Abstract
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 backstepping. We will elaborate on the design procedure (applicable to generic nth‐order systems) of each method and the corresponding stability and convergence theorems.
Keywords
- rotary system
- disturbance rejection
- robust adaptive control
- repetitive control
- iterative learning control
1. Introduction
Rotary systems play important roles in various industry applications, e.g., packaging, printing, 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 architecture 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 dynamics being nonlinear. Uncertainties mainly come from structured/unstructured uncertainties (also known as parametric uncertainty/unmodeled dynamics) and disturbances. For tackling nonlinearities, conventional techniques, e.g., feedback linearization and backstepping, 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 disturbance 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. However, 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 nonperiodic/time‐varying ones. This often results in a design either with complicated time‐varying feature or with degraded performance.
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 reformulation 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.,
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 backstepping. We will elaborate on the design procedure (applicable to generic
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 disturbance 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 stabilization 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.
2. Spatial‐based output feedback linearization robust adaptive repetitive control (OFLRARC)
Consider the state‐variable model of an
where
(1)
Here, band‐limited disturbances are signals with Fourier transform or power spectral density being zero above a certain finite frequency. The number of distinctive spatial frequencies and the spectrum distribution are the only available information of the disturbances.
(2)
(3)
Consider an alternate variable
will ensure that
where we denote
(1) can be rewritten as
Equation (3) is an nonlinear position‐invariant (NPI; as opposed to the definition of time‐invariant) system with the
This definition will be useful for describing the linear portion of the overall control system.
Drop the
where terms involving unstructured uncertainty are merged into
The state variables have been specified such that the angular velocity
It can be verified that (4) has the same relative degree in
where
where
(1)
(2) (4) is exponentially minimum phase, i.e., the zero dynamics is exponentially stable;
(3) The output disturbance is sufficiently smooth [i.e.,
(4)
(5) The reference command
With Assumption 2, the design of a nonlinear state observer may focus on the external dynamics of (5), i.e.,
2.1 State observer design
In this section, we show how to establish a state observer for the transformed NPI system (5). Because
where
where
Equation (8) can be further written in the form
where
By properly choosing
where
Define the state estimated error as
Here, we further assume that
(9)
Equation (10) or (11) cannot be readily implemented due to the unknown parametric vector
It can be easily verified that (13) is equivalent to (11). Hence, (13) may replace the role of (11) for providing the state estimate. With
where
where
2.2 Output feedback robust adaptive repetitive control system
In this section, we show how to incorporate the state observer established in the previous section into an output feedback adaptive repetitive control system. The control configuration consists of two layers. The first layer is the adaptive feedback linearization, which tackles system nonlinearity and parametric uncertainty. The second layer is a repetitive control module of a repetitive controller and a loop‐shaping filter. This layer not only enhances the ability of the overall system for rejection of disturbance, sensitivity reduction to model uncertainty, and state estimated error but also improves the robustness of the parametric adaptation. Although inclusion of the state observer relieves the design of the need of full‐state feedback, it actually introduces extra dynamics into the system. Hence, the stability of the resulting system needs to be further justified.
Suppose that (4) has relative degree
Substituting the
To put the previously developed state observer into use, we substitute the first equation of (15) into (17) and arrive at
Define the estimated parametric vector of
The control law using the estimated system parameters and states is
where we introduce two designable inputs,
where
where
Neglecting the details of
where
Define
Equation (24) becomes
where
Because
Substituting (26) into (27), we obtain
where
where
In the following, we present stability theorem for the proposed spatial‐based OFLRARC system. The theorem extends the results in the literature [33,34] to take into account the addition of the repetitive control module. It will be seen that the overall OFLRARC system will stay stable and the tracking error will be bounded as long as a stable and proper loop‐shaping filter stabilizes a certain feedback system.
Proof: Follow the same steps for proof of Theorem 3.1 in [24].
Proof: Follow the same steps for proof of Theorem 3.2 in [24] with some differences.
3.Spatial‐based output feedback backstepping robust adaptive repetitive control (OFBRARC)
Consider the same NPI model (3), which is transformed from the NTI model (1), under the same set of assumptions (Assumptions 2.1 and 2.2). The NPI model will be used for the subsequent design and discussion.
3.1 Nonlinear state observer
Drop the
where terms involving unstructured uncertainty are merged into
In addition, we have
The state variables have been specified such that the angular velocity
Because
By properly choosing
such that
where
3.2 Spatial domain output feedback adaptive control system
To apply adaptive backstepping method, we first rewrite the derivative of output
With the second equation in (32), (33) can be written as
where
In view of designing output feedback backstepping with K‐filters, we need to find a set of K‐filter parameters, i.e.,
To apply adaptive backstepping to (34), a new set of coordinates will be introduced
where
Consider a Lyapunov function
Define the estimates of
where
where
Consider a Lyapunov function
The derivative of
where
With respect to the new set of coordinates (35), the third equation of (34) can be written as
The overall Lyapunov function may now be chosen as
where
Specify the control input as
where
Substituting (41) into
From (42), we may specify the parameter update law to cancel the term
where
The tracking error
where we have chosen
where
Consider the control law of (41) and (45) employed to a nonlinear system with unmodeled dynamics, parametric uncertainty, and output disturbance given by (30). Suppose that
Proof: Refer to [36].
