Controllability and Observability
Before a state variable feedback control or observer based state variable feedback are used, it must be established that the system is controllable and observable. There are various methods to establish the above concept but the modal analysis is used in this chapter because it is more suitable with use of standard software than other methods. The interested readers are referred to more advanced books on this subject.
If observer based control strategy has to be used, both controllability and observability must be investigated. Because an observable system might not be controllable and a controllable system might not be observable.
If state variable feedback control strategy is used, only controllability must be studied. First a single input and a single output state equation is considered as
In Eq. (3.31), λ is a diagonal matrix with diagonal terms being the eigenvalues of the system. Now the controllability and observability can be discussed. For a single input the system equation is n uncoupled first order differential equations which have a simple solution. The input u can be used to change the eigenvalues λ. The system is controllable if all eigenvalues can be influenced by the input u. The matrix U−1 B for single input is a vector and for controllability all values should be nonzero meaning that that all eigenvalues can be influenced by the input u. If u is a vector then U−1 B is a matrix of n × m. The system is then controllable if U−1 B has no rows consisting entirely of zero elements. Most physical system is controllable by a single input variable even for multi-input control systems.
The system is observable if the row vector CU has no zero elements. For multi- output control system the matrix CU must have no columns consisting of entirely of zero elements. In this way, the output will have contribution from all state variables Z.