# L1 Adaptive Fractional Control Optimized by Genetic Algorithms with Application to Polyarticulated Robotic Systems

Authors: Boutheina Maalej; Rim Jallouli Khlif; Chokri Mhiri; Mohamed Habib Elleuch; Nabil Derbel
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Recently, an adaptive control approach has been proposed. This approach, namedL1 adaptive control, involves the insertion of a low-pass filter at the input of the Model Reference Adaptive Control (MRAC). This controller has been designed to overcome several limitations of classical adaptive controllers such as (i) the initialization of estimated parameters, (ii) the stability problems with high adaptation gains, and (iii) the appropriate parameter excitation. In this paper, a new design of the filter is presented, used for L1 adaptive control, for which the desired performances are guaranteed (appropriate values of the control during start-up, a high filtering of noises, a reduced time lag, and a reduced energy consumption). Parameters of the new proposed filter have been optimised by genetic algorithms. The proposed L1 adaptive fractional control is applied to a polyarticulated robotic system. Simulation results show the efficiency of the proposed control approach with respect to the classical L1 adaptive control in the nominal case and in the presence of a multiplicative noise.


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## Body

## 1. 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 L1 adaptive controller was proposed in [9, 10]. The basic part of the L1 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 L1 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 L1 adaptive control is the time lag. Researchers in [13–15] have proposed to add different supplementary controllers aiming to eliminate the appearing time lag. In this paper, the proposed contribution is the L1 adaptive fractional control based on fractional calculus [16–19]. Then, 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, L1 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.The 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 L1 adaptive fractional controller. Moreover, the parameters of these filters have been optimized by genetic algorithms [27, 28].The rest of this paper is organized as follows. In Section2, a background on fractional-order systems is presented. Section 3 presents a background on L1 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. Then, 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.


## 2. Background on Fractional-Order Systems

Oustaloup’s recursive approximation algorithm is presented in this section. This 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]. The aim is to synthesize the rational function in the frequency range [ωL, ωH]:(1)Ds=sωLr,r∈−1;0,using a recursive distribution of zeros and poles as:(2)D˜s=∏k=−NN1+s/ωk′1+s/ωk,where −ωk′,−ωk is the pair of the zero and the pole of order k expressed as(3)ωk′=ωLωHωLαk−0.5r2N+1,(4)ωk=ωLωHωLαk+0.5r2N+1,where αk=k+N+0.5, with the following:ωL is the high transitional frequencyωH is the low transitional frequencyand 2N+1 is the total number of recursive poles and zeros. The gain of the filter is equal to the unity for ω<ωL.For a good approximation of the fractional-order filter, two or three decades at least should be considered betweenωL and ωH.


