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This paper investigates the problem of robust stability analysis for systems with time‐delays and time‐varying structured uncertainties. The variable augmentation approach and the free‐matrix‐based inequality are employed to retain the linear property of the estimation of the derivative of the Lyapunov–Krasovskii functionals. Moreover, with the use of the variable augmentation approach related to the time‐varying structured uncertainties, the time‐varying S‐Procedure combined with the time‐varying delay and its derivative is employed. As a result, less‐conservative robust stability conditions are derived. In addition, the merits of the augmentation of the time‐varying structured uncertainties over the traditional methods are theoretically demonstrated. Finally, two numerical examples are given to verify the benefits of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of feedback control for a class of affine T–S fuzzy models using piece-wise Lyapunov functions. Although a large number of works on the issue have been published, several crucial problems still remain open. First, the paper shows what problems arise when using the affine T–S fuzzy model to design a controller, and in turn by employing the S-procedure, what kind of quadratic inequalities are required to help solve the resulting LMIs. It turns out that by partitioning the state space into certain cells based on the information of the antecedents of fuzzy rules, the required quadratic inequalities can be formularised. Taking advantage of the cell partition, a fuzzy controller is proposed using piece-wise Lyapunov functions, in which ensuing problems such as continuity functions used in the piece-wise Lyapunov functions and control input chattering also are addressed. Finally, examples are provided to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Copyright of Operations Research Transactions / Yunchouxue Xuebao is the property of Editorial office of Operations Research Transactions and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article proposes novel methods for the computation of state and output bounding sets for discrete‐time uncertain systems. The systems under consideration are formed from the interconnection of a nominal linear time‐invariant system and an uncertainty operator that is known to lie within a prespecified set. The set of allowable uncertainties is described by a pointwise integral quadratic constraint (IQC), that is, a quadratic constraint that holds for all time‐steps. Examples of pointwise IQCs characterizing common uncertainty sets are provided. The exogenous input to the system is assumed to lie in a given polytope or ellipsoid for all time‐steps. The proposed methods are illustrated via an example. [ABSTRACT FROM AUTHOR]

In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied to an explicit model assumption made on the functions to be learned. Working in the framework of Hilbert spaces, this article considers a model assumption based on approximability. It also incorporates observational inaccuracies modeled via additive errors bounded in ℓ 2 . Earlier works have demonstrated that regularization provides algorithms that are optimal in this situation, but did not fully identify the desired hyperparameter. This article fills the gap in both a local scenario and a global scenario. In the local scenario, which amounts to the determination of Chebyshev centers, the semidefinite recipe of Beck and Eldar (legitimately valid in the complex setting only) is complemented by a more direct approach, with the proviso that the observational functionals have orthonormal representers. In the said approach, the desired parameter is the solution to an equation that can be resolved via standard methods. In the global scenario, where linear algorithms rule, the parameter elusive in the works of Micchelli et al. is found as the byproduct of a semidefinite program. Additionally and quite surprisingly, in case of observational functionals with orthonormal representers, it is established that any regularization parameter is optimal. [ABSTRACT FROM AUTHOR]

For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters can be precomputed by semidefinite programming. This general framework can be applied to several scenarios, including the two-space problem and problems involving ℓ 2 -inaccurate data. It can also be applied to the problem of recovery from ℓ 1 -inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition. [ABSTRACT FROM AUTHOR]

When designing a controller based on a fuzzy model composed of fuzzy rules, in most cases controllers are designed only based on the consequent parts of the fuzzy rules, equally meaning the information of the antecedent parts is merely ignored. This paper first partitions all of the rules into two groups based on the information of the antecedent parts, in which one contains certain rules around the origin, and the other is a group of rules away from the origin. While being able to make the former arbitrarily small, each rule of the latter leads to a favorable quadratic inequality, which will be useful in controller design by the help of the S‐procedure. On the other hand, while the affine T‐S fuzzy model is much more effective for describing the dynamics of the system concerned in comparison with the regular T‐S fuzzy model, it encounters some difficulties when designing a controller based on the model because of the affine terms within. This paper makes clear that the difficulties can be overcame by using the quadratic inequalities based on the fuzzy rule partition. In addition, the effectiveness of the approach is demonstrated by applying it to the CSTR for an exothermic irreversible reaction. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

