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Lecture 10: Perspective

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Schedule :: Perspective :: Transfer Functions :: Laplace Transform :: Poles and Eigenvalues :: Time Domain :: Linear Transformations :: Special Transformation :: Controllability :: Discrete-Time

Transfer functions provide only an input/output perspective of what is going on in a system. There may be things going on physically that do not appear in a transfer function, due to cancellations, etc. On the other hand, state-space analysis provides a more complete representation. Furthermore, it can be generalized to time-varying systems, multi- input or output systems, and in some applications leads to very explicit design formulations. There is also much that can be done with nonlinear systems in state variable form.

We have seen that we can describe an LTIC system using a single differential equation. In state-space analysis, we deal with systems of equations, but make it so that all equations are first order . Sometimes this requires introducing some extra variables. The variables appearing in these equations (with respect to which we differentiate) are called the state variables. The idea behind the name is this: for a first order differential equation, if we know where we are initially (the initial condition), then this provides all of the information we need to determine where to go.

In circuits, it is common to choose the voltage across the capacitors and the current through the inductors as state variables. This provides our first example. \begin{example}
Circuit. Two state variables. \vspace*{2.5in}
\par
KCL:
\begin{...
...n terms of the state variables. Try a
few. Express in matrix form.
\end{example}
Note: we have

  1. First order differential equations.
  2. Each equation is expressed in terms of the state variables and the input.
For linear equations, we can put the equations in matrix form. Let

\begin{displaymath}\xbf(t) = \begin{bmatrix}x_1(t) \ x_2(t) \end{bmatrix}.
\end{displaymath}

Let $\xbfdot$ denote taking the derivatives individually:
\begin{displaymath}\xbfdot(t) = \begin{bmatrix}\xdot_1(t) \ \xdot_2(t) \end{bmatrix}\end{displaymath}

In the example above, let
\begin{displaymath}A = \begin{bmatrix}-25 & -5 \ 1 & -2\end{bmatrix}\qquad
B = \begin{bmatrix}10 \ 0 \end{bmatrix}\end{displaymath}

Then we can write

\begin{displaymath}\xbfdot(t) = A \xbf(t) + B f(t).
\end{displaymath}

Note: For nonlinear systems, we can still put them in state variable form, even when we cannot use a matrix for the representation.
\begin{example}
\begin{displaymath}\begin{aligned}
\xdot_1 &= x_2^2 + x_3 \cos(...
...)^2 \\
\xdot_3 &= x_3 \tan(x_2/x_1)
\end{aligned}\end{displaymath}\end{example}

\begin{example}
(Important) Consider the 3rd order equation
\begin{displaymath}...
...d{displaymath}\begin{displaymath}y = \cbf^T \xbf.
\end{displaymath}\end{example} In general, a set of state-variable equations can be written

\begin{displaymath}\xdot_i = g_i(x_1,x_2,\ldots,x_n,f_1,f_2,\ldots,f_j),\qquad i=
1,2,\ldots,n
\end{displaymath}

\begin{displaymath}y_i = h_i(x_1,x_2,\ldots,x_n,f_1,f_2,\ldots,f_j), \qquad
i=1,2,\ldots, k
\end{displaymath}

Note that this could be
  • Nonlinear
  • Multiple inputs
  • Multiple outputs
(But may not be!) This represents a considerable degree of flexibility.

A general $j$ -input, $k$ -output linear system with $n$ state variables can be written as

\begin{displaymath}\xbfdot = \begin{bmatrix}\xdot_1 \ \vdots \ \xdot_n \end{bmatrix}
= A \xbfdot + B \fbf
\end{displaymath}


\begin{displaymath}\ybf = C \xbf + D \fbf
\end{displaymath}

where $A$ is $\matsize{n}{n}$ , $B$ is $\matsize{n}{j}$ , $C$ is $\matsize{k}{n}$ and $D$ is $\matsize{k}{j}$ . (Write out the matrices.)

Have a student work the circuit on the board.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 08). Lecture 10: Perspective. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/10_1node1.html. This work is licensed under a Creative Commons License Creative Commons License