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
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
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
2.6.3 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
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
2
a
"
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 of unconsciousness in a patientThus, 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
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
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
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 andwhen 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
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
IE
where q is the forward shift operator such that q y(t)-y(t+k)
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 -JI B(q-1) b b -1 b b b ... 0 - + q + + q .... 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.
-1
The polynomial C(q ) 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;
-1
0(q ) is the moving average polynomial component;
2
eCt) is a white noise process with zero mean and variance q
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 " -1 B(q ), -1 A(q ), "n" on the orders n polynomial the of order 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
-1
and possibly C(q ).
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
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 -1 and R(q ) 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
-1
The polynomials P(q )
weighting function of the output variable as well as the control estimatas a (i - 1 "..." m)
" P (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
-1
where Pd(q ) is assumed to be stable and has degree nd
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) - b o [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) r 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
This is achieved by following the approach of (Clarke and
Gawthrop, 1975),
- l
where H(q )
- l
and E(q ) 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).
-It
from equation (2.3.21) that the system time delay q 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)
-1
co.t function similar to the one given in (2.3.11) with P(q )
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).
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
-1
The polynomial C(q ) 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
-1
B(q ) must be spectrally factored on line. This may be tae
characteristic polynomial tends to the first term in (2.3.26)
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) - 1
poles to positions determined by T(q ), 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
-1
to positions determined by T(q ) 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.
-1
origin of the q plane as in the case of the minimum variance
equation (2.3.28) ensures that some of the spare poles are shifted
-1
where Z(q ) 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:
2.3.2 / DESIGN FOR MULTIVARIABLE SYSTEMS:
2.3.2.1 / MIMO MINIMUM VARIANCE SELF-TUNING CONTB.OLI·ERS:
An ARMAX
representation
for a MIMO system may be represented byproc "" s to be controlled whicb has often been assUlDed to be the (2.3.29) (2.3.31a) (2.3.3lb) (2.3.30c) (2.3.30b) (2.3.30a) det B ,. 0 o
The vee tor e is a vee tor of
(positive definite);
-1 -k -1 -1
A(q ) y(t) - q B(q) u(t) + C(q )e(t)
A(q-l) - r + A q-1 + + A q-n A P 1 D A E{e(t» - 0 T E{e(t)e (t» - L the following equation:
-1
tbe backward sbift operator q . They may assuae tbe following input vector. The polynomial matrices A, Band C are functions of
form:
Furthermor., th. parameter k is the known time delay of the where y is a p dimensional output vector, and u is a q-dimensional
B(q-l) - B + B q-1 +
...
+ B q-n IOlD
I
same for all control signals.
covariance matrix L i.e.
normally distributed independent variables with zero mean and
where ý is a positive definite matrix and where T denotes the
vector/matrix
transposition.
r I
- l
h
The zeros of det C(q ) are assumed to be strictly outside t e
-l
unit circle in the q -plane.
are not unique.
Borison (1975) has derived the MV strategy for MIMO systems of
(2.3.35) (2.3.32) (2.3.33) (2.3.34) ;(0) - I. and -1 -1 -1 -le -1 C(q ) - A(q ) r(q ) + q G(q i . "(q-1) I F -1 r -(IE-1) ý - + q +
...
+ q 1 1E-1 G(q-1) _ G + G1q-1 +....
+ G q-(D-1) o D-1 - -1 -1 - -1 -1 r(q )G(q ) - G(q )l'(q ) - -1 -1 det F(q ) - det F(q )introduced
the identitySince A (-I in equation 2.3. 30a) is
non-singular.
th. matriceso
-1 -1
r(q ) and G(q ) are unique.
known
parameters.
For the system described by (2.3.29), he has- -1
Borrison has also
introduced
the polynomial matrices r(q ) andwhere
- -1
G(q ) such that:
cost function V given by:
2
- -1 --1
The polynomial matrices F(q ) and G(q ) always exist but they
where
v.
