New Iterative Learning Control Algorithm Using Learning Gain Based on σ Inversion for Nonsquare Multi-Input Multi-Output Systems

Model inversion Iterative Learning Control (ILC) for a class of nonsquare linear time variant/invariant multi-input multi-output (MIMO) systems is considered in this paper. A new ILC algorithm is developed based on σ-right inversion of nonsquare learning gain matrices to resolve the matrix inversion problems appeared in the direct model inversion of nonsquare MIMO systems. Furthermore, a sufficient and necessary monotonic convergence condition is established. With rigorous analysis, the proposed ILC scheme guarantees the convergence of the tracking error. To prove the effectiveness and to illustrate the performance of the proposed approach for linear time-invariant (LTI) and time-varying nonsquare systems, two illustrative examples are simulated.


Introduction
Iterative Learning Control is an intelligent control strategy to deal with repetitive processes.e aim of the ILC is to control systems which execute the same task over a finite duration, such as the industrial robot manipulator [1].e basic principle behind ILC is that the data generated from previous trial are used to adapt the control input for current trial.e control input in each trial is adjusted by using the tracking error obtained from the previous trial.As the iterations continue, the control system eventually learns the task and follows the desired trajectory with minimized tracking errors.e concept has been well developed in terms of both the fundamental theory and experimental applications that were accomplished in [2][3][4][5][6].
us, ILC research has considerably coped with mechanical systems such as multijoint hand-arm robots [7], the station stop control of train [8], and wafer stage motion systems [9].e most of ILC schemes in the literature focus on proportional type learning law [10] and Optimal Iterative Learning Control (OILC) [11][12][13][14] where in [13] the learning approach has been applied to a rapid thermal processing.
However, the majority of proposed algorithms were based on the notion of direct model inversion where the learning gain is obtained by Markov matrix inverse which represents the input-output map of the system [15].However, this kind of ILC cannot be developed for general nonsquare (rectangular) MIMO systems that are systems in which the numbers of inputs and outputs are unequal.Indeed, a problem of the matrix inversion is encountered.
Many industrial processes require nonsquare MIMO mapping, especially chemical plants such as crude distillation process [16], mixing tank process [17], three-tank systems [18], distillation column [19], and chemical mechanical planarization process [20].erefore, the need of control schemes treating this type of systems is of major interest.e model inverse design and its related applications in control design have been widely studied in [21][22][23][24].
In the literature, there are control works that treated the nonsquare MIMO systems such as a Proportional Integrator Derivator (PID) controller cited in [25] which has been applied to a voltage model of Proton Exchange Membrane Fuel Cell (PEMFC); Model Predictive Control (MPC) was studied in [26] and has been applied to a Shell Heavy Oil Fractionator (SHOF); Minimum Variance Control (MVC) which is an inverse model control was studied in [27] and other examples of control strategies applied to nonsquare systems [28].In the above cited works [25,26], the control schemes were achieved without a need to matrix inversion.However, in [27] the synthesis of the inverse model in the control procedure required a matrix inversion based on left and right inverses which are developed in [29][30][31].
However, all the above mentioned methods could not deal with repetitive systems that require a learning strategy based on inversion model to achieve the control of repetitive nonsquare MIMO systems.
erefore, developing an ILC procedure for nonsquare systems remains an open problem.
Based on nonsquare polynomial matrix inversions [32], we propose the design and the analysis of an ILC scheme based on the model inversion called I (n) -ILC for linear timeinvariant and varying-time nonsquare MIMO systems.e main contribution of this work is to prove the monotonic convergence of the proposed scheme where the tracking error trial-to-trial will converge to zero even though the system has the initial resetting state.rough the simulation results, we prove the e ectiveness of the proposed method.e rest of this paper is organized as follows: In Section 2, the problem formulation is presented.
e proposed inverse learning gain is developed in Section 3. A su cient and necessary monotonic convergence of the I (n) -ILC is established in Section 4. Further, the proposed I (n) -ILC law is extended to time-varying systems in Section 5. Simulation results are illustrated in Section 6 to prove the effectiveness of the scheme for nonsquare MIMO systems.Finally, conclusions and an outlook on future work are given in Section 7.

