create a zero-pole-gain system object - Maple Help

Home : Support : Online Help : Science and Engineering : Dynamic Systems : System Object : DynamicSystems/ZeroPoleGain

DynamicSystems[ZeroPoleGain] - create a zero-pole-gain system object

 Calling Sequence ZeroPoleGain(opts) ZeroPoleGain(sys, opts) ZeroPoleGain(tf, opts) ZeroPoleGain(z, p, k, opts) ZeroPoleGain(num, den, opts) ZeroPoleGain(a, b, c, d, opts) ZeroPoleGain(de, invars, outvars, opts)

Parameters

 sys - System; system object tf - algebraic or Matrix(algebraic); transfer function z - list(algebraic) or Matrix(list(algebraic)); zeros p - list(algebraic) or Matrix(list(algebraic)); poles k - algebraic or Matrix(algebraic); gain(s) num - list(algebraic) or Matrix (list(algebraic)); numerator coefficients den - list(algebraic) or Matrix (list(algebraic)); denominator coefficients a - Matrix; state-space matrix A b - Matrix; state-space matrix B c - Matrix; state-space matrix C d - Matrix; state-space matrix D de - equation or list(equation); diff-equations invars - name, anyfunc(name) or list of same; input variables outvars - name, anyfunc(name) or list of same; output variables opts - (optional) equation(s) of the form option = value; specify options for the ZeroPoleGain command

