### Browsing by Author "Fan, Michael K-H."

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Item An Algorithm to Compute the Structured Singular Value.(1986) Fan, Michael K-H.; ISRThe concept of structured singular value was recently introduced by Doyle as a tool for analysis and synthesis of feedback systems with structured uncertainties. It is a key to the design of control systems under joint robustness and performance specifications and it nicely complements the H_INFINITY approach to control system design. This report proposes an algorithm to compute the structured singular value.Item Aspects of Optimization-Based CADCS.(1988) Tits, A.L.; Fan, Michael K-H.; Panier, E.R.; ISRWith the recent dramatic increase in available computing power, numerical optimization has become an attractive tool for the design of complex engineering systems. Yet, generalized use of numerical optimization techniques in design has been hindered by (i) the difficulty to translate in a faithful manner the actual design problem into any kind of rigid mathematical optimization problem, (ii) the inability of classical optimization tools to efficiently take into account the many distinctive features of optimization problems arising in a design context, and (iii) the unavailability of software tools offering to the designer a powerful as well as congenial environment supporting such capabilities. In this paper, some aspects of these questions are touched upon and avenues are suggested to address them. In particular, a recently proposed interaction driven design methodology is briefly described and numerical optimization schemes satisfying two specific requirements of many design problems are sketched. As an example, the design of a controller for a copolymerization reactor using the Maryland developed CONSOLE system is considered.Item Characterization and Efficient Computation of the Structures Singular Value.(1985) Fan, Michael K-H.; Tits, A.L.; ISRThe concept of structured singular value was recently Introduced by Doyle (Proc. IEEE, vol. 129, pp. 242-250, 1982) as a tool for the analysis and synthesis of feedback systems with structured uncertainties. It is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the H^INFINITY approach to control system design. In this paper, it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems. Properties of these optimization problems are exhibited, leading to a fast algorithm that always yields the structured singular value for block-structures of size no larger than 3, and often does for block-structures of larger size.Item CONSOLE User's Manual.(1987) Fan, Michael K-H.; Koninckx, Jan; Wang, Li-Sheng; Tits, Andre L.; ISRThe CONSOLE tandem is a tool for optimization-based design of a large class of systems. The essential requirements are that a simulator be available for evaluating the performance of instances of the system under consideration and that the parameters to be optimally adjusted vary over a continuous (as opposed to discrete) set of values. Todate, CONSOLE has been used on problems as diverse as design of a controller for a flexible arm, an aircraft, or a copolymerization reactor. The manual is organized as follows. In Chapter 1, the ideas and principles upon which CONSOLE is constructed are outlined and the design methodology underlying CONSOLE is sketched. Chapter 2 introduces the novice user to CONSOLE by way of a simple tutorial example. This chapter is strongly recommended to new users as it leads them step by step through a CONSOLE session. Chapter 3 is entirely devoted to CONVERT. It includes a thorough description of the different data types, assignments and commands that form the CONVERT syntax. Chapter 4 discusses SOLVE. The essential features of the optimization algorithm are outlined and the operation of SOLVE SOLVE is discussed. Special attention is given to the interactive capabilities of SOLVE, in particuar the Pcomb display. In Chapter 5, the question of using an interface between SOLVE and simulators of the user's choice is discussed. A general structure is given. Finally, Chapter 6 presents two design examples. Appendices A and B consist in reference manuals, for CONVERT and SOLVE respectively.Item Design of a Flight Controller for an F14 Aircraft Using the DELIGHT.Mary.Lin Optimization-Based CACSD System.(1985) Fan, Michael K-H.; Walrath, C.D.; Lee, C.; Tits, A.L.; Rimer, M.; Grant, R.; Levine, William S.; ISRItem Generalized Numerical Range and the Computation of the Structured Singular Value.(1987) Fan, Michael K-H.; Tits, A.L.; ISRThe concept of structured singular value was recently introduced by Doyle (Proc. IEE, vol. 129, pp. 242-250, 1982) as a tool for the analysis and synthesis of feedback systems with structured uncertainties. In this paper an equivalent expression for the structured singular value is proposed, leading to an alternative algorithm for its computation. The new approach is based on the geometric properties of the generalized numerical range of certain matrices. Similar to previously considered schemes, the algorithm proposed here is proven to give the correct value for block-structures of size up to 3. For larger sizes, insight is gained in the question of the possible 'gap' between the structured singular value and its known upper bound.Item Geometric Aspects in the Computation of the Structured Singular Value.