DISCRETE SYSTEM SENSITIVITY
AND
VARIABLE INCREMENT OPTIMAL SAMPLING
By
ARCHIE WAYNE BENNETT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
December, 1966
ACKNOWLEDGMENTS
The author wishes to express his appreciation to
Dr. A. P. Sage. As chairman of the supervisory committee,
Dr. Sage's guidance and advice helped make this disser
tation possible.
The author wishes to express his deepest gratitude
to his wife, Shirley. In addition to typing the disser
tation, she has provided the inspiration needed for the
attainment of this degree.
He also wishes to acknowledge the support and encour
agement of his parents.
Finally, the author would like to express his grati
tude to the Graduate School of the University of Florida,
and to the National Aeronautics and Space Administration
for their financial assistance during the course of this
work.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ... . . . . . . . ii
LIST OF FIGURES . . . . . . . v
ABSTRACT . . ................ vii
Chapter
1. INTRODUCTION . . . . . . . . 1
Sensitivity Analysis In Automatic Control. 1
Research Objectives . . . . . . 2
Plan of the Dissertation . . . . . 2
Notation. . . . . . . . ... 3
References . . . . . . .. 5
2. A SURVEY OF SENSITIVITY ANALYSIS IN
OPTIMAL CONTROL . . . . . . .. 6
Introduction and Background . . . . 6
Sensitivity In Optimal Control . . .. 12
Summary . . . . . . . ... 21
References . . . . . . . .. 23
3. SENSITIVITY ANALYSIS FOR DISCRETE SYSTEMS. 28
Introduction . . . . . . ... 28
Parameter Sensitivity of Discrete Systems. 29
Perturbation matrix approach . . .. 30
Sensitivity vectorfunction approach . 34
Sampling Interval Sensitivity of
Sampled Systems . . . . . . . 36
Global sampling interval sensitivity . 36
Local sampling interval sensitivity 38
Example Problems . . . . . ... . 40
Problem 1 . . . . . . ... 40
Problem 2 . . . . . . .. 45
Summary . . . . . . . . . 50
References . . . . . . . . 51
TABLE OF CONTENTS (Continued)
Page
Chapter
4. VARIABLE INCREMENT OPTIMAL SAMPLING .. 52
Introduction ....... . . . . 52
Performance Criteria for Variable
Increment Sampling . . . . . .. 53
Sampling Interval Sensitivity . . .. 54
Local state variable sampling interval
sensitivity .... . . ..... 55
Local error sampling interval
sensitivity . . . . . . . 56
Variable Increment Sampling Based on
Sensitivity . . . . . . .. .. 59
Linear system simulation with an exact
discrete model .. . . . ... 60
Linear system simulation with an
approximate discrete model .. .... 67
Nonlinear system simulation . .... 70
Calculation Procedure for System Simulation. 72
System Analysis for Variable Increment
Sampling . . . . . . . . 74
Variable Increment Sampling for Optimal
Control . . . . . . . ... 77
Example Problems . ... . . . . . 79
Problem 1 . . . . . . .. 79
Problem 2 . . . . . ... .. . 88
Problem 3 . . . . . .. .. . 96
Problem 4 . . . . . ... . 101
Problem 5 . . . . . . ... 104
Summary . . . . . ... . . 109
References . . . ... . . . . 111
5. CONCLUSIONS AND RECOMMENDATIONS . . .. 113
Conclusions .. . . . .. . . . 113
Recommendations ....... .. . . 116
BIOGRAPHICAL SKETCH . . . ... . . .. 117
LIST OF FIGURES
Figure Page
31 FirstOrder System .. . . ... . 42
32 Unit Step Response .. . . . . . 42
33 Generalized FirstOrder Difference
and Sensitivity . . . ... . . . 42
34 Exact and FirstOrder Change . . ... 44
35 Global and Local Sensitivity . . ... 44
36 SecondOrder System . . . . . ... 46
37 Unit Step Response . . . . . ... 46
38 Generalized FirstOrder Change ..... . 48
39 Sensitivity Vector Components . . . .. 48
310 Local Sensitivity . . . . . ... 49
311 Global Sensitivity . . . . . .. 49
41 ZeroOrder Data Reconstruction . . .. 63
42 FirstOrder Data Reconstruction . . .. 63
43 Approximate Model Response . . . .. 68
44 Calculation Procedure . . . . . . 68
45 System Diagram ... . . . . . . 75
46 FirstOrder System . . . . . . 80
47 Response and Sensitivity . . . . .. 80
48 Data Reconstruction Error: ZeroOrder Hold 80
49 Sampling Interval: ZeroOrder Hold ... 83
410 ZeroOrder Reconstruction Error vs. Number
of Intervals . . . . . . . .. 83
LIST OF FIGURES (Continued)
Figure Page
411 Reconstruction Error: FractionalOrder
Hold . . . . . . . . .. 85
412 Modeling Error of Approximate Model . 85
413 Reconstruction and Modeling Error ... . 87
414 Sampling Interval: Approximate Model . 87
415 SecondOrder System . . . . .. 90
416 Input and Response . . . . . ... 90
417 Sensitivity . . . . . . ... 91
418 Reconstruction Error . . . . ... 91
419 Reconstruction Error . . . . ... 93
420 Sampling Interval . . . . . ... 93
421 Reconstruction and Modeling Error .... . 95
422 Sampling Intervals . . . . . ... 95
423 Reactor Optimal Response . . . ... 98
424 Sensitivity ... . . . . . 98
425 Modeling Error . . . . . . . 100
426 Sampling Interval . . . . . .. 100
427 Response and Sensitivity . . . ... 103
428 Modeling Error . . . . . . ... 103
429 Sampling Interval . . . . . ... 103
430 Model for Van der Pol's Equation . . .. 106
431 Response of Van der Pol's Equation ... 106
432 Sensitivity for Van der Pol's Equation . 106
433 Modeling Error . . . . . . ... 107
434 Sampling Interval . . . . . ... 107
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
DISCRETE SYSTEM SENSITIVITY
AND
VARIABLE INCREMENT OPTIMAL SAMPLING
By
Archie Wayne Bennett
December, 1966
Chairman: Dr. A. P. Sage
Major Department: Electrical Engineering
Discrete system sensitivity is investigated and a
scheme presented for the optimal adjustment of the sam
pling rate of a sampleddata system. As background for
the sensitivity study, a survey of the historical develop
ment of sensitivity analysis is presented. The survey
includes recent developments toward a generalized ap
proach to sensitivity analysis. Also included are some
of the applications of sensitivity to optimal control.
The investigation of discrete system sensitivity
includes variations in system parameters and sampling
intervals. Two approaches to parameter sensitivity are
used. One method makes use of a perturbation matrix and
is only applicable to linear systems. The general ap
proach, using partial derivatives, is useful in linear
and nonlinear systems. Discrete sensitivity equations
are derived for both methods.
Sampling interval sensitivity is investigated for
global and local effects. For the global sensitivity
function, all sampling intervals are equal and undergo
the same variation. In local sampling interval sensi
tivity, only one interval is assured to change.
A method for the optimal adjustment of the sampling
rate of a sampleddata system is presented. The technique
uses local and error sampling interval sensitivity in
sampling interval formulas that constrain the magnitude
of the reconstruction and modeling errors of the discrete
system response. The resulting sample adjustment scheme
is suitable for realtime digital computer simulation.
The possibility of using the sampling interval adjust
ment scheme for optimal control is discussed. Several
example problems are used to illustrate the technique.
In each case, variable increment sampling improved sam
pling efficiency and computer utilization.
viii
CHAPTER 1
INTRODUCTION
Sensitivity Analysis In Automatic Control
A general definition of sensitivity analysis states
that it is the development and use of equations for the
partial derivatives of the system response with respect
to system parameters [1.1 Sensitivity was probably first
applied to control systems by Bode [2] in 1945. From 1945
until about 1960, only a few people were interested in
sensitivity theory and thus advances occurred rather
slowly. However, the advent of adaptive control and its
associated problems of identification and parameter adjust
ment brought a renewed interest in sensitivity. Interest
in sensitivity was also stimulated by the need to know
more about system dynamics and the effect of perturbations.
In the last few years, considerable progress has been
made concerning the theory of sensitivity analysis. It
now includes a wide variety of perturbations and has been
extended to optimal control. It is also being used much
earlier in the design process. The future of sensitivity
1
Bracketed numbers refer to the list of references
collected at the end of each chapter.
appears to be very promising, particularly in discrete and
hybrid systems where work is just beginning.
Research Objectives
This dissertation has three basic goals. The first
objective is to survey the existing applications of sensi
tivity to control systems. The second aim is to develop
sensitivity procedures for discrete systems. The third
objective is to use the discrete sensitivity techniques
that have been developed to implement variable increment
optimal sampling.
The project is computer oriented and the aim is to
develop algorithms for various sensitivity functions that
are convenient for digital simulation. All of the tech
niques will be illustrated and recommendations regarding
accuracy and convenience will be made.
Plan of the Dissertation
Chapter 1 is an introduction to the dissertation. It
includes a brief discussion of sensitivity analysis and
outlines the objectives of the research. Chapter 1 also
presents a summary of each chapter and gives the notation
to be used throughout the dissertation.
Chapter 2 surveys the existing work on sensitivity
analysis. A brief historical development is presented
first as an introduction and background for a more
detailed look at sensitivity in optimal control. The
various approaches that have been suggested are presented
and their uses and interrelations are discussed.
In Chapter 3, the application of sensitivity to
discrete systems is investigated. Discrete sensitivity
equations are developed for parameter and sampling interval
sensitivity. The chapter also includes example problems.
Chapter 4 illustrates the application of techniques
developed in Chapter 3. Sampling interval sensitivity
is used to implement variable increment optimal sampling.
The use of variable increment sampling in optimal control
is also discussed. Several example problems are also
included.
Chapter 5 contains the conclusions and recommendations
for discrete sensitivity and variable increment sampling.
Notation
Throughout this dissertation, vectormatrix notation
is used to represent system dynamics. Scalars are indi
cated by lower case Greek or Roman letters. The only
exception is the performance index J of an optimal control
system. Column vectors are indicated by underlined, lower
case Greek or Roman letters; e.g., x. Capital Greek or
Roman letters are used to denote matrices.
