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Sysquake Pro – Table of Contents
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Graphics for Dynamic Systems
Graphical commands described in this section are related to automatic control. They display the time responses and frequency responses of linear time-invariant systems defined by transfer functions or state-space models in continuous time (Laplace transform) or discrete time (z transform).
Some of these functions can return results in output arguments instead of displaying them. These values depend not only on the input arguments, but also on the current scale of the figure. For instance, the set of frequencies where the response of the system is evaluated for the Nyquist diagram is optimized in the visible area. Option Range of responseset can be used when this behavior is not suitable, such as for phase portraits using lsim. Output can be used for uncommon display purposes such as special styles, labels, or export. Evaluation or simulation functions not related to graphics, like polyval, ode45 or filter, are better suited to other usages.
bodemag
Magnitude Bode diagram of a continuous-time system.
Syntax
bodemag(numc, denc) bodemag(numc, denc, w) bodemag(numc, denc, opt) bodemag(numc, denc, w, opt) bodemag(Ac, Bc, Cc, Dc) bodemag(Ac, Bc, Cc, Dc, w) bodemag(Ac, Bc, Cc, Dc, opt) bodemag(Ac, Bc, Cc, Dc, w, opt) bodemag(..., style) bodemag(..., style, id) (mag, w) = bodemag(...)
Description
bodemag(numc,denc) plots the magnitude of the frequency response of the continuous-time transfer function numc/denc. The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
bodemag(Ac,Bc,Cc,Dc) plots the magnitude of the frequency response
jw X(jw) = Ac X(jw) + Bc U(jw) Y(jw) = Cc X(jw) + Dc U(jw)
With output arguments, bodemag gives the magnitude and the frequency as column vectors. No display is produced.
Examples
Green plot for
bodemag(1, [1, 2, 3, 4], 'g');
The same plot, between
scale([0,10]); bodemag(1, [1, 2, 3, 4], 'g');
See also
bodephase, dbodemag, sigma, responseset, plotset
bodephase
Phase Bode diagram for a continuous-time system.
Syntax
bodephase(numc, denc) bodephase(numc, denc, w) bodephase(numc, denc, opt) bodephase(numc, denc, w, opt) bodephase(Ac, Bc, Cc, Dc) bodephase(Ac, Bc, Cc, Dc, w) bodephase(Ac, Bc, Cc, Dc, opt) bodephase(Ac, Bc, Cc, Dc, w, opt) bodephase(..., style) bodephase(..., style, id) (phase, w) = bodephase(...)
Description
bodephase(numc,denc) plots the phase of the frequency response of the continuous-time transfer function numc/denc. The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options (such as time delay) can be provided in a structure opt created with responseset; fields Delay and Range are utilized. The optional arguments style and id have their usual meaning.
bodephase(Ac,Bc,Cc,Dc) plots the phase of the frequency response
jw X(jw) = Ac X(jw) + Bc U(jw) Y(jw) = Cc X(jw) + Dc U(jw)
With output arguments, bodephase gives the phase and the frequency as column vectors. No display is produced.
Example
Green plot for
bodephase(1, [1, 2, 3, 4], 'g');
See also
bodemag, dbodephase, responseset, plotset
dbodemag
Magnitude Bode diagram for a discrete-time system.
Syntax
dbodemag(numd, dend, Ts) dbodemag(numd, dend, Ts, w) dbodemag(numd, dend, Ts, opt) dbodemag(numd, dend, Ts, w, opt) dbodemag(Ad, Bd, Cd, Dd, Ts) dbodemag(Ad, Bd, Cd, Dd, Ts, w) dbodemag(Ad, Bd, Cd, Dd, Ts, opt) dbodemag(Ad, Bd, Cd, Dd, Ts, w, opt) dbodemag(..., style) dbodemag(..., style, id) (mag, w) = dbodemag(...)
Description
dbodemag(numd,dend,Ts) plots the magnitude of the frequency response of the discrete-time transfer function numd/dend with sampling period Ts. The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
dbodemag(Ad,Bd,Cd,Dd,Ts) plots the magnitude of the frequency response
z X(z) = Ad X(z) + Bd U(z) Y(z) = Cd X(z) + Dd U(z)
where
With output arguments, dbodemag gives the magnitude and the frequency as column vectors. No display is produced.
