Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
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Mathematical Approaches in Advanced Control Theories

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Research Article | Open Access

Volume 2012 |Article ID 610971 | 20 pages | https://doi.org/10.1155/2012/610971

Spatial Domain Adaptive Control of Nonlinear Rotary Systems Subject to Spatially Periodic Disturbances

Academic Editor: Baocang Ding
Received22 Feb 2012
Revised13 Jun 2012
Accepted14 Jun 2012
Published22 Jul 2012

Abstract

We propose a generic spatial domain control scheme for a class of nonlinear rotary systems of variable speeds and subject to spatially periodic disturbances. The nonlinear model of the rotary system in time domain is transformed into one in spatial domain employing a coordinate transformation with respect to angular displacement. Under the circumstances that measurement of the system states is not available, a nonlinear state observer is established for providing the estimated states. A two-degree-of-freedom spatial domain control configuration is then proposed to stabilize the system and improve the tracking performance. The first control module applies adaptive backstepping with projected parametric update and concentrates on robust stabilization of the closed-loop system. The second control module introduces an internal model of the periodic disturbances cascaded with a loop-shaping filter, which not only further reduces the tracking error but also improves parametric adaptation. The overall spatial domain output feedback adaptive control system is robust to model uncertainties and state estimated error and capable of rejecting spatially periodic disturbances under varying system speeds. Stability proof of the overall system is given. A design example with simulation demonstrates the applicability of the proposed design.

1. Introduction

Rotary systems play important roles in various industry applications, for example, packaging, printing, assembly, fabrication, semiconductor, robotics, and so forth. Design of control algorithm for a motion system often comes up with nonlinearities and uncertainties. Nonlinearities are either inherent to the system or due to the dynamics of actuators and sensors. Uncertainties are mainly caused by unmodeled dynamics, parametric uncertainty, and disturbances. For dealing with nonlinearities, common techniques, for example, feedback linearization and backstepping, are to utilize feedback to cancel all or part of the nonlinear terms. On the other hand, design techniques for conducting disturbance rejection or attenuation in control systems mostly originate from the internal model principle [1], for example, those incorporating or estimating the exosystem of the disturbances [2โ€“6]. Conventional controllers are mostly time-based controllers as they are synthesized and operate in temporal or time domain. Several researches [7โ€“9] have started studying spatial domain controllers ever since a repetitive controller design was initiated by Nakano et al. [10]. In the design of Nakano et al., the repetitive control system has its repetitive kernel (i.e., ๐‘’โˆ’๐ฟ๐‘  with positive feedback) synthesized and operated with respect to a spatial coordinate, for example, angular position or displacement. Hence its capability for rejecting or tracking spatially periodic disturbances or references will not degrade when the controlled system operates at varying speed. All existing studies propose design methods starting with a linear time-invariant (LTI) system. After reformulation, a nonlinear open-loop system is obtained in spatial domain. Subsequently, the open-loop system is either linearized around an operating speed or regarded as a quasi-linear parameter varying (quasi-LPV) system and then adjoined with the spatial domain internal model of the tracking or disturbance signal. Design paradigms based on linear (robust) control theory are then applied to the resulting augmented system. However, presuming the open-loop system to be LTI and resorting to design paradigm of linear control will inevitably restrict the applicability and limit the achievable performance of a design method. Chen and Yang [11] introduced a new spatial domain control scheme based on a second-order LTI system with availability of state measurements. To achieve robust stabilization and high-performance tracking, a two-module control configuration is constructed. One of the modules utilizes adaptive backstepping with projected parametric adaptation to robustly stabilize the system. The other module incorporates a spatial domain internal model of the disturbances cascaded with a loop-shaping filter to improve the tracking performance.

This paper extends the work of Chen and Yang [11, 12]. The control scheme has been generalized such that it is applicable to a class of nonlinear systems (instead of just LTI systems). Moreover, the major shortcoming in Chen and Yangโ€™s design [11], that is, which requires full-state feedback, is resolved by incorporation of a nonlinear state observer. Various types of nonlinear state observers have been developed and put into use in the past (e.g., [13, 14]). This paper will study the feasibility of incorporating a ๐พ-filter-type state observer [13] into the proposed design. The proposed system incorporating the state observer can be proved to be stable under bounded disturbance and system uncertainties. An illustrative example is given for demonstration and derivation of the control algorithm. Simulation is performed to verify the feasibility and effectiveness of the proposed scheme. Compared to the preliminary work in [12] (which is only applicable to second order systems), the results have been generalized to be suited for ๐‘›th order systems. Specifically, the design and stability proof are more comprehensive and rigorous than those presented in [12].

Recently, there have been emerging design techniques based on adaptive fuzzy control (AFC), which may cope with nonlinearities and uncertainties with unknown structures [15โ€“17]. The major differences between those techniques and the proposed one are as follows: (1) design being time based (AFC) versus spatial based (the proposed approach); (2) assuming less information about the nonlinearities/uncertainties (AFC) versus more information about the nonlinearities/uncertainties (the proposed approach). Note that the spatial-based design is not just a change of the independent variable from time to angular displacement. A nonlinear coordinate transformation is actually involved. Therefore, the systems under consideration in AFC and the proposed method are different. Next, the capability of the design approaches suggested in AFC for tackling systems subject to a more generic class of nonlinearities/uncertainties lies in the usage of a fuzzy system to approximate those nonlinearities/uncertainties. It is actually not clear regarding the complexity of the fuzzy system (i.e., number of membership functions) that should be used to achieve the required control performance. It is also not clear whether or not the control effort is reasonable. In general, when characteristics of the uncertainties or disturbances are known, such information should be incorporated as much as possible into the design to enhance performance, avoid conservativeness, and result in sensible control input. Hence, instead of assuming the disturbances to be generic (probably just being bounded as by AFC), the proposed design is aiming at a type of disturbances specific to rotary systems and utilizes the spatially periodic nature of the disturbances to establish a well-defined control module integrated into the overall control configuration.

This paper is organized as follows. Reformulation of a generic nonlinear rotary system with respect to angular displacement will be presented in Section 2. Design of the state observer is described in Section 3. Section 4 will cover derivation and stability analysis of the proposed spatial domain output feedback control scheme. Simulation verification for the proposed scheme will be presented in Section 5. Conclusion is given in Section 6.

