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thesis entitled

Optimal Control System Design:
The Predictive Sampline Problem

presented by

Uhi Ahn

has been accepted towards fulfillment
of the requirements for

Doctoral degree in Systems Science

 

 

@Mam

Major professor

Robert A. Schlueter

 

Date February 16, 1978

 

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OPTIMAL CONTROL SYSTEM DESIGN:
THE PREDICTIVE SAMPLING PROBLEM

BY
Uhi Ahn

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering and System Science

1978

ABSTRACT

OPTIMAL CONTROL SYSTEM DESIGN:
THE PREDICTIVE SAMPLING PROBLEM

By
Uhi Ahn

The principal contributions of this thesis are the
formulation and solution of the optimal predictive sampling
criterion for a sampled-data control system and the develop-
ment of the optimal control system design methodology for the
optimal predictive sampling problem.

The system performance index is formulated with a
control performance index that measures actual performance of
the control rather than error due to the sample and hold device
as in the formulation of previous adaptive sampling criteria.
The control performance index measures control performance over
both the sampling interval over which the control is held con-
stant and a future interval where the control is permitted to
be continuous. Thus, only one sampling interval at a time is
chosen and is based on the estimate of this performance index
which in turn is based on past measurement of outputs of the
system and knowledge of system inputs, system dynamics, and
disturbance, initial conditions, and measurement noise statis-
tics. A cost of implementation is included in the system.per-

formance index and is a specified constant if the predictive

Uhi Ahn
sampling criterion is being used to perform control on a
specified set of hardware and is a function of the length
of the sampling interval if the objective is to design and
select the computer hardware, computational algorithms,
and computer software to implement the predictive sampling
criterion.

The results of the optimal adaptive sampled-data
control with predictive sampling criterion shows that the
optimal predictive sampling criterion is indeed adaptive for
on-line control if future performance can be precisely predicted
as in the deterministic system but is periodic if future per-,
formance cannot be predicted as in the stochastic systems
Mbreover, the optimal predictive sampling criterion performs a
control function because the control performance is improved
over that of the continuous-time control for the deterministic
system.

Optimal control system design methodology is further
refined in this thesis. This optimal control system design is
broken down into the optimal control design where the para-
meters of the control performance index are optimally tuned so
that the resulting control meets the control performance objec-
tives, and the optimal system design where the hardware to be
implemented is optimally determined. The optimal system de-
sign procedure determines a precise cost of implementation as
a function of the computational algorithms, computer software
implementing that algorithm, and the hardware and then deter-
mines the optimal selection of hardware, computational algorithm,

and computer software by a tradeoff of control performance

Uhi Ahn
and cost of implementation. Thus, optimal control system
design really completes the design problem for an optimal
control system because it not only tunes the control
performance index to obtain acceptable control but also
determines a precise cost of implementation and then
selects a hardware, computational algorithm, and software
option based on the control performance and cost specifi-
cations.

An example problem.is chosen and is the linear
second-order type two system which has been used extensively
in the past research related to the optimal sampling problem.
The control performance for the optimal sampled-data
control with predictive sampling criterion is compared to
the periodic sampled-data control and continuous control.
The actual hardware cost for optimal predictive sampling
problem.is developed for this particular system. The para-
'meters of the control performance index are then tuned for
this system based on control objectives. A particular hard-
ware, algorithm, and computer software option is then
selected for this system.based on a tradeoff of performance

and cost.

To my wife, Jay-Bum,
and Bobby Jae-Hong

ii

ACKNOWLEDGMENTS

This author wishes to express his sincere appreciation
to his academic advisor, Dr. Robert A. Schlueter, for his
guidance and numerous suggestions during the course of this
work and also for his painstaking review of this manuscript.

Other thesis guidance committee members, Drs. Jhon B.
Kreer, Gerald L. Park, Robert O. Barr, and Albert N. Andry, are
also warmly thanked for their concern and interest in this work.
A special note of thanks is due Dr. Gerald L. Park for his
valuable comments on this manuscript.

The financial support of Consumers Power Company in
Jackson in Michigan and the Division of Engineering Research at
Michigan State University are gratefully acknowledged.

The understanding, patience and encouragement of the

author's wife, Mee-Eyun, is sincerely appreciated.

iii

CHAPTER

LIST OF

LIST OF
1.

TABLE OF CONTENTS

TABLES .
FIGURES
INTRODUCTION .

Historical Development .

Optimal Control System Design .

Optimal Aperiodic Sampling Problem for
Control . . . .
Optimal Predictive Sampling Problem .
Important Results and Contributions .

PROBLEM FORMULATION

OPTIMAL CONTROL SYSTEM DESIGN (OCSD)
METHODOLOGY .

3.1 Optimal Control Design (OCD) Problem
3.2 Optimal System Design (OSD) Problem .

HH HHH
up UNH

OPTIMAL CONTROL DESIGN (OCD) FOR PREDICTIVE
SAMPLING PROBLEM . . . . .

4.1 Effects of Changes in Performance Index

Parameters . . . .
4.2 Selection of Control Performance
Parameters . . . . . . . .
4.3 Summary .

OPTIMAL SYSTEM DESIGN (OSD) FOR PREDICTIVE
SAMPLING PROBLEM . . . . . .

Selection of Algorithm

Hardware Options . .

Optimization of Software . . .
Development of Cost of Implementation .
Optimal Selection of Hardware . .

CONCLUSIONS .

UIU'IUIUIUI
U'lvPUDNH

iv

. 28

. 31
. 34

.37

38
62

°. 70

. 73

. 73
. 77
. 79
. 82
. 87

. 100

TABLE
4-1.

4-3.

5-3.

5-4.
5-5.

LIST OF TABLES

Cumulative control performance with a,F11
variations for unit step input when
A1.tf"t i and T Mi -0. 02 o o o 0

Cumulative control performance and terminal
error for unit step input with A variations
for medium system . . . . . . . . . . . .

Cumulative control performance and terminal
error for unit step input with A variations
for fast system . . . . . . . .

Cumulative control performance for variation
in a, A, and F on fast stochastic system
with samplef quition‘wj (t) and }N
Selected minicomputers and {‘Eij 1‘1
specifications . . .

Total number of equivalent additigns to find
T1 using DSC algorithm for each Ti

Cgmputation time for each computer for each
Ti. 0 o o o o o o o o o o o o 0

Set of feasible computers for each Ti

Cumulative control performance and terminal
error with Tmin variations for fast system .

. S9

. 6O

. 72

. 78

. 89

. 89
. 90

. 99

FIGURE

4-10.

4-11.

5-1.

LIST OF FIGURES

System.block diagram for the optimal pre-
dictive sampling problem . .

Flow chart of computational procedure for
solving the predictive sampling problem

J(To) vs T0 with a variations for medium
system.when Fll=O.l and Ai=tf-ti .

J(T0) vs T0 with F11 variations for medium
system when Ai-tf-ti .

Output trajectories for a and F11 vari-
ations for medium system.when Ai-tf-ti . .

Output trajectories for a and F11 vari-
ations for fast system when Ai=tf-ti .

J(TO) vs T0 with A variations for medium
system when 01-200 and 1711-0 .

Output trajectories for A variations for
medium system when a=200 and F11=0 .

J(TO) vs T0 with A variations for medium
system when a-ZOO and Ell-0.1 .

Output trajectories for A variations for
'medium.system.when a-ZOO and F11-0.1 .

Output trajectories for A variations for
medium system when a-l and F11=0.1 . . .

Output trajectories for Q22 variations for

medium system when aal, A=O.88 and Ell-0.1 .

Cumulative control performance Jc vs tf

‘with F-O for medium system .
J(T0) vs integral step size emax for

several time intervals To .

vi

. 43

. 44

. 47

. 48

52

. 53

54

57

. 58

61

. 65

88

LIST OF FIGURES (Continued . . .)

FIGURE
5-2.

5-3.

5-4.

Cost of implementation C(Ti) vs sampling
time interval Ti . . . . . . . . . . . .

Cumulative control performance J

(T . ) vs
den for unit step input operatifig Ehfidition .

Output trajectories for adaptive sampling
‘with Tmin-0.0105, periodic sampling, and

continuous control for fast system .

vii

. 98

CHAPTER 1. INTRODUCTION

1.1 Historical Development

Periodic sampling criteria have often been used be-
cause of the ease of design and analysis using transform
techniques. Adaptive sampling criteria [1-11] have been
developed to vary the sampling rate in proportion to the rate
of change of some output or error signal. The first attempt
at placing an analytic framework under the design of sampling
criteria was made by Hsia [7]. In this work a large class of
adaptive sampling rules was derived analytically from a con-
tinuous time integral performance index which measured the
squared error introduced by sampling the error or output sig-
nals of a feedback control system. The performance index was
augmented by a cost for sampling which was inversely propor-
tional to the sampling interval length and represented the
costs for measuring, transmitting and storing the sample.

The formulation of the optimal control problem has al-
ways included a control performance measure but has seldom
included cost of implementation. Thus, the optimal control
design is either impractical or must be modified to incorp-
orate practical constraints imposed by costs of implementing
the optimal control. An optimal control system design formu-

lation [12] would directly impose the cost of implementation

1

2
constraints by adding appropriate cost terms to the control
performance index as in the formulation of these adaptive
sampling criteria [7]. Practical control system.design
could thus be obtained directly. Although almost every
aspect of control system design could be included in this
formulation, the only aspects that have been investigated are
the control and sampling problem [12] and the optimal sampling
problem [13]. The number and the lengths of sampling inter-
vals and the levels of each control element over each inter-
val were therefore the variables optimized.

The control and sampling problem and the classical
optimal sampling problem were chosen for investigation using
this optimal control system design formulation [12] because
the previous work on sampling in control systems [1-16]
suggested such formulations. The classical formulation of the
optimal sampling problem has a performance index that measures
both the errors in sampled signals caused by the sampling cri-
terion and the costs for implementing this criterion. The
control law was specified and the sampling times were not con-
sidered control variables but rather design parameters that
could be used to make the sampled-data control better approx-
imate a continuous-time control. The optimal sampling problem
was developed more carefully and then solved by Van Wieren and
Schlueter [13]. In this work, the length of each sampling
interval and the number of sampling times were selected to-
gether rather than selecting sampling intervals sequentially

as in adaptive sampling. The system.performance index for this

3
classical optimal aperiodic sampling problem [13] was defined
over the entire control interval rather than just one sampl-
ing interval and the cost of implementation was not just
chosen to have a convenient form but was chosen to model the
actual costs of implementing an aperiodic sampling criterion.
Furthermore, the model of the system dynamics, input disturb-
ances, initial conditions, and control inputs were all assumed
known and were used to make the system performance index
dependent on this information.

The formulation of the optimal control and sampling
problem used a performance index that strictly measured con-
trol performance. The control law was not specified so that
both a sampling interval sequence and a control vector sequence,
which specified the level of each piecewise constant control
signal and the length of the sampling interval over which it
is held, were chosen optimally. This optimal sampled-data
control problem [12] was formulated to obtain an optimal con-
trol with a sampling criterion that could provide better con-
trol performance than an optimal control with any periodic or
arbitrary aperiodic criterion.

An efficient computational algorithm was developed for
this optimal sampled-data control problem for the special case
where the optimal control sequence can be determined as a
unique function of the particular sampling intervals sequence
chosen. For this special case, the performance index can be
determined as a function of this sampling interval sequence.
The optimal sampling interval sequence can be found by minimiz-

ing this derived performance index. The optimal sampled-data

4

control problem.oould thus be separated into the problem of
finding an optimal control law for any sampling interval
sequence and a problem.of determining the optimal sampling
criterion for this optimal control law. Thus, the optimal
sampled-data control problem can be considered an optimal
control and sampling problem when the Optimal control can be
determined as a function of the sampling interval sequence.

This algorithm was applied to compute the optimal
sampled-data control law for the regulator problem with con-
strained [14], state dependent [15], and adaptive sampling
[16] criteria. The excellent control performance obtained
with very few control changes indicates that the computer
‘memory and computer-communication system.required to store and
transmit the control can be significantly reduced if the
sampling intervals are determined optimally rather than speci-
fied a priori.

This control and sampling problem was not formulated
with a cost of implementation term in the performance index
because it was formulated as a traditional optimal control
problem. Although the sampling intervals sequence were con-
sidered control variables, the number of samples was not con-
sidered a control variable and was specified since the theory
indicated the solution when both the number and lengths of
sampling intervals is optimized is trivial (i.e. Nson and 379).

Recent result on observability and controllability
of sampled-data control system [17,18] have shown that the

observability and controllability of the continuous system

5

can only be preserved in general if the number and the
lengths of the sampling intervals are control variables.
Thus, this theory suggests selection of a sampling rule may
provide the same kind of performance improvement that the
selection of a control law can. This hypothesis was shown to
be true in the recent papers that established

(1) that the selection of an optimal sampling rule
can proceed as an independent optimization problem from the
determination of an optimal control if the optimal control
can be uniquely specified for any sequence of sampling time
chosen [12]; and

(2) that the selection of an optimal aperiodic sampl-
ing criterion can cause a remarkable reduction in data require-
ments to achieve the same performance value as observed using
periodic sampling [12]. These results were established for
the sampled-data control problem.where the levels of each
piecewise constant control element over each sampling interval
and the number and lengths of the sampling intervals were the
control variables to be optimized in order to specify the

optimal control law.

1.2 Optimal Control System Design

Since the number and lengths of sampling intervals are
control variables and since the solution to the optimal con-
trol and sampling problem approximates the continuous-time
control when there is no cost of implementation and no upper
bound on the number of samples [12, Theorem 3], a cost of

implementation should be included along with a control performance

6
index in any general formulation of the control system design
problem. Since the actuators, sensors, communication links,
and computer hardware and software depend on the number and
lengths of these sampling intervals and determine the cost of
implementation as a function of these control variables,
this hardware and software that go into implementing a con-
trol law along with the number and lengths of sampling inter-
vals must be considered part of the control system design
rather than part of plant being controlled as in the tradi-
tional optimal control formulation.