4. Spatial‐based adaptive iterative learning control of nonlinear rotary systems with spatially periodic parametric variation
Consider an NTI system described by
where
in the
4.1 Definitions and assumptions
In this section, we list and present the definitions and assumptions to be used in the subsequent sections.
where
the nonlinear functions
4.2 Spatial‐based adaptive iterative learning control
For tidy presentation, the
where the output
The system (50) is valid within the set
where
where
According to Assumption 3.2, we may rewrite (52) as
where
Consider a reference trajectory
We may form a state space model, which produces the reference trajectory, as
where
where
Next, specify an LKF as
where
Substituting the error dynamics (55) into
where
where
This will simplify
Using the periodicity of
According to the following algebraic relationship,
where
Therefore, we may specify a periodic parametric update law as
Recall that
With (60) and (64), we conclude that
The objective is achieved. The main results are summarized in the following theorem.
5. Conclusion
Adaptive fuzzy control (AFC) has been investigated for coping with nonlinearities and uncertainties of unknown structures [38–40]. The major distinctions between AFC techniques and the ones described in Sections 2 and 3 are (a) time‐based (AFC) versus spatial‐based design (OFLRARC/OFBRARC) and (b) less information assumed on the nonlinearities/uncertainties (AFC) versus more information on the nonlinearities/uncertainties (OFLRARC/OFBRARC). Because, in spatial‐based design, a nonlinear coordinate transformation is conducted to change the independent variable from time to angular displacement, the systems under consideration in AFC and OFLRARC/OFBRARC are distinct. Next, AFC design techniques claim being able to tackle systems with a more generic class of nonlinearities/uncertainties, which relies on incorporating a fuzzy system to approximate those nonlinearities/uncertainties. It is not clear how to determine the required structure complexity of the fuzzy system (e.g., number of membership functions) to achieve desired control performance with reasonable control effort. Generally speaking, known characteristics of the uncertainties or disturbances should be incorporated as much as possible into the control design to improve performance, avoid conservativeness, and produce sensible control input. Therefore, instead of assuming the disturbances to be of generic type (as done by AFC), the methods presented in this chapter aim at a category of disturbances prevalent in rotary systems and explore the spatially periodic nature of the disturbances to design a specific control module and integrate into the overall control system.
Acknowledgments
The author gratefully acknowledges the support from the Ministry of Science and Technology, R.O.C. under grant MOST104-2221-E-005-043.
References
- 1.
Ding Z. Adaptive disturbance rejection of nonlinear systems in an extended output feedback form. Control Theory & Applications, IET. 2007;1(1):298–303. - 2.
Priscoli FD, Marconi L, Isidori A. A new approach to adaptive nonlinear regulation. SIAM Journal on Control and Optimization. 2006;45(3):829–55. - 3.
Francis BA, Wonham WM. The internal model principle of control theory. Automatica. 1976;12(5):457–65. - 4.
Kravaris C, Sotiropoulos V, Georgiou C, Kazantzis N, Xiao M, Krener AJ. Nonlinear observer design for state and disturbance estimation. Systems & Control Letters. 2007;56(11):730–5. - 5.
Chen W‐H. Disturbance observer based control for nonlinear systems. Mechatronics, IEEE/ASME Transactions on. 2004;9(4):706–10. - 6.
Ding Z. Asymptotic rejection of asymmetric periodic disturbances in output‐feedback nonlinear systems. Automatica. 2007;43(3):555–61. - 7.
Liu ZL, Svoboda J. A new control scheme for nonlinear systems with disturbances. Control Systems Technology, IEEE Transactions on. 2006;14(1):176–81. - 8.
Tang G‐Y, Gao D‐X. Approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances. Nonlinear Analysis: Theory, Methods & Applications. 2007;66(2):403–14. - 9.
Teoh J, Du C, Xie L, Wang Y. Nonlinear least‐squares optimisation of sensitivity function for disturbance attenuation on hard disk drives. Control Theory & Applications, IET. 2007;1(5):1364–9. - 10.
Bullinger E, Allgöwer F, editors. An adaptive high‐gain observer for nonlinear systems. In Decision and Control, Proceedings of the 36th IEEE Conference on; 1997; San Diego. California: IEEE. - 11.
Marine R, Santosuosso GL, Tomei P. Robust adaptive observers for nonlinear systems with bounded disturbances. Automatic Control, IEEE Transactions on. 2001;46(6):967–72. - 12.
Vargas JAR, Hemerly E, editors. Nonlinear adaptive observer design for uncertain dynamical systems. IEEE Conference on Decision and Control; 2000: Citeseer. - 13.
Kanellakopoulos I, Kokotovic P, Morse A, editors. Adaptive output‐feedback control of a class of nonlinear systems. Decision and Control, Proceedings of the 30th IEEE Conference on; 1991: IEEE. - 14.
