Iterative Learning Control for High-Order Systems With Arbitrary Initial Shifts

In this paper, two iterative learning control methods are proposed for the different high-order systems with arbitrary initial shifts. The tracking errors caused by nonzero initial shifts are easily detected when applying conventional learning algorithms. But this defect is overcome through applying a step-by-step rectifying controller with initial rectifying action introduced in a small interval. It demonstrates the improvement of tracking performance and shows the robustness with respect to the stochastic initial shifts. Finally, simulation results are presented to illustrate the effectiveness of the stated algorithms.


I. INTRODUCTION
Iterative learning control (ILC) is suitable for the repeated operation in a limited interval. It can improve the task execution in the next iteration by the observation of the past attempts and achieve error-free tracking [1], [2]. Because of its simple design and small online computation, ILC is applied in industrial robot control [3], [4], chemical process control [5] and other occasions [6]- [8].
Iterative learning control has a strict condition toward the robustness analysis, which requires that the actual initial state must be same to the desired initial state at each iteration. In fact, this condition is not available. This strict condition limits the application of iterative learning control.In order to solve this problem, scholars have done a lot of beneficial researches [9]- [25]. [11]- [13] are early research results. they present that systems only achieve asymptotic tracking. Up to now, the strict requirement has been relaxed because of introducing the initial rectifying action. The algorithm presented in [14] ensures that the system with relative degree can achieve tracking completely from a pre-specified moment. [15] presents a new controller that can make the high-order multi-agent systems achieve the uniform tracking in a given The associate editor coordinating the review of this manuscript and approving it for publication was Yu-Huei Cheng .
interval through initial states rectifying actions. But [14], [15] are only for the fixed initial shifts. A new method proposed in [16] is different from others. It is a useful attempt for the nonlinear continuous system with arbitrary initial shifts. But the approach can't make the system achieve tracking completely through initial rectifying action. There is a steady error after finishing the initial rectifying action during the tracking process. So it only applies to the first-order systems. Hitherto, it is an open and important issue that the iterative learning control is applied to dynamical systems with non-fixed initial shifts by means of contraction mapping method.
Lyapunov-like method has brought great convenience in the process of convergence analysis. It has been used to focus on the ILC problem of dynamical systems with arbitrary initial shifts recently, and some results have been achieved [17]- [25]. The time-varying boundary layer method is presented in [20]. Because of the boundary layer monotonously decreasing with time and converging to zero, the tracking error decided by boundary layer converges to zero through learning. References [21]- [23] introduce initialrectified-attractor, and achieve complete tracking over a specified interval. Reference [24] summarizes the convergence results about five types of initial conditions. Reference [25] presents a discrete adaptive ILC method aiming at a class of VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ discrete time-varying uncertain systems with arbitrary initial shifts and non-identical trajectories. Initial rectifying actions introduced in the algorithms are combined into differential-differential(D-D) type feedback ILC technique for two classes of linear systems with non-fixed initial shifts. They can also guarantee the converged system output to achieve a desired trajectory with a smooth transient. Moreover, fast convergence of system errors is due to the application of compression mapping technology. At last, two numerical examples are given to illustrate the effectiveness of the proposed method.

II. PROBLEM FORMULATION
Consider the following class of high-order linear systems performing identical tasks repeatedly over a finite time interval, i.e., t ∈ [0, T ].
where k denotes the kth repetitive operation; x i,k (t) ∈ R m (i = 1, 2, · · · , n) are the states, and u k ∈ R r (r ≥ m), y k (t) ∈ R p are control input and output vectors, respectively. A, B, C are matrices with appropriate dimensions.