- E { Ily(t+k)
Uý} (2.3.36)
process time delay measured in sampling periods.
where Q is a positive definite weighting matrix and where k is the
(2.3.39)
(2.3.40) (2.3.38) (2.3.37)
(2.3.41a) this .int.ua variance with error e "
{
.-1
E!
r
t.-o v -:5 e (t) - F(q -l)e(t) " - -1 - -1 -1 G(q )y + F(q )B(q )u(t) - 0 t. - 1 -1 Q_(q ) -01 +Qq + +01 q -0 -1 -Q 01 -1 -1 yet) + ý(q )y(t-k) -ý(q )u(t-k) + f(t) ..ymptotic control TheFurthermore, after the initial transient period this strategy also
In order to minimize the function V asymptotically, one can show z
that the control law must be of the fora:
strategy is given by
For .yste.. with unknown par... ters, Borison considers the following model
- 1 -1
where ý(q ) and ý(q ) are polynomial matrices given as
-nf3 l!(q-1) - fJ + fJ q-1+ + fJ q
-0 -1 -nfJ (2.3.41h)
Borison seeks a recursive estimation scheme such that the error is
one column at a time.
directly. (2.3.42) These were A A ý(q-l) u(t) _ ý(q-l)y(t) , -[ , , """"""" ,] -[ a , a , """"" a ,fJ ,fJ ,
""""
fJ ] t (2.3 .43) -1 -p a 1 Da a 1 DfJgives the miný variance strategy under the assumption that the
as small as possible in the least squares sense.
Borison discusses conditions which guarantee that the algorithm equation
estiaated parameters converge.
The control is computed at every time step using the following
It i. shown that the MV strategy is obtained if the disturbances Thi. indicate. that the controller parameters are estimated
The least square. algorith. e.timate. the parameter .. tr1z
the following cost function:
- 1
are white noise (C(q ) - I), provided that na and
nfJ are "large
enough". It should also he noted that this regulator suffers the
same disadvantages met by its SISO counterpart. described in the preceding section.
J - E { Il y(t+k) -Yr 112 } (2.3.44) -1 -1 -1
where T(q ), Seq ) and Il(q ) are polynomial matrices.
equation (2.3.29).
the optimal control law for the case of known parameters can be
(2.3.46) (2.3.45) (2.3.47) (2.3.48) -1 -1 -It -1 a(q ) - Il (q ) + q Il (q ) 1 2 UO (t) - B-1 C 1'-1 Y - B-1 G 1'-1 yet) r -1 -1 -1
y(t+k) - T(q ) yet) + Seq ) u(t) + Il(q ) e(t)
By using the identity (2.3.32) and after substantial manipulation, parameters were chosen such that p-q ; L
-
l ; A(q-l),B(q-l) and
C(q-1) are all invertible polynomial matrices in the system
based on the observations (y(t), y(t-l), ...u(t-l), u(t-2) ... ). The vector y is the desired constant reference vector. The
r
shown to be given by:
For unknown parameters, Keviczky and Hetthessy suggested the uae of a dead-time transformation (Hetthessy et al, 1975) so that the
The optimal control law was found to be of the form:
Keviczky and Hetthessy have introduced the separation equation system model would have the following form
I
Since the matrices in the above equation are unknown, the authors used the control strategy
are + k -1 (2.3.49) (2.3.50) - n B -1 ý(q ) n a and where
Bayoumi and El-Bagoury (1979) have shown that - B. (q-l)e(t+k). 1 -1 -1 y(t+k) -ý(q )y(t) +
I!
(q )u(t) + c(t+k) c(t+k)the control in order to reduce fluctuations and in order to
moderate the excessive fluctuations in control signals compared
resulting controller.
as well as non-minimum phase systems can be handled by the to multivariable systems. The cost function included a penalty on
with the MVSTR of Borison (1979). Time-varying reference signals
The cost function to be considered is of the following form:
every time step.
polynomial matrices of degrees
the orders na and np given by the authors are in general incorrect for polynomial matrices and that in fact these values should be The control signal is determined by solving equation (2.3.49) at
respectively. where
cho.en to be "large enough" to have a reasonable confidence level
in the approximation of the unknown rational matrices,
S-l
aa-I
1
and
S-l[
T + a2aý1] in equation (2.3.48).Koivo (1980) extended the procedure of Clarke and Gawthrop (1975)
Thi. invariably gives the expected output at tiae (t+k) as
!
