Quadratic Lyapunov Function Approach Research Articles

Composite quadratic Lyapunov function event-triggered control of vertical take-off and landing aircraft systems with actuator saturation

This paper deals with the problem of event-triggered control (ETC) for vertical take-off and landing (VTOL) aircraft systems under actuator saturation. By introducing the VTOL system with saturated actuators, a composite quadratic Lyapunov function (CQLF) approach is first proposed to describe the problem of the event triggering mechanism. The proposed transfer mechanism not only further saves communication resources, but also further suppresses the conservatism of the attraction domain estimation. Then, sufficient conditions are established via linear matrix inequality (LMI) by incorporating new event triggering mechanism condition and prescribed passive performance metrics under the closed-loop system that is asymptotically stable and strictly passive, respectively. Finally, the method proposed in this paper can effectively reduce the number of triggers and expand the domain of attraction. The validity and feasibility of the obtained results are demonstrated by two example simulations.

Combined longitudinal and lateral dynamics regulation of autonomous vehicles via induced ℒ2$$ {\mathcal{L}}_2 $$‐gain linear parameter varying control strategies based on parameter‐dependent Lyapunov functions

AbstractThis article considers a path tracking control problem for autonomous vehicles constrained to maintain a safe distance from the next vehicle in the lane. The control design problem is solved by considering a mathematical model of the vehicle that jointly describes the lateral and longitudinal dynamics. Specifically, the optimal controller is designed aimed at minimizing the orientation and position tracking errors with respect to a given reference path. Moreover, the controller is able to ensure that the vehicle can both follow a speed profile and automatically adjusts the vehicle speed to maintain a proper distance from the next vehicle. The lateral dynamic is regulated by a 2‐DOF feedforward/feedback controller, where the feedforward law is implemented by exploiting an inverse kinematic model of the vehicle. On the other side, the feedback action, along with the action of the controller in charge to regulate the longitudinal dynamics, is the optimal gain‐scheduling state‐feedback controller. In particular, a parameter‐dependent quadratic Lyapunov function approach is used to reduce the conservativeness of the solution and a suitable convex LMI optimization problem is formulated for the synthesis. The vehicle is described by a seven‐degrees‐of‐freedom (7DOF) full‐car model, with the tire‐generated forces described by the classical Pacejka's Magic Formula. In order to assess the performance of the proposed control strategy, simulations have been undertaken in Matlab/Simulink by considering an autonomous vehicle driving into an urban scenario.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

Robust synchronization analysis of delayed fractional order neural networks with uncertain parameters

<abstract><p>This paper is concerned with the robust synchronization analysis of delayed fractional order neural networks with uncertain parameters (DFNNUPs). Firstly, the DFNNUPs drive system model and response system model are established. Secondly, using multiple matrix quadratic Lyapunov function approach and inequality analysis technique, the robust synchronization conditions are derived in the form of the matrix inequalities. Finally, the correctness of the theoretical results is verified by an example.</p></abstract>

New result on robust stability of switched systems with all subsystems unstable

In this study, the stability problem of continuous-time switched systems composed fully of unstable subsystems is considered. Unlike the hybrid conditions derived in previous literature, this study used a novel time-dependent quadratic Lyapunov function approach and a convex sufficient condition for switched linear systems is proposed in the framework of bounded maximum average dwell time (BMADT). The resulting condition in this study is enforced using the sum of squares programming. Then, the condition for switched linear systems is extended to uncertain systems and the robust stability condition is derived. Moreover, a new stability result of continuous-time switched systems with all subsystems unstable is investigated based on the BMADT conditions. The simulation shows that it is better than the results in previous literature. Two numerical examples are proposed to illustrate the authors' approach.

Stability Analysis of Switched Linear Systems with Uncertainty and Delays

Combining the common quadratic Lyapunov functional approach and free-weighting matrix approach, this paper is devoted to the stability analysis of continuous-time switched linear systems (CTSLS) with uncertainty and time-delays. A particular class of matrix inequalities, the so-called Lyapunov–Metzler inequalities are introduced for the CTSLS to investigate the stability and performance in the presence of the uncertainty and delays. We provide sufficient conditions in terms of the Linear Matrix Inequality criterions to guarantee delay-dependent asymptotically stability under the uncertainty of the CTSLS. The combination of switching rule and switching output feedback controllers which will be designed to stabilize the CTSLS and satisfy a prespecified $${\mathcal{L}}_{2}$$ gain performance. A example used jitterbug tools is provided to illustrate the effectiveness of the proposed method.

Asynchronous control for 2-D switched systems with mode-dependent average dwell time

This work is concerned with asynchronous control for a class of two-dimensional switched systems with mode-dependent average dwell time. The systems are formulated by the famous Fornasini–Marchesini local state-space model, and the switching signal of the switched controller involves time delays which lead to the asynchronism between controllers and subsystems. By extending mode-dependent average dwell time with the switched quadratic Lyapunov functional approach, the stabilization condition is established for 2-D asynchronously switched systems. Based on the stabilization result, mode-dependent state-feedback controllers are designed to guarantee H∞ performance and asymptotic stability of the corresponding closed-loop system. Finally, two examples are provided to illustrate the effectiveness of the proposed design scheme.

