Jianxiang Zhang, Baotong Cui, Xisheng Dai, and Zhengxian Jiang

Abstract—In this paper, an open-loop PD-type iterative learning control (ILC) scheme is first proposed for two kinds of distributed parameter systems (DPSs) which are described by parabolic partial differential equations using non-collocated sensors and actuators. Then, a closed-loop PD-type ILC algorithm is extended to a class of distributed parameter systems with a non-collocated single sensor and m actuators when the initial states of the system exist some errors. Under some given assumptions, the convergence conditions of output errors for the systems can be obtained. Finally, one numerical example for a distributed parameter system with a single sensor and two actuators is presented to illustrate the effectiveness of the proposed ILC schemes.

I. Introduction

IN practice, most systems can be described by a partial differential equation or a partial integral equation, referred to as distributed parameter systems. The states of distributed parameter systems are dependent on time and spatial position.Therefore, these systems are more suitable to describe system dynamics. At the same time, this has attracted many researchers to study the control and estimation of distributed parameter systems in a number of fields, most recently in [1]–[3].Since the sensors and actuators are low-cost and low energy,the distributed parameter systems using sensors and actuators have been extensively studied by many specialists. Demetriou[4] considered a law for the guidance of a mobile collocated actuator/sensor for the enhanced control of spatially distributed processes. Accordingly, he suggested an algorithm to replace the full state information from a scalar multiple of the output measurement in finite horizon linear quadratic regulator control of DPSs in [5]. Meanwhile, Muet al.[6] considered a scheme aimed at guiding the moving actuator/sensor pair for enhanced control and estimation of the distributed parameter systems. Jianget al.[7] proposed an even-driven observer-based control for DPSs based on a mobile sensor and actuator.

Iterative learning control (ILC) is an intelligent control method which particularly suits systems working in a fixed time interval with a repetitive model. ILC aims to find proper learning control schemes of the controlled system for the actual output signal to track the given desired output signal over a finite interval time. At the same time, the constructed learning control sequences can converge to a desired control.An effective ILC algorithm can promote tracking accuracy by adjusting the system input signal according to error observations from every iteration even when the system has incomplete knowledge. Initially, ILC was proposed in 1984 by Arimotoet al.[8] that mainly involved a class of ILC algorithm for robots to obtain better control performance.Since then, ILC has been established as a separate field of control theory [9]–[14]. This methodology has been given consideration in various industrial applications, including industrial robots [15], health care systems [16], batch processes [17], and so on [18]. Nowadays, ILC is extensively employed in distributed parameter systems [19]–[21]. In particular, Daiet al.[22] proposed a closed-loop P-type iterative learning law for uncertain linear DPSs. In addition,he considered ILC for second-order hyperbolic DPSs with uncertainties [23]. A D-type ILC law for a type of distributed parameter systems with collocated sensors and actuators is considered in [24]. In many industrial processes, the sensors and actuators are always non-collocated. Hence, a type of linear parabolic distributed parameter system based on noncollocated sensors and actuators is proposed. No research papers have taken into account the problem of a PD-type ILC for this system.

The distributed parameter system based on non-collocated sensors and actuators is a complex system since it depends on time and spatial position. Furthermore, ILC can be better in controlling dynamic systems with complex modelling,uncertainty and with strong non-linear coupling effects. As such, we can obtain good control performance of a distributed parameter system by using ILC schemes. As discussed above,there is no existing research that has been carried out using ILC for distributed parameter systems using non-collocated sensors and actuators. Thus, we first propose an open-loop PD-type ILC scheme for a distributed parameter system with non-collocated single sensor andmactuators. After that, we consider the distributed parameter system based on non-collocatedmsensors as well asmactuators, which include numerous industrial processes, such as heat exchangers, industrial chemical reactors, and agricultural irrigation processes. Lastly,we present a closed-loop PD-type ILC algorithm for the distributed parameter system using a single sensor and multiple actuators when some errors exist in the initial states of the system.

In distributed parameter systems with non-collocated sensors and actuators, the sensors are capable of gathering information from the systems in real time. At the same time, the actuators can perform various tasks. When the states change, an input is imposed to control the output of the actuators. However, the actual output of the systems may not represent the desired output in the running of actuators. In this case, it is crucial to use ILC schemes to learn the output error of the systems. This facilitates the actual output in tracking the desired output.Therefore, this work improves the performance of systems. At the same time, it significantly closes the existing theoretical gap.

The remainder of this paper is as follows: In Section II, we first discuss the system and problem formulation. Next, the open-loop PD-type and closed-loop PD-type ILC schemes are presented in a distributed parameter system with a sensor andmactuators. In addition, the proposed ILC schemes are extended to a class of distributed parameter systems using non-collocatedmsensors andmactuators in Section III. The effectiveness of the proposed methods are illustrated through numerical simulation in Section IV and conclusions follow in Section V.

