overshoot, and so on, and in the frequency domain in (the regulator problem ) or
in the time domain in terms of rise time, settling time, the system responds. For linear systems, it can be specified
2. Transient response. Roughly this is concerned with how fast including boundedness of inputs, outputs, and states.
such as limits in the magnitude of the allowable control this thesis. The design of a controller may be different depen-
tmportant considerations: stability, transient response, tracking Nonlinear, distributed parameters systems are out of the scope of
dins on which kind of system we are dealing with.
2) to follow a time varying reference value ( the servo problem ).
In any control system design problem one can distinguish five 1) to achieve stationary control around a fixed reference value The control objective may either be:
I I
l
on. These factors ultimately place an upper limit on the achievable performance.
5. Robustness. This is concerned with the degradation of the performance of the system depending on certain contingencies, such as unmodeled dynamics, parameter variations, component failure, and so On.
One of the purposes of this thesis is to present a new algorithm
in order to compute the interactor matrix which may be useful for the design of MIMO self-tuning controllers when the system to be
An important characteristic.of a system is its time-delay. In the case of single-input single-output systems the delay structure is very transparent once we know the transfer function. In the' multivariable case it turns out that the delay structure of the tran.fer function matrix can be specified in terms of a polynomial matrix called the interactor matrix. Most of the work where this matrix is used suppose that this matrix is diagonal or is known.
However, it was shown by Chan and Goodwin (1982) that a general nondiagonal form of the interactor (resulting from certain linear dependences arising during successive extraction of the delay structure from the transfer matrix) had to be considered for robu.t minimum prediction errors (MPE) controllers.
Therefore. it is necessary to use an online algorithm to compute this matrix. This algorithm has to be fast and does not require too much memory storage. Not too much attention has been brought for this matter.
A review controlled possess different input-output time delays.
of the techniques used to design such a controller would be necessary.
constraints. In this case, the control objective may have to be adjuated so that adaptive decoupling may be achieved under constrained control signals.
Another objective of this thesis is to extend the work of Tade BayoWli and Bacon (1986) to decouple multivariable systems. In that work the control objective was to decouple systems with unknown parameters in an adaptive way. It was noted, however that
these In many the available violate
is a limit to achieved may
the control effort could assume very large magnitudes.
practical situations where there control signals, the results
The problem of decoupling multivariable systems is a growing area of research in the control field. The goal of decoupling a linear multivariable system is to reduce the system to a set of
"""" ntially non interacting loops. Controller design can then be carried out using single loop techniques.
Considerable attention has been directed to decoupling procedures in the recent years.
Th. organization of this thesis goes as follows. Chapter two presents a review of most of the relevant literature where standard methods and design techniques for self-tuning controllers are described. Chapter three presents the concept of the interactor matrix and the new algorithm that has been developed to compute such a matrix. This is then followed by some simulation
l
ex-.ple ". The extenalon of the adaptlve decouplln, al,orldba 1.
de.crlbed ln chapter four. Chapter flve lnclude. a ."
re.ult. of thl "" tudy and conclualona arl.ln, fro. lt.
"
CHAPTER 2
REVIEW OF ST ANDARD METHODS AND TECHNIQUES FOR
SELF-
TUNING CONTROLLER DESIGN.2.1 I INTRODUCTION:
There are many ways of classifying control systems. One such cl""" ification may be achieved by distinguishing the following type. of control problems:
*
Deterministic control (when there are no disturbances and th""ystem model can be described in a deterministic way. Moreover, the model is assumed to be known).
*
Stochastic control (when there are stochastic disturbances and when models are available for the system and for the di.turbances)*
Adaptive control (when there may be disturbances and the models whos. mathematical models may not be complet.ly specified).Stochastic adaptive controllers can be classified into dual and non-dual controllers based on the information pattern and the performance index.
If the performance index takes into account only the previous measurements and does not assume any future information to be available, the controller is said to be non-dual.
On the other hand the performance index can also be dependent on expected future observations. In such a case, we talk about dual controllers.
For a dual control, the idea is that the system input has a dual role: learning and regulation.
*
In regards to learning, the input introduces perturbations which yield information about the system dynamics and thus allows the parameter uncertainty to be reduced.*
Concerning regulation. the input tries to keep the output atthe desired value.
Often the two roles of the input may be conflicting and thus the controller must achieve an optimal compromise between learning
(which may require large perturbations) and regulation (which may only need relatively small signals).
