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Inexpensive personal computers (IBM PC/XT/AT and compatibles) can do very respectable dynamic-system simulations at remarkable speed, as shown in this brief technical note. The direct-executing ENHANCED DESIRE system solves up to one hundred ordinary differential equations typed and edited on the CRT screen, without noticeable delay for program translation. Monochrome or color graphics are produced.Many simulation professionals may not realize that ordinary, quite inexpensive personal computers can do very respectable number-crunching if they are equipped with 8087 or 80287 math co-processors. Also, one needs software carefully designed to utilize the co-processors stack-machine architecture. This is especially true for newer PC and AT clones equipped with 7 or 8 MHz clocks. Such machines, which may retail for less than $2000, can match the simulation speed of a DEC PDP-11/44 minicomputer with hardware floating-point arithmetic and can solve much larger problems. Figure 1 depicts an example. Each link of this textbook delayline circuit may be described by a pair of first-order differential equations defining the delay-line capacitances and inductances. The straightforward computer program (see Appendix) lists the one hundred differential equations for a 50-link delay line in the DYNAMIC segment and specifies circuit parameters and a step-input amplitude vS = 1 Volt. The program then calls for a run-and-reset (drunr) with the termination resistance rL = 50 ohms, equal to the delay-line characteristic impedance, and Figure 1. Classical delay-line circuit (see, e.g., G.A. Korn, and JV. Wait, Digital Continuous-system Simulation, Prentice-Hall, Englewood, New Jersey, 1978).Figure 2. Epson screen prints for the delay-line response with matched and unmatched termination, taken from an ordinary 640 by 200 monocolor PC display.

Author's reply ... Your reviewer has suggested that it might be advisable to obtain sensitivities by subtracting successive solutions obtained for small changes in the system parameters. He asserts that the solution would be obtained more cheaply and more quickly in this way. These judgments are incorrect.A major advantage of the methods indicated in the paper is that they can provide the sensitivity information for less cost and effort than by the direct comparison method. This is primarily due to the fact that quantities which appear in the calculation of a particular measure of the dynamic performance also appear in the calculation of the sensitivities. These quantities (operators may be a more appropriate term), which were needed to determine the original solution, may then be used directly in the calculation of the sensitivities and need not be recalculated. I think that this point is particularly clear in the case of the frequency-response calculation. The operator which is calculated to give the frequency response G(s) is -[A -sI] -1 (see equation 13). The only additional effort required to obtain the frequency-response sensitivities is to multiply three matrices which are either fixed or previously calculated in computing the frequency response. This is indicated in equation (16) of the paper:The matrix -[A -sI] -1 was determined in the frequency-response calculation. The matrix Aij is exactly known. The matrix G(s) is simply the frequency-response vector, which we have available from the frequency-response calculation. It is true that in some applications all components of the vector G(s) may not be required, but we have found that we can calculate the whole frequency response vector about as fast using these methods as we can calculate a single component of the vector using other methods which we have tried. One should note that the operator -[A -sI] -1 would be different if the sensitivities were calculated by the comparison method and would necessarily be reevaluated for each change.A similar situation exists in the case of the transient response and its sensitivity. A number of methods may be used to solve a differential equation such as equation (1) in the paper. We use a method based on the matrix exponential, which we have found to be quite efficient. The solution of is s and the solution of is s if f is constant. If we have an arbitrary, nonconstant f, we can build the solution as follows: Note that the same operators [e,4A' and (e-4A' -I)A-1] are used at each time-step. This procedure can be used to solve the sensitivity equation to give Note that the operators used to obtain the solution to the sensitivity equation are exactly the same as the ones used in the solution of the system equation. This means that we do not have to recalculate these operators, but can use directly those calculated in obtaining the transient response.We can also obtain a semiquantitative estimate of the advantage of the method proposed in the paper for calculating pole sensitivities. In equation (30), we found that it ...