Controllability and Observability.

Controllability and observability are two very important things related to state space analysis. There are many tests for checking controllability and obervability and these tests are very essential during the design of a control system using state space approach.

Controllability verifies whether is state variable is useful or not. It checks whether a state variable can be manipulated for obtaining the required output. If a state variable is not controllable then there is no meaning in selecting it for any operation. If a particular state variable is found uncontrollable , then it is left untouched and any other state variable which is controllable is selected for operations.

A system is said to be completely state controllable if it is possible to change the system from any initial stage X(t0) to any required stage X(td) using a control vector U(t). Kalman’s test and Gilberts test are the two common methods used for testing controllability.

Gilbert’s method for checking controllability is done under two cases.

1)When the eigen values of the system matrix are distinct.

In this case the system matrix can be diagonalized and can be converted to the canonical form by giving a transformation X=MZ. M is the modal matrix derived from the system matrix and Z is the transformed state variable matrix.

Consider a system with state model represented by the equations

X = AX + BU

Y = CX + DU

The model is transformed into the canonical form as follows,

Z = ?Z + B˜U

Y = C˜Z + DU

Where ?= MA¯¹M,   B˜ = M¯¹B and C˜ = CM

The system is completely state controllable if the matrix B doesnot have any row with all zeros.

2) Eigen values of the system matrix are repeated.

In this situation it is impossible to diagonalize the system matrix and it can be converted to Jordan canonical form.

Consider a system with state model represented by the equations

X = AX + BU

Y = CX + DU

The model is transformed into the Jordan canonical form as follows,

Z = JZ + ?˜U

Y = C˜Z + DU

Where J = M A¯¹M,   B˜ = M¯¹B and C˜ = CM

The system will be completely state controllable if elements of any row of B that correspond to the last row of each Jordan block are not all zero and the rows corresponding to other state variables must not have all zeros.

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