Robust Repetitive Control Design for a Class of Linear Systems with Time-varying Uncertainties

This paper presents a novel design method for discrete-time repetitive control systems (RCS) based on two-dimensional (2D) discrete-time model. Firstly, the 2D model of an RCS is established by considering both the control action and the learning action in RCS. Then, through constructing a 2D state feedback controller, the design problem of the RCS is converted to the design problem of a 2D system. Then, using 2D system theory and linear matrix inequality (LMI) method, stability criterion is derived for the system without and with uncertainties, respectively. Parameters of the system can be determined by solving the LMI of the stability criterion. Finally, numerical simulations validate the effectiveness of the proposed method.

This paper shows that design of a robustly stable repetitive control system is equivalent to that of a feedback control system for an uncertain linear time-invariant system satisfying the well-known robust performance condition. Once a feedback controller is designed to satisfy the robust performance condition, the feedback controller and the repetitive controller using the performance weighting function robustly stabilizes the repetitive control system. It is also shown that we can obtain a steady-state tracking error described in a simple form without time-delay element if the robust stability condition is satisfied for the repetitive control system. Moreover, using this result, a sufficient condition is provided, which ensures that the least upper bound of the steady-state tracking error generated by the repetitive control system is less than or equal to the least upper bound of the steady-state tracking error only by the feedback system

The problem of designing output error controllers to track periodic reference signals for a class of uncertain linear time-delay systems subject to periodic disturbances is addressed. A repetitive controller, which is a type of output error controller, is developed for these systems and a sufficient condition for stability of the closed-loop system is derived by applying the Lyapunov–Krasovskii functional and linear matrix inequalities (LMIs). In order to relax the stability condition, the theory of the modified repetitive control (RC) system is extended to the systems represented by irrational transfer functions. Based on the extended theory of the modified RC system, a modified repetitive controller, another type of output error controller, is developed for these systems and sufficient conditions for stability of the closed-loop system are derived in terms of a frequency-domain criterion and LMI. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.

The tracking/rejection of periodic signals constitutes a wide field of research in the control theory and applications area and Repetitive Control has proven to be an efficient way to face this topic; however, in some applications the period of the signal to be tracked/rejected changes in time or is uncertain, which causes and important performance degradation in the standard repetitive controller. This thesis presents some contributions to the open topic of repetitive control working under varying frequency conditions. These contributions can be organized as follows: One approach that overcomes the problem of working under time varying frequency conditions is the adaptation of the controller sampling period, nevertheless, the system framework changes from Linear Time Invariant to Linear Time-Varying and the closed-loop stability can be compromised. This work presents two different methodologies aimed at analysing the system stability under these conditions. The first one uses a Linear Matrix Inequality (LMI) gridding approach which provides necessary conditions to accomplish a sufficient condition for the closed-loop Bounded Input Bounded Output stability of the system. The second one applies robust control techniques in order to analyse the stability and yields sufficient stability conditions. Both methodologies yield a frequency variation interval for which the system stability can be assured. Although several approaches exist for the stability analysis of general time-varying sampling period controllers few of them allow an integrated controller design which assures closed-loop stability under such conditions. In this thesis two design methodologies are presented, which assure stability of the repetitive control system working under varying sampling period for a given frequency variation interval: a mu-synthesis technique and a pre-compensation strategy. On a second branch, High Order Repetitive Control (HORC) is mainly used to improve the repetitive control performance robustness under disturbance/reference signals with varying or uncertain frequency. Unlike standard repetitive control, the HORC involves a weighted sum of several signal periods. With a proper selection of the associated weights, this high order function offers a characteristic frequency response in which the high gain peaks located at harmonic frequencies are extended to a wider region around the harmonics. Furthermore, the use of an odd-harmonic internal model will make the system more appropriate for applications where signals have only odd-harmonic components, as in power electronics systems. Thus an Odd-harmonic High Order Repetitive Controller suitable for applications involving odd-harmonic type signals with varying/uncertain frequency is presented. The open loop stability of internal models used in HORC and the one presented here is analysed. Additionally, as a consequence of this analysis, an Anti-Windup (AW) scheme for repetitive control is proposed. This AW proposal is based on the idea of having a small steady state tracking error and fast recovery once the system goes out of saturation. The experimental validation of these proposals has been performed in two different applications: the Roto-magnet plant and the active power filter application. The Roto-magnet plant is an experimental didactic plant used as a tool for analysing and understanding the nature of the periodic disturbances, as well as to study the different control techniques used to tackle this problem. This plant has been adopted as experimental test bench for rotational machines. On the other hand, shunt active power filters have been widely used as a way to overcome power quality problems caused by nonlinear and reactive loads. These power electronics devices are designed with the goal of obtaining a power factor close to 1 and achieving current harmonics and reactive power compensation.

