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 [26]. Conventional controllers are mostly time-based controllers as they are synthesized and operate in temporal or time domain. Several researches [79] 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., 𝑒𝐿𝑠 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 Yang’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 𝐾-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 𝑛th 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 [1517]. 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, 𝑓̇𝑥(𝑡)=𝑡𝑥(𝑡),𝜙𝑓+Δ𝑓𝑡𝑥(𝑡),𝜙𝑓+𝑔𝑡𝑥(𝑡),𝜙𝑔+Δ𝑔𝑡𝑥(𝑡),𝜙𝑔𝑢(𝑡),𝑦=Ψ𝑥(𝑡)+𝑑𝑦(𝑡)=𝑥1(𝑡)+𝑑𝑦(𝑡),(2.1) where 𝑥(𝑡)=[𝑥1(𝑡)𝑥𝑛(𝑡)]𝑇, Ψ=[100], and 𝑢(𝑡) and 𝑦(𝑡) correspond to control input and measured output angular velocity of the system, respectively. 𝑑𝑦(𝑡) 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. 𝑓𝑡(𝑥(𝑡),𝜙𝑓) and 𝑔𝑡(𝑥(𝑡),𝜙𝑔) are known vector-valued functions with unknown but bounded system parameters, that is, 𝜙𝑓=[𝜙𝑓1𝜙𝑓𝑘] and 𝜙𝑔=[𝜙𝑔1𝜙𝑔𝑙]; Δ𝑓𝑡(𝑥(𝑡),𝜙𝑓) and Δ𝑔𝑡(𝑥(𝑡),𝜙𝑔) represent unstructured modeling inaccuracy, which are also assumed to be bounded. Instead of using time 𝑡 as the independent variable, consider an alternate independent variable 𝜃=𝜆(𝑡), that is, the angular displacement. Since by definition 𝜆(𝑡)=𝑡0𝜔(𝜏)𝑑𝜏+𝜆(0),(2.2) where 𝜔(𝑡) is the angular velocity, the following condition: 𝜔(𝑡)=𝑑𝜃𝑑𝑡>0,𝑡>0,(2.3) will guarantee that 𝜆(𝑡) is strictly monotonic such that 𝑡=𝜆1(𝜃) exists. Thus all the variables in the time domain can be transformed into their counterparts in the 𝜃-domain, that is, 𝜆̂𝑥(𝜃)=𝑥1𝜆(𝜃),̂𝑦(𝜃)=𝑦1,𝜆(𝜃)̂𝑢(𝜃)=𝑢1,𝜆(𝜃)𝑑(𝜃)=𝑑1,𝜆(𝜃)𝜔(𝜃)=𝜔1,(𝜃)(2.4) where we denote ̂ as the 𝜃-domain representation of . Note that, in practice, (2.3) can usually be satisfied for most rotary systems where the rotary component rotates only in one direction. Since 𝑑𝑥(𝑡)=𝑑𝑡𝑑𝜃𝑑𝑡𝑑̂𝑥(𝜃)𝑑𝜃=𝜔(𝜃)𝑑̂𝑥(𝜃).𝑑𝜃(2.5) Equation (2.1) can be rewritten as 𝜔(𝜃)𝑑̂𝑥(𝜃)=𝑓𝑑𝜃𝑡̂𝑥(𝜃),𝜙𝑓+Δ𝑓𝑡̂𝑥(𝜃),𝜙𝑓+𝑔𝑡̂𝑥(𝜃),𝜙𝑔+Δ𝑔𝑡̂𝑥(𝜃),𝜙𝑔𝑑̂𝑢(𝜃),̂𝑦(𝜃)=Ψ̂𝑥(𝜃)+𝑦(𝜃)=̂𝑥1𝑑(𝜃)+𝑦(𝜃).(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 𝜃 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 ̂𝑔(𝜃) in the angular displacement domain as 𝐺(̃𝑠)=0̂𝑔(𝜃)𝑒̃𝑠𝜃𝑑𝜃.(2.7) This definition will be useful for describing the linear portion of the overall control system.

