If the state space model is of the form say:
dx/dt = Ax + Bu*e^t
y = Cx + Du*e^t
y(t) = 0 for all t>=0
and Cx + Du*e^t = 0 for all t>=0, x(t) = p*e^t
How do we find a solution to the generalized Eigen Value problem?
I am not sure if what you want is to find a solution to x(t) from such state space given that u(t) is an exponential function. The formulation of the eigenvalue problem from the state space representation remains the same independently of the input.
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Adrian-Josue Guel-Cortez Thank you for your response.
I was actually trying to derive the values of initial states and simulate the system response with the initial conditions. I tried to approach it using the formulation of the eigen value problem but to no avail.
to solve your state space system, have you tried using something like eq (4.5) in the attached pdf?
Ref LINEAR SYSTEM THEORY AND DESIGN, Chi-Tsong Chen
Dear Siddhanth Sunil Shah
A standard state-space model is something like this:
dx/dt = Ax + Bu
y = Cx + Du.
It seems you have considered an exponential term; e^t, is multiplied by the control-input; u.
By the way, you may use the standard state-space solutions, to find the time-histories of the state vector (state-signal), knowing the matrices A, B, C, D.
Please refer to the book "Modern Control Engineering", by Katsuhiko Ogata, page 727, solution of non-homogeneous state equations.
Warning: You have made a condition that for all t>=0, Cx + Du*e^t = 0. This is tantamount to say that, the out-put; y(t), is always zero, because for any control system, t>=0, always holds, since time is always a positive variable also could be zero. So you have discarded the out-put from your calculation by assuming it to be zero for all time intervals. Moreover, your state equation would be:
dx/dt = Ax + Bu*e^t,
therefore, knowing the matrices; A, and B, and as well the input function; u, as an explicit function of time, then you can solve the problem according to the process which has been mentioned in the book I have already cited.
Through another scope, if:
x(t) = p*e^t
then:
dx/dt = p*e^t=A*p*e^t + Bu*e^t,
which after cancelling out the exponential term; e^t, from both sides of the equation, could be rewritten as:
p=Ap+Bu,
if:
u=-Kp,
then the preceding equation would be translated to:
p=Ap-BKp,
or rearranged as:
(A-BK)*p=p,
or:
(A-BK-1)*p=0,
which is the eigenvalue problem and can be solved by the existing standard methods of eigenvalue analysis.
It's only a guess, but you can do a Taylor expansion to approximate the exponential form to a polynomial form (is not the same but close enough if the taylor expansion is more than second order), then you can do a eigenvalue solving.
Is for stability analysis in a state space equation in control systems?
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