We give a parameterization of all causal stabilizing repetitive learning controllers for single-input and single-output linear minimum phase systems. The past studies on the parameterization of a repetitive learning control system were to obtain the parameterization of all internally stabilizing controllers for the control system such that the plant is connected 1/(1-e/sup -sT/) in series. Since 1/(1-e/sup -sT/) has no zeros in the closed right half plane, also the serial connections of 1/(1-e/sup -sT/) and a minimum phase system have no zeros in the closed right half plane. That is, the parameterization of all stabilizing controllers for a repetitive learning control system may be obtained by using the parameterization for a minimum phase system in Glaria and Goodwin (1994). Using the above mentioned idea, at first, the parameterization of all stabilizing repetitive learning controllers for the biproper minimum phase system is carried out. Next, since the exact repetitive learning controllers to stabilize the strictly stable proper system do not exist, the parameterization of all modified repetitive learning controllers using a low-pass filter in the repetitive learning controller is given.

The structured singular value method is applied to continuous-time SISO repetitive control systems in order to determine their stability and performance robustness in the presence of structured parametric modeling error in the plant. Performance measures for repetitive control systems are introduced and the robust performance analysis using structured singular values is modified, taking the alternating magnitude behaviour of the sensitivity function of a repetitive control system into account. The analysis procedure is simplified considerably in the case of large time delay by noting that the upper and lower envelopes of the structured singular value plots are obtained from corresponding finite dimensional systems without any time delay. The analysis procedures developed are applied to an example on repetitive control of an electrohydraulic material testing machine, available in the literature, to demonstrate their usefulness.

The modified repetitive control system is a type of servomechanism for periodic reference inputs. That is, the modified repetitive control system follows the periodic reference input with a small steady state error, even if a periodic disturbance or uncertainty exists in the plant. Using previously proposed modified repetitive controllers, even if the plant does not include time-delay, transfer functions from the periodic reference input to the output and from the disturbance to the output have infmite numbers of poles. To specify the input-output and the disturbance attenuation characteristic easily, Yamada et al. proposed the concept of a simple repetitive control system such that the controller works as a modified repetitive controller and transfer functions from the periodic reference input to the output and from the disturbance to the output have fmite numbers of poles. In addition, Yamada et al. clarified the parameterization of all stabilizing simple repetitive controllers. Recently, the p...

Using a class of linear static controllers, we stabilize the Petersen open-loop two-dimensional linear system (Ref. 1), which consists of one time-varying uncertainty in the state matrixA and one timevarying uncertainty in the input matrixB. We show that the worst-case uncertainty strategy for the closed-loop system is a piecewise constant strategy of the angular state θ with three switches on the half-turn, −π/2≤θ≤π/2; it is unique with respect to a set of measure zero. Formulas are derived for the worst-case half-turn radius gainr HT as a function of the parameters of the class of stabilizing linear static controllers. Using the class of scalar-quadratic Lyapunov functions, we show that a necessary and sufficient condition for the closed-loop system to be robustly stable against all time-varying admissible uncertainties is thatr HT be less than unity. The bound on the time-varying real parameter uncertainties for the closed-loop system to be robustly stable is derived for the class of linear static feedback controllers. We obtain stabilizing linear static controllers such that the bound is as close to infinity as desired. The derived results are compared with numerical results obtained using commerical robust-control software.

