Page 385 - Mechanical Engineers' Handbook (Volume 2)

376 Mathematical Models of Dynamic Physical Systems

dx 1 dx 2
ƒ(x ,x ) ƒ(x ,x )
1
1
2
2
2
1
dt dt
Dividing the second equation by the first and eliminating the dt terms yield
ƒ(x ,x )
dx 2 2 1 2

dx 1 ƒ(x ,x )
1
2
1
This first-order equation describes the phase-plane trajectories. In many cases it can be solved
analytically. If not, it always can be simulated.
The phase-plane method complements the describing-function approach. A describing
function is an approximate representation of the sinusoidal response for systems of any order,
while the phase plane is an exact representation of the (free) transient response for first- and
second-order systems. Of course, the phase-plane method theoretically can be extended for
higher order systems, but the difficulty of visualizing the nth-order state space typically
makes such a direct extension impractical. An approximate extension of the method has been
8
used with some considerable success, however, in order to explore and validate the rela-
tionships among pairs of variables in complex simulation models. The approximation is based
on the assumptions that the paired variables define a second-order subsystem which, for the
purposes of analysis, is weakly coupled to the remainder of the system.
8.5 Discrete and Hybrid Systems
A discrete-time system is one for which the dependent variables are defined only at distinct
instants of time. Discrete-time models occur in the representation of systems that are inher-
ently discrete, in the analysis and design of digital measurement and control systems, and
in the numerical solution of differential equations (see Section 7). Because most control
systems are now implemented using digital computers (especially microprocessors), discrete-
time models are extremely important in dynamic systems analysis. The discrete-time nature
of a computer’s sampling of continuous physical signals also leads to the occurrence of
hybrid systems, that is, systems that are in part discrete and in part continuous. Discrete-
time models of hybrid systems are called sampled-data systems.

Difference Equations
Dynamic models of discrete-time systems most naturally take the form of difference equa-
tions. The I/O form of an nth-order difference equation model is
ƒ[y(k n), y(k n 1),..., y(k), u(k n 1),..., u(k)] 0

which expresses the dependence of the (k n)th value of the output, y(k n), on the n
preceding values of the output y and input u. For a linear system, the I/O form can be
written as

y(k n) a n 1 (k)y(k n 1) a (k)y(k 1) a (k)y(k)
0
1
b n 1 (k)u(k n 1) b (k)u(k)
0
In state-variable form, the discrete-time model is the vector difference equation
x(k 1) ƒ[x(k), u(k)]
y(k) g[x(k), u(k)]
where x is the state vector, u is the vector of inputs, and y is the vector of outputs. For a
linear system, the discrete state-variable form can be written as