Table Of ContentPergamon
Unified Engineering
Series
GENERAL EDITORS
Thomas F. Irvine, Jr.
State University of New York at Stony Brook
James P. Hartnett
University of Illinois at Chicago Circle
EDITORS
William F. Hughes
Carnegie-Mellon University
Arthur T. Murphy
Widener College
William H. Davenport
Harvey Mudd College
Daniel Rosenthal
University of California, Los Angeles
SECTIONS
Continuous Media Section
Engineering Design Section
Engineering Systems Section
Humanities and Social Sciences Section
Information Dynamics Section
Materials Engineering Section
Engineering Laboratory Section
Modem Control
Engineering
Maxwell Noton
University of Waterloo,
Ontario, Canada
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Preface
At the time of writing, control theory as taught at the undergraduate level
of most North American universities is dominated by the study of linear
feedback systems, usually single-loop. Such theory was well developed
by the 1950's, including the extension to sampled-data systems; it is
appropriate and usually adequate for the analysis and synthesis of
servomechanisms and elementary regulators. Emphasis is placed on
stability either by frequency response methods or by the root locus for
the time domain. The continued inclusion of such material in under-
graduate courses is assured, if for no other reason than the importance
of the Principle of Negative Feedback in applied science.
However, the limitations of classical control theory are now recog-
nized, for example the synthesis of nonlinear control systems, multi-
variable systems (multi-input and multi-output), the recognition of other
performance criteria such as profit, etc. Since 1960 there have been two
developments which together have changed dramatically the situation
with respect to the control of complex industrial processes, aerospace
and socio-economic systems. On the one hand digital computers have
come into widespread use, both for general purpose scientific computing
and as on-line computers. Digital computers have acted as a catalyst
for the second development, namely modern control theory which is
usually assumed now to include the related areas of system identification
and parameter estimation. It started with the application of Bellman's
dynamic programming, a revival of the classical calculus of variations
and some of Kalman's early work on linear systems using the state space
formulation. The developments since 1960 have been almost explosive
ix
x Preface
and their full impact on applications is far from realized. Modern control
theory is therefore likely to be introduced at progressively earlier stages
in applied science curricula.
This book has been written primarily for use as a first level graduate
course, although it seems possible that some of the material may appear
in the future in senior elective courses. Given that the student has
adequate preparation in matrix algebra, numerical methods and an
introduction to state variable characterization, then the book can be
used for a one-semester course by excluding say Chapter 5 on stochastic
control and estimation. On the other hand, for those students or readers
who need revision or re-enforcement in the above prerequisites, an
optional supplement has been provided. The inclusion of this preparatory
material plus Chapter 5 would then correspond more to a two-semester
course. By providing readers with this option it is hoped that the book
will also be used by engineers for self-study as its predecessor (written
in 1964) was used, largely in the U.K. and Western Europe. The present
text was incidentally initially conceived as a revision of "Introduction
to Variational Methods in Control Engineering,, but in fact little more
than five per cent of the original material remains.
Except for some results quoted from the earlier text, most of the
illustrative computing results were obtained by Susan Oates using the
University of Waterloo, IBM Computer, Systems 360-Model 75. The
"Powell-Zangwiir computer program was however provided by the
author's colleague Professor J. Vlach. The original manuscript (written
in 1970) was corrected, modified and amplified after using in a tutorial
manner with several graduate students. However the author is especially
indebted to Dr. W. A. Brown of Monash University (Australia) for his
careful and helpful review. Finally the author acknowledges the pains-
taking efforts of Margaret Adlington in typing most of the manuscript.
Waterloo, Ontario M.NOTON
1
State Representation
of Dynamical Systems
1.1 STATE EQUATIONS
A cornerstone of control theory, as developed since about I960, has
been the characterization of dynamical systems by means of state equa-
tions instead of transfer functions or frequency response functions.
Whereas the latter are of limited applicability to nonlinear systems, state
equations and state variables are equally appropriate to linear or non-
linear systems. They will be immediately recognizable to the student as
no more than the dependent variables of differential equations which
define the behaviour of a dynamical system, at least if those equations
are written in a certain form.
