Hitay Özbay

On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback

In this paper, it is shown that the problem of checking the solvability of a bilinear matrix inequality (BMI), is NP-hard. A matrix valued function, F(X,Y), is called bilinear if it is linear with respect to each of its arguments, and an inequality of the form, F(X,Y)>0 is called a bilinear matrix inequality. Recently, it was shown that, the static output feedback problem, fixed order controller problem, reduced order H/sup /spl infin// controller design problem, and several other control problems can be formulated as BMIs. The main result of this paper shows that the problem of checking the solvability of BMIs is NP-hard, and hence it is rather unlikely to find a polynomial time algorithm for solving general BMI problems. As an independent result, it is also shown that simultaneous stabilization with static output feedback is an NP-hard problem, namely for given n plants, the problem of checking the existence of a static gain matrix, which stabilizes all of the n plants, is NP-hard.

On the design of AQM supporting TCP flows using robust control theory

Recently it has been shown that the active queue management schemes implemented in the routers of communication networks supporting transmission control protocol (TCP) flows can be modeled as a feedback control system. Based on a delay differential equations model of TCPs congestion-avoidance mode different control schemes have been proposed. Here a robust controller is designed based on the known techniques for H/sup /spl infin// control of systems with time delays.

Rate-based flow controllers for communication networks in the presence of uncertain time-varying multiple time-delays

An H^~ based robust controller is designed for a rate-feedback flow-control problem in single-bottleneck communication networks. The controller guarantees stability robustness to uncertain time-varying multiple time-delays in different channels. It also brings the queue length at the bottleneck node to the desired steady-state value asymptotically and satisfies a weighted fairness condition. Lower bounds for stability margins for uncertainty in the time-delays and for the rate of change of the time-delays are derived. A number of simulations are included to demonstrate the time-domain performance of the controller. Trade offs between robustness and time-domain performance are also discussed.

Complexity issues in robust stability of linear delay-differential systems

In this paper the following stability problems concerning linear delay-differential systems are shown to beNP-hard: (i) asymptotic stability independent of delay, and (ii) robust asymptotic stability, when each delay is known to lie in an interval. The main results are based on theNP-hardness of complex bilinear programming over the polydisk n which also shows that the purely complexμ computation, analysis/synthesis problems areNP-hard even if there are no repeated blocks. Another side result of the paper is that checking robust nonsingularity, robust Hurwitz stability, and robust Schur stability of a disk matrix areNP-hard problems.

H/sup /spl infin// optimal and suboptimal controllers for infinite dimensional SISO plants

This paper presents a simple formula for H/sup /spl infin// optimal and suboptimal controllers for unstable SISO distributed plants, with rational weighting functions. The controller is expressed in terms of i) inner and outer parts of the plant, ii) a finite dimensional spectral factor obtained from the weighting functions, and iii) a rational function satisfying certain interpolation conditions. Under certain genericity assumptions, this rational function is of dimension less than or equal to n/sub 1/+l-1 (n/sub 1/+l in the suboptimal case), where l is the number of unstable poles of the plant and n/sub 1/ is the order of the sensitivity weighting function. There are 2(n/sub 1/+l) (2(n/sub 1/+l+1) in the suboptimal case) linear equations, which determine this rational function. These linear equations can be written directly from the structure of the controller. >

Stability Analysis of Switched Time Delay Systems

This paper addresses the asymptotic stability of switched time delay systems with heterogeneous time invariant time delays. Piecewise Lyapunov-Razumikhin functions are introduced for the switching candidate systems to investigate the stability in the presence of an infinite number of switchings. We provide sufficient conditions in terms of the minimum dwell time to guarantee asymptotic stability under the assumptions that each switching candidate is delay-independently or delay-dependently stable. Conservatism analysis is also provided by comparing with the dwell time conditions for switched delay-free systems. Finally, a numerical example is given to validate the results.

A skew Toeplitz approach to the H ∞ optimal control of multivariable distributed systems

In this paper the problem of the

On the synthesis of stable /spl Hscr//sup /spl infin// controllers

H^\infty

Brief On the strong stabilization and stable H∞-controller design problems for MIMO systems

optimization of multivariable distributed systems in the four block setting is studied. This work is based on several previous papers and employs the skew Toeplitz framework developed in [Operator Theory: Adv. Appl., 32 (1988), pp. 21–43], [Operator Theory: Adv. Appl.,32 (1988), pp. 93–112], [Operator Theory and Integral Equations, 11 (1988), pp. 726–767], [J. Functional Anal., 74 (1987), pp. 146–159], [SIAM J. Math. Anal., 19 (1988), pp. 1081–1091].

PID controller design for fractional-order systems with time delays

In this paper, a sufficient condition is derived for the existence of a stable suboptimal controller. It is shown that if a certain Algebraic Riccati Equation, determined from a parameterization of all suboptimal controllers, has positive definite stabilizing solution, then a stable controller of order can be constructed, where is the order of the generalized plant.A sufficient condition is derived for the existence of a stable suboptimal /spl Hscr//sup /spl infin// controller. It is shown that if a certain algebraic Riccati equation, determined from a parameterization of all suboptimal /spl Hscr//sup /spl infin// controllers, has a positive definite stabilizing solution, then a stable /spl Hscr//sup /spl infin// controller of order 2n can be constructed, where n is the order of the generalized plant.