## 3. Background onL1 Adaptive Control

TheL1 adaptive control [30] is designed as shown in the block diagram of Figure 1. A square system is considered with an input vector u∈ℝm and an output vector y∈ℝm. The desired output and the desired output velocity vectors are, respectively, yd and y˙d. The dynamic system is described as(5)y¨=a1y+a2y˙+bu+b′θ,with θ is a nonlinear function that gathers all the nonlinearities of the system and b is a regular matrix. a1 and a2 are square matrices with the appropriate dimension.Figure 1
Block diagram ofL1 adaptive control.The generalized tracking error is defined as follows:(6)ρ=y˙−y˙d+Λy−yd,with Λ∈ℝm×m being a positive definite diagonal matrix. The control input vector ut is(7)ut=b−1umt+uad,with(8)umt=b−1Amρt−a1y−a2y˙+y¨d−Λy˙−y˙d,whereAm∈ℝm×m is a Hurwitz matrix that characterizes the desired transient response of the systemuad∈ℝm is an adequate adaptive termThe time derivative of (6) leads to the following equation:(9)ρ˙=but+b′θ−y¨d+Λy˙−y˙d.Then,(10)ρ˙=Amρt+uadt+b′θt.The adaptive control is defined in order to cancel the nonlinearityθt. For that, in order to estimate and compensate the term θt, let us consider the following state predictor:(11)ρ^˙=Amρ^t+uadt+b′θ^t−Kρ˜t,whereθ^t is the estimation of the nonlinear term θθ˜t=θ^t−θt is the error of the nonlinear term θρ˜t=ρ^t−ρt is the prediction errorK∈ℝm×m is a designed matrix introduced to reject high-frequency noises(12)ρ˜˙=Aρ˜t+b′θ˜t,with A=Am−K being a Hurwitz matrix defining a dynamical behavior of ρ˜ at least two to three times faster than Am.The Lyapunov function associated with the system is(13)V=ρ˜TPρ˜+θ˜TΓ−1θ˜,where Γ is the positive diagonal high gain matrix and P is a definite positive matrix verifying(14)PA+ATP=−Q<0.Its differential with respect to time gives(15)V˙=ρ˜TPA+ATP︸=−Qρ˜+2θ˜Tb′TPρ˜+2θ˜TΓ−1θ˜.=−ρ˜TQρ˜+2θ˜Tb′TPρ˜+Γ−1θ˜.︸=0=−ρ˜TQρ˜≤0.Hence,(16)θ˜.=−Γb′TPρ˜.The estimate ofθt, θ^t is obtained using the projection operator which prevents the estimated values from exceeding their admissible range specified in the control design:(17)θ^.t=Projθ^t,−Γb′TPρt.The adaptive control term is as follows:(18)uads=−Csb′θ^s,whereθ^s is the Laplace transform of θ^t and Cs is a BIBO (bounded-input bounded-output) stable transfer function.


## 4. ProposedL1 Adaptive Fractional Control Applied to Polyarticulated Robotic Systems

### 4.1. Main Contribution

In general, the standard integer-order filters used for theL1 adaptive control generate(i) High values of the control during start-up(ii) High energy consumption(iii) A small filtering of noises(iv) An important time lagFor these reasons, the main proposition is to replace these filters by fractional-order ones that are realised by a sequence of first-order systems. The real order of the fractional system is equal to2N+1.This 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.The transfer functions of all considered filters in this paper will be implemented and compared. They are defined as(19)C−1s=11+τs,(20)C−2s=11+τs2,(21)Crs=1+s/ωL1+s/ωHr,r∈−2,−1∪−1,0,where C−1 and C−2 filters can be represented as Cr, for high values of ωH, and τωL=1.


### 4.2. 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:(22)Mqq¨+Fq,q˙=τ,whereMq∈ℝn×n is the inertia matrixFq,q˙∈ℝn is the vector of the Coriolis, centrifugal, gravitational, and contact forcesτ∈ℝn is the vector of torques generated by actuatorsq=q1q2,…,qnT∈ℝn is the position vectorq˙=q˙1q˙2,…,q˙nT∈ℝn is the velocity vectorq¨=q¨1q¨2,…,q¨nT∈ℝn is the acceleration vectorThe proposed control algorithm, namely, theL1 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).Table 1
Summary of the dynamic parameters of the lower limb exoskeleton.
ParametersValuesMass of the exoskeleton1 kgMean value of thigh length of the human limb0.282 mMean value of shank length of the human limb0.282 mMean value of thigh mass of the human limb1.082 kgMean value of shank mass of the human limb3.535 kgExoskeleton inertia1.24×10−3 kg m2Thigh inertia of human limb8.78×10−3 kg m2Shank inertia of human limb2.73×10−3 kg m2


