Chapter 6 State-Space Design
Xmath Control Design Module 6-14 ni.com
The optimal estimator and regulator problems illustrate the principle of
duality—that for any given system realization {A,B,C} there is a dual
realization {A',C',B'} with related controllability and observability. Refer
to the Duality and Pole Placement section.
regulator( )[Kr,ev,P] = regulator(Sys,Rxx,Ruu,{Rxu})
The regulator( ) function calculates the optimal gain matrix Kr for a
given dynamic system with specified state weighting, control weighting,
and (optionally) cross-weighting matrices.
Alternatively, Kr can be obtained through a call to riccati( ):
[P,resid,Kr,ev]=riccati(Sys,Rxx,Ruu,{S=Rxu})
The syntax for riccati( ) is discussed in the Riccati Equation section.
As shown in the diagram of a continuous-time regulator in Figure 6-3, the
state equation for the regulator is the following:
If you want the closed-loop system eigenvalues, compute them as the
eigenvalues of (A–BKr).
If numerical difficulties are encountered, the algorithm will attempt
to determine whether or not the problem is well posed. Checks are made
to determine stabilizability and the positive definiteness or
semipositive-definiteness of the cost functionals.
The most important design parameters are Rxx and Ruu, which need to be
chosen to reflect the real limitations on how much control can be provided,
or how problematic large state values can be. For an example of how to
design a regulator for the inverted pendulum, refer to Example6-5.
Example 6-5 Designing a Regulator for the Inverted Pendulum
A classic control design problem, the inverted pendulum, consists of a rod
(the pendulum) hinged to the top of a cart which can be moved freely in
either direction along a line. The goal of the controller is to supply an input
u such that the pendulum will be maintained in a vertical position (φ = 0,
in Figure 6-4).
x
·(ABK)r
–x=