Research paperA high-gain observer with Mittag–Leffler rate of convergence for a class of nonlinear fractional-order systems
Introduction
Fractional-order calculus is an emerging branch of mathematics that generalizes the classical calculus where the derivation and integration deal with non-integer order (rational, irrational, complex, and variable) [1], [2], [3]. In comparison to the classical calculus, fractional calculus describes better the memory effects and hereditary properties of various processes, employing operators like Caputo, Riemann–Liouville or Grünwald–Letnikov in mathematical models to describe the behavior of several physical systems with applications in many fields of engineering and science [4], [5], [6]. Fractional differential equations have gained popularity in science and engineering due to their theoretical results and experimental applications in chaos theory [7], [8], physics [9], electromagnetism and electronics [10], robotics [11], control theory and the design of PID controllers through fractional operators [12], [13], [14], among other [15]. Due to the variety of dynamic systems and their characteristics, in new applications, other operators like Marchaud [16] or Caputo-Fabrizio [17] have been used to model physical phenomena and analyze the stability of them.
One of the main problems in control theory is the estimation of the variables for dynamical systems. An estimator (or observer) is an algorithm that reconstructs the missing information from the state vector based on the output and control available measurements. The first advances on this theory were given by Luenberger for linear systems [18], and later extended to non-linear systems with applications such as fault detection [19], synchronization of chaotic systems [20], sliding modes [21], Kalman filter [22], adaptive estimation [23] or high-gain observers [24]. Recently, the estimation problem is being studied through fractional-order estimators with applications in fitting of experimental data [25], synchronization [26], [27], fault detection based on algebraic methods [28], nonlinear disturbance [29], sliding mode observers [30], [31], full and reduced-order observers [32], [33] among others [34], [35], [36]. The differences between all these techniques consist in the restrictions imposed on the system and the desired behavior for the estimation error dynamic, for example, initial overshooting, convergence on finite time, asymptotic precision, rate of convergence or the robustness with respect to external disturbances and measurement noises.
High-gain observers introduced by Gauthier [37], [38] have been extensively studied for integer-order systems. Several works related to this class of observers with applications due to their simple design, implementation, and the robustness in problems with external disturbances have been proposed [39], [40]. Some applications of high-gain observers involve adaptive control [41], [42], compensation of non-linearities [43], studies of robustness [44], convergence acceleration by commutation and increasing of gain [45], [46] or secure communications. The high-gain observer is robust in the presence of external disturbances or uncertainties like noise in the measurement. Besides, other advantages of the high-gain observers are the rapid exponential decay of the estimation error (exponential observers) and the acceleration of the speed of convergence by increasing the gain of the observer.
To the best of our knowledge, there are no results in the literature regarding high-gain observers for fractional-order systems despite the advantages of the high-gain observers of integer-order. Motivated by this, the first contribution consists of the solution of the estimation problem for a class of nonlinear fractional-order systems through the design of a high-gain observer and the adjustment of the convergence rate. We focus on a class of nonlinear systems expressed in the observer canonical form with a nonlinear Lipschitz vector. The structure of these dynamic systems allows obtaining a Riccati-like fractional differential equation whose solution is the gain matrix for the high-gain observer that depends on a parameter θ > 0. This parameter is essential since the convergence rate depends on it. We can control the speed of convergence varying (increasing or decreasing) the values of θ in the gain matrix, obtaining a faster (or slower) estimation of the states with the high-gain observer. Regarding the observation error, we obtain an explicit error bound that depends on a Mittag–Leffler function, similar to the bound obtained in our previous work [47], i.e., Mittag–Leffler decaying error and this shows that the high-gain observer belongs to the Mittag–Leffler observer family.
Determine if an estimation algorithm is a Mittag–Leffler observer, it is necessary to perform the stability analysis and verify that the observation error bound depends on a Mittag–Leffler function. Motivated by this, the second contribution of this paper is to establish a theoretical basis through some stability results that allow determining if an estimation algorithm belongs to the Mittag–Leffler observers class, and for specific cases, design Mittag–Leffler algorithms for estimation of variables in fractional-order systems. The rest of the article is organized as follows. In Section 2, some preliminaries about fractional calculus, and stability of fractional-order systems are presented. The Mittag–Leffler observers are defined in Section 3, and some results for the design of these types of observers based on a Lyapunov function are established. On the other hand, in Section 4, the synthesis of a high-gain observer for a class of nonlinear-fractional order systems is presented. Numerical simulations are carried out in Section 5 to illustrate the effectiveness of the Mittag–Leffler high-gain observer and the main conclusions are discussed in Section 6.
Section snippets
Basic elements of fractional calculus
The fractional calculus theory generalizes the usual calculus and it allows to consider the integration and derivation of arbitrary order . In this paper, the values of derivative with 0 < α < 1 are considered. The n-fold integral is calculated using the formulaOn the other hand, since the gamma functionis related to the factorial by it is natural to define the integration of a
Problem statement and Mittag–Leffler observers
Consider the commensurate fractional-order nonlinear systems with a single output described by:where 0 < α < 1, the state vector; the output; the control input; a vector function locally Lipschitz in and uniformly bounded in . Since then there are unknown state variables of this system. The problem is to design a fractional-order dynamic system, called observer for the system (18), such that the rest of unknown variables
A Mittag–Leffler high-gain observer
Consider a class of nonlinear fractional-order systems described in the observer canonical form [54]:where A is an upper shift matrix,1 and is a vector locally Lipschitz in a region D, i.e.: Assumption 4.1 In a region D including the origin with respect to uniformly in there exists a known positive scalar ψ that satisfies
We propose the following algorithm to estimate
Simple pendulum of fractional-order
A pendulum is a mass suspended from a pivot (without friction) with a string or rod of negligible mass so that it can swing freely [60] (Fig. 1). The differential equation that describes a simple pendulum is well known and can be generalized with fractional calculus theory. The mathematical model of a simple pendulum of fractional order is given by:where and . The physical interpretation of φ is the angle that the
Conclusions
In this paper, the family of Mittag–Leffler observers is characterized as the set of observers such that are asymptotic and with Mittag–Leffler decaying error. A simple method is proposed to design Mittag–Leffler observers, based on the Mittag–Leffler stability analysis for systems with a single output. We can see that in these observers, the convergence in the estimation can be accelerated by increasing value of the observer gain, due to the properties of the Mittag–Leffler function. Based on
Acknowledgement
The first author thanks the Consejo Nacional de Ciencia y Tecnología, Mexico for the financial support of PhD grant 295538.
References (61)
- et al.
FPGA implementation of two fractional order chaotic systems
AEU Int J Electron Commun
(2017) - et al.
FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald-Letnikov method
Commun Nonlinear Sci Numer Simul
(2019) Solving state feedback control of fractional linear quadratic regulator systems using triangular functions
Commun Nonlinear Sci Numer Simul
(2019)- et al.
A new collection of real world applications of fractional calculus in science and engineering
Commun Nonlinear Sci Numer Simul
(2018) - et al.
Fault tolerant system based on non-integers order observers: application in a heat exchanger
ISA Trans
(2018) - et al.
Observer-based fault detection for uncertain nonlinear systems
J Frankl Inst
(2018) - et al.
Kalman filter for adaptive learning of look-up tables with application to automotive battery resistance estimation
Control Eng Pract
(2016) - et al.
Filtered high gain observer for a class of uncertain nonlinear systems with sampled outputs
Automatica
(2019) - et al.
Synchronization of fractional order chaotic systems using active control method
Chaos Solitons Fract
(2012) - et al.
A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances
ISA Trans
(2016)