### 4.3. Parameters Optimization by Genetic Algorithms

The 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 [32–34]. For a minimization problem of a criterion J (chosen in our case, the tracking error), a genetic algorithm can proceed as follows [35]:(1)
Choice of a random generation ofNi individuals(2)
Evaluation of the individuals by computing their fitness translating the performance index to be optimized(3)
Cross of two parents with a probability to generate two children(4)
Mutation of the two parents with a probability(5)
Selection of bestNi solutions(6)
Evaluation of the new individuals(7)
Repetition of steps (3), (4), (5), and (6) until the new individuals are too close giving the same fitness or reaching the maximum number of iterations.The aim of this work is the determination of the filters’ parameters. The best parameters are those that give the lowest tracking error. A population ofNi=20 individuals and 30 generations is chosen. The initial populations of the different parameters are as follows: r: between 0.1 and 0.95ωL: between 1 and 100ωH: between 102×ωL and 105×ωLThe algorithm has been converged to the following results:r=−1 for C−1 (first-order filter)r=−2 for C−2 (second-order filter)r=−0.88 for C−0.88 (fractional-order filter)r=−1.87 for C−1.87 (fractional-order filter)FixedωL=19 rad/s and ωH=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 newL1 adaptive fractional controller. The 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 t1,t2 (e=qd−q denotes the tracking error). Moreover, it is very important to evaluate the energy consumption for the proposed controller. The evaluation criteria are defined as(23)IAE=∫t1t2edt,(24)IAER=∫t1t2edt∫t1t2qddt,(25)IAU=∫t1t2τdt,(26)ISU=∫t1t2τ2dt.


#### 4.4.1. 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:The intervalt1,t2=0,0.5 s describes the start-up duration, corresponding to the transient responseThe intervalt1,t2=1,2 s describes a steady-state period (a steady-state period describes the behavior of the system during one period of the steady-state response)The gait cycles of the hip and knee joints versus time are plotted in Figures2 and 3 for the L1 adaptive controller with different filters. Then, 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 L1 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).Figure 2
Evolution of the hip joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 3
Evolution of the knee joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 4
Evolution of the hip joint tracking errors versus time.Figure 5
Evolution of the knee joint tracking errors versus time.In the following, performance values regarding the hip joint are indexed by 1 and those of the knee joint by 2. Figure6 shows the integral of the absolute errors IAE and IAER versus the filter order r.The use of the integral fractional filterC−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.Figure 6
Clustered column chart ofIAE1, IAE2, and IAE1+IAE2, as well as IAER1, IAER2, and IAER1+IAER2 versus the filter order r.Table 2
Improvement quantification in % of tracking error while using fractional filters.
FiltersC−1C−2IAE1IAE2IAE1IAE2C−0.8885.23%43.38%90.31%69.82%C−1.870%0%29%46.04%Figures7 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 L1 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 and the steady-state periods, respectively. The improvement percentage of the energy consumption during both phases (start-up and steady-state periods) is illustrated in Tables 3 and 4. Indeed, there is an improvement of the integral of the absolute torques IAU, 12.77% and 13.64%, while using the C−0.88 filter with regard to the first-order filter and 38.24% and 31.06% of improvement with regard to the second-order filter, respectively, to the hip and knee joints. The integral of squared torques ISU is also improved (8.52% and 1.07% of improvement while using the C−0.88 filter with regard to the first-order filter and about 1.28% for the hip joint with regard to the second-order filter) during start-up. Moreover, there is an improvement of IAU, about 5.08% and 4.47%, with regard to the first-order filter and 24.56% and 16.61% of improvement with regard to the second-order filter. During the steady-state period, in terms of ISU, there are 7.24% and 7.59% of improvements with regard to the first-order filter and about 40.63% and 30.28% of improvements with regard to the second-order filter.Figure 7