3. Nonlinear State Observer

Drop the 𝜃 notation and note that (2.6) can be expressed as a standard nonlinear system: ̇̂𝑥=𝑓̂𝑥,𝜙𝑓+𝑔̂𝑥,𝜙𝑔𝑑̂𝑢+𝑠𝑑,̂𝑦=(̂𝑥)+𝑦𝑑=𝜔+𝑦,(3.1) where terms involving unstructured uncertainty are merged into 𝑑𝑠=Δ𝑓(̂𝑥,𝜙𝑓)+Δ𝑔(̂𝑥,𝜙𝑔)̂𝑢 with Δ𝑓̂𝑥,𝜙𝑓=Δ𝑓𝑡̂𝑥,𝜙𝑓̂𝑥1,Δ𝑔̂𝑥,𝜙𝑔=Δ𝑔𝑡̂𝑥,𝜙𝑔̂𝑥1.(3.2) In addition, we have 𝑓̂𝑥,𝜙𝑓=𝑓𝑡̂𝑥,𝜙𝑓̂𝑥1,𝑔̂𝑥,𝜙𝑔=𝑔𝑡̂𝑥,𝜙𝑔̂𝑥1,(̂𝑥)=𝜔=̂𝑥1.(3.3) The state variables have been specified such that the angular velocity 𝜔 is equal to ̂𝑥1, that is, the undisturbed output (̂𝑥). It is not difficult to verify that (3.1) has the same relative degree in 𝐷0={̂𝑥𝑛̂𝑥10} as the NTI model in (2.1). If (3.1) has relative degree 𝑟, we can define the following nonlinear coordinate transformation: 𝜓̂𝑧=𝑇(̂𝑥)=1𝜓(̂𝑥)𝑛𝑟(̂𝑥)𝐿(̂𝑥)𝑓𝑟1(̂𝑥)̂𝑧2̂𝑧1,(3.4) where 𝜓1 to 𝜓𝑛𝑟 are chosen such that 𝑇(̂𝑥) is a diffeomorphism on 𝐷0𝐷 and 𝐿𝑔𝜓𝑖(̂𝑥)=0,1𝑖𝑛𝑟,̂𝑥𝐷0.(3.5) With respect to the new coordinates, that is, ̂𝑧1 and ̂𝑧2, (3.1) can be transformed into the so-called normal form, that is, ̇̂𝑧2=𝐿𝑓𝜓||(̂𝑥)̂𝑥=𝑇1(̂𝑧)+𝑑𝑠𝑜Ψ̂𝑧1,̂𝑧2,̇̂𝑧1=𝐴𝑐̂𝑧1+𝐵𝑐𝐿𝑔𝐿𝑓𝑟1||(̂𝑥)̂𝑥=𝑇1(̂𝑧)𝐿̂𝑢+𝑟𝑓(̂𝑥)𝐿𝑔𝐿𝑓𝑟1|||||(̂𝑥)̂𝑥=𝑇1(̂𝑧)+𝑑𝑠𝑖,̂𝑦=𝐶𝑐̂𝑧1+𝑑𝑦,(3.6) where 𝑑𝑠𝑜 and 𝑑𝑠𝑖𝑑=[𝑠𝑖1𝑑𝑠𝑖𝑟]𝑇 come from 𝑑𝑠 going through the indicated coordinate transformation. ̂𝑧1=[̂𝑧11̂𝑧1𝑟]𝑟, ̂𝑧2𝑛𝑟, and (𝐴𝑐,𝐵𝑐,𝐶𝑐) is a canonical form representation of a chain of 𝑟 integrators. The first equation is called internal dynamics and the second is called external dynamics. Internal dynamics which is not affected by the control 𝑢. By setting ̂𝑧1=0 in that equation, we obtain ̇̂𝑧2=Ψ0,̂𝑧2,(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)𝑓(̂𝑥(𝜃),𝜙𝑓) and 𝑔(̂𝑥(𝜃),𝜙𝑔) are linearly related to those unknown system parameters, that is, 𝑓̂𝑥(𝜃),𝜙𝑓=𝜙1𝑓1(̂𝑥(𝜃))++𝜙𝑓𝑘𝑓𝑘𝑔(̂𝑥(𝜃)),̂𝑥(𝜃),𝜙𝑔=𝜙𝑔1𝑔1(̂𝑥(𝜃))++𝜙𝑔𝑙𝑔𝑙(̂𝑥(𝜃)).(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, ̇𝑑𝑦𝑑,,𝑦(𝑟) exist.(4)𝑑(𝑟1)𝑠𝑖1,𝑑(𝑟2)𝑠𝑖2̇𝑑,,𝑠𝑖𝑟1 exist, that is, the transformed unstructured uncertainty is sufficiently smooth.(5)The reference command ̂𝑦𝑚 and its first 𝑟 derivates are known and bounded. Moreover, the signal ̂𝑦𝑚(𝑟) is piecewise continuous.With assumption (2), we focus on designing a nonlinear state observer for external dynamics of (3.6), ̇̂𝑧1=𝐴𝑐̂𝑧1+𝐵𝑐𝐿𝑔𝐿𝑓𝑟1||(̂𝑥)̂𝑥=𝑇1(̂𝑧)𝐿̂𝑢+𝑟𝑓(̂𝑥)𝐿𝑔𝐿𝑓𝑟1|||||(̂𝑥)̂𝑥=𝑇1(̂𝑧)+𝑑𝑠𝑖.