Adaptive Fractional Control Optimized by Genetic Algorithms with Application to Polyarticulated Robotic Systems

University of Sfax, ENIS, Laboratory of Control & Energy Management, Sfax, Tunisia Clinical Investigation Center, Sfax, Tunisia Digital Research Center of Sfax, Sfax, Tunisia Higher Institute of Computer Science and Multimedia of Sfax, ISIMS, Sfax, Tunisia Department of Neurology, Hospital Habib Bourguiba, Sfax, Tunisia Neuroscience Laboratory, Faculty of Medicine, Sfax University, Sfax, Tunisia Research Unit: Les Pathologies de L’Appareil Locomoteur, Service de Medecine Physique et de Readaptation, CHU Habib Bourguiba, Sfax, Tunisia


Introduction
Conventional adaptive controllers such as the Model Reference Adaptive Controller [1] (MRAC) are lacking robustness. In fact, among the drawbacks of these controllers is the choice of the adaptation gains. A higher adaptation gain yields to a high gain controller with good performances and tracking properties but poor robustness to unmodeled actuator dynamics [2]. In [3,4] and [5], researchers have discussed the robustness of different architectures and schemes of adaptive controllers [6]. To overcome these limitations [7,8] that adaptive controllers are facing, the L 1 adaptive controller was proposed in [9,10]. e basic part of the L 1 adaptive control is the presence of the low-pass filter introduced in the control channel. It is the trade-off between performance and robustness. As a result, the L 1 adaptive control architecture [11] differs from the standard adaptive control by its particular architecture, where adaptation and robustness are decoupled [12]. Hence, even with an increasing adaptation gain, the closed-loop system can be enhanced without the degradation of the robustness. Among the drawbacks of the L 1 adaptive control is the time lag. Researchers in [13][14][15] have proposed to add different supplementary controllers aiming to eliminate the appearing time lag. In this paper, the proposed contribution is the L 1 adaptive fractional control based on fractional calculus [16][17][18][19]. en, the new idea is to carefully choose the filter. In fact, instead of using integer-order filters [20], this paper proposes the use of fractional-order filters [21]. Fractional-order filters [22,23] have many advantages; they tune the shaper bandwidth and ensure a better selectivity. Moreover, the use of fractional orders in a lot of applications [24,25], where the system is extremely complex, allows solving huge problems such as reducing high control values. In this paper, L 1 fractional adaptive control is implemented, and it has proved a reduction of the error (elimination of the time lag) and reduction of the energy consumption.
e new approach, based on fractional filters [26], is then implemented on a dynamical model of an exoskeleton and simulation results are presented. Good performances validate the L 1 adaptive fractional controller. Moreover, the parameters of these filters have been optimized by genetic algorithms [27,28]. e rest of this paper is organized as follows. In Section 2, a background on fractional-order systems is presented. Section 3 presents a background on L 1 adaptive control. Section 4 is dedicated to the validation of the main contribution by the application to a polyarticulated robotic system. First, the problem formulation is detailed. en, performance evaluation criteria are established for the case of an exoskeleton in the nominal case and then in the presence of a multiplicative noise in order to test the robustness of the proposed controller. Section 5 gives the final conclusion.

Background on Fractional-Order Systems
Oustaloup's recursive approximation algorithm is presented in this section. is method deals with the frequency space and it is based on a recursive distribution of negative real zeros and poles in order to guarantee a minimal phase behavior [29]. e aim is to synthesize the rational function in the frequency range [ω L , ω H ]: using a recursive distribution of zeros and poles as: where −ω k ′ , −ω k is the pair of the zero and the pole of order k expressed as where α k � k + N + 0.5, with the following: ω L is the high transitional frequency ω H is the low transitional frequency and 2N + 1 is the total number of recursive poles and zeros. e gain of the filter is equal to the unity for ω < ω L . For a good approximation of the fractional-order filter, two or three decades at least should be considered between ω L and ω H .

