Although the work above is very excellent, little research considered **robust** **stability** of quaternion-valued neural networks with delay and interval parameter uncertainties. Recently, Chen in [35, 36] considered the **robust** **stability** with diﬀerent kinds of delays for quaternion-valued neural networks, some suﬃcient **conditions** have also been obtained about the existence, uniqueness and global asymptotic **robust** **stability**. But in his judging criteria, the negative deﬁniteness of two matrices is needed, moreover, the entries in these matrices are all the maximal values, which are decided by the absolution of the upper and lower bounds of elements for the connection weight matrix, the sign of connection weight matrix is ignored. In this article, inspired by the methods of [20], we proposed a new approach to overcome this defects, in our unique criteria matrix, the given elements depend on not only the lower bounds but also the upper bounds of the interval parameters, which is diﬀerent from previous contributions and extends the relevant work in Refs. [19, 20, 22, 27, 35].

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A novel approach to design a **robust** controller, **robust** gain-scheduled controller and **robust** switched controller with arbitrarily switching algorithm is presented. The rate of the switching variable is assumed to be θ ˙ = ±∞ for ideal, and | θ| ˙ < ∞ for non-ideal switching. The proposed design procedure is based on the Bellman-Lyapunov equation, guaranteed cost, and **robust** **stability** **conditions** using parameter-dependent quadratic sta- bility, or for switched systems the multi-parameter-dependent quadratic **stability** approach. The obtained feasible **robust** **stability** and performance **conditions** for **robust**, **robust** gain- scheduled, and **robust** switched controller design are in the form of BMIs. Despite that the obtained solution is bilinear regarding the variables, the conservativeness of the introduced design procedure is markedly reduced compared with such approaches that use the multi- convexity lemma and/or its relaxations. In our case, the conservativeness is reduced due to the fact that the presented solution is directly convex regarding the scheduled and uncertain parameters. In addition, in the presented solution there is no need for major restrictions in system and/or controller matrices unlike in many references.

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Lyapunov **stability** is a general theory available for any differential equation. Over the past researches on time-delay systems, many useful approaches are applied to guarantee the **stability** or stabilization of systems. The application of Lyapunov-Krasovskii functional theory has first started for system without neither uncertainties nor control [2] [5], some **robust** **stability** **conditions** based on LMI approach are given. Then, the guaranteed cost control problem for neutral time delay system with feed-back control is investigated. Some papers are interested on **stability** and stabilization of this type of system where a linear–quadratic cost function is considered

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Abstract: - In this paper, design of pi/pid controllers for the gas turbine application is proposed. The gas turbine application considered here is a TITO system which has two inputs and two outputs. In most of the industrial applications controllers are of pid type. In this paper a new interval analysis method is proposed for the design of **robust** pi/pid controllers. This method uses the **robust** **stability** **conditions** to derive necessary and sufficient **conditions** for the **stability** of interval systems. These **conditions** are used to get the inequality constraints in terms of controller parameters. The constraints derived are solved using MATLAB. The optimum controller parameters are obtained using this method. The simulation results of the gas turbine application successfully verified and show the efficacy of the proposed method.

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Though LMI-based approaches have enjoyed great success and popularity, there still exist a large number of design problems that either cannot be represented in terms of LMIs, or the results obtained through LMIs are sometimes conser- vative. Recently, as a post-LMI framework, an SOS based approach has received a great deal of attention in control of nonlinear systems using polynomial fuzzy systems and con- trollers, which includes the well-known Takagi-Sugeno fuzzy systems and controllers as special cases. An SOS approach to polynomial fuzzy control system designs has first presented in [9]-[13]. It can be seen that SOS approaches [9]-[22] provide more extensive and/or relaxed results for the existing LMI approaches [2], [3], [23]-[35] to T-S fuzzy model and control. However, there exists a very few literature on SOS- based **robust** control designs for polynomial fuzzy systems with uncertainties. To the best of our knowledge, an SOS- based **robust** control design for polynomial fuzzy systems with uncertainties has been discussed only in [36]. The most important point of SOS-based design **conditions** is that, to obtain convex SOS design **conditions**, the existing SOS-based design **conditions** [9]-[20] utilize a typical transformation from non-convex SOS design **conditions** to convex SOS design **conditions**. However, the transformation often results in some conservative issues although no such conservatism exists in LMI transformation cases. In [36], the typical transformation is employed to obtain convex SOS **robust** stabilization condi- tions. Furthermore, not only the conservative issues but also other two difficulties are found in the existing SOS approach. One is a restrictive polynomial Lyapunov function setting that leads to conservative **stability** results. The other is that the **stability** does not generally hold globally in the existing SOS approach. These will be concretely discussed in Remarks 2 and 3. This paper gives new ideas to solve the conservative issues and the difficulties in the existing SOS approach.