Subscripts have a number of uses. Two subscripts
indicate the row and column of a component of a matrix;
4
e.g., xij. A single subscript is used to denote the com
ponent of a vector; e.g., xi. A single subscript also
means a particular column of a matrix; e.g., vi. In
discrete systems, subscripts also indicate a function
during a particular sampling interval; e.g., tk.
The arguments of functions are explicit, except in
two instances. The time argument on continuous functions
of time is often omitted for convenience; e.g., x(t) = x.
In the other instance, the sampling interval index (k) is
used in place to time tk.
REFERENCES
1. R. Tomovi6, "Modern sensitivity analysis," IEEE
Convention Record, vol. 13, pt. 6, pp. 818,
March 1965.
2. W. H. Bode, Network Analysis and Feedback Amplifier
Design, D. Van Nostrand Co., Inc., New York, N.Y.,
1945.
CHAPTER 2
A SURVEY OF SENSITIVITY ANALYSIS
IN OPTIMAL CONTROL
Introduction and Background
The sensitivity of a control system was perhaps first
mentioned by Bode [1] in his book published in 1945. His
definition of the sensitivity of the system gain, to vari
ation in a parameter, was
1 dp T (1)
SdT p'
where T is the system gain and p is a parameter of the
system. For almost ten years after its introduction, there
was very little written on sensitivity.
Beginning in 1955, work began to appear in which Bode's
definition of sensitivity was inverted and related to other
system characteristics. The principal contributors to this
early work were Horowitz [2], Truxal [2,3] and Mason [4].
During the years 1957 through 1962, sensitivity re
ceived increased attention. The polezero sensitivity was
studied [5,6,7], and sensitivity was related to rootlocus
properties [8]. The use of sensitivity analysis in linear
system theory was presented by a number of people [915].
It was extended to sampleddata systems [16,17], and a
number of articles were written on the use of the sensi
tivity coefficients for system identification and adaptive
control [1822]. It was during this period that the need
for a more general approach to sensitivity became more
apparent. As a result, even more articles on the appli
cation of sensitivity analysis began to appear.
A number of important developments took place in 1963.
Tomovi6 published the first book devoted entirely to sensi
tivity analysis [23]. The book contains a formulation of
theoretical and practical problems with the aim of encour
aging more work on a general theory of sensitivity analysis.
The close relationships, as well as the distinct differences
between stability and sensitivity, are discussed.
Tomovi6 considers a dynamic system with the mathemat
ical model
F(K,x,x,t,p) = 0, (2)
where x is the state of the system and p is a single system
parameter. (Note that the nomenclature used by Tomovi6 has
been changed for convenience.) The dynamic sensitivity co
efficient is defined as the change in the state x due to
variations in the parameter. This is expressed as
v(t,p) = dx(t,p) (3)
dp
The sensitivity equation is obtained by first taking the
partial derivative of equation (2) with respect to p,
aF 6x 6F 3* BF ax 6F
y 7 + J + Ti = 0,
and then substituting the relations
v= and = V.
This yields the sensitivity equation
SF .. 8F F F
3v + v +v = (5)
The sensitivity equation is a linear differential
equation and can be solved by analytical means. However,
the book deals only with machine and experimental solution
methods and gives several examples. One method is the
simultaneous solution of the system, equation (2), and the
sensitivity equation, equation (5), on an analog computer.
This makes use of the connections between the two equations.
Tomovi6 also notes that the structural similarity of the
two equations facilitates solution by a digital computer.
The book also includes a discussion on the solution
of the sensitivity equations by simulation. This method,
based on earlier work by Bihovski [24], assumes that the
dynamic system has been realized and the sensitivity co
efficients are to be obtained by direct measurement.
Bihovski's work is also the basis for the structural
method of obtaining the sensitivity coefficients of linear
systems.
The structural method has been presented in a number
of places [25] and its primary advantage is that the
sensitivity coefficients for a number of parameter variations
are obtained simultaneously. Another feature is the
simplicity of the analog model. The only portion of the
system that must be simulated in detail is the portion
that is to be studied closely.
The sensitivity coefficient (as a function of time)
about a particular parameter p is useful, but a knowledge
of its values in parameter space is even more useful. Such
problems as structural sensitivity and the effect of the
variations of a number of parameters can be studied by
means of the sensitivity coefficients in parameter space.
The problem of inverse sensitivity is also discussed
in the book. In this formulation, the variations in x are
known, and it is desired to determine the corresponding
variations in the parameter p. The book also includes
brief discussions on adaptive control of invariant systems
and performance adjustment of dynamic systems.
In 1963, another book written by Horowitz 126] gave
considerable space to the subject of sensitivity. Horowitz
used sensitivity in presenting the design and synthesis of
linear multivariable continuous and sampleddata systems.
Another indication of the growing importance of sensitivity
was a session on it at a circuit and system theory confer
ence [27]. It was also in 1963 that sensitivity was first
applied to optimal control systems [28,29]. The subject
of sensitivity in optimal control has received a great deal
of attention since its introduction and its development will
be discussed in the next section.
To see the development for a system more general than
that of equation (2), consider the system
k = f[x(t),p) x(0) = x (6)
where x is an n dimensional state vector and p is an m
dimensional parameter vector. A sensitivity equation and
sensitivity coefficients for parameters other than those
that change system order and initial conditions can be
derived [25]. For small changes in p, a firstorder
approximation for the corresponding change in x is
m
+' (7)
x = vp +., (7)
j=1 T
ap ... The
where v is the sensitivity vector v ... The
j j aPj p
th
i component of v is the sensitivity coefficient
j
ax
i (8)
ij apj aP = 0,
aPm = 0
which represents the variation in the it component of the
th
state vector x due to a change in the j component of the
parameter vector p. The sensitivity equation is
n af at
v vij + k (9)
kj axi ij opj
.=1
where k = 1,..., n, and j = 1,..., m, and the initial con
ditions are v (0) = 0.
kj
If the sensitivity to changes in initial conditions,
x(0), of equation (6) is desired, the initial conditions
for equation (9) must be vkj(0) = 1 [25]. In order to
handle parameters that change the order of the system, the
system equations
xi = fi(xl ...xn;xn+l ...xn4r;t) (i = 1,...,n)
(10)
n+s = f n (x ...x ;xn ,.. ".,x ;t) (s = l,.. ,r),
n+s n+s 1 n' n+1' 'n n+r .
are used. Note that X changes the order of the system and
when A = 0, equation (10) represents the original system.
The sensitivity equations are
n+r af
vk = v. (k = 1,...,n)
k axi 1
i=1
n n+s dXn+s
v+,r n+s i d (s = 1,.,r) (11)
Vn+s I x xi dt
i=l
with vh(0) = 0, where (h = l,...,n+r) [25]. Several other
papers on parameters that change the order of the system
have been published [30,31].
It should be noted that the sensitivity functions
can be determined by other methods. For example, the
method of undetermined coefficients, difference equations,
and asymptotic expansions can be used and do not require
equation (6) to be regular in p [32]. They also allow a
much wider class of perturbations to be treated. Some of
the other types of perturbations that have recently been
incorporated into the framework of sensitivity are: fre
quency of oscillation, time delay, sampling rate, inte
gration step, and "amount" of nonlinearity [32].
The rapid progress has been made possible, in part,
by drawing from other areas such as network theory, error
analysis in analog computers, and the theory of differential
equations. These recent developments in sensitivity analy
sis have been covered by several articles [25,32], and an
international symposium on sensitivity analysis [33]. Also,
sensitivity was the subject of several sessions at a con
ference on circuit and system theory [34].
Sensitivity In Optimal Control
Since 1963, when sensitivity was first applied to
optimal control, a number of uses for sensitivity in
optimal control have been presented. The sensitivity of
such quantities as the performance index, the state vector,
and the terminal state have been studied for variations in
the plant parameters, the state vector, and the control
vector. The following section will present some of the
results of the work in this area.
Dorato [28] used sensitivity to determine the variation
in the performance index
T
J = j F(x,u)dt, (12)
due to changes in the plant parameters. The plant state
due to changes in the plant parameters. The plant state
vector x(t) is related to the plant control vector u(t) by
the vector differential equation
x(t) = f{x(t),u(t),p (13)
where p represents the set of plant parameters. For this
case, the optimal closedloop control law is of the form
u(t) = Q{x(t),p;t (14)
For changes in p from the nominal p the change in the
performance index is
AJ = J(p ) J(p).
o
Dorato considers small variations in p and writes
AJ dJ = dpl +...+ dp (15)
8pl p m
or
8J
AJ dp, (16)
ap 
where 1J is the performance index sensitivity vector. This
ap
can be written as
6J JT 6F ax
J 2 tF dt, (17)
p J x .
where x is a matrix and is the solution to the sensitivity
ap
equation [35]. Using this notation, equation (9) can be
rewritten as
d rx .f 2 + f (18)
dt [p I x aJ xp 6p
Dorato suggested that sensitivity might be a useful
criterion to use in comparing openloop and closedloop
control. Cruz and Perkins have investigated this problem
and the more general problem of the sensitivity of multi
variable systems [3640]. They introduce a sensitivity
matrix S(s) which relates Eo(s) to Ec(s) by the relation
ship
Ec(s) = S(s)Eo(s), (19)
where E (s) is the output error due to plant parameter
variations of a closedloop realization of the system.
The output error of the openloop system is represented
by E (s). Note that in order to have a meaningful com
parison, the output of both the openloop and the closed
loop systems must be equal if there are no plant parameter
variations.
Cruz and Perkins have related the sensitivity matrix
S(s) to a matrix generalization of return difference for
multivariable, linear, timeinvariant feedback systems.
Also, for singleinput, singleoutput systems, the sensi
tivity matrix is compatible with the classical definition
of sensitivity. However, in the application of sensitivity
to optimal control systems, one of the most useful aspects
of their work has been the idea of a "comparative" sensi
tivity.
Another definition of sensitivity that is of a "com
parative" nature was introduced by Rohrer and Sobral [41].
