Hybrid FS–WNN approximator in indirect adaptive control of uncertain non‐linear MIMO systems

IET Control Theory & ApplicationsVolume 14, Issue 4 p. 539-548 Research Article Free Access Hybrid FS–WNN approximator in indirect adaptive control of uncertain non-linear MIMO systems Kheira Kahili, Corresponding Author kahili.kheira@gmail.com Mechatronics Laboratory (LMT), University of Mohamed Seddik Ben Yahia, Jijel, AlgeriaSearch for more papers by this authorOmar Bouhali, Mechatronics Laboratory (LMT), University of Mohamed Seddik Ben Yahia, Jijel, AlgeriaSearch for more papers by this authorNassim Rizoug, ESTACA' Lab, ESTACA School, Box 76121, Laval, 53000 FranceSearch for more papers by this authorFouad Khenfri, ESTACA' Lab, ESTACA School, Box 76121, Laval, 53000 FranceSearch for more papers by this author Kheira Kahili, Corresponding Author kahili.kheira@gmail.com Mechatronics Laboratory (LMT), University of Mohamed Seddik Ben Yahia, Jijel, AlgeriaSearch for more papers by this authorOmar Bouhali, Mechatronics Laboratory (LMT), University of Mohamed Seddik Ben Yahia, Jijel, AlgeriaSearch for more papers by this authorNassim Rizoug, ESTACA' Lab, ESTACA School, Box 76121, Laval, 53000 FranceSearch for more papers by this authorFouad Khenfri, ESTACA' Lab, ESTACA School, Box 76121, Laval, 53000 FranceSearch for more papers by this author First published: 01 March 2020 https://doi.org/10.1049/iet-cta.2019.0078Citations: 1 AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract In this study, a novel adaptive hybrid fuzzy system-wavelet neural network (FS-WNN) approximator is developed for control of uncertain non-linear multiple-inputs and multiple-outputs systems. The main idea is to use two different universal approximators to approximate each unknown non-linear function of the dynamic system. The first approximator is a FS and the second approximator is a WNN. Each approximator approximates the system's unknown functions independently. Then, the two approximations of the same function are combined using a modulation technique. The used modulation attributes to the first approximation a scaling factor and attributes its complementary factor to the second approximation to synthesise a hybrid optimal approximation of the function. Thus, system unknown functions are accurately approximated using the novel adaptive hybrid approximator that gathers the advantages of both approximators. The adaptation laws of the developed hybrid approximator are derived using Lyapunov's direct method to ensure the stability of the closed loop system. A simulation study is given to evaluate the performance of the proposed hybrid FS–WNN approximator in comparison with the FS and WNN approximators in indirect adaptive control of a two-link robot manipulator. 1 Introduction Control techniques of non-linear systems are an active field of research. Backstepping control and sliding mode control are non-linear advanced control techniques that depend on the system simplified mathematical model [1]. However, physical systems are complex by nature, with multiple inputs and multiple outputs (MIMO). The mathematical models of these systems are often imprecise due to the existence of parametric uncertainties and non-linearity uncertainties in the model [2]. Consequently, advanced control techniques cannot ensure good performance in the presence of uncertainties. The adaptive control technique was introduced as an effective method to handle parametric uncertainties [3]. Moreover, to overcome the problem of non-linearity uncertainties, researchers have emerged the adaptive control technique with intelligent modelling techniques such as fuzzy systems (FSs) and artificial neural networks (NNs) [4, 5]. FSs and NNs can reliably approximate dynamic models with parametric uncertainties and non-linearity uncertainties [6]. FSs approximate the functions of the system dynamic model using the available knowledge about the system behaviour [7]. In the absence of prior knowledge about the system, there is no systematic method to determine the required number of rules or the desired parameters of the membership functions to design a performant FS. However, artificial NNs can estimate the unknown functions without prior knowledge of the system behaviour. Usually, in the design of NNs, large size networks are trained using training algorithms [8]. Unfortunately, training NNs is a challenging and time consuming procedure. Therefore, researchers have proposed to combine the FSs and the NNs in one scheme to incorporate their advantageous and to overcome their individual limitations. The motivation