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In the theory of set-valued analysis, there is a famous theorem, namely Hörmander’s theorem, which establishes a relationship between the distance of two sets in Euclidean space and the maximum difference between their respective support functions over the unit ball of the same space, see Castaing and Valadier (1977, Theorem II-18). Here, we extend the theorem to the set of probability measures. One of the main reasons behind this extension is that in minimax distributionally **robust** optimization problems, the inner maximization of the worst expected value of a random function over an ambiguity set of probability distributions is indeed the support function of the random function over the ambiguity set. Therefore, in order to look into the difference between the worst expected values based on two ambiguity sets, it is adequate to assess the discrepancy between two support functions of the sets. We will come back to this in the next section. To this end, we need the concept of weak compactness of probability measures under the topology of weak convergence. Recall that a sequence of probability measures {P N } ⊂ P (Ξ) is said

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In this paper, we focus on solving the state feedback controller design problem of NCSs in discrete-time domain and under a general framework, where large network induced delay and arbitrary packet dropout are taken into account simultaneously. The mathematical model is proposed using similar technique in [8], [16]. It proposes a switched delay-based method to model the NCSs, and then the combined delay switching and parameter uncertainty based method is proposed. In terms of the given model, we give sufficient **conditions** for the existence of state feedback controller such that the closed-loop NCSs are **robust** asymptotically stable. Based on the obtained **stability** **conditions**, we further investigate the corresponding state feedback controller parameter uncertainty based method. Numerical examples are provided to demonstrate the effectiveness of the proposed approaches.

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We consider -solutions (approximate solutions) for a fractional optimization problem in the face of data uncertainty. Using **robust** optimization approach (worst-case approach), we establish optimality theorems and duality theorems for -solutions for the fractional optimization problem. Moreover, we give an example illustrating our duality theorems.

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In this paper we have investigated the **robust** **stability** for the linear time-varying implicit dynamic equations on time scale. Some characterizations for **robust** **stability** of IDEs subjected to Lipschitz perturbations are derived. Many previous results for **robust** **stability** of the time-varying ordinary differential and difference equations, the time-varying differential algebraic equations and the time- varying implicit difference equations are also unified and extended .

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• Advantages : Good theoretical properties: E.g.: Convergence with respect to Wasserstein metric (of order p ) is equivalent to the usual weak convergence of measures plus convergence of[r]

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Although much progress has been made in the field of fractional system **stability**, lin- ear time invariant fractional systems **robust** **stability** remains an open problem. Among the existing results and only for interval fractional systems, the **stability** issue was dis- cussed in [5-7]. As commented in [5,8], the result is rather conservative. To reduce the conservatism, in [8], a new **robust** **stability** checking method was proposed for interval uncertain systems, where Lyapunov inequality is used for finding the maximum

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The neutral stochastic systems can eﬀectively model a class of physical dynamical sys- tems, since the mathematical models of them include the time-delays of state and its derivative. These models have received considerable attention recently [21–25]. In fact, neutral stochastic systems are applied widely in automatic control, aircraft stabilization, lossless transmission lines, and system of turbojet engine [26–28]. Both the **stability** anal- ysis and synthesis of neutral stochastic systems have been extensively studied [29–36]. By using the generalized integral inequality and the nonnegative local martingale conver- gence theorem, the authors in [30] investigated the exponential **stability** and the almost sure exponential **stability** of neutral stochastic delay systems (NSDSs) with Markovian switching. The authors in [33] constructed a new sliding surface functional and consid- ered the H ∞ sliding mode control (SMC) for uncertain neutral stochastic systems with Markovian jumping parameters and time-varying delays. Using a delayed output-feedback control method, Karimi et al. in [36] designed a controller, which guarantees H ∞ synchro-

