**1. Introduction**

Rotary systems play important roles in various industry applications, e.g., packaging, print‐ ing, assembly, fabrication, semiconductor, and robotics. A conspicuous characteristic of such systems is the utilization of actuators, e.g., electric motor, to control the angular position, velocity, or acceleration of the system load. Depending on the occasion of application, simple or complicated motion control algorithm may be used. The increasing complexity in architec‐ ture and the high‐performance requirement of recent rotary systems have posed a major challenge on conceiving and synthesizing a desirable control algorithm.

Nonlinearities and uncertainties are common issues when designing a control algorithm for a rotary system. Nonlinearities are either intrinsic properties of the system or actuator and sensor

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dynamics being nonlinear. Uncertainties mainly come from structured/unstructured uncer‐ tainties (also known as parametric uncertainty/unmodeled dynamics) and disturbances. For tackling nonlinearities, conventional techniques, e.g., feedback linearization and backstep‐ ping, are to employ feedback to cancel all or part of the nonlinear terms. On the contrary, design techniques for conducting disturbance rejection or attenuation in control systems may be roughly categorized with respect to whether or not the techniques generate the disturbance by an exosystem. Representative techniques that resort to the exosystem of the disturbance are internal model design [1,2], which originates from the internal model principle [3], and observer‐based design [4,5]. Establishing a suitable mathematical description of the disturb‐ ance is an essential step for internal model design techniques. An internal model design for systems in an extended output feedback form and subject to unknown sinusoidal disturbances was addressed in [1]. For observer‐based design techniques, an observer is usually employed to estimate the states of the unknown exosystem. Chen [5] showed that the design of the observer can be separated from the controller design. For techniques that do not resort to the exosystem of the disturbance, disturbance observer [6,7] or optimization‐based control approaches [8,9] have been shown to work well. In [6], integral phase shift and half‐period integration operator were used together to estimate the periodic disturbances. Another type of disturbance observers was introduced in [7]. The proposed disturbance observer may estimate lumped disturbances that comprise unmodeled dynamics and disturbances. How‐ ever, the performance of the disturbance observer is very sensitive to the adaptation rate of the estimated disturbance components. If the output error of the disturbance observer does not converge sufficiently fast, instability or performance degradation is inevitable.

With measurement of the system states not available, a common technique is to establish a state observer that provides estimates of the states. Unlike state observer for linear systems, no state observer is applicable to general nonlinear systems. Most state observers for nonlinear systems are suited for systems transformable to a specific representation, e.g., normal form [10] or adaptive observer form [11]. One class of observers, known as adaptive state observers, are those having their own update laws adapt the estimated parameters [11,12] or the observer gain [10] to minimize the observer error, i.e., the error between the real states and the estimated states. Marine et al. [11] and Vargas and Hemerly [12] presented a state estimator design for systems subject to bounded disturbances. Bullinger and Allgöwer [10] proposed a high‐gain observer design for nonlinear systems, which adapts the observer gain instead of the estimated system parameters. The uncertainties under consideration are nonlinearities of the system. However, the observer error converges to zero only when persistent excitation exists or the disturbance magnitude goes to zero. Moreover, the update law for the observer might have an unexpected interaction with that of the control law. The other type of state observers, e.g., K‐filters [13,14] and MT‐filters [15,16], does not estimate the system states directly. Specifically, the update law for adapting estimated system parameters (which include both observer and system parameters) is determined from the control law to ensure desired stability and convergence property.

Temporal‐based motion control algorithms of various class have been in progress lately. Adaptive control is suited for systems susceptible to uncertain but constant parameters. Moreover, repetitive and iterative learning control [17–21] is capable of dealing with systems affected by periodic disturbances or in need of tracking periodic commands. Lately, adaptive control has been adopted to adapt the period of the repetitive controller [22,23]. Adaptive and iterative learning control has consolidated and been studied by researchers (see [17] and references therein). The integration immediately gains benefits, such as perfect tracking over finite time, dealing with time‐varying parameters, and nonresetting of initial condition. As indicated by Chen and Chiu [19] and Chen and Yang [24], most temporal‐based control algorithms for rotary systems of variable speed do not explore the characteristics of most uncertainties being spatially periodic. Analyzing and synthesizing such control system in time domain will mistakenly admit those spatially periodic disturbances/parameters as nonperi‐ odic/time‐varying ones. This often results in a design either with complicated time‐varying feature or with degraded performance.

