A system is described by a set of 8 non-linear ordinary differential equations with known disturbance profile and some dis-continuous functions. The state space is given as follows:
dx1/dt = k12*x2 -x1*x3 -F1 -F2 +F3 +D(t)
dx2/dt = -k12*x2 +x1*x3 +x2*x4
dx3/dt = -kb1*x3 + p1*x6
dx4/dt = -kb2*x4 + p2*x6
dx5/dt = -kb3*x5 + p3*x6
dx6/dt = -ke*x6 + p4*x7
dx7/dt = -ka*x7 + ka*x8
dx8/dt = -ka*x7 + u(t)
y=x1
where, [x1 x2 x3 x4 x5 x6 x7 x8]=state variables; y=x1 output; u(t):control input; D(t) = Disturbance;
[k12 kb1 kb2 kb3 p1 p2 p3 p4] are model parameters (constant) Discontinuous functions:
(1) if F3>=0 ; F3= 14.07(1-x3); otherwise, F3=0
(2) F1 = 12.98*x1/(x1+1)
(3) if x1>=7.75, F2 = 1.15(x1-7.75) otherwise, F2 = 0
The system has relative degree of 5 (differentiating until control input appears).
(1) How to transform this model to any other form of low relative degree so that control algorithm can be easily constructed for the system?
(2) How to handle the dis-continous functions F1,F2 and F3?
(3) The disturbance is given by D(T)=A*exp(-B*T). What non-linear control scheme is more appropriate for this system with this disturbance profile?
the partial system for x3 to x8 is linear inhogogeneous with
constant coefficients, hence you can solve it formally and obtain
its solution in depence of u(t) and the initial values.
Then you can insert this into the the first two equations.
I assume that you have some misprints and mean:
if x3>=0 then F3=14.07(1-x3) otherwise F3=0:
you can deal with this discontinuity by introducing an 'event option'
in the integrator: follow during (numerical) integration x3, predict
precisely its zero and restart the integrator exactly there.
(matlab ode-suite has this possibility, for example, but it is not too
hard to implement such a thing yourself).
Next F1 will hopefully stay away from -1. if there exist u(t)
there this is not the case, then this results in an indirect constraint on
u.
In the ode control solver you get a state constraint!
Finally, F2 isn't a large problem, since it is simply
F2=max(0,1.15(x1-7.75)), hence continuous. Normally,
a numerical integrator with step size control will not fail here, but
may reduce the stepsize drastically and you must provide for this by
proper choice of the codes parameters
hope this helps
@Peter Spellucci
thank you very much sir for your valuable advice.
can you provide any particular method or techinque or example how to solve the partial system for x3 to x8 with initial conditions and insert them into the first two equations.
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