# Discrete PID-Type Iterative Learning Control for Mobile Robot

Authors: Hongbin Wang; Jian Dong; Yueling Wang
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

Through studying tracking problems of the wheeled mobile robot, this paper proposed a discrete iterative learning control approach based on PID with strong adaptability, fast convergence, and small error. This algorithm used discrete PID to filter rejection and restrained the influence of interference and noise on trajectory tracking, which made it more suitable for engineering application. The PID-type iterative learning convergence condition and certification procedure are presented. The results of simulation reveal that the PID-type ILC holds the features of simplicity, strong robustness, and high repeating precision and can well meet the control requirement of nonlinear discrete system.


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## Body

## 1. Introduction

Iterative learning control (ILC) [1, 2] is an advanced control which can realize repetitive control and antidisturbance, tracking the trajectory with high precision independent of accurate mathematical models for its control. A substantial proportion of inchoate ILC algorithms was applied in the continuous-time systems [3, 4]. Some ILC algorithms have been developed for discrete-time systems, but they are restricted to linear systems [5, 6]. In an ordinary way, the ILC update rules are composed of P-type [7, 8], PD-type [9, 10], and high-order type [11].Many ILC algorithms [12, 13] have been used in motion controller system and a certain number of simulation results display the effectiveness of these algorithms. Besides, they can help the mobile robot to better track the desired trajectory iteratively. Even under the complicated conditions, the mobile control system could fulfill quite nicely the being stipulated tasks with the aid of ILC. However, for tracking problem of mobile robot, a multitude of researches are heavily dependent on the precision of the robot model and the developed control laws are often complex. Therefore, ILC would be the preferred method of tracking the given trajectories for mobile robot repeatedly, especially in the real complex environment. Ostafew et al. [14] present a mobile robot controller that combines ILC algorithm with a feedback controller to reduce path-tracking errors over repeated traverses along a reference path by the experimental results of mobile robot travelling through challenging terrain including steep hills and sandy turns. In the extreme environments without global localization system (GPS), for a path-repeating mobile robot, ILC can handle unmodelled terrain and rover dynamics. Furgale and Barfoot [15] apply an ILC scheme on the mobile robot to repeat long routes without the need for an accurate global reconstruction. In an urban area and in a planetary analog environment, it can perform path tracking by itself.Mobile robot is a complex automation system which combines environment sense, dynamical plan with motion control, and performance, whose requirement for the dynamical and static characteristics of mobile robot becomes high. There have been several research works on the control of mobile robots. Jiang and Nijmeijer [16] firstly apply adaptive control to the mobile robot. Buccieri et al. [17] use point stabilization for mobile robot in tracking control. Moosavian and Keymasi [18] designed a Lyapunov-based controller for a wheeled robot. A few articles [19] have showed the probability of realizing perfect tracking for mobile robots by the ILC algorithm. The closed-open-loop P-type iterative learning control scheme [20] for mobile robot system is proposed. Although it can reduce the system steady-state error and improve the control precision, the relative stability and convergence speed of control systems can be whittled down further. Yu and Weili [21] showed the open-loop D-type ILC laws in trajectory tracking of wheeled robot, but the anti-interference function of this system was not ideal. Panzieri and Ulivi [22] presented an ILC method which takes advantage of the possibility of transforming systems in chained form via feedback for mobile robots to improve the robustness properties of the controller.The contribution of this paper is applying PID-type ILC algorithm to the nonlinear discrete system of mobile robot. With the continual advances in control theory, PID controller is still the most commonly used controller in the controlling field [23, 24]. This is mainly attributed to its noticeable effectiveness, simple structure, and robustness. So we are keen on using the PID strategy in designing the iterative learning control schemes. PID have some specific advantage roles of each of modes in the ILC algorithm. The P-component has a stabilizer role in the ILC system and causes monotonic convergence and the I-component decreases the effect of nonzero initial errors and increases the convergence rate, while D-term can reduce the effect of disturbance inputs. The effectiveness and superiority of this algorithm are shown by simulation.This paper is organized as follows. Section2 gives the problem formulation. In Section 3, the controller design is proposed and its convergence is also discussed. Simulation results are shown in Section 4. Section 5 concludes the paper.


