## Continuous Time System:

In recent years, increasingly more attention is being paid to the question of digital implementation of the automatic generation control algorithms This is mainly due to the facts that digital control turns out to be more accurate and reliable, compact in size, less sensitive to noise and drift and more flexible. It may also be implemented in a time shared fashion by using the computer systems in load dispatch centre, if so desired. The ACE, a signal which is used for AGC is available in the discrete form, i.e., there occurs sampling operation between the system and the controller. Unlike the Continuous Time System, the control vector in the discrete mode is constrained to remain constant between the sampling instants. The digital control process is inherently a discontinuous process and the designer has thus to resort to the discrete-time analysis for optimization of the AGC strategies.

**Discrete-Time Control Model:**

The Continuous Time Dynamic System is described by a set of linear differential equations

where x, u, p are state, control and disturbance vectors respectively and A,B and F are constant matrices associated with the above vectors.

The discrete-time behaviour of the Continuous Time System is modelled by the system of first order linear difference equations:

where x(k), u(k) and p(k) are the state, control and disturbance vectors and are specified at t = kT, k = 0, 1, 2, … etc. and T is the sampling period. Φ, Ψ and γ are the state, control and disturbance transition matrices and they are evaluated using the following relations.

where A, B and F are the constant matrices associated with x, u, and p vectors in the corresponding Continuous Time Dynamic System. The matrix e^{AT} can be evaluated using various well-documented approaches like Sylvestor’s expansion theorem, series expansion technique etc..