Background on L 1 Adaptive Control
e L 1 adaptive control [30] is designed as shown in the block diagram of Figure 1. A square system is considered with an input vector u ∈ R m and an output vector y ∈ R m . e desired output and the desired output velocity vectors are, respectively, y d and _ y d . e dynamic system is described as with θ is a nonlinear function that gathers all the nonlinearities of the system and b is a regular matrix. a 1 and a 2 are square matrices with the appropriate dimension. e generalized tracking error is defined as follows: with Λ ∈ R m×m being a positive definite diagonal matrix. e control input vector u(t) is with where A m ∈ R m×m is a Hurwitz matrix that characterizes the desired transient response of the system u ad ∈ R m is an adequate adaptive term e time derivative of (6) leads to the following equation: en, e adaptive control is defined in order to cancel the nonlinearity θ(t). For that, in order to estimate and compensate the term θ(t), let us consider the following state predictor: where θ(t) is the estimation of the nonlinear term θ θ(t) � θ(t) − θ(t) is the error of the nonlinear term θ ρ(t) � ρ(t) − ρ(t) is the prediction error K ∈ R m×m is a designed matrix introduced to reject high-frequency noises with A � A m − K being a Hurwitz matrix defining a dynamical behavior of ρ at least two to three times faster than A m . e Lyapunov function associated with the system is where Γ is the positive diagonal high gain matrix and P is a definite positive matrix verifying Its differential with respect to time gives Hence, e estimate of θ(t), θ(t) is obtained using the projection operator which prevents the estimated values from exceeding their admissible range specified in the control design: e adaptive control term is as follows: where θ(s) is the Laplace transform of θ(t) and C(s) is a BIBO (bounded-input bounded-output) stable transfer function. For these reasons, the main proposition is to replace these filters by fractional-order ones that are realised by a sequence of first-order systems. e real order of the fractional system is equal to 2N + 1. is makes the system slower during its start-up and consequently generates appropriate values of the control during start-up, a high filtering of noises, and a reduced time lag.

Proposed
e transfer functions of all considered filters in this paper will be implemented and compared. ey are defined as where C −1 and C −2 filters can be represented as C r , for high values of ω H , and τω L � 1.

Problem Formulation of the Exoskeleton.
In this paper, we are interested in the control of polyarticulated robotic systems for which the dynamic model can be written in the following form: where M(q) ∈ R n×n is the inertia matrix F(q, _ q) ∈ R n is the vector of the Coriolis, centrifugal, gravitational, and contact forces τ ∈ R n is the vector of torques generated by actuators is the acceleration vector e proposed control algorithm, namely, the L 1 adaptive fractional control, is implemented on two-degrees-of-freedom lower limb exoskeleton [31], which is dedicated to the rehabilitation of cerebral palsy kids aged from two to ten years. Hence, the dynamic parameters are chosen as the mean values of lengths and masses (Table 1).

Parameters Optimization by Genetic Algorithms.
e choice of the filters' coefficients is the most important and essential part. In fact, they should be chosen to ensure the minimality of errors. For this purpose, an algorithm of optimization is proposed to determine these coefficients. Several numerical methods have been formulated from the minimization of a certain performance criterion to find an optimal solution such as genetic algorithms [32][33][34]. For a minimization problem of a criterion J (chosen in our case, r: between 0.1 and 0.95 ω L : between 1 and 100 ω H : between 10 2 × ω L and 10 5 × ω L e algorithm has been converged to the following results: Fixed ω L � 19 rad/s and ω H � 80000 rad/s for all cases 4.4. Performance Evaluation Criteria. One of our main objectives is to improve the precision and to increase the tracking accuracy of the lower limb exoskeleton through the proposed controller. Hence, let us define some performance indices in order to quantify the relevance of the proposed new L 1 adaptive fractional controller. e integral of the absolute error (IAE) and the relative integral of the absolute error (IAER) are accuracy evaluation tools used to evaluate the difference between the desired trajectory and the actual one during a specified time interval [t 1 , t 2 ] (e � q d − q denotes the tracking error). Moreover, it is very important to evaluate the energy consumption for the proposed controller. e evaluation criteria are defined as