dynamics being nonlinear. Uncertainties mainly come from structured/unstructured uncer‐ tainties (also known as parametric uncertainty/unmodeled dynamics) and disturbances. For tackling nonlinearities, conventional techniques, e.g., feedback linearization and backstep‐ ping, are to employ feedback to cancel all or part of the nonlinear terms. On the contrary, design techniques for conducting disturbance rejection or attenuation in control systems may be roughly categorized with respect to whether or not the techniques generate the disturbance by an exosystem. Representative techniques that resort to the exosystem of the disturbance are internal model design [1,2], which originates from the internal model principle [3], and observer‐based design [4,5]. Establishing a suitable mathematical description of the disturb‐ ance is an essential step for internal model design techniques. An internal model design for systems in an extended output feedback form and subject to unknown sinusoidal disturbances was addressed in [1]. For observer‐based design techniques, an observer is usually employed to estimate the states of the unknown exosystem. Chen [5] showed that the design of the observer can be separated from the controller design. For techniques that do not resort to the exosystem of the disturbance, disturbance observer [6,7] or optimization‐based control approaches [8,9] have been shown to work well. In [6], integral phase shift and half‐period integration operator were used together to estimate the periodic disturbances. Another type of disturbance observers was introduced in [7]. The proposed disturbance observer may estimate lumped disturbances that comprise unmodeled dynamics and disturbances. How‐ ever, the performance of the disturbance observer is very sensitive to the adaptation rate of the estimated disturbance components. If the output error of the disturbance observer does

100 Robust Control - Theoretical Models and Case Studies

not converge sufficiently fast, instability or performance degradation is inevitable.

convergence property.

With measurement of the system states not available, a common technique is to establish a state observer that provides estimates of the states. Unlike state observer for linear systems, no state observer is applicable to general nonlinear systems. Most state observers for nonlinear systems are suited for systems transformable to a specific representation, e.g., normal form [10] or adaptive observer form [11]. One class of observers, known as adaptive state observers, are those having their own update laws adapt the estimated parameters [11,12] or the observer gain [10] to minimize the observer error, i.e., the error between the real states and the estimated states. Marine et al. [11] and Vargas and Hemerly [12] presented a state estimator design for systems subject to bounded disturbances. Bullinger and Allgöwer [10] proposed a high‐gain observer design for nonlinear systems, which adapts the observer gain instead of the estimated system parameters. The uncertainties under consideration are nonlinearities of the system. However, the observer error converges to zero only when persistent excitation exists or the disturbance magnitude goes to zero. Moreover, the update law for the observer might have an unexpected interaction with that of the control law. The other type of state observers, e.g., K‐filters [13,14] and MT‐filters [15,16], does not estimate the system states directly. Specifically, the update law for adapting estimated system parameters (which include both observer and system parameters) is determined from the control law to ensure desired stability and

Temporal‐based motion control algorithms of various class have been in progress lately. Adaptive control is suited for systems susceptible to uncertain but constant parameters. Spatial‐based control algorithms have been studied by researchers recently. The initial step is to reformulate the given system model into the one in spatial domain. Because the reformu‐ lation renders those spatial uncertainties stationary in spatial domain, position‐invariant control design can be performed to achieve the desired performance regardless of the operating speed. A spatial‐based repetitive controller synthesizes its kernel (i.e., *e−Ls* with positive feedback) and operates in accordance with spatial coordinate, e.g., angular displacement. Therefore, its ability for spatially periodic disturbances or references rejection/tracking will not deteriorate as the system operates at variable speed. A regular repetitive controller is composed of repetitive (i.e., a kernel) and nonrepetitive (e.g., a stabilizing controller) parts. With the kernel synthesized with respect to spatial coordinate and given a time‐domain system, designing the nonrepetitive portion that interfaces with the repetitive kernel properly poses a challenge. For spatially periodic disturbance rejection, Nakano et al. [18] reformulated a given linear time‐invariant (LTI) system in an angular position domain. The resulting nonlinear system was linearized around an operating speed. Coprime factorization is then used to synthesize a stabilizing controller with repetitive kernel for the acquired linear model. A more sophisticated design based on linearization and robust control was proposed by Chen et al. [25]. Design approaches for linearized systems are straightforward. However, the overall system might lack the stability of operating at a variable speed or coping with large velocity fluctuation. For tracking of spatially periodic references, Mahawan and Luo [26] have vali‐ dated the idea of operating the repetitive kernel in angular domain and the stabilizing controller in time domain. Doing so does not require reformulation of the open‐loop system. For experimental verification, however, the approach involves solving an optimization problem to synchronize the hardware (time) and software (angular position) interruptions. To further limit the applicability, the mapping between time and angular position has to be known *a priori*. The problem formulation made by Nakano and Mahawan assumed the simplest scenario, i.e., the open‐loop system is LTI without nonlinearity and modeling uncertainty. Chen and Chiu [19] reported that a class of nonlinear models can be reformulated into a quasi‐ linear parameter varying (quasi‐LPV) system. An LPV gain‐scheduling controller was synthesized subsequently to address unmodeled dynamics, actuator saturation, and spatially periodic disturbances. The approach could lead to conservative design if the number of varying parameters rises, the parametric space is nonconvex, or the modeling uncertainties are significant. The restraint and conservatism of modeling uncertainties was relieved by Chen and Yang [24] by formulating a spatial‐based repetitive control system with the adoption of adaptive feedback linearization. However, this method is only applicable to systems with measurement of all states available in real time.

