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One advantage of the analog computer was that the flow of voltage signals through the computing elements closely resembled the flow of signals in the block diagram of the control system; the analog computer diagram and the block diagram of the control system looked nearly the same. In fact, this advantage has been retained in some of the digital computer simulation software that has been developed to solve control problems. The basic operation needed to solve control problems by either an analog computer or a digital computer is integration. The integration device, in the case of an analog computer or the simulation software in the case of a digital computer, is called an integrator. Some of the symbols used to represent integration are shown in Fig. 34.1. The operation performed by the integrator is r x dt + y(O) (34.1) i0 where y(O) is the initial value of i at t = 0. The symbol shown in Fig. 34. la is used in analog computing where sign inversion occurs. The symbol shown in Fig. 34.16 is used in block diagrams for state-space problems. The symbol in Fig. 34.1~ is used in digital computer simulation software. Since the focus of this chapter is on the digital computer, the method of achieving integration by means of an analog computer will not be considered further. The reader may consult Coughanowr and Koppel (1965) or other sources for this topic. In the branch of mathematics called numerical analysis, many routines or algorithms to perform integration have been developed. Perhaps the simplest method, which is often discussed in a course in calculus or differential equations, is the Euler method. The Euler method is easy to understand, but it has a large truncation error that makes it too inaccurate for general use. Many methods of numerical integration have been devised that are far more accurate than the Euler method; one of these is the Runge-Kutta method. In this chapter, only the fourth-order Runge-Kutta method will be used. This method is often used to solve sets of first-order differential equations. y=
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The Runge-Kutta method for solving a differential equation is often called a marching solution because the calculation starts at an initial value of the independent variable t and moves forward one integration step at a time.
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FIGURE 34-1 Symbols used to represent integrators.
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Consider the first-order differential equation
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dy dt = f(Y9 t>
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(34.2)
for which y = yo at t = to. In control problems, the initial time to is usually taken as zero. When the dependent variable y is defined in terms of a deviation variable, which is usually the case in control problems, the value of y at to is also zero. The Runge-Kutta method divides the independent variable t into increments of equal length At as shown in Fig. 34.2. The fourth-order Runge-Kutta method uses the following equations:
kl = f (yo> to)At k2 = f (yo + k1/2, to + At/2)At k3 = kq =
f (yo +
k2/2, to + Atl2)At (y. + k3, to + At)At
yl = yo + (kl + 2k2 + 2k3 + k4)/6 _ tl = to +At
(34.3) (34.4) (34.5) (34.6) (34.7) (34.8)
The equations just listed are applied during the first increment At from to to t 1. The values obtained at the end of the first increment (y 1, t 1) are then used as a new set of initial conditions in these equations to obtain a set of values of y and t at the end of the second interval. This procedure of computing y and t at the end of successive intervals generates the solution to the differential equation. The set of equations [Eqs. (34.3) to (34.8)] used to solve a single firstorder differential equation can be applied to each dependent variable in a set of differential equations. Consider the pair of differential equations
with the initial conditions yo, wo, to. The Runge-Kutta equations used to solve for y(t) and w(t) are given below.
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