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For the nonlinear parameter‐varying (NPV) model of unmanned surface vehicle with the consideration of the velocities on yaw and surge as well as wave disturbances, a robust H∞$$ {H}_{\infty } $$ control method is proposed based on extended homogeneous polynomial Lyapunov function (EHPLF) to regulate heading for the superior performance on the rapidity, accuracy, and robustness. First, a NPV model of heading error is established to design a general form of a state feedback controller with a robust H∞$$ {H}_{\infty } $$ performance. Second, a Lyapunov matrix with full states and varying parameter is constructed to derive the robust H∞$$ {H}_{\infty } $$ global exponential stability conditions by Euler's homogeneity relation for the NPV system, known as the EHPLF stability conditions. Third, since the EHPLF stability conditions consist of a set of nonlinear coupled inequalities that cannot be directly solved by sum of squares (SOS) toolboxes, they are decoupled with matrix transformations to obtain the EHPLF‐SOS stability conditions, which is solved for the parameters of the state feedback controller. Finally, the simulation results indicate that EHPLF method exhibits a superior performance on dynamic, steady‐state, and robustness. [ABSTRACT FROM AUTHOR]

In this paper, the consensus problem of a class of linear multi-agent systems is analysed. In our model, switching between subnetworks and competition between agents is considered simultaneously.We suppose that all subnetworks cannot achieve a consensus and switching signal has a minimum dwell time. By using the time-dependent quadratic Lyapunov function and the average dwell time method, a consensus condition of switched networks is obtained. By using the sum of squares (SOS) programming in the software package SOSTOOLS of Matlab, these resulting conditions can be implemented. The results obtained are also extended to uncertain linear multi-agent systems. Finally, a simulation example is given to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

This research addresses the problem of globally stabilizing a distinct category of fractional-order power systems (F-OP) by employing an observer-based methodology. To address the inherent nonlinearity in these systems, we leverage a Takagi–Sugeno (TS) fuzzy model. The practical stability of the proposed system is systematically established through the application of a sum-of-squares (SOS) approach. To demonstrate the robustness and effectiveness of our approach, we conduct simulations of the power system using SOSTOOLS v3.00. Our study contributes to advancing the understanding of F-OP and provides a practical framework for their global stabilization. [ABSTRACT FROM AUTHOR]

This paper deals with systematic approaches for the analysis of stability properties and controller design for nonlinear dynamical systems. Numerical methods based on sum-of-squares decomposition or algebraic methods based on quantifier elimination are used. Starting from Lyapunov's direct method, these methods can be used to derive conditions for the automatic verification of Lyapunov functions as well as for the structural determination of control laws. This contribution describes methods for the automatic verification of (control) Lyapunov functions as well as for the constructive determination of control laws. [ABSTRACT FROM AUTHOR]

In this paper, a fuzzy control issue of the nonlinear inverted pendulum system is discussed via Sum-Of-Square (SOS) technology. For describing the system, a Parameter-Dependent Polynomial Fuzzy (PDPF) model is constructed by combining Takagi–Sugeno (T–S) fuzzy model, polynomial representation and Linear Parameter Varying (LPV) description. Based on Parallel Distributed Compensation (PDC) method and gain-scheduled scheme, a PDPF controller is established to guarantee the stability. However, stability issue of PDPF model is often more complex and difficult than one of the traditional Takagi–Sugeno fuzzy model since the nonconvex problem caused by the coupling of variables. To avoid the nonconvex term, a parameter-dependent polynomial Lyapunov function is adopted to derive the stability criterion. Besides, the convex combination is employed to eliminate the restriction of time-varying parameters. Thus, some sufficient conditions are derived into the SOS form and solved by the corresponding toolbox efficiently. Finally, some simulation results are provided to verify the proposed design method. [ABSTRACT FROM AUTHOR]

