Periodic Reference Signals Research Articles

Exponential output tracking via a saturated RLC for a class of nonlinear systems under PSD of periodic references

This paper considers the same class of zero-relative-degree nonlinear systems for which a saturated Repetitive Learning Control (RLC) has been recently shown to ensure exponential – and not just asymptotical – convergence to zero of both the output and the input tracking errors in the case of periodic output reference signals with a known single period. The explicit role of the nonlinear unstructured uncertainties is here investigated within the more general scenario in which the output reference signal is multi-periodic. Special emphasis is provided to the generation of the input reference whose effect has to be exponentially nullified by the RLC. In particular, the following question is answered. Restrict the design to the sub-class of periodic output reference signals that admit a Periodic Signal Decomposition (PSD) (namely, the ones that can be written as a finite sum of periodic functions) with pairwise-commensurable periods (so that the knowledge of the multiple periods characterizing the input reference is preserved once the common multiple among the single periods is additionally included). Then, which robustness and convergence properties can be still achieved by the same output-feedback (definitely saturated) RLC, once it is intuitively generalized to transiently include an additional bank of learning estimation schemes corresponding to the single periods? Interestingly, the theoretical tools of this paper can be also used to successfully address identifiability and convergence issues regarding the identification of periodic components of general multi-periodic signals.

Periodic output regulation for 1-D wave equation subject to non-collocated control

In this paper, we focus on the periodic output regulation for a one-dimensional wave equation where the performance output is non-collocated with the control input. The exosystems considered here are multi-periodic time-varying, capable of generating periodic reference signals and disturbances within the domain and at the boundaries, each with distinct periods. Initially, a backstepping-based state feedback regulator is designed by tackling an ill-posed periodic regulator equation. Subsequently, leveraging an external matrix, we design a novel time-varying observer tailored for the transformed wave equation and the exosystems. An output feedback regulator is then obtained by replacing the states with their estimations. The tracking error is shown to decay exponentially and the closed-loop system is proved to be bounded. Some numerical simulations are presented to validate the results.

Quantifying the performance enhancement facilitated by fractional-order implementation of classical control strategies for nanopositioning

For most nanopositioning systems, maximizing positioning bandwidth to accurately track periodic and aperiodic reference signals is the primary performance goal. Closed-loop control schemes are employed to overcome the inherent performance limitations such as mechanical resonance, hysteresis and creep. Most reported control schemes are integer-order and combine both damping and tracking actions. In this work, fractional-order controllers from the positive position feedback family namely: the Fractional-Order Integral Resonant Control (FOIRC), the Fractional-Order Positive Position Feedback (FOPPF) controller, the Fractional-Order Positive Velocity and Position Feedback (FOPVPF) controller and the Fractional-Order Positive, Acceleration, Velocity and Position Feedback (FOPAVPF) controller are designed and analysed. Compared with their classical integer-order implementation, the fractional-order damping and tracking controllers furnish additional design (tuning) parameters, facilitating superior closed-loop bandwidth and tracking accuracy. Detailed simulated experiments are performed on recorded frequency-response data to validate the efficacy, stability and robustness of the proposed control schemes. The results show that the fractional-order versions deliver the best overall performance.

Disturbance attenuation for neutral Markovian jump systems with multiple delays

The work explores the study of a neutral Markovian jump system that contains multiple delays, external disturbances, and unknown non-linear uncertainties. To overcome the tracking problem, a modified repetitive controller infused with the higher equivalent-input-disturbance approach and the extended Smith predictor approach is suggested. To guarantee the mean-square asymptotic stability of the system, an appropriate mode-dependent loop-based Lyapunov–Krasovskii functional is used. The proposed controller explicitly forces the system output and accurately follows any supplied periodic reference signal. Subsequently, the efficiency of the recommended technique is demonstrated with the simulations that validate the outcomes by comparing them to the existing results.

