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MOTOROLA APR15
Motorola Digital Signal Processors
Implementation of Adaptive Controllers on the Motorola
DSP56000/DSP56001
byPascal Renard, Ph.D.Strategic Marketing – Geneva,
Switzerland
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Table of Contents
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SECTION 1
Introduction
MOTOROLA
SECTION 2
Numerical DomainRepresentation
For More Inf
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1.1 History of Adaptive Control 1-11.2 Theory of Adaptive
Control 1-2
ND
UC
2.1 Parametric Models 2-12.2 Adaptive Control Techniques 2-6
MIC
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SECTION 3
Adaptive Controland Adaptive
Controllers
SC
A
3.1 Adaptive Control Using Reference Models 3-1 3.1.1
Introduction 3-1
3.1.2 Closed-Loop System 3-2 3.1.3 Control Law 3-4
3.1.3.1 Known System Parameters 3-43.1.3.2 Unknown System
Parameters 3-9
3.1.4 Determination of Controller Parameters 3-10
3.1.5 Comment 3-14
3.2 Generalized Predictive Control 3-15 3.2.1 Introduction 3-15
3.2.2 Closed-Loop System 3-18 3.2.3 Control Law 3-19
3.2.3.1 Definition of Parametric Model 3-193.2.3.2 Definition of
System
Output Prediction 3-193.2.3.3 Determination of Polynomials
3-213.2.3.4 Determination of Control Law 3-21
3.2.4 Comment 3-25
LE S
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Table of Contents
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SECTION 4
Implementationand Simulation of
AdaptiveControllers UsingReference Models
iv
INDEXREFERENCES
For Mor
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4.1 Implementation 4-1
4.1.1 Simulation 4-24.2 Generalized Prediction Controllers
4-2
4.2.1 Implementation 4-2IC
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SECTION 5
Conclusion
5.1 Advantages of Adaptive Control 5-15.2 Advantages of
DSP56000/DSP56001
Architecture 5-3 SE
M
APPENDIX
The Least Mean-Square Principle
A.1 Equation Formulation A-1A.2 Estimation of Model Parameters
A-3A.3 The Least-Squares Estimator A-6 A.3.1 Is the LSE biased? A-6
A.3.2 How accurate is the LSE? A-7 A.3.3 Conclusion (on LSE
accuracy and bias) A-7A.4 Improving the “LSE” A-8
A.4.1 What is required to minimize LSE bias? A-8
A.4.2 What is required to minimize LSE variance? A-8
A.5 Conclusion A-9
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Index-1Reference-1
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Illustrations
MOTOROLA
v
Figure 1-1
Figure 2-1
Figure 2-2
Figure 2-3
Figure 2-4
Figure 2-5
Figure 3-1
Figure 3-2
Figure 4-1
Figure 4-2
Figure 4-3
Figure A-1
Basic principles of adaptive control 1-6
Parametric model description in terms of processinput and output
2-2
Adaptive controller in closed loop 2-6
Standard input and disturbance signals 2-7
Dynamic response in open loop 2-8
Desired dynamic response in closed loop 2-9
Adaptive control using reference models in closed loop 3-3
Generalized predictive control using closed loop 3-18
Simulation results for adaptive controller using
4-3parallel-serial reference models
Generalized Predictive Controller 4-4
Adaptive Controller 4-15
Parametric model description in terms of processinput and output
A-1
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MOTOROLA
“An adaptivecontrol system
measures acertain
performancerating of the
system (or plant)to be controlled.”
Introduction
SECTION 1
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This section shows how Motorola DSP56000/DSP56001 digital signal
processors can be used tosolve real-time digital control problems.
After review-ing the relevant basic theory of adaptive control,
welook at a number of implementations.
1.1 History of Adaptive Control
Computerized industrial process control has advancedby leaps and
bounds over the last ten years in hard-ware and methods. The
development of newmicrocontrollers and digital signal processors
(DSPs)has given rise to important changes in regulation sys-tem
design. The capabilities and low cost of the latestDSPs make them
ideal for a wide range of regulationapplications. Further, and
despite the fact that analogregulators still enjoy wide popularity,
DSPs offer higherperformance than their analog predecessors.
Very few of the microcontroller-based digital regula-tors
developed to date fully exploit the keyadvantages of microprocessor
technology. Most de-signers seem content to emulate the behavior
oftraditional PID analog regulators. Sad though it maybe, this is
indeed an accurate reflection of industrialreality, in many
cases.Unfortunately, conventional PIDcontrollers, whether analog or
digital, are only efficient
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1-2
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where the system to be controlled (the plant) — orrather the
model of that system represented within thecontroller — is
characterized by constant parametersapplicable at all operating
points. And yet, most com-plex industrial systems are characterized
byparameters that vary with the system operating point[REN-88],
thus failing to meet the basic assumptionjust stated. Two examples
are heat exchangers (suchas those used in the production of textile
fibers) and in-ternal-combustion engines. In such cases, a
controlsignal generated by a conventional PID controller (i.e.one
for which the parameters are computed once andfor all on the basis
of a constant-parameter systemmodel) will inevitably give rise to
progressively moredegraded operation of the overall control loop as
theerrors between controller and actual process parame-ters
increase. This can only be corrected by modifyingthe controller
coefficients . . . Which brings us to adap-tive control.
1.2 Theory of Adaptive Control
Together or separately, microcontrollers and DSPsenable us to
design higher performance regulationsystems using more
sophisticated digital control al-gorithms, many of which have
already beendeveloped under and tested in industrial
conditions[IRV-86]. Adaptive control represents an advancedlevel of
controller design. It is recommended for sys-tems operating in
variable environments and/orfeaturing variable parameters.
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Although adaptive control has only been around fora few years,
it has already been successfully em-ployed in a number of
industrial applications. Thebasic principles were published by
Kalman, in 1958,for the stochastic approach, and later by
Whitakerfor the deterministic approach. However, the tech-nique was
not viable for two reasons:
• The solutions proposed at the time were not very “robust”.
• The hardware (computers) required forimplementation were
either unavailable or fartoo expensive.
Currently, however, the technique is rapidly gainingnew
supporters. This is largely a result of recentwork that has
improved algorithm robustness[SAM-83 and IRV-83] and of the
development of mi-crocontrollers and/or DSPs which make it
possibleto support and implement the new algorithms.
