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Delay-dependent criterion for guaranteed cost control of neutral delay systems. (English) Zbl 1070.93039

The author considers a system described by a neutral functional differential equation

$\stackrel{˙}{x}\left(t\right)-C\stackrel{˙}{x}\left(t-\tau \right)={A}_{0}x\left(t\right)+{A}_{1}x\left(t-h\right)+Bu\left(t\right)$

with $\left({A}_{0}+{A}_{1},B\right)$ a controllable pair. To this system he associates the quadratic cost function

$J\left(u,\phi \right)={\int }_{0}^{\infty }\left({x}^{T}\left(t\right)Qx\left(t\right)+{u}^{T}\left(t\right)Su\left(t\right)\right)dt$

where $Q>0$, $S>0$ and $\phi \in {C}^{1}$ is the initial condition. The paper aims to find a control law $u\left(t\right)=-{B}^{T}Px\left(t\right)$ such that the resulting closed-loop system is exponentially stable with guaranteed quadratic cost $J\le {J}^{*}$, where ${J}^{*}>0$ is some number. The paper relies on the choice of a quadratic Lyapunov functional leading finally to some Linear Matrix Inequalities whose feasibility guarantees the problem’s solution.

##### MSC:
 93D15 Stabilization of systems by feedback 34K40 Neutral functional-differential equations 93C23 Systems governed by functional-differential equations 93D30 Scalar and vector Lyapunov functions 15A39 Linear inequalities of matrices
##### Software:
LMI toolbox; LMI Control Toolbox
##### References:
 [1] [2] [3] [4] [5] [6] · Zbl 0814.65078 · doi:10.1007/BF01935649 [7] · Zbl 0669.34074 · doi:10.1017/S0004972700027684 [8] · Zbl 0878.34063 · doi:10.1016/0096-3003(95)00301-0 [9] · Zbl 0969.34066 · doi:10.1016/S0016-0032(98)00040-4 [10] · Zbl 0947.65088 · doi:10.1023/A:1021781602182 [11] Ni, B., and Han, Q., On Stability for a Class of Neutral Delay-Differential Systems, Proceeding of the American Control Conference, Arlington, Virginia, pp. 4544-4549, 2001. [12] · Zbl 1060.34043 · doi:10.1023/A:1016087332235 [13] Park, J. H., Robust Stability of a Class of Uncertain Linear Neutral Systems with Time-Varying Delay, Journal of Systems Analysis Modeling Simulation, Vol. 43, pp. 741–748, 2003. [14] [15] · Zbl 0849.93055 · doi:10.1109/9.489272 [16] · Zbl 0807.93054 · doi:10.1080/00207179408923147 [17] · Zbl 0822.93057 · doi:10.1007/BF02192121 [18] · Zbl 0259.93018 · doi:10.1109/TAC.1972.1100037 [19] · Zbl 1041.93530 · doi:10.1016/S0005-1098(99)00007-2 [20] · Zbl 0976.93059 · doi:10.1006/jmaa.2000.7040 [21] [22] Gu, K., An Integral Inequality in the Stability Problem of Time-Delay Systems, Proceedings of 39th IEEE Conference on Decision and Control, Sydney, Australia, pp. 2805-2810, 2000. [23]