2. Problem Formulation

In this section, we show how a generic NTI model can be transformed into an NPI model by choosing an alternate independent variable (angular displacement instead of time) and defining a new set of states (or coordinates) with respect to the angular displacement. Note that the transformation described here is equivalent to a nonlinear coordinate transformation or a diffeomorphism. The NPI model will be used for the subsequent design and discussion, ๎€บ๐‘“ฬ‡๐‘ฅ(๐‘ก)=๐‘ก๎€ท๐‘ฅ(๐‘ก),๐œ™๐‘“๎€ธ+ฮ”๐‘“๐‘ก๎€ท๐‘ฅ(๐‘ก),๐œ™๐‘“+๎€บ๐‘”๎€ธ๎€ป๐‘ก๎€ท๐‘ฅ(๐‘ก),๐œ™๐‘”๎€ธ+ฮ”๐‘”๐‘ก๎€ท๐‘ฅ(๐‘ก),๐œ™๐‘”๐‘ข๎€ธ๎€ป(๐‘ก),๐‘ฆ=ฮจ๐‘ฅ(๐‘ก)+๐‘‘๐‘ฆ(๐‘ก)=๐‘ฅ1(๐‘ก)+๐‘‘๐‘ฆ(๐‘ก),(2.1) where ๐‘ฅ(๐‘ก)=[๐‘ฅ1(๐‘ก)โ‹ฏ๐‘ฅ๐‘›(๐‘ก)]๐‘‡, ฮจ=[10โ‹ฏ0], and ๐‘ข(๐‘ก) and ๐‘ฆ(๐‘ก) correspond to control input and measured output angular velocity of the system, respectively. ๐‘‘๐‘ฆ(๐‘ก) represents a class of bounded output disturbances which constitutes (dominant) spatially periodic and band-limited (or nonperiodic) components. Here we refer band-limited disturbances to signals whose Fourier transform or power spectral density is zero above a certain finite frequency. The only available information of the disturbances is the number of distinctive spatial frequencies and the spectrum distribution for band-limited disturbance components. ๐‘“๐‘ก(๐‘ฅ(๐‘ก),๐œ™๐‘“) and ๐‘”๐‘ก(๐‘ฅ(๐‘ก),๐œ™๐‘”) are known vector-valued functions with unknown but bounded system parameters, that is, ๐œ™๐‘“=[๐œ™๐‘“1โ‹ฏ๐œ™๐‘“๐‘˜] and ๐œ™๐‘”=[๐œ™๐‘”1โ‹ฏ๐œ™๐‘”๐‘™]; ฮ”๐‘“๐‘ก(๐‘ฅ(๐‘ก),๐œ™๐‘“) and ฮ”๐‘”๐‘ก(๐‘ฅ(๐‘ก),๐œ™๐‘”) represent unstructured modeling inaccuracy, which are also assumed to be bounded. Instead of using time ๐‘ก as the independent variable, consider an alternate independent variable ๐œƒ=๐œ†(๐‘ก), that is, the angular displacement. Since by definition ๎€œ๐œ†(๐‘ก)=๐‘ก0๐œ”(๐œ)๐‘‘๐œ+๐œ†(0),(2.2) where ๐œ”(๐‘ก) is the angular velocity, the following condition: ๐œ”(๐‘ก)=๐‘‘๐œƒ๐‘‘๐‘ก>0,โˆ€๐‘ก>0,(2.3) will guarantee that ๐œ†(๐‘ก) is strictly monotonic such that ๐‘ก=๐œ†โˆ’1(๐œƒ) exists. Thus all the variables in the time domain can be transformed into their counterparts in the ๐œƒ-domain, that is, ๎€ท๐œ†ฬ‚๐‘ฅ(๐œƒ)=๐‘ฅโˆ’1๎€ธ๎€ท๐œ†(๐œƒ),ฬ‚๐‘ฆ(๐œƒ)=๐‘ฆโˆ’1๎€ธ,๎€ท๐œ†(๐œƒ)ฬ‚๐‘ข(๐œƒ)=๐‘ขโˆ’1๎€ธ,๎๎€ท๐œ†(๐œƒ)๐‘‘(๐œƒ)=๐‘‘โˆ’1๎€ธ,๎€ท๐œ†(๐œƒ)๎๐œ”(๐œƒ)=๐œ”โˆ’1๎€ธ,(๐œƒ)(2.4) where we denote ฬ‚โ€ข as the ๐œƒ-domain representation of โ€ข. Note that, in practice, (2.3) can usually be satisfied for most rotary systems where the rotary component rotates only in one direction. Since ๐‘‘๐‘ฅ(๐‘ก)=๐‘‘๐‘ก๐‘‘๐œƒ๐‘‘๐‘ก๐‘‘ฬ‚๐‘ฅ(๐œƒ)๐‘‘๐œƒ=๎๐œ”(๐œƒ)๐‘‘ฬ‚๐‘ฅ(๐œƒ).๐‘‘๐œƒ(2.5) Equation (2.1) can be rewritten as ๎๐œ”(๐œƒ)๐‘‘ฬ‚๐‘ฅ(๐œƒ)=๎€บ๐‘“๐‘‘๐œƒ๐‘ก๎€ทฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘“๎€ธ+ฮ”๐‘“๐‘ก๎€ทฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘“+๎€บ๐‘”๎€ธ๎€ป๐‘ก๎€ทฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘”๎€ธ+ฮ”๐‘”๐‘ก๎€ทฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘”๎๐‘‘๎€ธ๎€ปฬ‚๐‘ข(๐œƒ),ฬ‚๐‘ฆ(๐œƒ)=ฮจฬ‚๐‘ฅ(๐œƒ)+๐‘ฆ(๐œƒ)=ฬ‚๐‘ฅ1๎๐‘‘(๐œƒ)+๐‘ฆ(๐œƒ).(2.6) Equation (2.6) can be viewed as a nonlinear position-invariant (as opposed to the definition of time-invariant) system with the angular displacement ๐œƒ as the independent variable. Note that the concept of transfer function is still valid for linear position-invariant systems if we define the Laplace transform of a signal ฬ‚๐‘”(๐œƒ) in the angular displacement domain as ๎๎€œ๐บ(ฬƒ๐‘ )=โˆž0ฬ‚๐‘”(๐œƒ)๐‘’โˆ’ฬƒ๐‘ ๐œƒ๐‘‘๐œƒ.(2.7) This definition will be useful for describing the linear portion of the overall control system.

