Page 125 - Modern Control of DC-Based Power Systems

Background 89

Any state can be represented as a point in the state-space which is
defined by the state variables. The state variables are the coordinate axes
of the state-space and their values indicate the position of a state in the
state-space. Although the state-space representation is not unique and any
linear system can be described by an infinite number of state-space repre-
sentations, all these representations consist of the same number of state
variables. This number is equal to the number of integrators in the system
because their outputs are the state variables of the system. Apart from
state variables, state-space analysis considers input and output variables. In
general, a linear time varying system can be described as follows in the
state-space:

(3.6)
_ x t ðÞ 5 A t ðÞx t ðÞ 1 B t ðÞu t ðÞ
y tðÞ 5 C tðÞx tðÞ 1 DðtÞuðtÞ
where A tðÞ is called the state matrix, B tðÞ the input matrix, C tðÞ the out-
put matrix, and DðtÞ the direct transmission matrix. x tðÞ is the vector of
state variables, u t ðÞ the vector of inputs, and y t ðÞ the vector of outputs. In
case of a time invariant system, the matrices describing the system are
constant. Since state-space models are described in vector and matrix
form, the analysis of multiinput multioutput is inherently supported.
From Eqs. (3.6), one major advantage of the state-space representation
can be directly concluded: in this equation, regardless of the order of the
system, the first derivate of each state variable is expressed in terms of
other state variables and the system inputs. Therefore, although the size
of the matrix A increases with an increase in the order of the system, the
state-space equations will still have the same form. This means that with
the increase in the system order, only the size of the equations increase,
but the same matrix approaches can be applied to tackle the system in the
state-space approach.
The existence of a solution to a control system described by a given
state-space representation is tied to its controllability and observability.
The concepts of controllability and observability are briefly described in
the following subsections.

3.4.1 Controllability

A system is controllable at time t 0 if there is an unconstrained input uðtÞ
which can transfer the system from any initial state xðt 0 Þ to any other state
xðt 1 Þ within a finite time as visualized in Fig. 3.2 for a second order