"V. Balakrishnan , S.P. Boyd","Global Optimization in Control System Analysis and Design","Control and Dynamic Systems",vol.53, pp.155, 1992  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we want to show that p(s; q) 6= 0 for all s 2 C with Re s 0, q 2 Q. To avoid dropping in degree we assume for simplicity throughout this paper that a 0 (q) 0 for all q 2 Q. y Author to whom all correspondence should be addressed. Unfortunately, most of the methods known from literature, e.g. [4, 5, 14, 15, 16, 18, 23, 32, 34, 35, 36], can only treat problems with polynomial dependency with only a few parameters and or polynomials of lower degree. The genetic algorithm [25] appears to be an exception. However, this algorithm seems not be fully tested for large control problems and gives no guarantee for finding the global ....

V. Balakrishnan and S. Boyd, "Global optimization in control system analysis and design," in Control and Dynamic Systems, vol. 53: High Performance Systems Techniques and Applications, C.T. Leondes, Ed., San Diego: Academic Press, 1992.

....n I rn ) with ffi i real scalars, and B(r) is the subspace of matrices which commute with every element of D(r) S(r) resp. G(r) denotes the set of positivedefinite (resp. skew symmetric) elements of B(r) Finally, the symbol Co denotes the convex hull. 2. LFRs of Rational Systems As shown in [8], system (1) admits the following Linear Fractional Representation (LFR) x = Ax Buu Bpp; q = Cqx Dquu Dqpp; p = Delta(x)q; Delta(x) diag (x1Ir 1 ; xnIrn ) 2) for appropriate nonnegative integers r 1 , r n and matrices A, B u , B p , C q , D qu and D qp . In [9] an ....

V. Balakrishnan and S. Boyd. Global optimization in control system analysis and design. In C. T. Leondes, editor, Control and Dynamic Systems: Advances in Theory and Applications, volume 53. Academic Press, New York, New York, 1992.

....to reduce conservatism. Specifically, if this test cannot establish AQS over the entire parameter box, the initial hyperrectangle can be divided into smaller hyperrectangles and the test reapplied to each of these hyperrectangles. For a complete discussion of branch and bound techniques, see [9, 10]. 4 Time Varying Uncertain Parameters We now turn to the case of time varying parameters i (t) with a bounded rate of variation. As shown below, this more general case can be handled by a minor modification of Theorem 3.1 and the resulting LMI conditions remain less conservative than the ....

Balakrishnan, V., and S. Boyd, "Global Optimization in Control System Analysis and Design," in Control and Dynamic Systems: Advances in Theory and Applications, 53, C. T. Leondes, Academic Press, New York, New York, 1992.

....to solve it exactly, it is highly relevant to study different methods that may provide approximate solutions. Moreover, if one is really interested in solving (1) globally, lower and upper approximation bounds on the global optimal value are required when implementing a branch and bound scheme [1, 18]. It is then necessary to have at hand well worked, widely spread approximation algorithms that feature polynomial complexity. In control engineering, the tool that best fits this description is undoubtedly semidefinite programming (SDP) see for instance [26] for a good overview. In this scope, ....

.... 1 n pC : Proof: First note that E[x 0 rand x rand ] E[u 0 VCV 0 C u] E[Tr(VCV 0 C uu 0 ) Tr(VCV 0 C E[uu 0 ] 13) Vector u can be anywhere on the unit sphere, thus E[u 2 1 ] Delta Delta Delta = E[u 2 n ] Recall that E[u 2 1 Delta Delta Delta u 2 n ] E[1] = 1, so E[u 2 i ] 1 n . Moreover, by symmetry E[u i u j ] 0 for i 6= j. The covariance matrix of u is then E[uu 0 ] 1 n I n and (13) becomes E[x 0 rand x rand ] 1 n TrVCV 0 C = 1 n TrV 0 C VC = 1 n TrXC : The desired equality then follows from LMI relaxation (5) ....

V. Balakrishnan and S. Boyd, "Global Optimization in Control System Analysis and Synthesis", in C. T. Leondes (Editor), "Control and Dynamic Systems", Volume 53, Academic Press, 1992.

....finding the global minimum is valuable, there are several heuristic methods for minimizing the likelihood of not finding the global solution, e.g. running the algorithm repeatedly from different starting points. There are also methods for global optimiza tion, e.g. branch and bound (see, e.g. [33, 9]) that are guaranteed to find the globally optimal design. These methods, however, are often orders of magnitude less efficient than the standard (local optimization) methods. The purpose of this paper is to show that by a change of variables, a wide variety of magnitude filter design problems ....

V. Balakrishnan and S. Boyd. Global optimization in control system analysis and design. In C. T. Leondes, editor, Control and Dynamic Systems: Advances in Theory and Applications, volume 53. Academic Press, New York, New York, 1992.

....can substantially reduce computation times, they do not alter the worst case combinatorial nature of the algorithm, as far as we know. The branch and bound algorithm may be readily applied towards the computation of many other quantities of interest for linear systems with parameters (see [BB91a, BB92] Appendix In the following, we show that the branch and bound algorithm converges in a finite number of steps, provided the bound functions Phi lb ( Delta) and Phi ub ( Delta) satisfy conditions (R1) and (R2) listed at the beginning of x2. We then show that the bounds for the MSD satisfy ....

V. Balakrishnan and S. Boyd. Global optimization in control system analysis and design. In C.T. Leondes, editor, Advances in Control Systems. Academic Press, New York, New York, 1992.

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"V. Balakrishnan , S.P. Boyd","Global Optimization in Control System Analysis and Design","Control and Dynamic Systems",vol.53, pp.155, 1992

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