(3.9) Since 𝑓(̂𝑥) and 𝑔(̂𝑥) are linearly related to system parameters, 𝐿𝑔𝐿𝑓𝑟1(̂𝑥) and 𝐿𝑔𝐿𝑓𝑟1(̂𝑥) can be written as 𝐿𝑟𝑓(̂𝑥)=Θ𝑇𝑊𝑓(̂𝑥) and 𝐿𝑔𝐿𝑓𝑟1(̂𝑥)=Θ𝑇𝑊𝑔(̂𝑥), where 𝑊𝑓(̂𝑥) and 𝑊𝑔(̂𝑥) are two nonlinear functions, and Θ=[𝜙𝑓1𝜙𝑓𝑘𝜙𝑔1𝜙𝑔𝑙]𝑇=[𝜙1𝜙]𝑇, where is the number of unknown parameters. Next, we adopt the following observer structure: ̇𝑧1=𝐴0𝑧1+𝑘𝑦+𝐹(𝑦,𝑢)𝑇Θ, where 𝑧1=[𝑧11𝑧1𝑟]𝑇 is the estimate of 𝑧1, and 𝑊𝑓(𝑦) and 𝑊𝑔(𝑦) are nonlinear functions with the same structure as 𝑊𝑓(𝑥) and 𝑊𝑔(𝑥) except that each entry of 𝑥 is replaced by 𝑦. Furthermore, 𝐴0=𝑘1𝑘𝑟𝐼(𝑟1)×(𝑟1)01×(𝑟1),𝑘𝑘=1𝑘𝑟𝑇,𝐹(𝑦,𝑢)𝑇=0(𝑟1)×𝑊𝑇𝑓(𝑦)+𝑊𝑇𝑔(𝑦)𝑢𝑟×.(3.10) By properly choosing 𝑘, the matrix 𝐴0 can be made Hurwitz. Define the state estimated error as 𝜀[𝜀𝑧11𝜀𝑧1𝑟]𝑇𝑧1𝑧1. The dynamics of the estimated error can be obtained as ̇𝜀=𝐴0𝜀+Δ, where Δ=𝑘𝑑𝑦+𝐵𝑐Θ𝑇[𝑊𝑔(𝑥)𝑊𝑔(𝑦)]𝑢+𝐵𝑐Θ𝑇[𝑊𝑓(𝑥)𝑊𝑓(𝑦)]+𝑑𝑠𝑖. To proceed, the role of the state observer is replaced by 𝑧1𝜉+Ω𝑇Θ and the following two 𝐾-filters: ̇𝜉=𝐴0𝜉+̇Ω𝑘𝑦,𝑇=𝐴0Ω𝑇+𝐹(𝑦,𝑢)𝑇,(3.11) such that 𝜉=[𝜉11𝜉1𝑟]𝑇𝑟 and Ω𝑇[𝑣1𝑣]𝑟×. Decompose the second equation of (3.11) into ̇𝑣𝑗=𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗,𝑗=1,2,,,   where 𝑒𝑟=[001]𝑟 and 𝜎𝑗=𝑤1𝑗+𝑤2𝑗𝑢 with 𝑤1𝑗 and 𝑤2𝑗 are the 𝑗th columns of 𝑊𝑇𝑓(𝑦) and 𝑊𝑇𝑔(𝑦), respectively. With the definition of the state estimated error 𝜀, the state estimate 𝑧1, and (3.11), we acquire the following set of equations which will be used in the subsequent design: 𝑧1𝑘=𝑧1𝑘+𝜀𝑧1𝑘=𝜉1𝑘+𝑗=1𝑣𝑗,𝑘𝜙𝑗+𝜀𝑧1𝑘,𝑘=1,,𝑟,(3.12) where 𝑗,𝑖 denotes the ith row of 𝑗.

4. Spatial Domain Output Feedback Adaptive Control System

To apply adaptive backstepping method, we firstly rewrite the derivative of output ̂𝑦 as ̇̇̂𝑦=̂𝑧11+̇𝑑𝑦=̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦=𝑧12+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦.(4.1) With the second equation in (3.12), (4.1) can be written as ̇̂𝑦=𝑧12+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦=𝜉12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦,(4.2) where 𝜔𝑇=[𝑣1,2𝑣1,20].