to combine FSs with NNs in one scheme is based on several insights: The individual limitations of both intelligent systems could be circumvented, e.g. the black box interpretability of the NNs, and the problem of selecting suitable membership function parameters for FSs [9]. This combination presents a solution to the tuning difficulties and the design problems of the FSs. The resulting network is more interpretable in the form of fuzzy linguistic rules. The FSs allow incorporation of the human expert knowledge in the design process [10]. In the literature, various combination schemes of the FSs and NNs have been proposed [11–16]. The most common and efficient combined intelligent scheme is the adaptive neuro fuzzy inference system (ANFIS) [14]. ANFIS is based on Takagi–Sugeno (T–S) fuzzy model. In contrary to the T–S fuzzy model, ANFIS uses the learning methods that are originally developed for training NNs, to tune the FS membership functions in order to achieve the desired performance [17]. However, ANFIS is constructed with a set of fuzzy rules with linear or constant consequent parts [18–20]. Chairez [15] proposed to replace the linear or the constant consequent part of ANFIS by a combination of sub-NNs, where the consequent part of each rule is composed of a weighted sum of sub-NNs, and the overall output is calculated by summing the outputs of all rules. In the same context, Tsai et al. [16] have integrated the wavelet theory with intelligent modelling techniques and designed a fuzzy wavelet neural network (FWNN) approximator. The FWNN assigns to the consequent part of each rule a single sub-wavelet neural network (WNN), and the overall output of the FWNN is calculated by summing the outputs of all rules. The fusion of FSs with NNs or WNNs in a single network is an effective method to overcome their individual limitations. However, the fused network of Chairez [15] and the FWNN of Tsai et al. [16] are expansions of ANFIS [21, 22], where different training algorithms are used to adjust the network structure and parameters. Therefore, these fused approximators inherit the difficulties of ANFIS. Qin et al. [18] indicated that the performance of ANFIS relies on the training dataset. Poor or noisy training dataset results in the system being improperly trained and causes large identification errors. In addition, several works have considered that a single network is not perfect because it often suffers from local minima [23]. Zhang [24] combined multiple NN approximators to get a more reliable approximator than a single NN. For that, three combination schemes were proposed and compared, including the average of the NNs outputs and the weighted average combination of the NNs outputs. The results showed that the combined networks outperformed the single network. On the other hand, Perrone and Cooper [23] showed that averaging in functional space guarantees an improved estimation and reduces the mean squared error (MSE). In addition, this study proved that averaging the outputs of estimators that are fallen into local minima and performing poorly in different regions of the function space, helps to manage the local minima sticky situation. Moreover, Cho [25] demonstrated mathematically that averaging (m) NNs reduces the MSE by a factor of (1/m), and precised that the combined networks must have different architectures. In summary, Golestaneh et al. [9], Li et al. [14], Walia et al. [17], Qin et al. [18], and Chairez [15] have fused a FS approximator and NN/WNN approximators in one network to design a single approximator that combines their advantages as shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint FWNN approximator In the works of Perrone et al. [23] and Zhang [24], multiple NN approximators have been averaged to obtain a more accurate combined intelligent approximator as shown in Fig. 2. Fig. 2Open in figure viewerPowerPoint Multiple NNs approximator To the best of the authors' knowledge, no previous work has implemented a FS approximator in parallel to a WNN approximator, and combined the outputs of the two approximators using the modulation technique that was proposed by Ghavidel and Kalat [26], Ghavidel [27], Dian et al. [28], Mao et al. [29] as shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint Proposed hybrid FS–WNN approximator The contribution of this study is the design of a novel adaptive hybrid FS–WNN approximator (Fig. 3). The novel hybrid approximator is the result of the hybridisation of a FS with a WNN. The FS and the WNN approximate independently each unknown function of the dynamic system. Then, the two