Intersample Behavior Research Articles

A lifting approach to learning-based self-triggered control with Gaussian processes

This paper investigates the design of self-triggered control for networked control systems (NCS), where the dynamics of the plant are initially unknown. Given the nature of the self-triggered control where state measurements are sent to the controller a-periodically, our proposal involves augmenting the continuous-time dynamics to a novel dynamical model that incorporates inter-event time as a supplemental input. Then, this new model is studied through Gaussian processes (GP) regression. Additionally, we propose a learning-based approach where a self-triggered controller is formulated by minimizing a cost function for the newly learned model, ensuring consideration of inter-sample behavior. The usage of the lifting approach facilitates the application of gradient-based policy updates, allowing for the effective optimization of both control and communication policies. Finally, we provide a numerical simulation to illustrate the effectiveness of the proposed approach.

Continuous-time approach of sampled-data systems

This paper studies the continuous-time behavior of sampled-data control systems. In these systems, a discrete-time control law is applied to a continuous-time system model using sampling and zero-order hold. Traditionally, sampled-data systems are modeled in discrete-time, and thus, the intersample dynamics cannot be examined. This paper shows that the classical methods (such as the d2c function of MATLAB) for obtaining continuous-time models can be inaccurate between the sampling instants, even when Shannon’s sampling theorem is satisfied. In this paper, a new method is also presented which can be used to characterize the continuous-time dynamics of sampled-data control systems. The new continuous-time system model including the intersample behavior is derived analytically from the discrete-time model, and the results are verified by simulations.

A multi-rate hierarchical fault-tolerant adaptive model predictive control framework: Theory and design for quadrotors

This paper presents the design and stability analysis of a dual-loop hierarchical controller for the trajectory tracking of quadrotors subject to unexpected actuator faults. The hierarchical controller, developed based on a dual-time-scale model decomposition, consists of two loops of control: the outer-loop translation control and the inner-loop rotation control. For translation control, we propose a novel fault-tolerant Lyapunov-based model predictive control strategy with the integration of an adaptive parameter estimator. For rotation control, feedback linearization and adaptive estimation are employed and a fault-tolerant control law is designed. The two loops of the developed hierarchical control system are both implemented in a sample-and-hold scheme with dual sampling rates — the outer-loop is sampled several times slower than the inner-loop. With the inter-sample behavior and the interconnection between the translational and rotational dynamics taken into account, the closed-loop stability of the dual-loop and dual-rate control system is rigorously proven using singular perturbation theory. Sufficient stability conditions are established, based on which control parameters and sampling periods can be tuned jointly such that a trade-off between computation efficiency and control accuracy can be attained. Results of numerical simulations are provided to demonstrate the effectiveness of the proposed control design in trajectory tracking and fault tolerance.

Emerging robust and data‐driven control methods for uncertain learning systems

Emerging robust and data‐driven control methods for uncertain learning systems

Beyond Nyquist in Frequency Response Function Identification: Applied to Slow-Sampled Systems

Fast-sampled models are essential for control design, e.g., to address intersample behavior. The aim of this paper is to develop a non-parametric identification technique for fast-sampled models of systems that have relevant dynamics and actuation above the Nyquist frequency of the sensor, such as vision-in-the-loop systems. The developed method assumes smoothness of the frequency response function, which allows to disentangle aliased components through local models over multiple frequency bands. The method identifies fast-sampled models of slowly-sampled systems accurately in a single identification experiment. Finally, an experimental example demonstrates the effectiveness of the technique.

Identifying Lebesgue-sampled Continuous-time Impulse Response Models: A Kernel-based Approach*

Control applications are increasingly sampled non-equidistantly in time, including in motion control, networked control, resource-aware control, and event-triggered control. Some of these applications use measurement devices that sample equidistantly in the amplitude domain. The aim of this paper is to develop a non-parametric estimator of the impulse response of continuous-time systems based on such sampling strategy, known as Lebesgue-sampling. To this end, kernel methods are developed to formulate an algorithm that adequately takes into account the output intersample behavior, which ultimately leads to more accurate models and more efficient output sampling compared to the standard approach. The efficacy of this method is demonstrated through a mass-spring damper case study.

An EM Algorithm for Lebesgue-sampled State-space Continuous-time System Identification*

This paper concerns the identification of continuous-time systems in state-space form that are subject to Lebesgue sampling. Contrary to equidistant (Riemann) sampling, Lebesgue sampling consists of taking measurements of a continuous-time signal whenever it crosses fixed and regularly partitioned thresholds. The knowledge of the intersample behavior of the output data is exploited in this work to derive an expectation-maximization (EM) algorithm for parameter estimation of the state-space and noise covariance matrices. For this purpose, we use the incremental discrete-time equivalent of the system, which leads to EM iterations of the continuous-time state-space matrices that can be computed by standard filtering and smoothing procedures. The effectiveness of the identification method is tested via Monte Carlo simulations.