In view of designing output feedback backstepping with 𝐾-filters, we need to find a set of 𝐾-filter parameters, that is, 𝑣,2,,𝑣1,2, separated from ̂𝑢 by the same number of integrators between ̂𝑧12 and ̂𝑢. From (3.11), we can see that 𝑣,2,,𝑣1,2 are all candidates if 𝑤2𝑗 are not zero. In the following derivation, we assume that 𝑣,2 is selected. Hence, the system incorporating the 𝐾-filters can be expressed as ̇̂𝑦=𝜉12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦,̇𝑣,𝑖=𝑣,𝑖+1𝑘𝑖𝑣,1̇𝑣,𝑖=2,,𝑟1,,𝑟=𝑘𝑟𝑣,1+𝑤1+𝑤2̂𝑢.(4.3) To apply adaptive backstepping to (4.3), a new set of coordinates will be introduced 𝑧1=̂𝑦̂𝑦𝑚,𝑧𝑖=𝑣,𝑖𝛼𝑖1,𝑖=2,,𝑟,(4.4) where ̂𝑦𝑚 is the prespecified reference command and 𝛼𝑖1 is the virtual input which will be used to stabilize each state equation. For simplicity, we define 𝜕𝛼0/𝜕̂𝑦1 for subsequent derivations.

Step 1 (𝑖=1). With (4.4), the first state equation of (4.3) can be expressed as ̇𝑧1=𝜉12+𝑧2𝜙+𝛼1𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦̇̂𝑦𝑚.(4.5) Consider a Lyapunov function 𝑉1=(1/2)𝑧21 and calculate its derivative ̇𝑉1=𝑧1̇𝑧1=𝑧1𝜉12+𝑧2𝜙+𝛼1𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦̇̂𝑦𝑚.(4.6) Define the estimates of 𝜙𝑖 as 𝜙𝑖 and Φ=[Φ1ΦΘ]=Θ, where 𝜙Θ=[𝑓1𝜙𝑓𝑘𝜙𝑔1𝜙𝑔𝑙]𝑇𝜙=[1𝜙]𝑇. Note that Θ is the “true” parameter vector while Θ is the estimated parameter vector. Design the virtual input 𝛼1 as 𝛼1=𝛼1/𝜙 and specify 𝛼1=1𝑧1𝑧1𝜉12𝑧1𝑧2𝜙𝑧1𝜔Θ+𝑧1̇̂𝑦𝑚𝑐1𝑧21𝑑1𝑧21𝑔1𝑧21=𝜉12𝑧2𝜙𝜔𝑇̇Θ+̂𝑦𝑚𝑐1𝑧1𝑑1𝑧1𝑔1𝑧1,(4.7) where 𝑐𝑖,𝑑𝑖,𝑔𝑖 are variables. Therefore, (4.6) becomes ̇𝑉1=𝑐1𝑧21𝑑1𝑧21𝑔1𝑧21+𝜏1Φ+𝑧1𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦,(4.8) where 𝜏1Φ=𝑧1𝑧2Φ+𝛼1Φ+𝑧1𝜔𝑇Φ.

Step 2 (𝑖=2,,𝑟1). With respect to the new set of coordinates (4.4), the second equation of (4.3) can be rewritten as ̇𝑧𝑖=𝑧𝑖+1+𝛼𝑖𝑘𝑖𝑣,1𝜕𝛼𝑖1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦+𝜕𝛼𝑖1𝐴𝜕𝜉0𝜉++𝑘̂𝑦𝜕𝛼𝑖1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑖1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑖1𝑗=1𝜕𝛼𝑖1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗)(4.9) Consider a Lyapunov function 𝑉𝑖=𝑖1𝑗=1𝑉𝑗+(1/2)𝑧2𝑖 and its derivative ̇𝑉𝑖=𝑖1𝑗=1̇𝑉𝑗+𝑧𝑖𝑧𝑖+1+𝛼𝑖𝑘𝑖𝑣,1𝜕𝛼𝑖1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦+𝜕𝛼𝑖1𝐴𝜕𝜉0𝜉++𝑘̂𝑦𝜕𝛼𝑖1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑖1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑖1𝑗=1𝜕𝛼𝑖1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗).(4.10) Specify 𝛼𝑖=1𝑧𝑖𝑧𝑖𝑧𝑖+1+𝑧𝑖𝑘𝑖𝑣,1+𝑧𝑖𝜕𝛼𝑖1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜕𝛼𝑖1𝐴𝜕𝜉0𝜉+𝑘+̂𝑦𝜕𝛼𝑖1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑖1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑖1𝑗=1𝜕𝛼𝑖1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗)𝑐𝑖𝑧2𝑖𝑑𝑖𝜕𝛼𝑖1𝜕̂𝑦2𝑧2𝑖𝑔𝑖𝜕𝛼𝑖1𝜕̂𝑦2𝑧2𝑖.(4.11) The derivative of 𝑉𝑖 becomes ̇𝑉𝑖=𝑖1𝑗=1𝑐𝑗𝑧2𝑗+𝑑𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝜏𝑖Φ𝑖1𝑗=1𝑧𝑗𝜕𝛼𝑗1𝜀𝜕̂𝑦̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦,(4.12) where 𝜏𝑖Φ=𝜏1Φ𝑖1𝑗=2(𝜕𝛼𝑗1/𝜕̂𝑦)(𝑧𝑗𝑣,1Φ+𝑧𝑗𝜔𝑇Φ).