Evolution of the control input torques versus time of the hip joint.Figure 8
Evolution of the control input torques versus time of the knee joint.Figure 9
Clustered column chart of the maximum of torques (U1max and U2max), IAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during start-up versus the filter order r.Figure 10
Clustered column chart ofIAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during steady-state period versus the filter order r.Table 3
Improvement quantification in % of energy during start-up period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.8812.77%13.64%8.52%1.07%38.24%31.06%1.28%0%C−1.870%0%34.04%40.55%18.52%14.31%28.82%21.36%Table 4
Improvement quantification in % of energy during steady-state period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.885.08%4.47%7.24%7.59%24.56%16.61%40.63%30.28%C−1.870%0%0%0%12.9%7.12%23.17%16.66%Besides, in the case ofC−1.87, the fractional filter shows an improvement during the start-up period with regard to the first-order filter (34.04% and 40.55% of improvement for the hip and knee joints in terms of ISU) and with regard to the second-order filter with 18.52% and 14.31% of improvement for the hip and knee joints in terms of IAU and about 28.82% and 21.36% in terms of ISU. Moreover, it shows an 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 second-order 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. The aim of this section is to test the robustness of the integral filter with −1<r<0 towards the standard filters (first- and second-order); 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 based on the integral of the absolute errors IAE and the integral of the absolute torques IAU.Figure 11
Evolution of the hip joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 12
Evolution of the knee joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 13
Evolution of the hip joint tracking error versus time in the presence of 5% of noise.Figure 14
Evolution of the knee joint tracking error versus time in the presence of 5% of noise.Moreover, it is obvious from Figure15 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 15
Clustered column chart ofIAE1sup and IAE2sup during start-up and IAE1p and IAE2p during steady-state period, for both joints, versus the filter order r.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.
FiltersC−1C−2IAE1supIAE2supIAE1supIAE2supC−0.880%21%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.
FiltersC−1C−2IAE1pIAE2pIAE1pIAE2pC−0.8890.87%79.39%93.10%85.38%Figure16 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).Figure 16
Clustered column chart of the maximum of torques (U1max and U2max), IAU1sup and IAU2sup during start-up and IAU1p and IAU2p during steady-state period, for both joints, versus the filter order r.Figure 17
Evolution of the control input torques versus time of the hip joint in the presence of 5% of noise.Figure 18
Evolution of the control input torques versus time of the knee joint in the presence of 5% of noise.Tables7 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 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.Table 7
Improvement quantification in % of energy while using the fractional filter in the presence of 5% noise for the hip joint during start-up and one period.
FiltersC−1C−2IAU1supIAU1pIAU1supIAU1pC−0.8811.42%7.12%38.04%17.4%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.
FiltersC−1C−2IAU2supIAU2pIAU2supIAU2pC−0.8811.96%5.58%30.8%9.4%