This paper discusses a passive control issue for Nonlinear Time-Varying (NTV) systems subject to stability and attenuation performance. Based on the modeling approaches of Takagi-Sugeno (T-S) fuzzy model and Linear Parameter-Varying (LPV) model, a Parameter-Dependent Polynomial Fuzzy (PDPF) model is constructed to represent NTV systems. According to the Parallel Distributed Compensation (PDC) concept, a parameter-dependent polynomial fuzzy controller is built to achieve robust stability and passivity of the PDPF model. Furthermore, the passive theory is applied to achieve performance, constraining the disturbance effect on the PDPF systems. To develop the stability criteria, by introducing a parameter-dependent polynomial Lyapunov function, one can derive some stability conditions, which belong to the term of Sum-Of-Squares (SOS) form. Based on the Lyapunov function, two stability criteria are proposed to design the corresponding PDPF controller, such that the NTV system is robustly stable and passive. Finally, two examples are applied to demonstrate the effectiveness of the proposed stability criterion. [ABSTRACT FROM AUTHOR]

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In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results. [ABSTRACT FROM AUTHOR]

This study investigates different observer architectures for the state estimation of linear aperiodic sampled-data systems and formally shows that a complex structure is not necessarily better than a simpler one with respect to the considered cost functional and to a class of timer-dependent Lyapunov functions. The work also shows how to design the state observer through sum of squares programming, presenting a numerical example. [ABSTRACT FROM AUTHOR]

Aim at the fore and aft asymmetry of unmanned surface vehicle (USV) and the loss of linear velocity, a static output feedback (SOF) H∞$$ {H}_{\infty } $$ control method is proposed by the sum of squares (SOS) based on two types of extended bounded real lemmas (EBRL) for heading regulation to guarantee the robustness and improve the rapidity. First, an asymmetric nonlinear heading error model with hydrodynamic parameters is established by the consideration of lack of linear velocity. Second, SOF controller with H∞$$ {H}_{\infty } $$ performance is presented for the asymmetric nonlinear USV model, and it's SOS conditions to solve controller parameters is deduced from EBRL by homogeneous Lyapunov function (HERBL) to maintain the system robustness. Further, the solution of SOF H∞$$ {H}_{\infty } $$ controller relies on the state feedback gain matrix, EBRL is deduced by quadratic Lyapunov function (QEBRL) to obtain the SOS conditions without the addition of an extended matrix to cut computing cost for the solution of the state feedback gain matrix. Finally, the heading control simulations for USV indicate that, QEBRL method takes less time for SOS conditions to solve the state feedback gain matrix, and SOF H∞$$ {H}_{\infty } $$ system of HERBL has a superior performance on the transient response and the robustness. [ABSTRACT FROM AUTHOR]

In this article, a collection of conditions to achieve asymptotic stability of switched nonlinear systems with guaranteed safety via a state-dependent switching law are provided, where the asymptotic stability of individual subsystem is not assumed. Multiple barrier functions and multiple Lyapunov functions (MLFs) are merged as a tool for analyzing the asymptotic stability with guaranteed safety control problem of switched nonlinear systems. Meanwhile, the state-dependent switching law is designed to refrain zeno phenomenon. Moreover, bilinear sum of squares methodologies are employed to construct corresponding MLFs for switched nonlinear systems. Finally, three simulation examples are introduced to prove the effectiveness of the provided method. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of polynomial fault detection filter design under an adaptive event-triggered scheme for continuous-time networked polynomial fuzzy model–based (PFMB) systems considering network transmission delays. The proposed adaptive polynomial event-triggered scheme is checked only at the sampling instant to eliminate the Zeno behavior as well as save the network bandwidth. With the consideration of the mismatched membership functions (MFs), the asynchronous problem between the physical plant and the polynomial fault detection filter (PFDF) is examined. A Lyapunov–Krasovskii (L-K) function is introduced to deal with the time delays caused by the network transmission and the zero-order holder (ZOH), and a proper line-integral Lyapunov function is also introduced to reduce the conservation of the design constraints, whose analytical procedure is rule-dependent. The design constraints are given in the form of sum of squares (SOS) to keep the PFMB fault detection system asymptotically stable with H ∞ performance γ. Finally, an inverted pendulum example together with a numerical example is given to verify the effectiveness and superiority of the proposed scheme in terms of transfer rate and conservatism. [ABSTRACT FROM AUTHOR]

