### Browsing by Author "Tits, A.L."

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Item Absolute Stability Theory, Theory, and State-Space Verification of Frequency-Domain Conditions: Connections and Implications for Computation(1997) Chou, Y.S.; Tits, A.L.; Balakrishnan, V.; ISRThe main contribution of the paper is to show the equivalence between the following two approaches for obtaining sufficient conditions for the robust stability of systems with structured uncertainties: (i) apply the classical absolute stability theory with multipliers; (ii) use the modern theory, specifically, the upper bound obtained by Fan, Tits and Doyle [IEEE TAC, Vol. 36, 25-38]. In particular, the relationship between the stability multipliers used in absolute stability theory and the scaling matrices used in the cited reference is explicitly characterized. The development hinges on the derivation of certain properties of a parameterized family of complex LMIs (linear matrix inequalities), a result of independent interest. The derivation also suggests a general computational framework for checking the feasibility of a broad class of frequency- dependent conditions, and in particular, yields a sequence of computable ﲭixed- -norm upper bounds , defined with guaranteed convergence from above to the supremum over frequency of the aforementioned upper bound.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 Avoiding the Maratos Effect by Means of a Nonrnonotone Line Search: I. General Constrained Problems.(1989) Panier, E.R.; Tits, A.L.; ISRAn essential condition for quasi-Newton optimization methods to converge superlinearly is that a full step of one be taken close to the solution. It is well known that, when dealing with constrained optimization problems, line search schemes ensuring global convergence of such methods may prevent this from occurring (the so called "Maratos effect"). Two types of techniques have been used to circumvent this difficulty. In the watchdog technique, the full step of one is occasionally accepted even when the line search criterion is violated; subsequent backtracking is used if global convergence appears to be lost. In a "bending" technique proposed by Mayne and Polak, backtracking is avoided by performing a search along an arc whose construction requires evaluation of constraint functions at an auxiliary point; along this arc, the full step of one is accepted close to a solution. The main idea in the present paper is to comWne Mayne and Polak's technique with a non-monotone line search proposed by Grippo, Lampariello and Lucidi in the context of unconstrained optimization, in such a way that, asymptotically, function evaluations are no longer performed at auxiliary points. In a companion paper (part II), it is shown that a refinement of this scheme can be used in the context of recently proposed SQP- based methods generating feasible iterates.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 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 Fast Feasible Direction Methods, with Engineering Applications(1991) Tits, A.L.; Zhou, J.; ISROptimization problems arising in engineering applications often present distinctive features that are not exploited, or not accounted for, in standard numerical optimization algorithms and software codes. First, in many cases, equality constraints are not present, or can be simply eliminated. Second, there are several instances where it is advantageous, or even crucial, that, once a feasible point has been achieved, all subsequent iterates be feasible as well. Third, many optimization problems arising engineering are best formulated as constrained minimax problems. Fourth, some specifications must be achieved over a range of values of an independent parameter (functional constraints).While various other distinctive features arise in optimization problems found in specific classes of engineering problems, this paper focuses on those identified above, as they have been the object of special attention by the authors and their co-workers in recent years. Specifically, a basic scheme for efficiently tackling inequality constrained optimization while forcing feasible iterates is discussed and various extensions are proposed to handle the distinctive features just pointed out.

