High Order Feedback-Feedforward Iterative Learning Control Scheme with a Variable Forgetting Factor

Abstract

In this work, we present a new iterative learning control (ILC) scheme for a class of non-linear systems with uncertain and non-repetitive disturbances, in order to achieve perfect tracking by proposing a high order feedback-feedforward ILC algorithm with a variable forgetting factor. The high order feedback-feedforward iterative learning controller can fully apply the previous control data to the system, which allows the system to track expectations more rapidly and precisely. Introducing a variable forgetting factor can weaken the former control output and its variance in the control law, while strengthening the robustness of the ILC. Through rigorous analyses, we demonstrate that uniform convergence of the state tracking error is guaranteed under this new ILC scheme. Simulation examples are also included to demonstrate the feasibility and effectiveness of the proposed learning controls.

1. Introduction

ILC is effective for handling repeated control processes. It has been widely used in industries for the control of repetitive motion, such as robotic manipulators, hard disk drives and chemical plants, because of its structural simplicity and effective learning ability in the controller design process [1–3]. ILC algorithms can asymptotically or exponentially improve the tracking performance to achieve perfect tracking with an increasing number of iterations by utilizing the repetitive nature of the learning process. As several surveys [4–6] have discussed the novel ideas and development of ILC methodology, we refer the reader to these references for more information on the main concepts of ILC.

ILC learns from past practice and experience to construct a control function for the current iteration. Learning performance can be expected to improve more as a greater number of previous iterations are used. This is called high order ILC, which was first proposed [7] for the tracking control of repetitive systems. A class of non-linear time-varying systems without uncertainty, disturbance and initialization error was discussed in previous studies for high order ILC algorithms. To overcome this problem, feedback-feedforward high order ILC [8] was proposed in the control system, despite uncertainties and disturbances. However, the control methods often achieved a poor effect upon practical applications when the system had non-repetitive disturbances and an initial error. Therefore, in this paper, we propose a high order feedback-feedforward ILC algorithm with a variable forgetting factor.

The contribution of this paper is the combination of high order feedback-feedforward ILC and a variable forgetting factor. Feedback control [9–11] enhances the anti-interference performance and improves the robustness of the system; thus, the tracking deviation decreases during the iterative process. By adding feedforward control [12–14], the system can avoid the high gain that occurs in the feedback control method and can eliminate the actuator's saturation. By introducing a variable forgetting factor [15], this method can filter the signal toward the direction of iteration, which can weaken the convergent influence of the system caused by the uncertain part of the system and non-repetitive disturbances. The combination is capable of achieving high performance trajectory tracking in simulation results, while maintaining good disturbance rejection.

The remainder of this paper is organized as follows. The problem statement for a class of repetitive non-linear time-varying systems is described in section 2. Section 3 presents the designed controller and the designed variable forgetting factor. In section 4, the convergence condition for the non-linear system is presented and the convergence analysis is provided. In section 5, a simulation is presented to verify the effectiveness of the proposed method. Finally, conclusions are drawn in section 6.

2. Problem Statement

Let us consider repetitive non-linear time-varying systems [16–17] with uncertainty and disturbances [18–19]:

$\begin{array}{l}{\dot{x}}_i(t)=f(x_i(t),t)+B(x_i(t),t)u_i(t)+w_i(x_i(t),t)\\ y_i(t)=C{x}_i(t)+v_i(t)\end{array}$

(1)

where i and t are the iteration index and continuous-time, respectively. $u_i(t)\in R^m$ is the control variable $x_i(t)\in R^m$ is the state variable, while $y_i(t)\in R^m$ is the output of the system at the i-th trial. $w_i(x_i(t),t)$ and $v_i(t)$ are the uncertain item and disturbance term, respectively. $C\in R^{m\times m}$ is a constant matrix. $f(\cdot )$ , $B(\cdot )$ and $w(\cdot )$ are non-linear functions

