Graduation date: 1991

Most of the existing research on mathematical morphology is restricted to the deterministic case. This thesis addresses the void in the results on the stochastic properties of morphological filters. The primary results include analysis of the stochastic properties of morphological operations, such as dilation, erosion, closing and opening. Two unbiased morphological filters are introduced and a quantitative description of the probability distribution function of morphological operations on independent, identically distributed random signals is obtained. Design of an optimal morphological filter in the sense of a criterion proposed here is also discussed. A brief, but systematic description of the definitions and properties of deterministic morphological operations on sets is presented to establish the necessary background for the analysis of the filter stochastic properties. / Graduation date: 1992

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 156-164).

This thesis is concerned with the dissipative control and filtering problems of singular systems. Four classes of singular systems are considered: delay-free singular systems, singular systems with constant time-delay, uncertain singular systems with time-varying delay and sensor failures, and singular Markovian jump systems with actuator failures. For delay-free singular systems, the system augmentation approach is employed to study the dissipative control and filtering problems. First, the approach is used to solve the dissipative control problem by static output-feedback for standard state-space systems which are the special cases of singular systems. For a continuous-time standard state-space system, the closed-loop system is represented in an augmented system form. Based on the augmented system, a necessary and sufficient dissipativity condition is proposed, which decouples the Lyapunov matrix and controller matrix. To further separate the Lyapunov matrix and the system matrices, an equivalent condition is obtained by introducing some slack matrices. Then, a necessary and sufficient condition for the existence of a static output-feedback controller is proposed, and an iterative algorithm is given to solve the condition. For discrete-time singular systems, by giving an equivalent representation of the solution set, a necessary and sufficient dissipativity condition is proposed in terms of strict linear matrix inequality (LMI) which can be easily solved by standard commercial software. Then a state-feedback controller design method is given based on the augmentation system approach. The method is extended to the static output-feedback control problem and the reduced-order dissipative filtering problem. For continuous-time singular time-delay systems, the problem of state-feedback dissipative control is considered. An improved delay-dependent dissipativity condition in terms of LMIs is established by employing the delay-partitioning technique, which guarantees a singular system to be admissible and dissipative. Based on this, a delay-dependent sufficient condition for the existence of a state-feedback controller is proposed to guarantee the admissibility and dissipativity of the closed-loop system. In addition to delay-dependence, the obtained results are also dependent on the level of dissipativity. Moreover, the results obtained unify existing results on H∞ performance analysis and passivity analysis for singular systems. For discrete-time singular systems with polytopic uncertainties, time-varying delay and sensor failures, the problem of robust reliable dissipative filtering is considered. The filter is designed by the reciprocally convex approach such that the filtering error singular system is admissible and strictly (Q, S, R)-dissipative. For singular systems with time-varying delay and sensor failures, a sufficient condition of reliable dissipative analysis is obtained in terms of LMIs. Then the result is extended to the uncertain case by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, a desired filter for uncertain singular systems with time-varying delay and sensor failures is obtained by solving a set of LMIs. For continuous-time singular Markovian jump systems with actuator failures, the problem of reliable dissipative control is addressed. Attention is focused on the state-feedback controller design method such that the closed-loop system is admissible and strictly (Q, S, R)-dissipative. A sufficient condition is obtained in terms of strict LMIs. Moreover, the results obtained unify existing results on H∞control and passive control on singular Markovian jump systems. / published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

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The Rational Chebyshev Function was first introduced by Bernstein (1926), used by Sharpe (1953), then later by Heldman (1955) to design elliptic-characteristic filters. Namely for a filter of order N, we have N/2 equal ripples in the passband and N/2 equal ripples in the stopband of the magnitude response. Here, the same mechanics are used but are now producing a new and different type of response. It has N/2 ripples in the passband but only one ripple in the stopband for all orders. As N increases from three, the result is a substantial saving in number of capacitors in the passive ladder realization of the above function as compared to that of traditional elliptic filters of the same order N. It also has been discovered that the above ladder's element values can be expressed as explicit expressions involving only the coefficients of the transfer function. These expressions can also be used for other types of filters. Numerically, the design can be carried out by a Fortran program or a set of programs on a programmable calculator. The design is termed automated because the user needs only to give the three specifications: the filter order N, the stopband zeros Z, and the passband ripple amount R(p). The program automatically selects the starting point for the given case and proceeds. The numerical results of the above programs over a range of specifications has led to a surprising and simple expression relating the above specifications to the minimum stopband attenuation. This is a useful relationship for the designer to estimate the zero position when using the programs.

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In this study, we address the factorization problem in the Hardy H('p) spaces, and provide a fast algorithm for its implementation with applications to some important engineering problems. The Thesis is presented in three autonomous papers. / In the first paper we lay down the technical foundation of the new approach in the scalar case. First, the factorization problem is formulated in the H('p) spaces. A formulation with sufficient generality to encompass practically all such engineering problems. Necessary and sufficient conditions for the existence of the spectral factors are derived, and a characterization of the class of functions admitting a canonical factorization is obtained. The reduction method is applied to certain Toeplitz equations in H('2) space to generate a sequence of approximate spectral factors. When the Laguerre basis is used in the reduction method the Toeplitz equation turns out to a Toeplitz set of linear equations. We also provide an error bound and an estimate for the speed of convergence. / In the second paper the matrix version of all the scalar results is provided and enriched with discussions and extension. In particular, we have shown that the factorization problem is associated with the solutions of certain Toeplitz equations in / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / spaces. The classical Gohberg-Krein factorization is re-examined within the framework developed here, and the connections between the outer-factorization, the canonical factorization, and inversion of certain Toeplitz operators have also been unveiled. / In Part III we generalize the Davis and Barry formula for the feedback gain in the LQR problems. The new setting, equipped with the spectral factorization method, provides fast and efficient algorithms for solving a wide class of LQR problems, rational matrix factorization, and positive polynomials factorization. Our parallel results for the discrete time case are given in brief together with many interesting computational properties.

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