Solution of the state vector differential equation

The exponential matrix e in equation (8.46) is ealled the state-transition matrix t) and represents the natural response of the system. Henee 

In equation (8.51) the first term represents the response to a set of initial eonditions, whilst the integral term represents the response to a foreing funetion. 

Using a state variable representation of a system, the eharaeteristie equation is given by 

State-space methods for control system design 241 

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Discrete-time solution of the state vector differential equation... 

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