Assumption 1. Functions $f$ , $B$ and $w$ are, uniformly, globally Lipschitz with respect to x on a compact set $\mathrm{\Omega }\in R^m\times R^m\times \left[1,T\right]$ ,

$\begin{array}{l}\Vert f(x_{i+1}(t),t)-f(x_i(t),t)\Vert \le k_f\Vert x_{i+1}(t)-x_i(t)\Vert \\ \Vert B(x_{i+1}(t),t)-B(x_i(t),t)\Vert \le k_B\Vert x_{i+1}(t)-x_i(t)\Vert \\ \Vert w(x_{i+1}(t),t)-w(x_i(t),t)\Vert \le k_w\Vert x_{i+1}(t)-x_i(t)\Vert \end{array}$

(2)

where k_f, k_B and k_w are the Lipschitz constants.

Assumption 2. The initial error of the system at the i-th trial satisfies the criterion:

$\Vert x_{i+1}(0)-x_i(0)\Vert \le b_{x_0}{}^{}{,}^{}\forall i$

(3)

Assumption 3. For $y_d(t)$ , there exist unique $u_d(t)$ and $x_d(t)$ , which satisfy the following equations:

$\begin{array}{l}\frac{\partial x_d(t)}{\partial t}=f(x_d(t),t)+B(x_d(t),t)u_d(t)+w(x_d(t),t)\\ y_d(t)=C{x}_d(t)\end{array}$

(4)

where $u_d(t)$ is the desired control input and $x_d(t)$ is the desired state.

It is important to introduce the Lambda norm. The Lambda norm for a function is defined as:

${\Vert \ast \Vert }_{\lambda }=\underset{t\in \left[0,T\right]}{\sup }\left\{e^{-\lambda t}\Vert \ast \Vert \right\}$

(5)

where $\lambda >0$ .

λ -norm is defined to simplify the formula as follows:

$\begin{array}{l}{b}_C\triangleq \sup {\Vert C\Vert }_{\lambda }, b_v\triangleq \max \{\underset{t\in \left[0,T\right]}{\sup }{\Vert v_l(t)\Vert }_{\lambda },\underset{t\in \left[0,T\right]}{\sup }{\Vert v_{l+1}(t)\Vert }_{\lambda }\}, \\ b_{\phi }\triangleq \underset{t\in \left[0,T\right]}{\sup }{\Vert {\phi }_k(t)\Vert }_{\lambda }, b_G\triangleq \underset{t\in \left[0,T\right]}{\sup }{\Vert G_k(t)\Vert }_{\lambda }, \\ b_B\triangleq \underset{\forall x\in R^n}{\sup } (\underset{t\in \left[0,T\right]}{\sup }{\Vert B\left(x_i\left(t\right), t\right)\Vert }_{\lambda })\end{array}$

Let $e_i=y_d-y_i=C\partial x_i-v_i$ , ${\dot{e}}_i={\dot{y}}_d-{\dot{y}}_i=C\partial {\dot{x}}_i-{\dot{v}}_i$ , $B_i\triangleq B(x_i(t),t)$ , $\partial u_i(t)\triangleq u_d(t)-u_i(t)$ , ${\dot{x}}_d(t)=f(x_d(t),t)+B(x_d(t),t)u_d(t)+w(x_d(t),t)\triangleq f_d+B_d{u}_d+w_d$ , $\partial x_i(t)\triangleq x_d(t)-x_i(t)$ , $\partial B_i(t)\triangleq B_d-B(x_i(t),t)$ , $\partial w_i(t)\triangleq w_d-w_i(x_i(t),t)$ , and $\partial {\tilde{x}}_l=\max \left\{sup{\Vert \partial x_l\Vert }_{\lambda },\sup {\Vert \partial x_{l+1}\Vert }_{\lambda }\right\}$ . We then have:

$\begin{array}{l}\partial {\dot{x}}_i={\dot{x}}_d-{\dot{x}}_i=f_d+B_d{u}_d+w_d-\hfill \\ -(f(x_i(t),t)+B(x_i(t),t)u_i(t)+w(x_i(t),t))\hfill \\ =\partial f_i(t)+B_d{u}_d-B(x_i(t),t)\left[u_d-\partial u_i(t)\right]+\partial w_i(t)=\hfill \\ =\partial f_i(t)+\partial B_i{u}_d-B(x_i(t),t)\partial u_i(t)+\partial w_i(t)\hfill \end{array}$

where x_d is the desired state of the system and ≜ indicates a formula that is defined as another formula.