Stability and stabilisation for time-varying polytopic quadratic systems

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

Mean-square admissibility for stochastic T–S fuzzy singular systems based on extended quadratic Lyapunov function approach

The mean-square admissibility problem of stochastic T–S fuzzy singular systems via an extended quadratic Lyapunov function is investigated in this paper. Comparing with the existing quadratic Lyapunov function method, the extended quadratic Lyapunov function method can relax stabilization conditions. Firstly, the sufficient condition is given for the mean-square admissibility of stochastic T–S fuzzy singular systems based on an extended quadratic Lyapunov function approach. Secondly, two sufficient conditions for mean-square admissibility of closed-loop systems via the parallel distributed compensation (PDC) fuzzy controller and non-parallel distributed compensation (non-PDC) fuzzy controller are proposed. Furthermore, through the extended quadratic Lyapunov function method and non-PDC fuzzy controller, the less conservative mean-square admissibility conditions on solving fuzzy controllers are derived in terms of linear matrix inequalities (LMIs). Finally, some simulation examples are given to show the effectiveness and merits of the proposed fuzzy controller design methodology.

Variable selective control for networked systems: A switched quadratic Lyapunov functional approach

This paper presents a new method for control of linear networked systems by combining the predictive control and the variable sampling period approaches. In this way, event-driven sensors are implemented, i.e. by construction the sensors are triggered to sample the outputs of the plant, when new control input signals are received by the actuators. In light of this formulation, at each sampling instant, total control loop delay will be equal to the sampling period which is unknown. In order to deal with the network effects associated with a range of pre-specified time delays, appropriate step invariant discrete-time models of the networked plant are calculated offline. Based on these, some stabilizing control signals are constructed online. The control signals are then packed in the control-side packet, transmitted back to the plant side and received by a time delay compensator module. A less conservative class of Lyapunov functions, called switched quadratic Lyapunov, is used here for stability analysis and stabilizing controller design. Simulation studies on well-known benchmark problems demonstrate the effectiveness of the proposed method.

Stabilization for a Class of Switched Nonlinear Systems With Novel Average Dwell Time Switching by T-S Fuzzy Modeling.

In this paper, the problem of switching stabilization for a class of switched nonlinear systems is studied by using average dwell time (ADT) switching, where the subsystems are possibly all unstable. First, a new concept of ADT is given, which is different from the traditional definition of ADT. Based on the new proposed switching signals, a sufficient condition of stabilization for switched nonlinear systems with unstable subsystems is derived. Then, the T-S fuzzy modeling approach is applied to represent the underlying nonlinear system to make the obtained condition easily verified. A novel multiple quadratic Lyapunov function approach is also proposed, by which some conditions are provided in terms of a set of linear matrix inequalities to guarantee the derived T-S fuzzy system to be asymptotically stable. Finally, a numerical example is given to demonstrate the effectiveness of our developed results.

Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings

In this paper, the problems of stability analysis and stabilization are investigated for discrete-time two-dimensional (2-D) switched systems, which are formulated by the well-known Fornasini–Marchesini local state-space model. Firstly, by using the switched quadratic Lyapunov function approach, a sufficient stability condition is established for such systems under arbitrary switching signal. Then, the extended average dwell time technique combining with the piecewise Lyapunov function approach is developed, and it is utilized for the stability analysis of the 2-D switched systems for the restricted switching case. Based on the stability analysis results, sufficient conditions are presented to stabilize the 2-D switched systems. Finally, two examples are provided to illustrate the effectiveness of the proposed new design techniques.

A Generalized Piecewise Quadratic Lyapunov Function Approach to Estimating the Domain of Attraction of a Saturated System

This paper revisits the problem of estimating the domain of attraction of a saturated system with an algebraic loop. A piecewise quadratic Lyapunov function of an augmented state vector composing of the system state and the saturated input has been proposed for use in estimating the domain of attraction for a saturated system. Considering the relationship between the system states and the saturation function, we propose in this paper a generalized piecewise quadratic Lyapunov function, which results from adding a term that characterizes the regional sector condition of the saturated input to the piecewise quadratic Lypunov function. The matrix associated with the generalized piecewise quadratic Lyapunov function is not required to be positive definite, and thus a set of less conservative stability conditions are established, from which a larger estimate of the domain of attraction can be obtained. Simulation results indicate that the effectiveness and superiority of the proposed method.

Stability analysis and failure tolerant control for discrete-time linear systems with controller failure

In this paper, the stability and failure tolerant control are investigated for discrete-time time-invariant linear systems with occasional controller failure due to physical or purposeful reason. By introducing the conceptions of minimal controller reliable time and maximal controller failure time, a novel time-varying piecewise quadratic Lyapunov function (TPQLF) approach is proposed to analyse the stability property of closed-loop system in presence of controller failure. The presented TPQLF method combines both merits of previous Lyapunov function based results such as common Lyapunov function and piecewise Lyapunov function approaches. Furthermore, based on TPQLF, the feasible region described by pairs of minimal controller reliable time and maximal controller failure time can be easily computed by an iterative algorithm. Then, on the basis of the analysis results, time-varying state feedback gains are formulated to construct failure tolerant controller. It is demonstrated that the time-varying structure of controller provides more failure tolerance capacity than time-invariant controller. Several numerical examples are given to illustrate our theoretic findings in this paper.