Adaptive control is a set of techniques for the au-tomatic,
on-line, real-time adjustment of control-loop regulators designed
to attain or maintain agiven level of system performance where the
con-trolled process parameters are unknown and/or time-varying. The
use of microcontrollers and/or DSPs incontrol loops offers the
following advantages:
• Wide range of alternative strategies for controllerdesign and
mathematical modelling,Freedom touse regulation algorithms that are
more complexand offer higher performance than PID
• Technique is suitable for process controlapplications
involving time delaysV
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• Automatic estimation of process models fordifferent operating
points
• Automatic adjustment of controller parameters
• Constant control system performance in thepresence of
time-varying process characteristics
• Real-time diagnostic capability
Adaptive control is based entirely on the following hy-pothesis:
the process to be controlled can bemathematically modelled and the
structure of thismodel (delay and order) is known in advance. The
de-termination of the structure of a parametric systemmodel is thus
a vital step before going on to design anadaptive control
algorithm. The identification tech-nique should be selected by a
specialist in automaticcontrol. The capabilities of the adaptive
control algo-rithm depends, to a large extent, on the
faithfulnesswith which the model represents the system and
itsbehavior. The chief advantage, in practical terms, ofadaptive
control appears to be the capability to ensurequasi-optimal system
performance in the presence ofa model with time-varying
parameters.
Once the model and its structure have been identi-fied, the next
step is to select a control strategy.This choice depends in part on
the nature of theproblem (regulation or tracking) and on the
systemcharacteristics (minimum phase or not). The num-ber of
options available depends on the extent ofour advance knowledge of
these characteristics.The aim is to select a strategy yielding a
satisfactorycontrol law in the case where the system model and
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its environment are fully determined. The strategiesmost
commonly encountered in adaptive control are:
• For minimum-phase systems (or for systems wherethe non-minimum
phase is fully determined):“minimum-variance control” or “control
usingreference models”
• For non-minimum-phase systems: “pole-placementcontrol” or
“quadratic-criterion optimal control”
The adaptive control algorithm is then designed inaccordance
with the structure of the system modeland the selected control
strategy. As a rule, theadaptive control algorithm can be seen as a
combi-nation of two algorithms. An identification algorithmuses
measurements made on the system and gen-erates information (a
succession of estimates) forinput to a control law computation
algorithm. Thissecond algorithm determines, at each instant,
theadaptive controller parameters and the control to beapplied to
the system. This type of adaptive con-trol is termed indirect .
However, the breakdowninto two parts is not always apparent. For
example,no control law computation algorithm is required atall if
the parameters characterizing the adaptivecontroller are directly
identified. This is known as di-rect adaptive control .
We will look first at adaptive control based on a di-rect scheme
using a reference model . There aretwo main reasons for this
choice: first, this type ofcontrol is relatively easy to implement;
second, it hasalready found practical applications in industrial
sys-tems [LAN-84], [DAH-82].V
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Comparisonsystem
Desiredperformances
Re
Figure 1-1 Basic princ
The performance ratinggoal. The adaptive sysin order to maintain
the
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A discussion follows on an adaptive control systembased on an
indirect scheme which, to date, judgingfrom our bibliographic
research, offers the best sys-tem response. This type of control
was introducedby Clarke [CLA-84]. It produces optimal control
overany system, with or without time delays and irre-spective of
whether the inverse is stable orunstable. This scheme is known as
generalizedpredictive control .
The basic principle underlying adaptive controlsystems is
relatively simple (see Figure 1-1). Anadaptive control system
measures a certain per-formance rating of the system (or plant) to
be
PlantAdaptivecontroller
ferenceinput
Performancerating
measure
Output
Disturbance
iples of adaptive control
of a system is measured and compared to the designtem modifies
the parameters of the adaptive controller performance rating close
to the desired value.
djustmentsystem
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controlled. Starting with the difference between thedesired and
measured performance ratings, theadjustment system modifies the
parameters of theadaptive controller (or regulator) and the
controllaw in order to maintain the system performancerating close
to the desired value(s).
Note that, in order to design and correctly adjust (ortune) a
good controller, we must specify the desiredperformance of the
regulation loop and determinethe dynamic process model describing
the relationbetween variations in control signals and output.This
means we must determine the representationmodel which, in turn,
means that we must establishthe system's order and time delay.
The literature on adaptive control includes hun-dreds of papers
on different approaches to theproblem. As a result, engineers who
are not special-ists in adaptive control theory often find it
verydifficult to determine which approach they shoulduse to solve a
given problem. The aim of this appli-cation note is to introduce
the reader to the twomain principles of adaptive control identified
to dateand to guide the design engineer in the selection ofcontrol
strategies applicable to a given situation.
The two principles selected for discussion were cho-sen on the
basis of the goal of any design project,namely the determination of
a real-time control lawapplicable to a given process. The total
number ofoperations required to parameterize the control lawis
assumed to be one of the criteria most important tothe design
engineer. It is true that for high-speed in-dustrial systems using
microcontrollers — such asV
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automotive Anti-lock Braking Systems (ABS) and ac-tive
suspensions, to name but two — the totalnumber of operations
assigned to the control algo-rithm cannot be very high. Given their
internalstructure (Von Neumann), conventional microcon-trollers
only have a limited real-time computationcapability, thus directly
limiting the complexity of thecontrol algorithms.
The architecture of Motorola DSP56000/DSP56001devices features a
multi-bus processor that is highlyparallel (extended Harvard
architecture) and spe-cially designed for real-time digital
signalprocessing. In view of their computational power,these
devices can be used to implement sophisti-cated control algorithms
and thus to control high-speed industrial systems.
Apart from the fact that digital signal processing isnow widely
employed, the chief advantages of Mo-torola DSP56000/DSP56001
controllers can besummarized as follows:
• Lower system component costs because a singleDSP56000/DSP56001
controller can replace notonly the microcontroller but also the
componentsrequired for 3-D lookup tables to digitally mapmodel
characteristics (as in the case of injectionsystems, active
suspensions, etc.).
• Further savings can be achieved by using digitaltechniques
(e.g. an observation model) toreplace expensive sensors.
• The computations performed by DSP56000/DSP56001 devices are
more accurate than
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microcontroller interpolation between the inputsof a 3-D LUT
(lookup table).
• A single DSP56000/DSP56001 controller cananalyze and process
several input parametersat a time.
