Asymptotic rejection of asymmetric periodic disturbances in output-feedback nonlinear systems

doi:10.1016/j.automatica.2006.10.005

Automatica

Volume 43, Issue 3, March 2007, Pages 555-561

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Abstract

This paper deals with asymptotic rejection of periodic disturbances which may have asymmetric basic wave patterns. This class of disturbances covers asymmetric wave forms in the half-period such as alternating sawtooth wave form, some disturbances which are generated from nonlinear oscillation such as Van de Pol oscillators, as well as disturbances with symmetric half-period wave forms such as sinusoidal disturbances and triangular disturbances etc. The systems considered in this paper can be transformed to the nonlinear output feedback form. The amplitude and phase of the disturbances are unknown. The novel concept of integral phase shift is introduced together with the newly introduced half-period integration operator to investigate the invariant properties of asymmetric periodic disturbances. They are used for the estimation of unknown disturbances in the systems, together with observer design techniques to deal with nonlinearity. The proposed control design with the disturbance estimation asymptotically rejects the unknown disturbance, and ensures the overall stability of the system.

Introduction

One of the common deterministic disturbances considered for asymptotic rejection in dynamic systems is sinusoidal disturbance (Bodson and Douglas, 1997, Bodson et al., 1994, Ding, 2003, Marino et al., 2003), and very often the internal model principle is used to generate the desired feedforward control input to reject the unknown disturbances. A related problem is formulated as output regulation, where the output measurement contains the disturbance (Chen and Huang, 2005a, Huang and Rugh, 1990, Isidori, 1995, Isidori and Byrnes, 1990, Pavlov et al., 2004). For sinusoidal disturbances, if the disturbance frequencies are available, the disturbance can be easily modelled as an output of a known linear dynamic model which is often referred to as the exosystem, and therefore a corresponding internal model can be designed (Ding, 2001, Huang and Rugh, 1990, Isidori, 1995, Isidori and Byrnes, 1990, Serrani and Isidori, 2000). If the frequencies are unknown, adaptive internal model can be used for disturbance rejection and output regulation (Ding, 2003; Serrani, Isidori, & Marconi, 2001).

Many periodic signals are not sinusoidal, and therefore cannot be modelled as an output of a linear exosystem. If we really force ourself to find a model for its generation, then a model can often be infinite dimensional, or nonlinear finite dimensional for a limited class of nonlinear disturbances. Limited results are available on output regulation with nonlinear exosystems (Chen and Huang, 2005b, Ding, 2006b, Priscoli, 2004), of which the periodic solutions can be viewed as smooth disturbances. Recently, a half-period integration method is proposed to characterize general periodic disturbances, and applied for asymptotic rejection of a class of general disturbances which have symmetric wave form in the half of the period, such as symmetric triangular waves and square waves. The half-period integration based disturbance rejection is demonstrated in a class of nonlinear output feedback systems which can be transformed to the output feedback form (Ding, 2006a).

In this paper, we deal with asymptotic rejection of more general disturbances than the disturbances with odd-function and symmetric wave form. In particular, we consider a class of periodic disturbances whose basic half-period wave forms may be asymmetric, and the second half-period form follows the first half-period with opposite sign. This class of periodic disturbances includes non-smooth disturbances such as alternating sawtooth wave forms, some disturbances which are generated from nonlinear oscillation such as Van de Pol oscillators, as well as disturbances with symmetric half-period wave forms such as sinusoidal disturbances and triangular disturbances etc. A new concept, integral phase shift, is introduced to tackle asymmetric wave patterns. The integral phase shift reflects the phase change of the basic wave pattern after half-period integration, and we introduce a new delay operator with the delay that depends on the integral phase shift. The half-period integration operator together with the integral phase shift is used to establish the invariant properties of the asymmetric periodic disturbances. A set of results for the class of disturbances are obtained and they are applied in control design for asymptotic disturbance rejection in nonlinear output feedback systems. With the information of the basic wave form, the phase and amplitude can be estimated by the proposed design. With the estimated disturbance, control design is then proposed for disturbance rejection with stability. The nice property of the estimate ensures the asymptotic rejection of general periodical disturbances under the proposed control for nonlinear systems in the output feedback form. A simpler control algorithm is proposed for linear systems. An example is included to demonstrate the proposed estimation and control algorithm for rejection of an alternating sawtooth disturbance.

Section snippets

Problem formulation

Consider a single-input–single-output nonlinear system which can be transformed into the output feedback form $\dot{x} = A_{c} x + ψ (y) + b (u - w), y = Cx$ with $A_{c} = [\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{matrix}], C = {[\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}]}^{T}, b = [\begin{matrix} 0 \\ ⋮ \\ 0 \\ b_{ρ} \\ ⋮ \\ b_{n} \end{matrix}],$ where $x \in R^{n}$ is the state vector, $u \in R$ is the control, $ψ$ , is a known nonlinear smooth vector field in $R^{n}$ with $ψ (0) = 0$ , $w \in R$ is a periodical disturbance.

Assumption 1

The disturbance can be expressed as $w (t) = {aw}_{b} (t + φ),$ where the unknown constants a and $φ$ are referred to as amplitude and phase, and $w_{b} (t)$ is a known function satisfying

Integral phase shift and half-period integration

Since the basic disturbance pattern is described by the function $w_{b} (t)$ , the disturbance can be reproduced if the amplitude a and phase $φ$ can be estimated. In this section, the periodic property and wave pattern properties described in Assumption 1 will be exploited to design estimation algorithms for a and $φ$ .