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BACKGROUND: Voltage source converter based HVDC using the insulated gate- bipolar transistor (IGBT) and pulse width modulation (PWM) techniques have been utilized in a number of installations which are presently working in many real system. OBJECTIVE: In this work, a **robust** and nonlinear control strategy are investigated in a power system inter connected with a VSC-HVDC system. The controller design is based on the H∞ control methodologies to deal with the nonlinearities introduced by requirements to power flow and line voltage. RESULTS: First, the steady state mathematical model of the VSC-HVDC system is developed and the relationship between the controlling variables is investigated. By use of H∞ control theory, constant DC voltage controller and constant AC voltage controller are designed. A **robust** H∞ control has been proposed to provide voltage support by means of reactive control at both ends; to damp out power oscillations and improve transient **stability** by controlling either active or reactive power, and to control the power flow through the HVDC link. CONCLUSION: The proposed control scheme is tested under several system disturbances like change in frequency at both ends, faults on the converter and inverter buses and change in short circuit ratio. Based upon the time domain simulations in MATLAB/SIMULINK environment, the proposed controller is tested and its better performance is shown compare with the PI controller whose gains are optimized with Genetic Algorithm Technique.

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A study done by Ackermann (1997) found that the yaw rate of the automotive vehicle is not only stirred by lateral acceleration in a way that the driver is used to, but also by disturbance torques resulting for example when a car encounters unexpected road **conditions**, such as a split-μ road, the tire slip angles. So, the vehicle slip angle may suddenly increase, which causes the vehicle to reach its physical limit of adhesion between the tires and the road. The driver has to compensate this disturbance torque by opposing at the steering wheel in order to provide disturbance reduction. This is the more hard task for the driver because the disturbance input comes as an abruptness to him; since most drivers have less experience operating a vehicle under this situation, they might at last lose control of the vehicle [30].

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Ibuprofen is a well-known nonsteroidal antiinflammatory of a product. The prime reasons for this popularity drug belonging to the family of propionic acid derivatives. include ease of accurate dosage, good physical and It can cause upper gastrointestinal damage, including chemical **stability**, competitive unit production cost and an lesion, peptic ulcers, bleeding and perforation. These elegant distinctive appearance resulting in high level of side effects are attributed to the presence of free – patient acceptability 5,6 . The present work was aimed to

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timator were given in terms of certain matrix inequalities based on the average dwell-time approach. The problem of **robust** reliable control for a class of uncertain switched neutral systems under asynchronous switching was investigated in []. A state feedback con- troller was proposed to guarantee exponential **stability** and reliability for switched neutral systems, and the dwelltime approach was utilized for the **stability** analysis and controller design. The exponential **stability** for a class of nonlinear hybrid time-delay systems was addressed in []. The delay-dependent **stability** **conditions** were presented in terms of the solution of algebraic Riccati equations, which allows computing simultaneously the two bounds that characterize the **stability** rate of the solution.

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1. Introduction. Systems in many branches of science and industry not only de- pend on the present state and the past ones but may also experience abrupt changes in their structures and parameters. Hybrid stochastic differential delay equations (SDDEs; also known as SDDEs with Markovian switching) have been widely used to model these systems (see, e.g., the books [23, 24] and the references therein). One of the important issues in the study of hybrid SDDEs is the asymptotic analysis of **stability** and boundedness (see, e.g., [3, 5, 13, 19]). In asymptotic analysis, **robust** **stability** and boundedness have been two of most popular topics. For example, Ack- ermann [1] gave a nice motivation of **robust** **stability**. Hinrichsen and Pritchard [7, 8] presented an excellent discussion of the **stability** radii of linear systems with struc- tured perturbations. Su [26] and Tseng, Fong, and Su [27] discussed **robust** **stability** for linear delay equations. In the aspect of robustness of stochastic **stability**, Hauss- mann [6] studied **robust** **stability** for a linear system and Ichikawa [11] for a semilinear system. Mao, Koroleva, and Rodkina [21] discussed the **robust** **stability** of uncertain linear or semilinear stochastic delay systems. Mao [20] investigated the **stability** of the stochastic delay interval system with Markovian switching. For more information on the **stability** and boundedness of hybrid SDDEs, please see, e.g., [12, 22, 23, 25]. However, all of the papers, up to 2013, in this area only considered these **robust**

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considered a nonlinear theory of co-prime factorisation. Here we highlight three con- tributions of particular relevance to the context of this paper. In [18], Verma defined a notion of co-prime factorisation for nonlinear mappings and, presented a **stability** result for a nonlinear system. In [2], Anderson, James and Limebeer generalised the linear theory of normalized co-prime factor robustness optimisation to the case of affine input nonlinear systems and presented a optimal robustness margin. In [10], a new definition of “normalized” was introduced for left representation for the graph of a nonlinear system and different gap metrics were studied. Many further pointers to a growing literature on nonlinear co-prime factorisation can be found in the monograph [14] and the references therein.

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