with Loop Transfer Recovery

The system model is the Remotely Piloted Vehicle (RPV) model found in the appendix of
*Multivariable Feedback Design*, by J.M. Maciejowski, Addison-Wesley, 1989. The model
has 6 states, 2 inputs, and 2 outputs. This example uses loop shaping and output recovery to
achieve robustness, following the procedure described in Chapter 5 of that text. First, a Kalman
filter is designed for the system, assuming identity matrices for the process disturbance and
measurement noise covariances. Next, integrators are added to each input channel, increasing
the system order to 8. This is done to improve steady-state performance. Then the process
disturbance covariance matrix is updated in three steps to obtain a desired frequency response
for the loop gain. With this approach, the covariance matrices are used as design parameters
instead of modeling actual noise covariances.

Once the Kalman filter's frequency response is set, the loop transfer recovery process begins. This involves adjusting the weighting matrices of the Linear Quadratic Regulator (LQR) until the frequency response of the complete system (LQG controller, augmentation dynamics, and actual system model) approaches that of the final Kalman filter design. The following figures illustrate the steps in the design.

Response with Kalman filter gain *K _{f}*

Response with Kalman filter gain *K*_{f2} for nominal
system augmented with 2 poles located at s = -0.001. Same covariance matrices as
*K _{f}*1. Disturbance input matrix for augmented system used.

Response with Kalman filter gain *K _{f}*

Response with Kalman filter gain *K*_{f4} gain for
augmented system to raise and to lower at
= 1 r/s to bring the crossover frequencies closer together. The new
process disturbance matrix is:

Response with Kalman filter gain *K _{f}*

This completes the design of the Kalman filter. Throughout the Loop Transfer Recovery
process, the Kalman filter gain will be *K _{f}*5.

Response of complete augmented system with LQR gain
*K*_{c1} with the following weighting matrices:

Response of complete augmented system with LQR gain
*K*_{c2} with the same *Q* weighting matrix and
= 10^{-2}.

Response of complete augmented system with LQR gain
*K*_{c3} with the same *Q* weighting matrix and
= 10^{-5}.

Response of complete augmented system with LQR gain
*K*_{c4} with the same *Q* weighting matrix and
= 10^{-8}.

Response of complete augmented system with LQR gain
*K*_{c5} with the same *Q* weighting matrix and
= 10^{-11}.

As goes to 0, recovery of the LQG/LTR design to the Kalman filter design
with *K _{f}*5 is more complete, so the performance and stability robustness
properties of the Kalman filter should be equaled by the LQG/LTR design. The bad news is that
the -20 db/decade roll-off of the Kalman filter at high frequencies is also recovered. This
leads to bad rejection of high frequency noise and unmodeled dynamics. Less recovery gives a
steeper roll-off, at least -40 db/decade, which is desirable; however, the performance and
stability robustness of the LQG/LTR design will then not equal the Kalman filter design. Thus,
there must be a tradeoff between stability robustness and low frequency performance versus
high frequency performance.

From the last two figures it can be seen that the singular values obtained with
*K _{c}*4 are nearly identical to those with

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*Lastest revision on 05/09/01 06:16 PM
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