Page 215 - Modern Control Systems

Section 3.7 The Time Response and the State Transition Matrix 189

_ R/(LC) _ R/{LC)
A(s) , R 1 '
L LC

which agrees with the result Equation (3.40) obtained from the flow graph model
using Mason's signal-flow gain formula. •

3.7 THE TIME RESPONSE AND THE STATE TRANSITION MATRIX

It is often desirable to obtain the time response of the state variables of a control sys-
tem and thus examine the performance of the system. The transient response of a
system can be readily obtained by evaluating the solution to the state vector differ-
ential equation. In Section 3.3, we found that the solution for the state differential
equation (3.26) was

x(/) = 4>(0x(0) + / ¢(/ - T)BU(T) dr. (3.80)
Jo
Clearly, if the initial conditions x(0), the input U(T), and the state transition ma-
trix ¢(/) are known, the time response of x(/) can be numerically evaluated. Thus
the problem focuses on the evaluation of ¢(0, the state transition matrix that
represents the response of the system. Fortunately, the state transition matrix can
be readily evaluated by using the signal-flow graph techniques with which we are
already familiar.
Before proceeding to the evaluation of the state transition matrix using signal-
flow graphs, we should note that several other methods exist for evaluating the
transition matrix, such as the evaluation of the exponential series
00 k k
A t
3 81
¢(/) = exp(A/) = 2 T T < - )
in a truncated form [2, 8]. Several efficient methods exist for the evaluation of ¢(/)
by means of a computer algorithm [21].
-1
In Equation (3.25), we found that ¢($) = [si - A] . Therefore, if ¢(5) is ob-
tained by completing the matrix inversion, we can obtain ¢(/) by noting that
l
¢(/) = $£~ {$(s)}. The matrix inversion process is generally unwieldy for higher-
order systems.
The usefulness of the signal-flow graph state model for obtaining the state tran-
sition matrix becomes clear upon consideration of the Laplace transformation
version of Equation (3.80) when the input is zero. Taking the Laplace transforma-
tion of Equation (3.80) when u(r) = 0, we have
X(s) = *($)x(0). (3.82)
Therefore, we can evaluate the Laplace transform of the transition matrix from the
signal-flow graph by determining the relation between a state variable Xfo) and the
state initial conditions [xj(0), x 2 (0),..., *„(())]. Then the state transition matrix is