Step 3. With respect to the new set of coordinates (4.4), the third equation of (4.3) can be written as ̇𝑧𝑟=𝑘𝑟𝑣,1+𝑤1+𝑤2̂𝑢𝜕𝛼𝑟1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦+𝜕𝛼𝑟1𝐴𝜕𝜉0𝜉+𝑘+̂𝑦𝜕𝛼𝑟1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑟1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑟1𝑗=1𝜕𝛼𝑟1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗).(4.13) The overall Lyapunov function may now be chosen as 𝑉𝑟=𝑟1𝑗=1𝑉𝑗+12𝑧2𝑟+12Φ𝑇Γ1Φ+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃𝜀,(4.14) where Γ is a symmetric positive definite matrix, that is, Γ=Γ𝑇>0. With the definition of state estimated error 𝜀, we can obtain that ̇𝑉𝑟=𝑟1𝑗=1̇𝑉𝑗+𝑧𝑟𝑘𝑟𝑣,1+𝑤1+𝑤2̂𝑢𝜕𝛼𝑟1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜀̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦+𝜕𝛼𝑟1𝐴𝜕𝜉0𝜉+𝑘+̂𝑦𝜕𝛼𝑟1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑟1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑟1𝑗=1𝜕𝛼𝑟1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗)+̇Φ𝑇Γ1Φ𝑟𝑗=114𝑑𝑗𝜀𝑇𝜀+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇.𝑃𝜀(4.15) Specify the control input as 1̂𝑢=𝑧𝑟𝑤2𝑧𝑟𝑘𝑟𝑣,1𝑧𝑟𝑤1+𝑧𝑟𝜕𝛼𝑟1𝜉𝜕̂𝑦12+𝑣,2𝜙+𝜔𝑇Θ+𝜕𝛼𝑟1𝐴𝜕𝜉0𝜉+𝑘+̂𝑦𝜕𝛼𝑟1𝜕Θ̇Θ𝑗=1𝜕𝛼𝑟1𝜕𝑣𝑗𝐴0𝑣𝑗+𝑒𝑟𝜎𝑗+𝑟1𝑗=1𝜕𝛼𝑟1𝜕̂𝑦𝑚(𝑗1)̂𝑦𝑚(𝑗)𝑐𝑟𝑧2𝑟𝑑𝑟𝜕𝛼𝑟1𝜕̂𝑦2𝑧2𝑟𝑔𝑟𝜕𝛼𝑟1𝜕̂𝑦2𝑧2𝑟+𝑧𝑟𝑅,̂𝑢(4.16) where 𝑅̂𝑢 is an addition input which will be used to target on rejection of uncertainties.
Substituting (4.16) into ̇𝑉𝑟, we have ̇𝑉𝑟=𝑟𝑗=1𝑐𝑗𝑧2𝑗+𝑑𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝜏𝑟1Φ𝜕𝛼𝑟1𝑧𝜕̂𝑦𝑟𝑣,1Φ+𝑧𝑟𝜔𝑇Φ+𝑧𝑟𝑅̂𝑢𝑟𝑗=1𝑧𝑗𝜕𝛼𝑗1𝜀𝜕̂𝑦̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦+̇Φ𝑇Γ1Φ𝑟𝑗=114𝑑𝑗𝜀𝑇𝜀+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇.𝑃𝜀(4.17) Write 𝜏𝑟Φ=𝜏𝑟1Φ(𝜕𝛼𝑟1/𝜕̂𝑦)(𝑧𝑟𝑣,1Φ+𝑧𝑟𝜔𝑇Φ) and we arrive at ̇𝑉𝑟=𝑟𝑗=1𝑐𝑗𝑧2𝑗+𝑑𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗+𝜏𝑟+̇Φ𝑇Γ1Φ+𝑧𝑟𝑅̂𝑢𝑟𝑗=1𝑧𝑗𝜕𝛼𝑗1𝜀𝜕̂𝑦̂𝑧12+𝑑𝑠𝑖1+̇𝑑𝑦𝑟𝑗=114𝑑𝑗𝜀𝑇𝜀+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇.𝑃𝜀(4.18) From (4.18), we may specify the parameter update law in order to cancel the term (𝜏𝑟+̇Φ𝑇Γ1)Φ. To guarantee the estimated parameters will always lie within allowable region 𝑤, a projected parametric update law will be specified as ̇Θ=Γ𝜏𝑇𝑟ifΘ𝑤0,𝑃𝑅Γ𝜏𝑇𝑟ifΘ𝜕𝑤,𝜏𝑟ΓΘperp>0,(4.19) where 𝑤 is the allowable parameter variation set (compact and convex) with its interior and boundary denoted by 𝑤0 and 𝜕𝑤, 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 𝑃𝑅() is employed to prevent the parameter vector