3. Nonlinear State Observer

Drop the ๐œƒ notation and note that (2.6) can be expressed as a standard nonlinear system: ฬ‡๎€ทฬ‚๐‘ฅ=๐‘“ฬ‚๐‘ฅ,๐œ™๐‘“๎€ธ๎€ท+๐‘”ฬ‚๐‘ฅ,๐œ™๐‘”๎€ธ๎๐‘‘ฬ‚๐‘ข+๐‘ ๎๐‘‘,ฬ‚๐‘ฆ=โ„Ž(ฬ‚๐‘ฅ)+๐‘ฆ๎๐‘‘=๎๐œ”+๐‘ฆ,(3.1) where terms involving unstructured uncertainty are merged into ๎๐‘‘๐‘ =ฮ”๐‘“(ฬ‚๐‘ฅ,๐œ™๐‘“)+ฮ”๐‘”(ฬ‚๐‘ฅ,๐œ™๐‘”)ฬ‚๐‘ข with ๎€ทฮ”๐‘“ฬ‚๐‘ฅ,๐œ™๐‘“๎€ธ=ฮ”๐‘“๐‘ก๎€ทฬ‚๐‘ฅ,๐œ™๐‘“๎€ธฬ‚๐‘ฅ1๎€ท,ฮ”๐‘”ฬ‚๐‘ฅ,๐œ™๐‘”๎€ธ=ฮ”๐‘”๐‘ก๎€ทฬ‚๐‘ฅ,๐œ™๐‘”๎€ธฬ‚๐‘ฅ1.(3.2) In addition, we have ๐‘“๎€ทฬ‚๐‘ฅ,๐œ™๐‘“๎€ธ=๐‘“๐‘ก๎€ทฬ‚๐‘ฅ,๐œ™๐‘“๎€ธฬ‚๐‘ฅ1๎€ท,๐‘”ฬ‚๐‘ฅ,๐œ™๐‘”๎€ธ=๐‘”๐‘ก๎€ทฬ‚๐‘ฅ,๐œ™๐‘”๎€ธฬ‚๐‘ฅ1,โ„Ž(ฬ‚๐‘ฅ)=๎๐œ”=ฬ‚๐‘ฅ1.(3.3) The state variables have been specified such that the angular velocity ๎๐œ” is equal to ฬ‚๐‘ฅ1, that is, the undisturbed output โ„Ž(ฬ‚๐‘ฅ). It is not difficult to verify that (3.1) has the same relative degree in ๐ท0={ฬ‚๐‘ฅโˆˆโ„๐‘›โˆฃฬ‚๐‘ฅ1โ‰ 0} as the NTI model in (2.1). If (3.1) has relative degree ๐‘Ÿ, we can define the following nonlinear coordinate transformation: โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐œ“ฬ‚๐‘ง=๐‘‡(ฬ‚๐‘ฅ)=1โ‹ฎ๐œ“(ฬ‚๐‘ฅ)๐‘›โˆ’๐‘Ÿ(ฬ‚๐‘ฅ)โ‹ฎ๐ฟโ„Ž(ฬ‚๐‘ฅ)๐‘“๐‘Ÿโˆ’1โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆโ‰œ๎‚ธโ„Ž(ฬ‚๐‘ฅ)ฬ‚๐‘ง2ฬ‚๐‘ง1๎‚น,(3.4) where ๐œ“1 to ๐œ“๐‘›โˆ’๐‘Ÿ are chosen such that ๐‘‡(ฬ‚๐‘ฅ) is a diffeomorphism on ๐ท0โŠ‚๐ท and ๐ฟ๐‘”๐œ“๐‘–(ฬ‚๐‘ฅ)=0,1โ‰ค๐‘–โ‰ค๐‘›โˆ’๐‘Ÿ,โˆ€ฬ‚๐‘ฅโˆˆ๐ท0.(3.5) With respect to the new coordinates, that is, ฬ‚๐‘ง1 and ฬ‚๐‘ง2, (3.1) can be transformed into the so-called normal form, that is, ฬ‡ฬ‚๐‘ง2=๐ฟ๐‘“๐œ“||(ฬ‚๐‘ฅ)ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง)+๎๐‘‘๐‘ ๐‘œ๎€ทโ‰œฮจฬ‚๐‘ง1,ฬ‚๐‘ง2๎€ธ,ฬ‡ฬ‚๐‘ง1=๐ด๐‘ฬ‚๐‘ง1+๐ต๐‘๎‚ƒ๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1||โ„Ž(ฬ‚๐‘ฅ)ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง)๎‚„โŽกโŽขโŽขโŽฃ๐ฟฬ‚๐‘ข+๐‘Ÿ๐‘“โ„Ž(ฬ‚๐‘ฅ)๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1|||||โ„Ž(ฬ‚๐‘ฅ)ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง)โŽคโŽฅโŽฅโŽฆ+๎๐‘‘๐‘ ๐‘–,ฬ‚๐‘ฆ=๐ถ๐‘ฬ‚๐‘ง1+๎๐‘‘๐‘ฆ,(3.6) where ๎๐‘‘๐‘ ๐‘œ and ๎๐‘‘๐‘ ๐‘–๎๐‘‘=[๐‘ ๐‘–1โ‹ฏ๎๐‘‘๐‘ ๐‘–๐‘Ÿ]๐‘‡ come from ๎๐‘‘๐‘  going through the indicated coordinate transformation. ฬ‚๐‘ง1=[ฬ‚๐‘ง11โ‹ฏฬ‚๐‘ง1๐‘Ÿ]โˆˆโ„๐‘Ÿ, ฬ‚๐‘ง2โˆˆโ„๐‘›โˆ’๐‘Ÿ, and (๐ด๐‘,๐ต๐‘,๐ถ๐‘) is a canonical form representation of a chain of ๐‘Ÿ integrators. The first equation is called internal dynamics and the second is called external dynamics. Internal dynamics which is not affected by the control ๐‘ข. By setting ฬ‚๐‘ง1=0 in that equation, we obtain ฬ‡ฬ‚๐‘ง2๎€ท=ฮจ0,ฬ‚๐‘ง2๎€ธ,(3.7) which is the zero dynamics of (3.1) or (3.6). The system is called minimum phase if (3.7) has an asymptotically stable equilibrium point in the domain of interest. To allow us to present the proposed algorithm and stability analysis in a simpler context, we will make the following assumptions for the subsequent derivation:(1)๐‘“(ฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘“) and ๐‘”(ฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘”) are linearly related to those unknown system parameters, that is, ๐‘“๎€ทฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘“๎€ธ=๐œ™1๐‘“1(ฬ‚๐‘ฅ(๐œƒ))+โ‹ฏ+๐œ™๐‘“๐‘˜๐‘“๐‘˜๐‘”๎€ท(ฬ‚๐‘ฅ(๐œƒ)),ฬ‚๐‘ฅ(๐œƒ),๐œ™๐‘”๎€ธ=๐œ™๐‘”1๐‘”1(ฬ‚๐‘ฅ(๐œƒ))+โ‹ฏ+๐œ™๐‘”๐‘™๐‘”๐‘™(ฬ‚๐‘ฅ(๐œƒ)).(3.8)(2)Equation (3.1) is minimum phase, and the internal dynamics in (3.6) is ISS (input-to-state stable).(3)The output disturbance is sufficiently smooth, that is, ฬ‡๎๐‘‘๐‘ฆ๎๐‘‘,โ€ฆ,๐‘ฆ(๐‘Ÿ) exist.(4)๎๐‘‘(๐‘Ÿโˆ’1)๐‘ ๐‘–1,๎๐‘‘(๐‘Ÿโˆ’2)๐‘ ๐‘–2ฬ‡๎๐‘‘,โ€ฆ,๐‘ ๐‘–๐‘Ÿโˆ’1 exist, that is, the transformed unstructured uncertainty is sufficiently smooth.(5)The reference command ฬ‚๐‘ฆ๐‘š and its first ๐‘Ÿ derivates are known and bounded. Moreover, the signal ฬ‚๐‘ฆ๐‘š(๐‘Ÿ) is piecewise continuous.With assumption (2), we focus on designing a nonlinear state observer for external dynamics of (3.6), ฬ‡ฬ‚๐‘ง1=๐ด๐‘ฬ‚๐‘ง1+๐ต๐‘๎‚ƒ๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1||โ„Ž(ฬ‚๐‘ฅ)ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง)๎‚„โŽกโŽขโŽขโŽฃ๐ฟฬ‚๐‘ข+๐‘Ÿ๐‘“โ„Ž(ฬ‚๐‘ฅ)๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1|||||โ„Ž(ฬ‚๐‘ฅ)ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง)โŽคโŽฅโŽฅโŽฆ+๎๐‘‘๐‘ ๐‘–.(3.9) Since ๐‘“(ฬ‚๐‘ฅ) and ๐‘”(ฬ‚๐‘ฅ) are linearly related to system parameters, ๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1โ„Ž(ฬ‚๐‘ฅ) and ๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1โ„Ž(ฬ‚๐‘ฅ) can be written as ๐ฟ๐‘Ÿ๐‘“โ„Ž(ฬ‚๐‘ฅ)=ฮ˜๐‘‡๐‘Š๐‘“(ฬ‚๐‘ฅ) and ๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1โ„Ž(ฬ‚๐‘ฅ)=ฮ˜๐‘‡๐‘Š๐‘”(ฬ‚๐‘ฅ), where ๐‘Š๐‘“(ฬ‚๐‘ฅ) and ๐‘Š๐‘”(ฬ‚๐‘ฅ) are two nonlinear functions, and ฮ˜=[๐œ™๐‘“1โ‹ฏ๐œ™๐‘“๐‘˜๐œ™๐‘”1โ‹ฏ๐œ™๐‘”๐‘™โ‹ฏ]๐‘‡=[๐œ™1โ‹ฏ๐œ™โ„“]๐‘‡โˆˆโ„โ„“, where โ„“ is the number of unknown parameters. Next, we adopt the following observer structure: ฬ‡โ€Œ๐‘ง1=๐ด0๐‘ง1+๐‘˜๐‘ฆ+๐น(๐‘ฆ,๐‘ข)๐‘‡ฮ˜, where ๐‘ง1=[๐‘ง11โ‹ฏ๐‘ง1๐‘Ÿ]๐‘‡ is the estimate of ๐‘ง1, and ๐‘Š๐‘“(๐‘ฆ) and ๐‘Š๐‘”(๐‘ฆ) are nonlinear functions with the same structure as ๐‘Š๐‘“(๐‘ฅ) and ๐‘Š๐‘”(๐‘ฅ) except that each entry of ๐‘ฅ is replaced by ๐‘ฆ. Furthermore, ๐ด0=โŽกโŽขโŽขโŽขโŽขโŽฃโˆ’๐‘˜1โ‹ฎโˆ’๐‘˜๐‘Ÿ๐ผ(๐‘Ÿโˆ’1)ร—(๐‘Ÿโˆ’1)01ร—(๐‘Ÿโˆ’1)โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ,๎‚ƒ๐‘˜๐‘˜=1โ‹ฏ๐‘˜๐‘Ÿ๎‚„๐‘‡,๐น(๐‘ฆ,๐‘ข)๐‘‡=โŽกโŽขโŽขโŽฃ0(๐‘Ÿโˆ’1)ร—โ„“๐‘Š๐‘‡๐‘“(๐‘ฆ)+๐‘Š๐‘‡๐‘”โŽคโŽฅโŽฅโŽฆ(๐‘ฆ)๐‘ขโˆˆโ„๐‘Ÿร—โ„“.(3.10) By properly choosing ๐‘˜, the matrix ๐ด0 can be made Hurwitz. Define the state estimated error as ๐œ€โ‰œ[๐œ€๐‘ง11โ‹ฏ๐œ€๐‘ง1๐‘Ÿ]๐‘‡โ‰œ๐‘ง1โˆ’๐‘ง1. The dynamics of the estimated error can be obtained as ฬ‡๐œ€=๐ด0๐œ€+ฮ”, where ฮ”=โˆ’๐‘˜๐‘‘๐‘ฆ+๐ต๐‘ฮ˜๐‘‡[๐‘Š๐‘”(๐‘ฅ)โˆ’๐‘Š๐‘”(๐‘ฆ)]๐‘ข+๐ต๐‘ฮ˜๐‘‡[๐‘Š๐‘“(๐‘ฅ)โˆ’๐‘Š๐‘“(๐‘ฆ)]+๐‘‘๐‘ ๐‘–. To proceed, the role of the state observer is replaced by ๐‘ง1โ‰œ๐œ‰+ฮฉ๐‘‡ฮ˜ and the following two ๐พ-filters: ฬ‡๐œ‰=๐ด0๐œ‰+ฬ‡ฮฉ๐‘˜๐‘ฆ,๐‘‡=๐ด0ฮฉ๐‘‡+๐น(๐‘ฆ,๐‘ข)๐‘‡,(3.11) such that ๐œ‰=[๐œ‰11โ‹ฏ๐œ‰1๐‘Ÿ]๐‘‡โˆˆโ„๐‘Ÿ and ฮฉ๐‘‡โ‰œ[๐‘ฃ1โ‹ฏ๐‘ฃโ„“]โˆˆโ„๐‘Ÿร—โ„“. Decompose the second equation of (3.11) into ฬ‡๐‘ฃ๐‘—=๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—,๐‘—=1,2,โ€ฆ,โ„“,โ€‰ โ€‰where ๐‘’๐‘Ÿ=[0โ‹ฏ01]โˆˆโ„๐‘Ÿ and ๐œŽ๐‘—=๐‘ค1๐‘—+๐‘ค2๐‘—๐‘ข with ๐‘ค1๐‘— and ๐‘ค2๐‘— are the ๐‘—th columns of ๐‘Š๐‘‡๐‘“(๐‘ฆ) and ๐‘Š๐‘‡๐‘”(๐‘ฆ), respectively. With the definition of the state estimated error ๐œ€, the state estimate ๐‘ง1, and (3.11), we acquire the following set of equations which will be used in the subsequent design: ๐‘ง1๐‘˜=๐‘ง1๐‘˜+๐œ€๐‘ง1๐‘˜=๐œ‰1๐‘˜+โ„“๎“๐‘—=1๐‘ฃ๐‘—,๐‘˜๐œ™๐‘—+๐œ€๐‘ง1๐‘˜,๐‘˜=1,โ€ฆ,๐‘Ÿ,(3.12) where โ€ข๐‘—,๐‘– denotes the ith row of โ€ข๐‘—.