A. THE MATRIX B IS INVERTIBLE
The given desired states are x i,d (t) ∈ R m , i = 1, 2, · · · , n. From these, the tracking state errors are denoted as In order to achieve complete tracking over the given interval, we present the following control law.
among them, 0 and 1 are controller gains.
N > n is a positive integer and t p is a given positive real number. Obviously, Remark 1: Obviously, this is a new D-D type feedback ILC rule with initial rectifying actions for high-order systems. Compared with the traditional ILC rule, the control law designed in this paper not only combines a feedback item, but also contains initial shifts rectifying actions. The purposes of the design are not only to accelerate convergence, but also to achieve tracking completely. In practice, it is difficult to obtain the high-order derivative of the current output, so the high-order derivative of the output at the previous moment can be used to replace the high-order derivative of the current moment output. Of course, the current derivative term can be discarded, that is, 1 There is a following theorem with respect to the function i (t). Theorem 1: For the any function i (t), when t 0 ∈ [h i,2 , T ], the following formula is true.
2) 0 < q < n − i + 1 In this case, it is easy to obtain the following result.
So the conclusion is true and the proof is completed. Remark 2: Theorem 1 indicates that each rectifying function i (t) has an effect similar to a pulse function after (n − i + 1)th integration. In fact, the pulse function don't exist, so we can only find a similar function instead of the pulse function.
We call this controller as step-by-step rectifying controller, because the controller can achieve a reference trajectory x n,d (t) at the time t p , x i,d (t) at the time 2 (n−i) t p , and so on x 1,d at the time 2 (n−1) t p , that is to say, the system completes the tracking perfectly at time 2 (n−1) t p .

B. THE PRODUCT OF MATRICES C AND B IS INVERTIBLE
The given desired trajectory is y d (t) ∈ R p , t ∈ [0, T ]. From this, we can define the tracking errors in the kth iterative as In order to achieve complete tracking over the given interval, we present the following control law.
among them, 0 and 1 are controller gains.
where the other variables are defined the same as before.
In order to facilitate the convergence analysis, we cite some definitions and lemmas in the reference [17] at first.
where · is some norm and its definition is following. If Z (t) is an n-dimensional vector and With respect to the λ-norm, there is a following lemma.
Lemma 1: Lemma 2: Supposing the series b k satisfy the following condition,

III. CONVERGENCE ANALYSIS
In this section, the convergence of the system (1) after using the control law (3) and (8) will be considered separately in what follows.

A. THE MATRIX B IS INVERTIBLE
Regarding to the system (1), the convergence results of control law (3) are provided in Theorem 2. Theorem 2: For the linear continuous system (1), if the initial states are arbitrary and bounded, and there exist matrices B(B is invertible), 0 , 1 such that then the step-by-step rectifying control law (3) can make the system (1) track the desired trajectory. Especially the system can track perfectly when t ∈ [2 n−1 t p , T ]. Proof: For the convenience of description, let's define The remaining proof consists of four steps. In step 1, we will obtain a conclusion that lim k→∞ e (n−1) x,k+1 (t) | t=t p = 0. In step 2, the following conclusions that lim k→∞ e will be obtained in turn due to the previous results. On the basis of step 2, we can draw the conclusions that lim k→∞ e (i−1) x,k (h 1,2 ) = 0, i = n, n−1, · · · , 1 in step 3, and the exact convergence of uniformly tracking error is analyzed when t ∈ [h 1,2 , T ] in step 4.
If (I + B 1 ) −1 (I − B 0 ) < 1, taking the λ-norm for the both sides of (37), and using Lemma 1 and 2, then the following result can be obtained.