Define the half-period integration operator $I$ and the delay operator $D (d)$ as $I \circ f (t) ≔ I (f (t)) = \int_{t - \frac{T}{2}}^{t} f (s) d s, D (d) \circ f (t) ≔ D (d, f (t)) = f (t - d),$ where $0 ⩽ d < T$ . For the convenience of notations, we often

Disturbance estimation

Similar to observer design, we have the following filter: $\dot{p} = (A_{c} + kC) p + φ (y) + bu - ky,$ where $p \in R^{n}$ , $k \in R^{n}$ is chosen so that $K (s) ≔ s^{n} - \sum_{i = 1}^{n} k_{i} s^{n - i} = B (s) (s^{ρ} + λ_{1} s^{ρ - 1} + \dots + λ_{ρ}) / b_{ρ}$ with $λ_{i}$ being positive real design parameters such that $(s^{ρ} + λ_{1} s^{ρ - 1} + \dots + λ_{ρ})$ is Hurwitz. An estimate of $w$ is given by $\hat{w} (t) = \hat{a} w_{b} ({\hat{φ}}_{1}),$ where $\hat{a} = \frac{I \circ | \bar{w} (t) |}{I \circ | w_{b, ρ} (t) |},$ ${\hat{φ}}_{1} (t) = \frac{1}{2} (I \circ sign (\bar{w} (t)) + \frac{T}{2}) sign (\bar{w} (t))$ with $\bar{w} (t) = Q \circ (p_{1} - y),$ $Q = D ({\bar{d}}_{ρ}) \sum_{i = 0}^{ρ} λ_{i} I^{i} {(1 - D (\frac{T}{2}))}^{ρ - i}$ and $p_{1}$ being the first element of p, and $\bar{d_{ρ}} = mod (\sum_{i = 0}^{ρ - 1} d_{i}, T)$ .

Theorem 4.1

If the disturbance in(1)satisfies the

Disturbance rejection with stabilization

After the estimation of unknown disturbances, the control design follows in the same way as the control design for asymptotic rejection of disturbances with symmetric wave forms (Ding, 2006a). We include the key steps in control design for completeness of presentation without the proofs.

A state observer is designed as $\dot{\hat{x}} = (A_{c} + kC) \hat{x} + ψ (y) + b (u - \hat{w}) - ky,$ Control design can then be carried out using backstepping based on (55). Finally the control input is given by $u = \hat{w} + \frac{α_{ρ} - {\hat{x}}_{ρ + 1}}{b_{ρ}},$ where $α_{i} = z_{i - 1} - c_{i} z_{i} - κ_{i} (\partial α_{i - 1})$

Example

Consider a nonlinear system in output feedback form ${\dot{x}}_{1} = x_{2} - y^{3} + (u - w), {\dot{x}}_{2} = (u - w), y = x_{1},$ where $w = {aw}_{b} (t + φ)$ is a periodic disturbance which satisfies Assumptions 1 and 2 with unknown a and $φ$ . It is easy to see that the system (58) are in the format of (1) with $φ (y) = [y^{3} 0]^{T}$ and $b = [1 1]^{T}$ . The system is minimum phase, and therefore Assumption 2 is satisfied. The control input is designed as $u = \hat{w} - c_{1} y - κ_{1} y - y^{3} - {\hat{x}}_{2} .$

Simulation study has been carried out for the estimation and control design shown in this example.

Conclusions

In this paper, we have introduced integral phase shift together with the half-period integration, and they have been successfully used to establish the invariant properties of a large class of periodic disturbances whose basic wave forms may not be odd functions nor symmetric in the half period. These invariant properties under the half-period integration with the delay of integral phase shift are instrumental in disturbance estimation together with a special observer designed to extract the

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Cited by (0)

Zhengtao Ding was born in 1964 in Jiangsu, China. He obtained his BEng degree from Tsinghua University, Beijing, in 1984, MSc and Ph.D. degrees in 1986 and 1989, respectively, from the Control Systems Centre, University of Manchester Institute of Science and Technology (UMIST), UK. Having worked in UK universities as a research fellow for a few years, he joined Ngee Ann Polytechnic, Singapore, in 1993, as a lecturer in the Department of Mechanical Engineering. Since September 2003, he has been a lecturer in control engineering with The University of Manchester, initially in the School of Engineering, and then in the Control Systems Centre, School of Electrical and Electronic Engineering, after the merge of the university and UMIST. Dr. Ding's main research interest is in control theory and applications, in particular, adaptive and robust control of uncertain nonlinear systems, disturbance rejection and output regulation.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publications in revised form by Associate Editor Alessandro Astolfi under the direction of Editor H.K. Khalil. This research was supported by UK EPSRC Grant EP/C500156/1.

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Brief paperAsymptotic rejection of asymmetric periodic disturbances in output-feedback nonlinear systems☆

Abstract

Introduction

Section snippets

Problem formulation

Integral phase shift and half-period integration

Disturbance estimation

Disturbance rejection with stabilization

Example

Conclusions

Automatica

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Systems & Control Letters

Harmonic generation in adaptive feedforward cancellation schemes

IEEE Transactions on Automatic Control

Brief paper
Asymptotic rejection of asymmetric periodic disturbances in output-feedback nonlinear systems☆