## 4.1. Main Contribution

In general, the standard integer-order filters used for theL1 adaptive control generate(i) High values of the control during start-up(ii) High energy consumption(iii) A small filtering of noises(iv) An important time lagFor these reasons, the main proposition is to replace these filters by fractional-order ones that are realised by a sequence of first-order systems. The real order of the fractional system is equal to2N+1.This 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.The transfer functions of all considered filters in this paper will be implemented and compared. They are defined as(19)C−1s=11+τs,(20)C−2s=11+τs2,(21)Crs=1+s/ωL1+s/ωHr,r∈−2,−1∪−1,0,where C−1 and C−2 filters can be represented as Cr, for high values of ωH, and τωL=1.


## 4.2. 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:(22)Mqq¨+Fq,q˙=τ,whereMq∈ℝn×n is the inertia matrixFq,q˙∈ℝn is the vector of the Coriolis, centrifugal, gravitational, and contact forcesτ∈ℝn is the vector of torques generated by actuatorsq=q1q2,…,qnT∈ℝn is the position vectorq˙=q˙1q˙2,…,q˙nT∈ℝn is the velocity vectorq¨=q¨1q¨2,…,q¨nT∈ℝn is the acceleration vectorThe proposed control algorithm, namely, theL1 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).Table 1
Summary of the dynamic parameters of the lower limb exoskeleton.
ParametersValuesMass of the exoskeleton1 kgMean value of thigh length of the human limb0.282 mMean value of shank length of the human limb0.282 mMean value of thigh mass of the human limb1.082 kgMean value of shank mass of the human limb3.535 kgExoskeleton inertia1.24×10−3 kg m2Thigh inertia of human limb8.78×10−3 kg m2Shank inertia of human limb2.73×10−3 kg m2