These results suggest the control system.design formm:
lation which differs from traditional optimal control in the
following two ways:

(1) A performance index is used that attempts to pre-
cisely model both the control performance and the cost of
implementation objectives for a particular application;

(2) Sensors, actuators, communication links, and com-
puter hardware and software as well as the number and lengths
of sampling intervals will be considered part of the control
system to be designed.

Optimal control system design has been developed as a
method of problem formulation and a design methodology not
only because of the above theoretical considerations but also
because

(1) the traditional optimal control formulation pro-
duced control laws that either could not be implemented or had

to be significantly modified because the formulation ignored

the cost of implementation;

(2) many important aspects of the control system design
problem could not be adequately formulated using the tradi—
tional optimal control formulations. Examples include

(a) the selection from.different control struc-
tures that range from completely centralized to completely
decentralized.

(b) the selection of different control laws
which range from the linear quadratic Gaussian (LQG) solution
to the classical single input single output (SISO) feedback
one,

(c) the cost versus performance tradeoff for
using a particular input or output in the control law, and

(d) the selection from among different analog or
digital hardware alternatives for actuators, sensors, communi-
cation links, and computer hardware;

(3) the optimal control system design formulation can
lead to improved design procedures and improved system evalu-
ation methodologies. An excellent example of the possible
improvement in design methodology and evaluation procedures is
found in [12,13] where optimal periodic, optimal aperiodic,
and optimal adaptive sampling criteria were designed based on
minimization of a system cost which is composed of a cost of
implementation and the control performance measure. The opti-
‘mal sampling criterion could then be selected based on the
criterion (optimal periodic, optimal aperiodic or optimal

adaptive) with the lowest system cost;

8

(4) the optimal control system design formulation is
the proper framework for developing a procedure for computer
aided design of control systems which could include a control

structure, control law, and the hardware-software combination.

1.3 Optimal Aperiodic Sampling Problem for Control

The recent results on controllability and observ-
ability [17,18], that indicated the lengths and number of
sampling intervals are control variables, and the control and
sampling problem, which indicated a dramatic 50:1 reduction
in data requirements were possible for the optimal sampled-
data control with optimal aperiodic sampling over the optimal
periodic sampled-data control with the same control perform-
ance index value, suggested that a study of optimal aperiodic
sampling for control be performed where the control law is
specified and the number and lengths of sampling intervals are
optimized. The system performance index measures the control
performance and the actual costs of implementation for a
sampled-data control law with Optimal aperiodic sampling. In
this optimal aperiodic sampling problem, the number and lengths
of each sampling interval were optimized together based on a
performance index defined over a specified control interval.
This problem extends work on optimal sampling for the optimal
tracking [12] and regulator [l4-l6] problems, but in this case

(1) the control sequence which specifies the level of
each control element over each sampling interval will not be

optimized;

9

(2) the control sequence will be determined based on
the values Of a specified continuous-time control law at the
sampling times.

Since the control sequence is uniquely specified by
the sampling interval sequence, the theory of optimal sampled-
data control indicates the optimal sampling problem can be
treated as separate Optimization problem from the determina-
tion Of the continuous-time control law. The Optimal number
and lengths of sampling intervals is determined by using a
nonlinear programming algorithm.to determine the Optimal
sampling interval sequence for each number of sampling inter-
vals Of interest and then plotting these Optimized system per-
formance index values to determine the Optimal number of
samples graphically.

The results Of this study of Optimal aperiodic sampl-
ing indicate that

(l) selecting an Optimal aperiodic sampling criterion
for a nonoptimal continuous-time control can dramatically
improve control performance over that Of the unsampled con-
tinuous-time control;

(2) Optimal aperiodic sampling can increase the speed
of response over that of the unsampled continuous-time control;

(3) the selection Of the Optimal sampling criterion
from.among optimal periodic, Optimal aperiodic, and optimal
adaptive depends on the terms included in the control perform-
ance and cost Of implementation;

(4) the control performance improvement due to Optimal

lO
aperiodic sampling is due to effective use of the delay
cause by the sample and hold device to meet the Objectives
measured by the control performance index;

(5) the Optimal aperiodic sampling interval sequence
depends on the specific control performance, cost of imple-
‘mentation, system.dynamics, inputs, and initial conditions for
the system considered.

The aperiodic sampling problem [13] considered input
uncertainty and random initial conditions, but did not con-
sider the case where measurement noise was present. The Opti-
mal aperiodic sampled-data stochastic control problem extends
these results to that case. The Optimal stochastic control
law is a piecewise constant vector control that is held over
sampling intervals. The level Of the control over any inter-
val is specified by a gain matrix multiplied by the estimate Of
the state at the sampling time at the beginning Of the parti-
cular sampling interval considered. The gain matrix may be
the gain Of the optimal or non-Optimal continuous-time control
law at that particular sampling time or the gain matrix Of the
Optimal sampled-data control law [14] for the particular
sampling interval sequence.

The control sequence that specifies the Optimal
sampled-data stochastic control law with Optimal aperiodic
sampling is closed loop because the level Of the control over
any interval depends on the state estimate which depends on
the sampled measurements Of the output at previous sampling
times. The sampling interval sequence for this Optimal sampled-

data stochastic control with Optimal aperiodic sampling is Open

ll
loop because the optimal number and lengths of sampling
intervals are determined based on the average performance
over all sample functions Observed on the system and is not
based on actual measurements and the actual sample functions
Of the processes Observed on that system.aver a particular

interval.

1.4 Optimal Predictive Sampling Problem

An Optimal predictive sampling problem will be formu-
lated in this thesis in order to produce a control law that
has both a closed loop control sequence and a closed loop
sampling interval sequence. The control law is identical to
that used for the Optimal sampled-data stochastic control
problem with Optimal aperiodic sampling but restricted to the
case where the gain matrix is specified by a continuous-time
Optimal or specified non-optimal control law at the particular
sampling time.

The performance index will be defined over the control
interval but is separated into a measure of control perform-
ance over the sampling interval to be Optimized, the control
performance over the remainder Of the control interval after
this sampling interval and a cost of implementation that
‘measures hardware cost for implementing this predictive sampl-
ing criterion. This system performance index is optimized to
produce a sampling interval. A sampling interval sequence is
thus Obtained by iteratively solving this predictive sampling

problem,

12

The sampling criterion is predictive because the con-
trol performance terms are predicted based on measurements Of
the system.autput at all previous sampling times, knowledge
Of system dynamics, control inputs, and the statistics Of
the input disturbances, initial conditions and measurement
noises. The predictive sampling problem assumes that the con-
tinuous-time control law is specified and may be Optimal or
suboptimal. Thus, the selection of the control is specified
by the specified continuous-time control and the sampling
times and is not selected optimally for the sampling times

sequence as for the control and sampling problem.

1.5 Important Results and Contributions

The main contributions of this thesis will be

(1) to formulate and solve the Optimal predictive

sampling problem;

(2) to extend the Optimal control system.design

methodologY; and

(3) to apply this methodology to the Optimal predic-

tive sampling problem,

In Chapter 2, the Optimal predictive sampling problem
for control is formulated for a linear time invariant system
‘with a known input and disturbance statistics and a specified
continuous-time control law. This control law is based on a
state estimate which is in turn based on sampled noisy measure-
ments Of the outputs at previous sampling times. The system
performance index chosen measures control performance and

cost of implementation. The control performance index proposed

13
measures performance over the next sampling interval where
this control is held constant and over a future interval
where the control is permitted to be continuous-time. The
value Of this control performance can be estimated for any
sampling interval length based on the sampled output measure-
ments Obtained at previous sampling times and the knowledge
Of system.inputs, system dynamics, and disturbance, initial
condition, and measurement noise statistics. The cost Of
implementation measures the precise cost Of implementation
as a function of computer hardware, computational algorithms,
and computer software for computing the Optimal sampling in-
terval on-line.

In Chapter 3, optimal control system design methodol-
ogy is developed as a formal procedure for the predictive
sampling problem. Optimal control system design is shown to
consist of a two step Off-line procedure; Optimal control
design, which determines the control performance index Optim-
ally, and Optimal system design, which determines the cost Of
implementation and the Optimal selection of hardware to be
implemented by a tradeoff of control performance and cost of
implementation. Traditional Optimal control design problem
corresponds tO this Optimal control design problem but ignored
the Optimal selection Of computational algorithm, computer
software, and computer-communication-instrumentation hardware
which corresponds to this optimal system design problem.

In Chapter 4, Optimal control design for predictive

sampling problem is developed in detail for a particular

l4
example problem. It is shown that the sampled-data control
with a predictive sampling criterion outperforms the periodic
sampled-data control with the same number of control changes
and even outperforms the continuous-time control if the
system is deterministic. It is also shown that the best
predictive sampling criterion for the stochastic control sys-
tem.is periodic which indicates that the selection Of sampling
intervals cannot improve control performance when the future
control performance as a function of this sampling interval
cannot be accurately predicted. Thus, the optimal predictive
sampling does perform a control function for the deterministic
control system by holding a control with a larger absolute
magnitude than the continuous-time control; thus improving
speed of response and terminal error.

In Chapter 5, optimal system design for the predic-
tive sampling problem is developed in detail for the same
deterministic example problem chosen in Chapter 4. Cost of
implementation is developed only for the hardware cost term
in the cost Of implementation because the hardware is dedi-
cated for this problem and because communication and instru-
mentation hardware are assumed chosen. An appealing computer
hardware cost function is Obtained by Optimizing computer
algorithms, computer software and computer hardwares. Optimal
selection of hardware is also performed using two distinct
‘methods that tradeoff control performance against cost of
implementation. Control performance Obtained from Optimal

predictive sampling criterion with this optimally selected

15
hardware is dramatically improved are the control perform-
ances for periodic sampled-data control and for continuous
control.

Conclusions are then presented in Chapter 6.

CHAPTER 2. PROBLEM FORMULATION

Consider a computer control system where the plant
can be modeled as a linear, time-invariant, observable and
controllable stochastic system

30:) - A 350:) + §_ 20:) + 110:) (1)
where x.is an n-dimensional state vector, u’is an r-
dimensional control vector, and w is an n-dimensional dis-
turbance vector.

E {g(t)} - 9n

E {3(t) w’(r)} - E 6(t-t)

(2)

where 5(-) is the impulse function, and E and ’ indicate ex-
pectation and transpose Operations respectively. The initial
time toe(-w,w) is fixed and the initial state is random and
satisfies

E {5(t°)} B 2(t0)

E {(§_(to) - 2(to)) (5(to) - 2(to))’} =- 2(to) (3)

E (50:0) w’(t)} - 9m ta(t°,tf)

The system is Observed by measuring the outputs y(t)

2(t) ' Q §(t) (4)

at (N-l) sampling times {ti}§;1 where the sampling intervals

0 s Tm 5 Ti 5 Tmax <5)
N-l
g(To,T1,....,TN_1) - i-o Ti - (tf - to) = 0 (6)

17

where N satisfies

Nmin _<_ N 5 Nmax (7)

The output measurements are corrupted with noise
fli such that
‘51 - Z(ti) +‘gi i=1,2, ....... ,N-l (8)
where $1 is an m-dimensional noise vector that satisfies
E {ii} - gm_
E {ii ivy-11:61.1

E {21210} - te(to.tf) (9)

0
—mn
E {5‘11 §’(to)} - 9m
where aij is the Kronecker delta function.
The control 2(t) is assumed to be a piecewise constant

vector function whose elements change value only at the

sampling times {ti}§;1 such that
2(t) ' 2(t a £1) tetti’ti‘l'l) (10)

and can depend on the previous measurements

:Ei - (gi,zé, ...... ,gi) for i=1,2, ........ ,N-l

The Optimal predictive sampling for control problem
can be stated formally as follows:

Given the linear system (1,4,8) with disturbances (2),
measurement noise (9), and initial conditions (3), and given
a control law (10) Of the form

2.0:) .. {Bi " P-(ti) ' 9. -;£<ti’ti) teEti’ti-I-l)

A (11)
131:) - 9 51th:) ts[ti+1,ti+Ai)

18

where h(t) is the input to be tracked and £(t/ti) is the
estimate of x(t) after measurement 51 is made, i.e.

act/q) = E {yo/11}

+ " + " + .