Notations:R, Rnand R+are the set of all real numbers,ndimensional space and the set of all positive real numbers.

The definition of theL2-norm of the functionW(x,t):[0,h]×[0,T]→Ris

II. The System and Problem Formulation

Consider the distributed parameter system with a noncollocated single sensor andmactuators as follows:

with the Neumann boundary conditions

and the initial condition

wherexandtare the spatial position and time which satisfyis a known continuous function ofx(?0is a constant).kdenotes thekth iteration of the repetitive operation of the system.qk(x,t) andyk(t) denote the state and output of the system at thekth iteration. When the system operates in thekth iteration,u(k,i)(t) is the associated control signal of theith actuator.denotes the spatial distribution of the actuating device of theith actuator anddenotes the centroid position of theith actuator.c(x) is the spatial distribution of the sensor. The sensor spatial distribution and the actuators spatial distribution satisfy

and

where δ and γ are constants. σ>0 is the spatial support of the actuators.

Throughout this paper, two lemmas and one assumption are first given as follows:

Lemma 1 [22]:Iff(t) andg(t) are two continuous nonnegative functions on [0,T], and there exist nonnegative constants ρ andMsatisfying

then

Lemma 2 [22]:If the constant sequence {dk}k≥0converges to zero, and the sequence {Zk(t)}k≥0?C[0,T] satisfies

then {Zk(t)}k≥0(k→ ∞) uniformly converges to zero, whereM>0 and 0 ≤θ<1 are constants.

Assumption 1:For a desired outputyd(t), a uniqueu(d,i)(t)exists such that

whereqd(x,0)=0.

In this paper, an open-loop PD-type ILC scheme is employed as follows:

whereek(t) is the output error ofkth iteration which satisfiesek(t)=yk(t)?yd(t). Γ and Υ are the open-loop ILC learning gains.

III. Convergence Analysis

In this section, we first prove the effectiveness of the openloop PD-type ILC for a distributed parameter system with non-collocated single sensor andmactuators. In addition, we extend the proposed scheme to the distributed parameter system with non-collocatedmsensors andmactuators. The following theorem is first given.

Theorem 1:Consider the open-loop PD-type ILC scheme(7) for the repetitive distributed parameter system (1) with the desired output satisfying Assumption 1. If the learning gain exists and satisfies (1+2mδγσΓ)2<1/2, then the output error converges to zero for allt∈[0,T] ask→∞, i.e.,∈[0,T].

Proof:The input erroru(k+1,i)(t)?u(d,i)(t) at the (k+1)th iteration can be expressed as

According to the state equation of system (1), we have

Applying integration by parts and using the boundary conditions for the third term on the right hand side of (9), we obtain

Substituting (10) into (9) yields

Based on the spatial distribution of the sensor (4) and actuators (5), (11) can be further rewritten as

Squaring both sides of (12) and using the definition of theL2-norm, we have

Integrating (14) with respect toxon [ 0,h], it satisfies

According to the spatial distribution of the actuators (5), and the definition ofL2-norm, the following gives

Integrating (17) with respect totand using the Bellman-Gronwall Lemma, we have

Substituting (18) into (13), we get

Because (1+2mδγσΓ)2<1/2, we can obtain(2(1+2mδγσΓ)2+4mσ(Υδγ)2/(λ?1))<1 if λ is chosen large enough. Hence,

According to (18), we have

And from the output equation of system (1), we readily conclude that

Remark 1:In engineering applications, there always need to be multiple sensors to finish complicated tasks. Hence, we consider the following distributed parameter system with noncollocatedmsensors andmactuators which exists the same boundary conditions and initial condition as system (1) in a repeatable environment

wheredenotes the spatial distribution of the sensing device of theith sensor andis the centroid position of theith sensor.

The spatial distribution of the sensors are assumed to be the boxcar function

and the spatial distribution of the actuators are also assumed to be a boxcar function

where β and α are constants. ε>0 and η>0 are the spatial support of the sensors and actuators, respectively.

Remark 2:In some practical engineering applications, the actuator needs to perform a task so the spatial distribution of actuating device is wider than sensing device. Therefore, we assume

According to the system (23), the following assumption is given.

Assumption 2:For a desired outputy(d,i)(t), a uniqueu(d,i)(t)exists such that

In this part, we consider the open-loop PD-type ILC scheme

wheree(k,i)(t) is the output error ofit h sensor duringkth iteration which satisfiese(k,i)(t)=y(k,i)(t)?y(d,i)(t). Υiand Γiareith number of open-loop PD-type learning gains.