approximations of the same function are combined using a modulation technique. Where a scaling factor α is attributed to the approximation of the FS and its complementary factor (1 − α) is attributed to the approximation of the WNN. The sum of these two scaled approximations yields an optimal hybrid approximation of the unknown function. The approximations of the system function obtained by the novel hybrid FS–WNN approximator gather the advantages of both approximators. In our case, the developed hybrid FS–WNN approximator is used in indirect adaptive control of a two-link robot manipulator system. The adaptation laws of the hybrid FS–WNN approximator are derived using Lyapunov's method to ensure the stability of the closed loop system. Simulation results and a comparative study of the proposed hybrid FS–WNN approximator in comparison with the FS and the WNN are given to evaluate the performance of the novel approximator. This study is organised as follows. Section 2 presents a formulation of the current problem. The adaptive FS and the adaptive WNN are presented in Sections 3 and 4, respectively. The novel hybrid FS–WNN is developed in Section 5. Simulation results and a comparative study of the FS, the WNN and the novel hybrid FS–WNN approximator in indirect adaptive control are given in Section 6. Finally, a conclusion is given in Section 7. 2 Problem formulation Consider the class of MIMO uncertain non-linear systems represented by [30] (1)where is a vector of state variables, is the control input. is the system output. , is the system's relative degree vector with and n is the system order. and are smooth unknown non-linear functions. is the vector of uncertainties. To design a control law such that the system output tracks the desired trajectory , it is necessary to make the following assumptions: Assumption 1.The vector of state variables is assumed to be observable [30]. Assumption 2.Uncertainties vector is assumed to be bounded [30]: for . Assumption 3.The desired reference signal is smooth and bounded and so are all its derivatives up to order , i = 1,…,m [30]. Let the tracking error be (2)and define the filtered tracking error as [31] (3)where , i = 1,…,r − 1 are chosen positive constant diagonal matrices such that the roots of the polynomials s = 0 are in the left half of the complex plane. The time derivative of (3) is given by (4)with . In case and are precisely known and the system is free of uncertainties, the optimal control law is designed as (5)where: (6) (7)with , , i = 1,…,m and sgn(.) is the sign function. Replacing (5) in (4), yields , which implies that as . Thus, and all its time derivatives up to . However, in most of the cases, system non-linearities and are unknown. The optimal control law cannot be implemented. Therefore, to design the control law u, and must be estimated using universal approximators. In the following section, the FS approximator is presented. 3 Fuzzy system (FS) A FS is a universal approximator that approximates system unknown dynamics using the available linguistic knowledge about the system behaviour. A MIMO FS is usually constructed of a set of IF–THEN rules of the form: Rl : IF is and is and…and is THEN is , is , …, is where is the FS input vector. hFS j is the FS j th output for j = 1,…, m and m is the number of outputs. θlj is an adaptive parameter connecting output hFS j with rule Rl for l = 1,…, L and L is the number of rules. are Gaussian membership functions , and are centre and width of membership function with respect to input . The firing strength of rule Rl is given as , while fuzzy basis functions are defined as . The structure of a FS is shown in Fig. 4. Fig. 4Open in figure viewerPowerPoint Fuzzy system (FS) The j th output of the FS is calculated as (8)where is a vector of fuzzy basis functions, is a vector of weights connecting rule l with network output hFS j. The overall output of the MIMO FS of Fig. 4 can be written as (9)with . The universal approximation theorem demonstrates that any continuous function can be approximated over a compact set with finite approximation errors. Referring to the theorem, theoretically, there exists an optimal parameters matrix of to approximate and to minimise the approximation error vector (10) Assumption 4.Approximation error vector is bounded. This assumption results from the universal approximation theorem. Thus, system functions and can be approximated using MIMO FSs as follows: (11) (12)Practically, ideal parameters and cannot be determined. Their estimates and are employed. Therefore, system unknown functions are approximated as (13) (14)Define the parameter approximation errors as and . It is known in fuzzy theory that FSs require prior knowledge in the form of fuzzy rules to be able to estimate the system dynamics. Meanwhile, WNNs do not require any prior knowledge to estimate the functions and . 