Robust Monotonically Convergent Iterative Learning Control for Discrete-Time Systems via Generalized KYP Lemma

and Applied Analysis 3 3. Problem Formulation 3.1. System Description. Consider the following MIMO discrete-time LTI system over [0, T]: x j (k + 1) = Ax j (k) + Bu j (k) , y j (k) = Cx j (k) + Du j (k) , x j (0) = x 0 , k ∈ N = {0, 1, . . . , N} , (9) where j is the iteration number, which denotes the jth repetitive operation of the system. The task interval [0, T] is finite and discretized in a set N that consists of sampled instances 0, 1, . . . , N. x j (k) ∈ R n is the state vector, u j (k) ∈ R m is the control input vector, and y j (k) ∈ R p is the output vector. A, B, C, and D are constant matrices of appropriate dimensions. x 0 is the initial condition for each iteration. The relative degree r (r ≥ 0) of system (9) can be defined by [34] (1) r = 0 ifD ̸ = 0; (2) r ≥ 1 if it holds that (a) CAr−1B ̸ = 0; (b) D = 0 and CAiB = 0 for all i < r − 1. Let y d (k) denote the reference vector, and then the tracking error on iteration j is e j (k) = y d (k) − y j (k) . (10) The control target is to design appropriate control signal u j (k) and present some LMI conditions such that the system output can converge monotonically to the reference trajectory y d (k) over a finite frequency range when the iteration number j tends to infinity, even if there exist system uncertainties. In order to complete the above control task, the following assumptions are imposed on system (9). Assumption 5. The initial resetting condition is satisfied; that is, e j (0) = 0, ∀j ∈ Z. Without loss of generality, it is considered that x j (0) = 0. Remark 6. Obviously, the transfer functionmatrix fromu j (k) to y j (k) can be expressed as y j (k) = G p (q) u j (k) , (11) where G p (q) = C(qI − A) −1 B + D =: [ A B C D ] (12) is usually obtained by discretizing the original continuoustime domain model using a sampling mechanism that consists of a sampler with the sampling interval T s and a zeroorder hold. Remark 7. The relative degree r is exactly the steps of delay in the output y j (k) in order to have the control input u j (k) appearing. Note that the relative degree of one, that is,D = 0 andCB ̸ = 0, is usually considered in the literature for discretetime ILC [5]. 3.2. Design of ILC. In this section, the ILC law is introduced as follows: u j+1 (k) = u j (k) + L (q) e j (k) , (13) where L(q) denotes anm × p polynomial gain operator to be designed. Subtract e j (k) from e j+1 (k) and then use (11) and (13) to obtain e j+1 (k) − e j (k) = y j (k) − y j+1 (k) = G p (q) [u j (k) − u j+1 (k)] = −G p (q) L (q) e j (k) (14)


