Chapter 6 State-Space Design
© National Instruments Corporation 6-21 Xmath Control Design Module
If you want the closed-loop system eigenvalues, compute them as the
eigenvalues of A–KeC. For an example of how to design a state estimator
for the inverted pendulum problem, refer to Example 6-6.
Example 6-6 Designing a State Estimator for the Inverted Pendulum Problem
Most systems have some level of internal process noise that affects
the value of the states. Returning to the inverted-pendulum plant of
Example 6-5, assume that internal disturbances enter the system with the
inputs. You thus can define a Qxx, which is a function of the input matrix:
Qxx = 0.25*B*B';
Similarly, allow for some disturbance noise affecting the accuracy of the
outputs you measure from the system. You have two output measurements
for this system, thus two separate sources of noise. Assume that the noise
affecting one output measurement does not affect that other, and that the
effects of measurement noise are rather small for this instance.
Qyy = diagonal([1e-6,3e-6]);
[Ke,ev,P] = estimator(ipsys,Qxx,Qyy);
Ke
Ke (a rectangular matrix) =
42.6012 -6.21395
965.35 -159.818
-18.6419 4.67597
-401.88 68.8522
Now that you have access to a set of augmented states for the system (found
with the differential equation for the continuous state vector shown in
Equation 6-5), you can find the optimal controller based on the assumption
of full-state feedback.
Linear Quadratic Gaussian Compensation
Many real-world control system design problems lend themselves to
solutions using a regulator, except that not all the states are available as
directly measured or computed outputs.
A compensator combines your ability to control a system using full state
feedback with our ability to estimate the system states given the system
output. You can design the controller and estimator separately and then
combine them to make the system respond as desired, based on the
measured output. The combination of system, controller, and estimator into