MATLAB: Determine IF system is stable

laplacestable

Hi! Im trying to determine if a system is stable for all positive values of L,C and R. We have the transferfuncion: 1/(Ls + R + 1/CS)
Regards

Best Answer

Use the Symbolic Math Toolbox, since without exact values, you can’t use the isstable function:
syms L R C s t 
assume((L > 0) & (R > 0) & (C > 0))
Ss = 1/(L*s + R + 1/(C*s));
Sspf = partfrac(Ss);
St = ilaplace(Sspf);
St = simplify(St, 'Steps',20)
St =
     (exp(-(R*t)/(2*L))*(cosh((t*((C*R^2)/4 - L)^(1/2))/(C^(1/2)*L)) - (C^(1/2)*R*sinh((t*((C*R^2)/4 - L)^(1/2))/(C^(1/2)*L)))/(2*((C*R^2)/4 - L)^(1/2))))/L
So since this term:
exp(-(R*t)/(2*L))
converges to zero for all positive values of ‘t’, and since the sinh and cosh terms have the same argument and so can be simplified as:
hyp_dif = simplify(expand( cosh(t) - sinh(t) ), 'Steps',20)
hyp_dif = 
          exp(-t)
that also converges to zero for all positive values of ‘t’, you can safely assume your system is stable. (It is actually globally stable.)
Q.E.D.
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