Inverter-based resources (IBRs) have become indispensable in power systems due to their numerous environmental and operational advantages, which can contribute to enhancing overall grid reliability and resilience. However, due to their stability limitations, IBRs commonly disconnect from the grid when a grid-side disturbance happens. Therefore, inverters' extensive integration introduces complexities and challenges to power system stability, necessitating the adoption of sophisticated stability assessment tools. One practical approach for assessing large-signal stability involves delineating the system's stability region. Traditionally, two primary methods have been employed for this purpose: time-domain simulation and direct methods. To tackle the stability assessment of IBRs and sensitivity analysis of stability region, this paper introduces a theoretical direct method based on the sum of squares (SOS) method. It provides a Hardware-in-the-Loop (HIL) simulation to verify the results. The SOS method offers a more precise and less conservative representation of the stability region. Through applying the SOS method, this paper endeavors to establish and investigate the stability region for grid-tied IBR while conducting sensitivity analysis of the inverter responses to grid-side disturbances. This sensitivity analysis specifically delves into the impacts of different load levels and different voltage-loop control parameters on the stability region of the IBR and inverter transit response. In other words, the study develops a theoretical approach using the nonlinear dynamic model and Lyapunov-based stability assessment to understand how a modification in the system can affect the stability region of the IBR and the dynamic response of the inverter for a grid-side fault. The SOS method is applied to construct an accurate Lyapunov function for the system, and the Lyapunov-based stability assessment is used to study system stability under a large disturbance. Besides the numerical sensitivity analysis of inverter dynamic response, the accuracy of the sensitivity analysis is validated through complementary techniques, including time-domain simulations of a grid-tied IBR and high-fidelity HIL simulations using a real-time digital simulation (RTDS) platform. The result proves the outcomes of numerical stability analysis and sensitivity assessment of the IBR using an SOS-based stability analysis. • Lyapunov function development via SOS method to construct an accurate stability region. • Lyapunov-based stability analysis to assess IBR potential in tolerating large contingencies. • Verification of stability analysis results using RTDS and HIL simulations. [ABSTRACT FROM AUTHOR]

This article deals with the problem of robust static output feedback H∞ control in finite frequency (FF) domain for polynomial Takagi–Sugeno fuzzy systems. By using generalized Kalman–Yakubovich–Popov lemma and polynomial Lyapunov functions, sufficient conditions are proposed for controllers design in terms of sum of squares. These conditions include neither transformation matrices nor equality constraints, which simplifies the numerical solution. The proposed method is designed in the FF domain to reduce the conservativeness generated by those designed in the full frequency domain. Some numerical examples are provided to demonstrate the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We investigate dual formulations for this approach and leverage these dual programs for developing new analysis tools for the JSR. We show that the dual of this convex problem searches for the occupations measures of trajectories with high asymptotic growth rate. We both show how to generate a sequence of guaranteed high asymptotic growth rate and how to detect cases where we can provide lower bounds on the JSR. All results of this paper are presented for the general case of constrained switched systems, that is, systems for which the switching signal is constrained by an automaton. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the polynomial stability of hybrid stochastic systems with pantograph delay (HSSwPD). By using the Razumikhin technique and Lyapunov functions, we establish several Razumikhin-type theorems on the th moment polynomial stability and almost sure polynomial stability for HSSwPD. For linear HSSwPD, sufficient conditions for polynomial stability are presented. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we investigate the consensus verification problem of nonlinear agents in a fixed directed network with a nonlinear protocol. Inspired by the classical Lipschitz‐like condition, we introduce a more relax condition for the dynamics of the nonlinear agents. This condition is motivated via the construction of general Lyapunov functions for achieving asymptotic consensus. Especially, for agents where dynamics are described by polynomial function of the states, our consensus criterion can be converted to a sum of squares (SOS) programming problem, solvable via semidefinite programming tools. Of interest is that the scale of the resulting SOS programming problem does not increase as the size of the network increases, and thus, the applicability to analyze consensus of large‐scale networks is promising. Finally, an example is given to illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a relaxed stabilization condition for polynomial-fuzzy-model-based control system by using sum-of-squares (SOS) approach. Full-block S-procedure is utilized to equivalently decouple the fuzzy weighting constraint into two constraints, which can be computed directly with the SOS approach. Besides, a free full-block multiplier imposed by this procedure also contributes to reducing the conservativeness of stability analysis. The copositivity verification relaxation performed on the membership-functions-related constraint can both obtain more relaxed results and reduce the computation complexity. The concept of polynomial Lyapunov function (PLF) with square matrical representation, which simplifies the stabilization condition by avoiding partial derivatives of a polynomial matrix, is also introduced to generalize the existing expression of PLFs. Then, the derived stabilization condition, which is represented in terms of SOS can be numerically solved. Three simulation examples are conducted to illustrate the effectiveness and applicability of the proposed method. These examples also demonstrate the advantages of the proposed method over the existing convex SOS approach. [ABSTRACT FROM AUTHOR]