At one extreme. by ignoring the uncertainty in the parameter estillates. one can design the control law as if the estimated parameters were the true system parameters. This approach is commonly called 'certainty equivalence' and involves the separation of the estimation and control problems.
Perhaps the best known certainty equivalence stochastic adaptive control law is the 'self·tuning' regulator.
The theory of self-tuning compromises the two aspects: Controller design and systell identification.
The general structure of a self·tuner is shown in Fig. 2.1.
Functionaly the self-tuner may be divided into the following three blocks:
1 di.t.urbane.
estimation methods. In fact:
and
,
extended squares,
minimum variance control, least
________ J
REGULATOR
e ontrol de.iln alaorith.
recur.iv. proce "" para "" t.r
"" ti "" tion
fia.2.1 Sebe.a of an adaptive controller.
approximation,
for the controller,
L/pu o/p y
r---ý---ýlplant.I---ýý--ýý
L
r
next control input.
of a suitable plant model.
input-output model.
parameters according to some prespecified design rule.
estimates and which synthesizes the appropriate controller
into the controller equations and are used to compute the
generalized least squares, instrumental variables, extended Kalman filtering and maximum likelihood.
stochastic
The regulator structure shown in Fig. 2.1 is very flexible
*
The plant can be described by a state space model or by an*
Many different parameter estimation schemes may be used e.g.,*
Finally,because it allows many different combinations of design and (ili) The updated coefficients of the controller are then inserted (il) A controller design algorithm that receives the parameter (1) A recursive estimator which is used to identify the parameters
generalized minimum variance control, pole placement, LQG
control, as well as control based on the phase and amplitude margins can be considered depending on the purpose of control.
on different kinds of controllers.
estimation of the parameters of an explicit process model.
(2.1.1)
Such a self-tuner is called an
Identification for a self-tuner may be explicit or implicit.
In an explicit algorithm. the identification phase deals with the
(ARKAX model representation).
that it can be described at discrete instants of tiSe by an
autoregressive moving average modal with exogenous variables estimation of an implicit process model.
In this thesis, we assume that the process is time-invariant, controllable and observable (Rosenbrock, 1970; Kailath, 1980) and
2.2 / SYSTEM MODELS:
taken at the sampling instant t.
Now let us expand on the system models, parameter estimation, and can b. expressed in terms of the regulator parameters. This gives
a significant simplification of the algorithm because the design
The SISO plant is represented by:
-1 -le -1 -1
A(q )y(t) - q B(q )u(t) + C(q )e(t) calculations are eliminated.
It is sometimes possible to reparameterize the process so that it
'iSplicit' self-tuning regulator/controller as it is based on the
where q is the forward shift operator such that q IEy(t)-y(t+k)
The time delay of this process is an integer number of sampling
intervals, ýl. This definition of k implies that b ý 0 , so that
o
polynomials A, Band C may have ehe form:
.ý
are deviationa from their steady state values;
(2.1.2b) (2.1.2a)
(2.L2c)
Oat-put yet) +
+
-D
-1 -1 -2 a
A(q ) - l + a1q + a q + + a q
2 D
a
-JI
C(q-1) _ l + c1q-l + + CD q 0
e
B(q-1) - b + b q-1 + + b q-JI b b ....... 0
o 1 Il 0
b
fla.2.2 Dl.crete .odel of a SISO ".,..te ".
Control input .ct.)
---+
the unie circle in the q-plane (stable polynomial)
uet)I---ý
multivariable equivalent include cases in which non stationary The disturbance e(t) is assumed to be a stationary sequence of
uncorrelated random variable with zero mean and variance q2.
The polynomial C(q-1 ) is assumed to have no roots outside or on The signal u(t) is the system input at time t; and y(t) and u(t)
The discrete time plant is shown in Fig. 2.2 _oorrelated
r_do. .equello e
It can be easily shown that the model (2.1.1) and its
Average) model.
disturbances in the process can be adequately modelled by the Box and Jenkins (1970) type of ARIMA (Autoregressive Integrated Moving
In general, this model is of the form:
-1 d -1
ý(q ) 'il D(t) - 0(q ) e(t) (2.1.3)
Where:
levels or trends in the time series data;
OCt) is the current value of the disturbance;
(2.1.4) (2.1.5)
-1 -1
C(q ) - 0(q ).