l - E { Il P ( q -1 ) Y ( t+k ) _ R ( q -1 ) Y / t ) Il 2 + Il Q' (q -l ) U ( t) 112
It}
(2.3.51) -1is a polynomial of degree (k -L) and Q' (q ) and
(2.3.54) (2.3.52)
(2.3.53)
- -1 - -1 - -1
H(q )u(t) + c' (q )y(t) + E(q )y (t) - 0
r -1 -1 -1 ý(t+k) - P(q )y(t+k) - R(q )y (t) + Q(q )u(t) r that n - n - n - n. ABC
equation (2.3.29), it is assumed that p - q (square system) and
R(q-1) are polynomial matrices of degrees nQ,and na respectively. Beside the basic assumption made that the system model is given by
to (2.3.35) the optimal control law can be shown to be given by:
Following the approach by Clarke and Gawthrop, ý(t) was introduced
By introducing identities similar to those in equations (2.3.32) liven by:
for non-minimum phase systems.
which is defined by a generalized vector output variable ý(t+k)
be appropriately chosen so that the ST controller may perform well
-1
The weighting polynomial matrix Q(q ) on the input vector should -1
This control law minimizes the cost function J given by:
J - E
{
i
.<t+kl!·}
Koivo has suggested the use of a square root algorithm which has a
computational advantage over the basic RLS scheme in which
non-square systems and is based on successive output estimation. (2.3.56)
(2.3.57)
Such rounding-off errors can
(2.3.55)
_
ý ý·(t+k)lt ý2 + E
{
IE (t+k)
In
where (ý·(t+k)lt) denotes the optimal prediction of ý(t+k) given
information up to time t. The vector E(t+k) is given by:
If the system parameters are unknown, a simplified model similar
models for systems with large time delays. This STC can deal with
property of being positive definite.
make the covariance matrix of the parameter estimator lose its
to that given by equation (2.3.55) can be used to estimate the
controller parameters recursively for the case where C(z-l) - I.
Another multivariable STC was developed by Bayoumi et al (1981), numerical problems can be faced because of rounding-off errors in
They considered a cost-function given by:
and El-Bagoury (1980) to overcome the requirement of high order Using these estimated values, the control action u(t) can then be obtained at every time step by solving equation (2.3.55) for u(t).
computers with short word lengths.
-1
where E(y(t+k)] is based on information available at time t and is evaluated using the estimated parameter matrix 8(t) which contains
- 1 - 1
the coefficients of A(q ) and B(q ) obtained by reparameterizing
RLS method.
vectors (e(t») are involved. Hence the estimated parameter vector
(2.3.58)
(2.3.59)
A
- Bu(t-l) + Bu(t-lIl) - A y(t+k-l)- .".".
1 " I c(t+k-l) A A A A u(t) - _ ( Bý Ql Ba + Q2)-1 Bý Ql c(t+k-l) , ,_(t) -[ ; , ; , ,; , _ý , _ý , _ý ] , o 1 " 1 2 n
... -A y(t+l) - A y(t) - ... -A y(t+k-n) .,.
y
It-I It D r
(2.3.60)
the system model (2.3.29) so that only white Gaussian noise
given by :
u(t) - 0 and is given by:
It can be shown that the control signal which minilllizes J is
t
where c(t+k-l) is the estimated deviation of y(t+k) from y if
are positive definite weighting matrices.
generalized likelihood ratio statistical test and where Q and Q
1 2
where m and n are appropriate degrees obtained by using the
The estimated parameters can be obtained using any variant of the
In extensive simulation studies, the performance of the above !(t) is defined by the following equation
control law was shown to be very good. Furthe rrnor e , Itwas
demonstrated by El-Bagoury (1980) that the computational requirement for this self-tuning scheme is far less than that
be to (2.3.61) assWled are - 1 A(q ). and - 1 A(q ). -It -1 -1 -1 -1 -1 -1
yet) - q A (q ) B(q ) u(t) + A (q ) C(q ) e(t)
Besides, the additive term Q in equation (2.3.60) ensures that
2
the different control loops since B is generally singular for o
is required.
required by most of the previously described MVST algorithms.
important characterization of coprimeness is explained by Wolovich
the control law can handle systems with different time delays in
is restricted to square systems and the a-priori knowledge of B
o
An equivalent model of the following form was considered:
are relatively left prime if their gcld is unimodular. A further
relatively left prime (RLP) polynomial matrices. Recall that two polynomial matrices (p(q-1), R(q-1)} with the same number of rows where
for the system model in equation (2.3.29) was used. The approach
such systems. However, the problem of choosing Q is still open.
