Iterative Learning Control for Remote Control Systems with Communication Delay and Data Dropout

Iterative learning control ILC is applied to remote control systems in which communication channels from the plant to the controller are subject to random data dropout and communication delay. Through analysis, it is shown that ILC can achieve asymptotical convergence along the iteration axis, as far as the probabilities of the data dropout and communication delay are known a priori. Owing to the essence of feedforward-based control ILC can perform trajectory-tracking tasks while both the data-dropout and the one-step delay phenomena are taken into consideration. Theoretical analysis and simulations validate the effectiveness of the ILC algorithm for networkbased control tasks.


Introduction
Iterative leaning control ILC is a control method that achieves perfect trajectory tracking when the system operates repeatedly.ILC has made significant progresses over the past two decades 1-3 and covered a wide scope of research issues such as continuous-time nonlinear system control 4 , discrete-time nonlinear system 5 , the initial reset problem 6, 7 , stochastic process control 8 , state delays 9 , and data dropout 10 .
On the other hand, the research on networked control systems has attracted much attention 11, 12 over the past few years.In network control, two frequently encountered issues are data dropout and communication delays, which are causes of poor performance of remote control systems.A central research area in remote control systems is to evaluate where "i" and "t" denote the iteration index and discrete time, respectively.x i t ∈ R n , u i t ∈ R p , and y i t ∈ R r for all t ∈ 0, T are system states, inputs, and outputs, respectively, at the ith iteration.A, B, and C are constant matrices with appropriate dimensions.The schematic diagram of the remote control systems under consideration is shown in Figure 1.
It should be noted that the open-loop system from the ILC input to the plant output is deterministic.The randomness occurs during the data transmission from the plant output to the ILC input.There are two approaches in analyzing the closed-loop system.The first approach is to treat the entire closed-loop system as a random or stochastic process.In such circumstances, the topology of the overall system keeps changing and the control process is either a Markovian jump process or a switching process.Another approach, which is adopted in this work, is to retain the essentially deterministic structure of the original openloop system, meanwhile model the random data dropout and communication delay into two random factors with known probability distributions.As a consequence, the signals used in ILC, y i t are the modulated plant output with the two random factors.
When the control process is deterministic, an effective ILC law for the linear system 2.1 is where u i 1 t and u i t are control inputs at the i 1 th and ith iterations, namely, the present trial and the previous trial, respectively.e i t 1 y d t 1 − y i t 1 is the output tracking error at the time t 1 th time instance of the ith iteration.L is a learning gain matrix.
Remark 2.1.Note that in the ILC law 2.2 , the control signal of the present iteration, u i 1 t , consists of both the pastcontrol input, u i t and the past error with one-step temporal advance, e i t 1 .The current-cycle feedback errors, such as e i 1 t , are not used.Since ILC does not require the current-cycle feedback nor the temporal stability, it is an effective control method for remote control systems problems with random data dropout and communication delay.
To facilitate the ILC design and convergence analysis, data dropout and one-step communication delay are formulated.First formulate the data-dropout problem.Denote γ t a stochastic variable with Bernoulli distribution taking binary values 0 and 1, where γ t 0 denotes an occurrence of data dropout and γ t 1 denotes a normal data communication.The probabilities of γ t are where γ > 0 is a known constant.Here, we assume that γ t is a stationery stochastic process, thus the data dropout rate is independent of the time t.In subsequent derivations, we treat γ as time invariant.
When the data dropout occurs in multiple communication channels, we can similarly define E γ j γ j > 0 for the jth communication channel.Thus, denote the corresponding mathematical expectation is where Γ > 0 is known a priori.Due to the data dropout, the plant output received by the controller at the i 1 th iteration is Generally speaking, the occurrences of data dropouts at two iterations are uncorrelated, thus independent.On the other hand, ILC law at the current iteration, the i 1 th iteration, uses only signals of the previous iteration, namely, ith iteration, as shown in 2.2 .Thus y i 1 t with the control input u i 1 t contains data dropouts upto the ith iteration.Therefore, Γ i 1 and y i 1 t are independent, that is,

2.7
Without the loss of generality, we assume E Γ i Γ, namely, the data dropout rate is invariant at different iterations.
Next formulate the one-step communication delay problem.Denote w t is a random delay factor with Bernoulli distribution, which takes binary values 0 and 1 that indicate, respectively, the presence and absence of an one-step communication delay.Here we assume that w t is a stationery stochastic process, thus the occurrence of the communication delay is independent of the time t.In subsequent derivations we treat w as time invariant.With multiple communication channels, we define matrix W diag w j , where w j denotes the occurrence of communication delay at the jth communication channel.Denote E w w and E W W. The plant output received by ILC with possible communication delay is formulated by where W i is the communication delay at the ith iteration.Without the loss of generality, we assume E W i W, namely, the probability of the communication delay is invariant at different iterations.Analogous to data dropout, assume that communication delay at any two iterations are independent, then W i 1 and W i are independent, so are W i 1 and y i 1 t because y i 1 t contains communication delays upto the ith iteration through the ILC law 2.2 .
It is worthwhile noting that stochastic variables γ and w are not completely independent.A delayed or nondelayed communication occurs only when γ 1, that is, no data dropout.Hence, we should have the condition probability for data transmission without delay Prob γ 1, w 1 and the condition probability for data transmission with one-step delay

2.10
As a consequence, we have

2.11
The relationship between data drop out and communication delay, 2.11 , can be extended to multiple channels at the ith iteration

2.12
At the ith iteration, the output signals perturbed by data dropout and one-step communication delay can be expressed as where I is a unity matrix of appropriate dimensions.The mathematical expectation of y i t can be derived using the independence property between Γ i , W i , and y i , as well as the relationship 2.12

2.14
The objective of control design is to seek an appropriate ILC law that can take into consideration data dropout and communication delay concurrently.The following ILC law is adopted where where δ t y d t 1 − y d t .