Adaptive tracking control under quantized observations and observation uncertainty with unbounded variance

AbstractIn this paper, we present a matrix‐type adaptive tracking control scheme for a stochastic regression system with multi‐threshold quantized observations and observation uncertainty. This observation uncertainty is described by an additive random noise, whose variance could be a polynomial‐type rate of increase and tends to infinity. Our method, first, to handle simultaneously parameter uncertainty, quantized observation, and observation uncertainty, incorporates an online stochastic approximation‐type parameter identification algorithm. Second, based on the above identification algorithm, we construct a matrix‐type adaptive tracking control law by the certainty equivalence principle. In addition, under suitable conditions, the above identification algorithm and matrix‐type adaptive tracking control law in the mean square sense can ensure that the estimations of the unknown parameters converge to the true values and achieve asymptotically optimal tracking of the periodic reference signal, respectively. To characterize the effect of observation uncertainty on the convergence speed of the identification algorithm in the mean square sense, a quantitative relationship between observation uncertainty and the convergence speed is proposed. Finally, the effectiveness of the proposed matrix‐type adaptive tracking control scheme is verified through a simulation.

Zero-Phase FIR Filter Design Algorithm for Repetitive Controllers

Repetitive controllers (RCs) are linear control structures based on the internal model principle. This control strategy is known for its ability to control periodic reference signals, even if these signals have many harmonic components. Despite being a solution that results in a good performance, several parameters of the repetitive controller need to be correctly tuned to guarantee its stability. Among these parameters, one that has high impact on the system performance and stability is the finite impulse response (FIR) filter, which is usually used to increase the stability domain of RC-based controllers. In this context, this paper presents a complete tutorial for designing the zero-phase FIR filter, which is often used to stabilize control systems that use RC-based controllers. In addition, this paper presents a Matlab® application developed for performing the stability analysis of RC systems and designing its FIR filter. Simulation and experimental results of a shunt active power filter are used to validate the algorithm and the Matlab® application.

Disturbance Observer-Based Repetitive Control System With Nonminimal State-Space Realization and Experimental Evaluation

This brief develops a disturbance observer-based repetitive control system using a non-minimal state-space (NMSS) realization where all state variables correspond to the system’s input and output variables and past values. Tracking a periodic reference signal or rejection of a periodic disturbance signal is achieved by including a disturbance observer to estimate an input disturbance containing the same frequency characteristics. This new approach differs from previously published designs because it separates the design procedure into two tasks: first, stabilization via state feedback control; second, independent incorporation of the periodic modes via estimation of the disturbance. Moreover, the new design naturally contains an anti-windup mechanism when the control signal reaches its maximum or minimum value. Results from the experimental evaluation are given, including a comparison against a design that constructs a minimal state controller using an observer. These results demonstrate that the new method can deliver significant performance improvement, with excellent disturbance rejection and reference tracking.

Ultraprecise Controller for Piezoelectric Actuators Based on Deep Learning and Model Predictive Control.

Piezoelectric actuators (PEA) are high-precision devices used in applications requiring micrometric displacements. However, PEAs present non-linearity phenomena that introduce drawbacks at high precision applications. One of these phenomena is hysteresis, which considerably reduces their performance. The introduction of appropriate control strategies may improve the accuracy of the PEAs. This paper presents a high precision control scheme to be used at PEAs based on the model-based predictive control (MPC) scheme. In this work, the model used to feed the MPC controller has been achieved by means of artificial neural networks (ANN). This approach simplifies the obtaining of the model, since the achievement of a precise mathematical model that reproduces the dynamics of the PEA is a complex task. The presented approach has been embedded over the dSPACE control platform and has been tested over a commercial PEA, supplied by Thorlabs, conducting experiments to demonstrate improvements of the MPC. In addition, the results of the MPC controller have been compared with a proportional-integral-derivative (PID) controller. The experimental results show that the MPC control strategy achieves higher accuracy at high precision PEA applications such as tracking periodic reference signals and sudden reference change.

Adaptive learning control for nonlinear systems: A single learning estimation scheme is enough

In this brief, continuous-time nonlinear systems with extended matching uncertainties are considered. The problem of designing a state-feedback adaptive learning control of reduced complexity — just including a single adaptive learning estimation scheme in the upper subsystem and a high-gain proportional action in the input channel — is addressed. By properly setting the control parameters, exponential output tracking of (sufficiently smooth) periodic reference signals with a known period is achieved. Fourier series expansions are used and estimates of the resulting Fourier coefficients are continuously adapted based on the persistency of excitation conditions that naturally hold due to the orthogonal nature of the sinusoidal basis functions.

