Filtering-Error Constrained Angle Tracking Adaptive Learning Fuzzy Control for Pneumatic Artificial Muscle Systems Under Nonzero Initial Errors

In this paper, a filtering-error constrained adaptive iterative learning control scheme is proposed to solve the angle tracking problem for a pneumatic artificial muscle-actuated mechanism. The adaptive learning controller is designed by a novel barrier Lyapunov function, and the filtering error of pneumatic artificial muscle system is ensured to be constrained during each iteration. The initial position problem of iterative learning control is solved by utilizing time-varying boundary layer method. Fuzzy logic system is applied to approximate the unknown nonparametric uncertainties in the pneumatic artificial muscle system, whose optimal weight is estimated by using difference learning approach. The approximation error of fuzzy logic system is tackled by robust control strategy. Simulation results show the effectiveness of the propose angle tracking adaptive learning fuzzy control scheme.


I. INTRODUCTION
Currently, by virtue of the excellent flexibility and compliance, soft robots have been widely used in rehabilitation robotics, prosthetic robots and the human-robot interaction. With the development of soft robots, soft actuators earn increasing interests as the formers' important components. Pneumatic artificial muscle (PAM) is a tube-like actuator and can largely mimic the functions of human muscles by inflating and deflating pressurized air through servo valves [1], [2]. PAM actuator may be seen as one of the most promising soft actuators for many advantages including rapid response, low cost and high power weight ratio. These merits have fueled the wide industrial applications for PAM actuators [3]- [6], [8]- [10], [44]. For the sake of existing inherent hysteresis, high nonlinearities and creep characteristics, the accurate The associate editor coordinating the review of this manuscript and approving it for publication was Bidyadhar Subudhi . modeling and control for PAM system is not an easy job, and the applicability of some existing control algorithms is limited. Therefore, in order to get better control performance, it is of urgent need to introduce new control design methods for decreasing the PAM-actuated mechanisms' sensitivity to dynamic uncertainties in practice.
Iterative learning control (ILC) is effective in handling repeated control processes [11]- [17], [17], [19], [20]. The ILC system can obtain excellent performances through gradual iterative learning, with little system model knowledge used [21]- [24]. Usually, PAM-actuated rehabilitation mechanisms are used to perform repeated tasks over a fixed period of operation time. Thus, ILC is a potential suitable control technique for obtaining accurate tracking. We will consider two important aspects of ILC algorithm designs for PAM systems in this work.
The first aspect is about the initial position problem of PAM systems. Many traditional ILC theoretical algorithms can perform well under zero initial error condition, but they perform poorly if this condition cannot be satisfied. More specifically, error divergence probably happens even if the initial error is very slight. Note that resetting the initial value of system state/output exactly to that of reference trajectory at each iteration is an impossible task in real applications. Hence, the zero initial error condition is too strict to be met in applications. The issue on how to develop iterative learning controller under nonzero initial condition, which is the so-called initial position problem of ILC, is a fundamental problem in ILC area. In order to solve this problem of ILC, some significant explorations and attempts have been made in the past two decades [25]- [30]. Up to now, the documents discussing the ILC approach for PAM systems are very few, and the documents have reported the result involving initial position problem of ILC for PAM systems are even fewer. In [31], a model-free discrete adaptive ILC method is proposed for nonlinear PAM systems. In [32], alignment condition is used to relax the initial position problem for PAM systems whose reference trajectory is smoothly closed, i.e., the proposed approach is suitable for the case that the initial state of reference trajectory is equal to the final state of reference trajectory. How to develop an effective control scheme for PAM systems whose reference trajectory is not closed is still unclear.
The second issue we will address in this work is about system constraints of PAM systems during operations. Due to the system specifications and safety considerations, there exist the requirements of constraining the system output, the system state, or the output tracking error in some situations. Some related results have been reported in literature, such as maximal output admissible set strategy [33], constrained model predictive control [34], reference governor approach [35], convex optimization strategy [36] and barrier Lyaponov function design method [37], [38]. In recent years, the system constrain problem in ILC design has attracted much interest. By referring to the state/output constraint technique in barrier adaptive control, the research results on barrier adaptive ILC have been reported in literature since the early 2010s. In [39], Jin et al. proposed an output-constrained adaptive ILC approach for a class of nonlinear systems which meets alignment condition. In [40], Yan and Sun presented an error-tracking iterative learning control method for a class of nonparametric uncertain systems to tackle simultaneously both the initial position problem and the state-constrained problem. In [41], Yu et al. developed an adaptive ILC algorithm for nonlinear uncertain systems with both state and input constraints. In [42], Jin investigated the joint position constrained ILC for robotic systems. Up to now, few literature has addressed the system constraint problem in the adaptive ILC for PAM systems.
Motivated by the above discussion, this work focuses on the adaptive ILC algorithm design for the angle tracking of PAM systems under nonzero initial errors with filtering error constraint. The main results and contributions are given as follows. 1) The filtering-error constraint in adaptive ILC for PAM systems is implemented by using a new type of barrier Lypunov function in ILC design. 2) Time-varying boundary layer and adaptive learning fuzzy logic system are jointly applied to deal with the initial position problem and uncertainties in PAM ILC systems. An adaptive ILC law is developed to make the angle tracking error converge to a tunable residual set as the iteration number increases. 3) In many existing related results, the state dependent control input gain in system model is assumed to be state independent, which is relaxed in this work. The paper is organized as follows. Section II introduces the control problem formulation and some preliminaries. In Section III, we propose the filtering-error constrained angle tracking adaptive learning fuzzy control scheme for PAM systems under nonzero initial errors via using the techniques of barrier Lyapunov function, time-varying boundary layer, and fuzzy logic system. The convergence analysis of closed-loop PAM systems is given in Section IV. In Section V, some simulation results are illustrated to verify the effectiveness of the proposed control scheme. Finally, Section VI concludes this work.

