Homogeneous Lyapunov Function Research Articles

Design of the second‐order robust exact differentiator: A describing function approach

AbstractThe paper proposes a design methodology for the second‐order robust exact differentiator in the presence of small time‐delays characterizing the time of signal processing. With this aim, the describing function is used to provide several sets of the gains ensuring high accuracy in the reconstruction of the first‐order derivative of smooth signals, that is, minimizing the main harmonic of chattering. The proposal fulfills the Loeb's criterion: if self‐excited oscillations appear in the trajectories of the delayed differentiator, they are orbital and asymptotically stable. Besides the global finite‐time stability of the error is proved for the non‐delayed differentiator under such gains settings by using a homogeneous Lyapunov function. Numerical examples are provided to illustrate the results.

Control of Heterogeneous Battery Energy Storage Systems-Based Microgrid Connected via Detail-Balanced Communication Topology

This letter proposes a distributed secondary control for heterogeneous battery energy storage systems (BESSs) to achieve finite-time consensus in frequency and active power while maintaining a balanced energy-level. The proposed scheme incorporates heterogeneity in electrical as well as control aspects and models heterogeneous BESS-based islanded AC microgrid as a multi-agent system with agents interacting according to a detail-balanced topology with heterogeneous edge weights. The finite-time stability of the closed-loop system, under the proposed controllers, is rigorously proved by considering a single composite homogeneous Lyapunov function, thereby calculating an upper limit on convergence time. Efficacy of the proposed controllers is illustrated by simulating a BESS-based microgrid in a Simulink environment.

Continuous Higher-Order Sliding Mode Control for a Class of n-th Order Perturbed Systems

In this brief, a general continuous higher-order sliding mode (HOSM) control algorithm is proposed for a class of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-order systems with Lipschitz disturbance, which can be regarded as a generalization of the continuous super-twisting algorithm for arbitrary relative degree with respect to the output. The proposed control algorithm only requires the information of the output and its derivatives up to the order <inline-formula> <tex-math notation="LaTeX">$n-1$ </tex-math></inline-formula>, and can compensate the Lipschitz disturbance exactly by using a discontinuous integral term, generating a continuous homogeneous control signal and achieving finite-time convergence to a sliding mode of order <inline-formula> <tex-math notation="LaTeX">$n+1$ </tex-math></inline-formula>. The convergence of the closed-loop system is proved rigorously by employing an explicit and directly constructed homogeneous Lyapunov function. The simulation results validate the effectiveness of the proposed method by comparing with the existing results.

Extended bounded real lemma based sum of squares for static output feedback H∞ heading control

AbstractAim at the fore and aft asymmetry of unmanned surface vehicle (USV) and the loss of linear velocity, a static output feedback (SOF) 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 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 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 system of HERBL has a superior performance on the transient response and the robustness.

Robust H∞ Fuzzy Observer-Based Fault-Tolerant Tracking Control for Nonlinear Stochastic System: A Sum of Square Approach

In this study, the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy is proposed for the stochastic polynomial fuzzy system (SPFS) under the effect of external disturbance, measurement noise, system/sensor fault signals and continuous/discontinuous internal random fluctuations. At first, the smoothed models of fault signals are constructed to describe their dynamic behavior. Then, by integrating the SPFS with the smoothed models of fault signals as one augmented system, the state/fault signal estimation problem can be transformed to a state estimation problem of augmented system by the proposed polynomial fuzzy observer. With the utilization of estimated state/fault signals and reference trajectory, a fuzzy polynomial fault-tolerant tracking controller can be implemented. To attenuate the effect of undesired external disturbance and measurement noise on the state/fault signal estimation and tracking control performance, the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy is proposed in this study. By utilizing the homogeneous Lyapunov function and Lipschiz condition, the Itô-Lévy formula is reformulated to relax the compensation terms of stochastic processes during the design. Then, the design conditions are derived in terms of interpolation function-dependent matrix inequality and consequently transformed to a two-step sum of square (SOS) condition design problem for the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy. A simulation example of double lane maneuvering task for autonomous ground vehicle (AGV) is provided to illustrate the effectiveness of proposed method. Index terms: Observer-based tracking control, polynomial fuzzy system, stochastic control, fault-tolerant control, sum of square (SOS).

Robust finite-time stabilisation of an arbitrary-order nonholonomic system in chained form

We present a robust finite-time stabilising algorithm for an arbitrary-order nonholonomic system in chained form. The state feedback is locally bounded, homogeneous and discontinuous. Its design follows from a generalised circle criterion based on the theory of homogeneous Lyapunov functions and a two-stage homogeneity-preserving strategy proposed by Nakamura.