4. Spatial Domain Output Feedback Adaptive Control System

To apply adaptive backstepping method, we firstly rewrite the derivative of output ฬ‚๐‘ฆ as ฬ‡ฬ‡ฬ‚๐‘ฆ=ฬ‚๐‘ง11+ฬ‡๎๐‘‘๐‘ฆ=ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ=๐‘ง12+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ.(4.1) With the second equation in (3.12), (4.1) can be written as ฬ‡ฬ‚๐‘ฆ=๐‘ง12+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ=๐œ‰12+๐‘ฃโ„“,2๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ,(4.2) where ๐œ”๐‘‡=[๐‘ฃ1,2โ‹ฏ๐‘ฃโ„“โˆ’1,20].

In view of designing output feedback backstepping with ๐พ-filters, we need to find a set of ๐พ-filter parameters, that is, ๐‘ฃโ„“,2,โ€ฆ,๐‘ฃ1,2, separated from ฬ‚๐‘ข by the same number of integrators between ฬ‚๐‘ง12 and ฬ‚๐‘ข. From (3.11), we can see that ๐‘ฃโ„“,2,โ€ฆ,๐‘ฃ1,2 are all candidates if ๐‘ค2๐‘— are not zero. In the following derivation, we assume that ๐‘ฃโ„“,2 is selected. Hence, the system incorporating the ๐พ-filters can be expressed as ฬ‡ฬ‚๐‘ฆ=๐œ‰12+๐‘ฃโ„“,2๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ,ฬ‡๐‘ฃโ„“,๐‘–=๐‘ฃโ„“,๐‘–+1โˆ’๐‘˜๐‘–๐‘ฃโ„“,1ฬ‡๐‘ฃ,๐‘–=2,โ€ฆ,๐‘Ÿโˆ’1,โ„“,๐‘Ÿ=โˆ’๐‘˜๐‘Ÿ๐‘ฃโ„“,1+๐‘ค1โ„“+๐‘ค2โ„“ฬ‚๐‘ข.(4.3) To apply adaptive backstepping to (4.3), a new set of coordinates will be introduced ๐‘ง1=ฬ‚๐‘ฆโˆ’ฬ‚๐‘ฆ๐‘š,๐‘ง๐‘–=๐‘ฃโ„“,๐‘–โˆ’๐›ผ๐‘–โˆ’1,๐‘–=2,โ€ฆ,๐‘Ÿ,(4.4) where ฬ‚๐‘ฆ๐‘š is the prespecified reference command and ๐›ผ๐‘–โˆ’1 is the virtual input which will be used to stabilize each state equation. For simplicity, we define ๐œ•๐›ผ0/๐œ•ฬ‚๐‘ฆโ‰œโˆ’1 for subsequent derivations.

Step 1 (๐‘–=1). With (4.4), the first state equation of (4.3) can be expressed as ฬ‡๐‘ง1=๐œ‰12+๐‘ง2๐œ™โ„“+๐›ผ1๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆโˆ’ฬ‡ฬ‚๐‘ฆ๐‘š.(4.5) Consider a Lyapunov function ๐‘‰1=(1/2)๐‘ง21 and calculate its derivative ฬ‡๐‘‰1=๐‘ง1ฬ‡๐‘ง1=๐‘ง1๎‚€๐œ‰12+๐‘ง2๐œ™โ„“+๐›ผ1๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆโˆ’ฬ‡ฬ‚๐‘ฆ๐‘š๎‚.(4.6) Define the estimates of ๐œ™๐‘– as ๎‚๐œ™๐‘– and ฮฆ=[ฮฆ1โ‹ฏฮฆโ„“๎‚ฮ˜]=ฮ˜โˆ’, where ๎‚๎‚๐œ™ฮ˜=[๐‘“1โ‹ฏ๎‚๐œ™๐‘“๐‘˜๎‚๐œ™๐‘”1โ‹ฏ๎‚๐œ™๐‘”๐‘™โ‹ฏ]๐‘‡๎‚๐œ™=[1โ‹ฏ๎‚๐œ™โ„“]๐‘‡โˆˆโ„โ„“. Note that ฮ˜ is the โ€œtrueโ€ parameter vector while ๎‚ฮ˜ is the estimated parameter vector. Design the virtual input ๐›ผ1 as ๐›ผ1=๐›ผ1/๎‚๐œ™โ„“ and specify ๐›ผ1=1๐‘ง1๎‚€โˆ’๐‘ง1๐œ‰12โˆ’๐‘ง1๐‘ง2๎‚๐œ™โ„“โˆ’๐‘ง1๐œ”๎‚ฮ˜+๐‘ง1ฬ‡ฬ‚๐‘ฆ๐‘šโˆ’๐‘1๐‘ง21โˆ’๐‘‘1๐‘ง21โˆ’๐‘”1๐‘ง21๎‚=โˆ’๐œ‰12โˆ’๐‘ง2๎‚๐œ™โ„“โˆ’๐œ”๐‘‡๎‚ฬ‡ฮ˜+ฬ‚๐‘ฆ๐‘šโˆ’๐‘1๐‘ง1โˆ’๐‘‘1๐‘ง1โˆ’๐‘”1๐‘ง1,(4.7) where ๐‘๐‘–,๐‘‘๐‘–,๐‘”๐‘– are variables. Therefore, (4.6) becomes ฬ‡๐‘‰1=โˆ’๐‘1๐‘ง21โˆ’๐‘‘1๐‘ง21โˆ’๐‘”1๐‘ง21+๐œ1ฮฆ+๐‘ง1๎‚€๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚,(4.8) where ๐œ1ฮฆ=๐‘ง1๐‘ง2ฮฆโ„“+๐›ผ1ฮฆโ„“+๐‘ง1๐œ”๐‘‡ฮฆ.

Step 2 (๐‘–=2,โ€ฆ,๐‘Ÿโˆ’1). With respect to the new set of coordinates (4.4), the second equation of (4.3) can be rewritten as ฬ‡๐‘ง๐‘–=๐‘ง๐‘–+1+๐›ผ๐‘–โˆ’๐‘˜๐‘–๐‘ฃโ„“,1โˆ’๎ƒฌ๐œ•๐›ผ๐‘–โˆ’1๎‚€๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚+๐œ•๐›ผ๐‘–โˆ’1๎‚€๐ด๐œ•๐œ‰0๐œ‰+๎‚+๐‘˜ฬ‚๐‘ฆ๐œ•๐›ผ๐‘–โˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘–โˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—)๎ƒญ(4.9) Consider a Lyapunov function ๐‘‰๐‘–=โˆ‘๐‘–โˆ’1๐‘—=1๐‘‰๐‘—+(1/2)๐‘ง2๐‘– and its derivative ฬ‡๐‘‰๐‘–=๐‘–โˆ’1๎“๐‘—=1ฬ‡๐‘‰๐‘—+๐‘ง๐‘–๎ƒฏ๐‘ง๐‘–+1+๐›ผ๐‘–โˆ’๐‘˜๐‘–๐‘ฃโ„“,1โˆ’๎ƒฌ๐œ•๐›ผ๐‘–โˆ’1๎‚€๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๐œ™โ„“+๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚+๐œ•๐›ผ๐‘–โˆ’1๎‚€๐ด๐œ•๐œ‰0๐œ‰+๎‚+๐‘˜ฬ‚๐‘ฆ๐œ•๐›ผ๐‘–โˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘–โˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—).๎ƒญ๎ƒฐ(4.10) Specify ๐›ผ๐‘–=1๐‘ง๐‘–๎ƒฏโˆ’๐‘ง๐‘–๐‘ง๐‘–+1+๐‘ง๐‘–๐‘˜๐‘–๐‘ฃโ„“,1+๐‘ง๐‘–๎ƒฌ๐œ•๐›ผ๐‘–โˆ’1๎‚ต๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๎‚๐œ™โ„“+โˆ’๐œ”๐‘‡๎‚ฮ˜๎‚ถ+๐œ•๐›ผ๐‘–โˆ’1๎‚ต๐ด๐œ•๐œ‰0๐œ‰+โˆ’๐‘˜๎‚ถ+ฬ‚๐‘ฆ๐œ•๐›ผ๐‘–โˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘–โˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘–โˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—)๎ƒญโˆ’๐‘๐‘–๐‘ง2๐‘–โˆ’๐‘‘๐‘–๎‚ต๐œ•๐›ผ๐‘–โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘–โˆ’๐‘”๐‘–๎‚ต๐œ•๐›ผ๐‘–โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘–๎ƒฐ.(4.11) The derivative of ๐‘‰๐‘– becomes ฬ‡๐‘‰๐‘–=โˆ’๐‘–โˆ’1๎“๐‘—=1๎ƒฉ๐‘๐‘—๐‘ง2๐‘—+๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—+๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—๎ƒช+๐œ๐‘–ฮฆโˆ’๐‘–โˆ’1๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1๎‚€๐œ€๐œ•ฬ‚๐‘ฆฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚,(4.12) where ๐œ๐‘–ฮฆ=๐œ1โˆ‘ฮฆโˆ’๐‘–โˆ’1๐‘—=2(๐œ•๐›ผ๐‘—โˆ’1/๐œ•ฬ‚๐‘ฆ)(๐‘ง๐‘—๐‘ฃโ„“,1ฮฆโ„“+๐‘ง๐‘—โˆ’๐œ”๐‘‡ฮฆ).