from leaving the variation set. With (4.19), (4.18) can be written as ̇𝑉𝑟=𝑟𝑗=1𝑐𝑗𝑧2𝑗+𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑖+12𝑑𝑗𝜀̂𝑧122+𝑧𝑟𝑅̂𝑢𝑟𝑗=1𝑧𝑗𝜕𝛼𝑗1𝑑𝜕̂𝑦𝑠𝑖1+̇𝑑𝑦𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇𝑃𝜀𝑟𝑗=1𝑐𝑗𝑧2𝑗+𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122+𝑧𝑟𝑅̂𝑢𝑟𝑗=1𝑧𝑗𝜕𝛼𝑗1|||𝑑𝜕̂𝑦𝑠𝑖1+̇𝑑𝑦|||𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇.𝑃𝜀(4.20) Add and subtract terms 𝑟𝑗=1(1/4𝑔𝑗𝑑)|𝑠𝑖1+̇𝑑𝑦|2; we have ̇𝑉𝑟𝑟𝑗=1𝑐𝑗𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122+𝑧𝑟𝑅̂𝑢𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝜕̂𝑦2𝑧2𝑗𝑟𝑗=1𝑧𝑗𝜕𝛼𝑗1|||𝑑𝜕̂𝑦𝑠𝑖1+̇𝑑𝑦|||𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇𝑃𝜀𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟.(4.21) Moreover, we obtain ̇𝑉𝑖𝑟𝑗=1𝑐𝑗𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑖𝜀𝑇𝑃Δ+Δ𝑇𝑃𝜀𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑧𝑟𝑅.̂𝑢(4.22) As shown in Figure 1, the tracking error 𝑍1(̃𝑠) and the control input 𝑈𝑅(̃𝑠) are related by 𝑈𝑅(̃𝑠)=𝑅(̃𝑠)𝐶(̃𝑠)𝑍1(̃𝑠),(4.23) where we have chosen 𝑅(̃𝑠) as a low-order and attenuated-type internal model filter, that is, 𝑅(̃𝑠)=𝑘𝑖=1̃𝑠2+2𝜁𝑖𝜔𝑛𝑖̃𝑠+𝜔2𝑛𝑖̃𝑠2+2𝜉𝑖𝜔𝑛𝑖̃𝑠+𝜔2𝑛𝑖,(4.24) where k is the number of periodic frequencies to be rejected, 𝜔𝑛𝑖 is determined based on the ith disturbance frequency in rad/rev, and 𝜉𝑖 and 𝜁𝑖 are two damping ratios that satisfy 0<𝜉𝑖<𝜁𝑖<1. We can adjust the gain of 𝑅(̃𝑠) at those periodic frequencies by varying the values of 𝜉𝑖 and 𝜁𝑖.

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 ̂𝑦𝑚,̇̂𝑦𝑚,,̂𝑦𝑚(𝑟) (where 𝑟 is the relative degree) and 𝑑𝑦,̇𝑑𝑦𝑑,,𝑦(𝑟) are known and bounded, 𝑑(𝑟1)𝑠𝑖1,𝑑(𝑟2)𝑠𝑖2̇𝑑,,𝑠𝑖𝑟1 are sufficiently smooth, 𝑓,𝑔,,𝐿𝑟𝑓,𝐿𝑔𝐿𝑓𝑟1 are Lipschitz continuous functions, at least one column of 𝑊(̂𝑦) is bounded away from zero. Furthermore, suppose that a loop-shaping filter 𝐶(̃𝑠) 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 Θw0 needs to be considered).
Denote by ̇Θperp the component of ̇Θ perpendicular to the tangent plane at Θ so that ̇Θ=𝑃𝑅(̇̇ΘΘ)+perp. Since Θ𝑤 and 𝑤 is convex, we have (ΘΘ)𝑇̇Θperp0. Choose Lyapunov function 𝑉(Φ)=Φ𝑇Φ and use the parameter update law as defined in (4.19). When Θ𝑤0, we have ̇𝑉=Φ𝑇̇Θ. When Θ𝜕𝑤, we have ̇𝑉=Φ𝑇𝑃𝑅̇Θ=Φ𝑇̇̇ΘΘperp=Φ𝑇̇Θ+Φ𝑇̇ΘperpΦ𝑇̇Θ,(4.25) where we use the fact that Φ𝑇̇Θperp=ΘΘ𝑇̇Θperp=ΘΘ𝑇̇Θperp0.(4.26) Thus, we only have to consider the scenario corresponding to Θ𝑤0 in the sequel.