This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach. [ABSTRACT FROM AUTHOR]

A decentralized verification procedure is proposed to verify stability and performance for systems with multiple controllers designed by noncooperative agents. The key concept is to use Lyapunov functions as measure of the influence of each controller on the system. Sum of squares programming is used to verify Lyapunov derivative conditions, thus providing a sufficient condition for the specification. Dual decomposition is then used to decentralize the algorithm with the guarantee that decentralized verification is equivalent to centralized verification. The verification procedure is extended to decentralized control synthesis. To reduce the conservatism caused by improper choice of Lyapunov candidate, a perturbation algorithm is proposed to modify the Lyapunov candidate such that it better suits the purpose of verification. [ABSTRACT FROM AUTHOR]

Abstract This work generalises the line-integral Lyapunov function in (Rhee & Won, 2006) for stability analysis of continuous-time nonlinear models expressed as fuzzy systems. The referred result applied only to Takagi–Sugeno representations, and required memberships to be a tensor-product of functions of a single state; these are generalised here so that membership arguments can be arbitrary polynomials of the state variables; in this way, systems for which earlier results cannot be applied are now covered. Both the modelling and the integral terms appearing in the Lyapunov functions are generalised to a fuzzy polynomial case. Illustrative examples show the advantage of the proposed method against previous literature, even in the TS case. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm estimating the regions of attraction of power systems based on the Lyapunov function approach where a sublevel set of a Lyapunov function for a target system is used as the estimate. In particular, we focus here on the algorithm based on sum of squares (SOS) programming, which has been recently proposed, and aim to develop a simpler algorithm for the practical use. For this aim, we present an algorithm overcoming the difficulty of the SOS programming problem addressed in the existing study, i.e., the bilinear constraints, in a simpler way. In the proposed algorithm, two SOS programming problems are iteratively solved, and the number of the problems solved at each iteration is reduced to half of that in the existing algorithm. In addition, we theoretically analyze the proposed algorithm, and show the convergence under certain conditions. The performance of our algorithm is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

A homogeneously state-dependent polynomial Lyapunov function for an observed-state feedback polynomial fuzzy control system is proposed. From which an advanced sufficient sum of squares (SOS) condition formulated in terms of state-dependent matrix inequalities to achieve fuzzy stabilization is derived. Three main features, compared with existing nonhomogeneously polynomial methods, are as follows: 1) homogeneous Lyapunov function and its time derivative are nicely linked via the Hessian matrix; 2) removing the nonconvex obstacles on $\dot{P}(x)$ when deriving a nonhomogeneous Lyapunov stabilization condition $\dot{V}(x)$ ; 3) separation principle holds; since common $P$ falls out as a special case, the proposed homogeneous SOS approach is placed ahead of the existing SOS-based methods found in polynomial fuzzy stabilization controls. To verify the problem-solving theories of the homogeneous polynomial method announced, four examples are demonstrated to show the promising features of the approach adduced. Finally, an Appendix illustrating the solving procedure is added for clarification on how the examples are solved via SOS. [ABSTRACT FROM AUTHOR]