Each Motorola DSP56000/DSP56001 device isboth a high-speed
microcontroller and a powerfuldigital signal processor. The
DSP56001 programRAM can accommodate 512 words of 24 bits. TheRAM
can be loaded, following a clear, from a 2K x8-bit EPROM or from a
host processor. For mass-produced products, the DSP56000 offers a
pro-gram ROM of 3.75K words of 24 bits which can befactory
programmed for stand-alone applications.
Apart from the amount of memory space allocatedto the program
field, the DSP56000 and DSP56001controllers are identical, with two
separate memoryspaces for data. A further feature is
multiplicationwith accumulation of previous values, a
capabilitymuch used by real-time control algorithms.
ADSP56000/DSP56001 controller can multiply two24-bit numbers, add
the 48-bit result to the contentsof the 56-bit accumulator, and
simultaneously ac-cess the two data memory fields, all in a
singleinstruction cycle.
For fast input/output, DSP56000/DSP56001 devic-es feature three
peripheral devices in the samepackage, namely: a host processor
interface (HI), asynchronous serial interface (SSI), and a serial
com-munications interface (SCI). When the SCI is notrequired for
communications, the baud rate genera-tors can be used as timers.
Depending on the wayV
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in which the peripherals are configured, DSP56000/DSP56001 can
offer up to 24 I/O lines. These fea-tures make DSP56000/DSP56001
controllers idealfor a wide range of real-time control
applicationswhere their processing power can be used to advan-tage.
Applications include: disk drives, motorcontrol, automotive active
suspensions, active noisecontrol, robotics, etc. ■
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MOTOROLA
Numerical Domain Representation
dnY t( )
dtn----------------- α1
dn –
dtn----------+
SECTION 2
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2.1 Parametric Models
For any continuous, mono- or multi-variable physicalsystem, the
search for a suitable parametric model —whether by empirical
methods or on the basis of experi-mental data — leads to the use of
linear differentialequations to represent the process to be
identified. Theseequations are of the form:
Eqn. 2-1
In nature, no system is rigorously linear in the mathe-matical
sense. However, most processes approachlinear behavior over a
limited operating range.
Contrary to non-parametric models (finite impulse re-sponse),
parametric models depend on a specificstructure. The parametric
model characterizes the dy-namic behavior of a physical system in
terms of itstransmittance or transfer function. This may be
de-duced using a z-transform. Applying such a transformto
expression Eqn. 2-1, we obtain:
1Y t( )1–
--------------- … αnY t( )+ + β0dmU t( )
dtm------------------- … βmU t( )+ +=
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2-2
G z( ) Y z( )U z( )-----------=
U(k)
Figure 2-1 Parametric
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Eqn. 2-2
where:
• (a1,. . .,an) and (b0,. . .,bm) represent theparameters of the
sampled model
• d represents the time delay (for i ≤ d then, bi = 0)
• n determines the order of the model (n ≥ m)
• U (z) is the model input
• Y (z) is the model output
The most widely used parametric model is illustrat-ed in Figure
2-1:
z d– b0 … bmzm–+ +( )
1 a1z1– … anz
n–+ +
+-------------------------------------------------------------
z
d– B z 1–( )
A z 1–( )--------------------------= =
b(k)
Y(k)q d– B q 1–( )A q 1–( )
--------------------------
model description in terms of process input and output
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006
with:
• q-1 is the time delay operator.
• A (q-1) = 1 + a1q-1 +. . .+anq-1
• B (q-1) = b0+. . .+bmq-1
• b(k) represents all noise sources expressed interms of their
equivalent effect on output.
The model described by equation Eqn. 2-2 is knownin the
literature as the polynomial parametric model.Expression Eqn. 2-2
is solely in terms of the pro-cess input and output. The model can
also berepresented as a first-order differential equation
byconverting expression Eqn. 2-1. This representa-tion is known as
the parametric state model and isdefined in accordance with
equation Eqn. 2-3.Throughout the remainder of this application
notewe will assume that polynomial B(q-1) is of thesame degree as
polynomial A(q-1).
Eqn. 2-3
where:
• Xk is the state vector of dimension ((n+d) x 1)
• P is the state matrix of dimension ((n+d) x (n+d))
• Q is the input vector of dimension ((n+d) x 1)
• C is the output vector of dimension (1 x (n+d))
• n is the order of the system
Xk 1+ P Xk⋅ Q Uk⋅+=
Yk C Xk⋅=
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The relation between these two representations ofthe parametric
model is given by:
Eqn. 2-4
While it is true that the parametric model approxi-mates the
behavior of the physical system, one mustbe cautious when it comes
to the physical interpre-tation of the parameters contributing to
the model'sstructure.
The purpose of the parametric model is to approxi-mate as
closely as possible the behavior of thesystem by ensuring the
closest possible match be-tween predicted and observed output. This
is done,moreover, within the limits of an accuracy vs. sim-plicity
trade-off that the automatic control specialistdefines when
choosing the parametric model togenerate the control law.
The advantages of the parametric model approachlie in its
structure:
• It enables us to describe, sufficiently accurately,the
dynamics of an arbitrary physical processusing fewer parameters
that are required by thenon-parametric model (finite impulse
response).
0
0
01
10
0
0
0
0
0
0
bn
d
0 0P = Q = C = 1b0
-a1
-an
} (vector from Eqn. 2-2)FR
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… an Y k n–( )⋅– e k( )+
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006
• It is relatively simple to implement on thecontroller. Using a
well-known property of the z-transform (time delay theorem), we can
proceedfrom the polynomial parametric model to thedifference
equation of the following form:
Eqn. 2-5
with: • e (k) representing the generalized or residual noise
• e (k) = b (k).(1 + a1q-1 +. . .+anq-1)
Given that we now have the time-history of the inputand output
signals, we can readily predict the modeloutput values. This
important point is widely used inmodern regulation theory.The state
parametricmodel is useful for describing multivariable systems.
The chief drawback of the parametric model is thedifficulty of
determining the order of the system. Ifthe designer underestimates
the process order,model predictions will not match actual system
be-havior. On the other hand, if the designeroverestimates the
order, the increased complexity ofthe model will mean longer
computation times. Thissame comment also applies to the estimation
of puretime delays. The automatic control specialist musttherefore
pay careful attention to this phase of themodelling procedure. With
most industrial systems,we do not have access to the states values,
which isa major handicap for the state parametric model.There are
state observer techniques allowing the
Y k( ) b0 U k d–( )⋅ … bn U k n– d–( )⋅ a1 Y k 1–( )⋅– –+ +=
HIV
ED
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2-6
Figure 2-2 Adaptive
This system will be useas it works to maintaindisturbance.
