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DynamicSystems

 StepProperties
 compute properties of a step response

 Calling Sequence StepProperties( sys, opts )

Parameters

 sys - System; a system object opts - (optional) equation(s) of the form option = value; specify options for the StepProperties command

Options

 • parameters = set(name = complexcons) or list(name = complexcons)
 Specifies numeric values for parameters in sys. These values override those specified by the parameters field of the system object, which in turn override the settings in in SystemOptions(parameters). The numeric value on the right-hand side of each equation is substituted for the name on the left-hand side in the expressions that define the model. No checking is done during the substitution to determine whether the substituted value is valid. For example, a complex value can be substituted for the coefficient of a polynomial. If the complex value had been originally assigned to the model at creation, a warning would be generated.

Description

 • The StepProperties command calculates the unit-step response properties of a system object, sys, where this object must be linear and stable. Both continuous and discrete objects are accepted.
 • It returns an expression sequence: (final-value, 10%-point, 33%-point, 67%-point, 90%-point, peak-point, settling-point).
 • The first element in the sequence (final-value) is a single value. Each of the remaining elements is a list of two values, the time or index of the point and the associated value of the step response.

Returned Properties

This list describes each of the properties returned.

 • Final-value: the steady-state value of the step response.
 • 10%-point: where response first reaches 1/10 to the final value.
 • 33%-point: where response first reaches 1/3 to the final value.
 • 67%-point: where response first reaches 2/3 to the final value.
 • 90%-point: where response first reaches 9/10 to the final value.
 • peak-point: where response first reaches the peak value. If the peak does not occur, [undefined, undefined] is returned.
 • settling-point: the settling point of the step response. If the settling time is not found within the given interval, [undefined, undefined] is returned.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$

Sampling frequency

 > $\mathrm{Ts}≔0.05$
 ${\mathrm{Ts}}{≔}{0.05}$ (1)

Duration of simulation

 > $\mathrm{dur}≔10$
 ${\mathrm{dur}}{≔}{10}$ (2)

Number of samples

 > $\mathrm{Ns}≔\mathrm{round}\left(\frac{\mathrm{dur}}{\mathrm{Ts}}\right)$
 ${\mathrm{Ns}}{≔}{200}$ (3)

Continuous System

 > $\mathrm{sys}≔\mathrm{TransferFunction}\left(\frac{1}{{s}^{2}+s+10}\right)$
 ${\mathrm{sys}}{≔}\left[\begin{array}{c}{\mathbf{Transfer Function}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\end{array}\right$ (4)
 > $\mathrm{prop}≔\mathrm{StepProperties}\left(\mathrm{sys}\right)$
 ${\mathrm{prop}}{≔}{0.100000000000000}{,}\left[{0.146150201098349}{,}{0.0100000000000000}\right]{,}\left[{0.278988775168172}{,}{0.0333333333333333}\right]{,}\left[{0.419998798778775}{,}{0.0666666666666667}\right]{,}\left[{0.512928287463024}{,}{0.0900000000000000}\right]{,}\left[{1.00611486305525}{,}{0.160467906604390}\right]{,}\left[{7.31709090935268}{,}{0.102000000000000}\right]$ (5)

Final value

 > ${\mathrm{prop}}_{1}$
 ${0.100000000000000}$ (6)

10% to the final value

 > ${\mathrm{prop}}_{2}$
 $\left[{0.146150201098349}{,}{0.0100000000000000}\right]$ (7)

33% to the final value

 > ${\mathrm{prop}}_{3}$
 $\left[{0.278988775168172}{,}{0.0333333333333333}\right]$ (8)

67% to the final value

 > ${\mathrm{prop}}_{4}$
 $\left[{0.419998798778775}{,}{0.0666666666666667}\right]$ (9)