This paper proposes methods to handle the problem of delay range stability analysis for a linear coupled differential–difference system (CDDS) with distributed delays subject to dissipative constraints. The model of linear CDDS contains many models of linear delay systems as special cases. A novel Lyapunov–Krasovskii functional with non-constant matrix parameters, which are related to the delay value polynomially, is applied to derive stability conditions. By constructing this new functional, sufficient conditions in terms of robust linear matrix inequalities (LMIs) can be derived, which guarantee range stability of a linear CDDS subject to dissipative constraints. To solve the resulting robust LMIs numerically, we apply the technique of sum of squares programming together with matrix relaxations without introducing any potential conservatism to the original robust LMIs. Furthermore, the proposed methods can be extended to solve delay margin estimation problems for a linear CDDS subject to prescribed dissipative constraints. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methodologies. [ABSTRACT FROM AUTHOR]

This paper presents a piecewise polynomial Lyapunov function (PPLF) approach to stabilization and robust stabilization of polynomial fuzzy systems that are a generalized form of the well-known Takagi-Sugeno fuzzy systems. Both stabilization and robust stabilization conditions are formulated as sum of squares (SOS) optimization problems which can be solved by an SOS solver. A switching polynomial fuzzy controller based on switching information on the PPLF is designed to stabilize both polynomial fuzzy systems and polynomial fuzzy systems with uncertainties. In particular, by fully considering the properties of the piecewise and polynomial Lyapunov function, some relaxation are carried out in the derivation of SOS robust stabilization conditions for polynomial fuzzy systems with uncertainties. Four design examples are demonstrated to show the effectiveness of the proposed design approach in comparison with the existing approaches, i.e., linear matrix inequalities approaches and SOS stabilization and robust stabilization approaches. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach on controller design for a doubly fed wind generator based on input-to-state stability (ISS) theory. In order to guarantee the stability of the nonlinear system with external disturbances, a systematic methodology for constructing locally ISS stabilizing (LISS) control Lyapunov functions and designing the corresponding robust LISS control law is proposed. This method avoids the Hamilton-Jacobi-Isaacs (HJI) partial differential equation and the limitation of exact linearization. The proposed ISS control law is proved to be inverse optimal and can guarantee the stability of the wind generation system under external disturbances. The simulation studies verify the effectiveness of the proposed ISS controller. Compared with conventional proportional-integral (PI) controllers, and exact linearization-based nonlinear controllers, the proposed ISS controller has the ability to enhance the transient stability of the wind generation system during and after faults, and can significantly improve the dynamic performance of the system. [ABSTRACT FROM PUBLISHER]

Domain of attraction plays an important role in control systems analysis, which is usually estimated by sublevel sets of Lyapunov functions. In this paper, based on the concept of common Lyapunov-like functions, we propose an iteration method for estimating domains of attraction for a class of switched systems, where the state space is divided into several regions, each region is described by polynomial inequalities, and any region has no intersection among with each other. Starting with an initial inner estimate of domain of attraction, we first present a theoretical framework for obtaining a larger inner estimate by iteratively computing common Lyapunov-like functions. Then, for obtaining a required initial inner estimate of domain of attraction,we propose a higher-order truncation and linear semidefinite programming--based method for computing a common Lyapunov function. Successively, the theoretical framework is under-approximatively realized by using S-procedure and sum-of-squares programming, associated with a coordinatewise iteration idea. Finally, we implement our method and test it on some examples with comparisons. These computation and comparison results show the advantages of our method. [ABSTRACT FROM AUTHOR]

This paper presents a new Line-integral polynomial Lyapunov functional approach for observer-based control for a class of polynomial fuzzy systems with time delay and unmeasured state variables. To guarantee the global asymptotic stability and estimation error convergence, a design method is proposed. In this work we consider a Line-integral polynomial Lyapunov function in which the Lyapunov matrices are polynomial matrices depending not only of the estimated states but also of the estimated delayed states and we use the dual system to reduce the computational efforts. The design conditions are established in terms of Sum Of Squares (SOS) which can be numerically and symbolically solved via the recent developed SOSTOOLS and a Semi-Definite Program solver. Finally, numerical examples are proposed to show the validity and applicability of the proposed results. [ABSTRACT FROM AUTHOR]