Input
r(k)PropAdaCont
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state estimation, but having a heavy penalty interms of
computation time.
2.2 Adaptive Control Techniques
Consider the two adaptive control techniques ap-plied to a
closed-loop physical system as shown inFigure 2-2:
In these examples, G(z), the plant transfer function,is defined
as follows:
Eqn. 2-6
controller in closed loop
d to measure the performance of the adaptive controller (or
equal) the desired response in the presence of a
++
u(k)
PlantG(z)
osedptiveroller
Disturbanced(t)
Outputy(k)
G z( )z 1– b0 b1z
1–+( )⋅
1 a1z1– a2z
2–+ +------------------------------------------------ Y z( )
U z( )-----------= =
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-
ance signal: D(t)
ime (sec.)5 10 15
rbance signals
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The nominal values for the process parameters are:
b0 = 0.039
h = 0.031
a1 = -1.457
a2 = 0.527
The performance of the adaptive controls present-ed here will be
evaluated on the basis of thesystem's capacity to equal the
closed-loop re-sponse (Figure 2-2) with the desired
performance.Before going on to make the different comparisons,we
must first define the standard (input and output)signals to be used
with the simulated system andthe desired closed-loop performance.
The input anddisturbance signals are shown in Figure 2-3.
Note,these signals will be assumed to be fixed through-out the
remainder of this section.
Reference signal: R(t) Disturb
Mag
nitu
de
Mag
nitu
de
Time (sec.) T
2
1
0
-1
-2
0 5 10 15
2
1
0
-1
-2
3
0
Figure 2-3 Standard input and disturbance signals
Noise free representation of the reference input and
distuIVE
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In order to characterize the dynamic response ofthe uncorrected
(i.e. without regulation) simulatedsystem, we apply the input
signal defined in Figure2-3. The system dynamic response is
illustrated inFigure 2-4.
The function of the different regulators presented inthe
following pages is to improve the dynamic be-havior of the
simulated system. Two constraints areimposed on these regulators.
These constraints willbe used, at first, to determine the desired
perfor-mance of the closed-loop system. The constraintsare defined
as follows:
• In order to respond more rapidly to variations in thereference
value and/or the level of disturbance, werequire that the simulated
system have a settlingtime of no more than 2.5 seconds.
Output signal in open loop
Time (sec.)
Mag
nitu
de2
1
0
-1
-2
0 5 10 15
Figure 2-4 Dynamic response in open loop
Dynamic response of the uncorrected systemwhen it has been
excited by the reference signal ofFigure 2-3.
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• During the dynamic response, the variation in theoutput signal
of the simulated system shall be setfor an overshoot of 70%
relative to the final value.
The desired system dynamic response is illustratedin Figure 2-5
(without disturbance):
To illustrate and compare the performance of the dif-ferent
adaptive controllers, we introduce, for eachmethod of adaptive
regulation, a variation in the ref-erence value (R(t), Figure 2-3),
followed, as soon asthe closed-loop system response has stabilized,
bya disturbance (D(t), Figure 2-3). The impact of thedisturbance is
then monitored. ■
Desired output signal in closed loop
Mag
nitu
de
Time (sec.)
2
1
0
-1
-2
0 5 10 15
Figure 2-5 Desired dynamic response in closed loop
Dynamic response without disturbance of the closedloop
system.
IVE
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Adaptive Control and Adaptive Controllers
“The mainadvantage of
generalizedpredictive
control is thatthe control isalways stable
irrespective ofthe nature of the
system to beregulated (the
plant). “
SECTION 3
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3.1 Adaptive Control Using Reference Models
3.1.1 Introduction
An adaptive controller may be of conventional de-sign or it may
be more complex in structure,including adjustable coefficients such
that their tun-ing, using a suitable algorithm, either optimizes
orextends the operating range of the process to beregulated. The
different methods of adaptive controldiffer as to the method chosen
to adjust (or tune) thecontrol coefficients.
This section discusses adaptive control using paral-lel-serial
reference models which, along with self-tuning control, are the
only control schemes to havefound practical applications to date.
The adaptivecontrol scheme using parallel reference models
(i.e.located in parallel on the closed-loop system) wasoriginally
proposed by Whitaker in 1958. The versionproposed at the time
offered a solution to the trackingproblem, but not the regulation
problem. Note that atracking problem is defined when the reference
value
ES
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(r(k)) varies and when no disturbances (d(k)) arepresent in the
output (y(k)). A regulation problem isdefined when the reference
value is zero or steadyand when there is a disturbance in the
output suchthat its effect must be reduced by the control
(u(k)).
The parallel model structure is suitable for solvingthe tracking
problem and is demonstrated by thefact that the model requires
reasonable control sig-nals; the structure is not suitable for
solvingregulation problems and is demonstrated by thefact that, in
this case, the model requires unreason-able control signals. We
obtain unreasonablecontrol signals because the estimated error
(differ-ences between the output of the parallel referencemodel and
that of the system) converges to zeroduring a single sampling
interval. To attenuate thecontrol signal, a serial reference model
(i.e. in se-ries with the estimated error) can be added to
thegeneral structure. This imposes a converge-to-zerorequirement,
with a chosen dynamic response, thatis less severe than in the
previous case [IRV-85].Let us now look at this adaptive control
method us-ing parallel-serial reference models more closely.
3.1.2 Closed-Loop System
An adaptive control system comprises not only afeedback-type
control loop (or inner loop) includingan adaptive controller, but
also an additional, or out-er, loop acting on the controller
parameters in orderto maintain system performance in the presence
of
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-
ep -
+
y (k)
eferenceodel
losed loop
l loop that includes an controller to maintain
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006
variations in the process parameters. This secondloop also has a
feedback-loop-type structure, thecontrolled variable being the
performance of thecontrol system itself. The arrangement is
schemat-ically shown in Figure 3-1.
where: • ep represents the parallel estimated error
• es represents the serial estimated error
This type of adaptive scheme offers the advantageof being able
to accommodate separately bothtracking and regulation problems.
This is becausethe desired performance of the controlled systemare
defined by a parallel model for a tracking prob-lem and by a serial
model for a regulation problem.
AdaptiveController
PlantG(z)
yref(k)
es
r(k) u(k)
Serial RM
IdentificationAlgorithm
Figure 3-1 Adaptive control using reference models in c
Note that this system not only has a feedback type
controadaptive controller but also an outer loop that acts on
theperformance in the presence of disturbances.