Step 3. With respect to the new set of coordinates (4.4), the third equation of (4.3) can be written as ฬ‡๐‘ง๐‘Ÿ=โˆ’๐‘˜๐‘Ÿ๐‘ฃโ„“,1+๐‘ค1โ„“+๐‘ค2โ„“๎ƒฌฬ‚๐‘ขโˆ’๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๐œ™โ„“+โˆ’๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚ถ+๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐ด๐œ•๐œ‰0๐œ‰+โˆ’๐‘˜๎‚ถ+ฬ‚๐‘ฆ๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘Ÿโˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—)๎ƒญ.(4.13) The overall Lyapunov function may now be chosen as ๐‘‰๐‘Ÿ=๐‘Ÿโˆ’1๎“๐‘—=1๐‘‰๐‘—+12๐‘ง2๐‘Ÿ+12ฮฆ๐‘‡ฮ“โˆ’1ฮฆ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๐‘ƒ๐œ€,(4.14) where ฮ“ is a symmetric positive definite matrix, that is, ฮ“=ฮ“๐‘‡>0. With the definition of state estimated error ๐œ€, we can obtain that ฬ‡๐‘‰๐‘Ÿ=๐‘Ÿโˆ’1๎“๐‘—=1ฬ‡๐‘‰๐‘—+๐‘ง๐‘Ÿ๎ƒฏโˆ’๐‘˜๐‘Ÿ๐‘ฃโ„“,1+๐‘ค1โ„“+๐‘ค2โ„“๎ƒฌฬ‚๐‘ขโˆ’๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๐œ™โ„“+โˆ’๐œ”๐‘‡ฮ˜+๐œ€ฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚ถ+๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐ด๐œ•๐œ‰0๐œ‰+โˆ’๐‘˜๎‚ถ+ฬ‚๐‘ฆ๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘Ÿโˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—)+ฬ‡ฮฆ๎ƒญ๎ƒฐ๐‘‡ฮ“โˆ’1ฮฆโˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๐œ€+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ.๐‘ƒ๐œ€(4.15) Specify the control input as 1ฬ‚๐‘ข=๐‘ง๐‘Ÿ๐‘ค2โ„“๎ƒฏ๐‘ง๐‘Ÿ๐‘˜๐‘Ÿ๐‘ฃโ„“,1โˆ’๐‘ง๐‘Ÿ๐‘ค1โ„“+๐‘ง๐‘Ÿ๎ƒฌ๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐œ‰๐œ•ฬ‚๐‘ฆ12+๐‘ฃโ„“,2๎‚๐œ™โ„“+โˆ’๐œ”๐‘‡๎‚ฮ˜๎‚ถ+๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ต๐ด๐œ•๐œ‰0๐œ‰+โˆ’๐‘˜๎‚ถ+ฬ‚๐‘ฆ๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๎‚ฮ˜ฬ‡๎‚ฮ˜โ„“๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•๐‘ฃ๐‘—๎€ท๐ด0๐‘ฃ๐‘—+๐‘’๐‘Ÿ๐œŽ๐‘—๎€ธ+๐‘Ÿโˆ’1๎“๐‘—=1๐œ•๐›ผ๐‘Ÿโˆ’1๐œ•ฬ‚๐‘ฆ๐‘š(๐‘—โˆ’1)ฬ‚๐‘ฆ๐‘š(๐‘—)๎ƒญโˆ’๐‘๐‘Ÿ๐‘ง2๐‘Ÿโˆ’๐‘‘๐‘Ÿ๎‚ต๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘Ÿโˆ’๐‘”๐‘Ÿ๎‚ต๐œ•๐›ผ๐‘Ÿโˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘Ÿ+๐‘ง๐‘Ÿ๎๐‘…๎ƒฐ,ฬ‚๐‘ข(4.16) where ๎๐‘…ฬ‚๐‘ข is an addition input which will be used to target on rejection of uncertainties.
Substituting (4.16) into ฬ‡๐‘‰๐‘Ÿ, we have ฬ‡๐‘‰๐‘Ÿ=โˆ’๐‘Ÿ๎“๐‘—=1๎ƒฉ๐‘๐‘—๐‘ง2๐‘—+๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—+๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—๎ƒช+๐œ๐‘Ÿโˆ’1ฮฆโˆ’๐œ•๐›ผ๐‘Ÿโˆ’1๎‚€๐‘ง๐œ•ฬ‚๐‘ฆ๐‘Ÿ๐‘ฃโ„“,1ฮฆโ„“+๐‘ง๐‘Ÿ๐œ”๐‘‡ฮฆ๎‚+๐‘ง๐‘Ÿ๎๐‘…โˆ’ฬ‚๐‘ข๐‘Ÿ๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1๎‚€๐œ€๐œ•ฬ‚๐‘ฆฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚+ฬ‡ฮฆ๐‘‡ฮ“โˆ’1ฮฆโˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๐œ€+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ.๐‘ƒ๐œ€(4.17) Write ๐œ๐‘Ÿฮฆ=๐œ๐‘Ÿโˆ’1ฮฆโˆ’(๐œ•๐›ผ๐‘Ÿโˆ’1/๐œ•ฬ‚๐‘ฆ)(๐‘ง๐‘Ÿ๐‘ฃโ„“,1ฮฆโ„“+๐‘ง๐‘Ÿ๐œ”๐‘‡ฮฆ) and we arrive at ฬ‡๐‘‰๐‘Ÿ=โˆ’๐‘Ÿ๎“๐‘—=1๎ƒฉ๐‘๐‘—๐‘ง2๐‘—+๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—+๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—๎ƒช+๎‚€๐œ๐‘Ÿ+ฬ‡ฮฆ๐‘‡ฮ“โˆ’1๎‚ฮฆ+๐‘ง๐‘Ÿ๎๐‘…โˆ’ฬ‚๐‘ข๐‘Ÿ๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1๎‚€๐œ€๐œ•ฬ‚๐‘ฆฬ‚๐‘ง12+๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๐œ€+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ.๐‘ƒ๐œ€(4.18) From (4.18), we may specify the parameter update law in order to cancel the term (๐œ๐‘Ÿ+ฬ‡ฮฆ๐‘‡ฮ“โˆ’1)ฮฆ. To guarantee the estimated parameters will always lie within allowable region ๐‘ค, a projected parametric update law will be specified as ฬ‡๎‚๎‚ปฮ˜=ฮ“๐œ๐‘‡๐‘Ÿ๎‚ifฮ˜โˆˆ๐‘ค0,๐‘ƒ๐‘…๎€ทฮ“๐œ๐‘‡๐‘Ÿ๎€ธ๎ifฮ˜โˆˆ๐œ•๐‘ค,๐œ๐‘Ÿฮ“๎‚ฮ˜perp>0,(4.19) where ๐‘ค is the allowable parameter variation set (compact and convex) with its interior and boundary denoted by ๐‘ค0 and ๐œ•๐‘ค, respectively. If the current estimated parameter vector lies within the allowable parameter variation set, normal update law is employed. If the current estimated parameter vector lies on the boundary of the allowable parameter variation set, projected update law denoted by ๐‘ƒ๐‘…(โ‹…) is employed to prevent the parameter vector from leaving the variation set. With (4.19), (4.18) can be written as ฬ‡๐‘‰๐‘Ÿ=โˆ’๐‘Ÿ๎“๐‘—=1๎ƒฉ๐‘๐‘—๐‘ง2๐‘—+๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—๎ƒชโˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘–+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2+๐‘ง๐‘Ÿ๎๐‘…โˆ’ฬ‚๐‘ข๐‘Ÿ๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1๎‚€๎๐‘‘๐œ•ฬ‚๐‘ฆ๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ๎‚โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ๐‘ƒ๐œ€โ‰คโˆ’๐‘Ÿ๎“๐‘—=1๎ƒฉ๐‘๐‘—๐‘ง2๐‘—+๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—๎ƒชโˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2+๐‘ง๐‘Ÿ๎๐‘…โˆ’ฬ‚๐‘ข๐‘Ÿ๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1|||๎๐‘‘๐œ•ฬ‚๐‘ฆ๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ.๐‘ƒ๐œ€(4.20) Add and subtract terms โˆ‘๐‘Ÿ๐‘—=1(1/4๐‘”๐‘—๎๐‘‘)|๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|2; we have ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐‘๐‘—๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2+๐‘ง๐‘Ÿ๎๐‘…โˆ’ฬ‚๐‘ข๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๎‚ถ๐œ•ฬ‚๐‘ฆ2๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘ง๐‘—๐œ•๐›ผ๐‘—โˆ’1|||๎๐‘‘๐œ•ฬ‚๐‘ฆ๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||โˆ’๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธโˆ’๐‘ƒ๐œ€๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ.(4.21) Moreover, we obtain ฬ‡๐‘‰๐‘–โ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐‘๐‘—๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘–๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธโˆ’๐‘ƒ๐œ€๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘ง๐‘Ÿ๎๐‘….ฬ‚๐‘ข(4.22) As shown in Figure 1, the tracking error ๐‘1(ฬƒ๐‘ ) and the control input ๎๐‘ˆ๎๐‘…(ฬƒ๐‘ ) are related by ๎๐‘ˆ๎๐‘…๎๎(ฬƒ๐‘ )=โˆ’๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘1(ฬƒ๐‘ ),(4.23) where we have chosen ๎๐‘…(ฬƒ๐‘ ) as a low-order and attenuated-type internal model filter, that is, ๎๐‘…(ฬƒ๐‘ )=๐‘˜๎‘๐‘–=1ฬƒ๐‘ 2+2๐œ๐‘–๐œ”๐‘›๐‘–ฬƒ๐‘ +๐œ”2๐‘›๐‘–ฬƒ๐‘ 2+2๐œ‰๐‘–๐œ”๐‘›๐‘–ฬƒ๐‘ +๐œ”2๐‘›๐‘–,(4.24) where k is the number of periodic frequencies to be rejected, ๐œ”๐‘›๐‘– is determined based on the ith disturbance frequency in rad/rev, and ๐œ‰๐‘– and ๐œ๐‘– are two damping ratios that satisfy 0<๐œ‰๐‘–<๐œ๐‘–<1. We can adjust the gain of ๎๐‘…(ฬƒ๐‘ ) at those periodic frequencies by varying the values of ๐œ‰๐‘– and ๐œ๐‘–.