Hit/ti) = E {(§_(t) - 35(t/ti)) (50:) - §(t/ti)) /_Z_i}
where the initial conditions are

5(to/to) =- yto)

into/to) - 111:0)
Determine the Optimal sampling interval T: that satisfies (5)

and minimizes a system.performance index:
S(Ti) - J(T1) + qC(Ti) (12)

where J(T1) is the control performance and C(Ti) is the cost
Of implementation, and where the control performance has the

form:

J(Ti) - 1/2 E{( a [(h_(ti+Ai) - 1(ti+Ai))’ g Q1_(ti+Ai) - z(ti+Ai))

+ t1+A1 . .
f {(116) - 202)) Q (110:) - 102)) + 3 (t) E; u(t)} dt]

t
1+1 (13)
‘1‘”
+ f t {Quit} - NOV 9. (bit) - 209) + 1130:) 32m} dt)
1
'/_Z_i}

Mauices E and Q are positive semidefinite symmetric, R is
positive definite symmetric matrix, a is positive constant, and
the expectation operator E is conditioned on measurements 21.
The first term.measures the terminal squared error and the

second and third terms measure the tracking and the control

19

square error over the time interval [ti'ti+1) and [ti+l'ti+Ai)
respectively. The parameter a weights future performance over
[ti+l'ti+Ai) against that of the immediate sampling interval
[t1,t1+1). The parameter Ai represents the length Of the
interval over which the control performance is predicted and
this parameter (A1) can be chosen to be constant (Ai-A) or can
be chosen based on a fixed terminal time and thus A1 = tf - ti‘

This control performance index can be expressed as a
function Of Ti as follows:

J(Ti) - 1/2 O[(h_(ti+A1) - g grim/thy g (l_1_(ti+Ai) - 9321:ng

up)

+ Tr {g’ _P_‘_ Q 1(ti-i-Ai/tpn

ti'mi " + " +
+ 1/2 aft [aim - g £(t/ti))’ g @(t) - g §(t/ti))
1&1

+ Tr {_c_’ 9, 9 much}
, : + (14)
+ (110:) - 31:51-1th 3, (110:) - g §(t/ti))] dt

1: A A
+ 1/2 réfltqgt) - _C_ gt/t’p)’ 9, (131:) - gage/:1»

+ '1‘: {g’ Q g y_(t/t1')}] dt
+ 1/2 ritgep - g 21:11:»; 5 cacti) - gag/1:13)]
by stbstituting (4), (11), and taking expectation term by term.

The conditional mean §(t/tI) and variance 2(t/tI)

satisfy:

20
£0:th a i<tvti) £(tI/ti) + .1: 30:3) 3; d1 [_Mti) - 2331:17th

i
t8[tetra>
git/:9 - yum) ism/c1) + I: gm) 2 [11(1) - 2 amp] d:
1&1
“um; '51“? (15)
2(t/ti) - anti) Kai/t1”) {(1:11) + 1: 10m) 11 9:11;») at
i
ta[ti,ti+Ai)

Ablock diagram of the system is shown in Figure 2-1. It
should be noted that the value Of performance J(Ti) is pre-
dicted based on measurements _Z_1 that includes measurement _z_i
at t1 and the assumption that grit-{MIL Z<tilti)’ system model
(1,4,8), control (11), and statistics (2,3,9) are all given.

Once T: is determined by minimizing S(Ti) with respect
to T1 satisfying (5), the computer must wait until ti +1 - t1 +
T: occurs. At t1+1 the computer triggers sampling Of measure-

ment _z_i+1 and then computes
3219141, ti+1) " iii-“fill ti) "' git-(3+1) [5(t1+1) ’ 9- g<ti+1l ti.”
Y.(ti+1/ti+1) " X<t1+1/ ti) " 5am) 9- Y.“1+1/ti) (15)
stag - Y-(t1+1/ .13 9.19. zeal/t? 9: + 11*.
The ocuputer is then ready to begin the cycle again by computing
T:+1 by using a search algorithm that requires repeated evalu-
ation of control (11), control performance J(Ti) (l4), and the

cost of implementation for several values of Ti+1 where in

this case T1+1 - Ti’ ti+1 - ti’ ti+2 = ti+l in these equations.

21

.EOHAOHQ MQHHQEOO o>wuoavoum Huaaumo Onu Hom_amumuav xOOHn Emummm Hum mmsuHm

 

Hit—"IN: MGH m5 AH+HU7 \ HITHU VN
was Aa+au\a+wuvm muagfioo

 

 

+ < I
£059 .HB+HO HHHO mu: BOO AHuxwuvx
1' I. Cu MHQ¢HUQHWUH A .va um“ .I. I l.. IRIHHHIJ
.3: £2qu Basso Mu :u

 

 

 

TOME

 

 

 

 

22

 

Initial Settings:
X(O),i 2(0) 9' H i! to: tf

 

 

i - O

 

 

 

1

Given Tmin and Tmax’
Optimize sampling interval IICOmpute Perform-
e.—

 

 

Ti using DSC algorithm. ance Index S(Ti)

 

 

'with search step size*AT using Simpsons
to find t - t + T Rule with step
1+1 i i size 6

 

 

 

 

NO ‘ Solve state
stop equation 2
- and X(t/ti)
V(t/ti) using Runge-

Ye: Kutta formula

 

 

 

 

Compute state estimate
x<ti+llti+l) and variance Random Gaussian
V<ti+llti+l) using Kalman generator to get

2.
filter equation r+1

cpl.

 

 

 

 

 

 

 

FIGURE 2-2 Flow chart of computational procedure for solving
the predictive sampling problem,

' 23
The control (11) can be a state variable feedback one since
g(t:/t:) must be available at ti so that J(Ti) can be pre-
dicted. The control law (11) must be closed lOOp in order to
reduce the effects of disturbances, system parameter variation,
and measurement noise. The predictor equations (15,16) are
derived in [19-21]. The flow chart Of the computational pro-
cedure for solving this predictive sampling problem.over the
control interval [to, tf] is shown in Figure 2.2.

This control performance index has the same form as
that used in the aperiodic sampling problem except that the
performance is only defined over [ti’ti+Ai) rather than [to,tf)
and can only be Optimized over Ti rather than (N,TO,T1,....,
TN_1). The problem is formulated so that the control per-
formance over [ti+l’ti+Ai) can be neglected if a is zero. In
this case, the optimal sampling interval T: would depend
strictly on the control performance over [ti’ti+l)’ the cost
Of hardware capable Of computing t1+1 - t1 + T: in an interval
less than Ti, and the cost Of communication and instrumenta-
tion hardware that could handle a sampling rate f1 - l/TI.

If a is not zero then the Optimal sampling interval T: must be
chosen based on: i

(a) the control performance over [ti'ti+l) that
generally increases with T1;

(b) the control performance over [ti+1,ti+Ai) which
is generally a convex function Of Ti; and

(c) the cost Of implementation that is a constant if

the hardware is already selected and the adaptive sampling

24
criterion is being implemented on-line. The cost of imple-
mentation is a decreasing function Of Ti if its selection
will be based on this performance versus cost tradeoff.
This would be the case for an Optimal system design of the
control and the hardware to implement it.

It should be noted that in the predictive sampling
problem the control performance is averaged over all sample
functions of 3(t) and {gi}§;i and the performance is condi-
tioned on knowing past measurements E1 of the output. These
differences between the aperiodic sampling problem [13] and
the predictive sampling problem being formulated here are in-
tentional because the Optimal sampling interval T: is to be
computed on-line after measurement 5i is taken rather than
Off-line without any measurements at all as in the aperiodic
sampling problem.

Observability and controllability of the sampled-data
system [17,18] need not be assumed to assure the existence of
T: for each 1. However, if the Optimal adaptive sampling
criterion is to provide acceptable control performance for
this Optimal adaptively sampled control system, sampled-data
controllability and Observability can be assured if and only
if the continuous-time system.is controllable and observable
when the number and lengths Of sampling intervals are control
variables which can be chosen [17,18]. Sampled-data controll-
ability and Observability can be obtained with only q(order
Of minimal polynomial of the system) sampling times or more

if these sampling times are taken so that no information on

25
or control over the controllable and observable continuous-
time system is lost by choice of these sampling times. If
tf is sufficient long or the adaptive sampling problem has
no specified terminal time, then at least q sampling inter-
vals will eventually be Obtained and thus the system will
become sampled-data controllable and observable if these
optimal sampling intervals do not cause loss of information
about or control over the system. The optimal sequence of
adaptive sampling intervals should not cause either loss of
information or control because the control performance index
would be degraded if such loss of control or information were
to occur and these sampling intervals are chosen to minimize
control performance. Thus, the optimal adaptively sampled
control system should be sampled-data observable and con-
trollable for all time after the initial q sampling intervals
are taken.

The cost of implementation can now be discussed since
the application of this predictive sampling problem has been
discussed. The cost of implementation in the optimal systems
control and sampling problem should include the hardware cost
which measures the cost Of additional instrumentation, com-
munication and computer hardwares required to implement a
criterion, the computation cost which measure the cost for
designing or tunning a sampling criterion in order to
achieve its best possible performance, and the communication
cost which measure the cost for communication on a time shared
communication link for the data from the computer to actuator

and from sensor to the computer. Thus, the cost of

26
implementation has the form:

2

The hardware cost (C1) is constant if the sampling
criterion has been implemented on an existing Or specified
set Of hardwares as in the on-line Optimal systems control
problem, However, if the instrumentation, computer and com-
‘munication hardware which is to be implemented as a part of
the control system will depend on the sampling criterion
selected, this hardware cost will depend on Ti' Although
the functional form of this cost term may vary for different
applications, the hardware cost would generally be a monotonic-
ally decreasing function Of the sampling interval Ti and
monotonically increasing function of the on-line memory and
computational requirements for a particular criterion.

Since the Optimal predictive sampling problem is on-
line control problem the hardwares for computer, communication
and instrumentation should be purchased or dedicated due to
the variable and yet unknown sampling rate. Therefore, the
cost of implementation for the Optimal predictive sampling
measures only the hardware cost (Cl). Thus, the computation
cost (C2) for the time shared use of computer facilities to
compute If for each N and the communication cost (C3) for the
time shared use of the communication link to transmit
{3&1} ,t:+l}l::;l
aperiodic sampling problem need not be included in the cost of

used in the cost of implementation for the

implementation for the predictive sampling problem, The cost

terms C2 and C3 were developed for the optimal aperiodic

27
sampling problem in a recent paper [13].

The hardware cost (C1) in this cost of implementation
will have one of the following two forms:
C1(Ti) = f0(Ti) + f1(Ti) + f2(Ti) (18)
or

Clai) " f0(Tmin) + fl(Tmin) + f2(Tmin) (19)

where fo(-) is the computer hardware cost, f1(-) is the com-
'munication hardware cost, and f2(-) is the instrumentation
hardware cost, respectively. The first form would be used
when the hardware to be used has not been selected and would
be implemented based on selection of T:. The second form
would be used when the hardware has already been selected and
has the capability Of sampling at a maximum rate of f - l/Tmin
samples per second.

The functions fk(') used in both expressions would be
identical and would actually measure the minimum hardware
cost required to implement a sampling criterion with sampling
rate fi - l/Ti samples per second. The hardware cost function
will be developed in detail in Chapter 5.

CHAPTER 3. OPTIMAL CONTROL SYSTEM DESIGN(OCSD)
METHODOLOGY

The purpose of this chapter is to discuss and extend
the optimal control system.design (OCSD) methodology [12]
and relate the concepts to the optimal predictive sampling
problem, The design methodology assumes that the plant to be
controlled and the statistics of disturbances and measurement
noise have both been modeled and the control performance and
the cost of implementation Objectives have been clearly
stated. It should be noted that this chapter is concerned
with:

(l) the Optimal control design (OCD) problem.which
selects the parameters (a,Ai) and matrices (9,3,2) that
specifies the control performance index. Given the parameters
and matrices to be determined in this step, an Optimal
sampling interval Tij can be computed that minimizes the con-
trol performance index (14) subject to the constraint (5)

for a particular set of Operating conditions

hd(t) t€[t0’ti]

23(t) tSEtorti] (20)
i

{ikj}k-1

where j specifies the particular set of inputs h(t) and sample
functions of the processes {ik}R=1 and‘wflt) with statistics
' 28

29
(2,9) that produce the sampled measurement sample function,
gij - (553.5%, ..... ,_z_j’_j). The determination of TL. is called
the optimal control (OC) problem and the determination of
the parameters and matrices will be called the optimal con-
trol design (OCD) problem, This terminology is used to
differentiate these two aspects of modern control theory from
the extension proposed in the optimal system design (OSD)
problem.

(2) the optimal system.design (OSD) problem where the
control performance and cost of implementation objectives are
used in conjunction with a list of Options for computational
algorithms and computer-communication-instrumentation hardware.

i) to determine the cost of implementation C(Ti)
as a function Of the sampling interval by determining the
best computer algorithm-hardware option for each sampling
interval Ti;

ii) to optimally select computational algorithms,
software implementation of these algorithms, and computer-
communication-instrumentation hardware based on an Optimal
tradeoff of control performance and the cost of implementation
over a number of sampling intervals (1) and operating condi-
tions (j).

The Optimal control design (OCD) and Optimal system
design (OSD) problem is a two-step off-line procedure for
selecting the control performance and the cost of implementa-
tion, and for performing an optimal tradeoff of control per-

formance and cost Of implementation that not only determines

30
the hardware to be implemented but also the optimal control
(selection of T:) which it implements. This two-step design
problem is called the optimal control system design (OCSD)
problem. Since the OCSD is performed Off-line and determines
the control performance index and the hardware to be used,
the Optimal control (OC) problem (which determines the Opti-
mal sampling interval for a particular set of measurements
from the system, the control performance index determined in
the OCSD problem, and the hardware selected based on the
OCSD problem) can thus minimize a system performance index
that includes control performance index and a constant cost
Of implementation specified by the OCSD problem.rather than a
control performance alone. This view of the OC problem
suggests it uses the same performance index used in the OCSD
problem but with a fixed cost Of implementation because the
hardware is specified.

The traditional 0CD problem has ignored the optimal
selection of computational algorithm, computer software, and
computer- comrmmication- instrumentation hardware. This
research is thus aimed at providing a foundation for incorp-
orating these aspects into the OCSD methodology. This OCSD
‘methodology can be applied to far more general control prob-
lems than the Optimal predictive sampling problem and will be
applied to such problems in the future.

The OCD and OSD subproblems will now be discussed in

detail in the next two subsections.