Parallel ReferenceModel
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3.1.3 Control LawThe dynamic behavior of the simulated system
isdefined by a parametric model. We recall that itsgeneral
structure is given by the relation:
Eqn. 3-1
where:
• n = 2 (order)
• d = 1 (time delay)
• A(q-1) = 1 + a1.q-1 + a2.q-2
• B(q-1) = b0 + b1.q-1
The order of polynomials A(q-1) and B(q-1) andalso the time
delay of the parametric model enableus to correctly dimension the
control law. To bringus nearer to the formulation of the adaptive
controllaw, we first consider the case where the systemparameters
are known.
3.1.3.1 Known System Parameters
With the objectives of tracking and regulation beingindependent,
we can formalize their respectiveequations as: A: Regulation (r(k)
= 0).
The problem here is to determine a control (u(k))that will
eliminate an initial disturbance (d(k)) with adynamic response
defined by the relation:
Eqn. 3-2
with:
• d = 1
• n = 2 (order)
A q 1–( ) Y k( )⋅ q d– B q 1–( ) U k( )⋅ ⋅=
Ar q1–( ) Y k d+( )⋅ 0=
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Ar(q-1) = 1 + ar1.q-1 + ar2.q-2 Eqn. 3-3
The polynomial Ar(q-1) is determined by the designengineer to be
asymptotically stable for order n.The polynomial represents the
serial (or regulation)model. B: Tracking (d(k) = 0).
The problem here is to determine a control (u(k))such that the
system output (y(k)) satisfies a rela-tion of the form:
Eqn. 3-4
where:
• n = 2 (order)
• d = 1 (time delay)
• Ap(q-1) = 1 + ap1.q-1 + ap2.q-2
• Bp(q-1) = bp0 + bp1.q-1
This corresponds to tracking a trajectory defined bythe
following reference model:
Eqn. 3-5
In general, one may assume that there is some linkbetween the
tracking dynamic response Ap(q-1)and the regulation dynamic
response Ar(q-1). How-ever, in this application note, and for the
sake ofsimplicity, we shall assume identical dynamic re-sponse to a
variation in either load or referencevalue, i.e. we shall assume
Ap(q-1) = Ar(q-1). We
Ap q1–( ) Y k d+( )⋅ Bp q
1–( ) R k( )⋅=
Gp q1–( )
q d– Bp q1–( )⋅
Ap q1–( )
-----------------------------------=
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shall further assume a reference model such thatthe output is
described by the relation:
Eqn. 3-6
Under these conditions, the aims to be achieved bythe control
signal can be expressed in the form:
Eqn. 3-7
The control law, with a parallel-serial referencemodel, can be
deduced by minimizing the followingquadratic criterion:
Eqn. 3-8
In the case of unit time delay (d = 1), we can deter-mine the
control law directly by minimizing criterionEqn. 3-8 relative to
u(k). The problem may be differ-ent, however, if the pure time
delay of the controlledsystem is equal to or greater than twice the
sam-pling period. In order to obtain a causal regulator,i.e. one
such that u(k) is of the form:
Eqn. 3-9
We must first rewrite the process output predictionin terms of
the quantities measurable at time k andprior to time k. The
prediction can be expressed inthe form:
Ap q1–( ) Yref k d+( )⋅ Bp q
1–( ) R k( )⋅=
es k d+( ) Ap q1–( ) Y k d+( ) Yref k d+( )–[ ]⋅ 0= =
J k d+( ) es2 k d+( ) Ap q
1–( ) Y k d+( ) Yref k d+( )–[ ]⋅[ ]= =
U k( ) Fu Y k( ) Y k 1–( ) ……… U k 1–( ) …, , , ,( )=
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) …, )
k 1–( )
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Eqn. 3-10
In the literature, an expression of this form is knownas a
“d-step-ahead predictive model”. An expres-sion such as Eqn. 3-10
can be obtained directlyusing the general polynomial identity:
Eqn. 3-11
where: • S(q-1) = 1 + s1.q-1 + . . . +sd-1.q-d+1
• R(q-1) = r0 + r1.q-1 + . . . + rn-1.q-n+1
This relation yields a unique solution for polynomi-als S(q-1)
and R(q-1) when the degree of S(q-1) isd–1. Polynomials S(q-1) and
R(q-1) can be ob-tained either recursively or by dividing
polynomialAp(q-1) by polynomial A(q-1). Polynomial S(q-1)then
corresponds to the quotient while q-d.R(q-1)corresponds to the
remainder. Multiplying bothsides of Eqn. 3-11 by y(k+d) and taking
into accountexpression Eqn. 3-1, we obtain:
Eqn. 3-12
This can be rewritten in the form:
Eqn. 3-13
where: B(q-1).S(q-1) = b0 + q-1.Bs(q-1)
Ap q1–( ) Y k d+( )⋅ Fy Y k( ) Y k 1–( ) ……… U k( ) U k 1–(, , ,
,(=
Ap q1–( ) A q 1–( ) S q 1–( )⋅ q d– R q 1–( )⋅+=
Ap q1–( ) Y k d+( )⋅ R q 1–( ) Y k( )⋅ B q 1–( ) S q 1–( ) U k(
)⋅ ⋅+=
Ap q1–( ) Y k d+( )⋅ R q 1–( ) Y k( )⋅ b0 U k( )⋅ Bs q
1–( ) U⋅+ +=
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3-8
J k d+( ) R q 1–( ) Y(⋅[=
b0 R q1–( ) Y k( )⋅ b0+[⋅
J k d+( )δU k( )δ
----------------------=
U k( )
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Substituting Eqn. 3-13 into criterion expressionEqn. 3-8, we
obtain:
Eqn. 3-14
The criterion can now be minimized by determiningthe control
u(k) for which:
Eqn. 3-15
Combining this with expression Eqn. 3-14, weobtain:
Eqn. 3-16
Now, using expression Eqn. 3-6, we obtain the re-quired control
in the form:
Eqn. 3-17
where polynomials Bp(q-1), R(q-1) and Bs(q-1) aredefined by:
Bp(q-1) = bp0 + bp1.q-1
R(q-1) = (ap1 - a1) + (ap2 - a2).q-1 = r0 + r1.q-1
Bs(q-1) = b1 = bs0
k) b0 U k( )⋅ Bs q1–( ) U k 1–( )⋅ Ap q
1–( ) Yref k d+( )⋅–+ + ] 2
J k d+( )δU k( )δ
---------------------- 0=
U k( )⋅ Bs q1–( ) U k 1–( )⋅ Ap q
1–( ) Yref k d+( )⋅–+ ] 0=
1b0------ Bp q
1–( ) R k( )⋅ R q 1–( ) Y k( )⋅– Bs q1–( ) U k 1–( )⋅–[ ]⋅=
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1) U k 1–( )⋅ ]
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The control expressed in relation Eqn. 3-17 thushas the property
of reducing criterion Eqn. 3-8 tozero while independently meeting
the requirementsof both tracking and regulation.