90% to the final value

 > ${\mathrm{prop}}_{5}$
 $\left[{0.512928287463024}{,}{0.0900000000000000}\right]$ (10)

peak value

 > ${\mathrm{prop}}_{6}$
 $\left[{1.00611486305525}{,}{0.160467906604390}\right]$ (11)

settling value

 > ${\mathrm{prop}}_{7}$
 $\left[{7.31709090935268}{,}{0.102000000000000}\right]$ (12)
 > $\mathrm{plots}[\mathrm{display}]\left(\mathrm{plot}\left(\left[{\mathrm{prop}}_{2..7}\right],\mathrm{style}=\mathrm{point},\mathrm{symbol}=\mathrm{cross},\mathrm{color}=\mathrm{black},\mathrm{symbolsize}=30\right),\mathrm{plot}\left(\left[\left[0,{\mathrm{prop}}_{1}\right],\left[\mathrm{dur},{\mathrm{prop}}_{1}\right]\right],\mathrm{color}=\mathrm{gray}\right),\mathrm{ResponsePlot}\left(\mathrm{sys},\mathrm{Step}\left(\right),\mathrm{duration}=\mathrm{dur},\mathrm{numpoints}=200\right)\right)$

Discrete System

 > $\mathrm{sysz}≔\mathrm{ToDiscrete}\left(\mathrm{sys},\mathrm{Ts}\right)$
 ${\mathrm{sysz}}{≔}\left[\begin{array}{c}{\mathbf{Transfer Function}}\\ {\mathrm{discrete; sampletime = .5e-1}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({z}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({z}\right)\right]\end{array}\right$ (13)
 > $\mathrm{propz}≔\mathrm{StepProperties}\left(\mathrm{sysz}\right)$
 ${\mathrm{propz}}{≔}{0.0999999989357504}{,}\left[{3}{,}{0.00760283829914389}\right]{,}\left[{6}{,}{0.0324850042198717}\right]{,}\left[{9}{,}{0.0677171304079914}\right]{,}\left[{11}{,}{0.0926758273948584}\right]{,}\left[{21}{,}{0.160324379799413}\right]{,}\left[{148}{,}{0.101735627624127}\right]$ (14)

Final value

 > ${\mathrm{propz}}_{1}$
 ${0.0999999989357504}$ (15)

10% to the final value

 > ${\mathrm{propz}}_{2}$
 $\left[{3}{,}{0.00760283829914389}\right]$ (16)

33% to the final value

 > ${\mathrm{propz}}_{3}$
 $\left[{6}{,}{0.0324850042198717}\right]$ (17)

67% to the final value

 > ${\mathrm{propz}}_{4}$
 $\left[{9}{,}{0.0677171304079914}\right]$ (18)

90% to the final value

 > ${\mathrm{propz}}_{5}$
 $\left[{11}{,}{0.0926758273948584}\right]$ (19)

peak value

 > ${\mathrm{propz}}_{6}$
 $\left[{21}{,}{0.160324379799413}\right]$ (20)

settling value

 > ${\mathrm{propz}}_{7}$
 $\left[{148}{,}{0.101735627624127}\right]$ (21)
 > $\mathrm{tfs}≔\mathrm{subs}\left(\mathrm{Ts_}=\mathrm{Ts},x→\left[\left({x}_{1}-1\right)\mathrm{Ts_},{x}_{2}\right]\right)$
 ${\mathrm{tfs}}{≔}{x}{→}\left[{0.05}{}{{x}}_{{1}}{-}{0.05}{,}{{x}}_{{2}}\right]$ (22)
 > $\mathrm{plots}[\mathrm{display}]\left(\mathrm{plot}\left(\mathrm{map}\left(\mathrm{tfs},\left[{\mathrm{propz}}_{2..7}\right]\right),\mathrm{style}=\mathrm{point},\mathrm{symbol}=\mathrm{cross},\mathrm{color}=\mathrm{black},\mathrm{symbolsize}=30\right),\mathrm{plot}\left(\left[\left[0,{\mathrm{prop}}_{1}\right],\left[\mathrm{dur},{\mathrm{prop}}_{1}\right]\right],\mathrm{color}=\mathrm{gray}\right),\mathrm{ResponsePlot}\left(\mathrm{sysz},\mathrm{Step}\left(\mathrm{sampletime}=\mathrm{Ts},\mathrm{samplecount}=\mathrm{Ns},\mathrm{discrete}=\mathrm{true}\right)\right)\right)$