Step 2. Substituting (4.23) back into (4.22), we havė𝑉𝑟𝑟𝑗=1𝑐𝑗𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇𝑃𝜀𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟𝑧𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧1.(4.27) Using the definition of tracking error 𝑧1=̂𝑦̂𝑦𝑚=(𝑧11̂𝑦𝑚)+𝜀̂𝑧11𝑑𝑦, (4.27) can be written as ̇𝑉𝑟𝑟𝑗=1𝑐𝑗𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇+||𝑧𝑃𝜀𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧11̂𝑦𝑚||+|||𝑧𝑟𝜀𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦|||.(4.28) Use the following equality: ̂𝑧𝑟𝜀𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦𝛾2̂𝑧2𝑟+1𝜀2𝛾𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦2,𝛾>0isdesignable.(4.29) Equation (4.28) becomes ̇𝑉𝑟𝑟1𝑗=1𝑐𝑗𝑧2𝑗𝑐𝑟𝑧2𝑟𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇+||𝑧𝑃𝜀𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧11̂𝑦𝑚||+1𝜀2𝛾𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦2𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟,(4.30) where 𝑐𝑟=𝑐𝑟𝛾2>0. Moreover, the positive designable parameters 𝑐𝑖 can be written as 𝑐𝑗=𝐶𝑗+𝑗𝑐,𝑗=1,,𝑟1,𝑟=𝐶𝑟+𝑟,(4.31) where 𝐶𝑗,𝐶𝑟 and 𝑗,𝑟>0. Thus, (4.30) can be written as ̇𝑉𝑟𝑟1𝑗=1𝐶𝑗+𝑗𝑧2𝑗𝐶𝑟+𝑟𝑧2𝑟𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇+|||𝑧𝑃𝜀𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧11̂𝑦𝑚|||+1𝑅𝐶𝜀2𝛾(̃𝑠)(̃𝑠)̂𝑧11𝑑𝑦2.(4.32) Utilizing the fact that 1||𝑧1||𝑟||𝑧𝑟||2=𝑧21+𝑟𝑧2𝑟1𝑟|||𝑧𝑟𝑧11̂𝑦𝑚+𝑧𝑟𝜀̂𝑧11𝑑𝑦|||,(4.33) we have ̇𝑉𝑟𝑟1𝑗=2𝐶𝑗+𝑗𝑧2𝑗𝐶1𝑧21𝐶𝑟𝑧2𝑟1||𝑧1||𝑟||𝑧𝑟||21𝑟|||𝑧𝑟𝑧11̂𝑦𝑚+𝑧𝑟𝜀̂𝑧11𝑑𝑦|||𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇+||𝑧𝑃𝜀𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧11̂𝑦𝑚||+1𝜀2𝛾𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦2.(4.34) To design 1,,𝑟 (or 𝑐1,,𝑐𝑟 ), 𝑑𝑗 and 𝑔𝑗 such that 𝑟𝑗=1𝑗𝑧2𝑗𝑟𝑗=1𝑑𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑑𝑗𝜀̂𝑧122𝑟𝑗=1𝑔𝑗𝜕𝛼𝑗1𝑧𝜕̂𝑦𝑗+12𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2𝑟𝑗=114𝑑𝑗𝜀2̂𝑧11+𝜀2̂𝑧13++𝜀2̂𝑧1𝑟1||𝑧1||𝑟||𝑧𝑟||21𝑟|||𝑧𝑟𝑧11̂𝑦𝑚+𝑧𝑟𝜀̂𝑧11𝑑𝑦|||+||𝑧𝑟𝑅(̃𝑠)𝐶(̃𝑠)𝑧11̂𝑦𝑚||0,(4.35) We arrive at ̇𝑉𝑟𝑟𝑗=1𝐶𝑗𝑧2𝑗+1𝜀2𝛾𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗𝜀𝑇𝑃Δ+Δ𝑇.