PI-generalizing saturated repetitive learning control for a class of nonlinear uncertain systems: Robustness w.r.t. relative degree zero or one

The class of single-input, single-output, nonlinear, time-invariant systems with unknown output-dependent nonlinearities, unknown parameters and relative degree ρ ∈ { 0 , 1 } is considered. A priori knowledge of the system includes globally Lipschitz nature of the nonlinearities and positive sign of the high-frequency gain, besides the minimum-phase property. Single-input, single-output, observable, minimum phase, linear, time-invariant systems are included as a very special case. The aim is to track, via the output error feedback only, periodic output reference signals with known period. Here we originally show that the same saturated repetitive learning control that constitutes the most natural generalization of the PI (Proportional–Integral) control, concurrently ensures the following properties for any initial condition of the system: asymptotic convergence to zero of the output tracking error is guaranteed if ρ = 1 ; exponential convergence to zero is achieved for both the output and the input tracking errors if ρ = 0 .

Preview repetitive control for polytopic nonlinear discrete-time systems.

In this study, a preview repetitive control (PRC) strategy was developed for uncertain nonlinear discrete-time systems subjected to a previewable periodic reference signal. The proposed preview repetitive controller was designed such that the system output tracked a previewable periodic reference signal even with model uncertainties and nonlinear terms. An augmented two-dimensional (2D) model was constructed based on the 2D model approach and state augmented technique. Second, considering the state unmeasured and periodic tracking reference signal, a static output PRC law was designed using the linear matrix inequality (LMI) techniques. Finally, the effectiveness of the proposed controller was verified through two illustrative examples.

Periodic tracking for piezo‐actuated displacement stage based on disturbance rejection proportional integral derivative and repetitive control

AbstractPiezo‐actuated displacement stage is often used to perform periodic scanning tasks, but the conventional control schemes may not satisfactorily track periodic reference signals. Our solution is a disturbance rejection proportional‐integral‐derivative (DR‐PID)‐based repetitive control (RC) scheme in which the hysteresis nonlinearity with the same period of reference input can be well suppressed by RC law. Stability criterions for the closed‐loop system are developed based on small‐gain theorem. A simple and effective design algorithm is obtained based on the stability conditions, such that DR‐PID and RC can be designed separately. Finally, the validation and superiority of the proposed method are demonstrated by simulation studies.

Output tracking for a class of non-minimum phase nonlinear systems: A two-point boundary value problem formulation with a hybrid regulator

The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem.

Observer‐based preview repetitive control for uncertain discrete‐time systems

AbstractIn this paper, the preview repetitive control (PRC) problem is considered for uncertain discrete‐time systems subject to a previewable periodic reference signal. First, an augmented system including future information on the previewable reference signal is constructed. Second, considering the state unmeasured and periodic tracking reference signal, state observer and preview repetitive controller (PRCr) are designed in both state feedback and output feedback settings. To achieve signal tracking performance, a two‐dimensional (2D) model approach and some LMI technique are presented to describe the control and learning actions of the PRC system by using the lifting technique. Then, by Lyapunov–Krasovskii functions, sufficient conditions for asymptotic stability of the 2D system are derived in the form of a set of LMIs, and meanwhile, the design of a repetitive controller with preview actions is proposed through the LMI approach. Finally, two examples illustrate the effectiveness of the results.

Disturbance observer-based predictive repetitive control with constraints

ABSTRACT This paper develops an observer-based predictive repetitive control system to track periodic reference signals or reject disturbances with bandlimited frequency content. The new design complements existing approaches to predictive control, where a model of the periodic disturbance is embedded in the controller. In particular, the new design, based on a novel combination of repetitive control and a disturbance observer, results in a significant improvement in design transparency and implementation simplicity. Although the design is undertaken using a state-space-model, frequency response analysis based on the sensitivity and complementary functions is used to demonstrate the characteristics of the repetitive control system for disturbance rejection, reference following and measurement noise attenuation. Moreover, operational constraints can be included in the design for applications where this feature is required. Simulation studies are given to highlight the closed-loop performance achievable in the presence of constraints. Experimental validation results from application to a two-joint robotic arm are also given and discussed.

