Linear Quadratic Gaussian
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 Kf1 for nominal system with both covariance matrices = I and process disturbance input matrix = control input matrix, = B.

Response with Kalman filter gain Kf2 for nominal system augmented with 2 poles located at s = -0.001. Same covariance matrices as Kf1. Disturbance input matrix for augmented system used.

Response with Kalman filter gain Kf3 for augmented system to boost by + 20db at = 0.001 r/s. The new process disturbance matrix is:

Response with Kalman filter gain Kf4 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 Kf5 for augmented system to make of the return difference function equal to +20 db at = 1 r/s. The new process disturbance matrix is:

This completes the design of the Kalman filter. Throughout the Loop Transfer Recovery process, the Kalman filter gain will be Kf5.

Response of complete augmented system with LQR gain Kc1 with the following weighting matrices:

Response of complete augmented system with LQR gain Kc2 with the same Q weighting matrix and = 10-2.

Response of complete augmented system with LQR gain Kc3 with the same Q weighting matrix and = 10-5.

Response of complete augmented system with LQR gain Kc4 with the same Q weighting matrix and = 10-8.

Response of complete augmented system with LQR gain Kc5 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 Kf5 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 Kc4 are nearly identical to those with Kc5 up to a frequency of 100 r/s. Both of these results are also nearly identical to the Kalman filter designed with Kf5, which is the target of recovery. At 100 r/s, the maximum singular value equals -20 db, so the closed-loop bandwidth will be lower than this frequency (probably about 15 r/s). Above 100 r/s, the curves with Kc4 roll off with a slope of -60 db/decade, while that slope is not achieved with Kc5 until approximately 400 r/s. Thus, Kc4 will provide better attenuation of high frequency disturbances than Kc5 without much loss of stability robustness.

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