Convergence Analysis for Left-Invertible Systems r ≥ p
In this section, we derive the convergence property of the ILC 2.15 in the presence of data dropout and communication delays.
In ILC, the learning convergence can be derived in terms of either the output tracking error, e i t , or the input tracking error, Δu i t .In this section, we prove the learning convergence property of Δu i t .Assumption 3.1.For a given output reference trajectory y d t , which is realizable, there exists a unique desired control input u d t ∈ R p such that where u d t is uniformly bounded for all t ∈ 0, T .It is assumed that for all i ∈ Z , x i 0 is a random variable with E x i 0 x 0 x d 0 .
Define the input and state errors then from 2.1 and 3.1 , we have From 2.15 , using the relationship 2.12 , we have

3.4
Theorem 3.2.Suppose that the update law 2.15 is applied to the networked control system and satisfied the Assumption 3.1.If there exist ρ satisfying then the input error along the iteration axis, E Δu i t , converges to a bound that is proportional to the factor δ t .
Proof.First, subtracting u d t from both sides of the ILC law 2.15 yields Applying the ensemble operator E • to both sides of 3.6 and substituting the relationship 3.4 with e i t CΔx i t , we obtain

3.7
Substituting the state error dynamics 3.3 into 3.7 leads to the following relationship:

3.8
Define ρ I p − LΓWCB .Now let us handle the second term on the right hand side of 3.8 , which is related to Δx i t .Applying the ensemble operation to the following relationship: Substituting the relation 3.9 into 3.8 , taking the norm • 2 on both sides, the following relationship is derived:

3.10
where a ≥ A and in this work we choose a > 1 if A ≤ 1.

Mathematical Problems in Engineering
In order to handle the exponential term with a t in 3.11 , we introduce the λ norm.From Assumption 3.1, multiplying both sides of 3.10 by a −λt and taking the supermum over 0, T yield

3.12
Substituting the properties of Lemma A.1 into 3.11 yields

3.13
Since 0 ≤ ρ < 1, it is possible to choose λ sufficiently large such that

3.14
Therefore we can rewrite 3.13 as Note that μ is proportional to δ t , namely, the maximum difference between y d t 1 − y d t in t ∈ 0, T , which is bounded and small when the reference trajectory is smooth or the sampling interval is sufficiently small.When the probability associated with the data communication delay, W, is known a priori, we can further revise the reference trajectory to an augmented one, such that the resulting δ t 0.
Γ j W j e j t 1 Γ j I − W j e j t .

3.17
Comparing the above expression with 2.16 , we conclude that δ t 0, subsequently μ 0, which implies a zero-tracking error according to 3.16 .

Convergence Analysis for Right-Invertible Systems r ≤ p
In this section, we prove the learning convergence property of e i t .then the tracking error along the iteration axis, E e i t , converges to a bound that is proportional to the factor δ t .
Proof.First note the relationship: Substituting ILC law 2.15 , 2.16 , and 4.3 into 4.2 yields Mathematical Problems in Engineering Applying the ensemble operator E • to both sides of 4.4 and substituting the relationship 4.2 , we obtain

4.5
Taking the norm • 2 on both sides of 4.5 , the following relationship is derived where a ≥ A and in this work we choose a > 1 if A ≤ 1.
In order to handle the exponential term with a t in 4.5 , we introduce the λ norm.Multiplying both sides of 4.5 by a −λt and taking the supermum over 0, T yield sup t∈ 0,T a −λt E e i 1 t 2 ≤ sup

4.7
Substituting the properties of Lemma A.2 into 4.7 yields where Therefore, we can rewrite 4.8 as

Numerical Examples
Consider the following linear discrete-time system: 5.1 with the initial condition x i 0 0. The desired trajectory is y d t sin 2πt/50 .The tracking period is {0, 1, . . ., 49}.The control profile of the first iteration is u 0 t 0 for t 0, 1, . . ., 49.Two sets of probabilities for the data dropout rate and communication delay are considered, which are γ 0.9, w 0.9, γ 0.6, and w 0.6, respectively.The learning gain is L 0.5, which yields I − LγwCB 0.595 < 1, and I − LγwCB 0.82 < 1 with respect to the two sets of probabilities.The tracking performance of two ILC algorithms is given in Figure 2, where Max Error denotes the maximum absolute error of each iteration.

Conclusion
In this work, we address a class of networked control system problems with random data dropout and communication delay.D-type ILC is applied to handle this remote control systems problem with repeated tracking tasks.Through analysis, we illustrate the desired convergence property of the ILC.Although we focus on one-step communication delay in this work, the results could be extended to multiple delays, which is one of our ongoing research topics.In our future work, we will also explore the extension to more generic nonlinear dynamic processes.

Appendix
Lemma A.1.For all a > 1, for all λ > 1, for all i ∈ Z , the inequality:

A.2
Lemma A.2.For all a > 1, for all λ > 1, for all i ∈ Z , the inequalities ≤ E e i λ,a sup t∈ 0,T A.4

Figure 1 :
Figure 1: The schematic diagram of the remote control system.

Figure 2 :a 1 ≤ 1 −a
Figure 2:The tracking error profiles for the discrete-time linear system with data dropout and one-step communication delay.a is learning results with the data dropout rate γ 0.9 and communication delay rate w 0.9.b is learning results with the data dropout rate γ 0.6 and communication delay rate w 0.6.