Initial-rectification Barrier Iterative Learning Control for Pneumatic Artificial Muscle Systems with Nonzero Initial Errors and Iteration-varying Reference Trajectories

Pneumatic artificial muscle actuators possess great potential in compliant rehabilitation devices since they are flexible and lightweight. The inherent high nonlinearities, uncertainties, hysteresis and time-varying characteristics in pneumatic artificial muscle systems brings much challenge for accurate system modeling and controller design. The angle tracking problem based on iterative learning control technology is considered in this work. This research proposes a new initial-rectification adaptive iterative learning control scheme for a pneumatic artificial muscle-actuated device with nonzero initial errors and iteration-varying reference trajectories. A barrier Lyapunov function is used to deal with the constraint requirement. A new initial rectification construction method is given to solve the nonzero initial error problem. Nonparametric uncertainties in the system are approximated by using a neural network, whose optimal weight is estimated by using difference learning method. As the iteration number increases, the system states of angle and angular velocity can accurately track the reference trajectories over the whole interval, respectively. In the end, the simulation results show excellent trajectory tracking performance of the iterative learning controller even if the reference trajectories are non-repetitive over the iteration domain.


I. INTRODUCTION
As a kind of tube-like actuators, pneumatic artificial muscle(PAM) actuators can contract or extend like real human muscles by inflating and deflating pressurized air through servo valves. The innate compliance and muscle elasticity of PAM actuators provide safe and soft interactions. The characteristic of low weight and flexibility of PAM actuators make them suitable for reconfigurable, compact and portable applications [1], [2]. Traditional electric and hydraulic actuators exhibit high stiffness, but are too heavy and rigid for medical rehabilitation and wearable applications. Due to the inherent features of high nonlinearities, complex hysteresis, and time-varying characteristics, achieving high-precision trajectory tracking control performance for PAM systems is not easy such that PAM actuators have not been extensively used in robotics to date even if they possess the obvious advantages over conventional actuators. To obtain satisfactory control performance, attempts have been made in the past two decades [3]- [12]. Up to now, the high-precision control of PAM systems is still a challenging issue.
Iterative learning control (ILC) technology has been put forward in the early 1980s. According to the system errors in the previous iteration(s), the control precisions of ILC systems may be gradually improved by updating the leaning parameters or control inputs, cycle by cycle, and better control performances may be obtained even without using accurate system models [13]- [22]. The special working principle of ILC bring continuous attention during the past serval decades. So far, ILC has been regarded as one of the most effective control strategies in handling repeated tracking control tasks or rejecting periodic disturbances for nonlinear systems, and has been applied in numerous practical applications, such as servo motors, robot manipulators, batch chemical process and traffic flows [23]- [25]. In the field of ILC, adaptive ILC, which can be seen as a combination of ILC and adaptive control, has been a hot topic since this century.
We will consider three important aspects in the research of ILC on PAM systems. The initial position problem of PAM ILC systems is the first issue that we will discuss. Theoretically speaking, for an ILC system, through continuous iterations by using the error information in the previous iteration(s), the control performance may get better and better during the whole operation time interval. However, in most traditional ILC algorithms, the above-mentioned excellent control performance is based on the premise that the initial error of ILC systems in each iteration should be zero; if the premise can not be satisfied, system divergence may occur even if the initial error is very slight. For the limitations of physical resetting, the zero initial error cannot be realized in practical applications. Consequently, how to design ILC controllers under nonzero initial error conditions, is a fundamental research issue in the field of ILC, which is usually be called initial position problem of ILC. In the context of PAM systems, the research on the initial position problem is still preliminary at present. Guo et al. proposed a robust adaptive ILC scheme to solve the angle tracking problem for a kind of PAM-actuated mechanism, with alignment condition used to solve the initial position problem [26]. Yang et al. design an angle error-track adaptive ILC algorithm solve the angle tracking problem for a PAM system with nonzero initial errors [27]. Overall, the number corresponding results is very limited. The initial position problem of PAM ILC systems is an issue worthy to be further studied.
The repetitiveness of reference trajectories for PAM ILC systems is another concern that we want to address. In most traditional ILC algorithms, a controlled system is assumed to perform a same specific control task during all iterations, i.e., the reference trajectory for a control system must be iterationinvariant. However, in practical applications, there exists the requirements on tracking iteration-varying reference trajectories to improve the efficiencies or match the technical processes. The earlier studies on iteration-varying trajectory tracking were reported in [28] on contraction-mapping ILC with slow iteration-varying reference trajectories, and in [29] on adaptive ILC with iteration-varying reference trajectories, respectively. On the basis of above works, some scholars continued in-depth research on adaptive ILC with iterationvarying reference trajectories [30]- [35]. In the context of PAM systems, the research on this issue is still a research blank at present.
In addition, the constraint requirement in ILC systems is also an interesting issue. For the purpose of system specifications and safety considerations, the system output, the system state, or the output tracking error should be constrained in some situations. Inspired by the development in barrier Lyapunov function-based adaptive control [36], [37], Jin and Xu carried out the earlier investigation on stateconstrained adaptive ILC [38] and output-constrained ILC [39]. Later on, some further results on barrier ILC have been reported, such as barrier error-tracking state-constrained ILC [40], state and input-constraint ILC [41], constrained datadriven optimal iterative learning control [42], joint position constrained robotic ILC [43], constrained spatial adaptive ILC [44], [45] . None of these works consider the state/output constraint ILC for PAM systems. How to develop an effective ILC algorithm to deal with PAM system under nonzero initial errors, as well as to meet the requirement of iteration-varying trajectory tracking and system constraint during operation, has not been addressed yet.
In this work, we present a novel barrier adaptive IL-C scheme for a PAM system with nonzero initial errors, iteration-varying reference trajectories and constraint requirements on angle/angle velocity tracking error. The main results and contributions of this work can be summarized as follows.
(1) A news construction method of rectification reference trajectories is presented to deal with initial position problem of PAM ILC system.
(2) The constraint requirement on angle/angluar velocity is implemented by using barrier Lyapunov function approach during the ILC design for the PAM system.
(3) By constructing a novel Lyapunov−Krasovskii functional, an adaptive ILC law is developed to address the iteration-varying trajectory tracking for the PAM ILC system. The rest of this paper is organized as follows. The problem formulation is introduced in Section II. The detailed procedure of controller design is addressed in Section III. The convergence analysis of closed-loop PAM systems is given in Section IV. In Section V, the simulation results are illustrated to verify the effectiveness of the proposed control scheme. Finally, Section VI concludes this work.