Example
dbodemag(1,poly([0.9,0.7+0.6j,0.7-0.6j]),1);
See also
bodemag, dbodephase, dsigma, responseset, plotset
dbodephase
Phase Bode diagram for a discrete-time system.
Syntax
dbodephase(numd, dend, Ts) dbodephase(numd, dend, Ts, w) dbodephase(numd, dend, Ts, opt) dbodephase(numd, dend, Ts, w, opt) dbodephase(Ad, Bd, Cd, Dd, Ts) dbodephase(Ad, Bd, Cd, Dd, Ts, w) dbodephase(Ad, Bd, Cd, Dd, Ts, opt) dbodephase(Ad, Bd, Cd, Dd, Ts, w, opt) dbodephase(..., style) dbodephase(..., style, id) (phase, w) = dbodephase(...)
Description
dbodemag(numd,dend,Ts) plots the phase of the frequency response of the discrete-time transfer function numd/dend with sampling period Ts. The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
dbodephase(Ad,Bd,Cd,Dd,Ts) plots the phase of the frequency response
z X(z) = Ad X(z) + Bd U(z) Y(z) = Cd X(z) + Dd U(z)
where
With output arguments, dbodephase gives the phase and the frequency as column vectors. No display is produced.
Example
dbodephase(1,poly([0.9,0.7+0.6j,0.7-0.6j]),1);
See also
bodephase, dbodemag, responseset, plotset
dimpulse
Impulse response plot of a discrete-time linear system.
Syntax
dimpulse(numd, dend, Ts) dimpulse(numd, dend, Ts, opt) dimpulse(Ad, Bd, Cd, Dd, Ts) dimpulse(Ad, Bd, Cd, Dd, Ts, opt) dimpulse(..., style) dimpulse(..., style, id) (y, t) = dimpulse(...)
Description
dimpulse(numd,dend,Ts) plots the impulse response of the discrete-time transfer function numd/dend with sampling period Ts.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
dimpulse(Ad,Bd,Cd,Dd,Ts) plots the impulse response of the discrete-time state-space model (Ad,Bd,Cd,Dd) defined as
x(k+1) = Ad x(k) + Bd u(k) y(k) = Cd x(k) + Dd u(k)
where
With output arguments, dimpulse gives the output and the time as column vectors. No display is produced.
Example
dimpulse(1, poly([0.9,0.7+0.6j,0.7-0.6j]), 1, 'r');
See also
impulse, dstep, dlsim, dinitial, responseset, plotset
dinitial
Time response plot of a discrete-time linear state-space model with initial conditions.
Syntax
dinitial(Ad, Bd, Cd, Dd, Ts, x0) dinitial(Ad, Cd, Ts, x0) dinitial(..., opt) dinitial(..., style) dinitial(..., style, id) (y, t) = dinitial(...)
Description
dinitial(Ad,Bd,Cd,Dd,Ts,x0) plots the output(s) of the discrete-time state-space model (Ad,Bd,Cd,Dd) with null input and initial state x0. The model is defined as
x(k+1) = Ad x(k) + Bd u(k) y(k) = Cd x(k) + Dd u(k)
where
Since there is no system input, matrices Bd and Dd are not used. They can be omitted.
The simulation time range can be provided in a structure opt created with responseset. It is a vector of two elements, the start time and the end time. Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time.
The optional arguments style and id have their usual meaning.
With output arguments, dinitial gives the output and the time as column vectors. No display is produced.
See also
initial, dimpulse, responseset, plotset
dlsim
Time response plot of a discrete-time linear system with arbitrary input.
Syntax
dlsim(numd, dend, u, Ts) dlsim(Ad, Bd, Cd, Dd, u, Ts) dlsim(Ad, Bd, Cd, Dd, u, Ts, x0) dlsim(..., opt) dlsim(..., style) dlsim(..., style, id) dlsim(..., opt, style) dlsim(..., opt, style, id) (y, t) = dlsim(...)
Description
dlsim(numd,dend,u,Ts) plots the time response of the discrete-time transfer function numd/dend with sampling period Ts. The input is given in real vector u, where the element i corresponds to time (i-1)*Ts. Input samples before 0 and after length(u)-1 are 0.
dlsim(Ad,Bd,Cd,Dd,u,Ts) plots the time response of the discrete-time state-space model (Ad,Bd,Cd,Dd) defined as
x(k+1) = Ad x(k) + Bd u(k,:)' y(k) = Cd x(k) + Dd u(k,:)'
where the system input at time sample k is u(k,:)'. For single-input systems, u can also be a row vector.
dlsim(Ad,Bd,Cd,Dd,u,Ts,x0) starts with initial state x0 at time t=0. The length of x0 must match the number of states. The default initial state is the zero vector.