31

3.1 Optimal Control Design (OCD) Problem

The OCD determines the parameters (a,Ai) and matrices
(3,3,2) assuming cost of implementation for all controls are
zero and that no constraints must be placed on Tmin due to
hardware constraints. These matrices and parameters thus can
be determined based on the control objectives and specifica-
tions for the particular system to be controlled. These
matrices and parameters are determined using the same intera-
tive procedure used to design an optimal closed loop control
law [22]. Specifically, this design procedure requires that
the parameters (a,Ai) and matrices (Q,§,§) be modified until
the performance of the system with closed loop optimal
sampling sequence {Tij}N_1 satisfies all system design Objec-
tives and specifications as determined from the simulation of
Optimal system state trajectory §(t) and sampled-data control
(11) for that optimal sampling interval sequence {T:j}§=1.

The parameter a is a positive constant and is used to
weight the future performance over [ti+1’ti+Ai) with respect
to the performance over [ti'ti+l)‘ This weighting is desir-
able because the selection Of Ti has a dramatic effect on
performance at any instant in interval [ti+Ti,ti+Ai) because
it determines x(ti+Ti) and thus x(t) and‘u(t) over [ti+Ti'ti+
A1), butTi has no effect on the performance at any time
instant in [ti’ti+Ti) because the control £1 = 2(ti) is com-
pletely specified there. _Thus, a should be large enough so
that the future control performance over [ti+l’ ti+Ai)

dominates the selection of the Optimal sampling time interval

32
T: if the sampling criterion is to be adaptive.

It can be easily seen that the selection of the
sampling interval depends completely on the control perform-
ance over [ti'ti+l) in the optimal predictive sampling prob-
lem.if O-O. Since this performance index would generally be
a strict monotone increasing function of T1 for Ti 3 Tmin’
when Tmin is sufficiently small, the sampling criterion would
in most cases be periodic with sampling period T: = Tmin'
Thus, if future performance is neglected (a-O) by the design
Objectives the on-line predictive sampling problem gives a
periodic sampling criterion at a sampling rate that is the
maximum allowed by the computation algorithm, computer soft-
ware, and computer-communication-instrumentation hardware
Option selected in the off-line system design problem.

If a is greater than zero and large enough, the future
performance over [ti+Ti’ti+Ai)’ which measures terminal error,
speed Of response, and overshoot, will dominate selection of
Tij' Since this future performance index keeps changing as
index 1 increases, the sampling intervals Tij will produce an
adaptive sampling criterion. The sampling interval Tij is
selected so that the speed of response is increased and the
settling time and terminal error are reduced, but not so long
that the control performance over [ti+l' ti+Ai) becomes large.

If a was chosen equal to zero in this OCD problem, it
is clear that the designer had made a choice of implementing
hardware, computer algorithms, and software knowing that a

periodic rather than adaptive sampling criterion would result.

33
Thus, the selection of a is a selection of control structure
for the predictive sampling problem not only because it
dictates the hardware, algorithms and software implemented
but also because of it will be shown later that adaptive
sampling is a closed loop sampling process where periodic
sampling is an Open loop sampling process.

The parameter A1 is used to determine the future
performance time interval and is dependent on the desired
speed of response in the control Objectives and is determined
using the same iterative procedure used to determine the
matrices (g,§,§) and parameter a. The parameter Ai obviously
does not affect the control performance if a is zero.

Two options are possible in selection of A1. In the
first case, Ai - tf - ti where tf is a known fixed terminal
time and the sampling interval [ti’ti+l) is chosen to effect
control performance over [ti,tf]. In the second case, A1 is
constant (Ai-A) whether tf is known fixed terminal time or
unknown and the selection of sampling interval [ti'ti+1) is
based on performance over a fixed interval [ti'ti+A)’ The
length of this interval will thus also determine how adaptive
this predictive sampling criterion will be.

The effects of selecting parameters (a,Ai) and
matrices (Q,§,§) will be discussed for a specific example in
the next chapter. Although the selection of a and A1 on the
adaptability of the sampling criterion is clear, the effects
Of selection of Q, R and §.on the adaptability and perform-
ance is not clear and will be shown to be contrary to intuition

in some cases.

34
3.2 Optimal System Design (OSD) Problem

The control performance index, determined in the
OCD problem, will be used in the optimal system design (OSD)
problem which first develops a cost of implementation as a
function of Ti and then selects the computational algorithm,
computer software, and computer-communication-instrumentation
hardware through an optimal tradeoff between control perform-
ance and cost of implementation for several Operating condi-
tions. The OCD and OSD problems are apparently treated as
completely separate. However, these problems are not really
separable because a choice of 1min in OCD problem affects the
control performance that can be obtained tuning the para-
meters of the control performance index and also because Tmin
is the minimum computation time of the hardware, algorithm,
and computer software Option selected in the OSD problem.
Since the hardware and the associated Tmin value are chosen in
the OSD problem by a tradeoff between cost of implementation
and a control performance index, whose parameters are chosen
in the OCD problem and depends on Tmin’ the OCSD problem
requires the OCD and OSD problem be considered iteratively
until hardware and associated Tmin satisfied both control
performance and cost of implementation objectives.

The following procedure for determining the cost of
implementation C(Ti) assumes the communication and instrumenta-
tion hardware has been specified and that only the computer
hardware, computational algorithm, and computer software need

be selected:

35
(1) Enumerate the computational algorithm options (p).
(2) Enumerate the computer hardware options (8) for
each computational algorithm option (p). The cost of the

computer hardware, C for each hardware and computational

s, .
algorithm.option (s,p)pmust be noted.

(3) Optimize the computer programming to minimize CPU
time, IS,P(T1), for each hardware-computer algorithm.option
(s,p) to compute Ti - Ti'

(4) Determine the set of feasible computer hardware-

computational algorithm options for each Ti' i.e.
9(Ti) - {(s,p) : Ts,p(Ti) 5 Ti} (21)

where this condition requires the CPU time, rs p(Ti)’ for any
feasible computer-computational algorithm option be able to

T].

* * *
compute T1 in less than Ti seconds for each T1 = TiEETmin’ max

(5) Determine the cost of implementation function
_ ‘min
O‘Ti’ <s.p>en<r,> {03,p} (22’

where the lowest possible cost option is selected for each Ti'
The second step in this OSD problem is to select the
hardware option to be implemented for the predictive sampling
criterion and the parameter q selected to weight cost of
implementation against control performance. This selection
of hardware procedure is
(1) repeatedly optimizing system performance

5(11) - J(Ti) + qC(Ti) (23)

i
for several operating conditions (hj(t),‘wj(t), {ikj}k=1) for
j-l,2,....,M and several sampling intervals i=0,l,....,N-1

36
for each operating condition to obtain a set of Optimal

sampling intervals

M }N"].

*

The control performance index is determined in the OCD
problem and the cost of implementation index is determined
in the first part of the OSD problem.

(2) select the hardware based on the maximum
cost over the optimal set of solutions P

* *
C - C T .
mix ( 13)

(25)
Tijsr

where F includes optimal sampling intervals chosen for
different operating conditions (j) and different sampling
intervals (1). It will in general be necessary to repeat
this OSD procedure for several values of parameter q until
the hardware and control variables meet the control pera

formance and cost of implementation objectives.

CHAPTER 4. OPTIMAL CONTROL DESIGN (OCD) FOR
- PREDICTIVE SAMPLING PROBLEM

The objectives of this chapter are

(l) to investigate the effects of changing parameters
(a,A1) and matrices (Q,§,§) on the control performance
achieved with predictive sampling on a particular example
system;

L (2) to determine a set of parameters (a,Ai) and
matrices (Q,§,§) that provide the best control performance
possible with predictive sampling for each operating condition;

(3) to show that the sampled-data control with predic-
tive sampling can outperform the periodic sampled-data with
the same number of control changes and even outperform the
continuous time control law if the system is deterministic so
that the performance of selecting any particular sampling
interval can be accurately predicted;

(4) to show that the best predictive sampling criterion
is a periodic sampling criterion for a stochastic system be-
cause future performance due to selection of a sampling inter-
val can not be accurately predicted. The parameters (a,Ai)
and matrices (9,3,2) selected to provide the best control per—
formance with predictive sampling for a stochastic system will

be shown to result in a periodic sampling criterion.

37

38

The effects of changing parameters (a,Ai) and
elements of matrices (9,5,E) for an error feedback control
of a second order system.is given in Section 4.1. The
selection of parameters and matrices for both deterministic
and stochastic inputs and the resulting performance of the
sampled-data control with a predictive sampling criterion is
given in Section 4.2. These results are obviously dependent
on the example system.and error feedback control law used,
but the qualitative behavior should hold for a great many

systems with error feedback controls.

4.1 Effects of Changes in Performance Index Parameters

The example system used in this section is a second-
order type two system which has been used extensively in the
literature [1-13] on evaluating performance of adaptive and
aperiodic sampling criteria. This particular system is

chosen not only because of the extensive results obtained on

it with different sampling criteria but also because it is un-

stable without feedback and thus provides an excellent basis
for determining the performance of a sampling criterion. The

system to be considered is deterministic (w(t) - 9, ii 8 Q)

X t 0 1 t 2
.l( ) - xl( ) + cgn u(t)

yl(t) 1 0 xl(t)
y2(t) 0 1 X2(t)

with initial conditions

and is written by

(26

39

- (27)
x2(0) O
The control law is specified as a closed form as
h (t ) - x (t ) te[t ,t. )

The system is said to be a "fast" responding when “n = 10
and "medium" responding when on = 5 for C - 0.5 in both
cases. These are two of the specific cases considered in
[13] for evaluation of Optimal aperiodic sampling. A block

diagram.of the system.is shown below.

 

 

hct) uct) Izcwns+ «f; (t)
. PCP- T L 82 J

I.-- -—

 

x1(t) 1' Control
LComputer r

 

 

 

 

 

The control objectives for the optimal predictive
sampling problem are:

(l) to increase speed of response;

(2) to reduce terminal error;

(3) to reduce overshoot.
A general form for a control performance index that can meet

these objectives is:

40

( ) /2 h1(ti+Ai)-x1(ti+Ai) ’ F11 o h1(ti+Ai)-x1(ti+Ai)
J'T =*a
1 h2(ti+Ai)-x2(ti+Ai) o o h2(ti+Ai)-x2(ti+Ai)

+A
. i
t - t ’ 0 (t - t
”’2 {1:102:10 [Q31 [11f ”10 “+Ruz(t)}dt

(t)- (t) (t)- (t
‘1‘": h2 "2 h2 "2 (29)
ti'Fri .
"‘ 1’2] {Wu-XI“) [Q11 0] fits-XI“) + 31120;) m:
ti h2(t)’32(t)

0 Q2 h2(t)*X2(t

where h2(t) - dh1(t)/dt. The off diagonal terms in Q and E
are assumed zero for ease of analysis. The coefficient F22
for the error rate at the end of the control interval,
(h2(ti+Ai) - x2(ti+Ai)' is also set to zero because this
error derivative would not seem to effect the speed of res-
ponse, overshoot, or terminal error for a sampled-data
control with a predictive sampling criterion. Q11 is set
equal to 0.1 arbitrarily and R is set equal to 0.02 again
arbitrarily because Q11 and R weight the same signal

(h1<t> - x1<t)>.

The initial time and terminal time are set equal
to zero and one respectively and A1 = tf - ti is set as the
time remaining in the control interval [0,1] unless other-
wise specified.

The sampling interval constraint

Imin 5 Ti 5 Tmax

are chosen to place very little restriction on the choice of

sampling intervals for the OCD problem because in this chapter

41
the objective is to determine the maximum improvement in con-
trol performance which can be achieved through selection of
sampling intervals optimally. In Chapter 5, the minimum
sampling interval Tmin will be selected when the hardware
to be implemented is selected in the OSD problem based on a
tradeoff of control performance and cost of implementation.
The minimum sampling period is chosen as 0.02 in this section
because it is smaller than one would expect to select for a
system with fast (wn = 10) or medium (wn = 5) speed of res-
ponse. Tmax is chosen as 0.6 seconds for the medium system
(wn - S) and 0.3 seconds for the fast system (wn = 10) which
is the Nyquist sampling period for such systems when the sys-
tem.is assumed bandlimited to w .

n
The input h1(t) is selected as a step input

{1 :30

h1<t> - <30)
0 t < 0

rather than a stochastic disturbance because the general
effects of the parameter changes a, F11, and later Ai - A
can be more easily determined for the deterministic step in-
put than a ramp, parabolic, or stochastic input.

Since the system, control law, performance index,
and sampling constraints have been defined, the effects of
changes parameter a, P1]. and A1 can be determined. The
effect of increasing a is to increase T; as shown in Figure
4-l(a) if a is less than five and then any further increase in

*
a has no effect on To. This can be understood by analyzing

42
the shape of the two components of performance index J(Ti),

i. e.

‘51
(31)
+ 0.02 u2(ti)} dt

J1‘13.) " “’2 Fll[h1(ti+Ai) " "1(ti+“1)32

ti+Ai 2 2
+ a/z f {0.1 [h1(t) - x1(t)] + Q22 [h2(t) - x2(t)]
t1+T1 (32)

+ 0.02 u2(t)} dt
The comment J0(TO) is a unnotone increasing function of T0 when
T0 is sufficiently small because the integrand is non-
negative and is a decreasing function of the integration
argument when T0 is sufficiently small. 31(To) is a convex
function as can be seen from Figure 4-1(b) when a is very
large. Thus, when a is above five J(To) closely approximates
J1(T0) and the Optimal T; is unaffected by changes in a.
However, when a is less than five, a decrease in a makes
J0(Ti) relatively more important in determining T3 and since
JO(TO) is monotonically increasing T3 should decrease as a
decreases as Observed.

