Design of H 2 Controllers for Sampled-Data Systems with Input Time Delays

Using the bilinear and tuning bilinear approximation together with the interval arithmetic operation, the discrete-time uncertain model and robust digital redesign of sampled-data uncertain system with input time delay are presented. The system matrices characterizing the stale-space representation of the original uncertain system are assumed to be interval matrices. The developed discrete-time interval model tightly encloses the exactly discretized continuous-time input time-delay uncertain system, and it can be utilized for digital simulation. Based on bilinear approximation, a tuning bilinear approximation is presented to find the digitally redesign controller of the digitally controlled sampled-data uncertain system with input time delay. The digitally redesigned predictor controller is able to closely match the states of the digitally redesigned uncertain sampled-data system with those of the original continuous-time predictor controlled uncertain system.

The optimal tracking design and its corresponding digital redesign for sampled-data systems with input time delay under state and control-input constraints have been proposed in this paper. The multi-objective evolutionary programming (MOEP) is utilized for achieving the desired goals, and the inter-sampling behavior of the designed sampled-data system via the suboptimal digital redesign approach is also considered. To satisfy the pre-specified state and control-input constraints, the MOEP is employed for dynamically tuning the LQ-tracker and the sampling period.

In this paper a robust dynamic controller design algorithm for nonlinear systems with unmatched uncertainties and an input time delay is provided. For many practical systems, the existence of time delay is common in control practice. Therefore the problem of controlling such systems becomes more complex and sometimes degrades the control performance. We will propose a feasible method to deal with the problem of unmatched uncertainties and an input time delay. Based on the Lyapunov theory and the concept of functional analysis, the conditions of the robust stability of the overall closed-loop system with unmatched uncertainties and an input time delay will be derived. Finally, some examples and simulation results are provided to illustrate the versatility and performance of the proposed method.

Designing active flutter suppression for high-dimensional aeroelastic systems involving a control delay

In this article, input-to-state stability (ISS) and stabilization are examined for sampled-data systems under deterministic aperiodic sampling and random sampling, respectively. Using the direct design method, the sampled-data systems are transformed into switched systems with switched time-varying delays. First, the ISS definition and criterion appropriate for these systems are provided. Based on this, the ISS criterion on sampled-data systems under deterministic aperiodic sampling is given. Second, after the stochastic ISS (SISS) definition and criterion are provided for switched nonlinear systems with randomly switching delays, the SISS criterion for the sampled-data systems under random sampling is provided. All of the ISS and SISS definitions are given in the form of $\mathcal {KL}$ function that is quite elegant and easy to work with. Then, sufficient conditions for input-to-state stabilization are obtained for sampled-data linear systems under deterministic aperiodic sampling and random sampling, respectively, via the Lyapunov-Krasovskii method. Finally, based on the criteria, a piecewise controller is designed by the matrix inequality approach for a sampled-data linear time invariant system, and simulation results are provided to illustrate our design method. The main conclusion of this article is that sampling intervals will affect the controller design of the systems, and the ISS properties are maintained using a piecewise controller.

This paper investigates the stability analysis and state feedback controller design of linear time-invariant (LTI) systems with input time delay. By investigating the real part and the imaginary part of the transcendental equation, the characteristic equation can be transformed into an equation with two vectors related to the system parameters and the feedback gain. The characteristic roots can be determined exactly and concisely that the feedback gain can be designed directly under a given time delay condition. Based on this approach and the pole placement method, preservation of the closed-loop control performance, the state feedback gain and the maximum allowable value of input time delay can be derived by solving an optimization equation. Numerical examples are provided to illustrate the proposed algorithm.

Nonlinear Sampled-data Systems—A lifting framework

In this paper, a reinforcement learning-based adaptive control algorithm is proposed to solve the tracking problem of a discrete-time (DT) nonlinear state and input time delayed system of the wheeled mobile robot (WMR). With the typical model of the WMR transformed into an affine nonlinear DT system, a delay matrix function and appropriate Lyapunov-Krasovskii functionals are introduced to overcome the problems caused by the state and input time delays, respectively. Furthermore, with the approximation of the radial basis function neural networks (NNs), the adaptive controller, the critic NN, and action NN adaptive laws are defined to guarantee the uniform ultimate boundedness of all signals in the WMR system, and the tracking errors convergence to a small compact set to zero. Two examples of simulation are given to illustrate the effectiveness of the proposed algorithm.