Velocity‐free fixed‐time attitude cooperative control for spacecraft formations under directed graphs

AbstractThis article addresses the problem of velocity‐free fixed‐time attitude cooperative control for multiple spacecraft under directed communication graphs and in the presence of bounded disturbances. A distributed fixed‐time state observer is designed to estimate the leader's attitude states and a fixed‐time extended‐state observer is presented to obtain accurate estimates of both unmeasurable angular velocities and bounded disturbances. By use of the fixed‐time observers, a distributed fixed‐time attitude cooperative control scheme is proposed to make the states of all spacecraft simultaneously track a time‐varying reference attitude even when the reference attitude is available only to a subset of the group members. Homogeneous Lyapunov function and homogeneity property are applied to demonstrate that the control scheme can guarantee the attitude tracking errors converging to the neighborhood of the origin in fixed time which is independent of initial conditions. Finally, numerical simulations are presented to show the effectiveness of the control law derived here.

Design of a proportional integral derivative‐like continuous sliding mode controller

AbstractA describing function‐based design for the continuous sliding mode controller in the proportional integral derivative (PID) form (PID‐CSMC) is provided. Two sets of the PID‐CSMC gains are suggested. The Harmonic Balance is used to predict the amplitude and frequency of the main harmonic of chattering caused by the presence of fast‐parasitic dynamics in the closed‐loop. Predicted values of amplitude and frequency allow to compute the average power needed to maintain the trajectories of the system into real sliding modes. The methodology for selection of suboptimal PID‐CSMC gains consists of amplitude of chattering and average power minimization, taking into account the presence of a critically damped actuator parameterizing the effects of parasitic dynamics. A novel homogeneous Lyapunov function proves that suggested sets of the PID‐CSMC gains ensure, in theory, finite‐time stability of the perturbed double integrator without parasitic dynamics, and exact compensation of Lipschitz perturbations. On the other hand, the Loeb's criterion allows to ensure that the PID‐CSMC with suggested gains generates orbitally asymptotically stable fast‐oscillations in the presence of fast‐actuators.

Lyapunov‐based consistent discretization of stable homogeneous systems

SummaryIn this article, we propose a discretization scheme for asymptotically stable homogeneous systems. This scheme exploits the information provided by a homogeneous Lyapunov function of the system. The main features of the scheme are: (1) the discretization method is explicit and; (2) the discrete‐time system preserves the asymptotic stability, the convergence rate, and the Lyapunov function of the original continuous‐time system.

Exponential stability analysis for singular switched positive systems under dwell-time constraints

The article addresses the exponential stability analysis for singular switched positive systems(SSPSs) under dwell-time constraints which include mode-dependent minimum dwell time(MDMDT), mode-dependent constant dwell time(MDCDT) and mode-dependent ranged dwell time(MDRDT) constraints. For SSPSs in both delay-free case and time-varying delay case, a sufficient exponential stability condition is proposed with MDMDT, MDCDT and MDRDT constraints, respectively, and the exponential decay rate can be set as a free parameter based on diverse circumstances. To analyze the dwell-time stability, a novel discretized linear copositive Lyapunov function(DLCLF) approach is introduced in the article, and compared with the general copositive and homogeneous Lyapunov function approach, the main advantage of the DLCLF technique is itself affine dependence on conditions of system matrix, which will be extended and utilized to systems with uncertainties or/and time-varying parameters quite easily. Meanwhile, the proposed condition of exponential stability under MDMDT(or MDCDT or MDRDT) constraint will be degenerated into the one under minimum dwell time(or constant dwell time or ranged dwell time) constraint for some specific situations, which implies that the considered MDMDT(or MDCDT or MDRDT) case is more general and practical. Finally, the validity and significance of the results are illustrated by seven examples.

Reaction wheel pendulum control using fourth‐order discontinuous integral algorithm

SummaryThe fourth‐order model of the reaction wheel pendulum is considered and a fourth‐order discontinuous integral algorithm is used for stabilization and tracking of the system, using a continuous control signal. The states reach the origin or a reference signal in finite‐time, even in presence of uncertain control coefficient and a kind of matched and unmatched uncertainties/disturbances. A homogeneous Lyapunov function is designed to ensure local finite‐time stability of the system, which can be used for designing the controller gains. Simulations and experimental results illustrate the performance and advantages of the presented algorithm.

Stability and stabilisation of heterogeneous switched systems with arbitrary switching constraints

Heterogeneous switched systems are a class of special hybrid systems containing different dimensional subsystems. To research the stability and stabilisation of heterogeneous switched systems, a novel approach is presented to design static output feedback controllers for asymptotically stabilising heterogeneous continuous-time and discrete-time switched systems under arbitrary switching rules. This approach, which has less computation complexity, includes a main diagonal square block common Lyapunov function for stability analysis of heterogeneous switched systems and a main diagonal square block common homogeneous polynomial Lyapunov function for stabilisation of heterogeneous switched systems. In the study of stabilisation of heterogeneous switched systems, a two-step iterative method including two convex optimisation functions is established by sums of squares to express existing properties condition and solvable properties condition for determining block controller. At last, a practical example and two numerical examples are presented to show the feasibility of the proposed results.

Homogeneous Lyapunov functions for homogeneous infinite dimensional systems with unbounded nonlinear operators

The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for a class of asymptotically stable homogeneous infinite dimensional systems with unbounded nonlinear operators.