-1
V is the differencing operator ( 1 - q );
ý(q-l) is the autoregressive polynomial component;
0(q-1 ) is the moving average polynomial component;
eCt) is a white noise process with zero mean and variance q 2
Identification of the process through the model (2.1.1) can be including identification techniques are reviewed.
d is the degree of differencing required to remove varying mean
In the remaining part of this chapter, standard self-tuning many practical systems.
the output and input variables, that is, From equation (2.1.3)
;(q-l) D(t) _ 0(q-l) V-d e(t)
controller design for both scalar and mu1tivariab1e systems then yet) and u(t) in eq. (2.1.1) represent differenced forms of By choosing:
Therefore, the model (2.1.1) provides very general descriptions of so that eq. (2.1.1) is of the form:
A(q-l) Vd y·(t) _ q-k B(q-l) Vd u·Ct) + C(q-l) eCt)
d "
yet) - V Y (t) and,
d "
u(t) - V u (t)
carried out in two steps:
a) Determination of the time delay k, the polynomial degrees n"
n ,and n
b c
3) The locations with respect to the unit circle of the roots of (Box
of the and n
c
and
n
" b
B(q-1 ),
A(q-1 ),
"n" on the orders n
polynomial the
order of
dynamics,
1) An upper bound
2) The magnitude k of the time delay,
-1 -1
b) Estimation of the parameters in the polynomials A(q ), B(q )
zeros of the system, and hence stability of the corresponding and possibly C(q-1 ).
lie inside the unit circle in the q-p1ane are termed minimum of B(q-1), or zeros of det S(q-1) for mu1tivariahl e sys tems , the polynomials A, S, and C which specify the poles and
transfer functions. In particular, systems in which the zeros the
identification, namely the parameter estimation, usually draws a
n " n .n ) may involve step response tests in the open loop mode
" b c
great deal of attention.
ilIportance to prevent erroneous conclusions regarding the In general, for self-tuning purposes, the second phase of of operation. Correlation analysis may also be used to determine
parameter estimates. Such structural information may include:
The first phase of identification 1. e. the determination of (k,
The choice of a proper model structure is thus of paramount Jenkins 1970)
be reviewed later.
TUNING CONTR.OI.J ERS :
2.3
n - n,
e
phase systems while non-minimum phase systems are those where
-1 -1
one or more zeros of B(q ). or det B(q ). 11e outside the unit circle in the q-plane.
A frequent motivation for KV control i. that by reducing the
the set point closer to the target. This is illustrated in Fig.
2/ Future noise components.
1/ Present and past noise components
2.3.1 / CONTROLLER DESIGN FOR SCALAR SYSTEMS:
into two parts:
2.3.1.1 / S1S0 mintmuM variance self-tuning controllers:
variance of a given output variable, it is then possible to move
The basic idea proposed by Astrom and Wittenmark (1973) was to use
a predictive model form obtained by separating the noise dynamics 2.3 / MINIMUM VAllIANCE AND DETUNED MINIMUM VAl.IANCE SELF-
Consider the system given by equation (2.1.1) with n - n
" b
The estimation of the parameters in any given structural form will
It can then be claimed that the input and output components at any Rtime are independent of the future noise component and thus the MV
strategy involves setting the optimal prediction at any time (given information up to that time) to zero.
determined simply by the least squares method ( Astrom and Eykhoff; 1971 ). The model (2.l.5) ia therefore referred to a. a
lea.t .quare. ýdel.
conditions:
.,d.lled.
The.e condition. are important. If the residual. are correlated,
(2.3.6) Thi. viii then yield tinuoua information about th. proce"" par... ter.).
3) The input i. p.r.i.tently exciting (that I., It provide. con- 1) Model (2.3.5) i. a good repre.entation of th "" y.te. b.ina
"equ.nce (e(t».
2) Th. re.idual. (e(t» are independent.
4) The input .equence (u(t» is independent of th. dl.turbanc "
th. l.ast squares estimates will be biased.
The LSE will converge to the true parameter. under th. following
If th. input .equence (u(t» depends on (e(t» it .. y not b. po-
equation (2.3.6)
y(t+k) + a(q-1) yet) - /Jo /J(q-1) u(t) + f(t+k)
The derivation of equation (2.3.6) is given in appendix A
-1 -1
Th. polynomials a(q ) and /J(q ) have the form
Th. probl_ of cOlÇuting the control par... t.r. of F and G frOia
""lbl. to deteraine all the par.. eter".
