Skip to main content
Engineering LibreTexts

6.2: System Representations

  • Page ID
    24263
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    There are two general representations of a dynamic model that we shall be interested in, namely behavioral and input-output description.

    6.2.1 Behavioral Models

    This a very general representation, which we have actually taken as the basis for our initial definition of a dynamic model. In this representation, the system is described as a collection of constraints on designated signals, \(w_{i}\). Any combination \(w(t)=\left[\begin{array}{lll}
    w_{1}(t), & \cdots & w_{\ell}(t)
    \end{array}\right]\) of signals that satisfies the constraints is a behavior of the model, \(w(t) \in \mathbb{B}\), where \(\mathbb{B}\) denotes the behavior. An example of such a representation is Example 6.1.

    Linearity

    We call a model linear if its behavior constitutes a vector space, i.e. if superposition applies:

    \[w_{a}(t), w_{b}(t) \in \mathbb{B} \Longrightarrow \alpha w_{a}(t)+\beta w_{b}(t) \in \mathbb{B}\ \tag{6.15}\]

    where \(\alpha\) and \(\beta\) are arbitrary scalars. Example 6.1 is evidently linear.

    Time- Invariance

    We call a model time-invariant (or translation-invariant, or shift-invariant) if every possible time shift of a behavior - in which each of the signals is shifted by the same amount - yields a behavior:

    \[w(t) \in \mathbb{B} \Longrightarrow \sigma_{\tau} w(t)=w(t-\tau) \in \mathbb{B}\ \tag{6.16}\]

    for all valid \(\tau\), i.e. \(\tau\) for which \(\mathbb{T}-\tau \subset \mathbb{T}\), with \(\sigma_{r}\) denoting the \(\tau\) -shift operator. Example 6.1 is evidently time-invariant.

    Memoryless Models

    A model is memoryless if the constraints that describe the associated signals \(w( \cdot )\) are purely algebraic, i.e., they only involve constraints on \(w(t_{0})\) for each \(t_{0} \in \mathbb{T}\) (and so do not involve derivatives, integrals, etc.). More interesting to us are non-memoryless, or dynamic systems, where the constraints involve signal values at different times.

    6.2.2 Input-Output Models

    For this class of models, the system is modeled as a mapping from a set of input signals \(u(t)\) to a set of output signals, \(y(t)\). We may represent this map as

    \[y(t)=(S u)(t)\ \tag{6.17}\]

    (i.e., the result of operating on the entire signal \(u( \cdot )\) with the mapping \(S\) yields the signal \(y( \cdot )\), and the particular value of the output at some time \(t\) is then denoted as above). The above mapping clearly also constitutes a constraint relating \(u(t)\) and \(y(t)\); this fact could be emphasized by trivially rewriting the equation in the form

    \[y(t)-(S u)(t)=0 \ \tag{6.18}\]

    The definitions of linearity, time-invariance and memorylessness from the behavioral case therefore specialize easily to mappings. An example of a system representation in the form of a mapping is Example 6.5.

    Linearity and Time-Invariance

    From the behavioral point of view, the signals of interest are given by \(w(t) = [u(t) y(t)]\). It then follows from the preceding discussion of behavioral models that the model is linear if and only if

    \[\left(S\left(\alpha u_{a}+\beta u_{b}\right)\right)(t)=\alpha y_{a}(t)+\beta y_{b}(t)=\alpha\left(S u_{a}\right)(t)+\beta\left(S u_{b}\right)(t)\ \tag{6.19}\]

    and the model is time-invariant if and only if

    \[\left(S \sigma_{\tau} u\right)(t)=\left(\sigma_{\tau} y\right)(t)=y(t-\tau)\ \tag{6.20}\]

    where \(\sigma_{r}\) is again the \(\tau\)-shift operator (so time-invariance of a mapping corresponds to requiring mapping to commute with the shift operator).

    Memoryless Models

    Again specializing the behavioral definition, we see that a mapping is memoryless if and only if \(y(t_{0})\) only depends on \(u(t_{0})\), for every \(t_{0} \in \mathbb{T}\):

    \[y\left(t_{0}\right)=(S u)\left(t_{0}\right)=f\left(u\left(t_{0}\right)\right)\ \tag{6.21}\]

    Causality

    We say the mapping is causal if the output does not depend on future values of the input. To describe causality conveniently in mathematical form, define the truncation operator \(P_{T}\) on a signal by the condition

    \[\left(P_{T} u\right)(t)=\left\{\begin{array}{ll}
    u(t) & \text { for } t \leq T \\
    0 & \text { for } t>T
    \end{array}\right.\ \tag{6.22}\]

    Thus, if \(u\) is a record of a function over all time, then \((P_{T} u)\) is a record of \(u\) up to time \(T\), trivially extended by 0. Then the system \(S\) is said to be causal if

    \[P_{T} S P_{T}=P_{T} S \ \tag{6.23}\]

    In other words, the output up to time \(T\) depends only on the input up to time \(T\).

    Example 6.6

    Example 6.5 shows a system represented as an input-output map. It is evident that the model is linear, translation-invariant, and not memoryless (unless \(h(x, y) = \delta (x, y)\)).

    Note

    For much more on the behavioral approach to modeling and analysis of dynamic systems, see

    J. C. Willems, "Paradigms and Puzzles in the Theory of Dynamic Systems," IEEE Transactions on Automatic Control, Vol. 36, pp. 259{294, March 1991.


    This page titled 6.2: System Representations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.