II. PROBLEM FORMULATION
Consider the angle tracking problem of a PAM-actuated device as shown in Fig. 1. The main component of this device includes an air compressor, two proportional valves, two PAM actuators, an angle sensor and a computer. In this PAM system, the control commands which are dictated from the computer can be sent to the two proportional valves. The computer queries the deflection angle of pulley through the sensor in real time. By opening and closing of the two valves, the force may be generated by the pressurized air inside PAM actuators. The two control variables of this system may be described as in which u o (t) is the initial preloaded control voltage, c u is the voltage distribution coefficient, and u(t) is the control input. The internal air pressures of actuators are determined by the property as follows: where P 1 (t) and P 2 (t) are the internal pressures of two PAM actuators, respectively; P 0 is the preloaded internal pressure, and ∆P is the variation of pressure. The relationship between the pulling forces and the internal air pressures of actuators may be expressed as follows: where F 1 (t) and F 2 (t) are two pulling forces of PAM actuators, c 1 -c 4 are four parameters, and 1 (t) and 2 (t) are derived according to (4).
where θ(t) and r are the deflection angle and the radius of pulley, respectively; 0 and l 0 are the initial shrinking rate and initial length of PAM actuators, respectively. The driving moment T p (t) of the device may be deduced from the following equation as where J p is the moment of inertia, b p is the damping coefficient, and d p (θ(t),θ(t), t) denotes unmodeled dynamics. Substituting (1)-(4) into (5) yields In the following of this paper, d p (θ(t),θ(t), t) is abbreviated as d p , and function arguments are sometimes omitted when no confusion arises. In real situations, the defection angle θ(t) is very small, such that 2c 1 (rθ(t)l −1 0 ) 2 ≈ 0 [4] and then (6) can be rewritten as Let (7), we get the state-space model of PAM systems during the kth iteration as The control task of this work is to let y k accurately track y d over [0, T ] while e e e k (0) = 0 cannot be guaranteed, as the iteration index k increases.

III. CONTROLLER DESIGN
In order to achieve the control objective, our control strategy is to make x x x k (t) follow the initial-rectification reference , which is formed as follows: where According to the construction in (9)-(10), x rk possesses the following properties:

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Define e e e r,k = [e r1,k , e r2,k ] T = x x x k − x x x r,k . Note that e e e r,k (0) = 0 holds, which is of significance to carry out the controller design in the next step. From (8) Let s r,k = λe r1,k +e r2,k with λ > 0. Obviously, s r,k (0) = 0.

