Page 727 - Mechanical Engineers' Handbook (Volume 2)

718 State-Space Methods for Dynamic Systems Analysis

of digital computers both for analysis and control synthesis and for implementation of the
controllers has been an important factor underlying the growing use of state-space-based
methods.
4. State-space-based methods have the potential to improve the performance of con-
trolled systems if such systems can be modeled accurately. They have been less success-
ful in cases where system models are characterized by significant uncertainty. Classical
transform-based techniques have been and continue to be widely used in such cases. In fact,
one of the more encouraging trends in control systems development has been the establish-
ment of links between state-space-based methods and transform-based methods. 5

Though the concept of state has been invoked by a number of methods of classical
mechanics and is implicit in the phase-plane concept used for nonlinear system stability
analysis, the effective application of state-space-based methods for analysis and control of
dynamic system behavior has occurred only over the last three decades. Pioneering theoret-
ical work by Kalman 8–11 and others and the availability of digital computers for performing
analysis and design computations have been important underlying factors. State-space meth-
ods have been most successful in aerospace control applications and less so in a variety of
industrial control applications. Among the factors favoring increased emphasis in the future
on state-space methods are:
1. The emphasis on controlled system performance improvement resulting from imper-
atives such as improved efficiency of energy utilization and improved productivity
2. The increasing availability of inexpensive but powerful digital computers for off-line
analysis and design computations and online control computations

In Sections 2–7, methods for analysis of dynamic systems using state-space methods
are described. Even though most of the results presented in the literature use the continuous-
time formulation, the fact that digital computers will be increasingly used for controller
implementation implies that discrete-time formulations have significant practical importance.
Hence, both continuous-time and discrete-time formulations are presented to the fullest extent
possible.

2 STATE-SPACE EQUATIONS FOR CONTINUOUS-TIME
AND DISCRETE-TIME SYSTEMS
The differential equations describing the input–output behavior of an nth-order, continuous-
time, nonlinear, time-varying, lumped-parameter system can be written in the form of a first-
order vector ordinary differential equation and a vector output equation:

x(t) f[x(t), u(t), t] t t (1)
0
y(t) g[x(t), u(t), t] t t (2)
0
where x(t) n-dimensional state vector
y(t) p-dimensional output vector
u(t) r-dimensional input vector
f, g vectors of appropriate dimension whose elements are single-valued nonlinear
functions of the arguments noted