Complex $nk + m$ Repetitive Controller Applied to Space Vectors: Advantages and Stability Analysis

Repetitive controllers represent an attractive solution for systems having periodic reference or disturbance signals with high harmonic content. In order to obtain some performance improvements for three-phase plants, as grid-connected converters, complex repetitive controllers were proposed later in literature. In this article, the advantages of complex repetitive control systems over the real ones are analyzed. In addition, sufficient conditions for the input-output stability of complex repetitive control systems and modified complex repetitive control systems are obtained from the small gain theorem. These stability conditions are then used to derive the stability domains for complex repetitive control systems. Further, a relative stability analysis based on the sensitivity function is introduced. An experimental application based on an active power filter is used to exemplify the effectiveness of the proposed stability analysis.

Discrete Output Regulator Design for the Linearized Saint–Venant–Exner Model

This manuscript addresses the regulator design in the discrete-time setting for the unstable linearized Saint–Venant–Exner model, which describes the dynamics of a sediment-filled water canal. The proposed regulator ensures the closed-loop stability and proper tracking of polynomial and periodic reference signals using output feedback in a sample-data setting. To design this regulator, the system discrete representation is achieved by the application of the structure-preserving Cayley-Tustin time discretization and the direct relation with the regulator in the continuous-time setting is shown. The regulator design in the continuous-time setting is developed using the backstepping methodology ensuring the closed-loop stability and the observer design, while the Sylvester equations are solved to achieve proper tracking. Finally, the numerical simulation results are presented to show the performance of the regulator.

Adaptive parameter identification based nadir-pointing control of spacecraft with misaligned rotational component

The kinematics and dynamics of a nadir-pointing spacecraft with rotational component are derived in the presence of both the parallel and angular misalignments. Based on the derived model, the nadir-pointing control problem is formulated and an adaptive attitude control law with parameter identification is developed. In the proposed control law, periodic angular reference signal is exerted firstly to tune the adaptive parameters, and the misalignments of the rotational component are identified simultaneously. Using the identified parameters, high precision nadir-pointing control is achieved. The stability and effectiveness of the proposed method are verified through Lyapunov method and Monte Carlo analysis.

Modified Repetitive Control Design for Nonlinear Systems With Time Delay Based on T–S Fuzzy Model

In this paper, based on the Takagi-Sugeno fuzzy model approach, a novel output feedback control strategy is developed for a class of continuous-time nonlinear time delay systems via modified repetitive control scheme. The main objective of this paper is to design a fuzzy rule-based modified repetitive controller such that the system output can track any given periodic reference signal even in the presence of periodic disturbances. For this purpose, by employing fuzzy-dependent Lyapunov-Krasovskii functional together with the free-weighting matrix technique, a new set of sufficient conditions is presented in terms of linear matrix inequalities to guarantee the robust stability of the resulting closed-loop system. Based on these conditions and using internal model principle, it is shown that the system output tracks the given reference input within the desired error. Finally, two application-oriented examples are presented to validate the obtained theoretical results in which the proposed control design scheme gives better performance over the existing control scheme.

The Tracking Problem in Tank Gun Control Systems With Periodic Reference Signals

In this paper, the position tracking control for tank gun control systems with periodic reference signal is studied. On the basis of corresponding system modeling, a novel repetitive controller is developed by using Lyapunov synthesis. During the controller design, signal replacement mechanism is used to deal with the nonparametric uncertainties under Lipschitz-like continuous condition, and repetitive learning laws are developed to estimate the unknown periodic parameters. Meanwhile, robust learning approach is used to compensate the sum of random disturbances, whose upper bound is estimated according repetitive learning mechanism. Hyperbolic tangent function, rather than sign function, is applied to design a robust feedback term to release the occurrence of chattering phenomenon. Numerical simulations demonstrate the effectiveness of the proposed repetitive control scheme.