Item Feasible Sequential Quadratic Programming for Finely Discretized Problems from SIP(1997) Lawrence, Craig T.; Tits, A.L.; ISRA sequential Quadratic Programming algorithm designed to efficiently solve nonlinear optimization problems with many inequality constraints, e.g. problems arising from finely discretized Semi-Infinite Programming, is described and analyzed. The key features of the algorithm are (i) that only a few of the constraints are used in the QP sub-problems at each iteration, and (ii) that every iterate satisfies all constraints.Item 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 Globally Convergent Algorithm with Adaptively Refined Discretization for Semi-Infinite Optimization Problems Arising in Engineering Design.(1988) Panier, E.R.; Tits, A.L.; ISROptimization problems arising in engineering design often exhibit specific features which, in the interest of computational efficiency, ought to be exploited. Such is the possible presence of 'functional' specifications, i.e., specifications that are to be met over an interval of values of an independent parameter such as time or frequency. Such problems pertain to semiinfinite optimization. While most of the algorithms that have been proposed for the solution of these problems make use, at each iteration, of a set of local maximizers over the range of the independent parameter, the question of suitably approximating such maximizers is generally left aside. It has been suggested that this issue can be addressed by means of an adaptively refined discretization of the interval of variation of the independent parameter. The algorithm proposed in this paper makes use of such a technique and, by means of a certain memory mechanism, avoids the potential lack of convergence suffered by an existing algorithm.Item Globally Convergent Algorithms for Robust Pole Assignment by State Feedback(1995) Tits, A.L.; Yang, Y.; ISRIt is observed that an algorithm proposed in 1985 by Kautsky, Nichols and Van Dooren (KNV) amounts to maximize, at each iteration, the determinant of the candidate closed-loop eigenvector matrix X with respect to one of its columns (with unit length constraint), subject to the constraint that it remains a valid closed-loop eigenvector matrix. This interpretation is used to prove convergence of the KNV algorithm. It is then shown that a more efficient algorithm is obtained if det (X) is concurrently maximized with respect to two columns of X, and that such a scheme is easily extended to the case where the eigenvalues to be assigned include complex conjugate pairs. Variations exploiting the availability of multiple processors are suggested. Convergence properties of the proposed algorithms are established. Their superiority is demonstrated numerically.Item Globally Convergent Algorithms for Semi-Infinite Optimization Problems Arising in Engineering Design.(1987) Panier, E.R.; Tits, A.L.; ISROptimization problems arising in engineering design often exhibit specific features which, in the interest of computational efficiency, ought to be exploited. Such is the possible presence of 'functional' specifications, i.e., specifications that are to be met over an interval of values of an independent parameter such as time or frequency. While problems involving such specifications could be handled by general purpose nondifferentiable optimization algorithms, the particular structure of functional constraints calls for specific techniques. Suitable schemes have been proposed in the literature. Global convergence is typically achieved by making use of some kind of adaptively refined discretization of the interval of variation of the independent parameter. One previously proposed algorithm exploits the regularity properties of the functions involved to dramatically-reduce the computational overhead incurred once the discretization mesh becomes small. In this paper examples are given that show however that, if the initial discretization is coarse, convergence to a nonstationary point may occur. The cause of such failure is investigated and a class of algorithms is proposed that circumvent this difficulty.Item Interaction, Specification Refinement, and Tradeoff Exploration in Optimization-Based Design of Engineering Systems.(1985) Tits, A.L.; Ma, Z.; ISRThe classical nonlinear programming problem formulation is often too rigid to adequately describe an engineering design problem. Recently a design methodology was developed, based on a problem formulation featuring three types of specifications: hard constraints, soft constraints and objectives. The corresponding mathematical problem is a sequence of constrained 'minimax' optimization problems. Such minimax problems highlight the competition between various specifications and facilitate tradeoff exploration. Accordingly, the new methodology emphasizes designer's intuition and man-machine interaction. In this paper, we demonstrate the crucial role of interaction in an engineering design context. As meaningful interaction is greatly facilitated if suitable sensitivity information is available, we then investigate ways of computing such information , of graphically conveying it to the user.Item Linear Fractional Transformations for the Approximation of Various Uncertainty Sets(1991) Lee, Li; Tits, A.L.; ISRRecently, it was shown that the structured singular value framework can be extended to the case when information on the phase of the uncertainty is available, and a computable upper bound on the corresponding "phase sensitive structured singular value" was obtained. Here we show that the same bound can be obtained via an entirely different approach, using a family of linear fractional transformations. Extension to various uncertainty "shapes" follows.Item Maximal Range for Generalized Stability- Application to Two Physically Motivated Examples.(1989) Saydy, L.; Tits, A.L.; Abed, Eyad H.; ISRSome recent results on guardian maps and their application to generalized robust stability are reviewed and a characterization of the maximum stability range is obtained. This framework is then applied to the analysis of robust stability in two physically motivated examples.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 Multiobjective Optimization of a Leg Mechanism with Various Spring Configurations for Force Reduction(1996) Shieh, W-B.; Azarm, Shapour; Tsai, L-W.; Tits, A.L.; ISRIn this paper, the design of a two degree-of-freedom leg mechanism is accomplished by a two-stage optimization process. In the first stage, leg dimensions are optimized with respect to three design objectives: minimize (i) leg size, (ii) vertical actuating force, and (iii) peak crank torque for an entire walking cycle. Following the optimization of leg dimensions, in the second stage, spring elements with various placement configurations are considered for further reduction of the actuating force and crank torque. Several tradeoff solutions are obtained and a comparison between variously spring configurations is made. It is shown that the inclusion of spring elements can significantly reduce the actuating force and crank torque.Item A New Class of Six-Bar Mechanisms with Symmetrical Coupler Curves(1996) Shieh, W-B.; Azarm, Shapour; Tsai, L-W.; Tits, A.L.; ISRA new class of six-bar mechanisms with symmetrical coupler-point curves is presented. This class of mechanisms is made up of a four-bar linkage with an additional dyad to form an embedded regular or skew pantograph. Because the coupler curve generated at an output point is amplified from that of a four-bar, a compact mechanism with a relatively large coupler curve can be obtained. In addition, due to their structure arrangement, the analysis and synthesis of such mechanisms can be easily achieved. It is shown that the admissible range of transmission angle for such mechanisms is smaller than that of a four-bar mechanism. It is also shown that mechanisms with an embedded skew pantograph exhibit better design flexibility than those with an embedded regular pantograph. Finally, an example mechanism from this class is illustrated and compared with a four-bar linkage with the same coupler curve.Item A New Formula for the Structured Singular Value.(1985) Fan, Michael K-H.; Tits, A.L.; ISRincluded In TR 85-2Item Nonlinear Equality Constraints in Feasible Sequential Quadratic Programming(1995) Lawrence, Craig T.; Tits, A.L.; ISRA simple scheme is proposed for handling nonlinear equality constraints in the context of a previously introduced sequential quadratic programming (SQP) algorithm for inequality constrained problems, generating iterates satisfying all constraints. The key is an idea due to Mayne and Polak (Math. progr., vol. 11, pp 67- 80, 1976) by which nonlinear equality constraints are treated as ﳣﱠtype constraints to be satisfied by all iterates, thus precluding any positive value, and an exact penalty term is added to the objective function which penalizes negative values. Mayne and Polak obtain a suitable value of the penalty parameter by iterative adjustments based on a test involving estimates of the KKT multipliers. We argue that the SQP framework allows for a more effective estimation of these multipliers, and we provide convergence analysis of the resulting algorithms. Numerical results, obtained with the FSQP/CFSQP code, are reported.

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