System stability in control system engineering

The mathematical model of a systemis given by:

x ̈ +(x^2- η) x ̇ +w^(2 )x=0

For w=1

Show that a stable equilibrium point becomes unstable as the parameter 𝜂 is varied from -1 in +1 using phase plane analysis.

At what value of 𝜂 does the instability occur?

What happens to the system after the equilibrium point becomes unstable?

[Sam Chak]

Hi @Rahul

Could you review this stability problem? The only MATLAB function I can think of is isstable(). You may guide @Chike on how to solve the problem based on the lecture notes (to be provided by the OP). The lecture notes should have described the stability property so that the OP can apply some MATLAB tools to find or prove them. The lecture notes must have also laid out the behavior or condition when the system becomes unstable.

However, please note that your solution to the Van der Pol oscillator has not been reviewed or accepted by the same OP. There is a high chance that if you post a full solution here, the review or acceptance may go unnoticed as well.

https://www.mathworks.com/matlabcentral/answers/2147084-van-der-pol-equation?s_tid=srchtitle

help isstable

 ISSTABLE True for stable filter
    FLAG = ISSTABLE(B,A) returns a logical output, FLAG, equal to TRUE if
    the filter specified by numerator coefficients B, and denominator
    coefficients A, is stable. Input vectors B, and A define a filter with
    transfer function:
 
                jw               -jw              -jmw
         jw  B(e)    b(1) + b(2)e + .... + b(m+1)e
      H(e) = ---- = ------------------------------------
                jw               -jw              -jnw
             A(e)    a(1) + a(2)e + .... + a(n+1)e
 
    FLAG = ISSTABLE(SOS) returns TRUE if the filter specified using the
    second order sections matrix, SOS, is stable. SOS is a Kx6 matrix,
    where the number of sections, K, must be greater than or equal to 2.
    Each row of SOS corresponds to the coefficients of a second order
    filter. From the transfer function displayed above, the ith row of the
    SOS matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].
 
    FLAG = ISSTABLE(D) returns TRUE if the digital filter, D, is stable. 
    You design a digital filter, D, by calling the designfilt function.
 
    % Example 1:
    %   Create an unstable filter and verify its instability.
 
    b = [1 2 3 4 5 5 1 2];      % numerator coefficients
    a = [4 5 6 7 9 10 4 6];     % denominator coefficients
    flag = isstable(b,a)        % determine if the filter is stable
    zplane(b,a)                 % zero-pole plot for filter
              
    % Example 2:
    %   Create a filter and determine its stability for different 
    %   coefficient data types and tolerances.  
 
    b = [1 -.5];                                % numerator coefficients
    a = [1 -.999999999];                        % denominator coefficients
    act_flag1 = isstable(b,a)                   % determine if its stable
    act_flag2 = isstable(single(b),single(a))   % becomes unstable due to 
                                                % precision
    zplane(b,a)                                 % zero-pole plot for filter
  
    % Example 3:
    %   Design a Butterworth highpass IIR filter using second order sections 
    %   and determine its stability.
 
    [z,p,k] = butter(6,0.7,'high');
    SOS = zp2sos(z,p,k);    
    flag = isstable(SOS)        % determine if the filter is stable
    zplane(z,p)                 % zero-pole plot for filter
 
    See also FILTERANALYZER, FILTORD, ISALLPASS, ISLINPHASE, ISMAXPHASE,
    ISMINPHASE

    Documentation for isstable
       doc isstable

    Other uses of isstable

       dsp.AllpassFilter/isstable    DynamicSystem/isstable