## 4.3. Parameters Optimization by Genetic Algorithms

The 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 [32–34]. For a minimization problem of a criterion J (chosen in our case, the tracking error), a genetic algorithm can proceed as follows [35]:(1)
Choice of a random generation ofNi individuals(2)
Evaluation of the individuals by computing their fitness translating the performance index to be optimized(3)
Cross of two parents with a probability to generate two children(4)
Mutation of the two parents with a probability(5)
Selection of bestNi solutions(6)
Evaluation of the new individuals(7)
Repetition of steps (3), (4), (5), and (6) until the new individuals are too close giving the same fitness or reaching the maximum number of iterations.The aim of this work is the determination of the filters’ parameters. The best parameters are those that give the lowest tracking error. A population ofNi=20 individuals and 30 generations is chosen. The initial populations of the different parameters are as follows: r: between 0.1 and 0.95ωL: between 1 and 100ωH: between 102×ωL and 105×ωLThe algorithm has been converged to the following results:r=−1 for C−1 (first-order filter)r=−2 for C−2 (second-order filter)r=−0.88 for C−0.88 (fractional-order filter)r=−1.87 for C−1.87 (fractional-order filter)FixedωL=19 rad/s and ωH=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 newL1 adaptive fractional controller. The 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 t1,t2 (e=qd−q denotes the tracking error). Moreover, it is very important to evaluate the energy consumption for the proposed controller. The evaluation criteria are defined as(23)IAE=∫t1t2edt,(24)IAER=∫t1t2edt∫t1t2qddt,(25)IAU=∫t1t2τdt,(26)ISU=∫t1t2τ2dt.


### 4.4.1. 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:The intervalt1,t2=0,0.5 s describes the start-up duration, corresponding to the transient responseThe intervalt1,t2=1,2 s describes a steady-state period (a steady-state period describes the behavior of the system during one period of the steady-state response)The gait cycles of the hip and knee joints versus time are plotted in Figures2 and 3 for the L1 adaptive controller with different filters. Then, 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 L1 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).Figure 2
Evolution of the hip joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 3
Evolution of the knee joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 4
Evolution of the hip joint tracking errors versus time.Figure 5
Evolution of the knee joint tracking errors versus time.In the following, performance values regarding the hip joint are indexed by 1 and those of the knee joint by 2. Figure6 shows the integral of the absolute errors IAE and IAER versus the filter order r.The use of the integral fractional filterC−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.Figure 6
Clustered column chart ofIAE1, IAE2, and IAE1+IAE2, as well as IAER1, IAER2, and IAER1+IAER2 versus the filter order r.Table 2
Improvement quantification in % of tracking error while using fractional filters.
FiltersC−1C−2IAE1IAE2IAE1IAE2C−0.8885.23%43.38%90.31%69.82%C−1.870%0%29%46.04%Figures7 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 L1 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 and the steady-state periods, respectively. The improvement percentage of the energy consumption during both phases (start-up and steady-state periods) is illustrated in Tables 3 and 4. Indeed, there is an improvement of the integral of the absolute torques IAU, 12.77% and 13.64%, while using the C−0.88 filter with regard to the first-order filter and 38.24% and 31.06% of improvement with regard to the second-order filter, respectively, to the hip and knee joints. The integral of squared torques ISU is also improved (8.52% and 1.07% of improvement while using the C−0.88 filter with regard to the first-order filter and about 1.28% for the hip joint with regard to the second-order filter) during start-up. Moreover, there is an improvement of IAU, about 5.08% and 4.47%, with regard to the first-order filter and 24.56% and 16.61% of improvement with regard to the second-order filter. During the steady-state period, in terms of ISU, there are 7.24% and 7.59% of improvements with regard to the first-order filter and about 40.63% and 30.28% of improvements with regard to the second-order filter.Figure 7
Evolution of the control input torques versus time of the hip joint.Figure 8
Evolution of the control input torques versus time of the knee joint.Figure 9
Clustered column chart of the maximum of torques (U1max and U2max), IAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during start-up versus the filter order r.Figure 10
Clustered column chart ofIAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during steady-state period versus the filter order r.Table 3
Improvement quantification in % of energy during start-up period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.8812.77%13.64%8.52%1.07%38.24%31.06%1.28%0%C−1.870%0%34.04%40.55%18.52%14.31%28.82%21.36%Table 4
Improvement quantification in % of energy during steady-state period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.885.08%4.47%7.24%7.59%24.56%16.61%40.63%30.28%C−1.870%0%0%0%12.9%7.12%23.17%16.66%Besides, in the case ofC−1.87, the fractional filter shows an improvement during the start-up period with regard to the first-order filter (34.04% and 40.55% of improvement for the hip and knee joints in terms of ISU) and with regard to the second-order filter with 18.52% and 14.31% of improvement for the hip and knee joints in terms of IAU and about 28.82% and 21.36% in terms of ISU. Moreover, it shows an 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 second-order 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. The aim of this section is to test the robustness of the integral filter with −1<r<0 towards the standard filters (first- and second-order); 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 based on the integral of the absolute errors IAE and the integral of the absolute torques IAU.Figure 11
Evolution of the hip joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 12
Evolution of the knee joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 13
Evolution of the hip joint tracking error versus time in the presence of 5% of noise.Figure 14
Evolution of the knee joint tracking error versus time in the presence of 5% of noise.Moreover, it is obvious from Figure15 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 15
Clustered column chart ofIAE1sup and IAE2sup during start-up and IAE1p and IAE2p during steady-state period, for both joints, versus the filter order r.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.
FiltersC−1C−2IAE1supIAE2supIAE1supIAE2supC−0.880%21%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.
FiltersC−1C−2IAE1pIAE2pIAE1pIAE2pC−0.8890.87%79.39%93.10%85.38%Figure16 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).Figure 16
Clustered column chart of the maximum of torques (U1max and U2max), IAU1sup and IAU2sup during start-up and IAU1p and IAU2p during steady-state period, for both joints, versus the filter order r.Figure 17
Evolution of the control input torques versus time of the hip joint in the presence of 5% of noise.Figure 18
Evolution of the control input torques versus time of the knee joint in the presence of 5% of noise.Tables7 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 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.Table 7
Improvement quantification in % of energy while using the fractional filter in the presence of 5% noise for the hip joint during start-up and one period.
FiltersC−1C−2IAU1supIAU1pIAU1supIAU1pC−0.8811.42%7.12%38.04%17.4%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.
FiltersC−1C−2IAU2supIAU2pIAU2supIAU2pC−0.8811.96%5.58%30.8%9.4%