II. PROBLEM FORMULATION A. SYSTEM DESCRIPTION
In this work, the angle trajectory tracking problem is considered for a PAM-acutated mechanism [2], which is used to perform repeated tasks. The control system structure of the mechanism is shown in Fig. 1.
As shown in this figure, the main components include an computer, an air compressor, two proportional valves, two PAM actuators and an angle sensor. Two PAM actuators are parallel to each other. Through opening and closing of two pressure proportional valves, the charging or discharging of two PAM actuators are controlled by the computer. The signals of deflection angles are transmitted to the computer through an angle sensor. The two control variables of the PAM-actuated mechanism are presented as where t ∈ [0, T ], α u is the coefficient of voltage distribution, u o is the preloaded voltage, u(t) is the control input, and u l (t) and u r (t) are two input control voltages of pressure proportional valves, respectively. Here, T represents the interval of operation time at each iteration. The internal pressures of two PAM actuators are given as where α 0 is the proportional coefficient of the control voltage and output pressure. P 0 denotes the preload internal pressure of actuators, P(t) represents the variation of pressure, and P 1 (t) and P 2 (t) are two internal pressures of PAM actuators. The relationship between the pulling forces and the internal pressures of PAM actuators may be described as where F 1 (t) and F 2 (t) are two pulling forces of PAM actuators, α 1 , α 2 , α 3 and α 4 are four parameters, 1 Here, θ(t) denotes the deflection angle of the mechanism, 0 and l 0 represent the initial shrinking rate and the initial length of PAM actuators, respectively. The driving moment of this mechanism is shown as where J is the moment of inertia, b υ is the damping coefficient, r is the radius of pulley, and d υ (θ,θ, t) denotes the unknown external disturbances and unmodeled dynamics. By substituting (1)-(3) into (4), we havë Then, the state-space model of the PAM system at the kth iteration may be got as follows: where The control task of this work is to make x 1,k (t) accurately track its reference signal x 1,d (t) over [0, T ], as the iteration index k increases. For the sake of brevity, the arguments in this paper are sometimes omitted when no confusion is likely to arise.
Remark 1: In many results, the term α 1 (rθ(t)l −1 0 ) 2 u(t) in (5) is assumed to be zero or bounded before controller design. In this work, we give up this usual assumption.

B. FUZZY LOGIC SYSTEMS
All is well known, fuzzy logic system (FLS) or neural network-based controller has become an important approach for adaptive control [8]- [44], [44]- [47] and adaptive learning control while the nonlinearity in nonlinear plants cannot be linearly parameterizable as in [25], [48], [49]. A typical FLS consists of four main components: knowledge base, fuzzifier, fuzzy inference engine and defuzzifier. The fuzzy inference mechanism uses the fuzzy IF-THEN rules to perform a mapping from an input vector 1 be the fuzzy sets defined on the universe of discourse of the ith input for j i = 1, 2, · · · , N i , then the fuzzy logic system is characterized by several if-then rules: By implementing the strategy of singleton fuzzification, center-average defuzzifcation and product inference, the output of fuzzy logic system can be formulated as wherep l is the point at which the fuzzy membership function of G l achieves its maximum value. Let (7) can be rewritten as If Gaussian functions are adopted as the membership, then the following lemma holds.
Lemma 1: For amy real continuous function f (χ χ χ) defined on a compact set in R n , there exists a FLS (9) and an optimal parameter vector p p p * such that with ε * being a arbitrary positive constant [43]. 41830 VOLUME 10, 2022