The high order feedback-feedforward ILC algorithm with a variable forgetting factor applies to non-linear time-varying systems. The objective is to design an iterative learning controller u_n, such that the output tracking error between the desired output trajectory y_d and system output y_n is in an error bound, which can be predetermined. As $i\to \infty$ , the bound of the tracking error converges to a small neighbourhood of the origin. If ${\Vert \partial x_i(0)\Vert }_{\lambda }$ and b_v tend towards zero, the bound of the tracking error asymptotically reaches zero by ILC.

3. Designed Controller and Variable Forgetting Factor

The high order feedback-feedforward ILC controller with a variable forgetting factor is constructed as follows:

$\begin{array}{l}{u}_{i+1}(t)=u_{i+1}^s(t)+u_{i+1}^c(t)\\ u_{i+1}^s(t)=a\left(i,t\right)u_0(t)+\left(1-a\left(i,t\right)\right)u_i(t)+{\displaystyle \sum _{k=1}^N{\phi }_k(t)e_l(t)}\\ u_{i+1}^c(t)={\displaystyle \sum _{k=1}^N{G}_k(t)e_{l+1}(t)}\end{array}$

(6)

where i indicates the iteration number, $l=i-k+1$ , $0\le a\left(i,t\right)<1$ is the variable forgetting factor, a is used instead of $a\left(i,t\right)$ to simplify the formula, $u_0^{}(t)$ is the initial value of the input, ${\phi }_k(t)$ and $G_k(t)$ are respectively the feedforward and feedback gain matrices, and $e_l=y_d-y_l$ .

The tracking ability of the system is stronger for smaller values of the forgetting factor a, and vice versa. We generally use a fixed forgetting factor, but a cannot vary with changes in the system features. Therefore, a variable forgetting factor a, which can vary automatically according to changing system deviation, is introduced as follows:

$\begin{array}{l}a=a_{\min }+(1-a_{\min })5^L\\ L=-\max \left[r{e}_i^2(t)\right]\end{array}$

(7)

where $1<r<10$ is sensitive to the gain and r is used to control the rate at which a approaches 1. If r is small, L becomes large, and the convergence speed is reduced. If r is large, the system stability decreases. When $e_i(t)\to \infty$ , the minimum value of a is $a_{\min }$ , $0<a_{\min }<0.5$ . When $e_i\left(t\right)\to 0$ , $a=1$ .

4. Convergence Analysis

In this section, the convergence condition of the controllers for the non-linear systems with uncertain and non-repetitive disturbance is given and proven.

Lemma 1 [20]. Assume that there is a positive real sequence ${\left\{a_n\right\}}_1^{\infty }$ and that the condition $a_n\le {\tilde{\rho }}_1{a}_{n-1}+{\tilde{\rho }}_2{a}_{n-2}+\dots +{\tilde{\rho }}_N{a}_{n-N}+\tilde{\delta }$ , $(n=N+1,N+2,\cdots )$ ; $\tilde{\delta }\ge 0$ is satisfied with ${\tilde{\rho }}_i\ge 0^{}{,}^{}(i=1,2,\cdots ,N)$ . If $\tilde{\rho }\le {\displaystyle \sum _{i=1}^N{\tilde{\rho }}_i}<1$ , then $\underset{n\to \infty }{\lim }{a}_n\le \tilde{\delta }/\left(1-\tilde{\rho }\right)$ .

