Parameter-dependent Lyapunov Function Research Articles

Robust α-exponential estimate for polytopic uncertain positive discrete-time systems with bounded time-varying delay

In this paper, the problem of finding a robust α-exponential estimate for a class of perturbed positive discrete-time time-delay systems with polytopic uncertainties will be addressed. By proposing a linear parameter-dependent Lyapunov function, a condition for the existence of robust α-exponential estimates will be introduced. Some optimisation techniques are also incorporated to obtain a more accurate estimate. In particular, a procedure to reformulate a class of fractional programming problems into a linear programming problem which can be solved efficiently. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Coordinated management of oxygen excess ratio and cathode pressure for PEMFC based on synthesis variable-gain robust predictive control

In this research endeavor, a novel synthesis variable-gain robust model predictive control (SVGRPC) method using the min-max technique with parameter dependent Lyapunov functions is introduced to achieve the desired trajectory tracking of oxygen excess ratio (OER) and cathode pressure in the polymer electrolyte membrane fuel cells (PEMFCs) system. Initially, a simplified control-oriented model of the air supply system is developed and then transformed into a polytopic form to accommodate the inherent uncertainties in the PEMFC. Subsequently, the polytopic model is integrated with the reference trajectory to construct an augmented state-space model for deriving an error state representation. In the framework of robust model predictive control (RPC) utilizing a parameter-dependent Lyapunov function, a series of variable feedback gains are employed to mitigate conservatism. Additionally, the online solution process imposes a significant computational burden, presenting challenges for the real-time implementation of RPC-based control strategy. Therefore, the offline computing is introduced to significantly reduce the computational burden, resulting in the development of the SVGRPC strategy proposed in this paper. Subsequently, the SVGRPC strategy computes explicit linear state-feedback control laws by solving linear matrix inequalities (LMIs) using an offline control algorithm. The effectiveness of the proposed controller is then validated through assessments conducted under three distinct operating conditions of the PEMFC system. The outcomes of this study affirm that the proposed controller improves the requirements of control performance when compared to the online RPC with a quadratic Lyapunov function while significantly alleviating the computational workload.

Design of event-triggered natural second-order observers for second-order vector systems

The problem of designing event-triggered natural second-order state observers for second-order vector systems subject to external disturbances is considered in this paper. In this work, state observer is constructed using discrete measurements at instants when an event-triggered condition is satisfied. Due to the errors that arise by using the event-triggered mechanism and the external disturbance, the method in this paper only robustly estimates the state position and the state velocity of the second-order vector observers instead of estimating them asymptotically. Instead of using the method based on a parameter-dependent Lyapunov function in investigating the stability of the dynamic error system of natural second-order observers, a convex optimization problem is established in this paper to give an event-triggered mechanism and minimized levels in evaluating the boundedness of the dynamic error system of the event-triggered natural second-order observers. An automobile suspension and two numerical examples demonstrate the effectiveness of the proposed method.

Sliding Mode Control for Uncertain 2-D FMII Systems Under Stochastic Scheduling.

In this article, the sliding mode control (SMC) problem is addressed for two-dimensional (2-D) systems depicted by the second Fornasini-Marchesini (FMII) model. The communication from the controller to actuators is scheduled via a stochastic protocol modeled as Markov chain, by which only one controller node is permitted to transmit its data at each instant. A compensator for other unavailable controller nodes is introduced by means of previous transmitted signals at two most adjacent points. To characterize the features of 2-D FMII systems state recursion and stochastic scheduling protocol, a sliding function associated with the states at both the present and previous positions is constructed, and a scheduling signal-dependent SMC law is designed. By constructing token- and parameter-dependent Lyapunov functionals, both the reachability of the specified sliding surface and the uniform ultimate boundedness in the mean-square sense of the closed-loop system are analyzed and the corresponding sufficient conditions are derived. Furthermore, an optimization problem is formulated to minimize the convergent bound via searching desirable sliding matrices, meanwhile, a feasible solving procedure is provided by using the differential evolution algorithm. Finally, the proposed control scheme is further demonstrated via simulation results.

Enhanced robust adaptive flight control for a convertible VTOL UAV

In this paper, we propose less conservative formulations for candidate controllers of robust adaptive mixing control (RAMC) strategies. The RAMCs are employed to solve the trajectory tracking problem throughout the full flight envelope, with guaranteed stability, of a convertible plane (CP) vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). Initially, the multi-body nonlinear dynamic model of the CP VTOL UAV is obtained using the Euler–Lagrange formalism, from which a linear parameter-varying (LPV) model is derived to be used in the design of the RAMC candidate controllers. The new formulations of the candidate controllers are proposed considering two approaches: a (i) Parallel Distributed Compensation (PDC); and (ii) a parameter-dependent Lyapunov function. Hardware-In-the-Loop (HIL) experiments are performed in a high-fidelity flight simulator to verify the fulfillment of real-time constraints, ensuring a computationally lightweight control strategy ready for implementation in a low-cost embedded computational system.