## 2. Problem Formulation

Figure1 shows system model of mobile robot with two wheels [20]. The mobile robot moves in two-dimensional space and Pk represents the mobile robot’s current position. In global coordinates, the current position of Pk is determined by the coordinates xPk and yPk and θPk is the orientation angle of the robot. So, the generalized coordinates of Pk are defined as xPk,yPk,θPk. Defining the linear and angular velocities of Pk in the robot coordinate as νPk and ωPk, we can describe the kinematic motion of the mobile robot in the discrete-time domain as(1)xPk+1yPk+1θPk+1=xPkyPkθPk+ΔTcos⁡θPk0sin⁡θPk001νPkωPk,where ΔT is the sampling time. Defining the robot states and velocities, qk=xPk,yPk,θPkT and uPk=νPk,ωPkT, we can express kinematic motion (1) as(2)qk+1=qk+Bqk,kuPk,where Bqk,k=ΔTcos⁡θPk0;sin⁡θPk0;01.Figure 1
Mobile robot configuration.As shown in Figure1, the dotted line is desired trajectory.P d ( k ) = x d ( k ) , y d ( k ) , θ d ( k )   ( 1 ≤ k ≤ n ). The trajectory tracking control problem of mobile robots is to determine u(k)=ν(k),ω(k)T which makes xP(k)→xd(k), yP(k)→yd(k), and θP(k)→θd(k). If ILC is applied to the mobile control system, the velocity errors can be uniformly bounded under bounded uncertainties and disturbances that occur in the mobile system dynamics [20, 25, 26]. So we try to find a novel ILC algorithm to make perfect tracking.Considering state disturbancesβ(k) and output measurement noises γ(k), the kinematic model of mobile robot at the ith iteration can be described as(3)qik+1=qik+Bqik,kuik+βik,(4)yik=qik+γik,where i indicates the number of iterations and k∈1,n is the discrete-time index. For all k∈N, qik,uik=νi(k),ωi(k)T, and yik=xik,yik,θikT, βik and γik are the states, inputs, outputs, state disturbances, and output noises at the ith iteration. If state disturbances βi(k)=0 and output measurement noises γi(k)=0 (k∈N), the desired trajectory equation can be written as(5)qdk+1=qdk+Bqdk,kudk,(6)ydk=qdk.Furthermore, the matrix Bqik,k is bounded as Bqik,k≤bB and meet Lipschitz conditions:(7)Bq1,k-Bq2,k≤LBq1-q2,where LB and bB are positive constants. Meanwhile, formulas (5) and (6) satisfy the following assumptions:(8)max1≤k≤n⁡udk≤bud,max1≤i≤∞⁡max1≤k≤n⁡βik≤bβ,max1≤i≤∞⁡max1≤k≤n⁡γik≤bγ,qd0-qi0≤bq0,where bud, bβ, bγ, and bq0 are positive constants.


## 3. Controller Design and Convergence Analysis

Now we propose a PID-type iterative learning rule as(9)ui+1k=uik+Gpkeik+1+Gdkeik+1-eik+GIk∑l=0keil+1,for the ith iteration, where ei(k)=yd(k)-yi(k) are the output tracking errors, Gp(k), Gd(k), GI(k) are the proportional, derivative, and integral gain matrices that satisfy Gpk≤bGp, Gd(k)≤bGd, and GI(k)≤bGI for all k∈N and for some bGp>0, bGd>0, and bGI>0.Define 1.
Theλ-norm is defined for a positive real function f:N→R as(10)f·λ=supk∈N⁡1λkfkforλ≥1.Theorem 1.
Consider the time-varying discrete-time system (3) and (4) with the iterative learning rule (9). Let us assume that the assumptions (8) are satisfied and the inequality(11)I-Gpk+Gdk+GIkBqik,k≤ρ<1holds for all qi(k),k∈Rn×N. Given a bounded output yd(k) and in the absence of state disturbances, output noises, and initial state errors, ui(k), qi(k), and yi(k), respectively, converge to ud(k), qd(k), and yd(k) for all k∈1,n, as i→∞. In the presence of disturbances, noises, and errors, uik-ud(k), qik-qd(k), and yi(k)-yd(k) will be ultimately bounded with bounds that are functions of bβ, bγ, and bq0.Proof.
Subtracting (5) from (3), we obtain the state errors as follows:(12)Δqik+1=qdk+1-qik+1=qdk+Bqdk,kudk-qik+Bqik,kuik+βik=Δqik+Bqdk,kudk-Bqik,kuik-udk+udk-βik=Δqik+Bqdk,k-Bqik,kudk-Bqik,kΔuik-βik.
Considering (7) and (8), taking norms on both sides of (12) gives(13)Δqik+1≤Δqik+LBbudΔqik+bBΔuik+bβ.
Leth1=1+LBbud. Then it follows from (13) that(14)Δqik+1≤h1Δqik+bBΔuik+bβfrom which we have (15)Δqik≤∑j=0k-1h1k-1-jbBΔuij+bβ+h1kbq0.
On the other hand, it follows from (9) that the control input errors become(16)
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Applying (7) and (8) to (16), we have(17)Δui+1k≤I-Gpk+Gdk+GIkBqik,kΔuik+bGp+bGd+bGI1+LBbud+bGd+bGIΔqik+bGI∑l=0k-2Δqil+1+bGp+bGd+bGIbB+bγ+bGI∑l=0k-1bγ+bGdbγ.
Lettingh2=(bGp+bGd+bGI)(1+LBbud)=(bGp+bGd+bGI)h1, h3=h2+bGd+bGI, and b1=bGp+bGd+bGIbB+bγ+bGI∑l=0k-1bγ+bGdbγ, we have(18)Δui+1k≤ρΔuik+h3Δqik+bGI∑l=0k-2Δqil+1+b1.
From (11) and (15), we get(19)Δui+1k≤ρΔuik+h3∑j=0k-1h1k-1-jbBΔuij+bβ+h3h1kbq0+bGI∑l=0k-2∑j=0lh1l-jbBΔuij+bβ+bGI∑l=0k-2h1l+1bq0+b1.
Multiplying both sides of (19) by 1/λk to compute the λ-norm, we have (20)Δui+1k1λk≤ρΔuik1λk+h1λkh3bq0+h3λ∑j=0k-1h1λk-1-jbBΔuij1λj+bβ1λj+1λk-lbGI∑l=0k-2∑j=0lh1λl-jbBΔuij1λj+bβ1λj+1λk-l-1bGI∑l=0k-2h1λl+1bq0+1λkb1.
Letm1=∑l=0k-21-h1/λl+1 and m2=∑l=0k-2h1/λl+1. Taking λ>max⁡1,h1 and noting that the λ-norm of a constant is itself the constant, we have(21)Δui+1kλ≤ρ+bBh31-h1/λnλ-h1+bBbGIm1λ-h2Δuikλ+bβλ-h1h31-h1λn+bGIm1+bGIbq0m2+b1.
Then (21) implies (22)Δui+1kλ≤ρ~Δuikλ+ε,where ρ~=ρ+bBh31-h1/λn/λ-h1+bBbGIm1/λ-h1 and ε=bβ/λ-h1h31-h1/λn+bGIm1+bGIbq0m2+b1.
Let us make recursive formula (22) and choose λ large enough so that ρ~≈ρ<1. Then we have (23)limi→∞Δuiλ≤ε1-ρ.
Similarly, multiplying both sides of (15) by 1/λk gives(24)Δqik1λk≤1λ∑j=0k-1h1λk-1-jbBΔuij1λj+bβ1λj+h1λkbq0.
Because ofbB1/λj≤bB and h1/λ<1, we have(25)Δqiλ≤bBΔuiλ+bβ1-h1/λnλ-h1+bq0.
Substituting (23) into (25), we obtain (26)limi→∞Δqiλ≤bB1-h1/λnλ-h1ε1-ρ+bβ1-h1/λnλ-h1+bq0.
Subtracting (4) from (6) gives(27)Δyik=ydk-yik=qdk-qik+γik=Δqik-γik.
Multiplying both sides of (27) by 1/λk results in (28)Δyiλ=Δqiλ+bγ.
Substituting (26) into (28), we obtain (29)limi→∞Δyiλ≤bB1-h1/λnλ-h1ε1-ρ+bβ1-h1/λnλ-h1+bq0+bγ.
Consequently, in the absence ofβi(k) and γi(k) and with perfect repeatability or with bβ=0, bγ=0, and bq0=0, the error bounds become zero. Otherwise, the errors become ultimately bounded with error bounds that are functions of bβ, bγ, and bq0 as given in (23), (26), and (29).


