یادگیری تکراری مقاوم برای کنترل حرکت بادقت بالا از طریق بازخورد تطبیقی ​​L1L1

The diversity of precision motion control applications and their demanding design specifications pose a large array of control challenges. Hence, precision motion control design relies on a variety of advanced control strategies developed to cope with specific problems present in control theory. A popular feedforward control technique for repetitive systems is iterative learning control (ILC). While ILC can decrease tracking errors up to several orders of magnitude, the achievable performance is limited by dynamic uncertainty. We propose the combination of L1L1 adaptive control (L1L1 AC) and linear ILC for precision motion control under parametric uncertainties. We rely on the adaptive loop to compensate for parametric uncertainties, and ensure that the plant uncertainty is sufficiently small so that an aggressive learning controller can be designed on the nominal system. We exploit the closed loop stability condition of L1L1 AC to design simple, robust ILC update laws that reduce tracking errors to measurement noise for time varying references and uncertainties. We demonstrate in simulation that the combined control scheme maintains a highly predictable, monotonic system behavior; and achieves near perfect tracking within a few trials regardless of the uncertainty present.

Iterative learning control (ILC) is a feedforward control strategy for systems that execute the same task repeatedly over a finite time horizon [1]. ILC is based on the idea that the tracking performance of such systems can be improved by using information from previous trials. Contrary to other learning type control strategies (e.g. adaptive control, neural networks, repetitive control), ILC modifies the input signal rather than the controller [2]. In a way, ILC is a form of feedback control over the iteration domain. Consistent with this property, iterative learning controllers offer simplicity, robustness and fast convergence to iteration domain equilibria with performance improvements up to several orders of magnitude over conventional control strategies. One of the essential challenges that motivates the field of ILC is dynamic uncertainty. Much as in feedback control, the main approaches for mitigating uncertainty can be roughly classified as robust or adaptive methods. Considerable research has been done on the synthesis of ILC algorithms that are robust to exogenous disturbances, stochastic effects, interval uncertainties, and high frequency modeling uncertainties (see [1] and [3] and references therein). Refs. [4], [5] and [6] provide good examples of H∞H∞ methods for finite and infinite horizon cases; an area in which much work has been done. In [7], the combination of H∞H∞ feedback control with ILC was analyzed, with the premise of bandwidth separated repetitive and nonrepetitive exogenous signals. One particular example that underlines parametric uncertainties from a robustness perspective is [8], in which stability of ILC to interval uncertainties in the impulse response is evaluated. The drawback to these methods is that while ILC convergence is guaranteed within the prescribed set of uncertainties, performance is often limited due to conservative designs. Additionally, the sensitivity of robust learning controllers to variations in the uncertainties is still an open question. Parametric uncertainties have similarly been studied extensively in the adaptive ILC setting with special attention to the application area of robotics, wherein iterative estimation schemes were used to augment the feedback controllers using Lyapunov like methods [9] and [10]. Iterative estimation was also used to reduce the model tracking error and improve transient response in model reference adaptive control (MRAC) [11], [12] and [13]. Other works showed how adaptive feedback control methods can be extended to ILC in a straightforward way [14], and proved universal adaptive ILC laws for single-input single-output (SISO) linear time invariant (LTI) systems with nonzero first Markov parameters [15]. While the adaptive nature of these systems signify high performance and reduced sensitivity to parametric variations, the robustness of adaptive ILC to unmodeled dynamics may be questionable, analogous to adaptive feedback control [16] and [17]. Most of the fundamental limitations and trade-offs of control theory can be observed to a greater extent in precision motion control due to complex, demanding design specifications. Key issues in the control of precision positioning systems include robustness to parameter variations, unmodeled high frequency dynamics, and the bandwidth-precision trade-off [18]. More complex process modeling can mitigate uncertainty issues to an extent, but this becomes unfeasible as complexity increases, specifically due to the fact that certain information about the process, such as external loads and/or parameters that are sensitive to exogenous effects, cannot be known a priori. Although adaptive feedback methods provide a good solution to the problem of robustness to parametric variation and increase precision, this often comes at the expense of reduced robustness to unmodeled dynamics [17] as fast estimation, which is desired from a performance standpoint, leads to high gain feedback. This problem essentially boils down to the fact that conventional adaptive control ignores Bode’s sensitivity integral [19] and [20], also known as the waterbed effect, by compensating for uncertainties throughout the whole frequency spectrum. Similarly, while ILC extends the available