Discretization and state feedback control for uncertain linear systems—A new approach considering linear multistep method theory

This paper proposes a novel procedure for discretizing uncertain time-invariant continuous-time linear systems in polytopic domains. The approach is based on multistep method theory and involves mixing models obtained with different multiples of a fixed step-size (sampling time), thereby increasing data availability. An optimization procedure is employed to determine the coefficients that minimize the residual discretization error, and then these coefficients are combined to construct an augmented structure for the discrete-time system. The discretization error is computed using a grid search and incorporated into the discrete-time system formulation. Using the resulting discrete-time system, a state feedback methodology is applied to synthesize digital controllers capable of stabilizing the original continuous-time system. The paper presents two sufficient Linear Matrix Inequality (LMI)-based conditions for designing robust controllers specific to the proposed structure. The first condition utilizes a constant Lyapunov function, while the second condition employs a parameter-dependent Lyapunov function. In both cases, the proposed conditions guarantee asymptotic stability for the continuous-time system in closed-loop. A numerical experiment is conducted to illustrate the validity and effectiveness of the proposed method.

A Full Envelope Robust Linear Parameter-Varying Control Method for Aircraft Engines

In order to solve the problem of full flight envelope control for aircraft engines, the design of a linear parameter-varying (LPV) controller is described in this paper. First, according to the nonlinear aerodynamic model of the aircraft engine, the LPV engine model for the controller design is obtained through the Jacobian linearization and fitting technique. Then, the flight envelope is divided into several sub-regions, and the intersection of adjacent sub-regions is not empty. The sub-region LPV controller is designed using the parameter-dependent Lyapunov function (PDLF)-based LPV synthesis method, while eliminating the dependence of the LPV controller on scheduling parameter derivatives. In order to ensure the stability and performance of the aircraft engine across the full flight envelope, a mixing LPV control method is proposed to design the LPV controller in the overall region. The effectiveness of the proposed method is verified by simulating a dual-spool turbofan engine on a nonlinear component level model and comparing the proposed method with the gain scheduling based on PI and H∞ point design.

Event-Triggered Control for LPV Modeling of DC–DC Boost Converter

This study presents the event-triggered control (ETC) for linear parameter varying (LPV) model of boost converters. We examine the nonlinear dynamics of boost converters in the LPV framework. The proposed controller is duty-ratio-dependent and provides better performance while requiring less computation. Using the parameter-dependent Lyapunov function (PDLF), we demonstrate the stablity analysis of the proposed approach. Furthermore, we demonstrate that the inter-event time is lower bound by a positive constant, which indicates Zeno behavior free performance. In comparison to earlier time-invariant synthesis techniques, the LPV formulation offers for increased robustness and performance properties. Simulation and experimental results validate the effectiveness of the proposed method.

H2/H∞ Robust Observed-State Feedback Control Based on Slack LMI-LQR for LCL-Filtered Inverters

A robust controller for an LCL-filtered grid-connected inverter (GCI) is presented in this paper in the context of the system model uncertainties and non-ideal grid environment. Under the occurrence of exogenous inputs from parameter drifts affecting the controller and observer, the H₂/H∞ approach of the linear quadratic regulator (LQR) control is developed to obtain the state-feedback gain and observer gain with the system stability guarantee by means of Lyapunov stability theory. Furthermore, the use of slack-linear matrix inequalities (LMIs) to handle parameter-dependent Lyapunov functions in synthesizing the robust controller can achieve a less conservative condition than the conventional quadratic stability. System robustness and stability are assessed through the closed-loop pole map and the discrete-time frequency analysis approaches. Hardware experiments are conducted for the LCL-filtered GCI to validate the feasibility and effectiveness of the proposed control method. Fair comparisons with the conventional schemes are also performed to highlight the superior performance of the proposed scheme.