In order to avoid having to completely specify the con
ditions associated with the normal or "absolute" definitions
of sensitivity, they define a relative sensitivity
SR[u(t),p] = Ju(t),p] J[u(t),p (20)
IJ[uo(t),p]I
where J[uo(t),p] represents the performance index asso
ciated with the system when driven by the optimum control,
u(t), for the given set of plant parameters p. J[u(t),p]
is the performance index when the control, u(t), is not
the optimum for the given set of parameters p. They use
the calculus of variations to show that,
SR u(t),p]J 6J[uo(t),u(t),p] (21)
Ji[uo(t),p~ l
for uo(t) interior to its allowable set U, and
sR[u(t),pl 62J[uo(t),u(t),p] (22)
IJ[uo(t),p]
when uo(t) is on the boundary of U. Note that relative
sensitivity approaches zero as a system approaches its
optimal performance. Also, it should be pointed out that
the relative sensitivity is a function of both the control
u(t) and the parameters p.
Rohrer and Sobral use relative sensitivity to define
a plant sensitivity
sM[u(t)] = Max[SR[u(t),p] (23)
peP 
which is the maximum value of relative sensitivity for all
which is the maximum value of relative sensitivity for all
parameters of the allowable set P. This definition of
plant sensitivity could be used as a design criterion
and the optimization would seek the control u (t) which
minimizes the plant sensitivity S M[u(t)]. Thus, if the
plant parameters were known to vary over a certain range,
the control u (t) would minimize the maximum deviation
from optimal performance for parameter variations over
the specified range.
Rohrer and Sobral have also defined a plant sensi
tivity that is useful in systems in which the plant
parameters are given as random variables. This definition
is based on the expected value of the relative sensitivity
and can be stated as
SE u(t)] = E [SR[u(t),p]), (24)
p P
where "E" indicates the expected value. Here the opti
mization would seek the control u (t) which minimizes
SE[u(t)]. This would minimize the average or expected
deviation from the optimal performance.
A similar optimization procedure using a game theory
approach has been formulated [42,43,44]. This method is
useful since both large and small plant parameter variations
can be considered and the controller structure need not be
fixed [42]. Dorato and Kestenbaum [43] consider a fixed
controller structure with a controller parameter pc. The
plant dynamics are given by
i(t) = f[x(t),u(t),p ),
 p
where pp is the plant parameter. If pc = ppthen the
controller generates the control
u(t) = Q[x(t),pc;t}, (26)
which is optimal and the performance index
T
J = t F(x,u)dt, (27)
is minimized. In the problem they formulate, all that is
known about p is that it lies somewhere in the range
pl pp 2a' and,therefore, the performance index is a
function of p and pc. The object of the optimization
is to determine the "best" value of p .
Since p is known only to range from p to p2, the
desirable controller parameter po should keep the per
c
formance index equal to or less than some value, J, for
all values of p in its expected range. This can be
expressed as
J(ppp)5J for pls pp P2. (28)
Also, the inequality
Jo J(pc,po), for pl Pc SP2 (29)
must also hold if Jo is to be as low as possible.
In the game theory interpretation, J(pc,Pp) is the
value of the game or the "payoff function" and the
players or "antagonists" are p and p The pair
p c
o o
c,(Pp) is an optimal or pure strategy and the type of
game is infinite or continuous. The conditions for optimal
strategies to be pure are [43],
Min Max J(pc,p ) = Max Min J(p ,p ) (30)
Pc Pp Pp Pc
and the existence of numbers po,pp, and J such that
c p
J(P,Pp)JOSJ(pCpp) (31)
Recently, Pagurek [45] has presented some interesting
results for linear systems. He formulates the sensitivity
of the performance index into the structure of the
HamiltonJacobi equation and shows that the open and
closedloop performance index sensitivity functions are
the same. This approach is useful in that sensitivity
analysis can be carried out by the same technique used
to obtain the optimal control law. His work has been
extended to the nonlinear case by Witsenhausen [46].
Siljak and Dorf [47] point out that most applications
of sensitivity to optimal control do not use sensitivity
as a criterion for determining the optimal control, but
determine sensitivity after the optimal control has been
synthesized. In order to avoid this, they use the time
domain sensitivity technique [19,48] and introduce a
general index of optimality which includes both sensitivity
and performance characteristics. Thus, the optimal
control synthesized satisfies sensitivity and optimality
requirements simultaneously.
In order to include sensitivity in a general index,
the usual index of performance
J = F(x,ut)dt, (32)
is altered to also include the sensitivity functions.
The resulting generalized index of optimality is
J = Jt G(x,u,vlZ ...' m,t)dt, (33)
ax
where the sensitivity functions are vI = ap and pi is
the ith variable parameter. The sensitivity functions,
v., should appear in the index as squares or magnitudes
to avoid the canceling effects of a change in sign. Also,
the authors indicate the usefulness of a weighting function
which would allow certain sensitivity functions to receive
different emphasis. The resulting control law uo(t)
optimizes both sensitivity and performance.
In addition to the sensitivity of the performance
index, the sensitivity of the terminal state of an optimal
control system has been studied. Gavrilovi6, Petrovi6,
and Siljak [29] investigated it by the adjoint method in
one of the first articles applying sensitivity to optimal
control. More recently, Holtzman and Horing [49] used
variational techniques to study the sensitivity of the
terminal conditions of both open and closedloop optimal
systems. An important part of their work is the inclusion
of sensitivity prior to optimization for the openloop
system. This allows the sensitivity of terminal con
ditions to be prespecified or constrained. The results
of their work confirm that the closedloop configuration
has superior sensitivity characteristics.
The sensitivity to variations in the plant parameters
has been the object of most of the work on sensitivity
in optimal control. However, Bl6anger [50] has investi
gated the effects of variations in the control. He points
out that this is useful in suboptimal control and also in
studies of the sensitivity of the computation of the de
sired control. Control variations have also been studied
by Gavrilovi6, Petrovi and Siljak [29]. They consider
variations in control by changes in the initial conditions
of the adjoint system instead of letting the control vary
directly.
B6langer considers both "weak" and "strong" variations
in the control. For the weak or "continual" variation,
the actual control differs from the desired control by an
infinitesimal amount E7(t). Tolerances on the control can
be set by limiting 7(t). The strong or "intermittent"
variation causes actual control to differ from the desired
control by large amounts, but only during infinitesimal
intervals of time. The continual variation is applicable
when the control is continuous and the intermittent var
iation is useful for "bangbang" control.
There are two effects of a variation in control. One
result of a variation in the control would be the failure
to hit a desired target or terminal state. The other
effect would be variations in the value of the performance
index. B6langer has considered both effects for the case
in which the actual target has been replaced by an ideal
target. This is useful since control tolerances necessary
to hit the actual'target can be determined and variation
in cost calculated. For example, if the target were a
small sphere, the control to hit a point at its center
would be calculated and then tolerances determined for
this control to insure that the sphere will always be
reached.
Summary
The first portion of this survey presents a historical
development of sensitivity analysis in automatic control.
The various sensitivity functions, vectors, and coeffi
cients are defined and several methods for their calcula
tion are discussed. Also, some of the recent contributions
toward a generalized approach to sensitivity analysis are
presented. With this background information, the appli
cation of sensitivity to optimal control is discussed next.
A variety of uses of sensitivity in optimal control
has been formulated in recent years. The discussion in
cludes the sensitivity of the performance index, the state
vector and the terminal state for variations in plant
parameters, controller parameters, the state vector, and
the control vector. Also included are some of the
22
optimization schemes which make use of sensitivity to
optimize system performance. The use of sensitivity in
establishing the control tolerance necessary for target
states is discussed.
In looking over the variety of methods and defi
nitions that has been used for the sensitivity analysis
of optimal systems, it is obvious that no single method
is completely satisfactory. The subject is still in the
early stages of its development and there is much need
of a general, comprehensive approach. There is a great
deal of interest in this subject and no doubt considerable
progress will be made in the near future.
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27
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January 1966.
CHAPTER 3
SENSITIVITY ANALYSIS FOR DISCRETE SYSTEMS
Introduction
The utility of sensitivity analysis in discrete
systems has been recognized [1] and a few papers have
been presented [2,3,4]. The early work on sensitivity
was influenced by the analog computer, and as a result,
most methods developed to determine sensitivity "factors"
are in the form of solutions of continuous equations.
In view of the wide spread use of discrete and hybrid
systems, a discrete approach to sensitivity analysis
would be useful. It would certainly be beneficial for
the sensitivity of discrete systems and might also be
useful in analysis of continuous systems.
The existing continuous sensitivity equations can
be solved on a digital computer. However, the discrete
version should be more convenient since an increasing
amount of system simulation is being done on digital
computers.
In developing a discrete approach to sensitivity,
discrete versions of the existing continuous factors
and equations will be formulated. Also, factors that
are only useful for discrete systems will be investigated.
Parameter Sensitivity of Discrete Systems
One of the most useful sensitivity factors is the
sensitivity of the state variable to changes in system
parameters. In the following discussion, stateparameter
sensitivity is developed by two methods. The change in
state is first expressed in terms of a perturbation matrix
and then developed by means of a sensitivityvector
function.
The most general representation of a lumped discrete
system is given by the equation
x(k) = f(x(kl),u(kl),p,tkk, (1)
where x(k) is a ndimensional state vector, u(k) is a
mdimensional control vector, and p is a constant rdimen
sional parameter vector. The sampling interval Tk is the
time between two consecutive sampling instants,
Tk = tk+1 tk (2)
For convenience, the arguments of x and u will not include
the time t explicitly. Instead, time will be designated
by the particular sampling interval. For example;
x(tk) = x(k).
If the discrete system is linear, it can be repre
sented by the equation
x(k) A(kl)x(kl) + B(kl)u(kl), (3)
where A(kl) and B(kl) are A(tkl,tk)and B(tkl,tk)
during the sampling interval Tk_1 = (tk tkl).
For linear systems, this is often more convenient than
the form given in equation (1). If the system of equation
(3) is also stationary, the matrices A and B are constant;
A(j) = A(i) = A and B(j) = B(i) = B.
In the following developments, some of the methods
are only applicable for linear systems, and equation (3)
will be used.
