How to solve Routh Hurwitz with constant K
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justin nabbs
am 26 Apr. 2015
Kommentiert: ami raa
am 8 Mai 2022
I am attempting to solve for system stab1ility for the equation: s^4+19*s^3+111*s^2+189*s+K*s+5*K=0
The constant K is throwing me off. I have an idea how to solve this with one variable "s" but need help on how to insert the K as a constant in matlab. thank you in advanced
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Julius
am 10 Dez. 2018
Bearbeitet: Julius
am 1 Nov. 2020
Hi, maybe a bit late, but anyway here is my solution using Matlab and Routh criterion for evaluation of K for stability (root locus does it perfectly in a graphical way by showing critical value of K if locus crosses jw axis or whatever)
syms Kp s
G = (5*s + 2)/(s*(s^2 + 3*s + 2)) % plant TF
Gc = (Kp*(s + 3))/(s + 5) % controller TF
chareq = 1+G*Gc==0
cheq = expand(simplify(chareq))
% haven't figure out how to extract char equation from symbolic, but you can simply copy coefs
% or adapt to you existing char.eq.
%% Update in 2020 - have figured out how to extract coefficients out of char eq
[n, d] = numden(cheq)
cheq = n == 0
cheq = collect(n,s) == 0
R = coeffs(n,s)
%% Now coefficients can be accessed from the vector R and put into Rouht Table
% RT = [R(1,5) R(1,3) R(1,1);
% R(1,4) R(1,2) 0]
% Routh table first two rows from coefs of char.eq. (from cheq)
RT = [1 17+5*Kp 6*Kp;
8 10+17*Kp 0];
% the rest of the table
b1 = (RT(2,1)*RT(1,2)-RT(1,1)*RT(2,2))/RT(2,1);
b2 = (RT(2,1)*RT(1,3)-RT(1,1)*RT(2,3))/RT(2,1);
b3 = 0;
c1 = (b1*RT(2,2)-RT(2,1)*b2)/b1;
c2 = (b1*RT(2,3)-RT(2,1)*b3)/b1;
c3 = 0;
d1 = (c1*b2-b1*c2)/c1;
d2 = (c1*b3-b1*c3)/c1;
d3 = 0;
% full Routh table
RT = [1 17+5*Kp 6*Kp;
8 10+17*Kp 0;
simplify(b1) b2 b3;
simplify(c1) c2 c3;
simplify(d1) d2 d3]
% coeficient Kp values for stability to satisfy condition when b1=0, c1=0 and d1=0
K1 = vpasolve(b1, Kp)
K2 = vpasolve(c1, Kp)
K3 = vpasolve(d1, Kp)
Haven't checked for limitations and what if there is row of zeros or zero in 1st column.
10 Kommentare
fatima mouffok
am 11 Mär. 2022
chareq = 1+G*Gc==0 remove ==0
chreq = 1+G*Gc like that it's gonna work!!
Weitere Antworten (1)
J. Carlos Aguado
am 9 Apr. 2018
I assume that the origin of this K is a proportional controller. Therefore, the simplest way (according to MatLab implementations, not to mathematical simplicity) to study its effect on stability is to use the root locus of the plant. MatLab will draw it for you and give you the gain value that marks the frontier with instability
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