𝑃𝜀(4.36) Equation (4.36) implies that ̇𝑉𝑟𝑟𝑗=1𝐶𝑗𝑧2𝑗12Φ𝑇Γ1Φ+𝑟𝑗=114𝑑𝑗𝜀𝑇+1𝑃𝜀2Φ𝑇Γ1Φ+𝑟𝑗=114𝑑𝑗𝜀𝑇+1𝑃𝜀𝑅𝐶𝜀2𝛾(̃𝑠)(̃𝑠)̂𝑧11𝑑𝑦2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗||𝜀𝑇𝑃Δ+Δ𝑇||𝑃𝜀2𝑘𝑣𝑉𝑟+𝐶,(4.37) where 𝑘𝑣min{𝐶1,,𝐶𝑟,𝜆min(Γ)}, 𝜆min(Γ) is the smallest eigenvalue of Γ and 1𝐶=2Φ𝑇Γ1Φ+𝑟𝑗=114𝑑𝑗𝜀𝑇1𝑃𝜀+𝜀2𝛾𝑅(̃𝑠)𝐶(̃𝑠)̂𝑧11𝑑𝑦2+𝑟𝑗=114𝑔𝑗|||𝑑𝑠𝑖1+̇𝑑𝑦|||2+𝑟𝑗=114𝑑𝑗||𝜀𝑇𝑃Δ+Δ𝑇||𝑃𝜀(4.38) is bounded since 𝜀̂𝑧11,𝑑𝑦 are bounded and Φ𝑇Γ1Φ is bounded due to the parameter update law specified in (4.19). We conclude that 𝑉𝑟𝑒2𝑘𝑣𝜃𝑉𝑟(0)+𝜃0𝐶𝑒2𝑘𝑣(𝜃𝜏)𝑑𝜏𝑒2𝑘𝑣𝜃𝑉𝑟(0)+1𝑒2𝑘𝑣𝜃𝐶2𝑘𝑣.(4.39) As 𝜃, we have 𝑉𝑟(𝐶)2𝑘𝑣,(4.40) which implies that the overall system is stable and the bound 𝐶/(2𝑘𝑣) can be decreased by increasing 𝑘𝑣 or increasing 𝛾. By (4.14), this implies that 𝑧,Θ,𝜀 are bounded. Since 𝑧1=̂𝑦̂𝑦𝑚, ̂𝑦 is also bounded. From (3.11), we can see that 𝜉 and 𝑣1,,𝑣 are bounded since 𝑊𝑓(̂𝑦) and 𝑊𝑔(̂𝑦) are bounded. Moreover, we conclude that the virtual inputs 𝛼 are bounded because they consist of bounded terms. Also, 𝑧1 is bounded from (3.12) and also ̂𝑧1 from the definition of 𝜀. With the ISS assumption and bounded ̂𝑧1, we conclude that the internal dynamics ̂𝑧2 is bounded. Finally, ̂𝑥 is bounded by diffeomorphism, that is, ̂𝑥=𝑇1(̂𝑧).

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 𝜃-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 ̇𝑑̂𝑥=𝑓(̂𝑥)+𝑔(̂𝑥)̂𝑢+𝑠𝑑,̂𝑦=(̂𝑥)+𝑦,(5.1) where 𝑓(̂𝑥)=𝑎1+̂𝑥2̂𝑥1𝑎00𝑏,𝑔(̂𝑥)=0̂𝑥1,(̂𝑥)=̂𝑥1,(5.2) with 𝑎0=5155, 𝑎1=1138, and 𝑏0=140368. For verification purpose, the output disturbance is assumed to be a low-pass rectangular periodic signal (with amplitude switching between −0.1 and 0.1) (see Figure 3), that is, 𝑑𝑦(𝜃)=0.10.0125̃𝑠+1𝑙=(1)𝑙+Π(𝜃1𝑙)10(0.005̃𝑠+1)2𝑁0,(5.3) where 1||𝜃||||𝜃||Π(𝜃)=<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, 𝑅(̃𝑠)=4𝑖=1̃𝑠2+2𝜁𝑖𝜔𝑛𝑖̃𝑠+𝜔2𝑛𝑖̃𝑠2+2𝜉𝑖𝜔𝑛𝑖̃𝑠+𝜔2𝑛𝑖,(5.5) where 𝜍𝑖=0.2,𝜉𝑖𝜔=0.0002,𝑛1=0.25𝜋,𝜔𝑛2=3×0.25𝜋,𝜔𝑛3=5×0.25𝜋,𝜔𝑛4=7×0.25𝜋,(5.6) Furthermore, the stabilizing filter is specified as 𝐶(̃𝑠)=100000(̃𝑠/100+1)(.̃𝑠/10000+1)(5.7) The parameters of the 𝐾-filter are set to 𝑘1=1600 and 𝑘2=100. The initial values of the estimated parameters are set to ̃𝑎0=1500, ̃𝑎1=500, and ̃𝑏0=1000000. The allowable parameter variation sets are ̃𝑎0Ω̃𝑎0̃𝑎0100̃𝑎0,10000̃𝑎1Ω̃𝑎1̃𝑎110̃𝑎1,̃𝑏100000̃𝑏Ω0̃𝑏0̃𝑏100000.10000000(5.8) Note that 𝑑𝑠(𝑡) 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 30 rev/s and then speed up to 35 rev/s and finally speed down to 25 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.