Theorem 1 [20]. The non-linear systems with uncertain and non-repetitive disturbance satisfy Assumptions 1 and 2 under the condition that:

$\tilde{\rho }={\displaystyle \sum _{k=1}^N{\rho }_k}<1$

(8)

Furthermore, there exists a sufficiently large constant λ, such that:

1. When i goes to infinity, the tracking error bound converges to a small neighbourhood of the origin. Meanwhile, the tracking error ${\Vert \partial y_i\Vert }_{\lambda }$ , initial state error ${\Vert \partial x_i(0)\Vert }_{\lambda }$ and bound of the output disturbance item b_v have a linear relationship.

2. When the conditions $b_v=0$ and ${\Vert \partial x_i(0)\Vert }_{\lambda }=0$ are satisfied, for any prescribed tracking error tolerance ${\delta }^{\ast }=f\left({\phi }_k(t),G_k(t)\right)$ , we can choose a group of parameters ${\phi }_k$ and $G_k$ to reach the conclusion of this theorem $\underset{t\in \left[0,T\right]}{\sup }{\Vert e_i\left(t\right)\Vert }_{\lambda }\le {\delta }^{\ast }$ , with $\forall i\ge M$ .

Proof: From (1) and (6), we obtain:

$\begin{array}{l}\partial u_{i+1}=u_d-u_{i+1}=u_d-a{u}_0-\left(1-a\right)u_i-\hfill \\ -{\displaystyle \sum _{k=1}^N{\phi }_k(t)e_l(t)}-{\displaystyle \sum _{k=1}^N{G}_k(t)e_{l+1}(t)}\hfill \\ =a\partial u_0-\left(1-a\right)\partial u_i-{\displaystyle \sum _{k=1}^N{\phi }_k(t)\left[C\partial {\mathrm{x}}_l-v_l\right]}-\hfill \\ -{\displaystyle \sum _{k=1}^N{G}_k(t)\left[C\partial {\mathrm{x}}_{l+1}-v_{l+1}\right]}\hfill \\ =\left(1-a\right)\partial u_i-{\displaystyle \sum _{k=1}^N{\phi }_k(t)}C\partial {\mathrm{x}}_l-{\displaystyle \sum _{k=1}^N{G}_k(t)}C\partial {\mathrm{x}}_{l+1}+\hfill \\ +{\displaystyle \sum _{k=1}^N{G}_k(t)}{v}_{l+1}+{\displaystyle \sum _{k=1}^N{\phi }_k(t)}{v}_l+a\partial u_0\hfill \end{array}$

(9)

Let us take the norm on both sides of (9) and define $b_{u_d}\triangleq \underset{t\in \left[0,T\right]}{\sup }{\Vert u_d(t)\Vert }_{\lambda }$ . Using Assumptions 1 and 2 yields the following:

$\begin{array}{l}{\Vert \partial u_{i+1}\Vert }_{\lambda }\le {\displaystyle \sum _{k=1}^N{p}_k}\left(1-a\right){\Vert \partial u_l\Vert }_{\lambda }+{\displaystyle \sum _{k=1}^N(b_c{b}_{\phi }+b_c{b}_G)}{\Vert \partial {\tilde{x}}_l\Vert }_{\lambda }+\hfill \\ +{\displaystyle \sum _{k=1}^N{b}_v(b_{\phi }+b_G)+}a{\Vert \partial u_0\Vert }_{\lambda }\hfill \\ ={\displaystyle \sum _{k=1}^N{p}_k}\left(1-a\right){\Vert \partial u_l\Vert }_{\lambda }+{\xi }_0{\displaystyle \sum _{k=1}^N{\Vert \partial {\tilde{x}}_l\Vert }_{\lambda }}+{\mu }_0\hfill \end{array}$

(10)

where ${\mu }_0=N\left(b_v{b}_{\phi }+b_v{b}_G\right)+a{\Vert \partial u_0\Vert }_{\lambda }$ , ${\xi }_0=b_c{b}_{\phi }+b_G{b}_c$ , $\partial u_l=u_d-u_l$ , we have:

