© National Instruments Corporation 6-1 Xmath Control Design Module
6
State-Space Design
The functions in this chapter are generally termed “modern control” tools.
They are based on the state-space linear system representation, and employ
methods which are generally applicable to both SISO and MIMO
problems. For a review of the state-space system representation, refer to
the State-Space System Models section of Chapter 2, Linear System
Representation.
The process of state-space control system design comprises several distinct
steps. First, you need to assess the controllability and observability of the
system. The designs discussed in this chapter are based on systems that are
both controllable and observable. When you have determined the
controllability and observability of the system, you can design a feedback
control law based on the set of state values. Next, you design an estimator
that estimates the state variable values based on the measured output.
Finally, you combine the controller and estimator to obtain a complete
compensator for the system.
In designing optimal control systems, you pick a performance index you
want to optimize for a given system. This performance index is a quadratic
function reflecting the physical constraints of the system and the
characteristics of any noise that may be present. When this performance
index is a quadratic, you solve mathematically for the optimal control law
and estimator as discussed in the Linear Quadratic Regulator section and
the Linear Quadratic Estimator section.
This chapter concludes with a discussion of system balancing. The
controllability and observability grammians provide a measure of how
controllable and observable a system is. They also can be used to transform
a system to its internally balanced form.

Controllability

Controllability is the property of being able to move the states of a system
arbitrarily in a finite time, given some control input to the system. Although
a particular physical system may be controllable by this definition, not all
state-space models describing that system may be controllable. For
example, if there exists a system eigenvector orthogonal to the input