Discrete-time Lyapunov Research Articles

Decentralized sampled-data [formula omitted] fuzzy filter for nonlinear large-scale systems

This paper presents a decentralized sampled-data H∞ fuzzy filter design method for nonlinear large-scale systems which are represented by a Takagi–Sugeno (T–S) fuzzy model. Based on the T–S fuzzy model, the error system between the nonlinear large-scale system and the filter is obtained. The discretization process of the error system is accomplished with the exact discrete-time approach to eliminate the exact-approximate mismatch. By using the discrete-time Lyapunov sense, the sufficient condition of the asymptotic stability for the error system is given and a prescribed level of the H∞ norm is ensured to guarantee the H∞ fuzzy filter performance. Finally, numerical examples are given to show the effectiveness of the proposed methods.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Development of a Recurrent Fuzzy CMAC With Adjustable Input Space Quantization and Self-Tuning Learning Rate for Control of a Dual-Axis Piezoelectric Actuated Micromotion Stage

This paper proposes a recurrent fuzzy cerebellar model articulation controller (RFCMAC) for a dual-axis micromotion stage powered by piezoelectric actuators. The proposed RFCMAC employs an adjustable input space quantization method with a learning rate capable of self-tuning. The proposed RFCMAC is aimed at dealing with the adverse effects due to unknown hysteresis nonlinearity, modeling uncertainty, and external disturbance that are commonly found in the piezoelectric actuator systems. The proposed RFCMAC mainly focuses on overcoming the drawbacks of the conventional CMAC control scheme when used to control a piezoelectric actuator system. In particular, the adjustable input space quantization method uses a simple repartitioning decision function to determine an appropriate input space quantization. In addition, the self-tuning law for the learning rate of the proposed RFCMAC is derived based on the discrete-time Lyapunov function so that convergence and faster learning can be guaranteed. Finally, in order to validate the effectiveness of the proposed control methodology, several contour-following experiments were conducted on a dual-axis piezoelectric actuated micromotion stage. Experimental results show that the proposed control scheme is feasible, and its performance is superior to that of the conventional CMAC control scheme.

Stability Analysis of Sampled-Data Systems Using Sum of Squares

This technical brief proposes a new approach to stability analysis of linear systems with sampled-data inputs or channels. The method, based on a variation of the discrete-time Lyapunov approach, provides stability conditions using functional variables subject to convex constraints. These stability conditions can be solved using the sum of squares methodology with little or no conservatism in both the case of synchronous and asynchronous sampling. Numerical examples are included to show convergence.

Roll control using discrete-time robust sliding hyperplanes and fast output sampling

Robustness to unmatched parametric uncertainty is prime requirement of roll control algorithm, especially when it is modelled in discrete time domain and implemented through on-board processor. Sliding mode control is a well established nonlinear control technique, which ensures a robust performance in presence of matched uncertainties and disturbances. In case of the discrete version of sliding mode control, due to finite operational sampling frequency, the system trajectories cannot be forced to slide on the switching manifold. The trajectories remain confined to certain domain around the sliding surface and this is known as Quasi Sliding Mode (QSM) motion. The bound of QSM decides the accuracy and performance of the discrete version of sliding mode. By design, the discrete-time sliding modes are robust to the matched bounded perturbations, however, unmatched perturbations directly affect the boundary layer width and hence the performance of the system. In the present paper, discrete time Lyapunov inequality based sliding hyperplane is designed, which enables robustness to unmatched perturbations arising due to uncertain system matrix A. Further, the requirement of full state-vector for the design of control and sliding surface is met through the multi-rate output feedback (MROF). This control strategy is then demonstrated with application to roll position control of missile with a bandwidth limited actuator.

Observer Design for Uniformly Observable Systems With Sampled Measurements

This work considers the problem of observer design for continuous-time systems with sampled output measurements (continuous-discrete time systems). In classical literature and in many applications, the continuous-discrete time extended Kalman filter (EKF) is used in order to tackle this problem. In this work, using a normal form characterizing the class of nonlinear uniformly observable single output nonlinear systems, it is shown that a particular stationary solution of a continuous discrete time Lyapunov equation can be used in order to design a constant high gain observer. Explicit conditions are given to ensure global convergence of the observer. Finally, an illustration of this result is given using an example of a biological process.