Geometric approach to robust stability analysis of Linear Parameter-Varying systems: Computational trade-offs between the exact and the simplex convex hulls

Robust stability is investigated for continuous-time, affine, Linear Parameter-Varying (LPV) models, with bounded variation rates of uncertainty. Towards this end, affine Parameter Dependent Lyapunov Functions (PDLF) are considered to certificate stability in the Lyapunov sense. In the literature, the search for PDLFs amounts to a non conservative Linear Matrix Inequality (LMI) problem, at expense of a factorial growth in its complexity. Remedies to overcome this complexity have been proposed recently, exploiting geometrical aspects of the problem, however, they can be conservative. This paper presents new stability tests that are a trade-off between the exact and the fastest solutions. We offer an analytical procedure to indicate when the proposed tests are prone to reduce conservativeness. Also, a simple procedure is introduced to possibly improve existing and new tests, without impacts on the computation effort. Numerical simulations illustrate the improvements of the proposed strategies.

New results on [formula omitted] control for interval type-2 fuzzy singularly perturbed systems with fading channel: The weighted try-once-discard protocol case

This article deals with the interval type-2 (IT2) fuzzy model-based H∞ control problem for networked singularly perturbed systems (SPSs) under fading channel. To prevent data collision, a weighted try-once-discard (WTOD) communication scheme is adopted to adjust the communication sequence of sensor nodes. Based on this transmission strategy, only one sensor with more considerable difference is allowed to be transmitted at each time instant. On the other hand, a time-varying stochastic process model is introduced to describe the behavior of the fading channels in unreliable networks. Based on fading state signal and known bounds, the imperfectly matched membership functions (MFs) are reconstructed in the controller. Accordingly, under the effect of fading channels and WTOD protocol, a new IT2 fuzzy controller is constructed for the networked IT2 fuzzy SPSs. Meanwhile, by constructing a singular perturbation parameter (SPP) dependent Lyapunov function, sufficient conditions are established to guarantee the mean square stability of networked IT2 fuzzy SPSs, and the numerical stiffness issue is avoided. Finally, the practicability of the proposed methods is proved through a tunnel diode circuit simulation.

Gain scheduling control of aero-engine based on mixing polynomial LPV synthesis

Abstract A full envelope LMI-based multi-region linear parameter-varying power controller is designed for a turbofan engine in this paper. According to the characteristics of aero-engine model, three scheduling variables are divided into two groups firstly, and then part of them are partitioned, rather than all scheduling variables are partitioned directly as the usual multi-region LPV control. The polynomial LPV model of aero-engine is established under a specific flight condition. An explicit LPV controller by gridding method based on parameter-dependent Lyapunov function is designed and we propose a method to eliminate the dependence of LPV controller on the derivative of scheduling parameter. The flight envelope of turbofan engine is divided into multiple sub-regions, and a mixing LPV control method with overlapping regions is proposed, which can guarantee stability and performance across the full envelope. Finally, the simulation results on the nonlinear component level model of a twin-spool turbofan engine verify our method.

Robust Optimized Current Controller Based on a Two-Step Procedure for Grid-Connected Converters

Proportional-resonant controllers are largely used for grid-connected converters with LCL filters to ensure sinusoidal reference tracking and harmonic disturbance rejection. However, the design of such controllers becomes more difficult when it is necessary to ensure stability and suitable performance for converters operating under uncertain grid conditions, from stiff to weaker, and also when the number of resonants increases. In this more challenging scenario, available design procedures may lead to a poor tradeoff between transient and steady-state performances. To overcome this problem, this paper proposes an off-line automated control design procedure to obtain multiple robust optimized proportional-resonant controllers. The proposed procedure consists of two steps, where a particle swarm algorithm is used to obtain: i) the gains of an inner loop ensuring optimal active damping for the LCL filter resonance; ii) the gains of multiple proportional-resonant controllers by minimizing a tracking error index, taking into account uncertain grid impedance and control signal limits. The robust stability of the closed-loop system with the control gains tuned by the metaheuristics is certificated through a parameter-dependent Lyapunov function, constructed based on linear matrix inequalities. Several results from control Hardware-in-the-loop are presented for a case study, revealing that the proposal provides better performance in terms of reference tracking and disturbance rejection, when compared to similar control strategies from the literature. Experimental results from a prototype connected to the utility grid attest the practical feasibility of the proposed procedure.

On Robust Stabilization of Discrete-Time LPV/LFR Systems

This technical article presents novel robust stabilization conditions for discrete-time linear parameter-varying (LPV) systems with linear fractional representation (LFR). The proposed conditions rely on the use of slack variables and decision matrices associated with the LFR approach to provide new controller designs. In addition, we address parameter-dependent Lyapunov functions and full-block multipliers to obtain less conservative synthesis conditions for discrete-time LPV/LFR systems. Design conditions are cast in the form of linear matrix inequalities to generate robust state-feedback and output-feedback controllers. Numerical examples demonstrate the effectiveness of the proposed method.