## 4.4.1. 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:The intervalt1,t2=0,0.5 s describes the start-up duration, corresponding to the transient responseThe intervalt1,t2=1,2 s describes a steady-state period (a steady-state period describes the behavior of the system during one period of the steady-state response)The gait cycles of the hip and knee joints versus time are plotted in Figures2 and 3 for the L1 adaptive controller with different filters. Then, 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 L1 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).Figure 2
Evolution of the hip joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 3
Evolution of the knee joint tracking versus time. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 4
Evolution of the hip joint tracking errors versus time.Figure 5
Evolution of the knee joint tracking errors versus time.In the following, performance values regarding the hip joint are indexed by 1 and those of the knee joint by 2. Figure6 shows the integral of the absolute errors IAE and IAER versus the filter order r.The use of the integral fractional filterC−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.Figure 6
Clustered column chart ofIAE1, IAE2, and IAE1+IAE2, as well as IAER1, IAER2, and IAER1+IAER2 versus the filter order r.Table 2
Improvement quantification in % of tracking error while using fractional filters.
FiltersC−1C−2IAE1IAE2IAE1IAE2C−0.8885.23%43.38%90.31%69.82%C−1.870%0%29%46.04%Figures7 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 L1 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 and the steady-state periods, respectively. The improvement percentage of the energy consumption during both phases (start-up and steady-state periods) is illustrated in Tables 3 and 4. Indeed, there is an improvement of the integral of the absolute torques IAU, 12.77% and 13.64%, while using the C−0.88 filter with regard to the first-order filter and 38.24% and 31.06% of improvement with regard to the second-order filter, respectively, to the hip and knee joints. The integral of squared torques ISU is also improved (8.52% and 1.07% of improvement while using the C−0.88 filter with regard to the first-order filter and about 1.28% for the hip joint with regard to the second-order filter) during start-up. Moreover, there is an improvement of IAU, about 5.08% and 4.47%, with regard to the first-order filter and 24.56% and 16.61% of improvement with regard to the second-order filter. During the steady-state period, in terms of ISU, there are 7.24% and 7.59% of improvements with regard to the first-order filter and about 40.63% and 30.28% of improvements with regard to the second-order filter.Figure 7
Evolution of the control input torques versus time of the hip joint.Figure 8
Evolution of the control input torques versus time of the knee joint.Figure 9
Clustered column chart of the maximum of torques (U1max and U2max), IAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during start-up versus the filter order r.Figure 10
Clustered column chart ofIAU1, IAU2, and IAU1+IAU2, as well as ISU1, ISU2, and ISU1+ISU2 during steady-state period versus the filter order r.Table 3
Improvement quantification in % of energy during start-up period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.8812.77%13.64%8.52%1.07%38.24%31.06%1.28%0%C−1.870%0%34.04%40.55%18.52%14.31%28.82%21.36%Table 4
Improvement quantification in % of energy during steady-state period while using fractional filters.
FiltersC−1C−2IAU1IAU2ISU1ISU2IAU1IAU2ISU1ISU2C−0.885.08%4.47%7.24%7.59%24.56%16.61%40.63%30.28%C−1.870%0%0%0%12.9%7.12%23.17%16.66%Besides, in the case ofC−1.87, the fractional filter shows an improvement during the start-up period with regard to the first-order filter (34.04% and 40.55% of improvement for the hip and knee joints in terms of ISU) and with regard to the second-order filter with 18.52% and 14.31% of improvement for the hip and knee joints in terms of IAU and about 28.82% and 21.36% in terms of ISU. Moreover, it shows an 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 second-order 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. The aim of this section is to test the robustness of the integral filter with −1<r<0 towards the standard filters (first- and second-order); 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 based on the integral of the absolute errors IAE and the integral of the absolute torques IAU.Figure 11
Evolution of the hip joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 12
Evolution of the knee joint tracking versus time in the presence of 5% of noise. Dashed line: desired trajectory; continuous line: actual trajectory.Figure 13
Evolution of the hip joint tracking error versus time in the presence of 5% of noise.Figure 14
Evolution of the knee joint tracking error versus time in the presence of 5% of noise.Moreover, it is obvious from Figure15 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 15
Clustered column chart ofIAE1sup and IAE2sup during start-up and IAE1p and IAE2p during steady-state period, for both joints, versus the filter order r.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.
FiltersC−1C−2IAE1supIAE2supIAE1supIAE2supC−0.880%21%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.
FiltersC−1C−2IAE1pIAE2pIAE1pIAE2pC−0.8890.87%79.39%93.10%85.38%Figure16 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).Figure 16
Clustered column chart of the maximum of torques (U1max and U2max), IAU1sup and IAU2sup during start-up and IAU1p and IAU2p during steady-state period, for both joints, versus the filter order r.Figure 17
Evolution of the control input torques versus time of the hip joint in the presence of 5% of noise.Figure 18
Evolution of the control input torques versus time of the knee joint in the presence of 5% of noise.Tables7 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 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.Table 7
Improvement quantification in % of energy while using the fractional filter in the presence of 5% noise for the hip joint during start-up and one period.
FiltersC−1C−2IAU1supIAU1pIAU1supIAU1pC−0.8811.42%7.12%38.04%17.4%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.
FiltersC−1C−2IAU2supIAU2pIAU2supIAU2pC−0.8811.96%5.58%30.8%9.4%


## 5. Conclusion

In this paper, a new solution to eliminate the time lag that appears while using theL1 adaptive control has been proposed. The new approach consists of replacing the standard integer filters, used in the L1 adaptive control, by fractional-order filters. The proposed approach has been tested on a polyarticulated system. Simulation results show clearly better performances of the new proposed L1 adaptive fractional control. In fact, the use of the integral fractional-order 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, L1 adaptive fractional controller will be implemented on a real exoskeleton.



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*Source: 5579541-2021-04-10.xml*