Description

 • The ZeroPoleGain command creates a zero-pole-gain (ZPK) system object. The frequency-domain behavior of the object is modeled by lists of zeros, poles, and gains of transfer functions.
 • The input can be specified as one of several representations: transfer function (TF), zero-pole-gain (ZPK), coefficients (Coeff), state-space (SS), or diff-equations (DE).
 • If no input is provided, a unity-gain ZPK system is created.
 • The optional parameter sys is a system object; it is converted to the ZPK representation. All options are ignored.
 • The optional parameter tf is the transfer function of a TF system. For a single-input/single-output system, tf is a rational function (ratpoly). For a multi-input/multi-output system, tf is a Matrix of rational functions. The indeterminate of the polynomials depends on whether the system is continuous or discrete; a continuous system typically uses s while a discrete system typically uses z as the indeterminate. The actual names are assigned by DynamicSystems[SystemOptions].
 • The optional parameters z, p, and k are the zeros, poles, and gain, respectively, of a ZPK system. For a single-input/single-output system, z and p are lists and k is an algebraic expression. For a multi-input/multi-output system, z and p are Matrices of lists and k is a Matrix of algebraic expressions.
 • The optional parameters num and den are the coefficients of the numerator and denominator, respectively, of a Coeff system. For a single-input/single-output system, num and den are lists, the first element being the coefficient of the highest order term. For a multi-input/multi-output system, num and den are Matrices of lists.
 • The optional parameters a, b, c, and d are the four state-space matrices, A, B, C, and D, respectively, of an SS system.
 • The optional parameter de is the difference/differential equation(s) of a DE system. A list is used to specify more than one equation.
 • The parameters invars and outvars specify the input and output variables of difference/differential equations. They are not required, but if either is not specified then the corresponding keyword parameter inputvariable or outputvariable must be assigned. If both positional and keyword parameters are specified, the keyword parameter take precedence.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{sys1}:=\mathrm{ZeroPoleGain}\left(\right):$
 > $\mathrm{PrintSystem}\left(\mathrm{sys1}\right)$
 $\left[\begin{array}{l}{\mathbf{Zero Pole Gain}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\\ {{\mathrm{z}}}_{{1}{,}{1}}{=}\left[\right]\\ {{\mathrm{p}}}_{{1}{,}{1}}{=}\left[\right]\\ {{\mathrm{k}}}_{{1}{,}{1}}{=}{1}\end{array}\right$ (1)
 > $\mathrm{sys2}:=\mathrm{ZeroPoleGain}\left(\frac{s}{{s}^{3}+5{s}^{2}+7s+6}\right):$
 > $\mathrm{PrintSystem}\left(\mathrm{sys2}\right)$
 $\left[\begin{array}{l}{\mathbf{Zero Pole Gain}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\\ {{\mathrm{z}}}_{{1}{,}{1}}{=}\left[{0}\right]\\ {{\mathrm{p}}}_{{1}{,}{1}}{=}{{}\begin{array}{l}{[}{-}\frac{{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{{6}}{-}\frac{{8}}{{3}{}{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{-}\frac{{5}}{{3}}{,}\\ {}\frac{{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{{12}}{+}\frac{{4}}{{3}{}{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{-}\frac{{5}}{{3}}\\ {}{+}\frac{{I}}{{2}}{}\sqrt{{3}}{}\left({-}\frac{{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{{6}}{+}\frac{{8}}{{3}{}{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}\right){,}\\ {}\frac{{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{{12}}{+}\frac{{4}}{{3}{}{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{-}\frac{{5}}{{3}}\\ {}{-}\frac{{I}}{{2}}{}\sqrt{{3}}{}\left({-}\frac{{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}{{6}}{+}\frac{{8}}{{3}{}{\left({388}{+}{36}{}\sqrt{{113}}\right)}^{{1}}{{3}}}}\right){]}\end{array}\\ {{\mathrm{k}}}_{{1}{,}{1}}{=}{1}\end{array}\right$ (2)
 > $\mathrm{sys3}:=\mathrm{ZeroPoleGain}\left(\left[1,2\right],\left[1,2,3\right]\right):$
 > $\mathrm{PrintSystem}\left(\mathrm{sys3}\right)$
 $\left[\begin{array}{l}{\mathbf{Zero Pole Gain}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\\ {{\mathrm{z}}}_{{1}{,}{1}}{=}\left[{-2}\right]\\ {{\mathrm{p}}}_{{1}{,}{1}}{=}\left[{-}{1}{-}{I}{}\sqrt{{2}}{,}{-}{1}{+}{I}{}\sqrt{{2}}\right]\\ {{\mathrm{k}}}_{{1}{,}{1}}{=}{1}\end{array}\right$ (3)
 > $\mathrm{sys4}:=\mathrm{ZeroPoleGain}\left(\left[\right],\left[-5+1I,-5-1I\right],1\right):$
 > $\mathrm{PrintSystem}\left(\mathrm{sys4}\right)$
 $\left[\begin{array}{l}{\mathbf{Zero Pole Gain}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\\ {{\mathrm{z}}}_{{1}{,}{1}}{=}\left[\right]\\ {{\mathrm{p}}}_{{1}{,}{1}}{=}\left[{-5}{+}{I}{,}{-5}{-}{I}\right]\\ {{\mathrm{k}}}_{{1}{,}{1}}{=}{1}\end{array}\right$ (4)
 > $\mathrm{ss_a}:=\mathrm{Matrix}\left(\left[\left[1,2\right],\left[0,4\right]\right]\right)$
 ${\mathrm{ss_a}}{:=}\left[\begin{array}{rr}{1}& {2}\\ {0}& {4}\end{array}\right]$ (5)
 > $\mathrm{ss_b}:=\mathrm{Matrix}\left(\left[\left[3,7\right],\left[9,6\right]\right]\right)$
 ${\mathrm{ss_b}}{:=}\left[\begin{array}{rr}{3}& {7}\\ {9}& {6}\end{array}\right]$ (6)
 > $\mathrm{ss_c}:=\mathrm{Matrix}\left(\left[\left[5,6\right],\left[5,2\right]\right]\right)$
 ${\mathrm{ss_c}}{:=}\left[\begin{array}{rr}{5}& {6}\\ {5}& {2}\end{array}\right]$ (7)
 > $\mathrm{ss_d}:=\mathrm{Matrix}\left(\left[\left[0,0\right],\left[0,0\right]\right]\right)$
 ${\mathrm{ss_d}}{:=}\left[\begin{array}{rr}{0}& {0}\\ {0}& {0}\end{array}\right]$ (8)
 > $\mathrm{sys5}:=\mathrm{ZeroPoleGain}\left(\mathrm{ss_a},\mathrm{ss_b},\mathrm{ss_c},\mathrm{ss_d},\mathrm{discrete},\mathrm{sampletime}=0.001,\mathrm{systemname}="Example discrete MIMO system"\right):$
 > $\mathrm{PrintSystem}\left(\mathrm{sys5}\right)$
 $\left[\begin{array}{l}{\mathbf{Zero Pole Gain}}\\ {\mathrm{discrete; sampletime = .1e-2}}\\ {\mathrm{systemname}}{=}{\mathrm{Example discrete MIMO system}}\\ {\mathrm{2 output\left(s\right); 2 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({z}\right){,}{\mathrm{u2}}{}\left({z}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({z}\right){,}{\mathrm{y2}}{}\left({z}\right)\right]\\ {{\mathrm{z}}}_{{1}{,}{1}}{=}\left[\frac{{8}}{{23}}\right]\\ {{\mathrm{p}}}_{{1}{,}{1}}{=}\left[{1}{,}{4}\right]\\ {{\mathrm{k}}}_{{1}{,}{1}}{=}{69}\\ {{\mathrm{z}}}_{{2}{,}{1}}{=}\left[{-}\frac{{4}}{{11}}\right]\\ {{\mathrm{p}}}_{{2}{,}{1}}{=}\left[{1}{,}{4}\right]\\ {{\mathrm{k}}}_{{2}{,}{1}}{=}{33}\\ {{\mathrm{z}}}_{{1}{,}{2}}{=}\left[\frac{{116}}{{71}}\right]\\ {{\mathrm{p}}}_{{1}{,}{2}}{=}\left[{1}{,}{4}\right]\\ {{\mathrm{k}}}_{{1}{,}{2}}{=}{71}\\ {{\mathrm{z}}}_{{2}{,}{2}}{=}\left[\frac{{92}}{{47}}\right]\\ {{\mathrm{p}}}_{{2}{,}{2}}{=}\left[{1}{,}{4}\right]\\ {{\mathrm{k}}}_{{2}{,}{2}}{=}{47}\end{array}\right$ (9)