As an example consider the motion of a rocket near the earth but free
of atmospheric forces. Assume that it has moved eastbound through an
angle of longitude Θ and is at radius r with reference to the centre of the
earth. Then the motion is governed by the following differential equations:
r = S sin ß + r(Ô + Ω)2 - g(rjr)2 ( l. I )
rè = Scosß- 2r(6 + Ω) ( l .2)
S is the dimensionless thrust of a rocket motor at angle β to the hori-
zontal, r the radius of the earth, g the acceleration due to gravity at the
0
surface and Ω the angular velocity of the earth. The differential equations
are nonlinear and present some difficulty for analytic solution. If they
were to be integrated numerically, say by means of a digital computer,
then the student is doubtless aware that some rearrangement is neces-
sary, to express the two second order equations as four first order
1
2 State Representation of Dynamical Systems
equations. Let us make certain substitutions in anticipation of future
needs. Put χ = r, x = 0, x = r and x = Θ yielding
λ 2 3 4
dx,
-^ = Ss'mß + x (x + Ω)2 - g(r /x )2
3 2 0 3
dx
^ = (Slx ) cos ß - 2x (x + Cl)lx
3 x 2 B
(1.3)
dx
3
dx
4
= x
~dt 2
The equations could now be integrated numerically by a standard com-
puter subroutine given the initial conditions on (x x , * , x ) and the
u 2 3 4
thrust angle β as a function of time. Equations (1.3) constitute the state
equations of the system; x x , x and x are the state variables or the com-
u 2 3 4
ponents of the state vectort x and, assuming that in this example the
rocket thrust angle β can be adjusted to vary the trajectory, β is the
control variable.
The importance of the state vector is that, in the case of a deterministic
system free of all unpredictable random effects, all future states are com-
pletely determined by an initial state and inputs to the system such as
known disturbances or control variables. By means of an example let
us examine the relationship between the transfer function of a linear
system and the state equations. An inductor L, resistor R and capacitor
C are connected in series and excited by a voltage source u(t) with all
initial conditions equal to zero. If the charge on the capacitor is q it is
elementary that summation of voltages around the circuit leads to
Lq + Rq + qlC = u (1.4)
Substitute q = Cv for the voltage v across the capacitor and, by taking
Laplace Transforms, we obtain the transfer function, i.e. the ratio of
the Laplace Transform of υ to the Laplace Transform of u. Thus
v(s)lu(s) =1/(1+ RCs + LCs2) ( 1.5)
Alternatively, in Eq. ( 1.4) with the substitution q = Cv
LCv + RCb + v = u (1.6)
tConsistent with modern practice, vectors are not underscored or indicated by bold
type. The reader should be ready to interpret any unsubscripted lower case letter as a
vector and an upper case letter as a matrix, depending on the text.
Linear State Equations 3
put JCJ = v and x = v and Eq. (1.6) can be expressed as two first order
2
differential equations, i.e. two state equations
dxjdt = x
2
(1.7)
dxjdt =(u-x - RCx )l(LC)
1 2
Unfortunately the definition of the state variables was not unique. Thus,
if we put Xi = v but x = RCv + v then, after a little manipulation we
2
obtain different state equations, namely
dxJdt=(x -x )l(RC)
2 1
(1.8)
dx ldt = -{\IRC)x + (\IRC-RlL)x + {RlL)a
2 x 2
The above example with a linear system has illustrated that, whereas the
transfer function between two variables is unique, the choice of state
variables and consequent form of the state equations is not.
At this stage some authors may take the reader through several
examples of deriving state equations from transfer functions but such
an exercise is deliberately avoided here, apart from noting that it is not a
unique procedure. The differential equations of a system are more funda-
mental than transfer functions and are of course equally applicable to
linear and nonlinear cases. Given the differential equations we proceed
along one route (transfer functions) for the application of classical con-
trol theory or via another route (state equations) anticipating modern
control theory.
1.2 LINEAR STATE EQUATIONS
Referring again to the above example of the elementary electric circuit,
put L, C and R all equal to one unit. Then Eq. (1.7) in matrix form
becomes
sen-? -:][»
(1.9)
It is a special case of the general form for linear systems, viz.
x = A(t)x + B(t)u (1.10)
where x is an «-component vector
u is an m-component vector
A is an n X n matrix
B is an n x m matrix.