A novel output feedback control law with disturbance compensation for nonlinear multi-input multi-output (MIMO) time-invariant systems under measurement noises is proposed. The designed control law, based on the noise and disturbance estimations, ensures the accuracy in steady state depending on the disturbance, only one component of noise vector and its first derivatives. Therefore, the proposed algorithm is efficient under noises with large magnitudes. Sufficient conditions in terms of linear matrix inequalities (LMIs) provide a practical exponential stability of the closed-loop system. The simulations illustrate an efficiency of the proposed method compared with some existing ones. [ABSTRACT FROM AUTHOR]

This letter studies an unmanned aerial vehicle (UAV)-enabled uplink communication system with turning and climbing angle constraints, and the communication process suffers from the attack from one ground jammer with imperfect location information. Under such setup, we aim to maximize the uplink throughput by jointly considering the UAV’s three dimensions (3D) trajectory design and the ground node (GN)’s transmit power control. However, the optimization problem is non-convex and is hard to be tackled. To overcome this difficulty, we develop an alternating algorithm by leveraging the block coordinate descent (BCD) method, successive convex approximation (SCA) technique, and ${\mathcal {S}}$ -procedure to acquire the suboptimal solution. Simulation results show that our proposed algorithm significantly improves the uplink throughput compared with the benchmark algorithms. [ABSTRACT FROM AUTHOR]

This paper investigates the problems of reachable set estimation and synthesis for periodic positive systems with two different exogenous disturbances. The lifting method and the pseudoperiodic Lyapunov function method are adopted for the estimation problem. The reachable set bounding conditions are proposed by employing Lyapunov-based inequalities and the S-procedure technique. Two optimization methods are used to minimize the bounding hyper-pyramids of the reachable set. In addition, the state-feedback controller design conditions that make the reachable set of closed-loop systems lie within a given hyper-pyramid are derived. Finally, numerical examples are presented to illustrate the validity of the obtained conditions. [ABSTRACT FROM AUTHOR]

Copyright of Control Theory & Applications / Kongzhi Lilun Yu Yinyong is the property of Editorial Department of Control Theory & Applications and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article is dedicated to the stability of one-dimensioned spatially interconnected systems. More precisely, it focuses on systems which results of the interconnection of a possibly large number of cells (continuous subsystems). This note is restricted to the case where cells are just distributed along a line. The global system can then be seen as a mixed continuous–discrete 2D Roesser system but with implicit discrete dynamics along the space dimension. Recent results on the stability of 2D Roesser models are exploited and adapted to derive a sufficient condition for such a system to be stable. The condition seems to be close to necessity if not necessary. It is tractable since it is expressed in terms of linear matrix inequalities. The novelty clearly lies in the reduction of the conservatism of the proposed analysis. [ABSTRACT FROM AUTHOR]

The control algorithm for nonlinear time-invariant systems with compensation of unknown disturbances under measurement noises is proposed. The dimension of noises is equal to the state vector dimension of the plant. Disturbances can be presented in any equation of the plant model. The novel control law is based on the noise estimator and the disturbance compensator. The accuracy in the steady state depends on the first derivative of disturbance and the smallest component of noise vector. A sufficient condition in terms of linear matrix inequality provides feasibility of the proposed algorithm. The simulations show the efficiency of the proposed method compared with some existing ones. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the problem on the minimum trace or volume ellipsoid covering intersection of ellipsoids. This problem can be solved by three major relaxation methods involving semi-definite programming relaxation, S-procedure relaxation and parameterized bounding ellipsoid relaxation. They are proposed from the different ideas or viewpoints, so that it is difficult to judge which relaxation method is the most suitable for the different practical implementations. To the best of our knowledge, the problem of proving the convexity of the optimization problem derived by the third relaxation method remains open. By rigorous analysis, we reveal the equivalence among the three relaxation methods, and address the open problem, i.e., the optimization problem derived by the third relaxation method can be equivalently transformed to a convex optimization problem. The computation complexity of these relaxation methods are also discussed, and the equivalent result is helpful to set membership filter and multisensor fusion obtaining a tighter ellipsoid to contain the state with lower computation complexity. Finally, the performance of each method is evaluated by several typical numerical examples in information fusion and filtering. [ABSTRACT FROM AUTHOR]

