Analysis, Control and Optimization of Complex Dynamic by El-Kébir Boukas, Roland P. Malhamé

By El-Kébir Boukas, Roland P. Malhamé

Research, keep watch over and Optimization of advanced Dynamic platforms gathers in one quantity a spectrum of advanced dynamic structures similar papers written via specialists of their fields, and strongly consultant of present learn developments. advanced structures current very important demanding situations, in nice half as a result of their sheer dimension which makes it tricky to know their dynamic habit, optimize their operations, or learn their reliability. but, we are living in an international the place, as a result of expanding inter-dependencies and networking of platforms, complexity has develop into the norm. With this in brain, the amount contains components. the 1st half is devoted to a spectrum of complicated difficulties of selection and keep watch over encountered within the region of creation and stock structures. the second one half is devoted to massive scale or multi-agent approach difficulties taking place in different components of engineering corresponding to telecommunication and electrical energy networks, in addition to extra widely used context.

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1994)) A linear matrix inequality where x E Rm is the variable and the symmetric matrices Fi = F: E Rnxn, i = 0 , . . ,m are given. e, v T ~ ( x ) v> 0 for all nonzero v E Rn. Linear matrix inequalities appear naturally in the formulation of sufficient conditions for Lyapunov stability of open-loop linear systems as the next example illustrates. 8). This example motivates the following definition. 9) with u ( t )E 0 , w ( t ) = 0 is said to be internally quadratically stable i f there exists a symmetric and positive definite matrix P > 0 , satisfying We now stake a very important Lemma that will be repeatedly used in this paper.

E. (1976). Abbreviated proof in the lost sales case. P. ), pp. 105-106 Academic Press, New York, NY. S. and zipkin, P. (1993). Inventory control in a fluctuating demand environment. Operations Research, 41:351-370. Sznadjer, R. A. (1992). Some comments on a theorem of Hardy and Littlewood. Journal of Optimization Theory and Applications, 75(1):201-208. Veinott, A. (1966). On the optimality of (s, S ) inventory policies: New conditions and a new proof. SIAM Journal on Applied Mathematics, l4(5):1067-1083.

3 Let y be a positive constant. 24) with Ki = KX-', i = 1 , 2 is stable and moreover the closed-loop system satisfies the disturbance rejection level of y > 0. 5) that must always be satisfied. Extra constraints should then be added to the previous system of LMIs to force the control law to always satisfy the bounds. For this purpose, we note first that when the initial condition is known (which is the case in our problem), we can add the constraint x ~ X - ' x o 5 1, which using Schur complement leads to the LMI 2 Inventory Control of Switched Production Systems: LMI Approach 35 Since the Lyapunov function is V = xTx-'x, and the level sets of the Lyapunov function are invariant sets for the system trajectories, this LMI guarantees not only that the initial state is inside the set but also that x(t) E 9 , Vt 2 0.

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