2
Another STe des ign procedure was proposed by Keviszky and Kumar
The control law was determined to minimize the cost function (1974).
v
-E{II liy(q-l)[ y(t+k)
-Yr] 112 + Il 'WUCq-l) [UCt)
-ur]
1121t}
(2.3.62)
5TC could not be assessed.
the corresponding optimal prediction to zero.
(2.3.63)
the desired constant reference
The approach uses the spectral is t· , - 1 -1 -1 B(q ) - B (q ) B (q ) 2 1 - 1 - 1
(q ) and li (q ) are assumed to be RLP MFDS.
u
factorization :
square multivariable systems.
were provided. Thus the performance of the resulting multivariable involved in that approximation. Besides. no simulation examples No mention was made about the orders of the polynomial matrices
Grimble and Fung (1981) followed the same procedure than the one
proposed an approximation to the non-linear prediction equation
obtained from the Smith canonical representation (Kailath, 1980; used for the 5150 case to develop an explicit WMVC self-tuner for For systems with unknown parameters, Keviczky and Kumar (1981) the cost function used by Gawthrop (1977) than by Clarke and
using a linear least squares model.
criterion can be considered as the multivariable generalization of available up to time
Gawthrop (1975). Thus the control law can be derived by equating Yr
vector and u is the corresponding set point vector. Hence this
r
where li
y
Kucera, 1979) of polynomial matrices where B (q-l) and B (q-l) are
2 1
the non-minimum and minimum phase spectral factors, respectively. The factorization does not require that the time delay k appear
explicitly in the system representation. Thus the approach can
where:
(2.3.64)
self-tuning (2.3.65)
the
is also required in the
discussed
(1981)
- 1
The polynomial matrix M(q ) is obtained
Prager " -n " -1 " t Tq + +Tq 1 D and
to (2.3.35) in the derivation of the control law.
implementation of the M1MO-DMVC. Besides the assumptions made by
implementation. This constraint is given by: .
h d of T*(q-l) constra1nt on t e or er
Following the S1SO procedure described in section 2.3.1.3, a
that
Borison (1979) which were discussed in section 2.3.2.1 a
\lellstead
2.3.2.2 / DETONED KYSTe FOR MIMO SYSTEMS:
by introducing identities similar to those in equations (2.3.32) minimizing the variance.
detuned MVC (DMVC) for multivariable systems can be designed. The DMVC introduces "moderating poles· into the closed loop system so
handle different time delay situations.
at the expense of making the control sub-optimal in the sense of
-k " -1
The determinant of ( I + q T (q » specifies the set of poles
n t * .$ n A - n c (2.3.66)
2.4 / POLE-ZERO PLACEMENT SELF-TUNING CONTROLLERS FOR SISO SYSTEMS:
give slower responses.
the allocation of the positions of the poles and/or zeros) to take
For such systems, a factorization
In multivariable systems, one further In particular, the basic algorithm cannot control desired response. For example, closed-loop poles close to the where nand n are the orders of the polynomial matrices A(q-l)
A C
- 1
and C(q ), in the system model (2.3.29) respectively.
magnitudes and maximum system response rates.
All the previously discussed self-tuning algorithms have the
advantage that the closed loop response can be specified (through
proper account of the form of permissible control signal
The pole/zero locations are arbitrarily assigned depending on the
origin give fast responses, while those closer to the unit circle
time delay (Wellstead et al., 1979b). This is a point of utmost
practical situations.
Multivariable MV self-tuners which were previously discussed suffer from disadvantages similar to their single-variable countreparts.
Moreover MV self-tuners are sensitive to variations in the system
behavior is the rule rather than the exception when we consider significance, since in discrete time systems, non-minimum phase
non-minimum phase systems.
control-law, and this increases the computational complexity.
disadvantage exists in that the minimum-variance design discussed
earlier demands that all loops must have the same time delay.
This requirement is not generally fulfilled by real systems.
section 2.3.1. 3;
in
(2.4.1)
(2.4.2)
e(t)
Such systems occur frequently
-1 -1 -It -1 -1
H(q )A(q ) + q B(q )G(q )
-1 -1
H(q )u(t) + G(q )y(t) - 0
y(t)
-ii) Pole-placement regulators (PPRs).
discrete time control.
i) Detuned minimum variance controllers, as was described in
derived for S1S0 systems considers a control law of the form with varying time delay.