The effects of increasing F11 when a is greater than
five is to decrease T; as shown in Figure 4-2. This can be
explained by noting that the longer control

u<c0) = h1<t0> - x1<t0> (33)
is held, the larger the overshoot of trajectory x1(t) and

43

F 1 1: 0-10
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8. + flLPHR=1.00 0 BLPHflazDO .00
-‘ A nLrnnsxooma x aunnuoomo

 

you "MLJJJPE _ofoo lion
i
“' 1

FIGURE 4-1 J (To) vs T0 with a variations for medium system

44

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45

thus the larger the error (h1(tf) - xl(tf)). Thus, in-
creasing F11 will more heavily weight this terminal error and
thus control performance index increase faster as To
increases. Although this analysis of the effects of changing
a and F11 was only performed for i=0, it will be shown to
hold for every i by Observing Figures 4-3 and 4-4.

The output response xl(t) of the system is plotted for
the medium.(wn - 5) and fast (an - 10) systems in Figure 4-3
and 4-4 respectively for the predictive sampling criterion
obtained using various values of a and F11. Sampling
instants are shown by special symbols on the trajectory.
The results indicate that speed of response and overshoot
all increase as a is increased or F11 is decreased on both the
fast and medium systems. i

The speed of response and overshoot increase as T:
(a increases and F11 decreases) since the difference between
the absolute magnitude of the sampled-data control

u(t) - h1(t) - x1(t) te[ti,ti+l) (34)

and the absolute magnitude of the continuous control in-
creases with (t - t1) and has the effect of accelerating the
reduction in error (h1(ti) - xl(ti))' Thus, increasing a
and decreasing F11 increase T: and thus increase speed Of
response and the overshoot which occurs due to this faster
reduction Of error.

Another measure of performance for a sampled-data

control is cumulative control performance

46

Jc<T3.TI......T§_1> =- 1/2 '[z(tf) =- g<cf>r itch) - gap]

N-l (,3 (35)
+ 2 J T
1-0 0 i

whflflzmemnmes the control performance over each sampling
interval in [to,tf] and the error energy at the terminal time
tf. The performance over intervals [tj'tj+Tj)' j=i+l, i+2,....,
N-l, depend on the selection of {Tj}§_o and thus this measure
of performance can be used to compare periodic, Optimal
aperiodic, and optimal predictive sampling criteria.
The matrix Ebis not identical to E used in the predic-

tive sampling performance index. This matrix is chosen as

0.05 0.0
-[0.0 0.0]

in this study so that terminal error is not considered as a

nun

major factor in assessing performance of a sampling criterion.
Table 4-1 tabulates the cumulative control performance Jc(T0,
T1,....,TN_1) and the terminal error (h1(tf) - x1(tf)) for
(l) predictive sampling, (2) periodic sampling with the same
number of sampling times as predictive, and (3) periodic
sampling criterion with a sampling period of 0.01 (N=100)
which approximates the performance of the continuous control.
The cumulative control performance and terminal error
for predictive sampling on the fast system is always con-
siderably better than for periodic sampling but always worse
than the continuous control. The lowest cumulative control
performance and terminal error occurs when a-l and F11=0.l
and the cumulative performance obtained closely approximate

the cumulative performance of the continuous control.

47

HLPHH:200-00 F11: 0.10
HLPHR:200.00 F11: 5.00

fiLPHfl:1.00 F11: 0.10
CONTINUOUS

21.00
I D GD)(

 

 

 

1.00

“ {‘n

M!

{LRDE

RHFL
on

0325

13.25

jms -9.so

 

 

T

'o.oo o'. 10 o‘. :0 also a .40 also 0‘.»
TIHE 1N SECONDS

1' I

am 0.» Eco

FIGURE 4-3 Output trajectories for a and F11 variations for
medium system when Ai =- t f'ti'

t‘.oo

48

HLPHH:200400 F11: 0.10
flLPHR:200400 F11: 5.00

HLPHR:1.00 F11: 0.10
CONTINUOUS

11.10
‘. 4 D G X

3 0” if“

0
I

“maggot o

O
l

11.00 14.20 0,00 0

19.18

 

 

 

O
1

'0.00 0210 0120 0100 0.40

0‘.00 0100 0110 3.00 0.00 1.00
TIHE 1N SECONDS

FIGURE 4-4 Output trajectories for a and F11 variations for
fast system when A1 = tf-ti.

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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50

The results for the medium system indicate the pre-
dictive sampling has lower cumulative control performance
and terminal error than periodic sampling except when
.a-200 and F11-0.l because in this case the sampling inter-
vals are so large that the error is not sampled at its peak
overshoot and does not reduce this overshoot as quickly as
it would Otherwise. This seems to be an isolated situation
where predictive sampling does not fully take advantage of
the control Opportunity because it is based on a single
interval performance measure. The lowest value of cumulative
control performance for predictive sampling for this medium
system is Obtained when a-l and F11=0.l which for this case
is lower than that Obtained for the continuous control. The
lowest terminal error is obtained when cal and Fll-S, but
terminal error in this case is not a good measure of perfomm-
ance because the control interval is short with respect to
the settling time for this medium system.

The effects Of setting Ai - tf-ti or setting Ai - A
for several values of A will now be investigated for the
medium system (wn'S). The values of a and Fllare set equal
to 200 and 0.0 or 0.01 respectively because the sampling
intervals are large and the effects of Ai are more easily
seen. The first case considered is 0-200 and F11=O.O, and
the results, shown in Figure 4-5, indicate that curves
J(T0,A) z J1(TO,A) increase with A and this effect occurs be-
cause the integrand is non-negative and the integration

interval for J1(T0,A) is (A - To). The decrease in J(TO,A)

51
for small T0 for any A is thus due to a decrease in the in-
tegration interval. However, when To becomes large, the
overshoot becomes larger as T0 increases and the curves be-
gin to increase. This increase in J(T0,A) with T0 is more
pronounced for larger A which is due to the fact that as A
increases more Of the interval where overshoot is experienced
is included in [tO+T0,t0+A]. The optimal sampling interval
T3 thus increases as A decreases because the performance
index has less concern for overshoot due to holding the
sampling interval too long. The trajectory x1(t), shown in
Figure 4-6, indicates the sampling interval T: increase for
all i as A decreases indicating the above analysis for i=0
holds for every interval.

The second case considered in this subsection is
included to indicate the effects of changing A when terminal
error is weighted slightly (Fll-O.l and 08200). The curves,
J(T0,A) = J1(T0,A), plotted in Figure 4-7 are quite different
from.the first case where terminal error was omitted from the
performance index because the terminal error can be very
large or be very sensitive to changes in To for particular
values of A. The error for A=O.44 has much larger values than
for A80.22, 0.88 or 1.0 and thus the J1(TO,O.44) curve is much
larger than the others there. The optimal sampling interval
T3(0.44) is thus quite small in order to optimally tradeoff
the reduction in the integral part Of performance index
J1(T0,0.44) with To and the rapid increase in terminal error
(h1(0.44) - x1(0.44)) with T0' The performance curves

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.00 02:0 0100 0.00 0240 0100 0200 0370 0.00 0200 1100
TIRE 1N SECONDS

FIGURE 4-6 Output trajectories for A variations for medium
system when “-200 and Fll-O.

54

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55

J(T0,0.22) for F11-0.0 and F11-0.l are identical when To
is small because the terminal error (h1(0.22) - x1(0.22))
is zero for the continuous control and is thus small when
To is small. However, as T0 becomes large this terminal
error increases rapidly and J1(To,0.22) increases rapidly
when F11-0.l but increases only slightly when F11=0.0. The
Optimal sampling interval T;(0.22) does decrease when
terminal error is weighted in the performance index. The
change in the curves J(To,0.88) and J(TO,1.0) and the change
in Optimal sampling intervals T;(0.88) and T3(l.0) are both
quite small due to inclusion of terminal error in the per-
formance index. The speed of response is again proportional
to T3(A) as it was when F11=0.0 but in this case T;(A) is not
inversely proportional to A but is dependent on the magnitude
of the terminal error and its sensitivity to changes To.
Thus, T;(0.44) is smallest followed by rgco.22), T;(l.0) and
T;(O.88). Since the speed of response and {T:(A)}§;i are
proportional to T3(A), the analysis of the effects of para-
meter change in the first interval hold for all other inter-
vals as shown in Figure 4-8.

Another set of trajectories xl(t), which indicate the
effects of changing A, is run when a is reduced from.200 to 1
thus reducing T:(A) and the speed of response but improving
cumulative control performance Jc(T3(A), TI(A),....,T;_1(A))
and terminal error as shown in Figure 4-9 and Table 4-2. The
performance and the trajectories x1(t) show comparatively

little change as a function of A for these values of a and

56

F11. However, the smallest T;(A) is still A=0.44 and the
slowest speed of response occurs for A=0.44 indicating the
sampling interval sequence {T:(A)}§;3 still depends on the
magnitude of the terminal error (h1(ti+A) - xl(ti+A)) and
thus on A just as when a-200 and F11-0.l.

The analysis of A variations for the fast system is
identical to that for the medium.systemm In this case, the
lowest cumulative control performance occurs when a-l,
F11-0.l and A-0.ll as shown in Table 4-3.

The speed of response can be increased by adjusting
a, A and FlllQ11 as indicated above but fast speed of res-
ponse results in a large peak overshoot in the transient
response. The peak overshoot of the output response can be
reduced by (l) reducing a, (2) reducing F11/Q11, (3) increas-
ing A, and possibly (4) increasing Q22. The effect of chang-
ing Q22 is investigated because 022 weights the tracking
error rate and could possibly reduce the peak overshoot by
minimizing this error rate. F22 is not considered because
the error rate at the terminal time would not appear to have
any effect on these control Objectives.

Results from.Figure 4-10 indicate

(1) increasing Q22 does increase damping and reduce
speed of response;

(2) the effect Of changing Q22 is very similar to
changes in a, A, or F11/Q11 because changing each of these
parameters also will increase speed of response at the expenses

of greater overshoot;

57

flLFHfl:200400 F11: 0.10

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0,00

+10 14.00 1.00

 

8
'0.00 01:0 01:0 0100 0140 0100 0100 0'710 0100 0100 R00
IHE 1N SECONDS .

 

FIGURE 4-8 Output trajectories for A variations for medium
system when a=200 and F11=0.l.

58

HLPI‘Iflal .00 F11: 0.10

0 0ELTH:0.22 + 0ELTH:0.00
A 0ELTR:O.44 - CONTINUOUS

0,00

0,10

0,00

 

0,00 ""01..on 0,10

0,00

«1.00

 

1m

 

6' 0” +0"

 

fl

V I v v v I v 1
.00 0.10 0.00 0.00 0. 0.00 0.10 0.00 0.00 1.00

40 0100
ms IN SECONDS

FIGURE 4-9 Output trajectories for A variations for medium
system when 091 and F11-0.'l.

59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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61

'RLFHR:1.00 F11: 0.10 DELTH:0.88

o 022: 1.00 + 022:.- 0.00
A 022: 0.005 - counuuous

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0100 0100
IHE IN SECONDS

FIGURE 4-10 Out ut trajectories for Q variations for
medlium system when and, A220.88 and F11. 0.1.

62

(3) The major difference between the effects of adjust-
ing a,A, and Fll/Qll and the effects of changing 022 is that Q22
causes the sampling rate to be proportional to acceleration
rather than velocity as with a,A, and Fll/Qll'

Since a sampling criterion that samples at a rate prOpor-
tional to acceleration for an error feedback control system is
contrary to intuition and since tradeoff between speed of res-
ponse and overshoot can be achieved through adjustment of a,A,

and Fll/Qll’ Q22 is set equal to zero.

4.2. Selection of Control Performance Parameters

The previous section discussed the effects of changing
performance index parameters a,A, Fll/Qll' and Q22 on the con-
trol performance of a predictive sampling criterion with an
error feedback control law for a second order example system.
The next step is to select or tune these parameters to achieve
acceptable control performance from the predictive sampling
criterion and this error continuous-time feedback control law
for a particular operating condition. This task is performed
in the OCD problem as described previously in Chapter 3.

Optimal control design will be performed for both a
deterministic and stochastic operating condition. The perform-
ance Objectives are to

(1) make the adaptive sampled-data control with the
predictive sampling criterion outperform the periodic sampled-
data control and the continuous-time control based on the
cumulative performance measure JC(T0, T1,....,TN_1);

(2) make the sampling criterion perform control of the

system based on predicted performance.

63

The results obtained in Table 4-1 for the medium.snd
fast deterministic systems with a step input indicate cal
and F11=O.l provides the best control performance according
to the above Objectives since the adaptive sampled-data con-
trol outperforms the periodic sampled-data control with the
same number of control changes. MOreover, for the medium
system.where the minimum time interval constraint Tmin=0.02
is smaller than that of the fast system in the sense of
settling time, the adaptive sampled-data control outperforms
the continuous-time control. The adaptive sampled-data
control evaluated with Tmin=0.02 for the fast system.does not
outperform the continuous-time control because the minimum
time interval constraint is relatively large compared to the
settling time and thus by the particular choice of Tmin=0.02
penalizes the faster speed of response and larger overshoot
Obtained with the adaptive sampled-data control law. If
Tmin'o'OI' the adaptive sampled-data control will outperform
the continuous-time control as shown from results in Chapter 5,
since the fast system predictive sampling problem would then
be a perfect time-scaled version of the medium system.pre-
dictive sampling problem in the sense of settling time.

The cumulative control performance is not affected by
changes in A when 0-1 and F11=O.l and thus a choice of A based
on maximizing improvement over a periodic sampled-data and
continuous-time control is not attractive. A can be chosen
based on maximizing the effective control exercised by a pre-

dictive sampling criterion which triggers the sample and hold

64
mechanism on the continuous-time control. The predictive
sampling criterion performs more control (holds sampled
control error longer for large control error values) when
A-0.22 or A-0.88 increasing the speed of response for both
positive or negative control error for the medium system as
seen from.Figure 4-9 and Table 4-2 and the cumulative control
performance is lowest when A-0.88. Thus, A-0.88 is chosen
for the medium.system.