Perturbation matrix approach
One method for determining the variation in the
state x(k) for changes in system parameters makes use
of a perturbation matrix. This method is applicable
only to linear systems and will be developed from
equation (3) and its solution, which is [5],
ki
x(k) = @(k,0)x(0) + (k,j+l)B(j)u(j), (4)
j=0
where the transition matrix is given by,
kI
i(k,j) = A(i) = A(kl)A(k2) A(j+l)A(j). (5)
ij
If the system is stationary, the transition matrix is
(k) = A AA.A.*; (6)
and the solution vector can be written as
ki
x(k) = (k)x(0) +X 4(j)Bu(kjl). (7)
j0
The derivation will be carried out for a stationary
system. The results will also be given for timevarying
systems. To determine the effects of a change in param
eters, let the matrix A of a stationary system change by
an amount EC. Equation (3) can be written for the perturbed
system [6]
3(k) = [A + EC]x(k1) + Bu(k1). (8)
If equation (8) is written for successive values of k,
and certain substitutions made, the system transition
equation can be derived for S(k) in terms of the control
over the interval, the initial value x(0), and the system
parameters. The resulting equation is
ki
3(k) = (k)x(0) +7 t(j)Bu(kjl)
j0
ki
+ Ej 4(kjl)CG(j)x(0)
j=0
k2 ki2
+ Ei I X (kji2)C(j)Bu(i) + 0(E2) (9)
i=0 j=0
The change in x(k), ax(k) = X(k) x(k), can be
determined by subtracting equation (7) from equation (9).
The firstorder change in x(k) is obtained by neglecting
the higher order terms. For a stationary linear system,
the firstorder change is
ki
Ax(k) E $(kjl)C (j)x(O)
j=0
k2 ki2
+ el (kji2)C4(j)Bu(i). (10)
i=0 j=0
A similar equation gives the firstorder change in x(k)
for a timevarying linear system.
k1
Ax(k) = E (k,j+l)C(j))(j,0)x(O)
j=0
j=O (11)
k2 ki2
+ C I t(k,j+i+2)C(j+i+l)i(j+i+l,i+l)B(i)u(i).
i=0 j0
The firstorder effects of EC can also be derived in
the form of a difference equation
Ax(k) = A(kl) Ax(kl) + (C(kl)x(kl), (12)
with Ax(0) = 0. Thus, the firstorder change at each
sampling instant can be calculated from the state and the
firstorder error at the preceding sampling instant.
An exact difference equation for the change in a
state x(k) due to the perturbations EC(k) is
Lx(k) [A(kl) + EC(kl)]Ax(kl) + EC(kl)x(kl),
where ax(0) = 0. Note that the exact change given by
equation (13) differs by only one term from the first
order change of equation (12). An equation of the form
of equation (11) could be derived for the exact change,
but it is too complex to be useful in computations.
Equation (12) is not so accurate as equation (13),
but it is more general. If equation (12) is divided by
e, the resulting difference equation is
Ax(k) = A(kl)a (kl) + c(kl)x(kl). (14)
C E
The vector x(k) can be calculated along with x(k) and
then the firstorder error for any small E can be deter
mined from Ox(k). This generalization cannot be made on
equation (13).
The equations derived in this section are not
sensitivity equations by definition. However, they do
give the change in the state due to known perturbations
of system parameters. Note that by carefully selecting
the elements of the matrix C, any number of the system
parameters of A can be varied. Equations (10) and (11)
give the firstorder change in any state in terms of
x(0) and the control over the interval. Equations (12)
and (13) give the firstorder and exact change in any state
in terms of the preceding state and change. These equa
tions will be illustrated in an example problem and then
will be used to check the accuracy of other sensitivity
methods.
Sensitivity vectorfunction approach
Another approach to the sensitivity of discrete
systems is very similar to the sensitivity vectors and
sensitivity functions of continuous systems [7]. This
method does not require the system to be stationary or
linear. A sensitivity vector can be defined as
v (k(k) axl(k ax) ax,( (15)
v i(k) 
1 api pi pi p Pi J
where the sensitivity functions are the components,
vij(k) = x(k) (16)
Spi 6pi = 0.
Apr = 0
There will be a sensitivity vector for each of the r
parameters of the parameter vector p. Using the sensi
tivity vectors, a firstorder approximation of the change
in x(k) can be written as
r
ax(k) = j(k)Pj. (17)
j=1
The method used to calculate the sensitivity co
efficients is of primary importance. In continuous
analysis, a differential equation is formulated and its
solution yields the sensitivity functions. For the
discrete approach, a difference equation would be desirable.
To formulate a sensitivity difference equation, take
the partial derivative of equation (1) with respect to pi.
This yields the equation
ax(k) af(k1) ax(kl) + 3f(k1) (18)
Spi ax(kl) api api
Note that the terms involving u are not present, as it is
assumed that the control is not dependent upon the changing
parameters. Substituting equation (15) into equation (18)
results in a difference equation for the sensitivity vectors
of a discrete system,
af(kl) af(k1)
iv(k) k ) v(k1) + (19)
i ax(k1) i api
where vi(0) = 0. The sensitivity functions are determined
from an equation for the components of equation (19),
n afj(k1) afj(kl)
v. (k) = sf(k1) s(k1) + f ) (20)
bxss(kl) vs api
s=l
where v..(0) = 0.
The sensitivity vector function approach of equations
(19) and (20) is in general more useful than the transition
matrix approach discussed in the previous section. The
primary advantage of the sensitivity vector function
approach is that the varying parameters are not restricted
to the elements of A. Both methods are illustrated by an
example problem in a later section.
Sampling Interval Sensitivity of Sampled Systems
The performance of a sampleddata system is usually
very sensitive to changes in the sampling period. There
fore, knowledge of the sampling interval sensitivity
should be quite useful. Sampling interval sensitivity
could be used as a basis for selecting the universal
sampling interval or for selecting individual sampling
intervals. These two applications are significant in
that they illustrate two distinct sampling interval
sensitivity functions. In selecting a universal sampling
interval, the fixed sampling rate that optimizes system
performance can be determined with the aid of a "global"
sensitivity function. Individual sampling intervals
would be selected by means of a "local" sensitivity
function [3]. In the following work, both "global" and
"local" sampling interval sensitivity functions are
defined, and the equations necessary for their calculation
will be presented.
Global sampling interval sensitivity
If a system has a fixed sampling interval, then
Ti = Tj in equations (1) and (3), and a global sampling
interval sensitivity function can be defined as
ST(k) lim x(k(T+AT)} x[k(T) (21)
T ATO AT
Equation (21) is an expression for the change in the state
at the kth sampling instant if every sampling interval is
changed by AT. Note that if K samples are required to
cover the interval of interest, then the size of AT must
obey the inequality
TTotal Time (22)
aTc (22)
K
th
This is necessary since the i sampling instant is
shifted by i'AT and the maximum shift K'AT must not move
the kth sampling instant into another sampling interval
[3].
If the state x(k) is a continuous function of T, the
global sampling interval sensitivity function of equation
(21) can be written as
T (k) (23)
T T
A discrete sensitivity equation for vT(k) can be
derived by taking the partial derivative of equation (1)
with respect to T. If parameter variations are not
considered,
Bx(k) f(k1) ax(kl) af(k1) bu(k1) af(k1)
6T ax(kl) 3T 3u(kl) 6T T (24)
Substituting with equation (23) yields the discrete
sensitivity equation
af(k1) af(k1) 6u(k1) af(k1)
vT(k) = vT(k1) + +
ax(kl) u(kl) 6T 8T
(25)
subject to the initial conditions v (0) = 0.
T
If system is linear, then equation (3) is used to
represent the system, and the sensitivity equation is
3A(kl)
v (k) A(kl)v (kl) + x(kl)
T T T
+ AB(k1) u(k1) +B(k1) yu(kl), (26)
6T aT
with the initial conditions v (0) = 0.
T
The ndimensional global sensitivity vector will have
a set of values for each sampling interval. The it com
ponent of v (k) indicates the effect of changes in every
T
sampling interval on the ith component of the state vector
x(k) at the kth sampling instant. Therefore, it can be
used to extrapolate solutions about a nominal sampling
interval T For small values of k, the first two terms
of a Taylor's series,
x[k(To + AT)] a x(kT) + vT(k)AT, (27)
will give good results. The accuracy of the approximation
deteriorates as k increases. This is characteristic of
sensitivity methods that involve frequency [3]. Global
sampling interval sensitivity will be illustrated in an
example problem.
If the sampling interval is not held constant, then
Local sampling interval sensitivity
the global sensitivity defined in the preceding section
does not apply. Therefore, a local sensitivity function
is required to determine the effect of a change in the
kth sampling period on the state at the (k+l)st sampling
instant. A local sampling interval sensitivity function
can be defined as
S lim x(tk+At) x(tk) (tk)
v(k)= (28)
Att0 At atk
To derive a sensitivity equation for local sampling
interval sensitivity, the system equation is first written
in the form
x(k) = f(x(kl),u(kl),tk_,tk) (29)
To determine v(k), differentiate equation (29) with respect
to tk,
v(k) = 3 {(kl),u(kl),tki,tk. (30)
8tk
If the system is linear, the sensitivity equation is
6A(tkl,tk) 6B(tkl,tk)
v(k) = x(kl) + u(kl). (31)
tk tk
Note that the terms A(tkl,tk) (k 1) and B(tk_,tk)aL(k1)
8tk atk
are missing from equation (31) because both ag(k1) and
atk
u(k1 ) are zero. This is true since x(k1) and u(kl)
atk
are insensitive to changes in the following kth sampling
instant.
Equation (31) gives the sensitivity of x(k) to changes
in Tk_ only. Similar to global sensitivity, local sensi
tivity can be used for extrapolation, but only about tk.
Also, additional insight into the relationship between
local and global sensitivity is gained by noting that for
a linear system, the global sensitivity is the sum of all
preceding local effects [3].
Example Problems
The purpose of this section is to illustrate the
calculation of the various sensitivity functions that
have been defined.
Problem 1
As a first example, consider the system of figure
31. The equation for the continuous system is
x(t) = ax(t) + u(t). (32)
The discrete version of the system is
x(k) = Ax(k1) + Bu(k1), (33)
where
A = exp[a(tk tkl)),
and
t
B = exp(a(tk))}d7. (35)
k1
For a sampling interval of 0.1 seconds and a = 1.0, the
coefficients are; A 0.90484 and B = 0.09516. The system
was simulated on a digital computer and then response for
a unit step is shown in figure 32.