"ubatltuted ln (2.3.5) vith C(q-l) - 1.
""tiaate. of A and B can be simplified if the Identity (2.3.4) i "
l
I
factors) and if the system order is not underestimated.
Another property of the algorithm is that lf the parameter
variance controller is that it aims at the minimization of the
aa:
(2.3.11) (2.3.9) (2.3.10)
are to be and R(q-1 )
1"..." t) converga as tý-
ý - k+l, ... , k+t+l
r - k+1, ... ,k+m
This generalized output includes a
Other suboptimal control strategies that can handle
is the set-point which is assumed to be known at E[y(t+ý)y(t)] - r (ý) - 0
7
E[y(t+ý)u(t)]- r (ý) - 0
yu
this case.
The main feature of the performance of the generalized minimum 2.3.1.2 I GENERALIZED MINIMUM VARIANCE CONTROLLER (GMVC)
At this .taga, it ls pertinant to note that the MY strategy is not
atrategy is very sensitive to even slight variations in the then the closed-loop system has the following properties
time t.
parameters and thus can create an unstable closed loop system in
appropriate for a non minimum phase system. This is because the
such aystems are reviewed in the following sections.
variance of a generalized output. ý(t) (Clarke and Gawthrop 1975 a,b and Gawthrop 1977).
variable, and the set-point. This generalized output is defined
where y (t)
r
The polynomials P(q-1 )
weighting function of the output variable as well as the control estimatas a (i - 1 "..." m) " P (i
i i
chosen by the designer.
Let:
(2.3.12)
(2.3.13) (2.3.lla)
ý (t+k) - P(q-l)y(t+k)
Y
A P " (t) - P B u(t-k) + P C e(t) (2.3.14)
d Y D ft
tim. t.
ttae (t+k) conditionel on all input/output data being known up to
Define the function. (t+k) as:
y
deterministic signals Qu(t) and Ry (t) are known at time (t+k).
r
square. prediction of a simplified form of equation (2.3.11).
Dropping the arguments of the polynomials for simplicity, the
ThWi the problem of predicting ý(t+k) is reduced to a least
It can be seen from equation (2.3.14) that " (t) follows the
y
relation can be obtained:
Using equation8 (2.1.1), (2.3.13), and (2.3.lla) the following
general process description as that given in equation (2.1.1).
where the expectation of the function .2(t+k) is taken at where Pd(q-1 ) is assumed to be stable and has degree nd
The cost function to be considered is:
An optimal control law can be obtained in the form of equation
mintmizes the more familiar cost function:
et al, 1980).
(2.3.19) (2.3.15)
(2.3.18) (2.3.17) (2.3.16)
B(O) - bo [FB+QC]
[C R Y (t) - ý y(t)]
r p.
and q' Q' (q-1)
Q(q-1) __ 0--...b _
o
u°(t) -
- q'
o
ý(t+k) - H u(t) + G y(t) + E Y (t) + t(t+k) and
Note that the cost function I app.ars to be th. same as th. usual linear quadratic Gaussian (LQG) cost function but this is not th.
Q' (0) where
rational transfer functions as in equation (2.3.11a), then a model If P, Q and R, in equation (2.3.11) are polynomials rather than case because of the use of the conditional expectation. (Grimble
of the form of equation (2.3.19) can be obtained
identity which is similar to that given by equation (2.3.4):
It CaD be shown Grimble (1981) that the control law (2.3.15) also
P C - A P F + -It G
D d q
-1 -1
where F(q ) is of degree k-l, and G(q ) is of degree n +n -l.
a d
-1 -1
The polynomials F(q ) and G(q ) are obtained from the following (2.3.15).
This is achieved by following the approach of (Clarke and Gawthrop, 1975),
- l
where H(q )
and E(q - l ) are related to the polynomials B, C, F, Q and R. The parameters in H(q-l) and G(q-l)
ANALYSIS:
(2.3.21)
(2.3.22) (2.3.20)
That is, for non-lIliniDlUDl phase systems, the GMVC
- 1 - 1
B(q ) C(q ) - 0
PB + QA - 0
can be estimated using the recursive least squares (RLS) technique.
equation
The control law is obtained at every time step by solving the
many of the disadvantages of the basic STR of Astrom and tU ttenmark.