Problem Formulation
e state-space representation of an LTI discrete-time system is given by (8): x j (k + 1) Ax j (k) + Bu j (k), where j ∈ I ≜ 1, 2, . . ., n j − 1 denotes the iteration numbers and n j is the total number of trials.k ∈ Λ ≜ 0, 1, . . ., N − 1 { } represents the number of the discrete-time sampling steps and N is the total number of discrete-time steps at each trial.x j (k) ∈ R p is the jth iteration vector of system states, u j (k) ∈ R n u is the jth iteration vector of the system inputs which will be recursively generated by an iterative learning algorithm, y j (k) ∈ R n y is the jth iteration of the system outputs and x j+1 (0) is a xed initial state.Further, A, B, and C are constant matrices with appropriate dimensions, and CB ≠ 0. erefore, this statespace system can be described as follows: where M is a Markov matrix of rank N and whose terms are Markov parameters of the plant as cited in [33] and e control objective is to nd a control sequence u j (k) with the ability to reduce tracking error for the whole trajectory based on the past tracking experiences until j → ∞, the system tracking error limit is where ∞ is the in nite norm.e main formulation of the ILC design problem is to achieve an update mechanism for the control input trajectory of a new cycle based on the information from previous cycles, so that the output trajectory converges asymptotically to the reference trajectory [34][35][36].is idea is depicted in the block diagram form in Figure 1, which shows the next trial's control input u j+1 to be calculated from the previous trials control input u j and transient output error e j .
In this paper, we focus on the nonsquare systems when the number of input variables n u is di erent from the number of output variables n y .
e goal is to make the system outputs track a given desired reference trajectory.
e rst-order ILC algorithms update the input trajectory u with the following equation: where we designate by H the learning gain and by e j the error between the set-point reference y c and the jth output as e most proposed algorithms were based on the direct model inversion, that is H M −1 , assuming that M −1 is invertible.M −1 represents the input-output map of the process [37,38].However, in nonsquare system case, the learning gain based on the model inversion cannot be calculated.In fact, the adaptation law in (5) cannot be satis ed for general nonsquare MIMO processes.To overcome this problem, a new method is proposed to design a realizable controller based on the direct model inversion 2 Modelling and Simulation in Engineering I-ILC for n u × n y linear nonsquare system process called Theorem 1 (stability of ILC).Let an LTI discrete-time system be described by ( 1) is asymptotically stable if and only if the polynomial matrix M is right invertible [39,40].
e ILC law ( 5) can be immediately written as where M R is the right inverse of Markov matrix M.

Inverses of Nonsquare Polynomial Matrices and I (n) -ILC Design
Several methods of right/left inverses of nonsquare polynomial matrices, such as the classical minimum-norm right/left inverse, called T-inverse and τ-σ-S-inverses methods, have been studied in [27,32].In this work, we focus on σ-inverse as follows: of full normal rank n y (or n u ) be arbitrary, including an arbitrary order r. e product M (q −1 )ψ T T � (q −1 ) must be of full normal rank n y .en a σ-inverse can be defined as For right invertible system (1), M R � denotes an infinite number of right inverse of M. Definition 1.Let M ∈ R n y ×n u and ψ ∈ R n y ×n u are a full rank n y < n u where M is right invertible and ψ is selected such that Using previous Definition 1, we can employ the σ-inverse in I (n) -ILC (7) to obtain Using ( 2) and ( 3), the tracking error ( 6) can be rewritten as Adding and subtracting CA k x j (0) into (11), we get

Convergence Analysis of the Proposed Learning Gain
e control design problem is to determine a new ILC law based on the model inversion such that the trial-to-trial error convergence occurs in j; that is, lim j→∞ ‖e j ‖ � 0 [41].
erefore, the series S j �  +∞ k�0 ‖e j (k)‖ 2 is convergent.en, we prove that lim j→∞ S j+1 � 0 by reduction to absurdity.Suppose that lim j→∞ S j+1 � S > 0. erefore, for τ 1 � S/2, there exists a finite positive integer j 0 so that, for all j > j 0 , S j+1 > S − τ 1 > (S/2).However, S j is convergent, so for all j > j 0 , the limit of the partial sequence S j+1 (r) �  r k�0 ‖e j+1 (k)‖ 2 exists.is means that, for a given constant τ 2 � S/4, there exists an integer N 0 so that, for all r > N 0 , we have in particular, Modelling and Simulation in Engineering 3 then, for k 0 ∈ [0, 2N 0 ], we obtain Hence, we have On the other hand, by recursion, (16) reduces to Although, by considering the assumption (15), we obtain is is contradictory to inequality (19). is contradiction means that the propositions ( 13) and ( 14) are true.
is proves the sufficiency assumption.
Proof (necessity).We assume that the inequality ( 15) is not always true.So, there exists at least a number k 0 such that e equality (16) gives thus, Finally, we obtain It is evident to choose u 1 (k) and y c (k) such that ‖e 1 (k 0 )‖ > 0, which contradicts to the assumption (14). is proves the necessity of the assumption.
e sufficient and necessary assumption for the monotonic convergence of the given I (n) -ILC algorithm shows that ILC trial-to-trial error convergence requires that all of the initial states and tracking errors are reset.