This paper presents the S-procedure characterization for the stabilization of positive switched linear systems and establishes the relationship between the S-procedure and its equivalent Lyapunov–Metzler inequalities. First, a piecewise linear co-positive Lyapunov function is constructed for positive switched linear systems. Under the Lyapunov function, the S-procedure stabilization for positive switched linear systems in the continuous-time context is explored under a min state switching law. The S-procedure conditions are formulated in the form of linear programming. Finally, an equivalence relationship between S-procedure and Lyapunov–Metzler inequalities is presented. [ABSTRACT FROM AUTHOR]

In this paper, we study some fundamental geometrical properties related to the S-procedure. Given a pair of quadratic functions (g, f), it asks when "g(x)=0⟹f(x)≥0" can imply "(∃λ∈R) (∀x∈Rn) f(x)+λg(x)≥0." Although the question has been answered by Xia et al. (Math Program 156:513-547, 2016), we propose a neat geometric proof for it (see Theorem 2): the S-procedure holds when, and only when, the level set {g=0} cannot separate the sublevel set {f<0}. With such a separation property, we proceed to prove that, for two polynomials (g, f) both of degree 2, the image set of g over {f<0},g({f<0}), is always connected (see Theorem 4). It implies that the S-procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over {f≤0}, but the extended results when g over {f≤0} is unbounded from below or bounded but unattainable. The robustness and the sensitivity analysis of an optimization problem involving the pair (g, f) automatically follows. All the results in this paper are novel and fundamental in control theory and optimization. [ABSTRACT FROM AUTHOR]

This work studies a new method to synthesise a robust observer to estimate the immeasurable state variables for an uncertain large‐scale Takagi–Sugeno (T–S) fuzzy system, which has not been considered in the previous studies. This system consists of a set of T–S fuzzy subsystems that are interconnected, but these systems have certain uncertainties. An observer relying on the unknown input method is synthesised such that not only the unknown states are estimated but also the effects of the uncertainties are eliminated completely, even though the upper bounds of the uncertainties are not known. The conditions to design the observer are obtained in the main theorems by employing Lyapunov methodology, S‐procedure, and linear matrix inequalities technique. Finally, the illustrative examples are provided to illustrate the advantages and success of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: This paper considers the bipartite leaderless and leader‐following synchronization in a network of coupled Lur'e systems with sector‐condition node dynamics and signed digraph topology, where the interaction digraph consists of both positive and negative communication links. Continuous‐time and sampled‐data control algorithms are, respectively, proposed to solve the bipartite synchronization problems for the signed Lur'e network. Under some assumptions on the network structure and node functions, some bipartite synchronization conditions are established for signed Lur'e network by using coordinate transform, linear matrix inequalities, and algebraic graph theories. For the continuous‐time control algorithms, it is shown that the bipartite synchronization problems of the signed Lur'e network can always be solved if the coupling strength is sufficiently large and the network topology satisfies some suitable conditions. For the sampled‐data control algorithms, an upper bound of the sampling intervals is estimated to ensure that the bipartite synchronization can be reached in the signed Lur'e network. Some simulation results are given to illustrate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Use of the semidefinite relaxation in the problem of sign-definiteness of the quadratic form under quadratic constraints enables one to establish from the duality conditions an S-procedure. However, the S-procedure giving the necessary and sufficient conditions for signdefiniteness of the relaxed problem provides only the sufficient conditions for sign-definiteness for the original problem for the case of two and more quadratic constraints. This property is called the deficiency of S-procedure. A method was proposed enabling one in some cases to establish the conditional sign-definiteness in the case where the S-procedure provides a negative result. This method give the necessary and sufficient conditions for sign-definiteness in the two-dimensional case. An example was given. [ABSTRACT FROM AUTHOR]