$\begin{array}{l}{\Vert \partial {\tilde{x}}_i\Vert }_{\lambda }={\Vert \partial {\tilde{x}}_i(0)+{\displaystyle {\int }_0^t\partial \dot{\tilde{x}}d\tau }\Vert }_{\lambda }\hfill \\ ={\Vert \partial {\tilde{x}}_i(0)+{\displaystyle {\int }_0^t\left[\partial f_i(t)+\partial B_i(t)u_d-B({\tilde{x}}_i(t),t)\partial u_i(t)+\partial w_i(t)\right]d\tau }\Vert }_{\lambda }\hfill \\ \le b_{x_0}+{\displaystyle {\int }_0^t\left(\widehat{e}{\Vert \partial {\tilde{x}}_i\Vert }_{\lambda }+b_B{\Vert \partial u_i\Vert }_{\lambda }\right)d\tau }\hfill \end{array}$

(11)

with $\widehat{e}\triangleq k_f+b_{u_d}{k}_B+k_w$ .

Taking the norm, we can obtain:

$\begin{array}{l}{\Vert {\displaystyle {\int }_0^t{\Vert \tilde{x}(\tau )\Vert }_{\lambda }}d\tau \Vert }_{\lambda }=\underset{t\in \left[0,T\right]}{\sup }{e}^{-\lambda t}{\displaystyle {\int }_0^t\Vert \tilde{x}(\tau )\Vert }\lambda d\tau =\hfill \\ =\underset{t\in \left[0,T\right]}{\sup }{e}^{-\lambda t}{\displaystyle {\int }_0^t{\Vert \tilde{x}(\tau )\Vert }_{\lambda }}{e}^{-\lambda \tau }{e}^{-\lambda \tau }d\tau \hfill \\ \le {\Vert \tilde{x}(\tau )\Vert }_{\lambda }\underset{t\in \left[0,T\right]}{\sup }{e}^{-\lambda t}{\displaystyle {\int }_v^t{e}^{\lambda \tau }}d\tau ={\Vert \tilde{x}(\tau )\Vert }_{\lambda }\underset{t\in \left[0,T\right]}{\sup }\left[\left(\text{1-}{e}^{-\lambda t}\right)/\lambda \right]\hfill \\ \le {\Vert \tilde{x}(\tau )\Vert }_{\lambda }○\left({\lambda }^{-1}\right)\hfill \end{array}$

(12)

with $○\left({\lambda }^{-1}\right)\triangleq \left(1-e^{-\lambda T}\right)/\lambda$ , we obtain ${\Vert \partial {\tilde{x}}_i\Vert }_{\lambda }\le b_{x_0}+\widehat{e}○\left({\lambda }^{-1}\right){\Vert \partial {\tilde{x}}_i\Vert }_{\lambda }+b_B○\left({\lambda }^{-1}\right){\Vert \partial u_i\Vert }_{\lambda }$ . This Equation means that:

${\Vert \partial {\tilde{x}}_i\Vert }_{\lambda }\le \frac{\left(b_{x_0}+b_B○\left({\lambda }^{-1}\right){\Vert \partial u_i\Vert }_{\lambda }\right)}{1-\widehat{e}○\left({\lambda }^{-1}\right)}$

(13)

Inserting (13) into (10) yields:

$\begin{array}{l}{\Vert \partial u_{i+1}\Vert }_{\lambda }\le {\displaystyle \sum _{k=1}^N{p}_k}\left(1-a\right){\Vert \partial u_l\Vert }_{\lambda }+{\xi }_0{\displaystyle \sum _{k=1}^N{\Vert \partial {\tilde{x}}_l\Vert }_{\lambda }}+{\mu }_0\hfill \\ \le {\displaystyle \sum _{k=1}^N{p}_k}\left(1-a\right){\Vert \partial u_l\Vert }_{\lambda }+{\xi }_0{\displaystyle \sum _{k=1}^N\frac{\left(b_{x_0}+b_B○\left({\lambda }^{-1}\right){\Vert \partial u_l\Vert }_{\lambda }\right)}{1-\widehat{e}○\left({\lambda }^{-1}\right)}}+{\mu }_0\hfill \\ ={\displaystyle \sum _{k=1}^N{\rho }_k}{\Vert \partial u_l\Vert }_{\lambda }+\delta \hfill \end{array}$