Event-triggered extended state observers design for dynamic positioning vessels subject to unknown sea loads

This paper presents a resource-aware state and uncertainty estimation method for dynamic positioning vessels subject to uncertain kinetics and unknown sea loads induced by wind, waves and currents. Specifically, an event-triggered extended state observer (ESO) is proposed such that unnecessary communications are avoided and sensor-to-observer communication cost are drastically reduced. A realizable event-triggering condition is established depending on the position-heading measurements only. It is proven that the observation errors are input-to-state stable (ISS), and Zeno behavior will not occur. Aimed at achieving finite-time convergence, an event-triggered finite-time convergent ESO is further designed. By resorting to a homogeneous Lyapunov function, the estimation errors are proven to be finite-time ISS. A salient feature of the proposed event-triggered ESOs is that a joint observation of the unmeasured velocities, unknown kinetics and unknown sea loads can be achieved with reduced communication cost. The proposed event-triggered ESOs can be applied to various motion scenario such trajectory tracking, path following, target tracking, and even formation control of marine vessels. Simulation results show that the communication times are largely decreased compared with the time-triggered ESOs.

On Homogeneous Lyapunov Function Theorem for Evolution Equations

The existence of homogeneous Lyapunov function for a stable homogeneous ordinary differential equation (ODE) is proven by V. Zubov in 1958 and refined by L. Rosier in 1992. The present paper proposes an extension of this result to evolution equations in Banach spaces.

A Consistent Discretisation method for Stable Homogeneous Systems based on Lyapunov Function

In this paper we propose a discretisation scheme for continuous and asymptotically stable homogeneous systems. This method is based on the dynamics of the system projected on a level surface of a homogeneous Lyapunov function. The discretisation method is explicit and preserves the convergence rate of the continuous-time system.

Stability Analysis of Aperiodic Sampled-Data Systems: A Switched Polytopic System Method

Stability of aperiodic sampled-data systems is investigated in this brief. A sampling interval discretization method is proposed to transform an aperiodic sampled-data system into a discrete-time switched polytopic system with help of the convex approximation technique. Using a high order time-scheduled switched homogeneous Lyapunov function (TSSHLF), a necessary and sufficient stability condition for the discrete-time switched polytopic system is proposed and presented in terms of linear matrix inequality (LMI). On the basis of this result, a sufficient stability criterion for the aperiodic sampled-data system is obtained. Finally, numerical examples are given to confirm the effectiveness of the proposed method and significant improvements over some existing ones.

Fixed‐time output‐feedback consensus tracking control for second‐order multiagent systems

SummaryThis paper is concerned with the problem of fixed‐time consensus tracking control for a class of second‐order multiagent systems under an undirected communication graph. A distributed output‐feedback fixed‐time consensus tracking control scheme is proposed to make the states of all individual agents simultaneously track a time‐varying reference state even when the reference state is available only to a subset of the group members and only output measurements are available for feedback. Homogeneous Lyapunov function and homogeneity property are employed to show that the control scheme can guarantee the consensus tracking errors converging the origin in finite time which is bounded by a fixed constant independent of initial conditions. Numerical simulations are carried out to demonstrate the effectiveness of the proposed control law.

Iterative Homogeneous Polynomial Lyapunov Functions Approach for Stabilizing Minimum Mode-Dependent Average Dwell Time Switched Systems via Output Feedback

A novel approach is proposed in this paper to design static output feedback controllers for asymptotically stabilizing continuous-time switched linear systems with minimum mode-dependent average dwell time (MMDADT). This approach adopts an iterative algorithm containing the solution of a traversal algorithm function and three optimization functions, the decision variables of which are established by the sum of squares (SOS) of matrix polynomials. The traversal algorithm function takes advantage of a polynomially parameter-bounded condition, allowing us to get the lower-bound of the Homogeneous Polynomial Lyapunov Functions’ (HPLFs) derivative. The first optimization function is a non-convex optimization function, which expresses a sufficient condition for stability analysis and computing MMDADT. The second and third optimization functions are both convex optimization functions, which are used to calculate the polynomially mode-dependent output feedback controllers. Two numerical examples are presented in order to show the feasibility of the proposed results.

State recovery and disturbance estimation of unmanned surface vehicles based on nonlinear extended state observers

This paper investigates the state recovery and disturbance estimation of unmanned surface vehicles in the presence of unknown disturbances as well as unmeasured surge, sway, and yaw velocities. The disturbances come from the internal parametric model uncertainties, unknown hydrodynamics, and external disturbances due to wind, waves and ocean currents. At first, a nonlinear extended state observer is developed for not only recovering the unmeasured velocities, but also estimating the disturbances. By using simultaneous Lyapunov inequalities, the observer error dynamics is proven to be input-to-state stable. Then, a finite-time convergent nonlinear extended state observer is proposed to estimate the unknown velocities and disturbances, and the finite-time input-to-state stability of the observer error dynamics is established based on a homogeneous Lyapunov function. By using the proposed nonlinear extended state observers, both velocity and disturbance information can be reconstructed from the position-heading measurement by using global positioning system and gyrocompass only. Simulation results are provided to validate the efficacy of the proposed state recovery and disturbance estimation method for both fully-actuated and under-actuated unmanned surface vehicles.