What are the various possible control schemes for controlling systems with long time-delay?
First at all , it can be interesting to specify if the delay is external ( i.e. in the input or output) or internal ( i.e. in the state). For instance, for a very simple solution to the problem , assume that it is internal equal to h, then the state equation can be ( n-state x; input u):
dx(t)/dt= Ax(t)+Cx(t-h)+D u(t)
Feedback controller
u(t)=Kx(t)+Gx(t-h)
Closed-loop system.
dx(t)/dt= (A+DK)x(t)+(C+DG)x(t-h)
a) Choose G if D is invertible so that C+DG=0. This cancells the delayed dynamics . Then , choose K with appropriate closed - loop pole placement so that the matrix A+DK=Am( eigenvalues of Am are the suitable closed-loop poles). This is always possible if the sate is of dimension one ( one variable only) and D is nonzero.
b) Assume that D is singular. Then fix G so that the norm of H= C+ DG be as small as possible. If D is full rank this can be solved by standard pseudoinverses ( otherwise D can be factorized into a product of matrices of the same rank). Now the closed-loop system is.
dx(t)/dt= (A+DK)x(t)+(C+DG)x(t-h)=Am x(t)+Hx(t-h)
The closed-loop poles are the zeros of
det( sI-Am-Hexp(-hs))=det(sI-Am)det (I-(si-Am)**(-1)Hexp(-hs))
( I is the identity matrix) if the norm of H is small enough so that for any pure imaginary numbers numbers s=iw,the second term in the determinant has gain less than one ( this is an heuristic way of speaking about H-infinity techniques) then the eigenvalues ( which are now infinitely many) of the closed-loop system , which include the delay effect, are all stable provided that Am( the delay-free part of the dynamics) is a stability matrix. This technique is an approximate pole -placement where the infinitely many modes of the delayed system are sufficiently close to the n modes of the delay-free part and in stable locations.
There are much more sophisticated solutions for the case of multiple internal delays , or alternative ones not being based on pole-placement or approximate pole-placement.
You can think of a smidt pridctor scheme if the delay and nominal plant model is known
... or Internal Model Control (IMC), which is a generalization of the Smith predictor.
Of course, there is no reason why you a can't use something like PID. Set the gains to give you healthy stability margins. The D term may help you there. See the attached.
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