You are here : Control
System Design  Index  Book Contents 
Chapter 4
4. Continuous Time Signals
Preview
The advantage of being able to cast a modeling problem into the form
of a linear approximation is that subsequent analysis, as well as
controller design, can draw on the wealth of information that is
available about the operation of linear systems. In this chapter we will
cover the fundamentals of this theory for linear models of
continuoustime processes. Specific topics to be covered include:
 linear high order differential equation models
 Laplace transforms, which convert linear differential equations to
algebraic equations, thus greatly simplifying their study
 methods for assessing the stability of linear dynamic systems
 frequency response.
Summary
 Ultimately, all physical systems exhibit some form of nonlinearity.
 Nevertheless, there is a strong incentive to find a model that is
 linear
 yet sufficiently accurate for the purpose
Incentives include the following properties of linear models:
 theory and techniques are significantly simpler and more
tractable
 there are more closed form solutions as well as easierto use
software tools
 superposition holds; for example, the response to
simultaneously acting setpoint changes and disturbances is equal
to the sum of these signals acting individually
 a powerful body of frequency domain properties and results
holds: poles, zeros, transfer functions, Bode and Nyquist plots
with their associated properties and results
 relative degree, inversion, stability and inverse stability
are easier to define and check
 Means to approximate physical systems by linear models include
 transformations, such as change of variable
 approximations, such as Taylor series in a
neighborhood, blackbox identification, etc.
 These points motivate the basic presumption of linearity used in
the next few chapters; later chapters introduce techniques for
systems which are inherently nonlinear.
 There are two key approaches to linear dynamic models:
 the, socalled, time domain, and
 the, socalled, frequency domain.
 Although these two approaches are largely equivalent, they each
have their own particular advantages and it is therefore important
to have a good grasp of each.
 In the time domain,
 systems are modeled by differential equations
 systems are characterized by the evolution of their variables
(output etc) in time
 the evolution of variables in time is computed by solving
differential equations
 Stable time responses are typically characterized in terms of:
Characteristic

Measure of

Steady state gain 
How the system, after
transients, amplifies or attenuates a constant signal 
Rise time 
How fast the system
reacts to a change in its input 
Settling time 
How fast the system's
transient decays 
Overshoot 
How far the response
grows beyond its final value during transients 
Undershoot 
How far initial
transients grow into the opposite direction relative to the
final value 
 In the frequency domain,
 modeling exploits the key linear system property that the
steady state response to a sinusoid is again a sinusoid of the
same frequency; the system only changes amplitude and phase of
the input in a fashion uniquely determined by the system at that
frequency
 systems are modeled by transfer functions, which capture this
impact as a function of frequency
 In the frequency domain, systems are typically characterized by
Characteristic

Significance 
Frequency response plots 
Graphical representation
of a systems impact on amplitude and phase of a sinusoidal input
as a function of frequency. 
Poles 
The roots of the transfer
function denominator polynomial; they determine stability and,
together with the zeros, the transient characteristics. 
Zeros 
The roots of the transfer
function numerator polynomial; they do not impact on stability but
determine inverse stability, undershoot and, together with the
poles, have a profound impact on the system's transient
characteristics 
Relative degree 
Number of poles minus
number of zeros; determines whether a system is strictly proper,
biproper or improper 
Strictly proper 
The system has more poles
than zeros; it is causal and therefore implementable, it has an
improper inverse and zero highfrequency gain 
Biproper 
The system has equal
number of poles and zeros; it is implementable, has a biproper
inverse and has a feedthrough term, i.e., a nonzero and finite
highfrequency gain 
Improper 
The system has more zeros
than poles; it is not causal, cannot be implemented, has a
strictly proper inverse and has infinite highfrequency gain. 
 Terms used to characterize systems in the frequency domain include
Characteristic

Measure of

Pass band 
Frequency range where the
system has minimal impact on the amplitude of a sinusoidal input 
Stop band 
Frequency range where the
system essentially annihilates sinusoidal inputs 
Transition band 
Frequency range between a
system's pass and stop bands 
Bandwidth 
The frequency range of a
system's pass band 
Cutoff frequency 
A frequency signifying a
(somewhat arbitrary) border between a system's pass and
transition band 
 Particularly important linear models include
 gain
 first order model
 second order model
 integrator
 a pure time delay (irrational) and its rational approximation
 The importance of these models is due to
 them being frequently observed in practice
 more complex systems being decomposable into them by partial
fraction expansion
 Evaluating a transfer function at any one frequency yields a
characteristic complex number:
 its magnitude indicates the system's gain at that frequency
 its phase indicates the system's phase shift at that frequency
 With respect to the important characteristic of stability, a
continuous time system is
 stable if and only if the real parts of all poles are strictly
negative
 marginally stable if at least one pole is strictly imaginary and
no pole has strictly positive real part
 unstable if the real part of at least one pole is strictly
positive
 nonminimum phase if the real part of at least one zero is
strictly positive
 The response of linear systems to an arbitrary driving input can be
decomposed into the sum of two components:
 the natural response, which is a function of initial conditions,
but independent of the driving input; if the system is stable, the
natural response decays to zero
 the forced response, which is a function of the driving input,
but independent of initial conditions
 Equivalent ways of viewing transfer function models include
 the Laplace transform of a system's differential equation model
 the Laplace transform of the system’s forced response to an
impulse
 a model derived directly from experimental observation
 In principle, the time response of a transfer function can be
obtained by taking the inverse Laplace transform of the output;
however, in practice one almost always prefers to transform the
transfer function to the time domain and to solve the differential
equations numerically.
 Key strengths of time domain models include:
 they are particularly suitable for solution and simulation on a
digital computer
 they are extendible to more general classes of models, such as
nonlinear systems
 they play a fundamental role in state space theory, covered in
later chapters
 Key strengths of frequency domain models (transfer functions)
include:
 they can be manipulated by simple algebraic rules; thus,
transfer functions of parallel, series or feedback architectures
can be simply computed
 properties such as inversion, stability, inverse stability and
even a qualitative understanding of transients are easily inferred
from knowledge of the poles and zeros
 Timedomain and frequency domain models can be converted from one to
the other.
 All models contain modeling errors.
 Modeling errors can be described as an additive (AME) or
multiplicative (MME) quantity.
 Modeling errors are necessarily unknown and frequently described by
upper bounds.
 Certain types of commonly occurring modeling errors, such as
numerical inaccuracy, missing poles, inaccurate resonant peaks or time
delays, have certain finger prints.
 One can generally assume that modeling errors increase with
frequency, the MME typically possessing a highpass character.