Introduction
The well-known iterative learning control (ILC) algorithm can effectively improve the transient responses and tracking performance for systems that execute the same task over a finite duration repetitively, the key idea of which is to iteratively reduce the tracking error by refining the control input signal based on the information from previous trials [1].As demonstrated in survey papers [2][3][4][5][6], ILC has attracted considerable research attention in many areas during the past few decades.Extensive applications of ILC have been used for many practical problems coming from, for example, batch processes [7][8][9], point-to-point control [10,11], and positioning control [12][13][14][15].
In fact, among all types of ILC research issues, both theoretical and practical, robustness and monotonic convergence have been studied as two major topics.Many uncertain factors such as model uncertainties, variable initial conditions, stochastic noises, and packet dropout need to be taken into consideration with regard to robust ILC design.For example, a kind of so-called adaptive ILC has been developed for local Lipschitz continuous (LLC) uncertain nonlinear systems with unknown parameters, and composite energy function (CEF) is usually constructed to facilitate the convergence analysis [2].Considering the inherent twodimensional (2 D) structure of every ILC process, 2 D system theory has been developed to design ILC based on linear repetitive processes [16][17][18], Roesser model [19,20], and Fornasini-Marchsini model [21,22].Moreover, the robust ILC has been particularly extended to networked control system [23,24] and switched systems [25,26].
To achieve good learning transients, the monotonic convergence is particularly important in ILC design problems.For example, first-order and second-order P-type ILC schemes are used for continuous linear time-invariant (LTI) systems, where the monotonic convergence of tracking error is guaranteed in the sense of Lebesgue-p norm [27].It is also noticed that the so-called super-vector formulation for discrete-time ILC has been prevalent for monotonic convergence analysis under different appropriate norm topology.In [28], the monotonic convergence analysis for interval ILC systems is presented for discrete-time systems.A gradientbased optimal ILC scheme is proposed for ensuring robust monotonic convergence [29].A new semisliding window ILC algorithm is developed for discrete-time LTI systems [30].Recently, by integrating the technique of linear matrix inequality (LMI), the well-established  ∞ norm has been used for deriving monotonical convergence conditions that Lemma 2 (see [36]).Assume , , and  =   are real matrices with appropriated dimensions.Then for any matrix Σ satisfying Σ  Σ ≤ , the following inequality: where   = ( 1 +  2 )/2,   = ( 2 −  1 )/2.
holds if and only if there exists a scalar  > 0 such that Lemma 3 (see [37]).Assume , , and  =   are real matrices with appropriated dimensions.There exists a matrix  such that the following inequality: holds if and only if the following two inequalities with respect to  are satisfied: Lemma 4 (generalized KYP lemma, [38]).For a discrete LTI system with transfer function () and frequency response matrix (  ) = C(   − A) −1 B + D, the following statements are equivalent: (1) the frequency domain inequality or holds for all   ∈ Λ(Φ, Ψ), where Π is a given real symmetric matrix and where Φ = [ −1 0 0 1 ] and Θ denotes the frequency ranges specified by Ψ as shown in Table 1.
(2) There exists Hermitian matrices ,  such that  > 0 and Abstract and Applied Analysis 3

Problem Formulation
where  is the iteration number, which denotes the th repetitive operation of the system.The task interval [0, ] is finite and discretized in a set N that consists of sampled instances 0, 1, . . ., .   () ∈   is the state vector,   () ∈   is the control input vector, and   () ∈   is the output vector., , , and  are constant matrices of appropriate dimensions. 0 is the initial condition for each iteration.The relative degree  ( ≥ 0) of system ( 9) can be defined by [34] (1)  = 0 if  ̸ = 0; (2)  ≥ 1 if it holds that (a)  −1  ̸ = 0; (b)  = 0 and    = 0 for all  <  − 1.
Let   () denote the reference vector, and then the tracking error on iteration  is The control target is to design appropriate control signal   () and present some LMI conditions such that the system output can converge monotonically to the reference trajectory   () over a finite frequency range when the iteration number  tends to infinity, even if there exist system uncertainties.
In order to complete the above control task, the following assumptions are imposed on system (9).Assumption 5.The initial resetting condition is satisfied; that is,   (0) = 0, ∀ ∈  + .Without loss of generality, it is considered that   (0) = 0. Remark 6. Obviously, the transfer function matrix from   () to   () can be expressed as where is usually obtained by discretizing the original continuoustime domain model using a sampling mechanism that consists of a sampler with the sampling interval   and a zeroorder hold.
Remark 7. The relative degree  is exactly the steps of delay in the output   () in order to have the control input   () appearing.Note that the relative degree of one, that is,  = 0 and  ̸ = 0, is usually considered in the literature for discretetime ILC [5].

Design of ILC.
In this section, the ILC law is introduced as follows: where () denotes an  ×  polynomial gain operator to be designed.Subtract   () from  +1 () and then use (11) and ( 13) to obtain which leads to where   () =  −   ()().

Convergence Analysis
4.1.Super-Vector Approach.Using the lifting approach, system ( 9) and ILC law (13) can be described respectively as where U  , Y  , and Y  , E  = Y  − Y  , are the supervectors which are lifted to contain  sampled points, and G p and L are two lower triangular block Toeplitz matrices.The elements of G p are the system Markov parameters (or the pulse response coefficients).Equation (16) gives where G e = −G p L. For more details of the developments on ( 16) and ( 17), refer to [5].
From (17), the monotonic convergence condition can be simply defined in an appropriate norm topology Remark 8. Clearly, when the state-space model matrices , , , and  have structured and polytopic-type uncertainties, it is difficult to derive learning gain matrix from condition (18).

