Differential-equation Models

Continuous-system simulation models delayed interactions of physical state
variables x1, x2, … with first-order ordinary differential equations (state
equations)
(d/dt) xi = fi(t; x1, x2,…; y1, y2,…; a1, a2,…) (i = 1, 2,…) (1-1a)
Here t represents the time, and the quantities
yj = gj(t; x1, x2, ...; y1, y2,…; b1, b2,…) (j = 1, 2,…) (1-1b)
are defined variables. a1, a2, …, and b1, b2, … are constant model
parameters.
Simulation programs exercise such models by solving the state-equation
system (1-1) to produce time histories of the system variables xi = xi(t) and
yj = yj(t) for t = t0 to t = t0 + TMAX, starting with given initial values t0 and
xi(t0). In Section 1-6 and Chapter 2, we shall add sampled-data operations
representing periodic inputs and outputs, sample-holds, and digital
controllers.

The state variables xi are system outputs. They start at t = t0 with given
initial values; subsequent values are produced by an integration routine
(Section 1-7) from the fi-values generated by the preceding execution
(derivative call) of the operations (1).
There are three kinds of defined variables yj:
1. system inputs (specified functions of the time t)
2. system outputs
3. intermediate results needed to compute the derivatives fi
It must be possible to sort the defined-variable assignments (1-1b) into a pro-
cedure that successively derives all the yj from state variables xi and/or the
time t without recurrence relations or “algebraic loops” (Section 1-9).
Some dynamic systems (e.g., systems involving interconnected mechani-
cal devices in automotive engineering and robotics) are modeled with differ-
ential-equation systems that cannot be explicitly solved for state-variable
derivatives as in Eq. (1-1). Simulation then requires solution of algebraic
equations at each integration step. Such differential-algebraic-equation
(DAE) systems are not treated in this book. References [1–4] describe suit-
able mathematical methods and special software.