State‐tracking iterative learning control in frequency domain design for improved intersample behavior

AbstractIterative learning control (ILC) yields perfect output‐tracking performance at sampling instances for systems that perform repetitive tasks. The aim of this article is to develop a framework for a state‐tracking ILC that mitigates oscillatory intersample behavior, which is often encountered in output tracking ILC. As a framework for the analysis, the stability of the iterative domain including the robustness filter and the asymptotic signal is formulated. In addition, as a framework for the design, the design method using frequency response data to reduce the modeling effort, the learning filter design based on inversion, and the specific design procedure of the robustness filter are presented. The designed method is successfully applied to a motion system and it is shown that the presented state‐tracking ILC provides better intersample behavior than the standard output‐tracking ILC.

Refined instrumental variable methods for unstable continuous-time systems in closed-loop

In continuous-time system identification, refined instrumental variable methods are widely used in open and closed-loop settings. Although their robustness and performance are well documented for stable systems, these estimators are not reliable for estimating unstable continuous-time models. The main difficulty we encounter in modeling unstable systems with refined instrumental variables is that the filtered regressor and instrument vectors, as well as the filtered output, become severely ill-conditioned if the model is unstable during the iterative process. In this work, we propose a solution to this problem by including a tailor-made all-pass filter in the prefiltering step. This approach is used for obtaining an extension of the least-squares state-variable filter method, as well as extensions for the refined instrumental variable method for continuous-time systems (RIVC) and its simplified embodiment (SRIVC), that admit the identification of unstable systems and are shown to minimise the prediction error upon convergence and as the sample size goes to infinity. In addition, several implementations of these methods are proposed depending on the intersample behaviour of the input (zero and first-order hold, multisine and arbitrary). The particular case when the plant has integral action is explicitly considered in this work. In an indirect system identification setting, an extension of the closed-loop version of the SRIVC method is also proposed and discussed in detail. Monte Carlo simulations are used to assess the performance of our methods.

Consistency analysis and bias elimination of the Instrumental-Variable-based State Variable Filter method

In this paper, we analyse the consistency of the Instrumental-Variable-based State Variable Filter (IVSVF) estimator by taking into account the intersample behaviour of the input and output signals. It is found that when only sampled input and output data are available for estimation, the IVSVF estimator is not consistent for any fixed sampling period due to the interpolation error that arises from constructing the filtered output. A Bias-Eliminated IVSVF (BEIVSVF) estimator is then proposed and shown to be consistent. The theoretical results developed in the paper are also discussed from a practical standpoint. Simulations are performed to verify the performance of the proposed method as well as to support the theoretical results.

Non-Causal State Estimation for Improved State Tracking in Iterative Learning Control

State-tracking Iterative Learning Control (ILC) yields perfect state-tracking performance at each n sample instances for systems that perform repetitive tasks, where n stands for the order of the system. By achieving perfect state-tracking, oscillatory intersample behavior often encountered in output-tracking ILC has been mitigated. However, state-tracking ILC only assures the estimated state error to converge to a significantly small value, meaning the accuracy of the state estimation takes a critical role. State estimation using a causal state observer has had an inevitable trade-off between the estimation delay and the noise sensitivity. By utilizing the non-causal operation of ILC, a non-causal state estimation can be designed. This non-causal state estimation performs beyond the trade-off of causal estimation, improving the estimation delay without compromising the noise sensitivity. The aim of this paper is to implement the non-causal state observer to state-tracking ILC, and present the improved state tracking by applying it to a second order system.

A QFT approach to robust dual‐rate control systems

AbstractA dual‐rate control system is a hybrid system composed of continuous‐time and discrete‐time elements with two sampling frequencies. In this work, a new frequency domain analysis and design approach, based on the quantitative feedback theory is developed, to cope with robust stability and robust tracking specifications. Tracking specifications are considered not only in the discrete‐time but also in continuous‐time, that allow a precise description of the intersample behavior (ripples), and characterization of frequencies below and beyond the Nyquist frequencies. Several illustrative examples and a case study has been developed.

Consistent identification of continuous-time systems under multisine input signal excitation

For many years, the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC) has been widely used for identification. The intersample behaviour of the input plays an important role in this method, and it has been shown recently that the SRIVC estimator is not consistent if an incorrect assumption on the intersample behaviour is considered. In this paper, we present an extension of the SRIVC algorithm that is able to deal with continuous-time multisine signals, which cannot be interpolated exactly through hold reconstructions. The proposed estimator is generically consistent for any input reconstructed through zero or first-order-hold devices, and we show that it is generically consistent for continuous-time multisine inputs as well. The statistical performance of the proposed estimator is compared to the standard SRIVC estimator through extensive simulations.

Robust Approximation-Based Event-Triggered MPC for Constrained Sampled-Data Systems

In this paper, an approximation-based event-triggered model predictive control (AETMPC) strategy is proposed to implement event-triggered model predictive control for continuous-time constrained nonlinear systems under the digital platform. In the AETMPC strategy, both of the optimal control problem (OCP) and the triggering conditions are defined in a discrete-time manner based on approximate discrete-time models, while the plant under control is continuous time. In doing so, sensing load is alleviated because the triggering condition does not need to be checked continuously, and the computation of the OCP is simpler since which is calculated in the discrete-time framework. Meanwhile, robust constraints are satisfied in a continuous-time sense by taking inter-sampling behavior into consideration, and a novel constraint tightening approach is presented accordingly. Furthermore, the feasibility of the AETMPC strategy is analyzed and the associated stability of the overall system is established. Finally, this strategy is validated by a numerical example.