In other words, in the case of regulation (r(t) = 0),criterion
expression Eqn. 3-8 represents a mini-mum-variance condition on the
process output.Physically, this criterion implies minimizing
themean energy of the “filtered error” expression in re-lation Eqn.
3-7.
The equations presented in this section were madepossible by the
fact that we knew the parameters ofthe controlled process. Let us
now look at the casewhere these process parameters are unknown.
3.1.3.2 Unknown System Parameters
In the adaptive case, the structure of the controlleris the same
as for known system parameters, ex-cept that we replace the fixed
parameters byvariable ones. With the role of the adaptive, or
out-er, loop being to determine the correct values ofthese
parameters, the self-tuning controller equa-tion can be derived
from Eqn. 3-17 and written as:
Eqn. 3-18
where: bo(k), ro(k),. . ., bs1(k), . . ., are the controller
parameter estimates at time k
U k( ) 1
b̂0 k( )------------- Bp q
1–( ) R k( )⋅ R̂ k q, 1–( ) Y k( )⋅– B̂s k q,–(–[⋅=
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3-10
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By defining the tuning vector θ(k) and the measure-ment vector
Ψ(k) by the following expressions:
Eqn. 3-19
The controller equation can be rewritten in the form:
Eqn. 3-20
The next step is to determine the recursive param-eter-vector
self-tuning algorithm.
3.1.4 Determination of Controller Parameters
The self-tuning controller parameters are deter-mined by
recursive minimization of a least-squarestype criterion starting
from asymptotic stability con-ditions dictated by the model-process
error. Theaim then is to estimate the parameter vector at timek in
such a way that it minimizes the sum of thesquares of the filtered
errors between the processand the model over a time-horizon of k
measure-ments. This is expressed by the relation:
Eqn. 3-21
θ̂T k( ) b ̂ 0 k ( ) b ̂ s0 k ( ) r ̂ 0 k ( ) r ̂ 1 k ( ) =
Ψ k( ) U k ( ) U k 1 –( ) Y k ( ) Y k 1 –( ) =
Bp q1–( ) R k( )⋅ θ̂T k( ) Ψ k( )⋅=
J1 k( ) es2 i( )
i 1=
k
∑ Ap q 1–( ) Y i( ) Yref i( )–( )⋅[ ] 2i 1=
k
∑= =
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This same condition can also be expressed in theform:
Eqn. 3-22
The values of θ(k) which minimize criterion Eqn. 3-22are
obtained by determining the value of θ(k) whichcancels in the
expression:
Eqn. 3-23
Applying relation Eqn. 3-23 to relation Eqn. 3-22,we obtain:
Eqn. 3-24
From equation Eqn. 3-24 we have:
Eqn. 3-25
J1 k( ) Bp q1–( ) R i( )⋅ θ̂T i( ) Ψ i( )⋅–[ ] 2
i 1=
k
∑=
J1 k( )δ
θ̂ k( )δ---------------- 0=
δJ1 k( )
δθ̂ k( )---------------- Ψ i( ) Bp q
1–( ) R i( )⋅ θ̂T i( ) Ψ i( )⋅–[ ]⋅[ ]i 1=
k
∑– 0= =
θ̂ k( ) Ψ i( ) ΨT i( )⋅i 1=
k
∑1–
Bp q1–( ) R i( ) Ψ i( )⋅ ⋅
i 1=
k
∑⋅=
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In the previous expression, we now let:
Eqn. 3-26
where:
Expression Eqn. 3-25 corresponds to the non-re-cursive
least-squares algorithm. To obtain arecursive algorithm, we
recompute the optimal val-ue of θ(k+1) for the minimization
condition J(k+1)and express θ(k+1) as a function of θ(k).
Thisyields:
Eqn. 3-27
where:
Here, F(k+1) represents the estimator tuning gain.This is an
important variable since it gives us an in-dication of the quality
of estimation (covariance ofparameter estimates).
It has been shown elsewhere [LJU-83] that if k (ex-periment
time) increases, the θ(k) estimates tendtowards constants. In this
case, the variance of theestimates tends towards zero (F(k+1) = 0).
The
θ̂ k( ) F k( ) Bp q1–( ) R i( ) Ψ i( )⋅ ⋅
i 1=
k
∑⋅=
F 1– k( ) Ψ i( ) ΨT i( )⋅i 1=
k
∑1–
=
θ̂ k 1+( ) θ̂ k( ) F k 1+( ) Ψ k( ) es k 1+( )⋅ ⋅+=
F 1– k 1+( ) F 1– k( ) Ψ k 1+( ) ΨT k 1+( )⋅+=
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least-squares algorithm briefly presented here hasprogressively
less effect on new measurement val-ues. This is acceptable if the
process is unvaryingin time. However, this is not the case in this
applica-tion note since the system parameters are explicitlyassumed
variable. This problem can be resolved bymodifying the J1(k)
criterion. We need to arrangefor the criterion to “forget” earlier
measurement val-ues by adding a suitable weighting factor. Whenthis
is done, the criterion to be minimized becomes:
Eqn. 3-28
where: λ represents the weighting, or “forgetting factor” (0
< l ≤ 1)
The thus modified least-squares algorithm is detailedin APPENDIX
A . The main difference between algo-rithms is which variables are
contained in vectorΨ(k). In the literature, this quantity is
referred to asthe “measurement vector” while es(k) is termedthe
“post-prediction tuning error”.
In order to ensure the stability of the overall system,the
recursive least-squares identification algorithmmust meet the
following three conditions:
• The rapid decrease in the prediction error(es(k)) must occur
during the periods when θ(k),the unknown parameter of the system to
beidentified, is constant.