To illustrate the perturbation matrix approach for
parameters variation, equation (14) for the generalized
firstorder change, was programmed on a digital computer.
The solution, ,x(k), for a unit step input is plotted in
figure 33. It should be noted that the response is not
continuous as shown in figure 33, but is actually a series
of discrete data points.
The sensitivity vector approach given in equation
(19) was also used. For variations in A, this first
order system will have only one sensitivity vector with
one component. The solution to equation (19) is identical
to that of equation (14). Thus, the sensitivity vector
and the perturbation matrix methods have the same solu
tion. Figure 33 represents the sensitivity of state x
to variations in the parameter A when determined by either
method.
To check the accuracy of both methods, the system
was simulated with A = 0.89584, B = 0.09516 and T = 0.1
seconds. This is equivalent to perturbing A by 0.009.
ZeroOrder Hold Plant
S sT 1
u(t) s (s + a) x(t) x(kT)
Figure 31 FirstOrder System
1.0
.75
.50
.25
0
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time in Seconds
Figure 32 Unit Step Response
10.0
7.5
5.0
Ax
e 2.5
0
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time in Seconds
Figure 33 Generalized FirstOrder
Difference and Sensitivity
Figure 34 contains a plot of the difference between the
original response x and the perturbed response x. For
comparison, the figure also includes a curve for the first
order change calculated by equation (12) with EC = 0.009.
The exact change, given by equation (13) with EC = 0.009,
is identical to the difference in x from the two sim
ulations. The firstorder change was calculated by
equation (10) for several values of k and the results
agree with those of equation (12).
The accuracy of the sensitivity vector approach was
checked by using equation (17) with p = 0.009. The cal
culated firstorder change in x is identical to that of
the perturbation matrix approach. Thus, the firstorder
error curve of figure 34 represents both methods.
The global sampling interval sensitivity function of
equation (26) is shown in figure 35 for a unit step input
and T = 0.1 seconds. The coefficients are; A = 0.90484
and B = 0.09516. Figure 35 also contains a plot of the
local sampling interval sensitivity function as given in
equation (31). The extrapolation by equation (27) of
Ax(k) for several values of TO and AT, was compared
with the actual change in x(k) when the system was sim
ulated with T = To+AT. The results were excellent as
long as inequality (22) held. It should be pointed out
that the Ax(k) predicted by global or local sensitivity
is not an error, but the change in x(k) due to a variation
in T or Tk.
Time in Seconds
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
.02
.04
ax
.06. Exact Change
.08
FirstOrder Change
.10
Figure 34 Exact and FirstOrder Change
4.0
3.0
vT I Global Sensitivity
2.0
v(k) ^
10
Local Sensitivity
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Time in Seconds
Figure 35 Global and Local Sensitivity
Problem 2
The sensitivity functions have also been calculated
for the system in figure 36. The discrete version of
the system is
x(k) Ax(kl) + bu(kl). (36)
The coefficient matrix A has the components:
all = exp(37)[cos(4t7) + .75sin(477)] (37)
a12 = (25/24)exp(37)sin(4T) (38)
a21 = 1.5exp(377)sin(47) (39)
a22 = exp(37l)[cos(47) .75sin(41)], (40)
where 1 = (tk tk_). The coefficient of u(kl) is a
vector b with the elements:
b = 6.25exp(36)sin(46)dT (41)
tk
Ik
b2 = 6exp(36)[cos(46) .75sin(46)]dT, (42)
t_l
where 6 = (tk T). For a sampling interval of 0.01
seconds, all = 0.99877, a12 = 0.040424, a21 = 0.058211,
a22 = 0.94056, bl = 0.0012251, and b2 0.058211. Figure
37 contains a plot of the response of x1(k) and x2(k)
for a unit step input.
ZeroOrder Hold
Figure 36 SecondOrder System
Time in Seconds
Figure 37 Unit Step Response
The perturbation matrix method was used to calculate
the effects of changing the components of the matrix A.
In this example, every element of A was changed the same
amount by setting all elements of the matrix C equal to
one. The resulting firstorder change in xl(k) and x2(k)
calculated by equation (14) for a step input are shown in
figure 38. The change in x(k) for a given value of e
can be determined from figure 38 by multiplying the
curves by e.
The accuracy for a specific perturbation was deter
mined by calculating the change in x(k) for various values
of ecij, and then simulating the system with the elements
of A actually changed by ecij. The actual changes in
xl(k) and x2(k) from the simulations and the firstorder
changes calculated by equation (12) are very close. The
exact changes given by equation (13) are identical to the
changes observed in the simulations.
The sensitivity vectors were calculated for each
element of A as a variable parameter. The elements of
the sensitivity vectors for changes in all are shown in
figure 39. To avoid confusion, the vectors for changes
in the other elements of A are not included. The changes
in xl(k) and x (k), due to changing all elements of A by
0.02, were calculated by equation (17). The results are
identical to the firstorder changes predicted by the
perturbation matrix approach.
Ax1
E 2.0
1.0
AX2
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2,
Time in Seconds
1.0
AX2
2.0
3.0
Figure 38 Generalized FirstOrder Change
3.0
V11
2.0
1.0
1 0.2 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.
S1.0
I T e i S.c o n d s l =
V12
2.0
3.0
Figure 39 Sensitivity Vector Components
v1(k)
v (k)
Time in Seconds
Figure 310 Local Sensitivity
1T (k)
Figure 311 Global Sensitivity
1.0.
2.0.
75
50
25
a
25
0
50
75
The global and local sampling interval sensitivity
vectors were calculated by equations (26) and (31) for a
step input and T 0.01 seconds. Figure 310 contains a
plot of each element of the local sensitivity vector. The
global sensitivity vector is shown in figure 311. Excel
lent results were obtained when the two vectors were
checked by extrapolating values of x(k) about T = 0.01
for AT = 0.005 and comparing them with values obtained
from a simulation with T = 0.015.
Summary
Two approaches to parameter sensitivity for discrete
systems have been presented. In each case, difference
equations for the sensitivity function were developed
and their calculation and use demonstrated.
The sensitivity to local and global changes in
sampling interval were investigated and formulas for
their calculation were given. Example problems were
worked and the results compared with the changes observed
when the sampling interval was actually changed.
Every method discussed was found to be accurate, and
no problems were encountered in calculating and using the
functions. Based on these observations, parameter and
sampling interval sensitivity should be quite useful in
the analysis and design of discrete systems.
REFERENCES
1. R. Tomovi6, "Modern sensitivity analysis," IEEE
Convention Record, vol. 13, pt. 6, pp. 8186,
March 1965.
2. R. Tomovi6, W. Karplus and J. Vidal, "Sensitivity of
discretecontinuous systems," Third IFAC Congress,
London, England, June 1966.
3. G. A. Bekey and R. Tomovi6, "Sensitivity of discrete
systems to variation of sampling interval," USCEE
Report 127, University of Southern California, School
of Engineering, Los Angeles, California, March 1965.
4. J. Vidal, W. Karplus and G. Keludjian, "Application
of sensitivity analysis to hybrid computations,"
Proc. Int'l. Symposium on Sensitivity Analysis,
Dubrovnik, Yugoslavia, September 1964.
5. H. Freeman, DiscreteTime Systems: An Introduction
to the Theory, John Wiley and Sons, Inc., New York,
N.Y., 1965.
6. R. C. Dorf, "System sensitivity in the timedomain,"
Proc. Third Allerton Conference on Circuit and System
Theory, University of Illinois, pp. 4662, October
7. R. Tomovic, Sensitivity Analysis of Dynamic Systems,
McGrawHill Book Co., Inc., New York, N.Y., 1963.
CHAPTER 4
VARIABLE INCREMENT OPTIMAL SAMPLING
Introduction
The purpose of this work is to develop a method for
optimal adjustment of the sampling intervals of a sampled
data system. The object of adjustment is to obtain a low
sampling rate, subject to a performance criterion that re
flects system fidelity. This is appealing for such appli
cations as timesharing of a digital computer, control of
discrete systems in which energy is to be conserved, and
selection of a set of optimal sampling instants.
Adjustable sampling has been investigated by several
approaches. Dorf, et al., [1] used the absolute value of
the derivative of the error signal in Type I and Type II
unity feedback systems to adjust the sampling rate. Gupta
[2] used the ratio of the first and second derivative of
the error signal to determine the sampling intervals. More
recently, Tomovic and Bekey [3] have used amplitude sensi
tivity to adjust sampling rate. The algorithm is based on
the sensitivity of the system output to changes in output
of a zeroorder data hold for the error signal. They have
also used the sensitivity of the system output to changes
in sampling interval to adjust the sampling rate [4]. The
related problem of quantization error in hybrid systems has
also been studied [5].
In this work, performance criteria for sampling inter
val adjustment will be selected, and error and state
variable sampling' interval sensitivity functions defined.
Also included will be sampling interval formulas that in
corporate sensitivity and the performance criteria to
determine sampling intervals that maintain desired system
fidelity.
Performance Criteria for Variable Increment Sampling
The performance criteria for sampling interval selec
tion should reflect the effects of sampling interval
variations on the discrete system state variables. The
criterion should also include the type of data reconstruc
tion, since this can affect the fidelity of the response.
Consider first the effects of sampling interval
variations on the discrete system state variables at the
sampling instants. If the discrete model of a continuous
system is exact, then the values of the state variables
are correct for any sampling interval size. However, if
the discrete model is not exact, then the error between
the continuous and discrete state variables at the sam
pling instants will, in general, depend upon the sampling
intervals. The performance criteria should include this
modeling error. This can be done analytically for linear
systems. For nonlinear systems, a combination of analytical
and experimental methods must be used.
When the reconstructed response must meet certain
specifications, then the error of the data reconstruction
device must also be included. For a linear continuous
system with an exact discrete model, the only error in
the response is that of the reconstruction device. If an
approximate discrete model is used, then both types of
errors are present and must be considered. Since the data
reconstruction device operates over a full sampling inter
val, a suitable criterion would be the magnitude or the
mean or integralsquare value of the difference between
the continuous and the reconstructed response over each
sampling interval. If the exact solution is unknown, the
response for very small sampling intervals could be used
for the comparison. The exact solution will, of course,
seldom be known in an online application.