## 4. Simulation

To investigate the validity of the proposed ILC algorithm and compare the performance of the proposed ILC algorithm with that of conventional PD-type ILC algorithm, we performed MATLAB simulations under the bounded disturbances and noises. The desired trajectory isPd=xd(t)yd(t)θd(t)T=cos⁡(πt)sin⁡(πt)πt+π/2T. The sample period ΔT is set to 0.001 s. The whole task period is i=1,…,300. The initial states are u00=0 and qk0=(0,0,0)T. The state disturbances term is βit=0.01[sin⁡40πt+0.05random⁡(0,8)sin⁡40πt+0.05random⁡(0,8)0.5sin⁡40πt+0.05random⁡(0,4)]T. The output measurement noises are Gaussian white noise (n(t)~N(0,δ2)), γi(t)=0.05n(t). According to convergence condition (12), we choose the gain matrices Gpk=0.11cos⁡θ(k)sin⁡θ(k)0;001,  GI(k)=0.001Gp(k), and Gd(k)=2.1Gp(k). Simulation results are shown in Figures 2 and 3.Figure 2
Desired and actual position trajectories.
(a) (b)Figure 3
Max absolute values of the tracking error.
(a) (b)As can be seen from Figure2, the output trajectory of the mobile robot converges to the given desired trajectory by increasing the iterations number. For more detailed comparison results, one can refer to Figure 3, from which it is easy to see that the provided PID-type ILC algorithm is better than the PD-type ILC algorithm for nonlinear discrete system in terms of tracking errors. Meanwhile, one can clearly see that the convergence rate of the presented ILC in this paper is faster than PD-type ILC.These simulation results illustrate that the output trajectory using PID-type ILC algorithm tends to the desired trajectory through less iterations than that using traditional PD-type ILC algorithm. However, due to the existence of disturbances, noises, and errors, tracking error is ultimately bounded and is only in a small range.


## 5. Conclusions

In this work, a PID-type ILC algorithm has been applied to the nonlinear discrete system of mobile robot. Conditions for guaranteeing the convergence and robustness of proposed ILC algorithm are presented. This algorithm is proposed to facilitate uniform tracking error convergence analysis under the effect of system disturbances and noises. A simulation result further verifies the theoretical results. The future work aims to apply the proposed algorithm to actual mobile robot control.



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