The contributions of this paper are threefold. We first explore the differences in uncertainty representations and control designs between continuous time systems and discrete time systems, particularly the high frequency interpretation does not work for discrete time systems. It is shown that uncertainties appeared in sampled data systems may be highly structured, however they do not fall into any category which is currently available in the literature. Secondly, we define a measure on these uncertainties which confines a neighborhood to the nominal model in terms of frequency content, and then we propose a control design method which guarantees the stability of the closed loop system in the face of uncertainties. Last, we consider a generalization of a classical continuous time approach in representing the multiplicative uncertainty to sampled data systems by using a slightly modified sampler and present a method to calculate a frequency bound on the uncertainty in sampled data systems.

In this work we are interested in the stability and ℒ2-gain of hybrid systems with linear flow dynamics, periodic time-triggered jumps and nonlinear possibly set-valued jump maps. This class of hybrid systems includes various interesting applications such as periodic event-triggered control. In this paper we also show that sampled-data systems with arbitrarily switching controllers can be captured in this framework by requiring the jump map to be set-valued. We provide novel conditions for the internal stability and ℒ2-gain analysis of these systems adopting a lifting-based approach. In particular, we establish that the internal stability and contractivity in terms of an ℒ2-gain smaller than 1 are equivalent to the internal stability and contractivity of a particular discrete-time set-valued nonlinear system. Despite earlier works in this direction, these novel characterisations are the first necessary and sufficient conditions for the stability and the contractivity of this class of hybrid systems. The results are illustrated through multiple new examples.

On the Discrete Adaptive Posicast Controller

Because the inter-sample behavior can be considered, the lifting technique now becomes the unique method for design and analysis of sampled-data systems. However, when we consider the robust stability problem by using the small gain theorem, the state transition operator is zero, and the current lifting algorithms can not be used directly. The lifting computation formulas for the case of the robust stability analysis are derived in detail. Further study shows that when the lifting technique is used for robust stability analysis, its result is conservative. A new discretization method for robust stability analysis is proposed in the paper, and it is not conservative when the condition of the small gain theorem is used. Two examples are given here to illustrate the above ideas.

This article studies the problem of global stabilization of discrete-time linear systems subject to input saturation and time delay. The considered time-delay systems are first transformed into delay-free systems based on prediction technique. Then, by utilizing saturation functions technique, the corresponding global stabilizing controllers are proposed for two special discrete-time linear systems-a chain of integrators and oscillators, and explicit conditions guaranteeing stability are also given. Both current and delayed feedback information are utilized in the controller design, and some free parameters are also introduced into these controllers. These advantages can help improve the control performance significantly. Subsequently, a systematic control design procedure for globally stabilizing general discrete-time linear systems subject to multiple inputs and/or multiple inputs delays is proposed. The design procedure is in an explicit and recursive way with explicit conditions guaranteeing stability being given, and thus is easier to use than the existing one. Finally, the effectiveness of the proposed approaches are illustrated by three numerical examples.

Basing on lifting technology, this paper researches a design method for multirate sampled-data control system in the background of flight control system design, which consummates the theory of multirate sampled-date control system design for further steps, and provides a method to design multirate sampled-data controller. As the reason of uncertainty in the aircraft model and the need of different sampling frequencies in different control loops, H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is considered. First, multirate sampled-data control system is outlined, and then the signal description are presented. Because of multirate sampled-data system is time-variant, the direct digital controller design becomes difficult. The equivalent discrete time-invariant systems can be converted from the linear time-variant systems through lifting technology. Then the H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> discretization of multirate sampled-data systems is researched. After the sampled-data control system was lifted, the equivalent L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> [0, h) -induced norm is proven. Base on the above, the method of H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> -optimal multirate sampled-data control is summarized. A height control is simulated using a simplified aircraft model to illustrate the design. The simulation shows that the multirate Sampled-data control system can achieve the satisfactory objective accurately and optimally.

Differential Linear Matrix Inequalities in Optimal and Robust Sampled-Data Filtering