4 Wavelet neural network (WNN) A WNN is a three-layer feed forward NN with wavelets as activation functions. The structure of a MIMO WNN is presented in Fig. 5, where is the WNN input vector. hWNN j is the WNN j th output for j = 1,…, m and m is the total number of outputs. for i = 1,…, nc are the activation functions of the hidden layer and nc is the total number of the hidden nodes. with is the Mexican hat wavelet function. Fig. 5Open in figure viewerPowerPoint Wavelet neural network (WNN) The expression of the Mexican hat wavelet is given by [9] as (15)with and are translation and dilation parameters of wavelet with respect to input . The j th output of the WNN is calculated as (16)while is the matrix of translation parameters with and is the matrix of dilation parameters with . Also is a vector of activation functions of the hidden layer and is a vector of weights connecting hidden node i with network output hWNN j. The overall output of the WNN of Fig. 5 is written in a compact form as (17)where . Referring to the universal approximation theorem, theoretically, there exist and optimal parameters matrices of and to approximate and minimise the approximation error vector (18) Assumption 5.Approximation error vector is bounded. System unknown functions and are approximated using MIMO WNNs as follows: (19) (20)Practically, optimal parameters and cannot be determined, their estimates and are used. Therefore, system unknown functions are approximated as (21) (22)Denote parameters approximation errors as , , , and . In the following section, the hybrid FS–WNN approximator that combines the FS and the WNN advantages is developed. 5 Hybrid FS–WNN We present in this section the novel hybrid FS–WNN approximator. Fig. 6 illustrates the architecture of the hybrid FS–WNN approximator in the closed loop system. Fig. 6Open in figure viewerPowerPoint Architecture of the hybrid FS–WNN approximator-based closed loop system The hybrid FS–WNN approximator combines the advantages of the FS and the WNN from the integration of knowledge and input–output data in one scheme, to combining several approximators to obtain an optimal hybrid approximator that surpasses local minima difficulty and surmounts individual limitations of each approximator. The principle of hybridisation of the two adaptive intelligent systems of Figs. 4 and 5 is as follows: each of the intelligent systems has a distinct function approximation of each unknown function of the non-linear system. The two approximations of the same function are combined using (α, I − α) and (β, I − β) modulation parameters, where α and β are diagonal square matrices with constant values selected by the designer, and I is the identity matrix. Attributing a different scaling factor to the approximation of each approximator permits balancing between the two estimators by adjusting the elements of α and β in the interval [0, 1]. The (α, I − α) and (β, I − β) factors have a dependency relationship. When the contribution of the first approximator increases by a specific amount, the contribution of the second approximator decreases by the same amount. Thus, the sum of the complementary factors remains equal to I. System functions and are approximated using the proposed hybrid FS–WNN approximator as follows: (23) (24)With and . Using (11), (12), (19), and (20) yields (25) (26)Practically, optimal parameters , , and cannot be determined, their estimates , , and are used. Therefore, using (13), (14), (21), and (22), system unknown functions are approximated as (27) (28)Adding and subtracting the term in (4) gives (29)Choosing the control law as (30)To avoid the chattering phenomenon caused by switching term in (30), a proportional–integral control law [32] is employed when the state is within a boundary layer as follows: (31)with K1, K2, K are positive diagonal gain matrices. v is a positive constant. Thus the control law (30) is rewritten as: (32)Substituting (32) in (29), one obtains (33)Replacing and by their estimates of (25) and (26) and using (27) and (28) yields (34)Developing (34) as (35)After simplification, (35) is written as follows: (36)Using Taylor linearisation [32] (37)where and is a vector of high-order terms, also (38)and Replacing (37) in (36), yields (39)Setting and in (39), yields (40)Rearranging (40) as (41)with . Assumption 6.The