The simulation time range can be provided in a structure opt created with responseset. It is a vector of two elements, the start time and the end time. Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time.
The optional arguments style and id have their usual meaning.
With output arguments, dlsim gives the output and the time as column vectors (or an array for the output of a multiple-output state-space model, where each row represents a sample). No display is produced.
Example
Simulation of a third-order system with a rectangular input
u = repmat([ones(1,10), zeros(1,10)], 1, 3); dlsim(1, poly([0.9,0.7+0.6j,0.7-0.6j]), u, 1, 'rs');
See also
dstep, dimpulse, dinitial, lsim, responseset, plotset
dnichols
Nichols diagram of a discrete-time system.
Syntax
dnichols(numd, dend) dnichols(numd, dend, w) dnichols(numd, dend, opt) dnichols(numd, dend, w, opt) dnichols(..., style) dnichols(..., style, id) w = dnichols(...) (mag, phase) = dnichols(...) (mag, phase, w) = dnichols(...)
Description
dnichols(numd,dend) displays the Nichols diagram of the discrete-time
transfer function given by polynomials numd and dend.
In discrete time, the
Nichols diagram is the locus of the complex values of the transfer function evaluated
at
The range of frequencies is selected automatically between 0 and
Further options can be provided in a structure opt created with responseset; fields NegFreq and Range are utilized. The optional arguments style and id have their usual meaning.
With output arguments, dnichols gives the magnitude and phase of the frequency response and the frequency as column vectors. No display is produced.
In Sysquake, when the mouse is over a Nichols diagram, in addition to the magnitude and phase which can be retrieved with _y0 and _x0, the normalized frequency is obtained in _q.
Example
scale('lindb');
ngrid;
dnichols(3, poly([0.9,0.7+0.6j,0.7-0.6j]))
See also
nichols, ngrid, dnyquist, responseset, plotset
dnyquist
Nyquist diagram of a discrete-time system.
Syntax
dnyquist(numd, dend) dnyquist(numd, dend, w) dnyquist(numd, dend, opt) dnyquist(numd, dend, w, opt) dnyquist(..., style) dnyquist(..., style, id) w = dnyquist(...) (re, im) = dnyquist(...) (re, im, w) = dnyquist(...)
Description
The Nyquist diagram of the discrete-time transfer function given by polynomials
numd and dend is displayed in the complex plane. In
discrete time, the
Nyquist diagram is the locus of the complex values of the transfer function evaluated
at
The range of frequencies is selected automatically between 0 and
Further options can be provided in a structure opt created with responseset; fields NegFreq and Range are utilized. The optional arguments style and id have their usual meaning.
With output arguments, dnichols gives the real and imaginary parts of the frequency response and the frequency as column vectors. No display is produced.
In Sysquake, when the mouse is over a Nyquist diagram, in addition to the complex value which can be retrieved with _z0 or _x0 and _y0, the normalized frequency is obtained in _q.
Example
Nyquist diagram with the same scale along both x and y axis and a Hall chart
grid (reduced to a horizontal line)
scale equal; hgrid; dnyquist(3, poly([0.9,0.7+0.6j,0.7-0.6j]))
See also
nyquist, hgrid, dnichols, responseset, plotset
dsigma
Singular value plot for a discrete-time state-space model.
Syntax
dsigma(Ad, Bd, Cd, Dd, Ts) dsigma(Ad, Bd, Cd, Dd, Ts, w) dsigma(Ad, Bd, Cd, Dd, Ts, opt) dsigma(Ad, Bd, Cd, Dd, Ts, w, opt) dsigma(..., style) dsigma(..., style, id) (sv, w) = dsigma(...)
Description
dsigma(Ad,Bd,Cd,Dd,Ts) plots the singular values of the frequency response of the discrete-time state-space model (Ad,Bd,Cd,Dd) defined as
z X(z) = Ad X(z) + Bd U(z) Y(z) = Cd X(z) + Dd U(z)
where
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
dsigma is the equivalent of dbodemag for multiple-input systems. For single-input systems, it produces the same plot.