These set of performance index parameters will provide
good performance on the deterministic system for a range of
operating conditions. As evidence of this, the cumulative
performance of the continuous, periodic, and adaptive
sampled-data controls was plotted versus tf for the medium
system with a parabolic input rather than a step input. The
parabolic input was chosen because the type two example system
(26) will have a non zero constant error (hl(t) - x1(t)) as t
becomes large. The results shown in Figure 4-11 indicate the
cumulative control performance has a large initial increase
due to control for the initial transient and then increases
with constant rate as tf increases. The adaptive sampled-
data control outperforms periodic and continuous for each tf
when the control performance for each control over a control
interval [0,tf] are compared.

The selection of predictive control performance para-
meters for the deterministic fast system is 081, F11=0.1, and
A-0.ll from Table 4-3. This system and control performance
selected here will be used in following chapter to solve the
OSD problem.

65

A‘. .. i

.aoummm :BHHOOE How Olh £33 mu m> OH. OOOOEHOMHOO Houucoo 0>Hu0HHEuHo HH...» guHm

Ozauuu H: us:
:30 8..~ 8.. 8.. 8w: 8.?

I P

8% 8.. 8... 8.» 8..

P #

 

oaoazzzoo X 39533.. O
uzhmcoc 9

2.0 "a: corouchaua be. Huczmac

530111303

‘0
.01.

0v‘0

 

31’0—

66
The Optimal control design, just performed for the
deterministic case, is now repeated for a stochastic system.
In order to obtain a meaningful comparison between determin-
istic and stochastic cases, the input h(t) is replaced by a

white noise process w(t) as shown below.

_ 0 pi

w(t) u(t) Zrmns + w: y(t1)+ 21
I %l- ' 32 f

" + + .--A c 1 _---

xl(ti/ t1) )4 0331552: I,

The state model for this example is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3210:)" 'o 1‘ 'xluz)‘ cmsnT 'wlm'
- + u(t) +
3‘2“). .0 0‘ L-x2(t)-l -wn_ _w2(t)-
. _ , (35)
Fw1(t) 21:0!“1
- w(t)
w. -
Z(ti) " Y(ti) + ‘91 [1 0] §<t1) + $1
”21“? t1) t5 [t1 ' ti+l)
(37)
u(t) = A

+
-x1(t/ti) ts[ti+l,ti+Ai)

67

0

E {3500) gap} - 10:0)

(38)
4§2m3 2cm;
E {3(t) g’(t)} - or2
2§w£ m;

E {01 0;} = w

The system.chosen is the fast system (c-0.5 and wn-lO). The
initial state covariance V(t0) is set to a null matrix so
that the initial state is assumed to be perfectly known.

The performance index for this system is

0(01) - a/Z F11 [;i(ti+Ai/t’i') + mug-sing);

t M A A
+ an I 1 humid/1:1) + vile/up] + 0.02 sin/1:13} dt
1:

{Fri
(39)
+-1/2 f {0'1[x1(t/ti)'+'V11(t/5I>J +-0.02 xi(ti/ti)} dt
t
1

The cumulative control performance over the interval [0,1]
* * * N'l *

Jc(TO,T1,....,TN_1) - 1:0 JO(T1) (40)
was computed for several adaptive sampled controls with predic-
tive sampling criteria determined based on performance index
(39) with several combinations of parameters 0, F11, and A.

The cumulative control performance for periodic sampled-data
controls were computed for comparison with the performance Of
the associated adaptive sampled-data control. The periodic

sampled-data control was in each case computed with the same

number of sampling intervals and the same sample functions

68 _
Q:j(t),{gij}§_1) as used with the adaptive sampling criterion
and both were constrained to a [0,1] control interval.

The standard deviation 0 was chosen as 0.033 to make
the state x1(t) have a maximum.excursion (3 standard
deviations) of t 0.1. The covariance of the measurement noise
was chosen as 0.001 so that the maximum measurement excursion
is ten percent of the actual output value. Thus, the system
is a random process with transient behavior and is thus an
excellent test case for the performance of an adaptive sampling
criterion which optimally selects the next sampling interval
based on performance prediction which is in turn based on
measurements of x1(t) at the last sampling time ti’ This
sampling criterion is closed loop since the selection of the
sampling interval is chosen based on measurements of a system
with random disturbances and measurement noise.

The results obtained with adaptive and its associated
periodic sampled-data controls, where the predictive sampling
criteria are computed based on different combinations of per-
formance index parameters, are shown in Table 4-4(a). The
results obtained with sample function 31(t) and {wil}§=l
indicate the adaptive sampled-data control will in general be
inferior to the companion periodic sampled-data control with
the same number of control changes. The two parameter combina-
tions, (a=l, F11=5, A=0.ll) and (cal, F11=0.l, A=O.ll), where
adaptive outperformed periodic were rerun with other sample

functions for processes w(t) and {$1}§-1- In these cases, the

periodic outperformed adaptive as shown in Table 4-4(b).

69

TABLE 4-4 Cumulative Control Performance for Variation in
a, A, and F11 on Fast Stochastic System with

. ~ . N
. 5.31”??? Wt??? 33".?) . in? $1.15 .}.i.-.1. . ._

 

 

 

 

 

 

 

 

 

 

 

Number Cumulative Performance<Jc)

a F11 A of
p'Samples Predictive Periodic
0.5 0.1 0.05 43 0.000449 0.000409
0.05 5.0 0.05 26 0.000584 0.000550
1.0 0.1 0.11 13 0.001659 0.001732
1.0 0.1 0.22 22 0.000815 0.000638
1.0 5.0 0.05 26 0.000607 0.000566
1.0 5.0 0.11 13 0.001679 0.001770
1.0 5.0 0.22 25 0.000613 0.000547
5.0 0.1 0.11 13 0.001656 0.001463
5.0 5.0 0.11 16 0.002083 0.001684
0.01 5.0 0.11 50 0.000198 0.000198

(a)
. Sample Number Cumulative Performance(Jc)

Parameters Function of
(j) Samples Predictive Periodic
1 13 1 .0.001659 0.001732

a 8 1.0
2 16 . .0.001013 0.000784
A - 0.11
3 15 0.002299 0.001868
4 12 0.001168 0.001133
1 0 ..1... .13 ........ 0-001679. ,.0.001770..
a - .
, 2 .716. 00.001033 0-000799
A - 0.11

.3.., .15 . .0.002299 0.001868
4 12 0.001183 0.001156

 

 

 

 

70
The results also show that when a becomes small the
adaptive criterion approaches the performance of the periodic.
The following analysis indicates that when a-O, the adaptive
criterion is periodic with period Tmin' The predictive
sampling problem when a=0 and Tmax is sufficiently small is
min J(Ti) - 'min 1/2 I i [gfit/tI)’
TiEEImin’Tmax] TiEETmin’Tmax]
Q ugh/ti)

+ mg Yin/1:13} + 31:1) 3 u(tifl dt

Since the integrand at any t is non-negative and not a function
of Ti’ the function J(Ti) = J0(Ti) is a monotone increasing
function of T1 with a minimum at T: a Tmin‘ Since Imin has
hopefully been chosen sufficiently small to cause no restric-
tion in control performance, the optimal periodic sampling
criterion could be determined by increasing Tmin until the

cumulative control performance

Jc(Tmin) . Jcitmin’rmin'°"°’1min) (42)
N' *
a z J<T) *-
1-0 0 i ITi Tmin

*
begins to increase significantly to obtain Tmin‘ This optimal
periodic criterion is based strictly on control performance.

An OCSD could also be performed based on minimizing
S(Tmin) - Jc(rmin) + q C(Tmin) (43)
where C(Tmin) is the cost of implementing the periodic sampled-

data control system with sampling period Tmin' The optimal

*
sampling interval Tmin would be based on a cost versus

71
performance tradeoff which would not only specify the
sampled-data control law but also the hardware required to
implement it.

The results of this subsection indicate that the
optimal predictive sampling criterion for the stochastic con-
trol system.based on a design objective that attempts to make
a sampled-data control with a predictive sampling criterion
perform as well as or better than periodic is periodic. The
results indicate a non-periodic adaptive sampling criterion
may outperform the periodic with the same number of sampling
intervals for some sample functions, but that on the average
a periodic sampling criterion is best for a completely sto-

chastic system.

4.3' Summary

The results of this chapter indicate an optimal predic-
tive sampling criterion outperforms periodic for deterministic
systems where future performance can be accurately predicted.
In this case, predictive sampling can dramatically outperform
the periodic sampled-data control and can ever outperfonm the
continuous-time control being sampled. Thus, predictive
sampling performs a control function by holding a control with
a larger absolute magnitude than the continuous-time control
thus improving speed of response and terminal error but in-
creasing overshoot.

The results on stochastic systems indicates the best

predictive sampling criterion is a periodic one that in a

72
sense makes no prediction at all. This result can be ex-
plained because the predicted performance is based on an
average performance and the predicted state and its co-
variance may not accurately describe the particular sample
function of the state. Thus, the predicted optimal sampling
interval may not perform anywhere near the intended control
on the particular sample function xfit) being observed at
that instant. Thus, the optimal predictive sampling criterion
will outperform the periodic sampling criterion for some
sample functions xfit) but will be outperformed by a periodic

sampling criterion on the average.

CHAPTER 5. OPTIMAL SYSTEM DESIGN (OSD) FOR
PREDICTIVE SAMPLING PROBLEM '

The purpose of this chapter is to determine the cost
of implementation and to select the optimal hardware for the
adaptive sampling criterion. Therefore, this chapter
illustrates the OSD problem.for the predictive sampling prob-
lem, The discussions for the cost of implementation will be
made by developing the computer hardware cost. The communi-
cation and instrumentation hardwares are not included in this
thesis and are a subject for future research. The steps used
to determine the cost of implementation and the selection of
computer hardware will be performed in Sections 5.1 to 5.4
and Section 5.5 respectively of this chapter for system (26)

with continuous control (27) and a closed loop bandwidth wn.

5.1 Selection of Algorithm

The first step in this procedure for developing a
cost of implementation C(Ti) is to select the algorithms which
could solve the following optimization problem

scr‘i‘) - min {JCTQ +qC (Tminfl. (44)

Ti€[Tmin’Tmax]

The cost of implementation for this on-line 0C problem is
specified since the hardware required has been assumed selected
in this OSD problem.based on a performance measure

73

74
sai) = J(Ti) + q C(Ti) (45)

which is optimized over several intervals i-Q,l,....,N-l
and several operating condition j=l,2,....,M to obtain r =
{{Tij}§;é}?;l. The minimum sampling interval

*
Tum = ‘31? {T11} <46)

the associated cost of implementation C(Tmin) and the hardware
8(Imin) - {s : C - C8(Tmin)} for the 0C problem are then deter-
mined based on determining Tmin from.(46) because
:2” «Tip - ca...> <47)
Tijsr
and because C(Ti) will be a monotone decreasing function. It
is obvious that the cost of feasible hardware options, com-
puter algorithms, and the efficient programming of these
algorithms will all affect the shape and magnitude of C(Ti).
Since Tmin is unknown because the hardware has not been
selected, a value of Tmin must be guessed at this point in
order to evaluate the performance of algorithms and hardware

options. Tmin is temporarily chosen to be a

A

Imin = 0.005 (21v)/wn (48)

which provides a rate two hundred times the system bandwidth
which is much faster than one would ever need to sample, and
is smaller than the minimum time needed to compute the optimi-
zation problem (44) by the fastest computer-algorithm option.
The value for amin. is chosen temporarily because Tmin

can only be determined after C(Ti) is determined. The use

of 1min < Tmin rather than actual Tmin to determine C(Ti)

75
introduces some error in the determination of the CPU time
TSP(T:j) require to compute Tij - T1 for a particular come
puter hardware and algorithm Option and thus the cost C(Ti).
This error is not significant and is in the direction which
would choose a slightly more capable computer than might
actually be necessary which leaves some room.for later modi-
fication or expansion of capability. The maximum sampling
period is determined by the stability consideration as in
the GOD problem and is

Tm - w/wn ‘ (49)

which is a rate twice the bandwidth and thus much slower
than one would generally wish to sample.

The optimization problem is a univariate search
over a relatively small closed bounded interval. Since the
Optimization to determine T: must be performed on-line in
less than T: seconds and since each function evaluation
requires relatively extensive computation due to integration
of differential equations (26) and the performance index,
the algorithms used should require very few function evalu-
ations. Four possible optimization algorithms are feasible
for this problem [23]; Fibonacci (p-l), Golden Section (p=2),
Powell (p-B), and Davies, Swann and Campey (DSC) (p=4). The
Powell algorithm was never evaluated because it was better
suited to multivariate search and because it was not as well
suited to a search over a small bounded interval. The
Fibonacci and Golden Section algorithms are suited to opti-

mization over a small bounded interval but require more

76
function evaluations than a DSC algorithm if S(Ti) is convex
and has a unique minimum. Thus, for the case where S(Ti) is
convex as shown in Figures 4-l,2,5, the DSC algorithm.(p-4)
will be used. This decision is made for all hardware
options (8) since the best algorithm is independent of the
computer used.