The approach of Wellstead et al (1979b) which was originally The PPR is very useful for regulating non-minimum phase systems placement (PZP) self-tuner.
Wellstead and Co-workers (1979 a,b) considered two versions of PZP
algorithms for self-tuning regulation purposes:
Through pole assignment we want the closed-loop performance to be:
Iy using this control law in the system representation (equation
These difficulties can be avoided by a multivariable pole zero
y(t) - e(t) (2.4.3)
where T(q-l) is a prespecified polynomial that defines the
controller parameters.
that equation (2.4.4) holds.
(2.4.4) Comparing equations (2.4.2)
forms of system models have been used. It has been shown that the
In the self-tuning literature (Wellstead; 1979a,b), two distinct
and hence equation (2.4.4) cannot be solved directly to get the
In self-tuning, the system parameters A, B and C are not known,
parameters Hand G.
then we can solve equation (2.4.4) to get the controller
n - n - 1 I " n - n + k - 1 (2.4.S) h li n:Sn +n +k-n-l to " b c
equation (2.4.1) may be described more conveniently by the system of equations (2.1.1) under the control scheme described by Hence, if the parameters of the polynomials A, Band C are known,
closed loop poles n must satisfy:
to
-1 -1
regulator polynomials H{q ) and G{q ) as well as the number of
For equation (2.4.4) to have a unique solution, the order of the
identity
H(q-l) A(q-l) + q-kB(q-l)G(q-l) _ T{q-l)C{q-l)
- 1 - 1
The problem is then to find the polynomials H{q ) and G(q ) so
location of the closed-loop poles.
and (2.4.3) the controller parameters can be determined from the
ý
.
, .
following two models:
a) Model (2.1.1) can be described by:
that i.
solving the following identity:
(2.4.8) (2.4.6)
This model is not a special case of
-It -1 -1t-1
y(t) - q B (q )U(t) + q A (q )y(t) + E (t)
1 1
- 1 - 1
b) yet) - B (q )u(t-l) + A (q )y(t-l) + e(t)
2 2
H(q-l)[ 1 _ q-ItA/q-l)] + q-ItS1(q-l)G(q-l) _ V(q-l)T(q-l)
(2.4.7)
-1
ý(t) - V(q )e(t)
model (2.4.6) with k - l, but rather the delay k has been absorbed
-1
vith C(q ) - 1 at every time step.
recur.ive scheme and then the regulator parameters are obtained by
where the orders of the polynomials A and Bare, (n - 1) and
22.
-1 -1
The parameters of A (q ) and B (q ) are estimated using any
1 1
where ý(t) i. a moving-average proce"" of order (k-L) in e(t),
The orders of the polynomials Al' B1 and V are (n.-l), (nb + k -1) and n - k - 1 respectively.
v
(n + k - 1) respectively. b
-l
into B(q ).
The polynomials A and B may be identified from equation (2.4.8),
2 2
whereas the polynomials Hand G are obtained from
the set point.
(2.4.9)
No stochastic incorporating digital suggested
been proposed which
that it focusses entirely on the servo problem.
procedure involves the cancellation of only those process zeros in tracking of the reference input is involved (Servo problem). The
disturbances were considered in deriving the controller. The
some "restricted stability region".
adaptive systems were made through their formulation. The major
disadvantage of this technique Is that factorization of 8(q-l) as However this work differs from that of Wellstead et al (1979) in integrators in the loop (Wellstead et al, 1979b) where the
Pole placement has also been considered by Astrom at al (1980).
The link between self-tuning design principles and model reference problem has been tackled by including servo compensation in the
effort is required. An al ternative and a more favored approach
controller may operate on the difference of the system output and once A (q-l) and B (q-l) have been estimated.
2 2
has
The pole placement concept presents some difficulties when
loops (Wellstead and Zanker, 1979) but the price paid for this
option that a significant increase in the needed computational
I
However. the me thad can eas Lly cope
with non-minimum phase systems using this decomposition.
appropriate.
set-point term in the control law does not excite the controller
This representation
A further requirement is
This is to ensure that the
Among some recent developments, a different model, namely the
state feedback concept is used to resolve the pole assignment
- - 1
that the factor B (q ) must be contained in the numerator of the
control problem. The well-known pole assignment technique via the state space representation, is used to cope with the self-tuning is referred to as an autoregressive moving average representation. interpreted as the backward shift operator.
identification problems. However, if the system has more than one noise source, the transfer function modelling may not be the most filter defining the set point.