approximation error is assumed to be bounded by [32]. To achieve the control objectives, it is necessary to adjust the hybrid FS–WNN approximator parameters in order to have as . Thus, the parameters adaptation laws are (42) (43) (44) (45) (46) (47)with , , , , , and are positive constant learning rates. Theorem 1.Consider the closed loop system of Fig. 6, with the uncertain non-linear MIMO system (1). The adaptive controller is chosen as (32) and the parameters adaptation laws (42)–(47). The adaptive hybrid FS–WNN-based controller guarantees that the closed loop system is asymptotically stable, and all signals are bounded. Proof of Theorem 1.Consider a Lyapunov function candidate as (48)The time derivative of (48) is given as (49)Substituting (41) in (49) gives (50)Rearranging all the terms to have (51)Finally, one obtains (52)with , , , , , and . Therefore, using (42)–(47) yields (53)using Assumptions 2 and 6 (54) (55)Since K and are selected by the designer's choice, K must be chosen such that . Therefore, it has been proved that the closed loop system is asymptotically stable. Thus, all signals are bounded which implies that the tracking error e and all its derivatives are bounded, also according to (3) s is bounded, i.e. . Integrating both sides of (55) yields (56)which signifies that based on the Barbalat lemma: , so . Thus, the tracking error will converge to zero as . By observing (56), we find that α and β are not present. Thus, we conclude that the stability of the closed loop system is independent of modulation parameters α and β, and whatever the value of α and β, the stability and performance of the control system is always guaranteed even at the extreme cases, i.e. when α = 0 and β = 0 or when α = I and β = I. □ 6 Simulation results A two-link robot manipulator of Fig. 7 is characterised by the following dynamic equations of movement [31]: is the joint positions vector and is the joints velocities vector. Fig. 7Open in figure viewerPowerPoint Two-link robot is the control torques vector applied to the system where is the positive definite inertia matrix, and are Coriolis and gravitational effects, respectively. and are uncertainties terms. The detailed procedure to derive the two-link robot model is given according to the Lagrange equation in [33]. The robot physical parameters are , , , and . The state vector is given as and output vector as . Let The design parameters are chosen as K1 = K2 = diag(8, 12), K = 20I2, the sliding surface parameters are selected from [31] as λ = I2. Initial conditions of the robot manipulator are given as (x1, x2, x3, x4) = (0.3, 1, 0.5, 0). The learning rates are chosen as , , , , , and . The MIMO FS has four inputs, two outputs, and 81 rules. For each input xi we associate three Gaussian membership functions, their translation and dilation parameters are initialised with a grid partitioning technique as and for k = 1,…, n, while the output weights of the FS are selected randomly in the interval [−1, 1] and tuned online. The WNN has similarly four inputs, two outputs, and 81 hidden nodes. The WNN translation and dilation parameters are also initialised using grid partitioning technique as c = (−0.5, 0, 0.5) and σki = 1.2 for k = 1,…, n, while the output weights are selected randomly in the interval [−1, 1]. Both the translation parameters and output weights of the WNN are tuned online. Two FS–WNNs have been designed to approximate system unknown functions and . Each FS–WNN was synthesised by combining a MIMO FS with a MIMO WNN using modulation technique, where the function approximation of generated by the FS was scaled by α and the function approximation of generated by the WNN was scaled by the complementary factor of α which is (I − α), the sum of these scaled approximations constitutes the hybrid approximation of system unknown function . The same procedure has been used to approximate , where modulation parameters (β , I − β) have been used to associate the different approximations of . The values of α and β are selected by trial and error as α = diag(0.01, 0.4) and β = diag(0.7, 0.8). The desired trajectories are selected as . To evaluate the performance of the hybrid FS–WNN in comparison with the WNN and FS, three indirect adaptive controllers were designed based on the three approximators. The three controllers undergo three different tests as in the literature [31], the first test concerns the robot system manipulating in nominal conditions, i.e. in the absence of disturbances and plant parameter variations. The second and third tests aim to explore the robustness of the three controllers, where