(14)

where ${\rho }_k=p_k\left(1-a\right)+{\xi }_0\frac{b_B○\left({\lambda }^{-1}\right)}{1-\widehat{e}○\left({\lambda }^{-1}\right)}$ and $\delta ={\xi }_0{\displaystyle \sum _{k=1}^N\frac{b_{x_0}}{1-\widehat{e}○\left({\lambda }^{-1}\right)}}+{\mu }_0$ .

By Theorem 1, we choose a sufficiently large constant λ to satisfy ${\tilde{\rho }}_k<1$ and $\tilde{\rho }={\displaystyle \sum _{k=1}^N{\rho }_k}<1$ . From Lemma 1, the following inequality holds:

$\underset{i\to \infty }{\lim }{\Vert \partial u_i\Vert }_{\lambda }\le \delta /\left(1-\tilde{\rho }\right)$

(15)

Inserting $\partial y_i=y_d-y_i=C{x}_d-C{x}_i-v_i=C\partial x_i-v$ , we obtain ${\Vert \partial y_i\Vert }_{\lambda }\le b_c{\Vert \partial x_i\Vert }_{\lambda }+b_v$ . Combining (13), (14) and (15), we find that the tracking error bound converges to a small neighbourhood of the origin, as i goes to infinity. Meanwhile, the tracking error ${\Vert \partial y_i\Vert }_{\lambda }$ , initial state error ${\Vert \partial x_i(0)\Vert }_{\lambda }$ and bound of the output disturbance item b_v have a linear relationship. If ${\Vert \partial x_i(0)\Vert }_{\lambda }$ and b_v tend towards zero, the tracking error bound asymptotically approaches zero by ILC.

When the conditions $b_v=0$ and ${\Vert \partial x_i(0)\Vert }_{\lambda }=0$ are satisfied, from (14), we get $\delta =0$ , from (15), we obtain $\underset{i\to \infty }{\lim }{\Vert \partial u_i\Vert }_{\lambda }=0$ . Then, as (13) gives $\underset{i\to \infty }{\lim }{\Vert \partial x_i\Vert }_{\lambda }=0$ , we have $\underset{i\to \infty }{\lim }{\Vert e_i(t)\Vert }_{\lambda }=\underset{i\to \infty }{\lim }{\Vert \partial y_i\Vert }_{\lambda }=0$ . According to the limit definition, for any prescribed tracking error tolerance ${\delta }^{\ast }=f\left({\phi }_k(t),G_k(t)\right)$ , its value is related to the selection of ${\phi }_k$ and $G_k$ . We can choose a group of parameters ${\phi }_k$ and $G_k$ to reach the conclusion of this theorem: $\underset{t\in \left[0,T\right]}{\sup }{\Vert e_i\left(t\right)\Vert }_{\lambda }\le {\delta }^{\ast }$ , with $\forall i\ge M$ .

5. Simulation

To demonstrate the effectiveness of the proposed algorithm of ILC, a comparable investigation is accomplished between the proposed ILC algorithm and traditional high order feedback-feedforward ILC. We consider the two manipulators of a robot described by (16).