By contrast the characteristic equation of the clo.ed-loop system weighting parameters. This can be obtained from the fact that the The controller proposed by Clarke and Gawthrop does not suffer corre.ponding polynomial.
where the circumflex denote. the e.ttmated value of the
could stabilize the control system for sOllle values of the
closed-loop characteristic equation using the GMVC is given by:
with the MV control is given by:
Thus by an appropriate choice of the polynomials p(q-l) and
....
- l
Q(q ), the closed-loop system using the GMVC can be stabilized (even for non-minimum phase systems). Furthermore, it is seen
(2.3.24) (2.3.23)
[ F B+ + Q C
]
UO(t) - --- -1
P(q ) - --- B-(q-1)
predictors. (Smith, 1959; Marshall, 1974; and Gawthrop 1979).
from equation (2.3.21) that the system time delay q-It is absent.
thi. predicament, a weighted minimum variance controller (WKVC)
taking the form
-- -1
B (q )
for non-minimum phase systems if the control weighting polynomial
Aa noted above, the GMVC can produce unstable closed-loop response Thus self-tuning least squares prediction can be linked to Smith
instead of the form given in (2.3.11a). It is assumed that B(q-l) was developed by Grimble (1981, 1982). The algorithm minimizes a
its zeros inside the unit circle in the q-plane. (i.e. B-(q) is the unstable part of B(q-l».
In equation (2.3.23). B-(q-1) represents the product q-Db B-(q) co.t function similar to the one given in (2.3.11) with P(q-1 )
Q tends to zero as can be seen from equation (2.3.21). To overcome
Tha optimal control law for the WKVC is of tha form (Grimble
+ -1 - -1 + -1
can be factorized as B (q ) B (q ) where B (q ) contains all
- -1
whara n is the degree of B (q ) and is the reciprocal polynomial
b
1981).
This is obtained by introducing the identity
s-
p c - PAF +q -k
B G
n d
(2.3.25)
user.
(2.3.26)
1 , respectively (assuming n + n S n + n
ý c pd "
If the plant is open loop stable and Q __. " then the given by
le.. t squares scheme may have to be used. Furthermore, the
-1 -1 -1
structures of P(q ), Q(q ) and R(q ) have to be assumed by the identification algorithms where for example the bilinear extended to be stable in both limiting situations.
+ k - 1 ) and have the same form as 2.3.3a and 2.3.3b.
and n - n + n
I " pd
stable.
consuming, and may lead in some situations to complicated The characteristic equation of the closed-loop system with YMVC is
The polynomial C(q -1 ) is stable and as Q --. 0 the final term goes to B(q-1)B+(q-1)p (q-1) and is stable if P is chosen to be
D D
which is again stable. Thus, the closed loop system is guaranteed
-1 -1
The degrees of the polynomials F(q ) and G(q ) are nt - k - 1
The disadvantage of the above self-tuner is that the polynomial B(q -1 ) must be spectrally factored on line. This may be tae characteristic polynomial tends to the first term in (2.3.26)
l
2.3.1.3 / SISO DETUNED (OR TAILORED) MINIMUM VARIANCE SELF-TUNING CONTllOJJ·ERS (DMVC):
In this regulator which was developped by Wellstead et al (1979a),
moderated.
(2.3.27)
(2.3.28) i.e.
yet) - --- e(t) T(q-1)
poles to positions determined by T(q- 1 ), the excessive control can often introduce large loop feedback gains so that the control
signals associated with minimum variance strategies may be signal excursions may be unacceptably large. And by shifting the to positions determined by T(q-1 ) instead of being located at the A strong similarity exists between the MV and the DMVC. In fact,
regulator.
The motivation for the DMVC is that the minimum variance regulator remaining terms in the quotient C(q-l)T(q-l)/A(q-1) and, T(q-1) is
a pre-selected polynomial which specifies the locations of the closed-loop poles of the system.
origin of the q-1 plane as in the case of the minimum variance equation (2.3.28) ensures that some of the spare poles are shifted where Z(q-1 ) is a kth order polynomial with coefficients equal to the intention is to force the output yet) to have the following
the first k terms in the quotient C(q-1)T(q-1)/A(q-1) C(q-1)T(q-1) N'(q-1)
Z(q-1) _ _ q-k _
A(q-1) A(q-1)
Here, N'(q-1) is a polynomial with coefficients equal to the form:
..