Sampling Interval Sensitivity
In order to select the sampling interval, Tk_1
tktk_, so that x(tk) satisfies a sensitivity performance
criterion, the effect of variations in Tkl on x(tk) must
be known. This can be determined from local sampling
interval sensitivity. If an approximate model is used,
it would also be helpful to know the effects of variations
in Tk1 on the error between continuous and discrete state
variables. This section will be devoted to an investigation
of these effects.
Local state variable sampling interval sensitivity
The sensitivity of x(tk) to variations in Tk_ is
given by the local sampling interval sensitivity function.
Variations in Tk1 = tkkl are obtained by perturbing
tk while tk1 is held constant. A definition for local
sampling interval sensitivity can be stated,
v(k) ,(t (1)
tk
where v(O) = 0.
For the system represented by the equation
x(tk) = f[x(tkl),u(tk ,tkl'tk1, (2)
the sensitivity, v(k), can be obtained by differentiating
with respect to tk,
v(k) = af(x(tkl),u(tkl),tkltk1 (3)
tk
If the system is linear, sensitivity can be calculated
from the equation
v(k) A(tkl,tk) xB(tk,tk)
v(k) x(tkl) + u(tk_l) (4)
It should be noted that local sampling interval sensi
tivity gives the effect on the state vector x(tk) of
changing only Tk_. It is not directly related to modeling
error at sampling instants nor reconstruction error between
sampling instants. Unlike parameter sensitivity, it is not
desirable to have sampling interval sensitivity approach or
equal zero, for this would mean that the state vector never
changes.
Local error sampling interval sensitivity
If the discrete model is approximate, it would be
desirable to base sampling interval selection on some
function of the error of the approximation. In order to
determine sampling intervals that maintain certain error
criteria, the sensitivity of the error to variations in
sampling intervals is needed. This can be determined
analytically for a linear system, using the exact discrete
model and the discrete approximation used to obtain the
approximate model.
As an example, consider the effect of using the
rectangular rule to derive the discrete model of the
linear continuous system,
x(t) = A(t)x(t) + B(t)u(t), (5)
where x(0) = xo. If the rectangular rule for integration
is used, an approximate discrete model for the system is,
k(tk) = [I + TkiA(tk,tkl)]t(tkl)
+ Tk,lB(tk,tk.k)u(tk1),
where Tk_ tktk1 and I is the identity matrix.
An exact discrete model can be determined by dis
cretizing the state transition equation,
x(t) = Q(t,to)x(to) + t Q(t,n)B(7)u(?)dn. (7)
For the interval Tk_l, equation (7) becomes,
t k
x(tk) = 0(tk,tkl)x(tkl) + it k (tk,7)B())u(77)dl.
k1 (8)
Equation (8) can be further simplified if u(t) and B(t)
are constant over the interval Tkl. This implies that
a zeroorder sample and hold is being used at the input,
and that B(t) is constant during sampling intervals. An
additional simplification can be made by defining,
tkl
(tk,tk1) = t k tk,l)dy. (9)
k1
Equation (8) can now be written,
x(tk) = S(tk,tk_l)(tkl)
+'((tk,tkl)B(ktk,tkl)u(tkl). (10)
The error, x(tk), in x(tk), due to the approximate
discrete model, is the difference,
(tk) x(tk) !(tk). (11)
Using equations (6) and (10), an equation for the error
is,
(tk) = [Q(tk,tk1) I TkiA(tk,tkl)T](tkkl)
+ Q(tk,tk1){(tkl)
+ [2(tk,tk_) Tki]B(tktkl)(tk). (12)
By varying tk with tk_1 constant, the sampling interval
Tkl is changed,and the sensitivity of the error in x(tk),
due to variations in the sampling interval, can be defined
as,
(tk) (tk (13)
S atk
At k = 0, the error, _(to), and the error sensitivity,
v(0), are zero since the initial conditions are identical
for both the exact and the approximate discrete models.
Equation (12) can be used to determine the equation for
the error sampling interval sensitivity,
= 1 (tk'tkl) A t(tktk1 (tkl)
TkI t + t (tkl)
+ (1tk,_tl atk (tkl)
a(tkstk1) I B(tk,tkl)u(tkl). (14)
L atk J
If the allowable error, 5(tk), is specified, equation
(12) could be used to determine the sampling intervals
required to keep x(tk) within tolerance. However, in this
work, sampling interval equations will be based on the
error sampling interval sensitivity of equation (14). It
should be noted that the error formulas were derived for an
approximate discrete model based on the rectangular inte
gration formula. Similar formulas can be derived for other
approximations.
Since it is not always possible to determine an exact
discrete model, the analysis of the effects of an approxi
mate discrete model must often be determined by experi
mental means.
Variable Increment Sampling Based on Sensitivity
The accuracy of the response of certain components of
the state vector might be more important than others, and
sampling interval calculation should reflect these require
ments. One way to do this is to weigh the components of
the sensitivity vector. Each component of the local sam
pling interval sensitivity vector, v(k), is the sensitivity
of the corresponding component in the state vector to
changes in sampling interval. Another way of incorporating
different performance requirements would be to specify a
performance criterion for each element of the state vector.
For each interval, a sampling increment could be calculated
from each element of the sensitivity vector and the
corresponding performance criterion. The desired interval
would be selected from the resulting set.
Sampling interval calculation and selection could
possibly be simplified somewhat by using the quadratic
form, v Qv, for sampling interval calculations. There
are several advantages. Only one sampling interval would
be calculated for each sampling period, and there would be
no need to select an interval from a set of n sampling
intervals. The different performance requirements on the
elements of the state vector can be maintained by selecting
the elements of the matrix Q to give different weighting to
the elements of the sensitivity vector.
Linear system simulation with an exact discrete model
Consider the simulation of a linear continuous system
with an exact discrete model derived from the state transi
tion equation. The error due to discretization of the in
put will be neglected. Under these conditions, the values
of the discrete system state variables at the sampling
instants are identical to the response of the continuous
system. The only error in the response of the discrete
model is between sampling instants,and it depends upon the
sampling interval and the type of data reconstruction used.
In this section, formulas that use sampling interval
sensitivity to calculate sampling intervals will be derived.
The sampling interval formulas will incorporate performance
criteria that constrain the difference between the con
tinuous system response and reconstructed response of the
discrete model.
To derive sampling interval formulas, consider first
zeroorder hold reconstruction. The zeroorder hold main
tains the value of the state at the preceding sampling
instant until a new value is obtained at the next sampling
instant. For an exact discrete model, the continuous and
sampled response for a zeroorder hold is shown in figure
41.
The derivation of sampling interval formulas will be
based on two approximations. First, it will be assumed
that the continuous response can be approximated by a
straightline over a single interval. This is shown in
figure 41. For this assumption, the error of zeroorder
hold data reconstruction is approximated by the shaded
area of figure 41. The other assumption is that x(tk)
can be determined from the first two terms of a Taylor's
series,
x(tk) x(tkl) + [v(k)](tk tkl), (15)
where v(k) is the local sampling interval sensitivity.
In order to maintain the error within specified limits,
the performance criterion should be related to the error
triangle of figure 41. The maximum difference, the inte
gral of the difference, or the average of the difference
squared, could be used. Since all three are based on the
shaded triangle of figure 41, they are identical. If the
performance criterion is the maximum difference, then the
sampling interval Tkl should be selected so that,
[xi(tk) xi(tkl) mi ( = 1,..,n). (16)
The elements, mi, of the performance vector m will be
selected to maintain the desired accuracy of the corre
sponding state variable. For the approximations used in
this derivation, the integral of the difference is related
to the maximum difference by idi = 1/2Tklmi. The maximum
difference is related to the average of the difference
squared by adsi 1/3Tkl(mi)2
A sampling interval equation that maintains the in
equality of equation (16) can be derived with the use of
equation (15). The ith sampling interval equation for
Tk1 is,
mi
T(kl)i i (k) (17)
where Tk_1 must be selected from the set of n intervals.
If the minimum value is used,
Tk1 min T(kl)i (i = 1,***,n), (18)
then all elements of the state vector should be within the
limits of the performance criteria. Different performance
requirements on certain elements of the state vector can
be handled by using different values for mi or by weighting
the elements of the sensitivity vector vi(k).
If the quadratic form v (k)Qv(k) is used, a different
approach must be used to derive a sampling interval equa
tion. To investigate the use of the quadratic form to
IL /X~i(tk)
Xi (tki)
xi Xi (tk2)
xi (tk3)
k3 Tk2 kl
tk3 k2 tk tk
Time
Figure 41 ZeroOrder Data Reconstruction
Xi (tk)
i(tkl)
xi xi (tk2)
k3 k2 Tk
k3 tk2 tk tk
Time
Figure 42 FirstOrder Data Reconstruction
determine sampling intervals, consider a secondorder system
and let Q be a diagonal matrix with diagonal elements qll
and q22. For these conditions, the quadratic form is
vT(k)Qv(k) = q11v2(k) + q22vi(k). (19)
If the approximation of equation (15) is used, a sampling
interval equation can be derived
T [qlll[x(tk) x(tk1)]2
k 1 L T(k)Qv(k)
1/2
Sq22[x2(tk) x2(tk1 (20)
v (k)Qv(k) (
Once the desired difference between xi(tk) and xi(tkl) is
established, equation (20) can be used to calculate the
sampling interval. The use of the quadratic form for sam
pling interval calculations will be illustrated with an
example problem.
If a firstorder hold is used for data reconstruction,
the response during a given interval is the straightline
extension of the sampled values from the two preceding
sampling instants. This is shown in figure 42. The
reconstructed sampled response during the interval
(tk tk1) is given by the equation
xi(tk ) [xi(tk1) xi(tk2)] () + xi(tkl),
Lk2 
where 7 (tk tk_).
In deriving sampling interval equations for first
order hold reconstruction, the same approximations used
in the zeroorder hold case will be assumed. That is, the
continuous response will be approximated by a straightline
over a single interval. Also, it is assumed that x(tk) is
given by the approximation of equation (15). The difference
between the assumed continuous response and approximate
response is represented by the shaded triangle of figure
42. Based on these assumptions, the maximum difference
will occur when 7 = (tk tk_). Thus, maximum error
occurs when the reconstructed response of equation (21) is
[xi(tk1) xi(tk2]T (22)
k) = [ Tk jTk + xi(tk1). (22)
The value of the maximum difference for the interval
(tk tk1) is xi(tk) xi(t).