Vehicle to everything (V2X) is emerging as a promising application scenario of next generation mobile communications. In V2X systems, a series of applications (information transmission, in-car entertainment, etc.) rely on the limited vehicle battery. However, most prior work limits to the automatic piloting, channel measurement/estimation and high speed transmissions, seldom study has been done on the battery-limited for V2X. In light of this, here we investigate the problem of the downlink beamforming design for signal-to-interference ratio balancing in a V2X-SWIPT environment with imperfectly estimated channels at the base station, where a practical non-linear energy harvesting model is considered. The problem is further transformed into a two-layer optimization problem, where we solve the inner-level problem by exploiting S-Procedure, and the outer-level problem through bisection approach. Finally, simulation results are carried out to confirm that the proposed robust beamforming design obtains a favorable result in applications of practical interest. [ABSTRACT FROM AUTHOR]

An algorithm was proposed to stabilize nonlinear systems with reduced level of impact of the measurement noise, parametric uncertainty, and external perturbation. Consideration was given to the noise of the measurements of dimensionality coinciding with that of the plant state vector. The parametric uncertainty and external perturbations can occur in any equation of the plant model. Conditions were obtained to calculate algorithm parameters in the form of solvability of the linear matrix inequality. Efficiency of the proposed scheme was illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the structural stability of a density dependent predator-prey system with ratio-dependent functional response. Starting with the geometrical analysis of hyperbolic curves, we obtain that the system has one or two positive equilibria under various conditions. Inspired by the S-procedure and semi-definite programming, we use the sum of squares decomposition based method to ensure the global asymptotic stability of the positive equilibrium through the associated polynomial Lyapunov functions. By exploring the monotonic property of the trace of the Jacobian matrix with respect to under the given different conditions, we analytically verify that there is a corresponding unique such that the trace is equal to zero and prove the existence of Hopf bifurcation, respectively. [ABSTRACT FROM AUTHOR]

This study proposes a novel sum of squares (SOS) decomposition‐based constrained guaranteed cost function controller for polynomial systems. The sufficient non‐linear controller design conditions are derived such that an upper bound of a quadratic cost function subjected to the input amplitude constraint is minimised and asymptotic stability of the polynomial system is guaranteed. First, the constrained optimisation problem is formulated by an alternative unconstrained optimisation problem with the vanishing disturbance input. Then, by considering the S‐procedure, the sufficient conditions are derived in terms of SOS decomposition. The main novelty of this study is that the stability issue of input saturated polynomial systems together with the non‐linear control law is considered. In addition, the other advantage of this approach over the recently guaranteed cost controller design methods is that the proposed conditions can be solved without any iterative techniques. Finally, to show the effectiveness of the proposed approach, it is applied to a piezoelectrically actuated clamped–clamped micro‐beam system and the results are compared with those of existing ones. [ABSTRACT FROM AUTHOR]

In order to estimate traffic densities in a large-scale urban freeway network in an accurate and timely fashion when traffic sensors do not cover the freeway network completely and thus only local measurement data can be utilized, this paper proposes a decentralized state observer approach based on a macroscopic traffic flow model. Firstly, by using the well-known cell transmission model (CTM), the urban freeway network is modeled in the way of distributed systems. Secondly, based on the model, a decentralized observer is designed. With the help of the Lyapunov function and S-procedure theory, the observer gains are computed by using linear matrix inequality (LMI) technique. So, the traffic densities of the whole road network can be estimated by the designed observer. Finally, this method is applied to the outer ring of the Beijing's second ring road and experimental results demonstrate the effectiveness and applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper deals with the tracking approach of a self-generated stable limit cycle for an underactuated mechanical system: The inertia wheel inverted pendulum (IWIP). Such system is subject to unilateral constraints limiting its swing motion. It is known that an interconnection and damping assignment passivity-based control (IDA-PBC) can be employed to control such pendulum to its upright position. In this work, we briefly show first that the IWIP can generate a stable period-1 limit cycle through a Hopf bifurcation by varying some gain parameter of the IDA-PBC. Thus, such self-generated limit cycle is used as a reference trajectory, which is chosen to be tracked by the IWIP. To achieve the tracking problem, a supplementary control input is added. Such tracking problem is reformulated as an asymptotic stabilization of the tracking error. Our fundamental approach hinges mainly on the use of the S-procedure to introduce the unilateral constraints, and the Schur complement and the matrix inversion lemma to transform bilinear matrix inequalities into linear matrix inequalities. Several simulations have been presented to corroborate the mathematical results and to show the efficiency of the proposed tracking scheme of the self-generated stable limit cycle of the controlled IWIP, even if it is subject to external disturbances, or in the presence of uncertainties in the friction parameters. [ABSTRACT FROM AUTHOR]