Frequency Domain Approach.
The following proposition will be helpful for developing frequency-domain monotonic convergence condition.
With Proposition 9, there exist matrices A, B, C, and D such that   () can be expressed by It is seen that the condition ( 19) can be resolved by combining a robust  ∞ control theory and the LMI technique.However, condition (19) requires control law design over the entire frequency and is a very strict condition.By utilizing the generalized KYP Lemma, this paper develops monotonically convergent ILC design restricted within a finite frequency range.Accordingly, ( 19) is replaced by following condition: where Θ denotes a finite frequency range.Moreover, condition ( 21) is replaced with However, in this case, inequality ( 22) is no longer a standard  ∞ problem.To this end, we denote  =  1  −1 2 for scalars  1 > 0 and  2 > 0. Then if holds,  ∈ (0, 1] follows immediately, and at the same time (22) equivalently becomes where   (  ) =  2   (  ).Thus, condition (24) can be viewed as an  ∞ problem that is subject to a linear constraint condition (23).Now with controlled system (15), let us further consider how to solve the condition (24) under the generalized KYP Lemma framework.Consider the frequency response matrix   (  ) and choose the matrix Π of Lemma 4 as We can get Obviously, inequality ( 26) is equivalent to condition (24).

Zero
Relative Degree ( = 0).Consider first that system (9) has a zero relative degree, resulting in Accordingly, the ILC law ( 13) is applied with the following gain operator: where  is an  ×  matrix to be determined.Moreover, it is easy to see that   () and   () satisfy (20) because it can be modeled by Now with Lemma 4 and ( 29), the following theorem can be presented.
Theorem 10.Consider the ILC system (9) and ( 13) satisfying  = 0 and Assumption 5, and the gain operator matrix () is defined by (28).Then, ‖E  ‖ 2 converges monotonically to zero over the low frequency range || ≤   when  → ∞, if there exist scalars  1 > 0,  2 > 0 and matrices P > 0, Q > 0, ,  satisfying (23) and the following LMI: If the LMIs of ( 23) and ( 30) are feasible, then the gain matrix  is given by Proof.Applying Lemma 4 gives that condition (26) holds if there exist symmetric matrices  and  such that  > 0 and where and Ψ is the only matrix whose block entries depend on the chosen frequency range.Without loss of generality, the low frequency range is considered; that is, || ≤   , which gives that Then, (32) becomes The condition of (34) cannot, however, be directly applied to control law design since it involves product terms A  B and B.
To separate the matrices  and  from the process model matrices, rewrite (34) as To apply the result of Lemma 3, we can set  = [ Then we can have The above inequality holds if and only if the diagonal blocks satisfy Hence  > 0 is required.Moreover, = , which is naturally set up according to asymptotic convergence of the ILC system.
Next, application of Lemma 3 gives that ( 35) and ( 36) are feasible if there exists a matrix  satisfying Applying the Schur's complement to (38) and inserting pre-and postmultiplying this last inequality by diag{  ,   , , } and diag{, , , } to obtain Finally, introduce the change of variables: giving immediately that (40) is equivalent to the LMI of (30) and the proof is complete.
Next it will be shown that Theorem 10 can be further developed to address system (9) with structured uncertainty matrices of the form: where Δ, Δ, Δ, and Δ represent admissible uncertainties which are assumed to satisfy where  1 ,  2 ,  1 , and  2 are known matrices of appropriate dimensions, and Σ is an unknown matrix satisfying In this case, the following robust result can be presented.
Proof.With Theorem 10 applied, this proof can be expressed as the requirement that where With Lemma 2 applied, one has that (46) holds for all Σ satisfying Σ  Σ ≤  if and only if there exists a scalar  3 > 0 such that It is noted that the above inequality can be rewritten to obtain Using the Schur complement Lemma, one knows that (49) is equivalent to Now pre-and post-multiplying (50) by diag{, , , ,  1/2 3 ,  1/2 3 } result in (45).This proof is completed.Furthermore, it will be demonstrated that Theorem 10 can be extended to address the case, where the matrices of system The following result enables robust ILC design when the model matrices of system (9) belong to a polytope-type uncertain domain Ω.