(1986) Fan, Michael K-H.; Tits, A.L.; ISRThe concept of structured signular value was recently introduced by Dolye (Proc. IEEE, vol.129, pp. 242-250, 1982) as a tool for the analysis and synthesis of feedback systems with structured uncertainties. In this paper an equivalent expression for the structured singular value is proposed, leading to an alternative algorithm for its computation. The new approach is based on the geometric properties of certain family of sets. Similar to previously considered schemes, the algorithm proposed here is proven to give the correct value for block-structures of size up to 3. For larger sizes, insight is gained in the question of the possible 'gap' between the structured singular value and its known upper bound.Item A Measure of Worst-Case HPerformance and of Largest Acceptable Uncertainty(1991) Fan, Michael K-H.; Tits, A.L.; ISRThe structured singular value (SSV or ) is know to be an effective tool for assessing robust performance of linear time- invariant models subject to structured uncertainty. Yet all a single analysis provides is a bound ݠon the uncertainty under which stability as well as Hperformance level of k/ݠare guaranteed, where k is preselectable. In this paper, we introduce a related quantity, denoted by v which provides answers for the following questions: (i) given ݬ determine the smallest with the peoperty that, for any uncertainty bounded by ݬ an H performance level ofItem A Method for Computing the Distance of a Stable Matrix to the Set of Unstable Matrices.(1989) Fan, Michael K-H.; Tsing, N.K.; ISRWe propose a method to compute the spectral norm distance from a given matrix A to the set of matrices having at least an eigenvalue on the imaginary axis. It is shown that the distance is one of the roots of a suitably constructed polynomial in one variable. Our method can be easily generalized to compute the distance from A to the set of matrices having at least an eigenvalue on any straight line or circle. Thus, it can be applied to compute the distance from a stable matrix to the set of unstable matrices in either continuous or discrete sense.Item Multiobjective Dynamic Optimization of Semibatch Free Radical Copolymerization Proceas with Interactive CAD.(1987) Butala, D.; Choi, Kyu Y.; Fan, Michael K-H.; ISROptimal open loop control strategies are developed for batch and semibatch free radical copolymerization of styrene and acrylonitrile using the interactive computer aided design (CAD) tool CONSOLE, which was developed at the University of Maryland in 1987. CONSOLE emphasizes control engineer's intuition and man- machine interaction and includes a classification of various design specifications. Conflict resolution among the different objectives is made by scaling of specification space based on designer's knowledge of the polymerization proceas. A feasible direction algorithm is used to obtain optimal monomer addition and reactor temperature policies.Item A New Formula for the Structured Singular Value.(1985) Fan, Michael K-H.; Tits, A.L.; ISRincluded In TR 85-2Item On Block Limiting Norm and Structured Singular Value.(1987) Fan, Michael K-H.; Fu, Jyun-Horng; ISRThe notion of limiting norm, introduced by Pokrovskii (Soviet Math. Dokl., vol. 20, pp. 1314-1317, 1979), is generalized to that of block limiting norm. A resemblance of inequalities shared by both the block limiting norm and the structured singular value, introduced by Doyle (Proc. IEE, vol. 129, pp. 245-250, 1982), motivates further investigation of their relationships. To that effect, the concept of generalized spectral radius of a set of linear operators is introduced. It is then shown that, for block-structure of size less than 4, the block limiting norm is equal to the structured aingular value and that, in the general case, the block limiting norm is always no less than the structured singular value. Finally, better bounds are obtained for both the block limiting norm and the structured singular value.Item On Robust Stability of Linear State Space Models.(1988) Fan, Michael K-H.; Doyle, John C.; Tits, A.L.; ISRThe structured singular value (MU), introduced by Doyle [1] allows to analyze robust stability and performance of linear systems affected by parametric as well as dynamic uncertainty. While exact computation of MU can be prohibitively complex, an efficiently computable upper bound was obtained in [2], yielding a practical sufficient condition for robust stability and performance. In this note, the results of [2] are used to study the case of state space models of the form x{WITH DOT ABOVE IT}=(A_0={SIGMA i=1 to m of DELTA_i * A_i}) where the A_i's are n X n real matrices and the DELTA_i's are uncertain real parameters. The case where the A_i's have low rank is given special attention. When the A_i's all have rank one, (1) is equivalent to the model used by Qiu and Davison [3], which itself generalizes that used by Yedavalli [4]. By means of two examples, we compare our bound to those proposed in [3] and [4].Item On the Generalized Numerical Range.