The system is described by:
dx1/dt = -ka1.x1 + kb1.x4
dx2/dt = -ka1.x2 + kb1.x4
dx3/dt = -ka1.x3 + kb1.x4
dx4/dt = -ke.x4 + ki.x5
dx5/dt = -ki.x5 + ki.x6
dx6/dt = -ki.x6 + u
state variables: [x1 x2 x3 x4 x5 x6]
Output: y=[1 1 1 0 0 0][x]
Input=u
Model Parameters: =[ka1 ka2 ka3 kb1 kb2 kb3 ke ki]
Hello,
In my opinion you can write these equations to be state-space equation like
dx/dt=Ax+Bu;
y=Cx+Du;
then, you can use the Matlab function “reduce” to reduce the order. There are at least five typical methods can be used such as “balance” and “hankel”.
Also, you can refer to the link below for details. Hope it helps you.
https://www.mathworks.com/help/control/ug/reduce-model-order-using-the-model-reducer-app.html
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We reduce the order of the system, to design low order controllers. The model in the question is already very simple. For example,
dx1/dt = -ka1.x1 + kb1.x4
dx2/dt = -ka1.x2 + kb1.x4
dx3/dt = -ka1.x3 + kb1.x4
have the same dynamics. If x4 is bounded, Ka1>0 the systems is stable. Then, the following two have the same structure as that of the above three. The stability condition is Ke> and Ki>0.
dx4/dt = -ke.x4 + ki.x5
dx5/dt = -ki.x5 + ki.x6
You have the control on the following only. You may define an objective based on x1,x2,x3,x4,x5,x6 and the constraint that the the control effort can directly affect the x6.
dx6/dt = -ki.x6 + u
Design a P or a PI controller to accomplish a performance on x6. The rest will indirect. Your system is under-actuated. The open loop system dynamics are over-damped. I do not see any need of model reduction in this syste.
@Muhammad Shafiq
Thanks a lot for your answer.
I am working on a highly nonlinear system with 8 different states. Out of which the first state is the output and last state is the input. The system has different sub-systems. The first two equations are nonlinear and I have omitted in the question which have x1,x2 and x3 in their dynamics.
Here is my complete model:
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]
Can you suggest how can I simplify the above system without loosing the properties of the original system. The system has a relative degree of three. Since my control law require five time differentiation of the output to get the control input, u.
Hi Anirudh,
Frankly, I doubt if such a system can be reduced by an approach that applies across the entire vector space ! I think loss of property is a risk that you face across many subspaces !!
First, I would not attempt reduction unless absolutely necessary for some implementation related issues.
If you simply have to reduce, I suggest you do so for limited range regulation problems, avoiding tracking problems as far as possible.
Do you have some specific equilibrium points in mind, at which you need to consider stability or dynamics ??
-Sanjay
Obtain the step response of the system. Then looking yo yhr reponse design thr pi controller. Then just show the closed loop stanility.
First find the eigenvalues of your system and check whether two (or more) time scale behavior exists. If so you can use singular perturbation methods.
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