High-gain fractional disturbance observer control of uncertain dynamical systems

https://doi.org/10.1016/j.jfranklin.2021.04.020Get rights and content

Abstract

Disturbance observer-based control allows to compensate unknown inputs, however, in most cases, requiring their integer-order differentiability. In this paper, a novel disturbance observer-based state feedback controller is proposed to compensate a more general class of fractional-, but not necessarily integer-order, differentiable unknown inputs. The proposed fractional PI-like structure yields precise conditions for feedback gain tuning. Remarkably, the resulting controller rejects non-differentiable disturbances with a smooth controller, guaranteeing robustness, an outstanding features for tracking tasks, under a prescribed practical stability regimen. A comparison to a fractional sliding mode observer is conducted via simulations to highlight the reliability of the proposed scheme.

Introduction

The high-gain concept was pioneered in [1], [2], [3] addressing observer design, assuring precise and fast state estimation, later extended into high-gain observer-based control in [4], [5], [6], [7], [8], [9]. These schemes arose as viable options for robust and fast estimation throughout increasing the feedback gains, as long as the controller remains within admissible bounds. Although high-gain controllers induce only practical stability [10], they facilitate implementation and tuning effort, in contrast to sound robust controllers, such as variable structure and model-based schemes, in which the theoretical asymptotic and finite-time stability are not achievable in practice.

Disturbance observer-based control is considered in [11], [12] based on a system nominal model without uncertainties nor disturbances. Frequency domain-based designs for linear systems have shown advantages in practice since relevant performance can be set, including bandwidth constraints. For nonlinear disturbance observers, lumped disturbances are considered as a typical approach [13], [14]. Recently, high-gain disturbance observers have been proposed for integer-order systems [15], [16], [17], [18], guaranteeing prescribed tracking. All these results assume that the unknown lumped disturbance is Lipschitz continuous, which for some advanced applications, could be a limitation. In response to this issue, fractional sliding mode control has been considered with a disturbance observer [19]. However, there remains a question on how to deal with simpler control structures with fractional disturbance observers. In this direction, Martínez-Fuentes and Martínez-Guerra [20] proposed a Mittag–Leffler stable state observer for fractional-order systems.

The contribution of this paper is then, the formulation of [20] in the realm of [21], [22] to provide a smooth controller able to precisely compensate continuous but not necessarily differentiable disturbances.

Fractional calculus constitutes an important and emerging tool to address advanced and complex phenomena [23], including a large variety of engineering systems subject to a wider class of dynamic effects [24]. Fractional-order controllers have been widely considered since they significantly enhance the dynamical performance of their integer-order counterparts [25], [26]. In [21], [22], sliding mode-based disturbance observers are studied to account for lumped Hölder continuous disturbances, that is, not necessarily integer-order differentiable functions, which possess well-posed fractional-order derivatives. Later Muñoz-Vázquez et al. [27] extended this idea to provide a theoretically finite-time exact estimation of Hölder continuous disturbances, nonetheless, the unavoidable chattering phenomenon affects the tracking precision, as well as the integrity of the system components in real-time applications. Other approaches have combined disturbance observers and fractional-order tools [28], [29], [30], [31], [32], [33], [34].

In contrast to the outstanding and inspiring mentioned contributions, the approach proposed in this paper stands for: A high-gain smooth PI-like structure to observe continuous but not necessarily differentiable disturbances, whose tracking precision can be adjusted with a single control parameter. In this sense, the proposed scheme provides a methodology that modulates constant gains through adjusting the ultimate upper bound of the tracking errors, thus providing further insights on the tuning procedure.

The rest of this paper is organized as follows. Section 2 presents basic preliminaries on fractional-order systems and stability analysis. Section 3 introduces the system definition and the high-gain observer based control design. Section 4 presents a simulation study. Finally, conclusive discussions are presented in Section 5.

Section snippets

On fractional-order systems

Fractional calculus deals with differentiation and integration of non-integer orders by means of differintegral operators. For x:ΩtRn a real-valued vector function and ΩtR a compact set, the differintegral operators of order α(0,1) are defined as follows [23], [35], [36]:

  • Riemann–Liouville integralIαx(t)=1Γ(α)0tx(τ)(tτ)1αdτ

  • Marchaud derivativeMDαx(t)=x(t)Γ(1α)(tt0)α+αΓ(1α)0tx(t)x(τ)(tτ)α+1dτ

  • Extended Caputo derivativeDαx(t)=x(t)x(0)Γ(1α)(tt0)α+αΓ(1α)0tx(t)x(τ)(tτ)α+1dτ

where Γ(ς)

System definition

Consider the following linear time invariant (LTI) system subject to a wide class of not necessarily integer-order differentiable disturbances, non-linearities and dynamic uncertainties,x˙(t)=Ax(t)+Bu(t)+Ed(t)where xRn is the state, which is assumed available for control purposes, uRnu is the control input, ARn×n and BRn×nu are constant matrices, dRnd is an unknown but continuous disturbance, with ERn×nd a constant matrix, and, without loss of generality, consider that nu,ndn. The

Simulations

A numerical study is programmed in Simulnk in Matlab, based on the Euler integrator running at 0.1 ms. The CRONE method was used with a transfer function of 10-order for a frequency range of [103,103] rad/s [47].

Since the purpose of this simulation is to assess the reliability of the proposed disturbance observer, consider the simple but representative nonlinear pendulum system,ml2q¨=Tmglsin(q)+φ,where l=0.5 m is the length, m=0.5 kg is the mass, g=9.81m/s2 is the gravity, T is the control

Conclusion

This study constitutes an evidence that robust control tools in combination with fractional-order techniques furnish high-end structures to address a large class of physical phenomena in uncertain dynamical systems. It is shown that high-gain schemes outperform non-smooth techniques, such as sliding mode ones, through the capacity of modulating the convergence accuracy by means of a smooth controller. In addition, it is worth to comment that the peaking phenomenon, inherent to any high-gain

Declaration of Competing Interest

We authors declare that we have no interest conflict regarding the publication of this paper.

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