Uniform search steps are used in the DSC algorithm
rather than acceleration steps in order to reduce the number
of function evaluations needed for a small bounded search
interval. The uniform steps in the search are continued until
the decrease in the performance index terminates and an
increase is noted on the last search step. A minimum T: is
thus known to have occurred in the last two intervals. A
single quadratic interpolation is performed to obtain T:
because the number of function evaluations is to be minimized
and because sufficient accuracy is obtained if the uniform
search step size is small enough. Minimizing function evalu-
ations reduces 18P(T:) and will reduce both C(Ti) for each Ti
and Tmin' A

The search is initiated at Tmin rather than Tmax in
order to cause the CPU time rsp(T:) required to compute T:
to be an increasing function of T: - Ti rather than a decreas-

ing function of Ti. Since the constraint
* * *
9(Ti) - {(s,p): TSP(T1) 5 Ti} (50)

requires that the computation be completed on any computer

* *
before trigger at that sampling instant t1+1 - ti + Ti is

77
necessary, the resulting cost of implementation

*
C(T ) - ‘mi (0 }
i (8.p)63(T:) SP (51)

will be defined for smaller values of T: resulting in a
lower value of Imin' Thus, the choice of algorithm.and the
direction of search for this algorithm will ultimately
effect the value of Tmin and the magnitude and shape of this

cost of implementation.

5.2 Hardware Options

The second step in this procedure is to determine a
set of computers that can handle this problem. .Attention was
restricted to minicomputers with

(l) at least 4K.words of memory size which is enough
memory for this particular problem;

(2) FORTRAN capability in order to make programming
easy;

(3) 16 bit word size in order to obtain the accuracy
required to compute T:.

It was assumed that multiplication and division
operations would be implemented using software since multipli-
cation and division hardware options were not always available
on every computer. The computation times for addition and
subtraction were assumed the same and the computation times
for multiplication and division were assumed to be eight
times per word as large as for addition and subtraction per

word on all computers considered [24],

78

TABLE S-l .Selected Minicomputersand Specifications

 

Memory Addition

 

8 Manufacturer Model Size Time per Cog;
(words) word(usec) (C)
(71) .
1 Digital Equipment PDP-ll/45 32K 0.3 38,000
2 Microdata Express I 32K 0.405 20,000
3 Data General 5/100 BR 0.6 9,200
4 Digital Computer D-6l6 4K 0.66 7,260
5 Data General NOVA 3/12 4K 0.7 3,600
6 Digital Computer MOD-5 4K 0.8 3,075
Controls
7 Interdata 6/16 4K 1.0 2,900
8 Interdata 5/16 4K 1.2 2,100
9 Digital Equipment PDP-ll/03 4K 3.5 1,995

 

79

Y1(S) = Y2(S)
(52)
Y3(8) = 74(8) = 8 71(8)

where Yl’ 72, Y3. and Y4 are addition time, subtraction time,
multiplication time, and division time per word respectively
for the computer 3.

A set of computers which met those specifications was
determined from.the 1976 DATAPRO REPORTS [25] and is shown in
Table 5-1 with the actual cost and computation time for addi-
tion for each selected computer shown. Mere specific data
should be required for a practical control problem such as
the proper hardware or software Options for each Of these
mathematical Operations. It is conceivable that the proper
hardware or software Option for any Operation on a particular
computer may be selected as part Of the design Of the Optimal
sampling interval in order to achieve a minimum cost Of

implementation C(Ti) for each Ti'

5.3 Optimization Of Software

The third step Of this procedure is to Optimize the
computer programming to minimize CPU time 18P(T:) for each
hardware-computer algorithm option (s,p) to compute T: for
the 0C problem, Since the computational algorithm was chosen
to be DSC algorithm.(p-4), the only consideration to Optimize
the computer programming is to minimize Ts(T:)' However, the
subscript p is retained because in general the algorithm may

not be selected at this point.

*
The computation time Tsp(Ti) for each computer s=l,2,....

9 and algorithm.p-l,2,3,4 is approximately

80

4

Sp(Ti) -k21 kacri> vk<s> fietr min rmax] (53)

where K pl’ sz, Kp3' and KP4 are the total number of addi-

tions, subtractions, multiplications, and divisions respect-
*

ively to compute T1 for each T1 for algorithm.p. The equa-

tion (53) can be rewritten as

Tap“? ' [Kplap + Km“? + 8 mp3”; + KP4(T:))]
71(3) (54)
= Kp<ri> v1<s>
by substituting (52).
The total number Of any particular Operation depends on
the number Of function evaluations NP(T:) to compute T: a T1

for pth

algorithm.and the total number Of integration steps
* * * *

NO(T1) and N1(T1) required to compute J0(Ti) and J1(Ti) res-

pectively. Thus, the number Of Operations Of any particular

kind for the pth algorithm can be expressed as

* * * *
ka(Ti) " MOk “0 “1’ + Mlk N1 “1’ + M2k Np(Ti) + M3k

(55)
k - l,2,3,4

*
where Ti - Ti' m is the integer index for function evaluation

and *
* N (Ti)
N£p<Ti) - mil szm. z - 0,1 (56)

Constant MOk and Mlk are the integer number Of Operations of
type k for each integration step in J0(Ti) and J1(Ti) respect-
ively. ‘M2k is the integer number Of operations Of type k

81
which must be performed for each function evaluation but is
independent of the number of integration steps per function
evaluation. Finally, M3k is the number of Operations Of
type k required to compute T: but not in the NP(T:) function
evaluations required to obtain T: - Ti' NOpm and Nlpm are
the integer number Of integration steps for themth function
evaluation required to compute J0(Ti) and J1(Ti) respectively
for algorithm p and thus N0p(T:) and N1p(T:) are the total
number of integration steps for all function evaluations
required to compute T: - T1 with algorithm p.

The total number of equivalent additions KP(T:) is a
simple notation which conveys the essential structure and in-
formation in expression (54). This number KP(T:) can be ex-
pressed as

* * * *

KP(T1) - M0 N0p(Ti) + M1 Nlp(Ti) +‘M2 NP(T1) +1M3 (57)

where

.- + .+ '=
M5 M11 + M32 8 (M133 M34) J 0,1,2,3

Thus,minimizing the computation time implies minimizing
* * *
Nb<Ti), N0p(Ti)' and Nlp(Ti) as well as minimizing the con-
stants M0, M1,'M2, and‘MB'by reducing the Operations in the
*
computer programming. Since NP(T1) is a decreasing function
f h if 1 T d N T* d N T*

O t e un orm.step s ze A , an Op( i) an 1p( 1) are
decreasing function Of AT and the integration step size a be-
cause NOpm and Nlpm are decreasing function of a, the choice
Of AT and e effect the magnitude and shape of the cost Of.

*
implementation C(Ti), the accuracy Of computation of J(Ti),

82
and the accuracy of the solution T: and thus should be in-
cluded in the OSD problem. The details of selecting AT and e
depend on the particular example problem chosen and thus are

considered in the next subsection.

5.4 Development Of Cost Of Implementation

This section will discuss the fourth and fifth steps of
the procedure to find the cost of implementation, which are
the determination Of the set Of feasible computer hardware-
computational algorithm options for each Ti and the deter-
mination Of the cost of implementation for each Ti'

The cost of implementation is Obtained from.the deter-
ministic system (26) of Chapter 4 where wn-IO. The output
dimension (m) is set equal to one rather than two as in
Chapter 4 because Q 2 was set equal to zero in the 0CD prob-

lem. The system description is repeated here for convenience

250:) ' at) + u(t)
o o 100

y<t> - xlm (58>
h(t ) - (t ) t < t < t
h(t) - x1(t) t:1+1 5 t 5 ti + A

From Chapter 4, the parameters and matrices in the control
performance index (29) are 01180.1, R-0.02, F11-0.l, d-l,
and A-0.11.

Now the cost Of implementation for this example problem
will be developed based on the following information:

(1) The Optimization algorithm is selected to be the DSC

algorithm with a uniform step size and forward search steps

83
fromTmin toward Tmax' Tmax is chosen to be 0.11 because
A-0.ll is less than Nyquist sampling period (0.3) for this
system, £min is temporarily chosen to be 0.003 by (48).
(2) The hardware Options are selected and listed in
Table 5-1 with addition time per word 71(3) and hardware
cost C8 for each hardware Option.
(3) The computer programming is Optimized to minimize
each type of Operation and the number of Operations {{Mjk}§-0}
:_1 are counted. The total number Of equivalent additions
becomes
T* - 449 T* + 497 N T* 4 T* + 9 N
KP-4( i) “04‘ 1) 14‘ 1) + 13 “4‘ 1) c
'+ 117 (60)
where the variable Nc is associated with the number of uni-

form search steps for [ti + Tmin’t + T ] and is Obtained

i max
from
No - min (N ; AT -= (Tmax - Tmin) / N 3 ATM) (61)
where N is a positive integer number and ATmax is maximum.

allowable constraint of AT. Since the number of uniform
search steps Nc is a positive integer value, Nc and AT are
determined simultaneously if ATmax were specified.

The determinations of N4(T:), No4(T:), N14(T:), ATmax’
and the maximum integration step size Emax will now be des-
cribed for DSC algorithmm

The number of function evaluations N4(T:) for the DSC

algorithm.(p-4) is

*
j + 3 T s[T +(j-1)AT,T +jAT]
N (T*) - ‘ i min min
N + 1
c TisETmifi+(Nc-1)AT,T ]

84
because (j+2) uniform step function evaluations are needed
to evaluate J(Tmifi+(j-1)AT), J(Tmifi+jAT), and J(Tmifi+(j+l)AT)
for quadratic interpolation formula for the Optimal sampling

solution

, ATEJam+O-1) Ar) - J<Tm+(j+l)AD] (63)

 

'3: - (T min+jzn') +
2[J(Tum+(j-1)AT)-2J(Tmn+jAT)+J(Tm+g+1)AT)]

when T: is in the interval [Tmin+(j'l)AT’Tmin+jAT]' Another
function evaluation is required to evaluate J(T:). For the
case where j-Nc, only (j+l) uniform step function evaluations
are required to determine J<Tmin+<Nc'2)AT)’ J<Tmin+(Nc-1)AT)’
and J<Tmin+NcAT) for quadratic interpolation formula

AT[J (T mn+(Nc-2) AT) - J('1‘mn+NcAT)] (64)

 

A*
T . +GN-JJAT+ '
14mm c 2[J(T] +0162) AT)-2J(T . +(Nc-1)A‘I)+J(Tm+NcAT)]

' *
when Ti is in the interval [Tmin+(Nc'1)AT'Tma§]' An additional
function evaluation is needed to evaluate J(Ti) for this case.
Simpson's integration formula [26,27] is used for inte-
gration to evaluate the control performance index and thus the
number Of integration steps, N04m and Nl4m’ for a function
evaluation must be a positive even integer. Since the total
*
number of function evaluations is N4(T1) and since the DSC
*
algorithm selected performs (N4(Ti)-l) uniform search steps Of
step size AT and a quadratic interpolation, the total numbers
* *
of integration steps, N04(Ti) and N14(Ti), for the DSC algorithm
*
for all function evaluagions N4(Ti) are

* N4(Ti)
N2(Ti) = Z sz’ 2 - 0,1 (65)

85
where the number of integration steps required to evaluate

J0(Tminf(mrl)AT) and J1(Imin+(m'1)AT) are
Tmin+ (m— 1) AT

 

 

 

 

min (2 L0 ; 60m - 2 L0 5 emax) m=l,...,N4-l
N -
04m . T .‘+(N -3)AT -
min (2 L s 8 min 4 5 emax) m N4
mama-(mam) s em> m=l,...,
min (2 L1 ; elm - 2L1 N4'I
N - a
14m .A-(T. . +(N -.4) AT) a
mi“ (2 L1 3 elN' iii? 4 5 an“) m N4
4 1

respectively. L0 and L1 are integer values, N4 = N4(T:), and
emax is maximum.allowable integration step size in these
expressions. 22m represents the integration step size for the
evaluation Of J£(Tmin+(msl)AT) and is chosen to make the posi-
tive even.number of integration steps (Ni4m) as small as poss-
2m must not exceed emax? This
number of integration steps, N24m for m < N4, is precisely

ible with the constraint that 6

determined by (66) and (67) because the integration time inter-
val [O’Imin+(m'1)AT] to evaluate J0(Thdn+(m-1)AT) and the inter-
val [Tmifi+(mrl)AT,A] to evaluate J1(Tmin+(m-1)AT) is known if
AT is known. The number of integration steps, N24N4’ to
evaluate J£(T:) is not precisely determined before T: is
Obtained by computer. Thus, the equations (66) and (67) for
m.- N4 were assumed to have the maximum number of integration
steps to evaluate J2(T:) which is the number Of integration
steps to compute J0(Tmin+(N4-3)AT) when i=0 and is the number
Of integration steps to compute J1<Imin+(Np'4)AT) when i=1.

86
The maximum uniform search step size is determined by

the error formula [26] Of the quadratic interpolation
* O*
J(Ti) - J(Ti) 5 1/6 J"'(t) (ATmax)3 (68)

* A*
where Ti and T1 are actual Optimal sampling time interval
and computed Optimal time interval respectively, and where t

is in the interval [Tmin+(N4-4)ATmax’Imin+(N4'2)AImax] which

5*
is the interval for quadratic interpolation to find Ti if

ax“ AT is

desired to be large in order to reduce N4(T:) and Nc and thus

uniform search step size is chosen to be ATm

also to reduce No4(T:) and N14(T:) but not tOO large so that
the control performance index error (J(T:) - J(T:)) and
solution error (T: - Ti) do not become large. A'I‘max is thus
chosen to be 0.05 because the control performance index is a
smooth and nearly quadratic so that the third derivatives Of
J(t) will be very small. The performance error (68) and
solution error (T: - T?) should be very small when ATmax-0.05.
Thus, from (61) Nc-3 and AT=0.035667 seconds.