Theorem 4.1. Consider the control law of (4.16) and (4.23) applied to a nonlinear system with unmodeled dynamics, parameter uncertainty and subject to output disturbance as given by (3.1). Assume that ฬ‚๐‘ฆ๐‘š,ฬ‡ฬ‚๐‘ฆ๐‘š,โ€ฆ,ฬ‚๐‘ฆ๐‘š(๐‘Ÿ) (where ๐‘Ÿ is the relative degree) and ๎๐‘‘๐‘ฆ,ฬ‡๎๐‘‘๐‘ฆ๎๐‘‘,โ€ฆ,๐‘ฆ(๐‘Ÿ) are known and bounded, ๎๐‘‘(๐‘Ÿโˆ’1)๐‘ ๐‘–1,๎๐‘‘(๐‘Ÿโˆ’2)๐‘ ๐‘–2ฬ‡๎๐‘‘,โ€ฆ,๐‘ ๐‘–๐‘Ÿโˆ’1 are sufficiently smooth, ๐‘“,๐‘”,โ„Ž,๐ฟ๐‘Ÿ๐‘“โ„Ž,๐ฟ๐‘”๐ฟ๐‘“๐‘Ÿโˆ’1โ„Ž are Lipschitz continuous functions, at least one column of ๐‘Š(ฬ‚๐‘ฆ) is bounded away from zero. Furthermore, suppose that a loop-shaping filter ๎๐ถ(ฬƒ๐‘ ) is designed such that the feedback system is stable. Then the modified parameter update law as given by (4.19) yields the bounded tracking error.

