This thesis presents a method which greatly reduces the conservativeness of conic sector analysis for sampled data feedback systems. The new method evaluates the stability and closed-loop performance of systems with structured uncertainty in the plant transfer function, including MIMO systems and those with multiple sampling rates. In contrast to most multirate analysis techniques, the sampling rates need not be related by rational numbers; this allows analysis when samplers are not strobed to a common clock.
The method is based on a theorem from P. M. Thompson which shows how to construct a conic sector containing a hybrid operator. Combining this theorem with the Structured Singular Value approach of J. C. Doyle, with its heavy use of diagonal scaling, provides an analysis framework for systems with multiple structured plant perturbations. Chapter 3 presents a theorem for the optimal conic sector radius in the SISO case; a MIMO extension of the the theorem completes the development of the new method. Chapter 5 gives three examples.
Chapter 6 presents a new method, based on the complex cepstrum, for synthesis of SISO rational functions to match given "target" transfer functions. The method offers complete control over stability and right half plane zeros. It solves directly for poles and zeros, avoiding the numerical sensitivity of methods which solve for polynomial coefficients. It can synthesize minimum phase functions to match a given magnitude or phase curve. In an example, it is used to synthesize a low- order digital replacement for an analog compensator which gives no degradation of stability margin or step response.
This thesis also presents a method for Kranc vector switch decomposition in state space; this is for stability analysis and input-output simulation of perturbed multirate systems. Moving the 30-year-old Kranc technique from the frequency domain to the state-space domain simplifies the analysis tremendously. Because the number of states is preserved, the dimensionality problems long associated with the Kranc method disappear. The new method is also useful for simulating intersample ripple behavior.