Extension to Time-Varying Systems
In this part, the proposed I (n) -ILC scheme is extended to linear time-varying systems as follows: where A(k), B(k), and C(k) are time-varying matrices with appropriate dimensions and C(k)B(k) ≠ 0. erefore, this state-space system can be described as follows: where . us the output can rewritten as Using the σ-inverse (8), the I (n) -ILC to control timevarying MIMO systems is similar to control law (10).Theorem 3.For the discrete linear varying-time system (28), the I (n) -ILC is chosen such that, for any constant 0 ≤ δ ≤ 1, sup erefore, the tracking errors e j (k) will converge to zero where j → ∞ if and only if the previous conditions ( 13) and ( 14) are proved.
Proof. e proof of eorem 3 can be completed identically as in the proof of eorem 2 such that the tracking error is given by Following Proof 1, we can conclude that lim j→∞ e j � 0 where all of initial states x j (0) are reset.

Illustrative Examples
In order to show the effectiveness of the proposed ILC based on the direct model inversion to deal with nonsquare systems, two examples are considered.
By applying the proposed I (n) -ILC law (10), the evolutions of transient outputs y 1 and y 2 pro les with di erent iteration numbers j 2, j 5, and j 25 are given in Figures 2 and 3, respectively.e evolutions of terminal inputs are shown in Figure 4.It is clear that the proposed scheme is well to be used for linear nonsquare MIMO systems to track time-varying reference.e I (n) -ILC approach converges quickly and the performance keeps well even the reference changes.Furthermore, the performance of the absolute initial and terminal tracking error pro les e 1,j and e 2,j versus the iteration numbers are illustrated in           where x j (0) [0, 0, 0, 0, 0, 0] T , j ∈ I ≜ 1, 2, . . ., 100 { }. k ∈ Λ ≜ 1, 2, . . ., 80 { }; thus N 80 and the initial input vector is chosen as u 0 [0, 0, 0] T .y c1 (dashed line) and the rst component of output y 1 (solid line) are depicted in Figure 9. y c2 (dashed line) and the rst component of output y 2 (solid line) are depicted in Figure 10.e performance of the absolute tracking errors e 1,j and e 2,j is illustrated in Figure 11, where the tracking errors will converge within 20 and 30 iterations.Moreover, Figures 12 and 13 show the tracking error pro les for 20th, 50th, and 100th iterations.It is clear that the tracking errors converge to zero in more than 70 iterations, where x j (0) [0, 0, 0, 0, 0, 0] T , j ∈ I ≜ 1, 2, . . ., 100 { }.

Conclusion
In this paper, a new model inversion Iterative Learning Control called I (n) -ILC is proposed to deal with nonsquare MIMO systems where the numbers of inputs and outputs are unequal.e proposed scheme is based on σ-right-inverse learning gain in order to resolve a major problem appeared in ILC based on inversion model (I-ILC).e convergence condition of the learning algorithm has been derived.It is shown that under some given conditions, the tracking error of I (n) -ILC law converges to zero through a su cient and necessary stability condition.en, the convergence properties are established.rough simulation results, we proved that the performances o ered by the proposed method in terms of tracking error convergence after few trials are achieved.erefore, robustness of the I (n) -ILC scheme to perturbations and parametric uncertainties, especially with less knowledge of plant model, remains a challenging topic which will be addressed in future works.
scheme is well to be used for linear nonsquare MIMO systems to track time-varying reference.e I (n) -ILC approach converges quickly and the performance keeps well even the reference changes.Furthermore, the performance of the absolute initial and terminal tracking error pro les e 1,j and e 2,j versus the iteration numbers are illustrated in Figures 5 and 6.It shows that the tracking e 1,j and e 2,j will converge within 15 and 20 iterations.Moreover, from Figures 7 and 8 which illustrated the rst and the second components of tracking error pro les at 5th, 10th, 15th, and 25th iterations.It is clear that the fast convergence of errors in the iteration domain is obvious for the proposed algorithm.

Example 2 :
Time-Variant System.In order to demonstrate the e ectiveness of our proposed algorithm, let the discretetime linear time-varying system as follows.

Figure 4 :
Figure 4: Evolutions of all inputs with trial length n j 25.

Figure 6 :
Figure 6: Terminal tracking errors e 1,j and e 2,j versus the iteration number j.

Figure 8 :
Figure 8: Tracking errors pro les of ILC e 2,j .

Figure 13 :
Figure 13: Tracking errors pro les of ILC e 2,j .