The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
With output arguments, dsigma gives the singular values and the frequency as column vectors. No display is produced.
See also
dbodemag, dbodephase, sigma, responseset, plotset
dstep
Step response plot of a discrete-time linear system.
Syntax
dstep(numd, dend, Ts) dstep(numd, dend, Ts, opt) dstep(Ad, Bd, Cd, Dd, Ts) dstep(Ad, Bd, Cd, Dd, Ts, opt) dstep(..., style) dstep(..., style, id) (y, t) = dstep(...)
Description
dstep(numd,dend,Ts) plots the step response of the discrete-time transfer function numd/dend with sampling period Ts.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
dstep(Ad,Bd,Cd,Dd,Ts) plots the step response of the discrete-time state-space model (Ad,Bd,Cd,Dd) defined as
x(k+1) = Ad x(k) + Bd u(k) y(k) = Cd x(k) + Dd u(k)
where
With output arguments, dstep gives the output and the time as column vectors. No display is produced.
Examples
Step response of a discrete-time third-order system
dstep(1, poly([0.9,0.7+0.6j,0.7-0.6j]), 1, 'g');
Step response of a state-space model with two outputs, and a style argument which is a struct array of two elements to specify two different styles:
A = [-0.3,0.1;-0.8,-0.4];
B = [2;3];
C = [1,3;2,1];
D = [2;1];
style = {Color='navy',LineWidth=3; Color='red',LineStyle='-'};
step(A, B, C, D, style);
See also
dimpulse, dlsim, step, hstep, responseset, plotset
erlocus
Root locus of a polynomial with coefficients bounded by an ellipsoid.
Syntax
erlocus(C0, P) erlocus(C0, P, sizes, colors)
Description
erlocus displays the set of the roots of all the polynomial whose coefficients are bounded by an ellipsoid defined by C0 and P. The polynomials are defined as C0 + [0,dC], where dC*inv(P)*dC' < 1.
If sizes and colors are provided, sizes must be a vector of n values and colors an n-by-3 matrix whose columns correspond respectively to the red, green, and blue components. The locus corresponding to dC*inv(P)*dC' < sizes(i)^2 is displayed with colors(i,:). The vector sizes must be sorted from the smallest to the largest ellipsoid. The default values are sizes = [0.1;0.5;1;2] and colors = [0,0,0;0,0,1;0.4,0.4,1;0.8,0.8,0.8] (i.e. black, dark blue, light blue, and light gray).
Warning: depending on the size of the figure (in pixels) and the speed of the computer, the computation may be slow (several seconds). The number of sizes does not have a big impact.
Example
Roots of the polynomial
scale('equal', [-2,2,-2,2]);
erlocus(poly([0.8, 0.7+0.6j, 0.7-0.6j]), eye(3));
zgrid;
See also
hgrid
Hall chart grid.
Syntax
hgrid hgrid(style)
Description
hgrid plots a Hall chart in the complex plane of the Nyquist diagram. The Hall chart represents circles which correspond to the same magnitude or phase of the closed-loop frequency response. The optional argument specifies the style.
The whole grid is displayed only if the user selects it in the Grid menu, or after the command plotoption fullgrid. By default, only the unit circle and the real axis are displayed. The whole grid is made of the circles corresponding to a closed-loop magnitude of plus or minus 0, 2, 4, 6, 10, and 20 dB; and to a closed-loop phase of plus or minus 0, 10, 20, 30, 45, 60, and 75 degrees.
Example
Hall chart grid with a Nyquist diagram
scale('equal', [-1.5, 1.5, -1.5, 1.5]);
hgrid;
nyquist(20, poly([-1,-2+1j,-2-1j]))
See also
ngrid, nyquist, plotset, plotoption
hstep
Step response plot of a discrete-time transfer function followed by a continuous-time transfer function.
Syntax
hstep(numd, dend, Ts, numc, denc) hstep(numd, dend, Ts, numc, denc, style) hstep(numd, dend, Ts, numc, denc, style, id)
Description
A step is filtered first by numd/dend, a discrete-time transfer function with sampling period Ts; the resulting signal is converted to continuous-time with a zero-order hold, and filtered by the continuous-time transfer function numc/denc.