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form V (x) = ‖x‖p Q := (x˕Qx) p/2, where the parameters are the positive definite matrix Q and the number p > 0. We give several examples of our proposed method and show how it improves previous results. [ABSTRACT FROM AUTHOR]

In this paper, controller design for discrete-time bilinear systems is investigated by using sum of squares programming methods and quadratic Lyapunov functions. The class of rational polynomial controllers is considered, and necessary conditions on the degree of controller polynomials for quadratic stability are derived. Next, a scalarized version of the Schur complement is proposed. For controller design, the Lyapunov difference inequality is converted to a sum of squares problem, and an optimization problem is proposed to design a controller, which maximizes the region of quadratic stability of the bilinear system. Input constraints can also be accounted for. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper concerns the simulation of a class of nonlinear continuous-time systems under a set of initial conditions described by an ellipsoid. By getting inspiration from Lyapunov theory, we show that it is possible to derive a procedure allowing the computation of a bounding ellipsoidal envelope, which will enclose all the states that can be reached from the set of initial conditions. This numerical procedure is based on convex optimization, and it makes it possible to set a guaranteed hard bound on the evolution of the state of the system for all the possible initial conditions. At the end of the paper, we show an application of the method through an academic example. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper proposes an LMI approach to model predictive control of nonlinear systems with switching between multiple modes. In this approach, at each mode, the nonlinear system is divided to a linearized model in addition to a nonlinear term. A sum of squares (SOS) optimization problem is presented to find a quadratic bound for the nonlinear part. The stability condition of the switching system is obtained by using a discrete Lyapunov function and then the sufficient state feedback control law is achieved so that guarantees the stability of the system and also minimizes an infinite prediction horizon performance index. Moreover, two other LMI optimization problems are solved at each mode in order to find the maximum area region of convergence of the nonlinear system inscribed in the region of stability. The performance and effectiveness of the proposed MPC approach are illustrated by two case studies. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability analysis of positive polynomial-fuzzy-model-based control systems with time delay. Each of the polynomial fuzzy model and the polynomial fuzzy controller is allowed to have its own set of premise membership functions, which can improve the design and realization flexibility. Conditions showing the positivity of the system are first obtained. Then, to deal with the difficulty resulting from the mismatched premise membership functions to the stability/stabilization analysis, approximated membership functions representing the original ones are constructed, and the information of membership functions is brought into the stability conditions. This way, the stability conditions can be relaxed, since the extra knowledge on the system is considered. Stability conditions are obtained by using the sum-of squares approach based on the Lyapunov stability theory. A simulation example is provided to verify the effectiveness of this method. [ABSTRACT FROM PUBLISHER]

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag's control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches. [ABSTRACT FROM PUBLISHER]

In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov-Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator-prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator-prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach. [ABSTRACT FROM AUTHOR]

There has been a recent interest in imitation learning methods that are guaranteed to produce a stabilizing control law with respect to a known system. Work in this area has generally considered linear systems and controllers, for which stabilizing imitation learning takes the form of a biconvex optimization problem. In this paper it is demonstrated that the same methods developed for linear systems and controllers can be readily extended to polynomial systems and controllers using sum of squares techniques. A projected gradient descent algorithm and an alternating direction method of multipliers algorithm are proposed as heuristics for solving the stabilizing imitation learning problem, and their performance is illustrated through numerical experiments. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability and stabilization of polynomial fuzzy systems with time delay. This study is based on a polynomial Lyapunov-Krasovskii functional, and delay-dependent results are described in terms of bilinear sum of squares (SOS) conditions. Two techniques are presented, in this paper, to solve the problem of non-convexity of SOS stability and stabilization conditions for polynomial fuzzy systems with time delay. The first technique presents the SOS stability conditions for a specific class of polynomial fuzzy systems with time delay. Delay-dependent conditions are derived throughout a predefined set of indices. This technique improves delay-independent results existing in the literature. The second method transforms the design problem on an optimization problem subject to SOS constraints. The proposed SOS approach is more innovative and effective than the existing linear matrix inequality in terms of conservatism reduction. The new convex conditions for polynomial fuzzy systems with time delay are shown in terms of SOS which can be solved via SOSTOOLS software. Numerical examples are given to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This article is concerned with the design of polynomial fuzzy controller for a class of polynomial fuzzy model subject to both state delay and actuator saturation. Based on polytopic model of the input saturation, two design methods are proposed. In the first method, a specific Lyapunov Krasovskii function is proposed to transform the nonconvex sum of squares (SOS) conditions into convex SOS ones. The second method overcomes the restriction of the first method by bounding the state derivative. The obtained results are formulated in terms of SOS matrices which can be symbolically and numerically solved via the SOSTOOLS and the SeDuMi. Moreover, an attractive region of initial states that ensures asymptotic stability of polynomial fuzzy model is determined. Two numerical examples are given to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

