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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]

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]

This paper is concerned with utilizing higher‐order derivatives of generalized Lyapunov‐like functions to infer asymptotic stability of the switched nonlinear system without requiring the first derivative of the Lyapunov function for each activated subsystem to be negative definite. In contrast to the existing switched systems literature, we allow generalized Lyapunov‐like functions to increase both during the continuous flows and the discrete jumps. Firstly, a set of sufficient conditions for asymptotic stability of switched nonlinear systems are derived under a state‐dependent switching law. Then, all the above‐mentioned conditions are transformed to a set of bilinearly diagonally dominant sum of squares constraints. Bilinearly diagonally dominant sum of squares programming is proposed as a tool to solve the conditions provided. Finally, two examples are introduced to prove the effectiveness of the above method. [ABSTRACT FROM AUTHOR]

This paper aims to propose a novel idea for the modeling and trajectory tracking control of a lower limb rehabilitation robot. A polynomial nonlinear uncertain model is established to deal with the difficulty of accurate modeling for the lower limb rehabilitation robot with complex dynamic characteristics. A high‐order nonlinear disturbance observer (HONDO) is utilized to estimate the lumped disturbance with fast time‐varying and the corresponding observer gain matrix is obtained by H∞ performance. Then an HONDO‐based composite tracking controller is designed to achieve good tracking performance of the lower limb rehabilitation robot. In addition, all solvability conditions can be directly handled by sum of squares programming, effectively avoiding the calculation problem. Finally, numerical simulations are conducted to verify the validity of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: This article is concerned with the polynomial filtering problem for a class of nonlinear stochastic systems governed by the Itô differential equation. The system under investigation involves polynomial nonlinearities, unknown‐but‐bounded disturbances, and state‐ and disturbance‐dependent noises ((x,d)‐dependent noises for short). By expanding the polynomial nonlinear functions in Taylor series around the state estimate, a new polynomial filter design method is developed with hope to reduce the conservatism of the existing results. In virtue of stochastic analysis and inequality technique, sufficient conditions in terms of parameter‐dependent linear matrix inequalities (PDLMIs) are derived to guarantee that the estimation error system is input‐to‐state stable in probability. Moreover, the desired polynomial matrix can be obtained by solving the PDLMIs via the sum‐of‐squares approach. The effectiveness and applicability of the proposed method are illustrated by two numerical examples with one concerning the permanent magnet synchronous motor. [ABSTRACT FROM AUTHOR]

Summary: In this article, without the help of predesigned dwell time constraints, a new state‐dependent switching law with guaranteed dwell time for switched nonlinear systems is studied. Some sufficient conditions for asymptotic stability of switched nonlinear systems are derived. Also, all the abovementioned conditions can be transformed to a set of sum of squares (SOS) constraints, which can be checked by using the bilinear SOS methodology. Meanwhile, an improved path following method is provided to solve a bilinear SOS problem. Finally, three simulation examples are given to demonstrate the effectiveness of the obtained results. [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]

Summary: A switched nonlinear system subject to disturbances is considered in this paper. The switching signal admits an average dwell time and a state feedback control depending on the system operating modes, detected with a maximum time delay, is applied to the system. In this framework, the input‐to‐state stability problem of the closed‐loop system is addressed. Based on some established existence conditions of mode‐dependent Lyapunov‐like functions, the values of the maximum time delay and the average dwell time that allow to achieve the input‐to‐state stability of the closed‐loop system are determined. In order to obtain more tractable results, the existence conditions of the mode‐dependent Lyapunov‐like functions are given in terms of sum‐of‐squares programming in the case of polynomial nonlinearities. In the linear case, they are expressed in terms of linear matrix inequalities and a procedure for the synthesis of the mode‐dependent controller is provided in this situation. The established theoretical results are illustrated through a control problem of a building ventilation system and a switched control problem of a vehicle suspension system. [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]

