Page 192 - Modern Control Systems

166 Chapter 3 State Variable Models

3.3 THE STATE DIFFERENTIAL EQUATION

The response of a system is described by the set of first-order differential equations
written in terms of the state variables (*i, x 2,. • •, x n) and the inputs (u h u 2,..., u m).
These first-order differential equations can be written in general form as

*i = «u*i + «12*2 + • • • + a lnx n + buUi + • • • + b lmu m,
X 2 = fl 2l*l + «22*2 + • • * + a 2„X n + Z>21"1 + • • • + &2 m M w ,

x„ — a nlXi + a n2x 2 + • • • + a nnx n + b n\U\ + • • • + b nmu m,(3.13)

where x = dx/dt. Thus, this set of simultaneous differential equations can be written
in matrix form as follows [2,5j:

*i «11 « 1 2 ' • • «1« *1 U\
'
d *2 «21 «22 '"' «2» *2 + ^11 •' b\m
dt (3.14)
b n\"'b nmj
_*"_ «nl «n2 * * * «/m_ _%n_

The column matrix consisting of the state variables is called the state vector and is
written as

*i
* 2
(3.15)

where the boldface indicates a vector. The vector of input signals is defined as u.
Then the system can be represented by the compact notation of the state differential
equation as

x = Ax + Bu. (3.16)

The differential equation (3.16) is also commonly called the state equation.
The matrix A is an n x n square matrix, and B is an n x m matrix^ The state
differential equation relates the rate of change of the state of the system to the state
of the system and the input signals. In general, the outputs of a linear system can be
related to the state variables and the input signals by the output equation

y = Cx + Du, (3.17)

'Boldfaced lowercase letters denote vector quantities and boldfaced uppercase letters denote matri-
ces. For an introduction to matrices and elementary matrix operations, refer to the MCS website and
references [1] and [2].