the second test investigates the effects of friction, introduced at t = 15 s. The third test investigates the effects of friction added to parameter change when the robot picks up a charge of 5 kg, introduced at t = 35 s. The tracking performance in nominal conditions for the first link y1 and second link y2 are given in Figs. 8 and 9, respectively. While Figs. 10 and 11 show the control signals for the two links, and the tracking errors are illustrated in Fig. 12 for the first link and Fig. 13 for the second link. Fig. 8Open in figure viewerPowerPoint Tracking performance of y1 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Fig. 9Open in figure viewerPowerPoint Tracking performance of y2 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Fig. 10Open in figure viewerPowerPoint Control signal u1 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Fig. 11Open in figure viewerPowerPoint Control signal u2 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Fig. 12Open in figure viewerPowerPoint Tracking error e1 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Fig. 13Open in figure viewerPowerPoint Tracking error e2 in nominal conditions for the three controllers (WNN, FS, FS–WNN) Figs. 8 and 9 show the tracking performances of system outputs y1 and y2 for the three controllers, i.e. the WNN-based controller, FS-based controller, and proposed hybrid FS–WNN-based controller. The three controllers could ensure high tracking performances, where system outputs y1 and y2 track accurately the desired trajectory y d. The control efforts of the three controllers in nominal conditions are given in Fig. 10 for the first link and in Fig. 11 for the second link. The WNN-based controller is showing an oscillatory control signal in the transient phase. In the case of the FS, the control signal is smooth. However, the FS and WNN control signals start with undesired high amplitude oscillations as shown in Fig. 11. In contrast, the hybrid FS–WNN is showing a control signal with small amplitude oscillations in the transient phase. However, these oscillations rapidly disappear. Figs. 12 and 13 show the tracking errors of the three controllers. Based on Figs. 12 and 13, it can be seen that the tracking error in the case of the FS is larger than the case of the WNN. Also, these figures show that the smallest and smoothest tracking error is obtained in the case of the hybrid FS–WNN adaptive indirect controller. To analyse precisely the performances of the three controllers in nominal conditions, their performances are compared in terms of their accuracy using the MSE function and the maximum value of the control effort. The MSE function is given as where N is the length of the error vector e. The maximum value of the control effort is given as Max(|u |). These criteria are considered in two cases, the first case is the transient period and the second case comprises only the permanent period from t = 8 s. The performances comparison in the absence of disturbances and plant parameter variations of the three controllers based on the MSE of their tracking errors is given in Table 1 for the first link and the second link in transient and permanent periods, while the comparison of the performances of these controllers based on the maximum value of their control efforts is given in Table 2 for both links in the transient and permanent periods in nominal conditions. Table 1. Performances comparison based on the MSE in the nominal conditions for the three controllers (WNN, FS, FS–WNN) Transient period Permanent period Link 1 Link 2 Link 1 Link 2 WNN 0.0059 0.0156 1.11 × 10−5 4.19 × 10−6 FS 0.0061 0.0169 3.84 × 10−5 8.83 × 10−5 FS–WNN 0.0057 0.0157 7.82 × 10−7 4.38 × 10−7 Table 2. Performances comparison based on the maximum value of control effort in nominal conditions for the three controllers (WNN, FS, FS–WNN) Transient period Permanent period Max |u1 | Max |u2 | Max |u1 | Max |u2 | WNN 114.91 91.48 100.92 44.54 FS 104.73 86.00 100.99 44.82 FS–WNN 114.63 54.87 102.73 45.32 Table 1 compares the performances of the three controllers based on the MSE of their tracking errors, while Table 2 compares the performances of these controllers based on the maximum value of their control efforts. By observing the MSE in the permanent period for both links, we see that the WNN presented a good tracking accuracy in the range of 10−6 in comparison with the FS that is in the range of 10−5. However, by combining the two approximators and by a proper choice of α and β, the hybrid FS–WNN minimised the tracking error to reach the range of 10−7. It is clear from Table 1 that the