If the performance criterion is based on limiting
the maximum difference, a sampling interval formula can
be derived for the criterion,
xi(tk) xi(tk)1 mi (i = 1, .,n). (23)
Using equations (15),(22) and (23), the sampling interval
is
m.
k2
k2
where i = l,...,n. From the approximation for the x(tk)
in equation (15), the sensitivity is
xi(tkI) xi(tk_2)
vi(kl) 1 (25)
Tk2
and equation (20) can be changed to
mTkl) (i = 1,*,n). (26)
Tk i vi(k) vi(k1)
The actual interval Tkl is selected from the set on n
intervals calculated from equation (26).
The quadratic form v (k)Qv(k) can also be used to
determine the sampling interval for an exact discrete
model with firstorder data reconstruction. Consider a
secondorder system and use equations (15),(19) and (22).
The resulting sampling interval equation is
2 2
[Tk ll[xl(tk)l(tkl)] xl(tk)xl(tk_)] 2
k1 T
L v(k)TQ(k) v (kl)Qv(ki)
q22([x2(tk)x2(tk1)]2 [x2(tk)x2(tkl)]2} 1/2
+ I
v(k)TQv(k) v (kl)Qv(k1) (27)
(27)
Note that in order to use equation (27), both the maximum
difference between the sampled and the continuous response,
and the maximum difference of the state variables,over an
interval, must be specified. For this reason, it may not
be as useful as other formulations.
A more general approach for determining the re
constructed response of a discrete system would be to use
a fractionalorder hold [6]. For a fractionalorder hold,
the reconstructed response during the interval (tk tk_),
is given by the equation
xi(tk1 + 7) = (F) xi(tk1) xi(tk (n) + xi(tk)
Tk2
(28)
where 7 = (tk tki) and 0 5 F 1 If F = 0, equation
(28) is for a zeroorder hold,and for F = 1.0, equation
(28) is identical to the firstorder hold of equation (21).
If the fractionalorder hold is used, the sampling interval
equation is
mi
T (29)
(k1) Ivi(k) Fvi(kl)2
where i = 1,,n, and O F 1. If the quadratic form is
used, equation (27) will have the terms qll[xl(tk)
xl(tk1)]2 and q22[x2(tk) X2(tk1)2 divided by F2.
Linear system simulation with an approximate discrete model
In a previous section, it was pointed out that both
modeling and reconstruction errors are present if an
approximate model is being used. In this section, sampling
interval formulas that constrain these errors will be
derived.
First, consider only the error in the discrete system
variables at the sampling instants. In figure 43, the
response for the discrete model is shown along with the
x x i(t k)
X(tk1) xitk)
S(tk1)
tk1 tk
Figure 43 Approximate Model Response
Figure 44 Calculation Procedure
exact or approximate continuous response. A suitable
criterion would be to constrain the magnitude of the error
by the relationship,
[xi(tik) mi. (30)
For a linear system, an equation for the error sensitivity,
similar to equation (13), could be derived for the approxi
mate discrete system. Then the approximation,
ii(tk) x i(tkl) + Tkliv(k), (31)
could be used to derive a relationship for the sampling
intervals,
xi(tk) + Tkli(k) j mi (i = l,,n). (32)
The error sensitivity could also be used to derive sampling
interval formulas that constrain the change in error,
i(tk) xi(tkl) S mi. (33)
The resulting sampling interval formula is
T(kl) mi (i =l,..,n) (34)
i Vi(k)I
If the difference between the reconstructed response
and the exact or approximate continuous response is to be
constrained, the sampling interval formulas must include
both data reconstruction and modeling errors. With zero
order hold reconstruction, figure 43 indicates that a
suitable criterion for sample interval selection would be
the constraint
[xi (tk) xi(tk1)] + xi(tk) m. (35)
Substitution of the approximations of equations (15) and
(31) into equation (35) gives
I[vi(k) + Vi(k)]Tkl + xi(tk_l)I mi. (36)
If firstorder data reconstruction is used, the per
formance criterion takes the form of the constraint
[xi(t) xi(tk)] + xi(tk) 5 mi. (37)
Equations (15),(22) and (31) can be used to determine an
approximate sampling interval relationship,
vi(k) vi(kl) + i(k)]Tk1 + i(tk1) mi. (38)
If a fractionalorder hold is used, the relationship is
I[(i(k) FOi(kl) + Vi(k)]Tkl + 2i(tkl)I mi, (39)
where 0 F 1.0.
Nonlinear system simulation
If the continuous system is nonlinear, the discrete
form of the system will, in general, be an approximate
model. Therefore, both modeling and reconstruction errors
will be present. Both types of simulation errors have
already been discussed for linear system simulation. State
variable sampling interval sensitivity was used to derive
sampling interval formulas that constrained the reconstruc
tion error. The modeling error of linear system simulation,
with an approximate model, was limited by sampling intervals
derived from an error sampling interval sensitivity.
Since state variable sampling interval sensitivity is
also applicable for nonlinear systems, it can be used for
adjusting sampling intervals to constrain the reconstruction
error. However, error sensitivity is not readily available
for nonlinear systems, and, therefore, it cannot be used to
determine sampling intervalsthat constrain modeling error.
Instead, state variable sampling interval sensitivity will
be used to calculate sampling intervals. The resulting
modeling error will be controlled by adjusting parameters
of the sampling interval calculator. This is not as
appealing as the variable increment sampling technique
worked out for linear system simulation. However, the
sampling efficiency can be improved without seriously
impairing the accuracy of the simulation.
For nonlinear system simulation, with an approximate
discrete model, equation (29) can be used to calculate
sampling intervals. The values of m and F are then
adjusted experimentally to achieve the desired system
fidelity. The process will be illustrated with example
problems.
Calculation Procedure for System Simulation
The order of the calculation sequence is very impor
tant. Several problems were encountered in applying the
formulas that have been derived. In order to provide
additional insight into sample adjustment, these problems
and their solutions will be presented along with the cal
culation procedure of simulation.
The first problem encountered was with calculating
the interval between x(kl) and x(k). Before any of the
formulas for sampling interval can be used to calculate
Tkl, the sensitivities, v(k), and _(k), must be known.
However, v(k), the sensitivity of x(k) to changes in
Tkl, and V(k), the sensitivity of 2(k) to changes in
Tkl, are functions of tk, which depends on Tk_1. The
easiest way to avoid this problem is to use values from
the preceding interval to calculate v'(k) and V'(k). For
aA(tk_2 ,tkl) aB(tk_2,tZk_)
example, A(t and 2 are used in place
atk1 6tkl
OA(tkltk) 2B(tkl,tk)
of and in equation (4). This pro
atk atk
cedure was used in actual simulation and gave good results
for some example problems.
An improvement can be obtained by using the predicted
values of Tkl to calculate values for aA(tkltk) and
atk
bB(tkltk), and then use these values to recalculate
btk
v(k), v(k), and Tk_. This predictorcorrector scheme
could be repeated to further improve the accuracy of TkI
An additional refinement can be obtained by comparing the
prediction with the correction. A large difference would
indicate a need to reduce the sampling interval.
The order of calculation for simulation is as follows:
o Calculate v(k) and v(k) using values from
the previous interval.
Use sensitivity to calculate Tk_1 as out
lined in the preceding paragraphs.
Calculate x(k) with equation (2).
o Repeat the cycle for the next interval.
The method is not selfstarting, and the first interval
must be preset to start the simulation.
Another problem encountered in developing the vari
able sampling technique was brought about by sudden changes
in the input. When the system is operating in a state of
low sensitivity, the sampling intervals will be large. If
a sudden change in input occurred just after a large sam
pling interval had been selected, a considerable portion
of the ensuing transient response might be missed. This
can be avoided by monitoring the input and sampling immedi
ately after any change in input greater than a suitable
threshold. Figure 44 contains a flow chart for the cal
culation procedure.
System Analysis for Variable Increment Sampling
If the techniques developed in the preceding sections
are used to determine sampling rate, the analysis of the
resulting system is quite involved. A block diagram for
the complete system is shown in figure 45. Block A rep
resents the actual system. The sensitivities are deter
mined in block B. Block B would also represent the com
parison of the prediction and correction if this is
included. Block C calculates the sampling interval from
the sensitivities (predicted and corrected), and block D
limits the size of the sampling intervals. The limits on
the size of the sampling interval will be discussed further
in the next paragraph. Block E provides To to start sim
ulation, begins correction if the predictorcorrector
scheme is used, and initiates sampling if a sudden change
in input occurs.
In discussing limits on sampling interval size,
consider first the lower limit. Since the purpose of
varying the sampling increment is to increase sampling
efficiency, the sampling rate is only as high as is
necessary to obtain an acceptably performing system. The
minimum value of sampling interval is that sampling in
terval beyond which further reduction in size is not
considered worthwhile. Certainly, the highest sampling
frequency (smallest sampling interval) must be several
times the highest frequency in the input and output of
the system [6]. In actual practice, the minimum is
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determined by the system bandwidth and the required accu
racy. Therefore, the lower limit on sampling interval
cannot be set in general.
The maximum allowable sampling interval for a closed
loop system is usually determined by steadystate sta
bility requirements. However, if an approximate model is
being used, the accuracy of the approximation will require
that the maximum allowable sampling interval be somewhat
below the stability limit. Thus, the upper limit on sam
pling interval can be set only after careful study of the
particular system.
The introduction of variable increment sampling com
pletely changes the stability considerations for the
sampleddata system. In addition to the stability of
the system of block A, the sampling interval selection
loop stability (blocks B, C, D, and E) must also be
studied. In addition, there is certainly coupling be
tween loops.
The system stability (block A) can be studied by
opening the sampling interval selection loop and deter
mining limits on the allowable values of sampling interval
for stability. For a linear system, the allowable values
of sampling interval can be determined by rootlocus
techniques. However, a procedure based on the "second
method" of Lyapunov is more general, since it applies to
nonlinear systems. The use of the "secondmethod" to
determine asymptotic stability of difference equations
has been studied [7,8,9], and it has also been applied to
frequency and pulsewidth modulated sampling systems [2,
10,11].