III. CONTROLLER DESIGN
By letting Let s k = λe 1,k + e 2,k and where λ > 0, µ > 0 and sat ·,· (·) represents the definition of saturation function. For a scalarâ, which is the estimation to a scalar a, where a andā are the lower bound and upper bound of the scalar a, respectively. For a vectorâ a a = [â 1 , Without loss of generality, we assume is the noncontinuous remainder with unknown bound. According to (11), the time derivative of s k may be obtained aṡ Let us choose a barrier Lyapunov function as By φ,k and calculating the derivative of V k with respect to t, according to (14), we havė By using (12), we have Substituting (17) into (16) yieldṡ where w w w : . According to Lemma 1, the unknown continuous function g −1 (20) where ϑ ϑ ϑ * (t) is the the optimal fuzzy weight vector and ϑ ϑ ϑ k (t) is the estimate of ϑ ϑ ϑ * (t). The approximation error is defined ). (21) On the basis of (21), we design the control law and learning laws as follows: where γ 1 > 0, γ 2 > 0, γ 3 > 0, γ 4 > 0, γ 5 > 0, ρ k is used to estimate ρ := g −1 1 g 2 , and k is used to estimate (t) := sup( F + g −1 1 |d 2 (x x x k , t)|). Remark 2: According to the definition of s φ,k (t) given in (12) and (13), s φ,k (0) = 0 holds, which is useful for the formula derivation from (47) to (52). As shown in (15), barrier Lyaupunov function, rather than a traditional Lyapunov VOLUME 10, 2022 function V k = 1 2 s φ,k (t), is used for controller design. The purpose is to let s φ,k (t) constrained during each iteration as s φ,k (t) < b s .
Remark 3: While the barrier Lyapunov function V 2,k = 1 2 is adopted in controller design, one should check or prove the nonnegativity of V 2,k in advance, which brings the inconvenience in controller design. This inconvenience may be removed by replacing V 2,k with (15).
Proof: The proof consists of two parts. In Part A, we will prove that |s φ,k | < b s hold for any k. Then, we will analyze tracking error convergence in Part B.
Part B. The convergence of tracking error On the basis of (31) and (32), we have While k > 0, from (46), we can derive Note that in (47), V k (0) = 0. By using the relationship (25), we obtain Similarly, from (27) and (28), we can obtain and respectively. Substituting (48)-(51) into (47) yields By using the recursive relation (52) and the definition of V k−1 , we can further obtain for k > 0. It can be seen from (40) that L 0 (t) is bounded for t ∈ [0, T ]. Therefore, there exists a positive constant m L , which satisfies the equality as follows: Combining (53) with (54), we can draw a conclusion that lim k→+∞ s φ,k (t) = 0 holds for all t ∈ [0, T ], which means By using the relationshiṗ from (56), we can obtain as the iteration number k increases. Therefore, by choosing proper positive numbers λ and µ, |e 1,k (t)| may exponentially converge with respect to t.
In this work, difference learning method is applied to estimate unknown parameters, the optimal weight of FLS, and the upper bound of the sum of approximation error and noncontinuous disturbance. Since the uncertain input gain contains angle state signal and unknown parameters, it is difficult to design iterative learning controller through direct estimating all contained unknown parameters. Robust control strategy is used to deal with this difficulty during controller design.
In the implementation, seven fuzzy sets are characterized by the following membership functions.
The control parameters and gains in control law (22) and learning laws (25)-(26) are set as follows: λ = 2, µ = 5,     Fig. 6. Fig. 7 gives the convergence history of s φ,k in the filtering-error constrained ILC, where J k max t∈[0,T ] |s φ,k |. From Fig. 7, we can see that the constraint s φ,k < b s holds for each iteraion.   To verify the effectiveness of our proposed ILC algorithm, two comparison examples are provided in the following part.
Comparison A: To verify the constraint property of our proposed ILC algorithm, the simulation applying non-constraint adaptive ILC (60)-(65) is studied for PAM system (6).
The control parameters here are set as λ = 2, µ = 5, γ 1 = 5, γ 2 = 1, γ 3 = 1, µ 4 = 0.01, µ 5 = 0.01 and T = 7s. Fig. 8 shows that |s φ,k | converges to a tunable residual set as the iteration increases, but the constraint inequality s φ,k < b s is violated. The definition of J k in Fig. 8 is the same as the one in Fig. 7. Comparing Fig. 7 with Fig. 8, we can see that our constraint adaptive ILC algorithm is effective in both error convergence and system error constraint. Comparison B: To show the necessity of dealing with initial position problem, the robust adaptive ILC algorithm based on time-invariant boundary layer is adopted for simulation.

VI. CONCLUSION
A filtering-error constrained adaptive learning fuzzy control scheme is proposed to solve the tracking problem of PAM systems in this paper. To achieve system constraint in iterations, a new type of barrier Lyapunov function is introduced in controller design. The initial position problem of ILC for PAM systems is handled by using the technique of time-varying boundary layer. The continuous nonparametric uncertainties in the PAM system is compensated by the adaptive leaning FLS, whose optimal weight is estimated by difference learning approach. It is shown that the angle tracking error and angular velocity tracking error can asymptotically converge to a tunable residual set as the iteration number goes to infinity.