6.7 Summary

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6.7 Summary

In this chapter, I discussed how to use the MATLAB Control System Toolbox to design optimal controller and observer gains. The LQR controller minimizes the performance index J shown in Eq. 6.1, which depends on the weighting matrices Q and R supplied by the designer. In most cases, these matrices will be diagonal, with appropriate positive weighting factors applied to each state variable and output signal. Because it is rarely obvious what values to use in the Q and R matrices, the design process generally involves iteratively tuning these matrices and testing the resulting controller. The product of the LQR design process is a controller gain K in the same format as was developed with the pole placement technique in Chapter 5.

The optimal observer described in this chapter is the steady-state Kalman filter. This filter minimizes the steady-state mean-squared state estimation error in the presence of process noise and measurement noise. In the development of the Kalman filter, a process noise model is required. This model represents the dynamic response of the system states and outputs to random inputs modeled by white noise. The process noise model is contained in the G and H matrices of Eq. 6.4. A simplified model of process noise assumes that noise terms are added to each plant input. In this model, G = B and H = D.

The variances of the process noise inputs are contained in the diagonal terms of the process noise covariance matrix Q_N. The measurement noise is also modeled as white noise with variances in the diagonal terms of R_N. The linear plant model (A, B, C, and D matrices), the process noise model (G and H matrices), and the noise covariance matrices (Q_N and R_N) are supplied to the kalman() command, which computes the optimal observer gain L. As with the controller gain K, this gain is in the same format as was used in the pole placement technique.

It is not mandatory that the optimal controller gain be combined with the optimal observer gain. For example, it might make sense to develop an optimal controller gain for a given system and use the pole placement technique to develop the observer. Although this approach results in an observer with less than optimal error characteristics, it also avoids the work of developing the process noise model and the noise covariance matrices. It is up to the designer to select the most appropriate techniques to use for designing controller and observer gains.