A Design of Repetitive Control Systems – high Accuracy Servomechanisms for Periodic Inputs

An LMI approach to guaranteed cost control of linear uncertain time-delay systems

In view of the surplus torque and complexity of controlled plant in passive electrical load system, a novel approach based on single neuron PID adaptive control and repetitive control for repetitive periodic load control system is proposed. Radial basis function (RBF) neural network is used to identify the system on-line for the single neuron PID controller to adjust its weights and PID parameters by self- learning and self-adapting based on the desired output. The dynamic state performance can be improved by the single neuron adaptive PID control and the steady state performance is also improved by modified repetitive control. Computer simulation results show that the force/position hybrid control system can effectively reduce the surplus torque and improve the loading precision, and also it has fine dynamic and steady state performance and good robustness. The reliability of whole system is further improved.

Repetitive control, a distinctive control method with a human-like learning capability, is of both theoretical and practical importance. This paper deals with the problem of designing a robust observer-based repetitive-control system that provides a given H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> steady-state disturbance-attenuation performance for a class of plants with time-varying structured uncertainties. A continuous-discrete two-dimensional hybrid model is built that accurately describes the features of repetitive control so as to enable independent adjustment of the control and learning actions. A sufficient condition for the repetitive-control system to have a steady-state disturbance-attenuation bound in the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> setting is given in terms of a linear matrix inequality. The condition is then used to obtain the parameters of the repetitive controller and the state observer. Finally, a numerical example demonstrates the effectiveness of the method.

To regulate uncertainties in system dynamics and external disturbances, a robust controller is proposed for a plant with time-varying uncertainty. The design is based on the concept of extended state observer (ESO) with repetitive controller (RC). In real time estimation and mitigation of the total effect of various uncertainties, ESO based RC is particularly effective against a wide range of parameter variations, model uncertainties, and large disturbances. With only two tuning parameters, the controller provides a simple and user friendly solution to complex engineering problems in practice. The comparison results verified the effectiveness of the ESO based RC over state observer (SO) based RC.

Effects of Non-Periodic Disturbances on Repetitive Control Systems

Repetitive control can be effective in improving current quality when used with grid-connected inverters. Mathematically, a repetitive controller is equivalent to a parallel combination of resonant controllers with high gain at the fundamental frequency and its harmonics, and accordingly good disturbance rejection can be achieved at these frequencies. To ensure stability, a low-pass filter needs to be incorporated within the repetitive controller to attenuate the high frequency resonant peaks of the controller gain (above the system's cross-over frequency), without significantly affecting the low frequency resonant peaks corresponding to significant grid harmonics that need to be rejected. The paper therefore argues that it is desirable that the system and the inverter's output filter should have a high bandwidth - higher than the most significant grid harmonics - which may be counter intuitive. The high bandwidth requirement dictates a high PWM switching frequency, which becomes challenging in high power systems as the maximum achievable switching frequency of power electronic devices reduces as their power rating increases. This limitation can be overcome in low voltage systems by using an interleaved inverter topology in which the power is shared between several 2-level half-bridge inverter legs connected in parallel. In addition to using low power devices capable of high switching frequency, the value of the LC output filter capacitance can be relatively very small thanks to the ripple cancellation feature of the interleaved inverter. Both of these features mean that the bandwidth of an interleaved inverter can be much higher than that achievable using a classical 2-level inverter with an LCL output filter. The paper discusses the design and practical implementation of a repetitive controller for an interleaved inverter with 6 half-bridge legs per phase. Simulation and experimental results are also presented to demonstrate the effectiveness of the proposed controller in improving the THD of the output current of the inverter.