B. THE PRODUCT OF MATRICES C AND B IS INVERTIBLE
In this case, there is a following theorem about system (1) and control law (8).
Theorem 3: For a given desired trajectory y d (t), if control law (8) is applied to system (1) under the Assumption 1, and there exist matrices C, B(The product of matrices C and B is invertible), 0 and 1 such that then the step-by-step rectifying control law (8) can guarantee that the tracking error e k (t) is bounded. Especially, the tracking error e k (t) can converges monotonically to zero when t ∈ [2 n−1 t p , T ] and k → ∞. Proof: Since the structure and proofs are similar to the previous one, this subsection contains only three steps, and step 3 in the last subsection is omitted.
Let's denote the virtual error as According to the definitions of i,k (h i,1 )(i = 1, 2, · · · , n − 1), and using (1) and (8), we can obtain According to the formula of integration, we have Insert (41) with (42), and multiply with the matrix C at the both sides of (41).
Cx n,k+1 (t) − Cx n,k (t) = C x n,k (0) + t 0 n (t 1 )CB n,k (0)dt 1 Substituting n,k (0), Simplifying (44), we can obtain According to the definition of e (n−1), * k (t), it is easy to obtain Step 2: The proofs of lim k→∞ e Subsequently, we can obtain among them, the definition of i,k is the same as before.
Inserting (48) with (8), Using the formula of integration, we can obtain (50) and (51), and multiply with the matrix C at the both sides of (49) In fact, we can obtain the following equation. where When k → ∞, using (60), we can get * i,k = 0 and ω 2, * i,k = 0. Then (62) can be further simplified.
If (I + CB 1 ) −1 (I − CB 0 ) < 1, taking the λ-norm for the both sides of (63), and using Lemma 1 and 2, then the result can be obtained.
That is to say, uniform convergence of the system output to the desired trajectory is ensured on [h 1,2 , T ] when k → ∞.
Remark 4: Control law (3) and control law (8) have a sequential order in rectifying state errors. x n,k has the highest priority and x 1,k has the lowest priority. In other words, both of them rectify the state error of x n,k at first, then the errors of x n−1,k , x n−2,k , · · · , x 2,k , x 1,k respectively, and the last rectifying actions are finished when time t = 2 n−1 t p . From the definition of the functions i , we can know that the actions of rectifying the errors of x j,k (j = i + 1, i + 2, · · · , n) have completed when rectifying the state x i,k , and the rectifying action of x i,k will be complete at t = h i,2 . Even though the rectifying action of x i,k has been finished, it doesn't mean that x i,k has tracked the desired state x i,d when t ∈ (h j,1 , h j,2 ), j = i − 1, i − 2, · · · , 1 respectively. Because the equation (1) is true only when t = h j,1 , j = i − 1, i − 2, · · · , 1 or t ∈ [2 n−1 t p , T ] for any j . Once rectifying action of x i,k is over, the effects upon x j,k , (j = i + 1, i + 2, · · · , n) are also over and x j,k , (j = i + 1, i + 2, · · · , n) return to the desired states x j,d (j = i + 1, i + 2, · · · , n) respectively. In one word, the essence of these algorithms is to add a pulse function for each x i,k , and there is an order when adding pulse functions. The first is for x n,k , then for x n−1,k , · · · , x 2,k , and the last is for x 1,k .
Remark 5: When rectifying the error of x i,k , the effects upon x j,k , j = i, i + 1, · · · , n − 1 are inevitable. Considering the limitations of control input, system rectifies the state errors in the following way. The smaller i is, the longer it takes when rectifying because the smaller i is, the more it effects. For example, rectifying action must have influenced on x 2,k , x 3,k , · · · , x n,k when rectifying x 1,k .
Remark 6: The rectifying functions are not unique as long as they satisfied Theorem 1. For example, we can select the following rectifying functions.

IV. SIMULATION RESULTS
In this section, we will present two examples to demonstrate the effectiveness of the proposed methods. There is an invertible system gain matrix B in the first example, and the other is that the product of matrices C and B is nonsingular.

A. THE MATRIX B IS INVERTIBLE
We first consider the system (1) with matrices given by The task interval of the system is [0, 4] and the corresponding initial shifts rectifying interval is [0, 0.8). The desired trajectories are such as In the control law (3), the control gains and the candidate functions i (t), i = 1, 2 are given as 0 = 9.0000 −9.0000 −1.8000 3.6000 4,4] In order to verify our ILC algorithm for a class of linear systems with stochastic initial shifts, we randomise the initial states of system (64) that x 1,k (0) = (1 − 0.2 * rand; −0.1 * rand) T and x 2,k (0) = (0.5 * rand; π 2 + 0.5 * rand), which rand function vary between 0 and 1 at different iterations of ILC process randomly. 10 iterations are performed during the simulation.
Via simulation, we obtain figures 1 − 5, which show that y 1,k and y 2,k completely track y 1,d and y 2,d in finite time which is specified in advance. The dashed, dashed−dotted and solid lines represent the system outputs as the control law (3) is iteratively executed 8, 9 and 10 times, respectively, and the dotted line represents the desired output in figures 1 and 3, and there are similar meanings in other figures. Figures 1 and 2 demonstrate that the control law (3) does't rectify the errors of the trajectory x 1,k in the interval [0, 0.4), so the actual trajectories y 1,k doesn't trend to the desired trajectory y 1,d in this process. the errors of x 1,k will be rectified in the second stage.     From figures 3 and 4, we can know that y 2,k has completely track y 2,d when t = 0.4, but y 2,k don't follow up y 2,d when t ∈ (0.4, 0.8). The control law has completed rectifying the error of x 1,k when t = 0.8s, and y 2,k completely track y 2,d when t ∈ [0. 8,4].
From figure 5, we can observe that two control inputs are continuous, because the functions i (t), i = 1, 2 are continuous.