$M\left({\theta }_i\right){\ddot{\theta }}_i+C({\theta }_i,{\dot{\theta }}_i){\dot{\theta }}_i+G({\theta }_i)+f({\theta }_i,t)=u_i{{}^{}}^{}(i=1,2,3,\cdots )$

(16)

where ${\theta }_i\in R^n$ is the vector of the state variables, $M({\theta }_i)\in R^{n\times n}$ is the inertia matrix, $C({\theta }_i,{\dot{\theta }}_i){\dot{\theta }}_i\in R^n$ is the vector of the Coriolis and centripetal torques, $G({\theta }_i)\in R^n$ is the gravitational term, $f\left({\theta }_i,t\right)\in R^n$ represents the interference terms, and $u_i\in R^n$ is the vector of the control input.

equation (16) can now be rewritten. If $x_{1i}={\theta }_i$ , $x_{2i}={\dot{\theta }}_i={\dot{x}}_{1i}$ , we then obtain:

$\begin{array}{l}{\dot{x}}_{1i}=x_{2i}\hfill \\ {\dot{x}}_{2i}=M^{-1}\left({\theta }_i\right)\left[u_i-C({\theta }_i,{\dot{\theta }}_i){\dot{\theta }}_i-G({\theta }_i)-f({\theta }_i,{\dot{\theta }}_i)\right]\hfill \\ =M^{-1}\left(x_{1i}\right)\left[u_i-C(x_{1i},x_{2i})x_{2i}-G(x_{1i})-f(x_{1i},x_{2i})\right]\hfill \end{array}$

(17)

If $X_i=[x_{1i},{x_{2i}]}^T$ , $y_i=X_i$ , equation (17) can then be rewritten as:

$\begin{array}{l}{\dot{X}}_i=\left[\begin{array}{c}{x}_{2i}\\ -M^{-1}(x_{1i})[C(x_{1i},x_{2i})+G(x_{1i})]\end{array}\right]+\left[\begin{array}{c}0\\ M^{-1}(x_{1i})\end{array}\right]u_i+\left[\begin{array}{c}0\\ -M^{-1}(x_{1i})f(x_{1i},x_{2i})\end{array}\right]\\ y_i=X_i\end{array}$

(18)

Simulations are conducted on a planar direct-driven two-joint robot. The matrices for the robot arm in the state space are:

$M\left({\theta }_i\right)=\left[\begin{array}{cc}a& bcos\left({\theta }_{2i}-{\theta }_{1i}\right)\\ bcos\left({\theta }_{2i}-{\theta }_{1i}\right)& c\end{array}\right], C\left({\theta }_i,{\dot{\theta }}_i\right)=\left[\begin{array}{l}-b{\dot{\theta }}_{2i}^{2}sin\left({\theta }_{2i}-{\theta }_{1i}\right)\\ -b{\dot{\theta }}_{2i}^{2}sin\left({\theta }_{2i}-{\theta }_{1i}\right)\end{array}\right]$

(19)

where a, b and c are three constants: a =5.76794 $kg\cdot m^2$ , b =1.473 $kg\cdot m^2$ and c =1.76794 $kg\cdot m^2$ .

The expected output trail is ${\theta }_d={\left[\begin{array}{cc}2.55t& 3.1t\end{array}\right]}^T$ . The interference term is $f(t)={\left[\begin{array}{cc}\cos (5t)& \sin (5t)\end{array}\right]}^T$ and the initial state is $x(0)={\begin{array}{cccc}[0& 3& 0.34& 3.12]\end{array}}^T$ .

The high order feedback-feedforward ILC algorithm with a variable forgetting factor is defined as:

$u_{i+1}(t)=a{u}_0(t)+\left(1-a\right)u_i(t)+{\displaystyle \sum _{k=1}^N{\phi }_k(t)e_l(t)}+{\displaystyle \sum _{k=1}^N{G}_k(t)e_{l+1}(t)}$

(20)

with $l=i-k+1$ .

The high order feedback-feedforward ILC algorithm is defined as

$u_{i+1}(t)=u_i(t)+{\displaystyle \sum _{k=1}^N{\phi }_k(t)e_l(t)}+{\displaystyle \sum _{k=1}^N{G}_k(t)e_{l+1}(t)}$

(21)

with $l=i-k+1$ .