(1986) Fan, Michael K-H.; Tits, A.L.; ISRLet A_k, k = 1, ...., m be n x n Hermitian matricies and let f: C^n --> R^m have components f^k(x) = x^H A_k(x), k = 1, ..., m. When n >= 3 and m = 3, the set W(A_1>,..., A_m) = {f(x): || x || = 1} PROPER subset of R_m is convex. This property does not hold in general when m > 3. These particuar cases of known results are proven here using a direct, geometric approach. A geometric characterization of the contact surfaces is obtained for any n and m. Necessary conditions are given for f(x) to be on boundary of W(A_1,..., A_2) or on the certain subsets of this boundary. These results are of interest in the context of the computations of the structured singular value, a recently introduced tool for the analysis and synthesis of control systems.Item On the Stability of Polynomials with Uncoupled Perturbations in the Coefficients of Even and Odd Powers.(1987) Panier, E.R.; Fan, Michael K-H.; Tits, A.L.; ISRIn this note, we present some results concerning the stability (in Hurwitz' sense) of a family of polynomials with even and odd coefficients subject to uncoupled perturbations. It is shown that the stability of an appropriate small subset of extreme polynomials guarantees the stability of the entire family. In particular, a polytope of polynomials with the even-odd uncoupling property is stable provided a certain small subset of its vertices is. For the case of an arbitrary subset of polynomials, this result gives a less conservative sufficient condition than that provided by Kharitonov's theorem.Item Optimization-Based Design of Nonlinear Systems Using CONSOLE and SIMNON.(1987) Fan, Michael K-H.; ISRThe most challenging task when designing a nonlinear system is that of coming up with an appropriate system 'structure.' This task calls extensively upon the engineer's ingenuity, creativity, intuition and experience. After a structure has been (maybe temporarily) selected, it remains to determine the 'best' values of a number of 'design parameters.' The input from the engineer is still essential here, as trade offs are likely to be encountered. However, except in the simplest cases, achieving anything close to optimal would be impossible without the support of numerical optimization and simulation tools. In this report, we present an interface between CONSOLE, an interactive, optimization-based design package, and SIMNON, an interactive program for simulation of systems governed by ordinary differential equations and difference equations. We show, by means of an example, how to set up a design problem for nonlinear systems by using CONSOLE and SIMNON.Item A Power Method for the Structured Singular Value.(1988) Packard, A.; Fan, Michael K-H.; Doyle, John C.; ISRThis paper presents an iterative algorithm to compute lower bounds for the structured singular value. The algorithm resembles a mixture of power methods for eigenvalues and singular values, which is not surprizing, since the structured singular value can be viewed as a generalization of both. If the algorithm converges, a lower bound for 11 results. We prove that MU is always an equilibrium point of the algorithm, however, since in general there are many equilibrium points, some heuristic ideas to achieve convergence are presented. Extensive numerical experience with the algorithm is discussed.Item Robustness in the Presence of Joint Parametric Uncertainty and Unmodeled Dynamics.(1988) Fan, Michael K-H.; Tits, A.L.; Doyle, John C.; ISRIt is shown that, in the case of joint real parametric and complex uncertainty, Doyle's structured singular value can be obtained as the solution of a smooth constrained optimization problem. While this problem may have local maxima, an improved computable upper bound to the structured singular value is derived, leading to a sufficient condition for robust stability and performance.Item Robustness under Uncertainly with Phase Information.(1989) Lee, Li; Tits, A.L.; Fan, Michael K-H.; ISRThe framework of Doyle's structured angular value is extended to take advantage of possibly available phase information on the dynamic uncertainty. A computable upper bound is obtained for this phase-sensitive structured singular value.Item Structured Singular Value and Geometry of the m-Form Numerical Range.(1987) Wang, J.C.; Fan, Michael K-H.; Tits, A.L.; ISRAlthough the question of the numerical evaluation of Doyle's structured singular value has been repeatedly addressed, it is not yet entirely resolved in the case of block structures of size larger than 3. It has been shown recently that this question can be reduced to that of iteratively computing the distance from the origin to the m-form numerical range of certain tuples of matrices. How to effectively compute such distances in the nonconvex case (which may arise when dealing with more than 3 blocks) is an open problem. In this paper, in an attempt to tackle this problem, the question of graphically displaying sections of the m-form numerical range is investigated.