2.5 / STATE SPACE SELF-TUNERS:
- - 1
modes corresponding to B (q ). If excitation occurs, an unstable
suffer from the bias problem of the previously discussed
2.5.1 / S1SO state-space STe:
algorithms since no noise is involved.
closed loop system can result. The estimation procedure does not
Most existing self-tuning algorithms are developed for systems represented by their transfer functions in the z-domain with q-l
The transfer function representation has advantages in solving the
self-tuning control problem. The approach can be equally applied
to control non-minimum phase and/or unstable systems without the need to introduce any modification. The main shortcoming is that
estimate the states of the dynamic system. However, the Kalman
pole-placement concept. (2.5.lb) (2.5.la) - E 6 " -1 iJ' T
and with covariance E[ý(i)
ý (j)
x(t+l)
-ý x(t) + ý u(t) + ý (t)
y(t) - At x(t) + v(t)
ý " A
- n x 1 input and output parameter vectors
ý(t) n x 1 white noise vector with zero mean
u(t) the input of the system
r n x n system matrix
x(t) - n x 1 state vector
y(t) the output of the system
where 6 is a kronecker delta.
iJ
v(t) - white noise process with zero mean and variance q2.
v
The signal v(t) is assumed to be independent of ý(t) for all t.
gain depends on the system parameters, the variances of the system Tsay and Shieh (1981) used the Kalman filter approach to optimally where :
fora:
they are usually more computationally involved than some of the
previously discussed self-tuning controllers based on the
noise as well as the observation noise. Thus a joint algorithm
was developed for bach the syscem identification and state
estimation. This is done by:
parameters.
canonical form (Kailath 1980) which can be converted to the
computation.
(2.5.2)
for parameter
n)
The major disadvantage
n - n
b c
However, the simulations presented
TA u(t) - K Y (t) - h x (t)
r r 0
The closed loop gain K is chosen to ensure that the
r
method (with b - 0, k - 0 and n
o "
autoregressive moving-average model.
matrices at every time step.
vector in the observable canonical form.
of the algorithm is the need to determine two transformation estimation through the recursive extended least squares
is, to prevent any offset); and
x
(t) is the estimate of the stateo
mean of the steady state output equals the reference input (that
matrices rand 0 only, a dependence which will simplify control
-0 -0
assigned.
T
where y (t) is the reference input and h is the feedback gain row r
The feedback control law was given by
vector which is determined from the location of the poles to be feedback control law was derived from the noise-free systems. In
other words, the state feedback control law depends on the system Furthermore according to the separation principle. the state (iii) Reconstructing the system state estimates using the estimated
(ii) Using the ARMAX representation (2.1.1)
by the authors demonstrated that the algorithm can stabilize both
unstable and non-minimum phase systems.
transition than the latter.
estimation of the innovations model in the controllable canonical
with that of Wellstead et al (l979b) through a simulation example
the
Secondly,
First, the Kalman gain
He used both state and output feedback for control
(2.1.1).
fora. This has two main consequences.
A modification of these algorithms was proposed by Omani and Sinha showed that the Warwick regulator tends to give a smoother
vector becomes explicitly parameterized; hence it may be estimated on line together with system parameters.
computation of the feedback gains becomes trivial.
was made where a variable time delay was assumed. The comparison
determination. A comparison of the performance of the regulator
dimension of the state vector by the time delay) for the system,
- 1
and C(q ) was chosen to be equal to 1 (white noise) in equation
An algorithm using an approach similar to the one discussed above
(Ljung and Soderstom, 1983) was used for joint state and parameter (1985), in that a recursive prediction error (RPE) algorithm
used an implicit delay state-space model (i.e. extending the basic difference between the two approaches is that the author
computation of a transformation matrix is required in the design
of the control law. In some cases (Warwick, 1981), this requires
an on-line matrix inversion which is undesirable.
He.keth (1982) require the on-line computation of transformation matrices. This in view of the large dimensions of the matrices
involved, results in excessive computation.