Gain-scheduling robust control with guaranteed stability for ball screw drives with uncertain load mass and varying resonant modes

This paper presents a gain-scheduling (GS) robust control strategy using the interpolation approach for ball screw drives while guaranteeing the system stability in the operating range. A set of linear-time-invariant (LTI) robust controllers is designed at selected table positions for the robustness against the uncertain load mass and the structural vibration suppression at those positions. By interpolating the LTI robust controllers, a GS robust controller is obtained to deal with the dynamic variations caused by the table motion. The closed-loop stability under the proposed GS robust controller is verified with the parameter-dependent Lyapunov function, which avoids tedious simulations to check the reliability of interpolating GS controllers before practical operations. The method presented in this paper enables the GS controller to consider the unmeasurable load mass in ball screw drives and simplifies the application complexity. The extensive experimental results demonstrate that the proposed GS robust controller significantly improves the tracking performance and the vibration suppression, in comparison to LTI controllers and GS controllers.

Static output feedback H∞ control for switched LPV time-delay systems with nonlinear constraints

This paper addresses the issues of static output feedback [Formula: see text] control for switched linear parameter-varying (LPV) time-delay systems using multiple parameter-dependent Lyapunov functions. First, a parameter and state-dependent switching strategy with dwell time is designed and a sufficient condition ensuring the stability of the unforced switched LPV time-delay systems is established with the aid of the switching law. Then, the piecewise parameter-dependent controller is designed, such that the closed-loop system is asymptotically stable and satisfies an [Formula: see text] performance index. Finally, two examples are provided to show the effectiveness of the results.

Stochastic leader-following consensus of discrete-time nonlinear multi-agent systems with multiplicative noises

This paper studies the stochastic leader-following consensus problem of discrete-time nonlinear multi-agent systems (MASs) with multiplicative noises. The measurement information obtained from agents’ neighbors is inevitably affected by communication uncertainties, where the multiplicative noise is one of the important communication uncertainties. Multiplicative noises together with the intrinsic nonlinear dynamics bring more difficulties in the consensus control design under the leader-following topology. To solve the problem, the parameter-dependent Lyapunov functions are constructed to analyze the consensus control of first-order and second-order MASs, respectively. Some sufficient conditions, explicitly related to control gains, intensity of multiplicative noises and the Lipschitz constant regarding nonlinear functions, are established for reaching the mean square (m.s.) and almost sure (a.s.) leader-following consensus. Specifically, the obtained conditions are some scalar inequalities, which are more convenient in engineering application. Numerical simulations are conducted to validate the theoretical results.

Fault Detection of Discrete-time LPV System Based on Event Triggering Mechanism

For discrete-time linear parameter varying systems with distributed time delays, the problem of fault detection under event triggering mechanism is studied. Based on the parameter dependent Lyapunov function method, the sufficient conditions for the system to be asymptotically stable and meet the H∞ performance criterion are given. The parameter matrix of fault detection filter is obtained by solving LMI technology. In addition, the event triggering strategy is introduced to save network resources.

Improved ℋ2 and ℒ2 control designs for discrete-time LPV/LFR systems

This paper addresses novel and norm-based controller designs for discrete-time linear parameter-varying (LPV) systems with linear fractional representation (LFR). Despite the importance of LPV/LFR systems to the control literature, there is a lack of works on this topic. The distinctive formulation of the proposed controller designs is given by using parameter-dependent Lyapunov functions in combination with slack variables and general full-block multipliers. Such proposed formulation leads to less conservative synthesis conditions characterised in terms of a finite number of linear matrix inequalities for both state-feedback and static output-feedback controllers subject to and performances for discrete-time LPV/LFR systems. Improvements obtained by the proposed controller design conditions over existing methods in the area are assessed by numerical examples.

Networked‐based H∞ control for discrete‐time slow sampling singularly perturbed systems via an improved event‐triggered approach

AbstractThis paper investigates the H∞ control problem for a class of slow sampling singularly perturbed systems (S3PSs) with an improved event‐triggered method (ETM). Compared with the conventional static ETM, an improved one with time‐varying threshold is exploited to enhance the dynamic performance for the S3PSs with certain communication frequency reduction. By adjusting the threshold with the triggering error, a faster converge speed can be achieved. Sufficient conditions are derived by constructing a singular perturbation parameter (SPP) dependent Lyapunov function, with which both asymptotic stability can be guaranteed for the closed‐loop S3PSs and the ill‐conditioned numerical issues can be avoided. By resorting to the matrix inequality techniques, an ETM‐based controller can be developed within the upper bound of the SPP. Further demonstration for the feasibility of the proposed algorithm is presented by designing an ETM‐based controller for an inverted pendulum system.