4 State Representation of Dynamical Systems
The (t) after A and B in Eq. (1.10) is a reminder that the coefficients of
the state variables may be functions of time.
Many important and powerful results can be deduced for linear sys-
tems, in contrast to the difficulties encountered with nonlinear systems,
and the remainder of this chapter is concerned with certain relevant
results of linear systems theory. Due to the tractability of linear systems,
control algorithms are often derived to apply to linearized systems, for
small perturbations about some nominal state. In fact, further considera-
tion of the first example in Section 1.1 serves to illustrate how study of an
equation of the type Eq. (1.10) may arise even though the state Eqs. (1.3)
are nonlinear.
Suppose it can be assumed that the rocket has been launched on
approximately the desired trajectory and it is planned to vary ß only
slightly about a previously computed schedule in time ß(t). Given the
initial conditions, Eqs. (1.3) are integrated with ß = ß(t) to generate the
"standard trajectory" Jc(/) where x = (*!, JC, Jc, JC). Introduce
2 3 4
Δ*, = *,-*, /= 1,2,3,4 (1.11)
where Ax(t) is considered as a small perturbation, and take a first order
{
expansion of both sides of Eqs. ( 1.3) about x(t) and ß(t), remembering that
those equations are satisfied exactly by the latter. We obtain
Ax = S cos ß . Aß + (x + Ci)2Ax + 2x (x + ü)Ax
dt 1 2 3 3 2 2
+ 2g(r 2lx./)Ax,
{)
-j-Ax = -(S/x ) sinß .Aß-(S/x 2) cosß . Δχ -2^0*2+ Ω)/*
t 2 3 3 3 3
(1.12)
— 2χΑχ Ιχ + 2χ (χ + Ω)Αχ /χ 2
λ 2 3 χ 2 3 3
■j Ax = Αχ,
t 3
Αχ = Αχ
dt 4 2
which in matrix notation becomes
0 α 0 b7\
λ «2
Ax = "3 a4 «5 0 Ax + b2\ Δ/3 (1.13)
dt 1 0 0 0 0
0 1 0 0 oj
Fundamental Matrices 5
where
α = 2x (x + Ω), b = S cos β
λ s 2 i
a = (x + Ω)2 + 2gr 2lx-\ b = - (S/x ) sin β
2 2 0 s 2 3
a = -2(x + il)/x (1.14)
3 2 3
a = - 2xjx
4 3
a = 2x (x2 + n)lxs2- (Slx 2) cos/3
5 1 3
Thus perturbations about the standard trajectory satisfy an equation of
the form Eq. (1.10), where A and B are functions of time evaluated on the
standard trajectory.
1.3 FUNDAMENTAL MATRICES
We consider first the homogeneous part of Eq. ( l. 10)
x = A(t)x (1.15)
where x is «-component vector. It can be shown(ll) that if the real
elements of A(t) are bounded in the interval [t , t] then a unique solution
0 f
always exists, dependent on the initial state vector x(t ). Furthermore, by
0
using different initial vectors x(t ), we can generate n and no more than n
0
linearly independent solutions. In other words, if jc(/) (/ = 1,2,...,«) are
f
n such solutions and x +i(t) is any other solution for some different initial
n
vector, then real scalars #,·(/= 1, 2,...,«+ 1) can be found such that
ΰΛ(0 + ΰΛ(0 + ··· + Α,ι+Λι+ι(0=0 (1.16)
for all times / in the closed interval [i , t],
0 f
Let us take any set of n linearly independent solution column vectors
and arrange them side by side to form a time dependent n X n matrix, thus
Φ(ί) = [*,(*) x(t) x(t)] Φ(ί ) = Ζ (1.17)
2 n 0
Φ(/) is called a fundamental matrix, corresponding to the initial condition
matrix Z which is obtained by putting the initial vectors Xi(t ) (i = 1,2,...,
0
n) side by side. Now, because x(t) are solution vectors
t
x^AXi i= 1,2,...,« (1.18)
or
[xx ,i„] =A[x x ,. - · ,*,,] (1.19)
x 2 u 2
Combination of Eqs. (1.17) and (1.19) leads to the important result
Φ(ί) = A(t) Φ(ί) (1.20)
