# Controllability and Observability.

## 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.

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.

### Gilberts method.

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.

X = AX + BU

Y = CX + DU

Z = ΛZ + B˜U

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

2) Eigen values of the system matrix are repeated.

Consider a system with state model represented by the equations

Y = CX + DU

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

Z = JZ + Β˜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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