### 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 *x*_{1}(*t*_{0}), *x*_{2}(*t*_{0}),…,*x*_{n}(*t*_{0}) that are specified at an initial time *t* = *t*_{0}, which together with the given inputs *u*_{1}(*t*), *u*_{2}(*t*),…, *u*_{m}(*t*) for *t* *t*_{0} determine the state at any future time *t* *t*_{0} [4, 11–14]. 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 *x*_{1}(*t*_{0}), *x*_{2}(*t*_{0}),…, *x*_{n}(*t*_{0}) define the initial state of the system ...

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