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Linear State-Space Control Systems
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Linear State-Space Control Systems Robert L. Williams II Douglas A. Lawrence
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1
INTRODUCTION
This chapter introduces the state-space representation for linear time-
invariant systems. We begin with a brief overview of the origins of
state-space methods to provide a context for the focus of this book. Fol-
lowing that, we define the state equation format and provide examples to
show how state equations can be derived from physical system descrip-
tions and from transfer-function representations. In addition, we show
how linear state equations arise from the linearization of a nonlinear state
equation about a nominal trajectory or equilibrium condition.
This chapter also initiates our use of the
MATLAB software package
for computer-aided analysis and design of linear state-space control sys-
tems. Beginning here and continuing throughout the book, features of
MATLAB and the accompanying Control Systems Toolbox that support each
chapter’s subject matter will be presented and illustrated using a Continu-
ing
MATLAB Example. In addition, we introduce two Continuing Examples
that we also will revisit in subsequent chapters.
1.1 HISTORICAL PERSPECTIVE AND SCOPE
Any scholarly account of the history of control engineering would have
to span several millennia because there are many examples throughout
1
LinearState-SpaceControlSystems.RobertL.WilliamsIIandDouglasA.Lawrence
Copyright
2007JohnWiley&Sons,Inc. ISBN: 978-0-471-73555-7
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2 INTRODUCTION
ancient history, the industrial revolution, and into the early twentieth
century of ingeniously designed systems that employed feedback mech-
anisms in various forms. Ancient water clocks, south-pointing chariots,
Watt’s flyball governor for steam engine speed regulation, and mecha-
nisms for ship steering, gun pointing, and vacuum tube amplifier stabiliza-
tion are but a few. Here we are content to survey important developments
in the theory and practice of control engineering since the mid-1900s in
order to provide some perspective for the material that is the focus of this
book in relation to topics covered in most undergraduate controls courses
and in more advanced graduate-level courses.
In the so-called classical control era of the 1940s and 1950s, systems
were represented in the frequency domain by transfer functions. In addi-
tion, performance and robustness specifications were either cast directly in
or translated into the frequency domain. For example, transient response
specifications were converted into desired closed-loop pole locations or
desired open-loop and/or closed-loop frequency-response characteristics.
Analysis techniques involving Evans root locus plots, Bode plots, Nyquist
plots, and Nichol’s charts were limited primarily to single-input, single-
output systems, and compensation schemes were fairly simple, e.g., a
single feedback loop with cascade compensation. Moreover, the design
process was iterative, involving an initial design based on various sim-
plifying assumptions followed by parameter tuning on a trial-and-error
basis. Ultimately, the final design was not guaranteed to be optimal in
any sense.
The 1960s and 1970s witnessed a fundamental paradigm shift from the
frequency domain to the time domain. Systems were represented in the
time domain by a type of differential equation called a state equation.
Performance and robustness specifications also were specified in the time
domain, often in the form of a quadratic performance index. Key advan-
tages of the state-space approach were that a time-domain formulation
exploited the advances in digital computer technology and the analysis
and design methods were well-suited to multiple-input, multiple-output
systems. Moreover, feedback control laws were calculated using analytical
formulas, often directly optimizing a particular performance index.
The 1980’s and 1990’s were characterized by a merging of frequency-
domain and time-domain viewpoints. Specifically, frequency-domain per-
formance and robustness specifications once again were favored, coupled
with important theoretical breakthroughs that yielded tools for handling
multiple-input, multiple-output systems in the frequency domain. Further
advances yielded state-space time-domain techniques for controller syn-
thesis. In the end, the best features of the preceding decades were merged
into a powerful, unified framework.
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STATE EQUATIONS 3
The chronological development summarized in the preceding para-
graphs correlates with traditional controls textbooks and academic curric-
ula as follows. Classical control typically is the focus at the undergraduate
level, perhaps along with an introduction to state-space methods. An in-
depth exposure to the state-space approach then follows at the advanced
undergraduate/first-year graduate level and is the focus of this book. This,
in turn, serves as the foundation for more advanced treatments reflecting
recent developments in control theory, including those alluded to in the
preceding paragraph, as well as extensions to time-varying and nonlinear
systems.
We assume that the reader is familiar with the traditional undergrad-
uate treatment of linear systems that introduces basic system properties
such as system dimension, causality, linearity, and time invariance. This
book is concerned with the analysis, simulation, and control of finite-
dimensional, causal, linear, time-invariant, continuous-time dynamic sys-
tems using state-space techniques. From now on, we will refer to members
of this system class as linear time-invariant systems.