Most discrete-time controllers are used with a zero-order hold and a continuous-time system. hstep can display the simulated output of the system when a step is applied somewhere in the loop, e.g. as a reference signal or a disturbance. The transfer function numd/dend should correspond to the transfer function between the step and the system input; the transfer function numc/denc should be the model of the system.
Note that the simulation is performed in open loop. If an unstable system is stabilized with a discrete-time feedback controller, all closed-loop transfer functions are stable; however, the simulation with hstep, which uses the unstable model of the system, may diverge if it is run over a long enough time period, because of round-off errors. But in most cases, this is not a problem.
Example
Exact simulation of the output of a continuous-time system whose
input comes from a zero-order hold converter
% unstable system continuous-time transfer function num = 1; den = [1, -1]; % sampling at Ts = 1 (too slow, only for illustration) Ts = 1; [numd, dend] = c2dm(num, den, Ts); % stabilizing proportional controller kp = 1.5; % transfer function between ref. signal and input b = conv(kp, dend); a = addpol(conv(kp, numd), dend); % continuous-time output for a ref. signal step scale([0,10]); hstep(b, a, Ts, num, den); % discrete-time output (exact) dstep(conv(b, numd), conv(a, dend), Ts, 'o');
See also
impulse
Impulse response plot of a continuous-time linear system.
Syntax
impulse(numc, denc) impulse(numc, denc, opt) impulse(Ac, Bc, Cc, Dc) impulse(Ac, Bc, Cc, Dc, opt) impulse(..., style) impulse(..., style, id) (y, t) = impulse(...)
Description
impulse(numc,denc) plots the impulse response of the continuous-time transfer function numc/denc.
Further options can be provided in a structure opt created with responseset; fields Delay and Range are utilized. The optional arguments style and id have their usual meaning.
impulse(Ac,Bc,Cc,Dc) plots the impulse response of the continuous-time state-space model (Ac,Bc,Cc,Dc) defined as
dx/dt = Ac x + Bc u y = Cc x + Dc u
where
With output arguments, impulse gives the output and the time as column vectors. No display is produced.
Example
impulse(1, 1:4, 'm');
See also
dimpulse, step, lsim, initial, responseset, plotset
initial
Time response plot for a continuous-time state-space model with initial conditions.
Syntax
initial(Ac, Bc, Cc, Dc, x0) initial(Ac, Cc, x0) initial(Ac, Bc, Cc, Dc, x0, opt) initial(..., style) initial(..., style, id) (y, t) = initial(...)
Description
initial(Ac,Bc,Cc,Dc,x0) plots the output(s) of the continuous-time state-space model (Ac,Bc,Cc,Dc) with null input and initial state x0. The model is defined as
dx/dt = Ac x + Bc u y = Cc x + Dc u
where
Since there is no system input, matrices Bd and Dd are not used. They can be omitted.
The simulation time range can be provided in a structure opt created with responseset. It is a vector of two elements, the start time and the end time. Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time.
The optional arguments style and id have their usual meaning.
With output arguments, initial gives the output and the time as column vectors. No display is produced.
Example
Response of a continuous-time system whose initial state is
[5;3]
initial([-0.3,0.1;-0.8,-0.4],[2;3],[1,3;2,1],[2;1],[5;3])
See also
dinitial, impulse, responseset, plotset
lsim
Time response plot of a continuous-time linear system with piece-wise linear input.
Syntax
lsim(numc, denc, u, t) lsim(numc, denc, u, t, opt) lsim(Ac, Bc, Cc, Dc, u, t) lsim(Ac, Bc, Cc, Dc, u, t, opt) lsim(Ac, Bc, Cc, Dc, u, t, x0) lsim(Ac, Bc, Cc, Dc, u, t, x0, opt) lsim(..., style) lsim(..., style, id) (y, t) = lsim(...)
Description
lsim(numc,denc,u,t) plots the time response of the continuous-time transfer function numd/dend. The input is piece-wise linear; it is defined by points in real vectors t and u, which must have the same length. Input before t(1) and after t(end) is 0. The input used for the simulation is interpolated to have a smooth response.
lsim(Ac,Bc,Cc,Dc,u,t) plots the time response of the continuous-time state-space model (Ac,Bc,Cc,Dc) defined as
dx/dt = Ac x + Bc u y = Cc x + Dc u
where the system input at time sample t(i) is u(i,:)'. For single-input systems, u can also be a row vector.
lsim(Ac,Bc,Cc,Dc,u,t,x0) starts with initial state x0 at time t=0. The length of x0 must match the number of states. The default initial state is the zero vector.