Performance comparisons of intelligent load forecasting structures and its application to energy-saving load regulation

This study mainly focuses on the development of intelligent forecasting structures via a similar time method with historical load change rates for the hourly, daily and monthly load forecasting simultaneously based on the basic frameworks of fuzzy neural network (FNN) and particle swarm optimization (PSO). In the regulative aspect of network parameters, conventional back-propagation (BP) and PSO tuning algorithms are used, and varied learning rates are designed in the sense of discrete-time Lyapunov stability theory. The performance comparisons of different intelligent forecasting structures including neural network (NN) structure with BP tuning algorithm (NN-BP), FNN structure with BP tuning algorithm (FNN-BP), FNN structure with BP tuning algorithm and varied learning rates (FNN-BP-V), FNN structure with PSO tuning algorithm (FNN-PSO) and newly-designed adaptive PSO (APSO) structure are verified by numerical simulations. In order to verify the effectiveness of the superior APSO forecasting structure in practical energy-saving load regulation, the load forecasting during every 15 min is also given, and its result is used to manipulate the scheduled unloading control of a real case in Taiwan campus.

Direct neural control of hypersonic flight vehicles with prediction model in discrete time

In this paper, the direct adaptive neural controller is investigated for the longitudinal dynamics of a generic hypersonic flight vehicle (HFV). The objective of the controller is to make the altitude and velocity to follow a given desired trajectory in the presence of aerodynamic uncertainties. Based on the functional decomposition, the adaptive discrete-time nonlinear controllers are developed using feedback linearization and neural approximation for the two subsystems. Different from the back-stepping design, the altitude subsystem is transformed into the explicit four-step ahead prediction model. With the prediction model, the controller is proposed without virtual controller design. Furthermore, only one direct neural network (NN) is employed for the lumped system uncertainty approximation. The controller is considerably simpler than the ones based on back-stepping scheme and the algorithm needs less NN parameters to be adjusted online. The semiglobal uniform ultimate boundedness (SGUUB) stability is investigated by the discrete-time Lyapunov analysis and the output tracking error is made within a neighborhood of zero. Accordingly, the NN controller is designed for velocity subsystem. The simulation is presented to show the effectiveness of the proposed control approach.

Robust sampled-data fuzzy control of nonlinear systems with parametric uncertainties: Its application to depth control of autonomous underwater vehicles

This paper presents a new direct discrete-time design methodology of a robust sampled-data fuzzy controller for a class of nonlinear system with parametric uncertainties that is exactly represented by Takagi-Sugeno (T-S) fuzzy model. Based on an exact discrete-time fuzzy model in an integral form, sufficient conditions for a robust asymptotic stabilization of the nonlinear system are investigated in the discrete-time Lyapunov sense. It is shown that the resulting sampled-data controller indeed robustly asymptotically stabilizes the nonlinear plant. To illustrate the effectiveness of the proposed methodology, an example, a sampled-data depth control of autonomous underwater vehicles (AUVs) is provided.

Stability Analysis for Sampled-Data Systems with Input Delay

This article proposes a novel approach to assess the stability of linear systems with delayed and sampled-data inputs. It proposes an extension of existing results on the stability of sampled-data systems to the case where a delay is introduced in the control loop. The method is based on a continuous-time modelling of the systems together with the discrete-time Lyapunov theorem, which provides easy tractable sufficient conditions for asymptotic stability. Those conditions cope the problem of stability under asynchronous samplings and time-varying delays. The period and delay-dependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency of the stability criteria.

On balancing a cart-pole system using T–S fuzzy model

This paper presents two easily implementable control schemes for balancing a cart-pole system using intelligent control tools. The proposed schemes use the Takagi–Sugeno (T–S) fuzzy model of a nonlinear system. The concept of network inversion is used to design the controller for such a system. In one of the control schemes, the control input, necessary to achieve a desired output, is computed directly through iterative inversion of the fuzzy model. In the other scheme, a parallel distributed compensator form is chosen for the controller and the parameters or the feedback gains of the controller are updated through network inversion. The updating laws are derived using both continuous and discrete time Lyapunov approaches. The inversion based parameter update avoids the need of any sufficient condition or prerequisite constraint as in existing T–S fuzzy model based control designs like LMI techniques and robust control techniques. The proposed controllers have been implemented on the cart pole system both in simulation and real time. A comparative performance analysis for different control algorithms is presented. The performance of the proposed controller is compared with the well established LQR control in real time. The proposed control algorithms work for a wide range of operating regions compared to LQR and are more robust in the sense that they can tolerate an output disturbance of higher magnitude.

