Dealing with transfer function in symbolic math toolbox.

I used the command syms to symbolize the letter 's' and use it to construct transfer functions. is there a way to bodeplot the TF or getting it's DC gain without having to manually copy the numbers into a Numerator and Denominator vectors to use the function TF(Numerator,Denominator)?

thanks in advance.

Answer 1

[Star Strider]

Yes.

Once you have the symbolic polynomials in s, use the sym2poly function to extract the coefficiinets. (It may be necesary to use the collect function fiirst on each of the numerator and denominator polynomials.)

Example —

syms s 
H(s) = (2*s^2 + 5*s + 1) / (s^3 + 7*s^2 +2*s + 7)                 % Arbitrary Transfer Function

H(s) = 

[nm,dn] = numden(H)

nm(s) = 

dn(s) = 

nmd = sym2poly(nm)

nmd = 1×3

2 5 1

dnd = sym2poly(dn)

dnd = 1×4

1 7 2 7

Hs = tf(nmd,dnd)

Hs =
 
     2 s^2 + 5 s + 1
  ---------------------
  s^3 + 7 s^2 + 2 s + 7
 
Continuous-time transfer function.

figure
bode(Hs)
grid

figure
step(Hs)
grid

At least my random transfer function appears to be stable!

.

[Hamza Harbi]

That helped alot, thanks @Star Strider

[Star Strider]

As always, my pleasure!

Answer 2

[Sam Chak]

Hi @Hamza Harbi

In MATLAB, the transfer function is preferred to be modeled as a tf-class System Object, which is used for simulating dynamic systems with inputs that change over time. System Objects can take advantage of many useful functions from the Control System, Robust Control, and System Identification Toolboxes for modeling, analysis, design, and simulation.

However, if a dynamic system is modeled as a Symbolic Function, then some extra work is required to mathematically compute the log-magnitude curve and the phase-angle curve. Two methods are shown below:

Method 1: When a dynamic system is modeled as a System Object:

% transfer function as a System Object
G = tf(12, [1 2 3])

G =
 
       12
  -------------
  s^2 + 2 s + 3
 
Continuous-time transfer function.

className = class(G)

className = 'tf'

get(G)

       Numerator: {[0 0 12]}
     Denominator: {[1 2 3]}
        Variable: 's'
         IODelay: 0
      InputDelay: 0
     OutputDelay: 0
       InputName: {''}
       InputUnit: {''}
      InputGroup: [1×1 struct]
      OutputName: {''}
      OutputUnit: {''}
     OutputGroup: [1×1 struct]
           Notes: [0×1 string]
        UserData: []
            Name: ''
              Ts: 0
        TimeUnit: 'seconds'
    SamplingGrid: [1×1 struct]

A = dcgain(G)

A = 4

figure(1)
bodeplot(G), grid on 

Method 2: When a dynamic system is modeled as a Symbolic Function:

% transfer function as a Symbolic Function
syms G(s)
G(s)   = 12/(s^2 + 2*s + 3)

G(s) = 

className = class(G)

className = 'symfun'

Gdc    = limit(G, s, 0)                 % calculate DC gain using limit as s → 0

Gdc = 

4

% Compute response against the frequency ω on a logarithmic scale with base 10.
omega  = logspace(-1, 2);   % create log scale for ω on x-axis from 10⁻¹ to 10²
Gjw    = vpa(subs(G, s, 1j*omega), 4);  % G(jω)
GjwMag = 20*log10(abs(Gjw));            % log-magnitude of G(jω)
GjwPha = rad2deg(angle(Gjw));           % phase-angle of G(jω)

figure(2)
tiledlayout(2, 1, 'TileSpacing', 'Compact');

nexttile()
semilogx(omega, GjwMag), , grid on      % plot the log-magnitude of G(jω)
ylabel('Magnitude (dB)'); 
title("Frequency Response of G(s) = "+string(G))
set(gca, 'XLim', [0 max(omega)]);

nexttile()
semilogx(omega, GjwPha), grid on        % plot the phase-angle of G(jω)
xlabel('Frequency (rad/s)');
ylabel('Phase (deg)');
set(gca, 'XLim', [0 max(omega)]);
set(gca, 'YLim', [-180 0]);
yticks(-180:45:0)

[Hamza Harbi]

thank you for your effort @Sam Chak, it seems a long way but i enjoyed trying it on my transfer function.

[Paul]

Seems pretty short, and can be made even shorter using Control System Toolbox for the plotting.

syms G(s) w
G(s)   = 12/(s^2 + 2*s + 3);
omega  = logspace(-1, 2);   % create log scale for ω on x-axis from 10⁻¹ to 10²
Gjw    = vpa(subs(G, s, 1j*omega), 4);  % G(jω)
bodeplot(frd(double(Gjw),omega));