The design of spatial‐based repetitive control has been sophisticated enough to cope with a class of uncertain nonlinear systems. On the contrary, existing spatial‐based iterative learning controls [27,28] are still primitive and aim at only linear systems. It is not apparent whether those methods can be generalized to be applicable for nonlinear and high‐order systems. Knowing that spatial uncertainties in rotary systems may be tackled as periodic disturbances or periodic parameters [29–31], treating the uncertainties as disturbances seem to be more prevalent in literatures.

This book chapter reviews and summarizes the recent progress in the design of spatial‐based robust adaptive repetitive and iterative learning control. In particular, the collection of methods aims at rotary systems that are subject to spatially periodic uncertainties and based on nonlinear control paradigm, e.g., adaptive feedback linearization and adaptive backstep‐ ping. We will elaborate on the design procedure (applicable to generic *n*th‐order systems) of each method and the corresponding stability and convergence theorems. The outline of the chapter is as follows.

Section 2 presents a spatial‐based robust repetitive control design that builds on the design paradigm of feedback linearization. This design basically evolves from the work of Chen and Yang [24]. The proposed design resolves the major shortcoming in their design, i.e., which requires full‐state feedback, by the incorporation of a K‐filter‐type state observer. The system is allowed to operate at varying speed, and the open‐loop nonlinear time‐invariant (NTI) plant model identified for controller design is assumed to have both unknown parameters and unmodeled dynamics. To attain robust stabilization and high‐performance tracking, we propose a two‐degrees‐of‐freedom control configuration. The controller consists of two modules, one aiming at robust stabilization and the other tracking performance. One control module applies adaptive feedback linearization with projected parametric adaptation to stabilize the system and account for parametric uncertainty. Adaptive control plays the role of tuning the estimated parameters, which differs from those methods (e.g., [22,23]), where it was for tuning the period of the repetitive kernel. The other control module comprises a spatial low‐order and attenuated repetitive controller combined with a loop‐shaping filter and is integrated with the adaptively controlled system. The overall system may operate in variable speed and is robust to model uncertainties and capable of rejecting spatially periodic and nonperiodic disturbances. The stability of the design can be proven under bounded disturb‐ ance and uncertainties.

Section 3 presents another spatial‐based robust repetitive control design that resorts to the design paradigm of backstepping. This design basically builds on the work of Yang and Chen [32]. The method has been extended to a category of nonlinear systems (instead of just LTI systems). Furthermore, the main deficiency of requiring full‐state feedback in Yang and Chen's design is resolved by incorporating a K‐filter‐type state observer. To achieve robust stabiliza‐ tion and high‐performance tracking, a two‐module control configuration is constructed. One of the module using adaptive backstepping with projected parametric adaptation to robustly stabilize the system. The other module incorporates a spatial‐based low‐order and attenuated repetitive controller cascaded with a loop‐shaping filter to improve the tracking performance. The overall system incorporating the state observer can be proven to be stable under bounded disturbance and system uncertainties.

Section 4 introduces a spatial‐based iterative learning control design that is suited for a generic class of nonlinear rotary systems with parameters being unknown and spatially periodic. Fundamentally, this design borrows the feature of parametric adaptation in adaptive control and integrates it with iterative learning. Note that the theoretical success of the integration is not immediate because the stability and tracking performance of the overall system is in need of further justification. Control input and periodic parametric tuning law are specified by establishing a sensible Lyapunov‐Krasovskii functional (LKF) and rendering its derivative negative semidefinite. The synthesis of the control input and parametric tuning law and stability/convergence analysis established for this design is distinct from that in [17]. Moreover, unlike a typical adaptive control, the proposed periodic parametric tuning law can cope with unknown parameters of stationary or arbitrarily fast variation.

Section 5 concludes the chapter and points out issues and future research directions relevant to spatial‐based robust adaptive repetitive and iterative learning control.
