Design of Decoupled PID Controllers for MIMO System

By-Rahul Deyee14mtech11033

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Abstract

• The design of PID controller for interacting system is discussed:

• The scheme proposed here is based on:• Modified PID design• Static decoupling• The controller is easily,implemented due to

simple configuration.

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Introduction• MPC i.e Model Predictive Control is the standard

technique for solving multivariable control problem in industries.

• MPC operates with PID controller in supervisory mode at lower level.

• Interaction is severe among,when lower level loops are closed.

• The main difficulty in dealing with the interaction at MPC level is,BW of MPC loop are limited i.e they operate in supervisory mode with sampling interval that are more than PID loop

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THEORY

• G is a linear stable process of the 2x2 transfer function of the form: G(s)=

The controller to be designed is a static decoupler combined with a decentralized PID controller with set point weighting.The control law is:

Where U is control signal,Y process output, is the reference.

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• The static decoupler is a constant matrix of the form:

• Here we have assumed that G(0) is non singular.• The transfer function of decoupled system is

Q(s) = G(s)D• From construction Q(0) is always a identity matrix.

Taylor series expansion of Q(s) for small s gives

for some constant .

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• The closed loop system is described by

• The eqn can written as:

where,

• The zeros of G is given by the solution to the eqn:

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• If we assume,|det G(s)|>>0 for all |s|<ωb,as det Q(s)=det G(s)*detD,so it can be said|q11(s)*q22(s)|>>|q12(s)*q21(s)|,since det Q(0) = 1.

• Then the matrix H is then approximated by:

• Here,we see that the structure of diagonal element of H is same as for SISO control design.

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Interaction indices• Here we will be elaborating on diagonal elements

of H

• Due to integral action in the controller, interaction is small at low frequencies.The maximum interaction is given when:

• Where ,are sensitivity of loop 1 & 2 respectively

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• Upper bound of the interaction terms is given by | ||

where, are maximum sensitivity for individual loop. For low frequencies,we have , & for low frequencies,are constant. hence ,the interaction indices are given by,

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Examples• The control design method in this section

illustrated on two examples from the literature.• ROSENBROCK’S SYSTEM The process G(s)=• The system looks very easy to control, but which has

fundamental limitations because it has a zero at s = 1.The compensated transfer function is

Q(s)=

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• The interaction indices are given by =4/3 and =0.• There is a limitation in the system as there is a RHP

zero at s=1.• Subject to the constraints that the maximum

sensitivity() & maximum complimentary sensitivity() are less than , gives =0.245 and =0.248

• Here the coupling required must be less than 0.1 so the maximum sensitivity() must be less than we find that the integral gain of the second loop must be less than=0.075

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• For designing the PI controller, we use direct pole placement method based on the model

• This gives & ,with we find .With the controller gain becomes & .

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WOOD-BERRY’S BINARY DISTILLATION COLUMN• It is a multivariable process. The process has

transfer function G(s) =

• The compensated system is given by G(s) =

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• Taylor series approximation for Q(s) gives Q(s)• The interaction is given by and .• The time constant of the approximated model are

and • Let us assume desired maximum sensitivity of

individual loop be • It is required here that interaction indices to be

less than 0.20 the integral gain should be and

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• The corresponding crossover frequencies can be calculated as follow

• Choosing equal to 1 we find the proportional gain

as

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• After some experimentation we found that a reasonable value is ,which gives the controller parameter for second loop as and

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Conclusion

• Suggested control scheme is quite easy to implement.• It can handle a large number of practical control

problem, with the help of regular PID controller.• The advantage of this method is that it gives

performance enhancement in frequency range that is not dealt with MPC.

• But here the main assumption is that loop interaction is not severe.

• Set point weighting is necessary for PID controller in multivariable system.

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Thank You