Robust Control of Continuous-Time Systems With State-Dependent Uncertainties and Its Application to Electronic Circuits

In this paper, the problems of robust stability and stabilization are investigated for a class of continuous-time uncertain systems. The uncertainties in the model are state-dependent and belong to a polytopic convex set, as can be found in many electronic circuits and some other applications. The global asymptotic stability conditions for such systems are first established by the classic common quadratic Lyapunov function approach. To reduce conservativeness, a particular class of nonquadratic parameter-dependent Lyapunov functions is introduced, by which improved robust stability conditions for the underlying systems are also derived. Based on the stability criteria, a static output feedback controller is then designed for the system. The effectiveness of the proposed approaches is illustrated by a numerical example, and the applicability of our theoretical findings is simultaneously demonstrated by modeling, analysis, and control design for a class of electronic circuits.

Delay-Dependent Stability of Cellular Neural Networks with Multiple Delays

Back propagation (BP) neural network is used to approximate the dynamic character of nonlinear discrete-time system. Considering the unmodeling dynamics of the system, the weights of neural network are updated by using a dead-zone algorithm and a robust adaptive controller based on the BP neural network is proposed. For the situation that jumping change parameters exist, multiple neural networks with multiple weights are built to cover the uncertainty of parameters, and multiple controllers based on these models are set up. At every sample time, a performance index function based on the identification error will be used to choose the optimal model and the corresponding controller. Different kinds of combinations of fixed model and adaptive model will be used for robust multiple models adaptive control (MMAC). The proof of stability and convergence of MMAC are given, and the significant efficacy of the proposed methods is tested by simulation. This paper deals with the problem of delay-dependent stability criterion of delay-difference system with multiple delays of cellular neural networks. Based on quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criteria are found to guarantee delay-dependent asymptotical stability of these systems. And one example illustrates the exactness of the proposed criteria.

Disturbance attenuation and rejection for discrete-time Markovian jump systems with lossy measurements

This study addresses the disturbance attenuation and rejection problem for discrete-time Markovian jump systems with lossy measurements and multiple disturbances. The measurements transmitted from the plant to the observer and the controller are assumed to be imperfect, and two stochastic variables are utilized to model the missing data separately. A composite disturbance observer-based control and H∞ control scheme is proposed for attenuating and rejecting the disturbances. This method is focused on the design of a new structure for the disturbance observer, dynamic output feedback controller, and composite controller, such that the composite system is stochastically stable and it satisfies scheduled performance requirements. The computation-oriented conditions of the disturbance observer gains and controller matrices for the controlled plant are provided based on the piecewise quadratic Lyapunov functional approach. Finally, a numerical example is provided to demonstrate the utility and applicability of the proposed theoretical method.

Delay-Dependent Asymptotical Stabilization Criterion of Recurrent Neural Networks

This paper deals with the problem of delay-dependent stability criterion of discrete-time recurrent neural networks with time-varying delays. Based on quadratic Lyapunov functional approach and free-weighting matrix approach, some linear matrix inequality criteria are found to guarantee delay-dependent asymptotical stability of these systems. And one example illustrates the exactness of the proposed criteria.

Switching PDC control for discrete-time T–S fuzzy system: A membership function ranking approach

In this paper, a new type of switched parallel distributed compensation (PDC) controller is presented, in which a novel switching scheme, namely, membership function ranking (MFR) scheme is proposed. The MFR scheme can be viewed as an improvement and extension of previous results named largest membership function (LMF) scheme concerned with piece-wise quadratic Lyapunov function (PQLF) approach, since it provides a denser subdivision of normalized membership function space and therefore essentially yields less conservative results. Moreover, it shows that MFR scheme covers LMF scheme as a particular case. Based on MFR scheme, a family of ℋ∞ controllers and a switching scheme is designed to ensure the closed-loop system asymptotically stable and has a required ℋ∞ performance. A design example is presented to illustrate the advantages of MFR scheme.

A New Approach For Stability Analysis Of Time‐Dependent Switched Continuous‐Time Linear Systems

AbstractIn this paper, a new approach for stability analysis of time‐dependent switched linear systems is proposed. System equivalence is the main idea in this new approach, which derives a switched discrete linear parameter‐varying system from the switched continuous‐time linear switched system with interval dwell time, and the stability properties of the two corresponding systems are proved to be equivalent. Then, by applying a quadratic Lyapunov function approach for the equivalent switched discrete system, the stability of the switched continuous‐time linear system can be established without checking any average dwell time condition. Finally the computation complexity is analyzed, and mode incidence matrix is introduced to reduce the computation cost.