Thus, the on-line
Also, the two methods, like the one proposed by
observable canonical form (Kailath, 1980).
2.5.2 I MULTIVARIABLE STATE-SPACE STC:
For multivariable systems, Hesketh (1982) has proposed a state
space pole-placement self-tuner which uses input-output data for
feedback instead of the state estimates (which is the case of the
algorithm reported above).
His approach involved a joint parameter and state estimation, followed by classical state feedback control. The main drawback
of this method is that the joint state and parameter estimation algorithms which were used require a system model in the
dimensions.
Bezanson and Harris (1984), Shieh et al (1982) have proposed MIMO STe' s which are generalizations of the SISO cases presented by Tsay and Shieh (1981) and Warwick (1981). The MIMO state space STe'. proposed above, represent the controlled system by various block canonical forms. The use of such "pseudo-canonical forms" often results in system matrices which have larger than minimal
Finally, self-tuning controller designs based on the LQG approach
for both SISO and MIMO systems have been proposed for example, by
usual form of the LQG controller involves a Kalman filter and
control gain matrix and these require the solution of two Riccati equations. The method presented by Grimble (1984a) involved the
Grimble (l984b) has shown that there are in fact two equivalent minimum variance problems to every LQG problem, involving either non-minimum phase or minimum phase related system descriptions. solution of two diophantine equations and spectral factorization. Grimble has also shown that for many systems only one diophantine equation must be solved and the spectral factorization stage can
be avoided if, in the nonoptimal case, the control closed loop
the
in comparison wi th
unstable or non-minimum phase.
the more common k steps ahead This approach tends to increase the
open-loop
case for plant is either
This is not the poles are prespecified.
optimal control laws.
bl Integral action may easily be introduced by appropriate choice of the cost function
cl Stochastic reference and disturbance inputs to the system are taken into account in the optimal controller design.
dl The time delay does not enter explicitly into the calculation of the controllers and thus the magnitude of the delay need not be known a priori.
self-tuning techniques described in other sections of this
chapter. However, the LQG have a number of advantages (Grimble,
Moir and Fung, 1982).
al The controllers are stable for all values of the control weighting parameters when the plant is known, even when the
on-line computational requirements
The solution of these minimum variance problems is identical to
that for the underlying LQG control problem.
2.6 / PARAMETER ESTIMATION:
Solving the parameter estimation problem requires information
about the following :
(i) Input-output data from the process,
(ii) A class of models,
(ill) A criterion.
On-line methods give estimates
In many cases off-line methods give the process is time-varying.
line and off-line schemes.
recursively and only updates are calculated as the measurements
are obtained. This may be seen as the only alternative if the
identification is going to be used in an adaptive controller or if
estimates with higher precision. In the off-line case, it is presumed that all the data are available prior to analysis. Consequently. the data may be treated as a complete block of information, with no strict time limit on the process analysis.
The input-output characteristics of a wide class of linear and
nonlinear deterministic dynamical systems can be described by a
model that may be expressed succintly in the following simple form
The parameter estimation problem can then be formulated as an optimization problem, where the best model to be found is the one that best fits the data according to the given criterion.
There is a large number of different identification methods
linear function of
The identification method and the model structure dictate the
f is the estimation error. The dimension of the vector P and x
(2.6.2) (2.6.1)
P(t-l) + &(t) x(t) f(t)
In the RLS methods, R is the inverse information
This is due to the relative simplicity of these
The most commonly used estimation algorithm for the
y(t) - P(t)x(t-l) + f(t)
A P(t)
x( t-l) denotes the information vector that is linear or
time t. The vector x denotes a regression or memory vector and Many on-line estimators can be described by a recursive equation
technique.
where P(t) denotes the vector of the estimated parameters at of the following form:
variants. matrix.
are both equal to N.
f(t) denotes a modeling error of some kind [e.g.. the model prediction error arising from the use of P(t-l) in calculating the
estimated inputs]. The matrix K denotes an algorithm gain.
design of self-tuning controllers is the RLS technique and its
quantities x and ý. while R depends on the particular estimation ý(t-l) -{ y(t-l). y(t-2) "... } 11(-1) -( u(t-l). u(t-2). }
P denotes the parameter row vector to be identified (unknown). where y(t) denotes the system output at time t