J2 k( ) λk i– es
2 i( )⋅i 1=
k
∑=
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3-14
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• Irrespective of any variations in the domainbounded by θ(k),
the adjusted parameter θ(k) ofthe identifier must remain within the
appropriatebounded domain.
• The variation θ(k) - θ(k-1) in the estimatedparameter must
decrease at the same time asthe prediction error es(k). If es(k) is
below acertain threshold, then θ(k) - θ(k-1) must bezero.
These conditions can only be met by making furtherchanges to the
recursive least-squares algorithm.Several authors [IRV-85 and
BOD-87] have alreadytackled this problem. We have used their
results toimprove the robustness of the controller
parameterestimation algorithm.
3.1.5 Comment
The use of a control strategy based on an output-signal
minimum-variance criterion theoretically re-quires that the system
to be regulated (the plant)have a stable inverse (i.e. b0 > b1).
It is thereforeimportant to have some prior knowledge of the
na-ture of the plant, and its behavior over its entireoperating
range. Parametric identification is used todetermine not only the
structure of the representa-tion model (order and time delays), but
also thenature of the system to be regulated (i.e. whether itis a
minimum-phase system or not). Note also thatthis type of controller
can be used to define trackingand regulation performance totally
independently.
The main disadvantage of a control strategy usinga
minimum-variance criterion applied to the variable
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to be regulated is that it always leads to a directadaptive
scheme. This presents a problem for theother control strategies
where the control law pa-rametering is broken down into two
distinct steps,namely:
• Estimation of parameters of the systemrepresentative model,
and
• Adjustment of controller parameters usingsystem
parameters.
This method of breaking down control law parame-tering leads to
an indirect adaptive scheme. Note,however, that it can be an
advantage to have ameans of monitoring system dynamic response
inreal time. Thus, the estimation of process parame-ters can be
used for diagnostics, monitoring, etc.Let us now look at this
indirect adaptive schememore closely.
3.2 Generalized Predictive Control
3.2.1 IntroductionThe adaptive control scheme presented in the
pre-vious section is useful when the system to becontrolled has a
stable inverse. This leads to inves-tigations to see if other
control schemes, associatedwith the least-squares identification
method, cangenerate stable control signals irrespective of
thenature of the system to be controlled. Given that theIV
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minimization of the mean tracking error energy(Eqn. 3-21) is not
sufficient to ensure control stabil-ity in the case of a so-called
“non-minimum phasesystem”, it seems fairly natural to investigate
whathappens if one introduces a control weighting terminto the
expression for the criterion to be minimized[SAM-83]. This is
expressed by the relation:
Eqn. 3-29
where: α is the strictly positive weighting term
An improvement in this criterion has been suggest-ed on the
basis of the following observation. A cardriver does not need to
have a complex mathemat-ical model in mind in order to be able to
drive. All heneeds is the ability to recall a set of images of
pos-sible trajectories produced by a corresponding setof control
actions on the car steering wheel. Giventhe driver's view of the
road to be followed, the hu-man control algorithm chooses the
control action(or signal) that will produce the vehicle
trajectoryclosest to the desired trajectory [IRV-85].
From this we conclude, in other words, that to ob-tain a robust
control scheme, we can use thepredictions obtained from the
identification of thesystem to be controlled and minimize a
least-squares criterion involving the difference betweenthe
predicted desired trajectory and the predictedtrajectories in
response to the control signals. This
J3 k( ) Ap q1–( ) Y i( ) Yref i( )–( )⋅[ ] 2 α U k( )2⋅+
i 1=
k
∑=
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∆ U k 1+( )2⋅
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criterion has been formulated by Clarke [CLA-84]and is expressed
in the form:
Eqn. 3-30
where:
• is the output prediction with Ny
• Yref is the predicted output of the reference model over
horizon Ny
• U(k+i) represents the predicted control over Nu
• Ny determines the horizon on the outputs
• Nu determines the horizon on the control
• α is the control weighting factor
• ∆ represents the differentiation operator (∆ = 1-q-1)
Thus, the control weighting term (α) ensures controlstability in
all cases where the system has an unsta-ble inverse, provided the
time delay is greater thanunity. The differentiation operator (∆)
enables us toobtain a control that is free of static error in the
vari-able to be controlled (Y) relative to the referencetrajectory
(Yref).
On the basis of our bibliographic research, general-ized
predictive control is considered to be the bestcontrol technique
currently available. This is whywe chose to discuss it in detail in
this application
J4 k( ) Yref yk i d+ +( ) Ŷ k i d+ +( )–[ ]2
i 0=
Ny 1–
∑ α ⋅i 0=
Nu 1–
∑+=
Ŷ k i d+ +( )
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3-18
Processing oController
Parameters
AdaptiveControllerr(k)
Figure 3-2 Generalize
Note the presence of aprocess. The advantagthe main
disadvantagecontrol law.
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note and why we made it the subject of our simula-tion studies.
An industrial application of thistechnique is described in
[LIM-89].
3.2.2 Closed-Loop SystemAs with all adaptive control systems,
the systemdiscussed in this section features not only a
con-ventional servo-type feedback loop, but also anadditional loop
designed to identify the on-line pro-cess and determine the
parameters to be adjustedon the basis of the process parameters.
The ar-rangement is schematically shown in Figure 3-2.
IdentificationAlgorithm
f
PlantG(z)
Parallel Reference Model
u(k) y(k)
yref(k)
e +
_
d predictive control using closed loop
n additional loop to perform system identification on thee is
that the process parameters would be accessible with that there is
an increase time in the computation of the
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The main advantage of this indirect adaptivescheme is that it
gives access to the process pa-rameters, which is important for
monitoring,diagnostics, and the like. The main disadvantage isthe
increased computation time required to param-eter the control
law.
3.2.3 Control Law
3.2.3.1 Definition of Parametric Model
The parametric model required to formulate thecontrol law was
defined earlier on. Recall that themathematical structure is of the
form:
Eqn. 3-31
where: A(k,q-1) and B(k,q-1) are the polynomials estimated by
the identifier at each samplinginterval.
In the remainder of this application note, we willsimplify the
mathematical notation by omitting the ^symbols (indicating
estimated variables) and the (k)portion of the different terms
indicating that the vari-able is estimated at each sampling
interval k.Te.
