dx1/dt = -x1.x3 + k12.x2 + EGP0(1 - x5) - F - Fr + u1
dx2/dt = x1.x3 - k12.x2 - x2.x4
dx3/dt = -ka1.x3 + kb1.x6
dx4/dt = -ka1.x4 + kb1.x6
dx5/dt = -ka1.x5 + kb1.x6
dx6/dt = -ke.x6 + ki.x7
dx7/dt = -ki.x7 + ki.x8
dx8/dt = -ki.x8 + u2
states: [x1 x2 x3 x4 x5 x6 x7 x8]
parameters: [k12 EGP0 ka1 ka2 ka3 kb1 kb2 kb3 ki ke]
dis-continuous functions: [1] Fr = 0.003(x1-9), when x1>=9
Fr = 0, [otherwise]
[2] F = 0.97 , when x1>=4.5
F = 0.97.x1/4.5 [otherwise]
Dear Anirudh,
your system is MIMO, so you must use a MIMO method to get a controllable canonical form. I suggest an attached file which is a sample treated.
Best regards
This system is not controllable (since for instance d/dt(x3-x4)=-ka1(x3-x4) and d/dt(x4-x5)=-ka1(x4-x5)), so cannot be put into controller canonical form. Please add some precisions about what motivates your question, maybe you need only a weaker property.
Dear Anirudh,
I apologize I did not check your system in a detailed way it emerges:
- First your system is nonlinear the fact that you have couplings between several states therefore of the nonlinearities at the level of the states and this is explicit in the equation of state 1 and 2, it is no longer a question of speaking of controllable Canonical form since the latter is designed only for linear systems.
As for the controllability of this system we cannot say much, can we use criteria dedicated to a weakly nonlinear system class such as the bilinear system precisely for your system? The question remains.
Best regards
Dear Mohamed-Mourad Lafifi,
I guess the original question referrred to Fliess' generalized controller canonical form, which holds for linear and nonlinear systems (cf the 1990 paper "Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics"). As for controllability, the system is as I said uncontrollable, since there are some combinations of variables which evolve independently of the inputs.
Dear Philippe Martin,
Having read this article, I fully agree with you that the question relates to the generalized canonical form of Fliess which is applicable to nonlinear systems through the use of the Feedback Liniearization of Nonlinear Dynamics. However, concerning the controllability of the system exposed and recalling the hypotheses of the canonical decomposition of Kalman, it is not said that the system is not controllable through U1 and U2 since the connections between the 8 states can be done in a way Or another.
Dear Mohamed-Mourad Lafifi,
The existence of combinations of variables evolving independently of the inputs does characterize uncontrollability (the existence of m such combination is equivalent to a rank deficiency of m in Kalman's criterion). In this system, we have for instance d/dt(x3-x4)=-ka1(x3-x4), which means x3(t)-x4(t)=exp(-k1a*t)*(x3(0)-x4(0)). Whatever the inputs, the combination x3(t)-x4(t) will lead its own life, and is cleary uncontrollable.
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