Summary: This paper concerns the consensus analysis of multiagent systems made of the interconnection of identical nonlinear agents interacting with one another through an undirected and connected graph topology. Drawing inspiration from the theory of linear "decomposable systems," we provide a method for proving the convergence (or consensus) of such multiagent sytems in the case of polynomial dynamics. The method is based on a numerical test, namely a set of linear matrix inequalities providing sufficient conditions for the convergence. We also show that the use of a generalized version of the famous Kalman‐Yakubovic‐Popov lemma allows the development of a linear matrix inequalities test whose size does not directly depend on the number of agents. The method is validated in simulation on three examples, which also show how the numerical test can be used to properly tune a controller. [ABSTRACT FROM AUTHOR]

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]

Summary: This paper concerns the simulation of a class of nonlinear discrete‐time systems under a set of initial conditions described by a bounding ellipsoid. We derive a procedure allowing the propagation of such ellipsoids through time, which makes it possible to set a guaranteed hard bound on the evolution of the state of the system for all the possible initial conditions. Two versions of this procedure are given, the second of which is slightly less general but less computationally demanding. At the end of this paper, we first show an application of the method in the domain of aerospace engineering. Subsequently, three academic examples of applications are presented, two of which come from the theory of fractals. [ABSTRACT FROM AUTHOR]

This paper studies the consensus problem of second-order discrete-time multi-agent systems with relative-state-dependent noises. Directed switching topologies are considered. Firstly, for a kind of switching topology with each digraph containing a spanning tree, we give a weak consensus result on the basis of the mode-dependent average dwell time method. Then, if all digraphs in a switching topology are strongly connected and the corresponding Laplacian matrices have a common left eigenvector for zero eigenvalue, we prove that the mean square and almost sure consensus can always be guaranteed for an arbitrary switching sequence with some constant distributed control gains, and we also give the statistic properties of the final consensus points. Numerical examples are presented to illustrate the effectiveness of our results. Copyright © 2017 John Wiley & Sons, Ltd. [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]

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]

This paper is concerned with the distributed control problem of second-order agents under directed network topology. The control input of each agent only depends on its own state and the states of its neighbors corrupted by white noises. By using the algebraic graph theory and stochastic analysis method, necessary and sufficient conditions are presented for mean square bounded tracking. Finally, several simulation examples are given to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper investigates the fault diagnosis problem for discrete-time networked control systems under dropouts in both control and measurement channel with no delivery acknowledgment. We propose to use a proportional integral observer-based fault diagnoser collocated with the controller. The observer estimates the faults and computes a residual signal whose comparison with a threshold alarms the fault appearance. We employ the expected value of the arriving control input for the open-loop estimation and the measurement reception scenario for the correction with a jump observer. The jumping gains are scheduled in real time with rational functions depending on a statistic of the difference between the control command being applied in the plant and the one being used in the observer. We design the observer, the residual, and the threshold to maximize the sensitivity under faults while guaranteeing some minimum detectable faults under a predefined false alarm rate. Exploiting sum-of-squares decomposition techniques, the design procedure becomes an optimization problem over polynomials. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non-convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY This paper presents a new nonlinear polynomial controller for wind turbines that assures stability and maximizes the energy produced while imposing a bound in the generated power derivative in normal operation (guarantees a smooth operation against wind turbulence). The proposed controller structure also allows eventually producing a transient power increase to provide grid support, in response to a demand from a frequency controller. The controller design uses new optimization over polynomials techniques, leading to a tractable semidefinite programming problem. The ability of the wind turbine to increase its power under partial load operation has been analysed. The aforementioned optimization techniques have allowed quantifying the maximum transient overproduction that can be demanded to the wind turbine without violating minimum speed constraints (that could lead to unstable behaviour), as well as the total generated energy loss. The ability to evaluate this shortfall has permitted the development of an optimization procedure in which wind farm overproduction requirements are divided into individual turbines, assuring that the total energy loss in the wind farm is minimum, while complying with the maximum demanded power constraints. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]