Proof.For all systems of the type of ( 9) falling within Ω, let P = ∑  =1   P , Q = ∑  =1   Q .Then the LMI of ( 30) can be achieved from the set of LMIs in (52).The remaining of this proof is omitted since it can follow the same lines of the proof of Theorem 10.
In order to compensate for the influence of , the ILC law ( 13) is considered with an anticipatory gain operator of the form: where   is an  ×  matrix.Moreover,   () and   () can still satisfy (20) as shown in the following: or Remark 13.Using the fact that ( − ) −1 =  + ( − ) −1  or ( − ) −1 =  + ( − ) −1 repetitively yields that   ( − ) −1 can be expressed as or Then   () can be derived as or Theorem 14.Consider the ILC system (9), (13), and (55) satisfying  ≥ 1 and Assumption 5, and the gain operator matrix () is defined by (54).Then, ‖E  ‖ 2 converges monotonically to zero over the low frequency range || ≤   when  → ∞, if there exist scalars  1 > 0,  2 > 0, and matrices P > 0, Q > 0, , and   satisfying (23) and the following LMI: If the LMIs of ( 23) and (61) are feasible, then the gain matrix   is given by Proof.This proof is omitted since it follows identical steps to that of Theorem 10.
Even if a higher-order relative degree exists, LMIs of ( 23) and (61) can still be obtained to achieve the monotonic convergence.Since LMI (61) contains product of matrices of the plant   (), Theorem 14 cannot be extended like that done in Corollaries 11 and 12.However, it is noted that, for the case  = 1, the results of Theorem 14 are feasible to deal with the norm-bounded and polytopic-type uncertainties.
Next let us consider only the case where the uncertain model is of the form: The matrices Δ and Δ represent admissible uncertainties that are assumed to satisfy where  1 ,  2 , and  1 are known constant matrices, and Σ is an unknown matrix satisfying Σ  Σ ≤ .Now the effects of norm-bounded uncertainty can be addressed via the following result.
Proof.This proof is omitted since it follows in an identical manner to that of Corollary 11.The following result is able to address the problems of polytope-type uncertainty.
Proof.This proof is omitted since it follows identical steps to that of Corollary 12.
Remark 17. Considering the dual transfer function   () that is described by (56), the following result can be obtained in an identical manner to that of Theorem 18.
Based on Theorem 18, the robust results with  = 1 are omitted since it follows identical steps to those of Corollary 15 and Corollary 16.Obviously, the considered system has a relative degree of one.Each iteration duration  is 1.5 s and the sampling frequency is set to 100 Hz.The reference trajectory is shown in Figure 1(a) and associated frequency spectrum in Figure 1(b).Inspecting the amplitudes in the frequency spectrum, it is shown that significant harmonics in the range from 0 to 10 Hz, which can be taken as the low frequency range.And hence   is chosen as   = 0.6284.
The simulation results are shown in Figures 2, 3, and 4. From these three figures, it is clearly demonstrated that the tracking error converges monotonically to zero along the iteration axis.LMIs ( 23) and (67) to obtain   = 0.3326.The simulation results are shown in Figures 5, 6, and 7, from which it is seen that the tracking error also decays monotonically to zero along the iteration axis.

Conclusion
This paper deals with tracking problem of uncertain MIMO discrete-time systems with a relative degree.Based on the idea of generalized Kalman-Yakubovich-Popov lemma, the proposed ILC scheme achieves robust monotonically convergent control law design over a finite frequency range, and sufficient conditions in terms of LMIs have been developed.The effectiveness of the controller design is validated through two numerical examples.

Figure 1 : 2 EFigure 2 :
Figure 1: (a) The reference trajectory and (b) the frequency spectrum for the reference trajectory.

Example 1 .5Figure 3 :
Figure 3: Example 1, process of the tracking error along both the iteration axis and the time axis.

Example 2 .Figure 6 :Figure 7 :
Figure 6: Example 2, process of the tracking error along both the iteration axis and the time axis.