IV. CONVERGENCE ANALYSIS
Theorem 1: For the closed-loop PAM ILC system composed of (8) and (17)- (20), the tracking performance and system stability are guaranteed as follows: (i) lim k→∞ e e e k (t) = 0 holds for t ∈ [t ω , T ]; (ii) |s r,k (t)| < b s holds during each iteration for t ∈ [0, T ].
(iii) All adjustable control parameters and internal signals are bounded ∀t ∈ [0, T ], ∀k. Proof 1: Define a barrier Lyapunov functional as follows: part A In this part, we will give the detailed calculation process of L k − L k−1 for subsequent analysis.
While k > 0, according to the definition of L k , we obtain Combining (17) with (16) yieldṡ V k ≤ − γσ k gs 2 r,k + σ k s r,k g˜ T k ξ ξ ξ k + σ k s r,k gw w w T k (t)ϕ ϕ ϕ(x x x k ) + σ k s r,k g˜ N,k + σ k s r,k g N,k − σ k s r,k g N,k tanh(σ k s r,k × N,k (k + 1) 2 ), By the property 0 ≤ |α| − α tanh( α ε ) ≤ 0.2785ε, we obtain Note that V k (0) = 0 holds because s r,k (0) = 0 is guaranteed according to the construction strategy of e e e r,k (t). Based on (23) and (24), calculating the integral ofV k from 0 to t, we have Then, substituting (25) into (22), we get From (18), we obtain g Combining (27) with (28), we have Similarly, from (20), we get g It follows from the above three inequations that Note that lim k→∞ j=k+1 j=1 holds. Further, we can get the recursive result of (31) as part B In this part, we will prove that b 2 From (21)-(24), we havė By the property of saturation function and (18), we have in which m is a proper positive number. Similarly, for a large enough m w > 0, by using the learning law (19), we obtain

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For a large enough m > 0, with the help of the learning law (20), we have Substituting (34)-(36) into (33) yieldṡ On the basis of which and L k (0) = 0, we get According to the definition of L k (t), we then deduce holds during each iteration for t ∈ [0, T ]. Note that s 2 r,k (0) = 0 for any k ≥ 0. Suppose that |s r,k (t)| may increase to b s− for any t ∈ (0, T ], then would happen, which is contrary to the inequality (39). Therefore, holds during each iteration. By Using the relationship λe r1,k +ė r1,k = s r,k , from (41), we can obtain On the other hand, according to (41), the boundedness of e r1,k , e r2,k , x r1,k and x r2,k may be deduced. Then, from (18)- (20), we can check that k , w w w k and k are bounded. Further, u k and all other signals in the closed-loop system may be verified to be bounded.
part C In this part, we will analyze the convergence of tracking error.
It is a direct result of (38) that Applying the conclusion given in (43), we have By using the nonnegativity of L k (t), (44) means From (45), we have lim k→+∞ |s r,k (t)| = 0.
In this work, barrier Lyapunov function approach is used to design the iterative learning controller. Through constraining s r,k , we implement the constraint to e r1,k and e r2,k during each iteration.
Figs. 2-12 express the simulation results. Figs. 2-3 give the profiles of angle position/angular velocity x 1 and their reference signals x 1,d and x 2,d during 29th iteration, respectively. Figs. 4-5 show the curves of e r1 and e r2 during 29th iteration, respectively. The control signal u q (t) during 29th iteration is depicted in Fig. 6. Figs. 7-8 give the profiles of angle position/angular velocity x 1 and their reference signals x 1,d and x 2,d during 30th iteration, respectively. Figs. 9-10 show the curves of e r1 and e r2 during 30th iteration, respectively. The control signal u q (t) during 30th iteration is depicted in Fig. 11. From Figs. 2-3 and Figs. 7-8, we can see that angle position/angular velocity states can accurately the reference signals for t ∈ [t ω , T ], respectively. As shown in Figs. 4-5 and Figs. 9-10, the rectification state error converges to zero over the interval [0, T ] as the iteration number increases. It can be observed from Fig. 12 that s r,k converges to zero, and |s r,k | is constrained between 0 and b s during each iteration, where J k max t∈[0,T ] |s r,k |. The above simulation results show that the tracking performance of closed-loop PAM ILC system improves progressively as the iteration number increases.

VI. CONCLUSION
An initial-rectification adaptive ILC scheme is proposed for a PAM system with nonzero initial errors and iteration-varying reference trajectories. The iterative learning controller is developed by using barrier Lyapunov function approach so as to constraint rectification filtering error during each iteration.