## State space analysis.

State space analysis is an excellent method for the design and analysis of control systems. The conventional and old method for the design and analysis of control systems is the transfer function method. The transfer function method for design and analysis had many drawbacks.

### Drawbacks of transfer function analysis.

- Transfer function is defined under zero initial conditions.
- Transfer function approach can be applied only to linear time invariant systems.
- It does not give any idea about the internal state of the system.
- It cannot be applied to multiple input multiple output systems.
- It is comparatively difficult to perform transfer function analysis on computers.

Any way state variable analysis can be performed on any type systems and it is very easy to perform state variable analysis on computers. The most interesting feature of state space analysis is that the state variable we choose for describing the system need not be physical quantities related to the system. Variables that are not related to the physical quantities associated with the system can be also selected as the state variables. Even variables that are immeasurable or unobservable can be selected as state variables.

### Advantages of state variable analysis.

- It can be applied to non linear system.
- It can be applied to tile invariant systems.
- It can be applied to multiple input multiple output systems.
- Its gives idea about the internal state of the system.

### State of a dynamic system.

The state of a system is the minimum set of variables (state variables) whose knowledge at time t=0, along with the knowledge of the inputs at time t≥ t0 completely describes the behaviour of a dynamic system for a time t >t0 . State variable is a set of variables which fully describes a dynamic system at a given instant of time.

Consider a system having a inputs, b outputs and c state variables. Then,

Output variables = Y1(t), Y2(t), Y3(t)…………………………..Yb(t)

Input variables = U1(t), U2(t), U3(t)……………………………Ua(t)

State variables = X1(t), X2(t), X3(t) …………………………….Xc(t)

Then the system can be represented as shown below.