Lecture Note 1

Additional Notes

1. Controllability:

We know, on application of any input if it is able to control that particular state then the system must be controllable otherwise it would not be. Literally we are forcing a system to a particular state by an input then detect its properties. In other words it can be defined as “In order to be able to do whatever we want with the given dynamic system under control input, the system must be controllable”.

Consider a dynamical system

………………………………………………………………………..……State Equation(1)

…………………………………………………………………………...Output Equation(2)

where

The state equation(1) is said to be completely state controllable or simply controllable if for any state x(0) and any final state x(N),there exists an input sequence u(k),k=0,1,…….N, which transfers the x(0)to x(N)for some finite N. Otherwise state equation(1) is uncontrollable .

Mathematically it can be illustrated by determining the State transition matrix (STM) then at infinite discrete time sequences what would be its output as time tends to infinity, value of STM approaches to Null Matrix then it would be Controllable otherwise not.

 

State Transition Matrix is given by

 

Since determination of state transition matrix is applicable to both autonomous (input exclusive) and non autonomous system (with applied input).

Example 1

Consider a system whose state equation is given by

 

Determine the system is controllable or not?