The stability theorem can be stated [10] as follows:
a sufficient condition for asymptotic stability (in the
large) of the vector difference equation, x(k) = f[x(kl)},
is the existence of V(x), a scalar function of the state
variables such that:
1. V(0) = 0
2. V(x) b 0 when x / 0
3. V[x(k)]
4. V(x) is continuous in x
5. V(x) oo when xco,
where V(x) is a Lyapunov function for the system.
The stability of the sampling interval selection loop
is more difficult to determine. Probably the best approach
is computer simulation.
Variable Increment Sampling for Optimal Control
The application of variable increment optimal sam
pling to an optimal controller is very appealing. Sam
pling interval sensitivity could be used to determine
the sampling interval for both the system and the optimal
control computer. A system of this type is shown in
figure 45. The optimal control computer (block F) is
shown with dotted connections.
The scheme in figure 45 could possibly be improved
by determining the sampling interval sensitivity of both
the optimal controller and the system. The two sensi
tivities could then be used to determine a sampling interval
for the entire system. The resulting sampling interval
should be the "best" for both the system and the optimal
controller.
There are several ways to determine a sampling interval
from the two sensitivity vectors. One method is to use
the quadratic form vT(k)Qva(k), where v (k) is a n + m
a a a
column vector constructed from the sensitivity vectors
of the system and the controller. Another method is to
weigh the elements of the sensitivity vectors. Both methods
have already been discussed. The calculation procedure is
as follows:
Determine the sampling interval sensitivity
vectors. There are n components for the
system and m for the optimal controller.
o From the n + m sensitivity components, cal
culate a sampling interval.
Use the resulting sampling increment for
the next interval of both system and con
troller.
Repeat the procedure.
Although this procedure requires m additional sensi
tivity calculations, the possibility of obtaining the
"best" sampling increment for both system and controller
may justify the extra calculations.
Example Problems
The techniques discussed in the preceding sections
will be illustrated in the following example problems.
Problem 1
For the system shown in figure 46, the form of the
difference equation is,
x(k) = bx(kl) + cu(kl).
If the state transition equation is discretized, the exact
(except for the effect of discretizing the input) model
will have the coefficients
b = exp[a(tk tkl)],
c = l/a[l exp(a(tk tkl)3.
The input selected for checking the variable increment
technique was
u(t) 1.0, 0 t a 5.0 and 6.0 t t A 7.0,
u(t) = 0,
for all other values of t. This is shown with a dashed
line in figure 47. This particular input was selected
ZeroOrder Hold Plant
esT 1
u(t) s (s+a) x(t) x(kT)
Figure 46 FirstOrder System
1.0   u 
x .2x
x .2 v
V C'V
1 2 3 4 58 10
u .2I
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1.0
Figure 47 Response and Sensitivity
PredictorCorrector Error
Predictor Error
   
Error Criteri
b 2\
0 1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 48 Data Reconstruction Error: ZeroOrder Hold
since it can eliminate the error of discretizing the input
and, at the same time, provide a rather severe test of the
sampling adjustment process. The response, x,and the
local sampling interval sensitivity, v, for an initial
condition x(0) = 0, are shown in figure 47.
For this exact discrete model, the only error is that
of data reconstruction. The magnitude of the reconstruction
error of a zeroorder hold is shown in figure 48. Sampling
intervals were determined from equation (17) with m = 0.05.
The figure includes curves for both predictor and predictor
corrector calculating methods.
The reconstruction error for the predictor scheme is
closer to the desired value, 0.05, than that of the pre
dictorcorrector method. The reason for this can be seen
in figure 47. Sensitivity is a decreasing function, and
if data from the preceding interval are used, a smaller
sampling interval is calculated. If the sensitivity is
corrected, the sampling interval will be larger, and as
a result, the error is increased. If the sensitivity were
an increasing function, the predictorcorrector scheme
would give smaller intervals and the reconstruction error
would be reduced.
The large initial error shown in figure 48 is due to
the size of the first sampling interval. The first interval
is preset, and, therefore, the initial error can be con
trolled. In figure 48, the initial sampling interval was
selected to give a large initial error to show that the
error is reduced at the next sampling interval. The sudden
decreases in the error at 5, 6 and 7 seconds are due to the
small sampling intervals selected by the sudden change
provision. Also,note that the sign of the reconstruction
error changes at 5, 6 and 7 seconds.
Figure 49 contains a plot of sampling interval versus
time for several different values of reconstruction error
magnitude. The discontinuities at 5, 6 and 7 seconds are
due to sudden changes in the input.
Figure 410 presents the tradeoff between the number
of samples required to cover 10 seconds of the response and
the magnitude of the reconstruction error. The curve shown
is for predictor sampling. The number of samples required
is also strongly dependent on the nature of the input.
Since the main objective of variable increment sam
pling is to save computer time, it is necessary to compare
variable and fixed increment sampling for the same magni
tude of reconstruction error. For a zeroorder hold and
variable increment sampling, 130 samples were required to
cover 10 seconds of the response. The maximum magnitude
of the reconstruction error was 0.02635. In order to main
tain the same maximum error with fixed interval sampling,
476 samples would be required to cover 10 seconds. The
ratio of the number of samples is 0.273.
The reduction in the number of sampling intervals
does not give a complete evaluation. Each iteration of
variable increment sampling requires more computer time
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0 1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 49 Sampling Interval: ZeroOrder Hold
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0 .04
.02
0
0 40 80 120 160 200
Number of Intervals
Figure 410 ZeroOrder Reconstruction Error vs. Number of
Intervals
than an iteration for fixed interval sampling. For this
particular system, the ratio of calculation time is
approximately 3.11. Thus, a more accurate figure of merit,
for the first 10 seconds, would be the product of the two
ratios, 0.273 x 3.11 0.848. If the same input is used,
and the first 20.0 seconds considered, variable increment
sampling is even more appealing. The combined figure of
merit drops to 0.438.
The effect of varying F in fractionalorder hold
reconstruction is shown in figure 411. For values of F
greater than 0.5, the sampling intervals were alternately
large and small and the results are not useful.
The system was simulated with an approximate model
based on the rectangular rule for integration. The
difference equation for the system is
x(k) = [la(tktkl)]x(kl) + (tktk_l)u(k1). (45)
The sampling interval sensitivity equation is
v(k) = ax(kl) + u(k). (46)
The difference equation for the error is
K(k) = [exp[a(7)} 1 + a(n)]x(k1)
+ exp[a())1x(kl)
+ [l/a(lexp[a(?)1) 7]u(k1),
11
0
3
F=0.3 F=0 F0.3
S F=0.55 F=0.5
o
0
0 1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 411 Reconstruction Error: FractionalOrder Hold
S3
S1
0 2 1 2 3 4  7 S 
TieinScod
0 1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 412 Modeling Error of Approximate Model
and the error sampling interval sensitivity is
V(k) = a[1exp[a())]3X(k1)
a[exp[a(7))]x(kl)
+ [exp[a(n)) l]u(kl), (48)
where 7 = tk tk_.
The error sensitivity of equation (48) was used in
equation (34) to calculate sampling intervals for the
approximate model. The magnitude of the resulting modeling
error is shown in figure 412 for m = 0.02. Figure 414
contains a plot of sampling interval size versus time.
Sampling interval size is varied by 20 to 1 to keep the
magnitude of the modeling error within the desired level,
0.02.
In figure 413, the magnitude of the algebraic sum
of the reconstruction and modeling errorsis shown for two
values of F, the fraction of the datahold circuit. Sam
pling intervals were determined from equation (39). The
discontinuities at t = 5, 6 and 7 seconds are due to sudden
changes in the input.
The corresponding curves for sampling interval size
are shown in figure 414. It is interesting to compare
the F = 0 sampling interval curve of figure 414 with the
m = 0.02 curve of figure 49. If the modeling and re
construction errors are of opposite sign, the sampling
interval curve of figure 414 gives a larger sampling
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1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 413 Reconstruction and Modeling Error
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/ / Error Magnitude Set
/ / /
/ at 0.02 /
// /
F=. / I F=
F0
0 1 2 3 4 5 6 7 8 9 10
Time in Seconds
Figure 414 Sampling Interval: Approximate Model
88
interval than the curve of figure 49. When both errors
are of the same sign, the curve of 414 gives lower values
of sampling interval.
Problem 2
Variable increment sampling has also been applied to
the system of figure 415. The exact difference equation
for the system is
x(k) = Ax(kl) + bu(kl), (49)
where the components of A are,
all = exp(3T)[cos(4T) + .75sin(4T)] (50)
12 = (25/24)exp(3T)sin(4T) (51)
a21 = 1.5exp(3T)sin(4T) (52)
a22 = exp(3T)[cos(4T) .75sin(4T)]. (53)
The elements of the vector b are
bl = 0.25[4.0 exp(3T)[3sin(4T) + 4cos(4T)}] (54)
and
b2 = 1.5exp(3T)sin(4T). (55)
The local sampling interval equation is
v(k) = Cx(kl) + du(k1), (56)
where the components of C are,
Cll = exp(3T)[6.25sin(4T)] (5;
c12 = exp(3T)[(25/6)cos(4T) (25/8)sin(4T)] (5i
c21 = exp(3T)[4.5sin(4T) 6cos(4T)] (5i
c22 = exp(3T)[6cos(4T) + 1.75sin(4T)]. (6(
The elements of d are
dI = 6.25exp(3T)sin(4T) (61
and
d2 = exp(3T)[6cos(4T) 4.5sin(4T)]. (6;
Variable increment sampling was investigated for the
input
u(t) = 1.0 0 t1.0 and 1.4 M t l1.6,
u(t) = 0,
for all other values of t. The input is shown with dashed
lines in figure 416. This input is similar to the input
used in problem 1. Figure 416 also contains the response
curves,x1 and x2,for the initial conditions xl(0) = 0
and x2(0) = 0. The sensitivities, vl and v2, are shown
in figure 417.
(t) x (kT)
__/1I
Figure 415 SecondOrder System
.5
Time in Seconds
Figure 416 Input and Response