The stability of uncertain periodic and pseudo-periodic systems with impulses is analyzed in the looped-functional and clock-dependent Lyapunov function frameworks. These alternative and equivalent ways for characterizing discrete-time stability have the benefit of leading to stability conditions that are convex in the system matrices, hence suitable for robust stability analysis. These approaches, therefore, circumvent the problem of computing the monodromy matrix associated with the system, which is known to be a major difficulty when the system is uncertain. Convex stabilization conditions using a non-restrictive class of state-feedback controllers are also provided. The obtained results readily extend to uncertain impulsive periodic and pseudo-periodic systems, a generalization of periodic systems that admit changes in the 'period' from one pseudo-period to another. The obtained conditions are expressed as infinite-dimensional semidefinite programs, which can be solved using recent polynomial programming techniques. Several examples illustrate the approach, and comparative discussions between the different approaches are provided. A major result obtained in the paper is that despite being equivalent, the approach based on looped functional reduces to the one based on clock-dependent Lyapunov functions when a particular structure for the looped functional is considered. The conclusion is that the approach based on clock-dependent Lyapunov functions is preferable because of its lower computational complexity and its convenient structure enabling control design. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, tracking control synthesis problem for nonlinear polynomial discrete-time systems are studied. Proposed controller drives the plant such that the state vector of the plant follows those of a stable reference model. The objective is to design a controller such that the energy gains from the exogenous signals that are the reference signal and the state vector of the reference model, to the tracking error to be less or equal to prescribe thresholds. The main difficulty in the problem of designing tracking nonlinear discrete-time control law for the polynomial discrete time systems is that in general this problem may not be formulated as a convex problem. With proper selection of Lyapunov function and based on Lyapunov theory and by using sum of square approach, sufficient conditions for existence of controller are presented in terms of a feasibility SOS programming problem that can be solved using numerical solvers such as SOSTOOLS. Finally, the performance of proposed approach will be shown using the simulation of several examples. [ABSTRACT FROM AUTHOR]

The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which satisfies certain positivity conditions. Unlike linear dynamical systems, there is no algorithmic method for constructing Lyapunov functions for general nonlinear systems. However, if the systems of interest evolve according to polynomial vector fields and the Lyapunov functions are constrained to be sum-of-squares polynomials then stability verification can be cast as a semidefinite (convex) optimization programme. In this paper we describe recent advances in sum-of-squares programming that facilitate advanced stability analysis and control design. [ABSTRACT FROM AUTHOR]

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H∞ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes sum-of-squares (SOS) methodologies for stability analysis and region-of-attraction (ROA) estimation for nonlinear systems represented by polynomial fuzzy models via piecewise polynomial Lyapunov functions (PPLFs). At first, two SOS-based global stability criteria are proposed by applying maximum-type and minimum-type PPLFs, respectively. It is known that less-conservative results can be obtained by reducing global stability to local stability, since it is usually the case for nonlinear systems that the stability cannot be reached globally. Therefore, based on the two types of PPLFs, two local stability criteria are further proposed with the algorithms that enlarge the estimated ROA as much as possible. The constraints for checking (global and local) stability and enlarging the estimated ROA are represented in terms of bilinear SOS problems. Hence, the path-following method is applied to solve the bilinear SOS problems in the proposed methodologies. Finally, some examples are provided to illustrate the utility of the proposed methodologies. [ABSTRACT FROM AUTHOR]