3.2.3.2 Definition of System Output Prediction
The prediction of the parametric model output —which is to say
the probable behavior of the processoutput between time k1 and some
future time kj —is deduced using a j-step-ahead prediction
model.
 k q 1–,( ) Y k( )⋅ q d– B̂ k q 1–,( ) U k( )⋅ ⋅=
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The general expression for such a model is:
Eqn. 3-32
where:
• j = 1..kj
• kj is the given future time-horizon
• Fj(q-1) = f0j + ... + f(kj-1)j.q-kj+1
• Ej = 1 + ... + ej-1.q-j+1
This expression is known as a Diophantine equa-tion. The
expression for the predicted model outputcan be deduced by
multiplying the two sides ofequation Eqn. 3-30 by Ej.∆, then
substituting the ex-pression for Ej.A(q-1).∆ from equation Eqn.
3-32.This gives:
Eqn. 3-33
The expression for the predicted model output cannow be
rewritten in the form:
Eqn. 3-34
where: Gj(q-1) = Ej(q-1).B(q-1)
The sequence of predicted parametric model out-puts can now be
represented by vector Y. Note thatfor all future sampling intervals
smaller than or equalto the system time delay (i.e. for j ≤ d), the
Y(k+j) val-ues can be computed using the input and outputdata
available up to time k. For sampling intervalsgreater than the
system time delay (i.e. for j > d),
1 Ej q1–( ) A q 1–( ) ∆⋅ ⋅ q d– Fj q
1–( )⋅+=
1 q j– Fj q1–( )⋅–( ) Y k( )⋅ q d– Ej q
1–( ) B q 1–( ) ∆ U k( )⋅ ⋅ ⋅ ⋅=
Ŷ k j+( ) Fj q1–( ) Y k( )⋅ Gj q
1–( ) ∆ U k d– j+( )⋅ ⋅+=
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we need to know the future control U(k+j). Theseassumptions form
the basis of generalized predic-tive control.
3.2.3.3 Determination of Polynomials
Fj(q-1) and Gj(q-1)
In Section 3.1.3.1 we showed that we could obtainthe polynomials
S(q-1) and R(q-1) by simple divi-sion. The disadvantage, however,
of this techniqueis that it is very time consuming. Clarke proposes
arecursive method for the determination of polynomi-als Fj(q-1) and
Ej(q-1). This is the solution we haveadopted. Readers interested in
this reformulation ofthe polynomials in recursive form should refer
to thebibliography, and particularly to [BOD-87], [CLA-85], and
[AST-84]. Note, the control algorithm en-coded in Motorola
DSP56000/DSP56001 digitalsignal processors is based on the same
solution.
3.2.3.4 Determination of Control Law
Above, we derived an expression (Eqn. 3-34) forpredicting the
behavior of the process output signal.The behavior of the reference
model output signal,on the other hand, is predicted by expression
(Eqn.3-35). We must now solve this equation from sam-pling time
(k+d) to the chosen time-horizon(k+d+N–1).
Eqn. 3-35
where:
Yref k d+( ) Ap* q 1–( )– Yref k d 1–+( )⋅ Bp q
1–( ) R k( )⋅+=
Ap* q 1–( ) a1 q
1–⋅ … an qn–⋅+ +=
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Determining the output of the parallel referencemodel at a
future time is not difficult since the poly-nomials Ap and Bp are
known and are constant ateach sampling time. The sequence of
referencemodel outputs from sampling time (k+d), can be ex-pressed
in vector form as follows:
Eqn. 3-36
Recall that the parallel reference model plays thesame role as
that defined for the adaptive controllerof a parallel-serial model
(Section 3.1 ). In order tocompare the performance of the two
adaptive regu-lators, we choose the same structure (n = 2 and d =1)
for the parallel model and the same dynamicresponse.
The prediction error at future time k+j is given by:
Eqn. 3-37
Proceeding as previously described, let us now de-fine the
prediction error vector:
Eqn. 3-38
Thus:
Eqn. 3-39
We saw earlier that at time k, certain elements ofvector Y are
functions of known and unknown data.Among the unknowns, we can
define the predicted
f Yref k d+( )……Yref k d N 1–+ +( )[ ]=
e k j+( ) Yref k j+( ) Ŷ k j+( )–=
e e k d+( )……e k d N 1–+ +( )[ ]=
e Yref Y–=
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control vector (U) as follows:
Eqn. 3-40
A clever decomposition of expression Eqn. 3-39 en-ables us to
separate the terms that depend onknown data at time k and those
that are unknown attime k, such as vector U. We thus obtain:
Eqn. 3-41
where: G is a triangular matrix of dimension N.N
The elements of G are generated by reformulatingthe Diophantine
equation in recursive form.
Eqn. 3-42
U ∆.U(k)……∆.U(k+N-1)[ ]=
e Yref G U f–⋅–=
G
g0 0 • • • • 0
g1 g0 • •
• • • • • • • • • • • • • • • • • • 0gN gN 1
– • • • • g0
=
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Fd q–(
Fd 1+ q1–( ) Y(⋅
Fd N 1–+ q1–( ) ⋅
Gd N 1–+[
f =
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The elements of vector f are the components of thepart of the
prediction depending on known data attime k. This vector can be
written in the form:
Eqn. 3-43
The control vector, U, can be determined by mini-mizing
criterion J4(k) as expressed in equation Eqn.3-40. In vector form,
this criterion is given by:
Eqn. 3-44
It can be shown that this criterion has a simple opti-mal
solution for:
Eqn. 3-45
Control u(k) is computed from
∆
u(k) using the fol-
lowing expression:
Eqn. 3-46
The power of generalized predictive control can begauged from
the fact that it allows us to reduce the
1) Y k( ) q Gd q1–( ) g0– ∆ U k 1–( )⋅ ⋅+⋅
k) q2+ Gd 1+ q1–( ) g0 g1– q
1–⋅– ∆ U k 1–( )⋅ ⋅ ⋅
Y k( ) qN.+
q 1–( ) g0– g1 q1– ……gN 1– q
N– 1+⋅⋅– ] ∆ U k 1–( )⋅ ⋅
J4 eT e⋅ α UT U⋅ ⋅+=
U GT G⋅ α I⋅+[ ] 1– GT Yref f–[ ]⋅ ⋅=
U k( ) U k 1–( ) ∆ U k 1–( )⋅+=
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control prediction time-horizon. The only difficultiesare the
mathematical problems posed by the inver-sion of matrix (GT.G +
a.I) of dimension N.N and theassociated computation times. But this
can be over-come by defining a control prediction time-horizonsuch
that Nu < N and an output prediction time-ho-rizon such that Ny
= N. This reduces the numberof colu