Theorem 2:If the open-loop PD-type gain Γiof the ILC scheme ( 27) satisfies ( 1+2αβεΓi)2<1/2, and the system with the desired output satisfies the Assumption 2 under the initial and boundary conditions, the output errors of ( 23) converge to zero when

Remark 3:In previous proof, we just consider that the initial condition is zero at every iterative learning process, however,there always exits some errors at the beginning in every iterative process. Hence, a more favorable initial condition is given as follows:

For the system (1), if we consider using a closed-loop PDtype ILC scheme to replace the open-loop PD-type ILC scheme (7), we can obtain the convergence conditions of tracking error. The closed-loop PD-type ILC scheme is employed

where Φ and Ψ are the P-type learning gain and D-type learning gain, respectively.

Proof:From the desired input and actual input, the following gives

whereand

From ( 30), We can get

According to the definition ofL2-norm, we have

Similar to the proof of Theorem 1, we can investigateby using the spatial distribution of the actuators and the definition ofL2-norm as follows:

Substituting ( 34) into ( 32), we have

Multiply both sides of (35) bye?t, and letwe can get

whereo1=2|1?2mΨδγσ|?2,o2=2|Φδ|2l?k+1/|1?2mΨδγσ|2ando3=|4Φ2δ2mγ2σ|/|1?2mΨδγσ|2.

From Lemma 1, we can have that

Multiply both sides of (37) byand letWe obtain

From the initial condition, we know ? ∈[0,1), henceo2→0, whenk→∞. And wheno1<1, we can obtainV(k,i)(t)→0 (k→∞) from Lemma 2. BecauseV(k,i)(t)=we have

Hence, we can obtain

IV. Numerical Simulations

Consider the following distributed parameter system with a sensor and two actuators in a repeatable environment

wherex∈[0,1],t∈[0,0.8] and ? (x)=0.01>0.

Assume that the sensor spatial distribution satisfies

and the actuators spatial distribution satisfy

and

In this example, we employ the open-loop and closed-loop PD-type ILC schemes, and assume Γ=?0.7, Φ=?0.3 and Υ=Ψ=?15. Thus, we can obtain (1+2mδγσΓ)2<1/2 and 1/|1?2mΨδγσ|2<1/2which satisfy Theorems 1 and 3,respectively. The desired trajectory is given asyd=sin(5πt).

Using the difference method for partial differential equations, the simulation results can be obtained which are shown in Figs. 1–6.

Figs. 1–3 are obtained using an open-loop PD-type ILC scheme. Fig. 1 shows the desired output and actual output of system atk=15,25,30, respectively. Fig. 2 shows the statesqk(x,t) of system atk=25, and it is seen that only whenx∈[0.3125,0.375]∪[0.5,0.5625], the statesFig. 3 shows a curve chart which describes the variation in the error of output with the number of iterations. Whenk=25, the maximum error of the output function is 1.2×10?3. The simulation results demonstrate the effectiveness of the proposed scheme.

Figs. 4–6 are obtained using a closed-loop PD-type ILC algorithm. Fig. 4 shows the desired output and actual output of system atk=15,25,30, respectively. Fig. 5 shows the statesqk(x,t) of system atk=25, and it is seen that only whenx∈[0.3125,0.375]∪[0.5,0.5625], the statesFig. 6 shows a curve chart which describes the variation in the error of output with the number of iterations. Whenk=25, the maximum error of the output function is 0.64×10?3. The simulation results demonstrate the effectiveness of the proposed algorithm.

Fig. 1. The desired output yd(t) and iterations for output function yk(t) when k = 15,25,30, respectively (open-loop PD-type ILC scheme).

Fig. 2. The states qk(x, t) of system when k = 25 (open-loop PD-type ILC scheme).

Fig. 3. The variation of maximum output error ek(t) along with iterative number (open-loop PD-type ILC scheme).

V. Conclusion

Fig. 4. The desired output yd(t) and iterations for output function yk(t) when k = 15,25,30, respectively (closed-loop PD-type ILC scheme).

Fig. 5. The states qk(x, t) of system when k = 25 (closed-loop PD-type ILC scheme).

Fig. 6. The variation of maximum output error ek(t) along with iterative number (closed-loop PD-type ILC scheme).

In this paper, we extended the open-loop and closed-loop PD-type ILC schemes for two types of the parabolic distributed parameter systems based on non-collocated sensors and actuators working in a repeatable environment. Firstly, we took into consideration the convergence condition of a type of distributed parameter system with non-collocated single sensor andmactuators by using two classes of ILC schemes.Thereafter, we discussed the convergence condition of a class of distributed parameter systems using non-collocatedmsensors andmactuators. Lastly, we presented a distributed parameter system based on one sensor and two actuators to illustrate the effectiveness of the proposed control. From Figs. 3 and 6, we know the maximum error of the output function are 1.2×10?3(open-loop PD-type) and0.64×10?3(closed-loop PD-type), respectively. Hence, the closed-loop PD-type ILC scheme is more effective.