Local Sampled-Data Gain-Scheduling Control of quasi-LPV Systems

Abstract This paper deals with the problem of sampled-data gain-scheduling control design for affine quasi-LPV systems. As the control implementation is based on sample-data and its update occurs only at specific sampling instants, the state-dependent scheduling functions of the controller are piecewise continuous. This characteristic causes a mismatch between the system model and the controller parameters, namely asynchronous scheduling functions of the controller with respect to the plant during the inter-sampling behavior. To cope with this phenomenon, a polytopic description of the inter-sampling is constructed based on bounding assumptions on the scheduling functions and their time derivatives. Then, regarding a time-delay approach, a constructive and numerically implementable LMI-based synthesis condition is derived, and a convex procedure is proposed for enlarging the estimate of the region of attraction of the origin of the closed-loop system. The time-derivative bounds of the scheduling functions are explicitly accounted into the local analysis to ensure the method’s implementation. A numerical example is provided to illustrate the proposed methodology.

Intersample Modeling of the Converter Output Admittance

The stability of the converter-grid interconnection can be studied by analyzing the product of the converter output admittance and the grid impedance. For reliable stability analysis, it has been of interest to obtain accurate converter output admittance models for a wide range of frequencies, ideally also around and above the Nyquist frequency of the converter system. This article presents a modeling method for the output admittance of power converters defined in the Laplace domain that takes into account the discrete nature of the control system. The modeling method is based on analyzing the intersample behavior of sampled-data systems, a class of systems that includes the modern digitally controlled power converters. The proposed method is compared to conventional admittance modeling methods, and its accuracy is validated by means of simulations and experiments.

MIMO multirate feedforward controller design with selection of input multiplicities and intersample behavior analysis

Inversion-based feedforward control is a basic method of tracking controls. The aim of this paper is to design MIMO multirate feedforward controller that improves continuous-time tracking performance in MIMO LTI systems considering not only on-sample but also intersample behavior. Several types of MIMO multirate feedforward controllers are designed and evaluated in terms of the 2-norm of the control inputs. The approach is compared with a conventional MIMO single-rate feedforward controller in simulations. The approach improves the intersample behavior through the optimal selection of input multiplicities with MIMO multirate system inversion.

Efficiency analysis of the Simplified Refined Instrumental Variable method for Continuous-time systems

In this paper, we derive the asymptotic Cramér–Rao lower bound for the continuous-time output error model structure and provide an analysis of the statistical efficiency of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC) based on sampled data. It is shown that the asymptotic Cramér–Rao lower bound is independent of the intersample behaviour of the noise-free system output and hence only depends on the intersample behaviour of the system input. We have also shown that, at the converging point of the SRIVC algorithm, the estimates do not depend on the intersample behaviour of the measured output. It is then proven that the SRIVC estimator is asymptotically efficient for the output error model structure under mild conditions. Monte Carlo simulations are performed to verify the asymptotic Cramér–Rao lower bound and the asymptotic covariance of the SRIVC estimates.

Improving Intersample Behavior in Discrete-Time System Inversion: With Application to LTI and LPTV Systems

Discrete-time system inversion for perfect tracking goes at the expense of intersample behavior. The aim of this article is the development of a discrete-time inversion approach that improves continuous-time performance by also addressing the intersample behavior. The approach balances the on-sample and intersample behavior and provides a whole range of new solutions, with stable inversion and multirate inversion as special cases. The approach is successfully applied to a linear periodically time-varying system in both simulations and experiments. The approach improves the intersample behavior through discrete-time system inversion and outperforms existing approaches.

Neutralizing zero dynamics attack on sampled-data systems via generalized holds

Zero dynamics attacks can be lethal to cyber–physical systems because they can be harmful to physical plants and impossible to detect. Fortunately, if the given continuous-time physical system is minimum phase, the attack is not so effective even if it cannot be detected. However, the situation can become unfavorable if one uses digital control by sampling the sensor measurement and using a zero-order hold for actuation because of the ‘sampling zeros.’ When the continuous-time system has a relative degree greater than two and the sampling period is small, the sampled-data system must have unstable zeros (even if the continuous-time system is minimum phase), so that the cyber–physical system becomes vulnerable to ‘sampling zero dynamics attack.’ In this paper, we present an idea to neutralize the zero dynamics attack for single-input and single-output sampled-data systems by shifting the unstable discrete-time zeros into stable ones. This idea is realized by employing the so-called ‘generalized hold’ which replaces a standard zero-order hold. It is shown that, under mild assumptions, a generalized hold exists which places the discrete-time zeros at desired positions. Furthermore, we formulate the design problem as an optimization problem whose performance index is related to the inter-sample behavior of the physical plant, and propose an optimal gain which alleviates the performance degradation caused by generalized hold as much as possible.