Sampled-data observer-based output-feedback fuzzy stabilization of nonlinear systems: Exact discrete-time design approach

This paper presents a new direct discrete-time design methodology of a sampled-data observer-based output-feedback fuzzy controller for a class of nonlinear system that is exactly modeled in Takagi–Sugeno's form at least locally. A fundamental yet challenging issue in this direction is the unavailability of the exact discrete-time model of the nonlinear plant in a closed form. In contrast to the earlier works that are based on an approximate discrete-time model thus the stability obtained in the design step is not preserved in the implementation step, we employ an exact discrete-time fuzzy model in an integral form. Sufficient asymptotic stabilization conditions are investigated in the discrete-time Lyapunov sense. We then show that the resulting sampled-data controller indeed asymptotically stabilizes the nonlinear plant. An example is provided to illustrate the effectiveness of the proposed methodology.

A novel stability analysis of linear systems under asynchronous samplings

This article proposes a novel approach to assess the stability of continuous linear systems with sampled-data inputs. The method, which is based on the discrete-time Lyapunov theorem, provides easy tractable stability conditions for the continuous-time model. Sufficient conditions for asymptotic and exponential stability are provided dealing with synchronous and asynchronous samplings and uncertain systems. An additional stability analysis is provided for the cases of multiple sampling periods and packet losses. Several examples show the efficiency of the method.

Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators

We obtain continuous-time and discrete-time Lyapunov operator inequalities for the exponential stability of strongly continuous, one-parameter semigroups acting on Banach spaces. Thus we extend the classic result of Datko (1970) [2] from Hilbert spaces to Banach spaces.

A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations

The squared Smith method is adapted to solve large-scale discrete-time Lyapunov matrix equations. The adaptation uses a Krylov subspace to generate the squared Smith iteration in a low-rank form. A restarting mechanism is employed to cope with the increase of memory storage of the Krylov basis. Theoretical aspects of the algorithm are presented. Several numerical illustrations are reported.

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.

SOS for sampled-data systems

AbstractThis article proposes a new approach to stability analysis of linear systems with sampled-data inputs. The method, based on a variation of the discrete-time Lyapunov approach, provides stability conditions using functional variables subject to convex constraints. These stability conditions can be solved using the sum of squares methodology with little or no conservatism in both the case of synchronous and asynchronous sampling. Numerical examples are included to show convergence.

Networked control: taking into account sample period variations and actuators saturation

AbstractThis paper exposes a novel method to cope with the stabilization of networked control systems with asynchronous sampling and actuators saturations. The constructive stabilization criterion is expressed in terms of linear matrices inequalities using the continuous-time model of the systems. However the stability analysis of the closed-loop system is based on the discrete-time Lyapunov Theorem. An example shows that the conservatism of the conditions has been reduced with respect to the literature.

Synthesis and operation of an FFT-decoupled fixed-order reversed-field pinch plasma control system based on identification data

Recent developments and applications of system identification methods for the reversed-field pinch (RFP) machine EXTRAP T2R have yielded plasma response parameters for decoupled dynamics. These data sets are fundamental for a real-time implementable fast Fourier transform (FFT) decoupled discrete-time fixed-order strongly stabilizing synthesis as described in this work. Robustness is assessed over the data set by bootstrap calculation of the sensitivity transfer function worst-case -gain distribution. Output tracking and magnetohydrodynamic mode m = 1 tracking are considered in the same framework simply as two distinct weighted traces of a performance channel output-covariance matrix as derived from the closed-loop discrete-time Lyapunov equation. The behaviour of the resulting multivariable controller is investigated with dedicated T2R experiments.

Discrete-time neuroadaptive control using dynamic state feedback with application to vehicle motion control for intelligent vehicle highway systems

Discrete time neuro-compensated dynamic state feedback control system for lateral and longitudinal control of intelligent vehicle highway systems (IVHS) is developed. A discrete time counterpart of the continuous time non-linear IHVS model is obtained in state-space form and the controller is analysed in three stages, with and without compensation mechanisms resulting in an implementation from low to high complexity. Gain parameters of the dynamic state feedback control are optimised with respect to a minimisation of a linear quadratic cost function. The weight convergence of the neuro-compensation algorithm is established in discrete time Lyapunov sense via a graphical method. The performance enhancement of each design stage of the controller is presented and compared with the aid of computer simulations.