Options can be provided in a structure opt created with responseset:
- 'Range'
- The range is a vector of two elements, the start time and the end time. Such an explicit time range is required when the response is not displayed in a plot where the x axis represents the time.
- 'tOnly'
- When opt.tOnly is true, lsim produces output only at the time instants defined in t. The logical value false gives the default interpolated values.
The optional arguments style and id have their usual meaning.
With output arguments, lsim gives the output and the time as column vectors (or an array for the output of a multiple-output state-space model, where each row represents a sample). No display is produced.
Example
Response of continuous-time system given by its transfer function
with an input defined by linear segments
t = [0, 10, 20, 30, 50]; u = [1, 1, 0, 1, 1]; lsim(1, [1, 2, 3, 4], u, t, 'b');
See also
step, impulse, initial, dlsim, responseset, plotset
ngrid
Nichols chart grid.
Syntax
ngrid ngrid(mag) ngrid(..., style)
Description
ngrid plots a Nichols chart in the complex plane of the Nichols
diagram
The whole grid is displayed only if the user selects it in the Grid
menu, or after the command plotoption fullgrid.
By default, only the lines corresponding to unit magnitude and to a phase
equal to
The closed-loop magnitude can be specified with an input argument, a scalar or an array of positive real values. If the style is also specified, it must follow the magnitue.
Examples
Plain Nichols chart grid for a Nichols diagram:
ngrid; nichols(7, 1:3);
Finer Nichols chart with dashed lines:
ngrid(logspace(-2, 1, 20), LineStyle='-');
See also
hgrid, nichols, plotset, plotoption
nichols
Nichols diagram of a continuous-time system.
Syntax
nichols(numc, denc) nichols(numc, denc, w) nichols(numc, denc, opt) nichols(numc, denc, w, opt) nichols(..., style) nichols(..., style, id) w = nichols(...) (mag, phase) = nichols(...) (mag, phase, w) = nichols(...)
Description
nichols(numc,denc) displays the Nichols diagram of the
continuous-time transfer function given by polynomials
numc and denc. In
continuous time, the
Nichols diagram is the locus of the complex values of the transfer function evaluated
at
The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; fields Delay, NegFreq and Range are utilized. The optional arguments style and id have their usual meaning.
With output arguments, nichols gives the phase and magnitude of the frequency response and the frequency as column vectors. No display is produced.
In Sysquake, when the mouse is over a Nichols diagram, in addition to the magnitude and phase which can be retrieved with _y0 and _x0, the frequency is obtained in _q.
Examples
Nichols diagram of a third-order system
scale('lindb');
ngrid;
nichols(20,poly([-1,-2+1j,-2-1j]));
Same plot with angles in degrees:
scale('lindb');
scalefactor([180/pi, 1]);
ngrid;
nichols(20,poly([-1,-2+1j,-2-1j]));
See also
dnichols, ngrid, nyquist, responseset, plotset, scalefactor
nyquist
Nyquist diagram of a continuous-time system.
Syntax
nyquist(numc, denc) nyquist(numc, denc, w) nyquist(numc, denc, opt) nyquist(numc, denc, w, opt) nyquist(..., style) nyquist(..., style, id) w = nyquist(...) (re, im) = nyquist(...) (re, im, w) = nyquist(...)
Description
The Nyquist diagram of the continuous-time transfer function given by polynomials
numc and denc is displayed in the complex plane. In
continuous time, the
Nyquist diagram is the locus of the complex values of the transfer function evaluated
at
The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; fields Delay, NegFreq and Range are utilized. The optional arguments style and id have their usual meaning.
With output arguments, nyquist gives the real and imaginary parts of the frequency response and the frequency as column vectors. No display is produced.
In Sysquake, when the mouse is over a Nyquist diagram, in addition to the complex value which can be retrieved with _z0 or _x0 and _y0, the frequency is obtained in _q.
Example
Nyquist diagram of a third-order system
scale equal; hgrid; nyquist(20, poly([-1,-2+1j,-2-1j]))
See also
dnyquist, hgrid, nichols, responseset, plotset
plotroots
Roots plot.
