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4. Continuous Time Signals

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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 continuous-time 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 easier-to 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, black-box 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, so-called, time domain, and
    • the, so-called, 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 high-frequency gain
Biproper The system has equal number of poles and zeros; it is implementable, has a biproper inverse and has a feed-through term, i.e., a non-zero and finite high-frequency gain
Improper The system has more zeros than poles; it is not causal, cannot be implemented, has a strictly proper inverse and has infinite high-frequency 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
Cut-off 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
    • non-minimum 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
       
  • Time-domain 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 high-pass character.