According to Theorem 1 and Lemma 1, the parameters used in the simulation are chosen as:

$a_{\min }=0.05$ , $\gamma =5$ , $u_0=\left[\begin{array}{c}0.1\\ 0.1\end{array}\right]$ , $N=2$ , $G_1=\left[\begin{array}{cc}1.36& 0\\ 0& 1.5\end{array}\right]$ , $G_2=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right]$ , ${\phi }_1=\left[\begin{array}{cc}0.2& 0\\ 0& 0.2\end{array}\right]$ , ${\phi }_2=\left[\begin{array}{cc}0.2& 0\\ 0& 0.02\end{array}\right]$ , ${\delta }^{\ast }=1.2$ .

By checking the convergence condition in Theorem 1 and Lemma 1, we can obtain the following:

The first algorithm:

$\tilde{\rho }={\displaystyle \sum _{k=1}^N{\rho }_k}={\displaystyle \sum _{k=1}^N{p}_k\left(1-a\right)+\left(b_c{b}_G+b_c{b}_{\phi }\right)\frac{b_B○\left({\lambda }^{-1}\right)}{1-\widehat{e}○\left({\lambda }^{-1}\right)}}=0.61<1$

(22)

The second algorithm:

$\tilde{\rho }={\displaystyle \sum _{k=1}^N{\rho }_k}={\displaystyle \sum _{k=1}^N{p}_k+\left(b_c{b}_G+b_c{b}_{\phi }\right)\frac{b_B○\left({\lambda }^{-1}\right)}{1-\widehat{e}○\left({\lambda }^{-1}\right)}}=0.67<1$

(23)

Thus, $\underset{t\in \left[0,T\right]}{\sup }{\Vert e_i\left(t\right)\Vert }_{\lambda }\le {\epsilon }^{\ast }$ is satisfied for any prescribed tracking error tolerance ${\epsilon }^{\ast }$ .

Figure 1.

Desired and actual position trajectories

Figure 2.

Maximum absolute tracking error in each iteration

Figure 3.

Output error in the last iteration

Figure 1 shows that the control law drives the system output through the desired trajectory as closely as possible. We show the iterative process of the formation learning error in Figure 2. The high order feedback-feedforward ILC arithmetic, with a variable forgetting factor, can significantly converge to a small neighbourhood of zero, in which the tracking error in the eighth iteration is less than ${\delta }^{\ast }=1.2$ and the range of error ${\delta }^{\ast }$ is related to the selection of ${\phi }_k$ and $G_k$ . From Figure 2 and Figure 3, ${\theta }_2$ in the conventional high order feedback-feedforward ILC algorithm has a small error and fast convergence rate, but ${\theta }_1$ has a large tracking error, which is larger than ${\delta }^{\ast }=1.2$ . This error is not allowable for engineering applications. Therefore, the high order feedback-feedforward ILC algorithm with a variable forgetting factor has small tracking errors, conforming to the requirements and fast convergence rate.

The system includes steady-state error due to the initial error. If the initialization error tends towards zero, then the final tracking error will also tend towards zero. From the simulation results, the high order feedback-feedforward ILC algorithm, with a variable forgetting factor, has the best performance and control effort from a practical perspective, in comparison with the conventional high order feedback-feedforward ILC algorithm, even though there are some steady-state errors, which could be neglected in practice. The results demonstrate the effectiveness of the proposed ILC scheme, as well as the design trade-off between the convergence speed and accuracy.

6. Conclusion

In this work, we propose a novel ILC scheme to address tracking problems for a class of non-linear time-varying system with an uncertain or interference component. By introducing a variable forgetting factor, this method weakens the convergent influence of the system caused by the uncertain part of the system and the non-repetitive disturbances. This algorithm is proposed to facilitate uniform tracking error convergence analysis under the effect of system uncertainties and non-repetitive disturbances. The simulation results further verify the theoretical results. Future work will aim to apply the proposed algorithm to actual robot control and biochemical reaction processes.