bandwidth [20] of the control channel for repetitive systems, thereby alleviating the bandwidth-precision trade-off, the achievable reduction in errors and monotonicity on the iteration axis depends largely on the level of uncertainty in the feedback stabilized plant. To address these issues, this work combines conventional ILC with L1L1 adaptive feedback control, and is an extension of our previous work in [21], [22] and [23]. L1L1 adaptive control (L1L1 AC) is a recent MRAC paradigm that bridges the gap between adaptive and robust control with a priori known, quantifiable transient response and robustness bounds [17]. The idea of combining ILC with L1L1 AC was first introduced in [21], wherein the adaptive loop was utilized to keep the plant sensitivity close to its nominal value for performance improvement through learning. Despite the displayed advantages of L1L1 AC over linear feedback, a trade-off was observed between the closed loop bandwidth and learning performance. More precisely, it was seen that higher closed loop bandwidths resulted in slower convergence and larger converged errors in the iteration domain. To resolve this problem, we proposed the augmentation of the L1L1 AC architecture with an arbitrary feedforward signal to accommodate learning, leading to an adaptation that considers changes in the nominal system behavior due to learning [22]. The resulting L1L1 AC-ILC (L1L1-ILC) scheme had predictable performance in both the time and iteration domains: The feedforward augmented closed loop preserved the a priori known quantifiable transients from L1L1 AC theory, and the learning controller displayed similar convergence behavior regardless of the uncertainty present in the system. It was also seen that increasing feedback bandwidths resulted in decreasing effects of uncertainty in the iteration domain, with faster convergence and lower converged errors. In [23], we presented design guidelines and showed the performance gains of the modified scheme over linear output feedback on a large range nanopositioner via simulation. The main differences between this work and our previous work include: 1. A generalized approach to L1L1-ILC for different classes of linear systems through vector space methods. 2. Extension of the robust monotonic learning convergence results to time varying parametric uncertainties. 3. Design guidelines for the L1L1-ILC scheme that link feedback-learning filter designs to classical control ideas, and show how the L1L1 AC stability condition can be satisfied for a given system. 4. Validation of the performance improvements of the proposed scheme in comparison with an LTI feedback based ILC, through extensive simulations on a precision positioning system subject to time varying parametric uncertainties. Our work differs from the existing literature in several ways: First, as we have mentioned, previous work on adaptive methods in learning have focused on adaptive ILC, wherein adaptive learning laws are considered with or without adaptive feedback. Second, adaptive feedback has not been used in a robust ILC setting before. Third, although the idea of combining ILC with advanced feedback methods to achieve better performance is not new, to the best of our knowledge, the combination of conventional ILC with adaptive feedback has not been employed before. In this paper, we demonstrate how ILC algorithms can be combined with L1L1 AC schemes to achieve robust, high precision motion control. We present feedforward augmented L1L1 AC architectures for state and output feedback cases (see Fig. 2 and Fig. 5)) to accommodate parallel ILC signals and show how this preserves the a priori known L1L1 AC transient bounds. We explain how these bounds, which imply arbitrary close tracking of linear reference models in the time domain, can be exploited for learning purposes in the iteration domain. We then show how the L1L1 AC stability condition relates directly to the robust monotonic convergence conditions of LTI learning laws, and how robust ILC algorithms can be designed in a simple, straightforward manner for different L1L1 AC architectures. The rest of the paper is organized as follows. Section 2 introduces some preliminaries for clarity of exposition. Section 3 gives a brief introduction to L1L1 AC and ILC, and presents our proposed method for the state feedback case. Section 4 extends the results to time varying uncertainties in output feedback. Simulation results are given in Section 5. Section 6 gives concluding remarks and summarizes our findings. For a streamlined presentation, we give certain intermediate results in Appendix A, proofs of our main results in Appendix B and several auxiliary variables in Appendix C.

In this paper we presented a combined L1L1-ILC scheme for robust precision motion control. L1L1 AC was utilized to reduce the effects of parametric variation and increase precision whilst preserving robustness against unmodeled dynamics. This reduction in parametric uncertainty enabled the use of aggressive ILC design to increase system bandwidth and improve tracking performance. The combined controller is robust against parametric uncertainties and unmodeled dynamics, with high tracking performance over a large bandwidth. Simulation results on a precision nanopositioner demonstrate that the well posed feedback controller helps us in extracting high performance from ILC and achieve near perfect tracking even with information, bandwidth and hardware constraints, which is especially important due to the complex requirements for high precision tracking even in the presence of parametric uncertainty.