### 2.21.  STATE-VARIABLE CONCEPTS

In the analysis of a system via the state-variable approach, the system is characterized by a set of first-order differential or difference [3, 5] equations that describe its “state” variables. System analysis and design can be accomplished by solving a set of first-order equations rather than a single, higher-order equation. This approach simplifies the problem and has several advantages when utilizing a digital computer for solution. It is also the basis of optimal-control theory.

What is meant by the state of a system? Qualitatively, a system’s state refers to the initial, current, and future behavior of a system. Quantitatively, it is defined as the minimum set of variables, denoted by x1(t0), x2(t0),…,xn(t0) that are specified at an initial time t = t0, which together with the given inputs u1(t), u2(t),…, um(t) for t t0 determine the state at any future time t t0 [4, 1114]. We can view the state of a system, therefore, as describing the past, present, and future behavior of the system.

What is meant by the term state variables? These are the variables which define the smallest set of variables which determine the state of a system. Physically, this means that a set of state variables x1(t0), x2(t0),…, xn(t0) define the initial state of the system ...

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