High-gain fractional disturbance observer control of uncertain dynamical systems
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 a real-valued vector function and a compact set, the differintegral operators of order are defined as follows [23], [35], [36]:
- •
Riemann–Liouville integral
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Marchaud derivative
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Extended Caputo derivative
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,where is the state, which is assumed available for control purposes, is the control input, and are constant matrices, is an unknown but continuous disturbance, with a constant matrix, and, without loss of generality, consider that . 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 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,where m is the length, kg is the mass, ms is the gravity, 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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