Numerical Simulation Results
(1) Scenario 1: Nominal Case. In this section, simulation results will be presented in the nominal case. Two specific intervals are considered, defined by the following: s describes a steady-state period (a steady-state period describes the behavior of the system during one period of the steady-state response) e gait cycles of the hip and knee joints versus time are plotted in Figures 2 and 3 for the L 1 adaptive controller with different filters. en, Figures 4 and 5 show the hip and knee joints tracking errors. It is clear from these figures that the time lag that appeared while using the classical L 1 adaptive control has been eliminated thanks to the fractional-order filter. In fact, for the hip and the knee joints, the error is too high while using the standard filters (first-or second-order filter); however, the error becomes small and does not exceed 1 ∘ while using integral fractional-order filters (in case of C −0.88 filter).
In the following, performance values regarding the hip joint are indexed by 1 and those of the knee joint by 2. Figure 6 shows the integral of the absolute errors IAE and IAER versus the filter order r. e use of the integral fractional filter C −0.88 shows an improvement of 85.23% and 43.38% for the hip and knee joints, respectively, in terms of the tracking precision with regard to the first-order filter (see Table 2) and 90.31% and 69.82% with regard to the second-order filter. Moreover, the fractional filter C −1.87 proves an improvement in terms of the tracking error about 29% and 46.04% for the hip and knee joints, respectively, with regard to the second-order filter. Figures 7 and 8 depict the generated control input torques of the hip and knee joints. Figure 9 presents the maximum values of the torques while using L 1 adaptive controller with different filters. In terms of energy consumption, Figures 9 and 10 show clearly that there is a considerable difference between filters during the start-up    Mathematical Problems in Engineering improvement during the steady-state period, but it is less than C −0.88 filter. In fact, with regard to the first-order filter, there is no improvement, and with regard to the secondorder filter, about 12.9% and 7.12% improvement in terms of IAU and about 23.17% and 16.66% improvement in terms of ISU are proved. As a result, the integral fractional filter with −1 < r < 0 is proved to be the filter thanks to the rate of improvement that has been shown.
(2) Scenario 2: robustness towards noise. e aim of this section is to test the robustness of the integral filter with −1 < r < 0 towards the standard filters (first-and secondorder); hence, a multiplicative measurement noise (5%) has been added to the system. Figures 11 and 12 show the tracking gait cycles of the hip and the knee joints, respectively. It is obvious in Figures 13 and 14 that the effect of the noise is more important for the first-and second-order filters than for the fractional-order filter with −1 < r < 0     Mathematical Problems in Engineering based on the integral of the absolute errors IAE and the integral of the absolute torques IAU. Moreover, it is obvious from Figure 15 that the integral of the absolute errors IAE is high for both standard filters. In fact, the use of the integral fractional filter C −0.88 shows an improvement of 21% for the knee joint (during start-up) and an improvement of 90.87% and 79.39% for the hip and the knee joints, respectively (during one period), in terms of tracking precision compared to the first-order filter. With regard to the second-order filter, it shows an improvement of 15.49% for the hip joint (during start-up) and an improvement of 93.10% and 85.38% for the hip and knee joints respectively (during one period) (see Tables 5 and 6). Figure 16 shows the maximum values of the torques for the different used filters. It is clear that the torques' values, during start-up, are comparable to previous results shown in the first scenario (Figures 17 and 18).
Tables 7 and 8 present the improvement quantification of the energy consumption while using an integral fractional order filter. In fact, in terms of the integral of the absolute   torques IAU, there is an improvement of 38.04% and 30.8% while using C −0.88 filter with regard to the second-order filter for the hip and the knee joints, respectively, during the start-up and an improvement of 17.4% and 9.4% during the steady-state period. Moreover, there is an improvement of 11.42% and 11.96% while using C −0.88 filter with regard to          Table 5: Improvement quantification in % of tracking error while using the fractional filter in the presence of 5% noise during start-up for the hip and knee joints.
15.49% 0% Table 6: Improvement quantification in % of tracking error while using the fractional filter in the presence of 5% noise during one period for the hip and knee joints.    the first-order filter for the hip and the knee joints, respectively, during the start-up; and there is an improvement of 7.12% and 5.58% during the steady-state period.

Conclusion
In this paper, a new solution to eliminate the time lag that appears while using the L 1 adaptive control has been proposed. e new approach consists of replacing the standard integer filters, used in the L 1 adaptive control, by fractionalorder filters. e proposed approach has been tested on a polyarticulated system. Simulation results show clearly better performances of the new proposed L 1 adaptive fractional control. In fact, the use of the integral fractionalorder filter enhances the performances in terms of tracking error (elimination of the time lag) and energy consumption for both cases (nominal case and in the presence of a multiplicative noise). Moreover, the filter parameters have been optimized by genetic algorithms. In future works, L 1 adaptive fractional controller will be implemented on a real exoskeleton.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.    Table 8: Improvement quantification in % of energy while using the fractional filter in the presence of 5% noise for the knee joint during start-up and one period.