Syntax
plotroots(pol) plotroots(pol, style) plotroots(pol, style, id)
Description
plotroots(pol) displays the roots of the polynomial pol in the complex plane. If this argument is a matrix, each line corresponds to a different polynomial. The default style is crosses; it can be changed with a second argument, or with named arguments.
Example
den = [1, 2, 3, 4]; num = [1, 2]; scale equal; plotroots(den, 'x'); plotroots(num, 'o');
See also
rlocus, erlocus, sgrid, zgrid, plotset, movezero
responseset
Options for frequency responses.
Syntax
options = responseset options = responseset(name1, value1, ...) options = responseset(options0, name1, value1, ...)
Description
responseset(name1,value1,...) creates the option argument used by functions which display frequency and time responses, such as nyquist and step. Options are specified with name/value pairs, where the name is a string which must match exactly the names in the table below. Case is significant. Options which are not specified have a default value. The result is a structure whose fields correspond to each option. Without any input argument, responseset creates a structure with all the default options. Note that functions such as nyquist and step also interpret the lack of an option argument as a request to use the default values. Contrary to other functions which accept options in structures, such as ode45, empty array [] cannot be used (it would be interpreted incorrectly as a numeric argument).
When its first input argument is a structure, responseset adds or changes fields which correspond to the name/value pairs which follow.
Here is the list of permissible options:
| Name | Default | Meaning |
|---|---|---|
| Delay | 0 | time delay |
| NegFreq | false | negative frequencies |
| Offset | 0 | offset |
| Range | [] | time or frequency range |
| tOnly | false | samples for specified time only (lsim) |
Option Delay is used only by continuous-time frequency-response and time-response functions; for frequency responses, it subtracts a phase of delay*w, where w is the angular frequency. Option Offset adds a a value to the step or impulse response.
Option NegFreq is used in Nyquist and Nichols diagrams, continuous-time or discrete-time; when true, the response is computed for negative frequencies instead of positive frequencies. Option Range should take into account the sampling period for discrete-time commands where it is specified.
Examples
Default options:
responseset Delay: 0 NegFreq: false
Nyquist diagram of
nyquist(1, [1,1], responseset('Delay', 1));
Complete Nyquist diagram of
nyquist(2, [1,2,2,1]);
nyquist(2, [1,2,2,1], responseset('NegFreq',true), '-');
See also
bodemag, bodephase, dbodemag, dbodephase, dlsim, dnichols, dnyquist, dsigma, impulse, lsim, nichols, nyquist, sigma, step
rlocus
Root locus.
Syntax
rlocus(num, den) rlocus(num, den, style) rlocus(num, den, style, id) branches = rlocus(num, den)
Description
The root locus is the locus of the roots of the denominator of the closed-loop
transfer function (characteristic polynomial) of the system whose open-loop transfer
function is num/den when the gain is between 0 and
The part of the root locus which is calculated and drawn depends on the scale. If no scale has been set before explicitly with scale or implicitly with plotroots or plot, the default scale is set such that the zeros of num and den are visible.
With an output argument, rlocus gives the list of root locus branches, i.e. a list of row vectors which contain the roots. Different branches do not always have the same numbers of values, because rlocus adapts the gain steps for each branch. Parts of the root locus outside the visible area of the complex plane, as defined by the current scale, have enough points to avoid any interference in the visible area when they are displayed with plot. The gains corresponding to roots are not available directly; they can be computed as real(polyval(den,r)/polyval(num,r)) for root r.
As with other plots, the id is used for interactive manipulation. Manipulating a root locus means changing the gain of the controller, which keeps the locus at the same place but makes the closed-loop poles move on it. Other changes are done by dragging the open-loop poles and zeros, which are plotted by plotroots. To change the gain, you must also plot the current closed-loop poles with the plotroots function and use the same ID, so that the initial click identifies the nearest closed-loop pole and the mouse drag makes Sysquake use the root locus to calculate the change of gain, which can be retrieved in _q (see the example below).