B. THE PRODUCT OF MATRICES C AND B IS INVERTIBLE
The second example is considered with the following form which the product of matrices C and B is nonsingular.
Transfer function of this system is G(s) = (2 * (20 * s − 1))/(s 2 * (50 * s 2 + 35 * s − 7)). Obviously, this system has a pole and a zero on the right half open plane of s. So it is a non-minimum phase system.One type of non-minimum phase system is a system containing non-minimum phase components or some local small loops are unstable systems; the other is a time-delay system. In the example, one of the eigenvalues of the system parameter matrix A is greater than zero. So the system is unstable.
In the simulation, the task interval of the system is [0, 8], and the corresponding initial shifts rectifying time interval is [0, 3.2]. The desired trajectories are given as In the control law (8), the control gains and the candidate functions i (t), i = 1, 2, 3 are given as 0 = 0.8; 1 = 0; Similarly, for the sake of examining the robustness of our ILC algorithm for this kind of linear systems with variable initial shifts, we let the initial conditions be reset to x 1,k (0) = (1 − 0.2 * rand; 0.1 * rand) T , x 2,k (0) = (−0.5 * rand; π/4 − 0.5 * rand) T and x 3,k (0) = (−(π/4) 2 + 2 * rand; 2 * rand). 20 iterations are performed during the simulation. 5156 VOLUME 8, 2020 FIGURE 6. Convergence performance of y 1,k in the presences of non-fixed initial shifts. We can conduct the simulation according to the above mentioned conditions and obtain figures 6 − 12, which show that y 1,k , y 2,k and y 3,k completely track y 1,d , y 2,d and y 3,d when t ∈ [3. 2,8]. Similarly, the dashed, dashedĺCdotted and solid lines represent the system outputs as the control law (8) is iteratively executed 18, 19 and 20 times, respectively, and the dotted line represents the desired output in figures 6, 8 and 10, and there are similar meanings in other figures. Figures 6 and 7 demonstrate that the control law (8) does't rectify the error of y 1,k until t ∈ [1.6, 3.2), so the actual trajectory y 1,k doesn't trend to the desired trajectory y 1,d when t ∈ [0, 1.6). The error of y 1,k will be rectified in the third stage.
From figures 8 and 9, we can know that the control law has completed rectifying the error of y 2,k when t = 1.6s, but y 2,k don't track x 2,d perfectly when t ∈ (1.6, 3.2).
From figures 10 and 11, we can know that the control law has completed rectifying the error of y 1,k when t = 3.2s, that is to say, the system has tracked perfectly after 15 iterations. The control law has completed rectifying the errors of y 2,k and y 3,k when t = 1.6s, but y 2,k (t) and y 3,k (t) don't track y 2,d (t) and y 3,d (t) perfectly when t ∈ (1.6, 3.2). Because it must effect tracking y 2,d (t) and y 3,d (t) when rectifying the state error e 1,k (t). In fact, y 2,k (t) and y 3,k (t) change not only very large but also very fast in these process. It's easy to understand, for instance, y 2,k (t) and y 3,k (t) actually represent   the speed and acceleration, rectifying the errors of y 1,k (t), i.e., changing displacement, it must need very large acceleration when spending little time. When t = 3.2s, rectifying the error of y 1,k (t) is completed certainly, and the system has complete tracking y 1,d (t), y 2,d (t) and y 3,d (t). Figure 12 demonstrates that the control input are very large when t ∈ (1.6, 3.2). Considering the limitations of system input in practice, it demands that the rectifying time can't be too short.
There are two examples presented in this paper to demonstrate that the proposed step-by-step ILC methods for two classes of linear systems with stochastic initial shifts are  very effective. they can overcome random initial shifts of ILC systems and can completely track the desired output trajectory over a small initial interval which is specified in advance.

V. CONCLUSION
The convergence and robustness properties of the step-bystep D-D type feedback iterative learning controllers have been presented in this paper. These controllers are suit for the high-order systems with arbitrary initial shifts, and the different controllers are designed for the different types of systems. Applying one of these controllers, there is an order when they rectify the states shifts. After finishing rectifying the each state shift, the systems can achieve tracking completely. Compared with the conventional ILC methods, the robustness performances of the proposed algorithms versus arbitrary initial shifts are greatly improved by the initial rectifying actions. However, the control schemes only solve any initial value problem in iterative learning control theoretically, and there is still a gap from the actual application.