The maximum integration step size emax is also desired
to be large in order to reduce N04m and N14m and thus to
reduce N04(T:) and N14(T:) but not so large that the integra—
tion error becomes significant. Control performance J(T0)
for the first interval (To) is shown as a function Of emax
for To - 0.03, 0.05, 0.08, and 0.11 seconds in Figure 5-1.
Since the first significant change in J(T0) is when a is

max

0.025 seconds for T0 = 0.05 seconds, emax is selected to be

0.025 seconds so that the integration error incurred by Emax

can be assured to be small enough for all T0 5 0.11. This

87

selection of emax holds for all Ti because this integration
error for the first interval is more significant than other
intervals in this example problem”

The total number of equivalent additions K4(T:) for this
DSC algorithm is shown for each T: = T1 in Table 5-2 from
(60) with AT-0.035667 and sum-0.025. Thus, 186:) is known
for each computer option from.(54) and is shown in Table 5-3.
The set of feasible computer options for each Ti can be found
by

0(Ti) = {s : 18(Ti) 3 Ti} (69)

and the results are shown in Table 5~4.

Finally, the cost of implementation can be formulated
as a piecewise constant monotone decreasing function of Ti
as shown in Figure 5-2 by choosing the computer for each Ti
which has a minimum cost among the feasible set of computers,

or

c 'r ) - ' {c }
< 1 86331) 8 (70)

The resultant cost of implementation is appealing because it
decreases very fast during the short time interval
[0.0045,0.0105] and it is almost constant with very small
change in cost over [0.0105,0.ll]. This cost of implementa-
tion will now be used to find an optimal selection of hard-

ware in the next section.

5.5 Optimal Selection Of Hardware
This section will describe the optimal selection of

hardware by a tradeoff of the control performance index and

88

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1 A TIHE INTERVFIL=0.05 X TII‘IE INTERVFIL=0.11

 

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intervals To.

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89

' *
TABLE 5-2 Total Number of Equivalent Additions to Find Ti
. 931.118. DSC. 41.8.0???“ .50? 939*! .Tifi. . . . . A A .

 

 

* * * * *
3 _ . _ T1 N4(T1.) No4<T1> .Nl4<Ti) K4(T1)
1 .003 - .038667 4 12 18 15,014
2 .038667 - .074333 5 20 16 17,746

3 .074333 - .ll 5 22 14 17,650

 

*
TABLE 5-3 lComputation Time for Each Computer for Each Ti

 

1,05 (118%)

 

 

J I:

.3 .405. .6 w -66 .7 .3* 1.0 1.2 3:5
1 @838367 .00450 .00608 .00901 .00991 .01051 .01201 .01501 .01802 .05255
2 '8332333 .00532 .00719 .01065 .01171 .01242 .01420 .01775 .02130 .06211
3 .074333-

11 .00530 .00715 .01059 .01165 .01236 .01412 .01765 .02118 .06178

 

90

 

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92

cost of implementation which have been determined in Chapter
4 and the previous section of Chapter 5 respectively. The
selection of hardware is the second stage of the OSD problem
which completes the OCSD problem described in Chapter 3.

The system performance index to be minimized for the
optimal selection of computer hardware problem

s01) -= mi) + q 0a,) (71)

includes cost of implementation with weighting factor q. The
selection of hardware requires the minimization of (71) for
several operating conditions (j) and several sampling inter-
vals (i) for some q to obtain a set of Optimal sampling
intervals

r -_-. {{Tij}?_1}§;(l) (72)
and then select the hardware based on the maximum cost over
the optimal set of solutions I

0* - Ea)! C(TIJ) (73)

Tijef
Conceptually, q is a conversion parameter from.the actual comp
puter cost to the equivalent control performance value and thus
can be determined by inverse of actual dollar benefit of the
performance improvement. However, the selection of q is
difficult to obtain because its choice determines the hardware
selected based on (71,72). If q can not be obtained easily,
the following alternative procedures make determination of
hardware, the associated hardware cost C(Tmin), and Tmin
easier.
The particular value of Tmin chosen will not only deter-

‘mine the hardware

93

s(T ) = {s : C8 = C(Tmin)} (74)

min
implemented but also the cumulative control performance index
evaluated for that Tmin over a set of intervals i a 0,1,...,
N-l and operating conditions j a 1,2,...,M

A ‘M * * *

JC(Tmin) I 121 ch(T0j,T1j, ..... ,TN-1'j,Tmin)

M (75)

N']. *
Z J (T )} ~1
i=0 0 ij
where Tij satisfies

*
Tmin 5 Tij 5 Tmax

Since C(Tmin) decreases very rapidly for Tmin < a and Jc(rmin)

increases very rapidly for Tmin > b, there is a feasible region

for Tmin

Imi

n 6 [a,b] (76)
Obviously a good design using the OSD methodology would choose

a q to obtain a TminEEa,b] because otherwise the cost would be
excessive or the control performance would be seriously degraded.
The particular choice Of :min in this region or the choice of q
that will produce the same 2min in the initial procedure, would
be based on the designers objectives. If the designer wanted
the lowest possible cost of implementation consistent with good
control performance the hardware 8*(b)‘W1th cost C(b) would be
selected. If the designers Objective is to minimize control

performance consistent with acceptable cost of implementation

*
the hardware 3 (a) with cost C(a) would be implemented.

94
This procedure to determine the optimal hardware
will now be applied to the example problem which is the
deterministic fast system (58,59) with control performance

index (29) where a-l, F -0.1, A-0.ll, and 02280, and a

11
cost of implementation shown in Figure 5-2. The minimum of
feasible region for Tmin’ "a", can easily be selected to be
0.0105 seconds from.Figure 5-2 because the cost of imple-
mentation is a rapidly decreasing function up to 0.0105
seconds and then a slowly decreasing function from.that point.
The maximum of feasible region for Tmin' "b", is chosen to
be 0.053 from Figure 5-3 which is the cumulative control per-
formance (75) with respect to Tmin with F-0.05 for an operat-
ing condition, h(t) - 1, for the OC problem, This figure
shows that the cumulative control performance is a slowly
increasing before Tmin-0.053 and a rapidly increasing after
Tmin-0.053. This feasible region for Tmin [0.0105,0.053] is
obtained based on just the unit step operating condition
(h(t)-1) because the results for other Operating conditions
(h(t)-t, h(t)-t2) are very similar to that for the unit step
input.

Thus, the optimal choice of Tmin is in the range of
0.0105 and 0.053, and corresponding optimal computer hardwares
are Data General NOVA 3/12, Digital Computer Controls MOD-5,
Interdata 6/16, and Interdata 5/16 from Table 5-4. The
choice from these four optimal computers is quite arbitrary

and is dependent solely on the designers priorities. Data
General NOVA 3/12 will be chosen if the control performance is

95

considered to be more important than the hardware cost.
The Interdata 5/16 will be chosen if the computer cost is
considered more important than the control performance.

An arbitrary choice of computer hardware, the Data
General NOVA 3/12, for implementation is made for this
example control problem. The state trajectories for the
sampled-data control with predictive sampling and this hard-
ware compared to the periodic and continuous control are shown
in Figure 5-4. The sampled-data control with predictive
sampling and this computer appears to significantly outperform
both the periodic sampled-data and continuous-time controls.
The precise values of the cumulative control performance in-
dex for predictive sampling with different values of Tmin
for unit step input are shown in Table 5-5 with the cumulative
control performance of the periodic sampling criterion
(constrained to have the same number of sampling times as
predictive sampling) and continuous control. The cumulative
control performance and terminal error for the sampled-data
control with Optimal predictive sampling are dramatically
improved over those of the periodic sampled-data control.
Mbreover, the sampled-data control with predictive sampling

criterion for :mi 3 0.012 outperforms the continuous control

n
which is the control being adaptively sampled by the predic-
tive sampling criterion. These results confirm the hypothesis
(page A? ) in Chapter 4 for the fast system that Tmin was
chosen too large so that the sampled-data control with pre-
dictive sampling did not outperform.the continuous control

being sampled.

96

The results thus indicate a predictive sampling
criterion does perform control because it enhances the
control performance over that of the continuous control for
both the fast and medium systems when Imin’ a, F11/Q11, and
A are chosen properly. Mbreover, the predictive sampling
criterion seems practical because it can be implemented with
fairly inexpensive minicomputers. The exact minicomputer
chosen is shown to depend on the designers priorities on

performance and cost.

97

0,21 3.24 0,27 .00 0,0: 0,30

E I104

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l

0,00 0,00 0,12 0.10

0,00

 

 

U U T

0.01 0.02 0.03 0'.0
TI

41.00
'8

4 0105 0100 0107 0100 0100 0.10
NE IN SECONDS

FIGURE 5-3 Cumulative control performance Jcamin) vs
Tmin for unit step input Operating condition.

98

RLPHH=I.OO F11: 0.10 DELTH=Ooll THIN: 0.0105

HDHPTIVE

PERIODIC
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TIHE IN SECONDS

 

 

 

fl

0710 0 .00 0100 1 .00

FIGURE 5-4 Output trajectories for adaptive sampling with
Tmin"0105’ periodic sampling, and continuous

control for fast system.

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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CHAPTER 6. CONCLUSIONS

This thesis has two principal contributions:

(1) the formulation and solution of the Optimal
predictive sampling criterion for a sampled-data control
system;

(2) the development of the Optimal control system
design methodology for the optimal predictive sampling prob-
lemm

The optimal sampled-data control problem with
predictive sampling criterion was motivated by the following
past developments:

(1) periodic sampling criterion which is commonly
used because of the ease of design and analysis using trans-
form.technique;

(2) adaptive sampling criteria [l-ll] , where the
sampling rate is varied in proportion to the change of
error rate. The objective of these criteria, as indicated
by the performance index used to derive the sampling rules,
is to make the sampled-data control approximate a continuous-
time control;

(3) optimal aperiodic sampling criterion [12,13]
where the system performance index measures the control per-
formance rather than the error introduced by sample and hold

100

101

device as in the adaptive sampling described above. This sys-
tem.performance index was also included an actual cost of
implementation. This system performance index was minimized
with respect to the number and the lengths of each sampling
interval to Obtain an optimal aperiodic sampling criterion.

These previous results are extended in this thesis by
formulating and solving the optimal predictive sampling prob-
lem" The system performance index is formulated with a con-
trol performance index that measures actual performance of
the control as in the formulation of optimal aperiodic sampl-
ing criterion rather than error due to the sample and hold
device as in the formulation of the adaptive sampling criteria.
The control performance index measures control performance
over both the sampling interval over which the control is held
constant and over a future interval where the control
is permitted to be continuous. Thus, only one sampling
interval at a time is chosen and is based on the estimate of
this performance index which in turn is based on past measure-
ment of outputs of the system.and knowledge of system inputs,
system dynamics, and disturbance, initial conditions, and
measurement noise statistics. A cost of implementation is
included and is a specified constant if the predictive
sampling criterion is being used to perform control on a
specified set of hardware and is a function of the length of
the sampling interval if the objective is to design and select
the computer hardware, computation algorithms, and computer

software to implement the predictive sampling criterion.

102

The results of the optimal adaptive sampled-data
control with predictive sampling criterion shows that the
optimal predictive sampling criterion is indeed adaptive for
on-line control if future performance can be precisely pre-
dicted as in the deterministic case but is periodic if future
performance cannot be predicted as in the stochastic case.
These results agree with the results for optimal aperiodic
sampling criterion which indicated that the Optimal sampling
criterion is aperiodic for the deterministic system and is
periodic for the stochastic system. MOreover, the adaptive
sampling criterion and aperiodic sampling criterion both per-
form.a control function because it has been shown in both
cases that the control performance is improved over that of
the continuous-thme control. The results on the optimal pre-
dictive sampling problem complete a theoretical foundation for
optimal sampling applied to control systems. Optimal predic-
tive sampling could also be applied to estimation and identi-
fication problems in both control and communication systems.

Optimal control system design methodology has been
further refined in this thesis. This optimal control system
design (OCSD) is broken down into the conventional optimal
control design (0CD) where the parameters of control perform»
ance index are optimally tuned so that the resulting control
meet the control performance objectives, and the Optimal
system design (OSD) where the hardware to be implemented is
optimally determined. The optimal system design procedure,

which has been proposed, determines a precise cost of

103
implementation as a function of the computational algorithms,
computer software implementing that algorithm, and the hard-
ware and then determines the optimal selection of hardware,
computational algorithm, and computer software by a tradeoff
of control performance and cost of implementation. Thus,
optimal control system.design really completes the design
problem of the optimal control system because it not only
tunes the control performance index to obtain acceptable con-
trol but also determines a precise cost of implementation and
then selects a computer hardware, computation algorithm, and
software Option based on the control performance and cost
specifications of the designer. The results obtained where
restricted to a cost of implementation based solely on come
puter hardware cost and did not consider communication and
instrumentation costs. MOreover, this Optimal control system
design was only performed for the predictive sampling problem.
Therefore, a development of the communication and instrumenta-
tion hardware cost for predictive sampling and a development
of the optimal control system design for more general control

problem was left for future research.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

REFERENCES

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Trggg. on Automatic Control, AC-7, 38-47 (January
19 .

Tomovic, R. and C.A. Bekey, "Adaptive sampling based on
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Ciscator, D. and L. Mariani, "0n increasing sampling
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Mitchel, J.R. and W;L. McDaniel, Jr., "Adaptive sampling
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Gupta, S.C., "Increasing the sampling efficiency for a
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Smith, M.J., "An evaluation of adaptive sampling,"
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Hsia, T.C., "Analytic design of adaptive sampling control
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Kalman, R.E. and J.E. Bertram, "A unified approach to the
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Schlueter, R.A., "The optimal linear regulator with con-
strained sampling times," IEEE Trans. on Automatic
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Schlueter, R.A. and A.H. Levis, "The Optimal linear
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Schlueter, RmA. and A.H. Levis, "The optimal adaptive
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