Proof. Step 1 (show that only ๎‚ฮ˜โˆˆw0 needs to be considered).
Denote by ฬ‡๎‚ฮ˜perp the component of ฬ‡๎‚ฮ˜ perpendicular to the tangent plane at ๎‚ฮ˜ so that ฬ‡๎‚ฮ˜=๐‘ƒ๐‘…(ฬ‡๎‚ฬ‡๎‚ฮ˜ฮ˜)+perp. Since ฮ˜โˆˆ๐‘ค and ๐‘ค is convex, we have (๎‚ฮ˜โˆ’ฮ˜)๐‘‡ฬ‡๎‚ฮ˜perpโ‰ฅ0. Choose Lyapunov function ๐‘‰(ฮฆ)=ฮฆ๐‘‡ฮฆ and use the parameter update law as defined in (4.19). When ๎‚ฮ˜โˆˆ๐‘ค0, we have ฬ‡๐‘‰=โˆ’ฮฆ๐‘‡ฬ‡๎‚ฮ˜. When ๎‚ฮ˜โˆˆ๐œ•๐‘ค, we have ฬ‡๐‘‰=โˆ’ฮฆ๐‘‡๐‘ƒ๐‘…๎‚€ฬ‡๎‚ฮ˜๎‚=โˆ’ฮฆ๐‘‡๎‚€ฬ‡๎‚ฬ‡๎‚ฮ˜ฮ˜โˆ’perp๎‚=โˆ’ฮฆ๐‘‡ฬ‡๎‚ฮ˜+ฮฆ๐‘‡ฬ‡๎‚ฮ˜perpโ‰คโˆ’ฮฆ๐‘‡ฬ‡๎‚ฮ˜,(4.25) where we use the fact that ฮฆ๐‘‡ฬ‡๎‚ฮ˜perp=๎‚€๎‚ฮ˜๎‚ฮ˜โˆ’๐‘‡ฬ‡๎‚ฮ˜perp๎‚€๎‚๎‚=โˆ’ฮ˜โˆ’ฮ˜๐‘‡ฬ‡๎‚ฮ˜perpโ‰ค0.(4.26) Thus, we only have to consider the scenario corresponding to ๎‚ฮ˜โˆˆ๐‘ค0 in the sequel.
Step 2. Substituting (4.23) back into (4.22), we haveฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐‘๐‘—๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธโˆ’๐‘ƒ๐œ€๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถโˆ’๐‘ง๐‘Ÿ๎๎๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘ง1.(4.27) Using the definition of tracking error ๐‘ง1=ฬ‚๐‘ฆโˆ’ฬ‚๐‘ฆ๐‘š=(๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š)+๐œ€ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ, (4.27) can be written as ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐‘๐‘—๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ+||๐‘ง๐‘ƒ๐œ€๐‘Ÿ๎๎๎€ท๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ||+|||๐‘ง๐‘Ÿ๎๎๎‚€๐œ€๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚|||.(4.28) Use the following equality: ฬ‚๐‘ง๐‘Ÿ๎๎๎‚€๐œ€๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚โ‰ค๐›พ2ฬ‚๐‘ง2๐‘Ÿ+๎‚ต1๎๎๎‚€๐œ€2๐›พ๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2,๐›พ>0isdesignable.(4.29) Equation (4.28) becomes ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿโˆ’1๎“๐‘—=1๐‘๐‘—๐‘ง2๐‘—โˆ’๐‘๎…ž๐‘Ÿ๐‘ง2๐‘Ÿโˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ+||๐‘ง๐‘ƒ๐œ€๐‘Ÿ๎๎๎€ท๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ||+๎‚ต1๎๎๎‚€๐œ€2๐›พ๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ,(4.30) where ๐‘๎…ž๐‘Ÿ=๐‘๐‘Ÿโˆ’๐›พ2>0. Moreover, the positive designable parameters ๐‘๐‘– can be written as ๐‘๐‘—=๐ถ๐‘—+โ„Ž๐‘—๐‘,๐‘—=1,โ€ฆ,๐‘Ÿโˆ’1,๎…ž๐‘Ÿ=๐ถ๐‘Ÿ+โ„Ž๐‘Ÿ,(4.31) where ๐ถ๐‘—,๐ถ๐‘Ÿ and โ„Ž๐‘—,โ„Ž๐‘Ÿ>0. Thus, (4.30) can be written as ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿโˆ’1๎“๐‘—=1๎€ท๐ถ๐‘—+โ„Ž๐‘—๎€ธ๐‘ง2๐‘—โˆ’๎€ท๐ถ๐‘Ÿ+โ„Ž๐‘Ÿ๎€ธ๐‘ง2๐‘Ÿโˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ+|||๐‘ง๐‘ƒ๐œ€๐‘Ÿ๎๎๎‚€๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )โˆ’๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎‚|||+๎‚ต1๎๐‘…๎๐ถ๎‚€๐œ€2๐›พ(ฬƒ๐‘ )(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2.(4.32) Utilizing the fact that ๎‚€โˆšโ„Ž1||๐‘ง1||โˆ’โˆšโ„Ž๐‘Ÿ||๐‘ง๐‘Ÿ||๎‚2=๐‘ง21+โ„Ž๐‘Ÿ๐‘ง2๐‘Ÿโˆ’โˆšโ„Ž1โ„Ž๐‘Ÿ|||๐‘ง๐‘Ÿ๎€ท๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ+๐‘ง๐‘Ÿ๎‚€๐œ€ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚|||,(4.33) we have ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿโˆ’1๎“๐‘—=2๎€ท๐ถ๐‘—+โ„Ž๐‘—๎€ธ๐‘ง2๐‘—โˆ’๐ถ1๐‘ง21โˆ’๐ถ๐‘Ÿ๐‘ง2๐‘Ÿโˆ’๎‚€โˆšโ„Ž1||๐‘ง1||โˆ’โˆšโ„Ž๐‘Ÿ||๐‘ง๐‘Ÿ||๎‚2โˆ’โˆšโ„Ž1โ„Ž๐‘Ÿ|||๐‘ง๐‘Ÿ๎€ท๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ+๐‘ง๐‘Ÿ๎‚€๐œ€ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚|||โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถ+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ+||๐‘ง๐‘ƒ๐œ€๐‘Ÿ๎๎๎€ท๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ||+๎‚ต1๎๎๎‚€๐œ€2๐›พ๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2.(4.34) To design โ„Ž1,โ€ฆ,โ„Ž๐‘Ÿ (or ๐‘1,โ€ฆ,๐‘๐‘Ÿ ), ๐‘‘๐‘— and ๐‘”๐‘— such that โˆ’๐‘Ÿ๎“๐‘—=1โ„Ž๐‘—๐‘ง2๐‘—โˆ’๐‘Ÿ๎“๐‘—=1๐‘‘๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘‘๐‘—๐œ€ฬ‚๐‘ง12๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=1๐‘”๐‘—๎‚ต๐œ•๐›ผ๐‘—โˆ’1๐‘ง๐œ•ฬ‚๐‘ฆ๐‘—+12๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||๎‚ถ2โˆ’๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎‚ต๐œ€2ฬ‚๐‘ง11+๐œ€2ฬ‚๐‘ง13+โ‹ฏ+๐œ€2ฬ‚๐‘ง1๐‘Ÿ๎‚ถโˆ’๎‚€โˆšโ„Ž1||๐‘ง1||โˆ’โˆšโ„Ž๐‘Ÿ||๐‘ง๐‘Ÿ||๎‚2โˆ’โˆšโ„Ž1โ„Ž๐‘Ÿ|||๐‘ง๐‘Ÿ๎€ท๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ+๐‘ง๐‘Ÿ๎‚€๐œ€ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚|||+||๐‘ง๐‘Ÿ๎๎๎€ท๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )๐‘ง11โˆ’ฬ‚๐‘ฆ๐‘š๎€ธ||โ‰ค0,(4.35) We arrive at ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐ถ๐‘—๐‘ง2๐‘—+๎‚ต1๎๎๎‚€๐œ€2๐›พ๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๎€ท๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡๎€ธ.๐‘ƒ๐œ€(4.36) Equation (4.36) implies that ฬ‡๐‘‰๐‘Ÿโ‰คโˆ’๐‘Ÿ๎“๐‘—=1๐ถ๐‘—๐‘ง2๐‘—โˆ’๎ƒฉ12ฮฆ๐‘‡ฮ“โˆ’1ฮฆ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๎ƒช+๎ƒฉ1๐‘ƒ๐œ€2ฮฆ๐‘‡ฮ“โˆ’1ฮฆ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๎ƒช+๎‚ต1๐‘ƒ๐œ€๎๐‘…๎๐ถ๎‚€๐œ€2๐›พ(ฬƒ๐‘ )(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—||๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡||๐‘ƒ๐œ€โ‰คโˆ’2๐‘˜๐‘ฃ๐‘‰๐‘Ÿ+๐ถ,(4.37) where ๐‘˜๐‘ฃโ‰œmin{๐ถ1,โ€ฆ,๐ถ๐‘Ÿ,๐œ†min(ฮ“)}, ๐œ†min(ฮ“) is the smallest eigenvalue of ฮ“ and 1๐ถ=2ฮฆ๐‘‡ฮ“โˆ’1ฮฆ+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—๐œ€๐‘‡๎‚ต1๐‘ƒ๐œ€+๎๎๎‚€๐œ€2๐›พ๐‘…(ฬƒ๐‘ )๐ถ(ฬƒ๐‘ )ฬ‚๐‘ง11โˆ’๎๐‘‘๐‘ฆ๎‚๎‚ถ2+๐‘Ÿ๎“๐‘—=114๐‘”๐‘—|||๎๐‘‘๐‘ ๐‘–1+ฬ‡๎๐‘‘๐‘ฆ|||2+๐‘Ÿ๎“๐‘—=114๐‘‘๐‘—||๐œ€๐‘‡๐‘ƒฮ”+ฮ”๐‘‡||๐‘ƒ๐œ€(4.38) is bounded since ๐œ€ฬ‚๐‘ง11,๎๐‘‘๐‘ฆ are bounded and ฮฆ๐‘‡ฮ“โˆ’1ฮฆ is bounded due to the parameter update law specified in (4.19). We conclude that ๐‘‰๐‘Ÿโ‰ค๐‘’โˆ’2๐‘˜๐‘ฃ๐œƒ๐‘‰๐‘Ÿ๎€œ(0)+๐œƒ0๐ถ๐‘’โˆ’2๐‘˜๐‘ฃ(๐œƒโˆ’๐œ)๐‘‘๐œโ‰ค๐‘’โˆ’2๐‘˜๐‘ฃ๐œƒ๐‘‰๐‘Ÿ๎€ท(0)+1โˆ’๐‘’โˆ’2๐‘˜๐‘ฃ๐œƒ๎€ธ๐ถ2๐‘˜๐‘ฃ.(4.39) As ๐œƒโ†’โˆž, we have ๐‘‰๐‘Ÿ(๐ถโˆž)โ‰ค2๐‘˜๐‘ฃ,(4.40) which implies that the overall system is stable and the bound ๐ถ/(2๐‘˜๐‘ฃ) can be decreased by increasing ๐‘˜๐‘ฃ or increasing ๐›พ. By (4.14), this implies that ๎‚๐‘ง,ฮ˜,๐œ€ are bounded. Since ๐‘ง1=ฬ‚๐‘ฆโˆ’ฬ‚๐‘ฆ๐‘š, ฬ‚๐‘ฆ is also bounded. From (3.11), we can see that ๐œ‰ and ๐‘ฃ1,โ€ฆ,๐‘ฃโ„“ are bounded since ๐‘Š๐‘“(ฬ‚๐‘ฆ) and ๐‘Š๐‘”(ฬ‚๐‘ฆ) are bounded. Moreover, we conclude that the virtual inputs ๐›ผ are bounded because they consist of bounded terms. Also, ๐‘ง1 is bounded from (3.12) and also ฬ‚๐‘ง1 from the definition of ๐œ€. With the ISS assumption and bounded ฬ‚๐‘ง1, we conclude that the internal dynamics ฬ‚๐‘ง2 is bounded. Finally, ฬ‚๐‘ฅ is bounded by diffeomorphism, that is, ฬ‚๐‘ฅ=๐‘‡โˆ’1(ฬ‚๐‘ง).