Examples
Root locus of
num = [1, 3, 2];
den = [1, 2, 3, 4];
scale('equal', [-4,1,-2,2]);
sgrid;
rlocus(num, den);
plotroots(num, 'o');
plotroots(den, 'x');
The second example shows how rlocus can be used interactively in Sysquake.
figure "Root Locus"
draw myPlotRLocus(num, den);
mousedrag num = myDragRLocus(num, _q);
function
{@
function myPlotRLocus(num, den)
scale('equal', [-3, 1, -2, 2]);
rlocus(num, den, '', 1);
plotroots(addpol(num, den), '^', 1);
function num = myDragRLocus(num, q)
if isempty(q)
cancel;
else
num = q * num;
end
@}
Caveat
The Laguerre algorithm is used for fast evaluation (roots and plotroots are based on eig and have a better accuracy, but their evaluation for a single polynomial is typically 10 times slower). The price to pay is a suboptimal precision for multiple roots and/or high-order polynomials.
See also
plotroots, plotset, erlocus, sgrid, zgrid
sgrid
Relative damping and natural frequency grid for the poles of a continuous-time system.
Syntax
sgrid sgrid(damping, freq) sgrid(..., style)
Description
With no numeric argument, sgrid plots a grid of lines with
constant relative damping and natural frequencies in the complex plane of
s
The whole grid is displayed only if the user selects it in the Grid menu, or after the command plotoption fullgrid. By default, only the imaginary axis (the stability limit for the poles of the Laplace transform) is displayed.
With one or two numeric arguments, sgrid plots only the lines for
the specified values of damping and natural frequency. Let
The style argument has its usual meaning.
Example
Typical use for poles or zeros displayed in the s plane:
scale equal; sgrid; plotroots(pol);
See also
zgrid, plotroots, hgrid, ngrid, plotset, plotoption
sigma
Singular value plot for a continuous-time state-space model.
Syntax
sigma(Ac, Bc, Cc, Dc) sigma(Ac, Bc, Cc, Dc, w) sigma(Ac, Bc, Cc, Dc, opt) sigma(Ac, Bc, Cc, Dc, w, opt) sigma(..., style) sigma(..., style, id) (sv, w) = sigma(...)
Description
sigma(Ac,Bc,Cc,Dc) plots the singular values of the frequency response of the continuous-time state-space model (Ac,Bc,Cc,Dc) defined as
jw X(jw) = Ac X(jw) + Bc U(jw) Y(jw) = Cc X(jw) + Dc U(jw)
The range of frequencies is selected automatically or can be specified in an optional argument w, a vector of frequencies.
Further options can be provided in a structure opt created with responseset; field Range is utilized. The optional arguments style and id have their usual meaning.
sigma is the equivalent of bodemag for multiple-input systems. For single-input systems, it produces the same plot.
With output arguments, sigma gives the singular values and the frequency as column vectors. No display is produced.
See also
bodemag, bodephase, dsigma, responseset, plotset
step
Step response plot of a continuous-time linear system.
Syntax
step(numc, denc) step(numc, denc, opt) step(Ac, Bc, Cc, Dc) step(Ac, Bc, Cc, Dc, opt) step(..., style) step(..., style, id) (y, t) = step(...)
Description
step(numc,denc) plots the step response of the continuous-time transfer function numc/denc.
Further options can be provided in a structure opt created with responseset; fields Delay and Range are utilized. The optional arguments style and id have their usual meaning.
step(Ac,Bc,Cc,Dc) plots the step response of the continuous-time state-space model (Ac,Bc,Cc,Dc) defined as
dx/dt = Ac x + Bc u y = Cc x + Dc u
where
With output arguments, step gives the output and the time as column vectors. No display is produced.
Example
Step response of the continuous-time system
step(1, 1:4, 'b');
See also
impulse, lsim, dstep, hstep, responseset, plotset
zgrid
Relative damping and natural frequency grid for the poles of a discrete-time system.
Syntax
zgrid zgrid(damping, freq) zgrid(..., style)
Description
With no numeric argument, zgrid plots a grid of lines with
constant relative damping and natural frequencies in the complex plane of
z
The whole grid is displayed only if the user selects it in the Grid menu, or after the command plotoption fullgrid. By default, only the unit circle (the stability limit for the poles of the z transform) is displayed.
With one or two numeric arguments, zgrid plots only the lines for
the specified values of damping and natural frequency. The damping
The style argument has its usual meaning.
Example
Typical use for poles or zeros displayed in the z plane:
scale('equal',[-1.2,1.2,-1.2,1.2]);
zgrid;
plotroots(pol);