This study presents a new approach for stability analysis of polynomial‐fuzzy‐model‐based (PFMB) control system using membership function information. For the purpose of extracting the regional information of the membership functions, the operating domain is partitioned into several sub‐domains. In each sub‐domain, the boundaries of every single membership function overlap term and the numerical relation among all the membership function overlap terms are represented as a group of inequalities. Through the S‐procedure, the regional membership function information is taken into account in the stability analysis to relax the stability conditions. The operating domain partition scheme naturally arises the motivation of constructing the PFMB control system with the sub‐domain fuzzy controllers. Each polynomial fuzzy controller works in its corresponding sub‐domain such that the compensation capability of controller is enhanced. The sum‐of‐squares (SOS) approach is proposed to obtain the stability conditions of the PFMB control system using the Lyapunov stability theory. The PFMB control system studied in this study has the feature that the number of fuzzy rules and the membership function shapes of the polynomial fuzzy controller can be designed independently from the polynomial fuzzy model. To verify the stability analysis result, a numerical example is given to demonstrate the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an OFDM radio link corrupted by oscillator phase noise in the receiver, namely the problem of estimating and compensating for the impairment. To lessen the computational burden and delay incurred onto the receiver, we estimate phase noise using only scattered pilot subcarriers, i.e., no tentative symbol decisions are used in obtaining and improving the phase noise estimate. In particular, the phase noise estimation problem is posed as an unconstrained optimization problem whose minimizer suffers from the so-called amplitude and phase estimation error. These errors arise due to receiver noise, estimation from limited scattered pilot subcarriers and estimation using a dimensionality reduction model. It is empirically shown that, at high signal-to-noise-ratios, the phase estimation error is small. To reduce the amplitude estimation error, we restrict the minimizer to be drawn from the so-called phase noise geometry set when minimizing the cost function. The resulting optimization problem is a nonconvex program. However, using the S-procedure for quadratic equalities, we show that the optimal solution can be obtained by solving the convex dual problem. We also consider a less complex heuristic scheme that achieves the same objective of restricting the minimizer to the phase noise geometry set. Through simulations, we demonstrate improved coded bit-error-rate and phase noise estimation error performance when enforcing the phase noise geometry. For example, at high signal-to-noise-ratios, the probability density function of the phase noise estimation error exhibits thinner tails which results in lower bit-error-rate. [ABSTRACT FROM PUBLISHER]

This paper investigates the stability and stabilization problem of fractional-order nonlinear systems for $$0<\alpha <2$$ . Based on the fractional-order Lyapunov stability theorem, S-procedure and Mittag-Leffler function, the stability conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with $$0<\alpha <2$$ are proposed. Finally, typical instances, including the fractional-order nonlinear Chen system and the fractional-order nonlinear Lorenz system, are implemented to demonstrate the feasibility and validity of the proposed method. [ABSTRACT FROM AUTHOR]

This paper develops the problems of local stability and stabilisation for time-varying polytopic quadratic systems. The state-space data is assumed to be dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates. These problems are solved by utilising the parameter-dependent quadratic Lyapunov function andS-procedure approach. Sufficient conditions for local stability and stabilisation are first formulated as optimisation problems with ‘quasi’ linear matrix inequalities. Simulation examples are then provided to confirm the effectiveness of the given approach. [ABSTRACT FROM AUTHOR]

This paper is designed to deal with the Lyapunov stability analysis of fractional-order nonlinear systems with impulses. Based on the theory of fractional calculus, impulsive differential equation and S-procedure, several sufficient criteria are established to guarantee the Mittag–Leffler stability for the addressed model with appropriate impulsive controller. Furthermore, two numerical examples are given to verify the validity and feasibility of the obtained results. [ABSTRACT FROM AUTHOR]