5. Illustrative Example

For realistic simulation, we set up a simulation configuration as shown in Figure 2, in which the controller and parametric adaptation operate in the ๐œƒ-domain whereas the open-loop system operates in the time domain. The proposed spatial domain output feedback adaptive control scheme is applied to a reformulated system in spatial domain expressed as ฬ‡๎๐‘‘ฬ‚๐‘ฅ=๐‘“(ฬ‚๐‘ฅ)+๐‘”(ฬ‚๐‘ฅ)ฬ‚๐‘ข+๐‘ ๎๐‘‘,ฬ‚๐‘ฆ=โ„Ž(ฬ‚๐‘ฅ)+๐‘ฆ,(5.1) where ๐‘“โŽกโŽขโŽขโŽขโŽฃ(ฬ‚๐‘ฅ)=โˆ’๐‘Ž1+ฬ‚๐‘ฅ2ฬ‚๐‘ฅ1โˆ’๐‘Ž0โŽคโŽฅโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽขโŽฃ0๐‘,๐‘”(ฬ‚๐‘ฅ)=0ฬ‚๐‘ฅ1โŽคโŽฅโŽฅโŽฅโŽฆ,โ„Ž(ฬ‚๐‘ฅ)=ฬ‚๐‘ฅ1,(5.2) with ๐‘Ž0=5155, ๐‘Ž1=1138, and ๐‘0=140368. For verification purpose, the output disturbance is assumed to be a low-pass rectangular periodic signal (with amplitude switching between โˆ’0.1 and 0.1) (see Figure 3), that is, ๎๐‘‘๐‘ฆ(๐œƒ)=0.1๎ƒฌ0.0125ฬƒ๐‘ +1โˆž๎“๐‘™=โˆ’โˆž(โˆ’1)๐‘™๎ƒญ+ฮ (๐œƒโˆ’1โˆ’๐‘™)10(0.005ฬƒ๐‘ +1)2๐‘0,(5.3) where โŽงโŽชโŽจโŽชโŽฉ1||๐œƒ||||๐œƒ||ฮ (๐œƒ)=<1,0.5=1,0otherwise.(5.4) Note that the disturbance has been low-pass filtered so that it is continuously differentiable. Parameters of the internal model filter are specified to target the fundamental frequency and the first three harmonic frequencies of the periodic disturbance, that is, ๎๐‘…(ฬƒ๐‘ )=4๎‘๐‘–=1ฬƒ๐‘ 2+2๐œ๐‘–๐œ”๐‘›๐‘–ฬƒ๐‘ +๐œ”2๐‘›๐‘–ฬƒ๐‘ 2+2๐œ‰๐‘–๐œ”๐‘›๐‘–ฬƒ๐‘ +๐œ”2๐‘›๐‘–,(5.5) where ๐œ๐‘–=0.2,๐œ‰๐‘–๐œ”=0.0002,๐‘›1=0.25๐œ‹,๐œ”๐‘›2=3ร—0.25๐œ‹,๐œ”๐‘›3=5ร—0.25๐œ‹,๐œ”๐‘›4=7ร—0.25๐œ‹,(5.6) Furthermore, the stabilizing filter is specified as ๎๐ถ(ฬƒ๐‘ )=100000(ฬƒ๐‘ /100+1)(.ฬƒ๐‘ /10000+1)(5.7) The parameters of the ๐พ-filter are set to ๐‘˜1=1600 and ๐‘˜2=100. The initial values of the estimated parameters are set to ฬƒ๐‘Ž0=1500, ฬƒ๐‘Ž1=500, and ฬƒ๐‘0=1000000. The allowable parameter variation sets are ฬƒ๐‘Ž0โˆˆฮฉฬƒ๐‘Ž0โ‰œ๎€ฝฬƒ๐‘Ž0โˆถ100โ‰คฬƒ๐‘Ž0๎€พ,โ‰ค10000ฬƒ๐‘Ž1โˆˆฮฉฬƒ๐‘Ž1โ‰œ๎€ฝฬƒ๐‘Ž1โˆถ10โ‰คฬƒ๐‘Ž1๎€พ,ฬƒ๐‘โ‰ค100000ฬƒ๐‘โˆˆฮฉ0โ‰œ๎€ฝฬƒ๐‘0ฬƒ๐‘โˆถ10000โ‰ค0๎€พ.โ‰ค10000000(5.8) Note that ๐‘‘๐‘ (๐‘ก) is set to 0 so that the system performance is not affected by the unstructured uncertainty. Suppose that a variable speed control task demands the system to initially run at 30โ€‰rev/s and then speed up to 35โ€‰rev/s and finally speed down to 25โ€‰rev/s (see Figure 4). To avoid getting infinite value when taking derivative, the reference command is specified to have smooth (instead of instant) change. Figure 5 compares the tracking performance of two scenarios. The figures on the left are for the pure output feedback adaptive backstepping design. The ones on the right are for the proposed output feedback design with internal model control. Without internal model control, the adaptive backstepping design has already shown superb tracking performance. We see that adding the internal model control further reduces the magnitude of the tracking error without noticeable increase in the control input.

6. Conclusion

This paper presents the design of a new spatial domain adaptive control system, which can be applied to rotary systems operating at varying speeds and subject to spatially periodic and band-limited disturbances and structured/unstructured parametric uncertainties. The proposed design integrates two control paradigms, that is, adaptive backstepping and internal model control. The overall output feedback adaptive control system can be shown to be stable and have bounded state estimated error and output tracking error. Feasibility and effectiveness of the proposed design are further justified by a numerical example. Future effort will be dedicated to implementation and verification of the proposed control design to a practical rotary system, for example, a brushless dc-motor-driven control system.

References

  1. B. A. Francis and W. M. Wonham, โ€œThe internal model principle of control theory,โ€ Automatica, vol. 12, no. 5, pp. 457โ€“465, 1976. View at: Google Scholar
  2. V. O. Nikiforov, โ€œAdaptive non-linear tracking with complete compensation of unknown disturbances,โ€ European Journal of Control, vol. 4, pp. 132โ€“139, 1998. View at: Google Scholar
  3. Z. Ding, โ€œAdaptive disturbance rejection of nonlinear systems in an extended output feedback form,โ€ IET Control Theory & Applications, vol. 1, no. 1, pp. 298โ€“303, 2007. View at: Publisher Site | Google Scholar
  4. L. Marconi, A. Isidori, and A. Serrani, โ€œInput disturbance suppression for a class of feedforward uncertain nonlinear systems,โ€ Systems & Control Letters, vol. 45, no. 3, pp. 227โ€“236, 2002. View at: Publisher Site | Google Scholar
  5. L. Gentili and L. Marconi, โ€œRobust nonlinear disturbance suppression of a magnetic levitation system,โ€ Automatica, vol. 39, no. 4, pp. 735โ€“742, 2003. View at: Publisher Site | Google Scholar
  6. C. Kravaris, V. Sotiropoulos, C. Georgiou, N. Kazantzis, M. Xiao, and A. J. Krener, โ€œNonlinear observer design for state and disturbance estimation,โ€ Systems & Control Letters, vol. 56, no. 11-12, pp. 730โ€“735, 2007. View at: Publisher Site | Google Scholar
  7. B. Mahawan and Z.-H. Luo, โ€œRepetitive control of tracking systems with time-varying periodic references,โ€ International Journal of Control, vol. 73, no. 1, pp. 1โ€“10, 2000. View at: Publisher Site | Google Scholar
  8. C. L. Chen, G. T. C. Chiu, and J. Allebach, โ€œRobust spatial-sampling controller design for banding reduction in electrophotographic process,โ€ Journal of Imaging Science and Technology, vol. 50, no. 6, pp. 530โ€“536, 2006. View at: Publisher Site | Google Scholar
  9. C. L. Chen and G. T. C. Chiu, โ€œSpatially periodic disturbance rejection with spatially sampled robust repetitive control,โ€ Journal of Dynamic Systems, Measurement, and Control, vol. 130, no. 2, pp. 11โ€“21, 2008. View at: Google Scholar
  10. M. Nakano, J. H. She, Y. Mastuo, and T. Hino, โ€œElimination of position-dependent disturbances in constant-speed-rotation control systems,โ€ Control Engineering Practice, vol. 4, no. 9, pp. 1241โ€“1248, 1996. View at: Publisher Site | Google Scholar
  11. C. L. Chen and Y. H. Yang, โ€œSpatially periodic disturbance rejection for uncertain rotational motion systems using spatial domain adaptive backstepping repetitive control,โ€ in Proceedings of the 33rd Annual Conference of the IEEE Industrial Electronics Society, pp. 638โ€“643, Taipei, Taiwan, November 2007. View at: Google Scholar
  12. Y. H. Yang and C. L. Chen, โ€œSpatially periodic disturbance rejection using spatial-based output feedback adaptive backstepping repetitive control,โ€ in Proceeding of the American Control Conference, pp. 4117โ€“4122, 2008. View at: Google Scholar
  13. G. Kreisselmeier, โ€œAdaptive observers with exponential rate of convergence,โ€ IEEE Transactions on Automatic Control, vol. 22, no. 1, pp. 2โ€“8, 1977. View at: Publisher Site | Google Scholar
  14. R. Marino, G. L. Santosuosso, and P. Tomei, โ€œRobust adaptive observers for nonlinear systems with bounded disturbances,โ€ IEEE Transactions on Automatic Control, vol. 46, no. 6, pp. 967โ€“972, 2001. View at: Publisher Site | Google Scholar
  15. S. C. Tong, X. L. He, and H. G. Zhang, โ€œA combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control,โ€ IEEE Transactions on Fuzzy Systems, vol. 17, no. 5, pp. 1059โ€“1069, 2009. View at: Publisher Site | Google Scholar
  16. S. Tong and Y. Li, โ€œObserver-based fuzzy adaptive control for strict-feedback nonlinear systems,โ€ Fuzzy Sets and Systems, vol. 160, no. 12, pp. 1749โ€“1764, 2009. View at: Publisher Site | Google Scholar
  17. S. Tong, C. Li, and Y. Li, โ€œFuzzy adaptive observer backstepping control for MIMO nonlinear systems,โ€ Fuzzy Sets and Systems, vol. 160, no. 19, pp. 2755โ€“2775, 2009. View at: Publisher Site | Google Scholar

Copyright © 2012 Yen-Hsiu Yang and Cheng-Lun Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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