Direct location of closed-loop singular values in the linear quadratic servo

In the last decade, a number of advances have been made in the application of modern control theory to process control problems. However, there are still a number of unresolved issues which hinder further applications of this theory.

One of these issues involves the modification of the conventional linear quadratic (LQ) regulator into a form suitable for command following and disturbance rejection. This modified form, called the LQ servo, has structure and properties which do not allow the singular values of its closed-loop transfer function to be located directly.

In this thesis, an attempt was first made to extend the theoretical tools which allow us to set the LQ regulator singular values into the domain of the LQ servo. However, the available theoretical tools specify only external properties of the LQ regulator, and cannot be extended to modify those internal properties which differentiate the LQ servo from the LQ regulator.

Next, several low-order state space models were developed in which the LQ servo was derived in pure algebraic form. The principal result of this effort was that the LQ servo and LQ regulator share similar s-domain transfer functions, but with different pole and zero locations, depending on the optimal control gain matrix G and, ultimately, on the state and control weighting matrices Q and R, respectively.

Finally, a six-state, two-input, two-output LQ servo was investigated for a wide range of control/state weightings. The results of this numerical study supported the principal result of the aforementioned algebraic study. In addition, the LQ servo demonstrated the following properties:

- For the particular case studied, the closed-loop transfer function singular value break points for both the LQ servo and the LQ regulator occurred at approximately the same frequencies.

- In contrast, singular value rolloff rates for the LQ servo and the LQ regulator were not consistent - a phenomenon which was particularly noticeable in the minimum singular value trends.

- Over the full range of control/state weightings examined in this study, the LQ servo proved remarkably effective in preserving a constant distance (condition number) between the maximum and minimum singular values as a function of frequency.