Coupling is the interaction between the controlled variables and manipulating variables. The decoupling controller is used to eliminate  the interaction.

Dear Engr Safeer Ullah,

coupling the view through the control theory:

The existence of interactions between the inputs and outputs of a MIMO system is a form of coupling when the system is represented by a transfer matrix G (s), also to a representation in the form of state interaction appears in the A matrix and well understood the link with the matrices B, C and D.

In practice, the coupling results in difficulty to able to control a given system, that is why we are in the need to decouple the system.

Decoupling can be relase by Feedforward Action,  Converting MIMO Problems to SISO Problems and Decoupling when the system is represented by a transfer matrix G (s). Also we can use ELMER G. GILBERT Algorithm for a class system represented by the state equations .

In all cases we can check on the transfer matrix having undergone the decoupling action the lack of interactions inputs outputs.

Please find attached the documents relative to decoupling.

Best regards

Safeer, 

That is a very general question! Let me try to put things together as briefly as possible !!

• Coupling between states is a situation when the time variation dxi/dt of the i-th state is in some way dependent on states other than the i-th.
• For nonlinear systems as always "the sky is the limit" ! But in linear systems, coupled states are easy to identify from the off-diagonal elements of the A-matrix being non-zero. Following from the earlier statement for example, if  dxi/dt depends on xjamounting to coupling between xi and xj, then the elements Aij and Aji will be non-zero.
• Decoupling helps in design, since it allows the designer to set the controller for each state independent of the other states.
• For a linear system with distinct eigenvalues, the Lure' canonical form is the perfectly decoupled structure.
• For a linear system with repeated eigenvalues, the Jordan canonical form is the most decoupled structure. Decoupling between the states with the same eigenvalue (within the Jordan block, that is) is not possible.
• Decoupling may or may not be possible in nonlinear systems - one can not generalise.

Hope that helped.

-Sanjay

In multi-variable control terminology:

If the transfer-function (Y-U model) matrix is non-diagonal, then coupling effect is present. This results in retaliating effect in closed loop, i.e., a change in one output affects another output and it again affects back the previous.  This makes the decentralized control difficult.

Some techniques for decoupling (proper pairing of input-output variables) are:

After proper pairing, decoupler could be incorporated in the loop to cancel any effect from other interacting loops.

Roughly speaking a system is input-output coupled if when you vary (say) input 1, not only one output changes but many. In consequence, a system is not coupled if when you change input 1, only output 1 changes.

Which are the state variables (elementary or true) of a system from a phenomenological point of view? These are probably the variables describing elementary processes of energy accumulation, in algebraically sense, that take place in the system. All other state variables associated to that system are more simplified or more sophisticated functions of the above mentioned state variables.

The coupling or decoupling issues address the connections between the individual inputs and the individual outputs of a system. (The discussion may be enlarged to groups of input variables and groups of output variables). Every path between an input and an output includes necessarily the state variables. Hence, the coupling and the decoupling are structural invariants. The existence of coupling is obvious for over-actuated systems and for under-actuated systems. Therefore, the decoupled structures should be discussed only for systems with the same number of inputs and outputs.

It is possible through control to compensate from the outside for the coupling interactions of a process. But, the coupling interactions within the process remain. Furthermore, the real decoupling through control is always questionable and very sensitive to identification errors, to nonlinearities and to assuring of the assumptions of separability associated to the practical connection making. Consequently, any theoretical design approach used in this context should clearly state the situations when it is applied.