Page 363 - Mechanical Engineers' Handbook (Volume 2)

354 Mathematical Models of Dynamic Physical Systems

kth time step as the initial value of the state for the (k 1)st time step, yields a discrete
˜
succession of approximations x(1) x(t ), x˜(2) x(t ),..., x˜(K) x(t ) spanning the
k
1
2
solution time interval.
The basic procedure for completing the single-variable, single-step problem is the same
regardless of the particular integration method chosen. It consists of two parts: (1) calculation
of the average value of the ith derivative over the time step as
x (k)
i
˜
˙ x (t*) ƒ[x(t*), u(t*)] ƒ (k)
i
i
i
t(k)
and (2) calculation of the final value of the simulated variable at the end of the time step as
˜ x (k) ˜x (k 1) x (k)
i i i
˜
˜x (k 1) t(k)ƒ (k)
i i
If the function ƒ [x(t), u(t)] is continuous, then t* is guaranteed to be on the time step, that
i
is, t k 1 t* t . Since the value of t* is otherwise unknown, however, the value of x(t*)
k
˜
can only be approximated as ƒ(k).
Different numerical integration methods are distinguished by the means used to calculate
the approximation ƒ (k). A wide variety of such methods is available for digital simulation
i
of dynamic systems. The choice of a particular method depends on the nature of the model
being simulated, the accuracy required in the simulated data, and the computing effort avail-
able for the simulation study. Several popular classes of integration methods are outlined in
the following sections.
Euler Method
The simplest procedure for numerical integration is the Euler method. The standard Euler
method approximates the average value of the ith derivative over the kth time step using the
derivative evaluated at the beginning of the time step, that is,
˜ ƒ (k) ƒ[˜x(k 1), u(t k 1 )] ƒ(t )
i
i
ik 1
i 1, 2,..., n and k 1, 2, . . . , K. This is shown geometrically in Fig. 25 for the scalar
single-step case. A modification of this method uses the newly calculated state variables in
the derivative calculation as these new values become available. Assuming the state variables
are computed in numerical order according to the subscripts, this implies
˜
ƒ (k) ƒ[˜x (k),..., ˜x i 1 (k), ˜x (k 1),..., ˜x (k 1), u(t k 1 )]
n
i
i
1
i
The modified Euler method is modestly more efficient than the standard procedure and,
frequently, is more accurate. In addition, since the input vector u(t) is usually known for the
entire time step, using an average value of the input, such as
1 t k
u(k) u( ) d
t(k) t k 1
˜
frequently leads to a superior approximation of ƒ (k).
i
The Euler method requires the least amount of computational effort per time step of
2
any numerical integration scheme. Local truncation error is proportional to t , however,
which means that the error within each time step is highly sensitive to step size. Because
the accuracy of the method demands very small time steps, the number of time steps required
to implement the method successfully can be large relative to other methods. This can imply