Page 194 - Modern Control Systems

168 Chapter 3 State Variable Models

which converges for all finite t and any A [2]. Then the solution of the state differential
equation is found to be

/
x(t) = exp(A0x(0) + f exp[A(f - T)]BU(T) dr. (3.24)
Jo
Equation (3.24) may be verified by taking the Laplace transform of Equation (3.16)
and rearranging to obtain

_1
X(s) = [si - A] x(0) + [si - ApBTJ(s), (3.25)
-1
where we note that [si — A] - <J>(s) is the Laplace transform of (0 = exp(Af).
¢
Taking the inverse Laplace transform of Equation (3.25) and noting that the second
term on the right-hand side involves the product ¢(^)611(^), we obtain Equation
(3.24). The matrix exponential function describes the unforced response of the sys-
tem and is called the fundamental or state transition matrix &{t). Thus, Equation
(3.24) can be written as

(3.26)

The solution to the unforced system (that is, when u = 0) is simply

*i(0 *n(0 <M')~ ~*i(0)"
¢2,1(0 x 2(0)
*2l(0
(3.27)
*«(0 <M0 <£««(0_ _*„(0)_

We note therefore that to determine the state transition matrix, all initial conditions
are set to 0 except for one state variable, and the output of each state variable is eval-
uated. That is, the term faj{t) is the response of the /th state variable due to an initial
condition on the jfth state variable when there are zero initial conditions on all the
other variables. We shall use this relationship between the initial conditions and the
state variables to evaluate the coefficients of the transition matrix in a later section.
However, first we shall develop several suitable signal-flow state models of systems
and investigate the stability of the systems by utilizing these flow graphs.

EXAMPLE 3.1 Two rolling carts

Consider the system shown in Figure 3.5. The variables of interest are noted on the
figure and defined as: M h M 2 = mass of carts,/?, q = position of carts, u = external
force acting on system, k\, k 2 = spring constants, and b h b 2 = damping coefficients.
The free-body diagram of mass M x is shown in Figure 3.6(b), where p,q = velocity
of Mi and M 2, respectively. We assume that the carts have negligible rolling friction.
We consider any existing rolling friction to be lumped into the damping coefficients,
b\ and b 2.