Yang Z‐J, Kunitoshi K, Kanae S, Wada K. Adaptive robust output‐feedback control of a magnetic levitation system by K‐filter approach. Industrial Electronics, IEEE Transactions on. 2008;55(1):390–9. - 15.
Marino R, Tomei P. Global adaptive observers for nonlinear systems via filtered transformations. Automatic Control, IEEE Transactions on. 1992;37(8):1239–45. - 16.
Marino R, Tomei P. Global adaptive output‐feedback control of nonlinear systems. I. Linear parameterization. Automatic Control, IEEE Transactions on. 1993;38(1):17–32. - 17.
Chi R, Hou Z, Sui S, Yu L, Yao W. A new adaptive iterative learning control motivated by discrete‐time adaptive control. International Journal of Innovative Computing, Information and Control. 2008;4(6):1267–74. - 18.
Nakano M, She J‐H, Mastuo Y, Hino T. Elimination of position‐dependent disturbances in constant‐speed‐rotation control systems. Control Engineering Practice. 1996;4(9):1241–8. - 19.
Chen C‐L, Chiu GT‐C. Spatially periodic disturbance rejection with spatially sampled robust repetitive control. Journal of Dynamic Systems, Measurement, and Control. 2008;130(2):021002. - 20.
Moore KL. Iterative Learning Control for Deterministic Systems. Springer Science & Business Media; 2012. - 21.
Xu J‐X, Tan Y. Linear and Nonlinear Iterative Learning Control. New York: Springer; 2003. - 22.
De Wit CC, Praly L. Adaptive eccentricity compensation. Control Systems Technology, IEEE Transactions on. 2000;8(5):757–66. - 23.
Tsao T‐C, Bentsman J. Rejection of unknown periodic load disturbances in continuous steel casting process using learning repetitive control approach. Control Systems Technology, IEEE Transactions on. 1996;4(3):259–65. - 24.
Chen CL, Yang YH. Position‐dependent disturbance rejection using spatial‐based adaptive feedback linearization repetitive control. International Journal of Robust and Nonlinear Control. 2009;19(12):1337–63. - 25.
Chen C‐L, Chiu GT‐C, Allebach J. Robust spatial‐sampling controller design for banding reduction in electrophotographic process. Journal of Imaging Science and Technology. 2006;50(6):530–6. - 26.
Mahawan B, Luo Z‐H. Repetitive control of tracking systems with time‐varying periodic references. International Journal of Control. 2000;73(1):1–10. - 27.
Ahn H, Chen Y, Dou H. State‐periodic adaptive cogging and friction compensation of permanent magnetic linear motors. Magnetics, IEEE Transactions on. 2005;41(1):90–8. - 28.
Moore KL, Ghosh M, Chen YQ. Spatial‐based iterative learning control for motion control applications. Meccanica. 2007;42(2):167–75. - 29.
Fardad M, Jovanović MR, Bamieh B. Frequency analysis and norms of distributed spatially periodic systems. Automatic Control, IEEE Transactions on. 2008;53(10):2266–79. - 30.
Al‐Shyyab A, Kahraman A. Non‐linear dynamic analysis of a multi‐mesh gear train using multi‐term harmonic balance method: period‐one motions. Journal of Sound and Vibration. 2005;284(1):151–72. - 31.
Young T, Wu M. Dynamic stability of disks with periodically varying spin rates subjected to stationary in‐plane edge loads. Journal of Applied Mechanics. 2004;71(4):450–8. - 32.
Yang Y‐H, Chen C‐L, editors. Spatially periodic disturbance rejection using spatial‐based output feedback adaptive backstepping repetitive control. American Control Conference; 2008: IEEE. - 33.
Sastry SS, Isidori A. Adaptive control of linearizable systems. Automatic Control, IEEE Transactions on. 1989;34(11):1123–31. - 34.
Peterson BB, Narendra KS. Bounded error adaptive control. Automatic Control, IEEE Transactions on. 1982;27(6):1161–8. - 35.
Khalil HK, Grizzle J. Nonlinear Systems: Prentice‐Hall, New Jersey; 1996. - 36.
Yang Y‐H, Chen C‐L. Spatial domain adaptive control of nonlinear rotary systems subject to spatially periodic disturbances. Journal of Applied Mathematics. 2012;2012. - 37.
Yang Y‐H, Chen C‐L, editors. Spatial‐based adaptive iterative learning control of nonlinear rotary systems with spatially periodic parametric variation. Asian Control Conference, ASCC 7th; 2009: IEEE. - 38.
Tong S‐C, He X‐L, Zhang H‐G. A combined backstepping and small‐gain approach to robust adaptive fuzzy output feedback control. Fuzzy Systems, IEEE Transactions on. 2009;17(5):1059–69. - 39.
Tong S, Li Y. Observer‐based fuzzy adaptive control for strict‐feedback nonlinear systems. Fuzzy Sets and Systems. 2009;160(12):1749–64. - 40.
Shaocheng T, Changying L, Yongming L. Fuzzy adaptive observer backstepping control for MIMO nonlinear systems. Fuzzy Sets and Systems. 2009;160(19):2755–75.