The techniques developed in this book are applicable to various types of
engineering (even nonengineering) systems, such as aerospace, mechani-
cal, electrical, electromechanical, fluid, thermal, biological, and economic
systems. This is so because such systems can be modeled mathematically
by the same types of governing equations. We do not formally address
the modeling issue in this book, and the point of departure is a linear
time-invariant state-equation model of the physical system under study.
With mathematics as the unifying language, the fundamental results and
methods presented here are amenable to translation into the application
domain of interest.
1.2 STATE EQUATIONS
A state-space representation for a linear time-invariant system has the
general form
˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t) +Du(t)
x(t
0
) = x
0
(1.1)
in which x(t) is the n-dimensional state vector
x(t) =
x
1
(t)
x
2
(t)
.
.
.
x
n
(t)
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4 INTRODUCTION
whose n scalar components are called state variables. Similarly, the
m-dimensional input vector and p-dimensional output vector are given,
respectively, as
u(t) =
u
1
(t)
u
2
(t)
.
.
.
u
m
(t)
y(t) =
y
1
(t)
y
2
(t)
.
.
.
y
p
(t)
Since differentiation with respect to time of a time-varying vector quan-
tity is performed component-wise, the time-derivative on the left-hand side
of Equation (1.1) represents
˙x(t) =
˙x
1
(t)
˙x
2
(t)
.
.
.
˙x
n
(t)
Finally, for a specified initial time t
0
,theinitial state x(t
0
) = x
0
is a
specified, constant n-dimensional vector.
The state vector x(t) is composed of a minimum set of system variables
that uniquely describes the future response of the system given the current
state, the input, and the dynamic equations. The input vector u(t) contains
variables used to actuate the system, the output vector y(t) contains the
measurable quantities, and the state vector x(t) contains internal system
variables.
Using the notational convention M = [m
ij
] to represent the matrix
whose element in the ith row and j th column is m
ij
, the coefficient
matrices in Equation (1.1) can be specified via
A = [a
ij
] B = [b
ij
] C = [c
ij
]
D = [d
ij
]
having dimensions n × n, n × m, p × n,andp × m, respectively. With
these definitions in place, we see that the state equation (1.1) is a compact
representation of n scalar first-order ordinary differential equations, that is,
˙x
i
(t) = a
i1
x
1
(t) + a
i2
x
2
(t) +···+a
in
x
n
(t)
+ b
i1
u
1
(t) + b
i2
u
2
(t) +···+b
im
u
m
(t)
for i = 1, 2,...,n, together with p scalar linear algebraic equations
y
j
(t) = c
j1
x
1
(t) + c
j2
x
2
(t) +···+c
jn
x
n
(t)
+ d
j1
u
1
(t) + d
j2
u
2
(t) +···+d
jm
u
m
(t)
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EXAMPLES 5
+
+
+
+
A
CB
D
u(t)
y(t)
x(t)
x(t)
x
0
∫
FIGURE 1.1 State-equation block diagram.
for j = 1, 2,...,p. From this point on the vector notation (1.1) will
be preferred over these scalar decompositions. The state-space descrip-
tion consists of the state differential equation ˙x(t) = Ax(t) + Bu(t) and
the algebraic output equation y(t) = Cx(t) + Du(t) from Equation (1.1).
Figure 1.1 shows the block diagram for the state-space representation of
general multiple-input, multiple-output linear time-invariant systems.
One motivation for the state-space formulation is to convert a cou-
pled system of higher-order ordinary differential equations, for example,
those representing the dynamics of a mechanical system, to a coupled
set of first-order differential equations. In the single-input, single-output
case, the state-space representation converts a single nth-order differen-
tial equation into a system of n coupled first-order differential equations.
In the multiple-input, multiple-output case, in which all equations are of
the same order n, one can convert the system of knth-order differential
equations into a system of kn coupled first-order differential equations.
1.3 EXAMPLES
In this section we present a series of examples that illustrate the construc-
tion of linear state equations. The first four examples begin with first-
principles modeling of physical systems. In each case we adopt the strat-
egy of associating state variables with the energy storage elements in the
system. This facilitates derivation of the required differential and algebraic
equations in the state-equation format. The last two examples begin with
transfer-function descriptions, hence establishing a link between transfer
functions and state equations that will be pursued in greater detail in later
chapters.
Example 1.1 Given the linear single-input, single-output, mass-spring-
damper translational mechanical system of Figure 1.2, we now derive the
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