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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 450241, 12 pages

http://dx.doi.org/10.1155/2014/450241
Research Article

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

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, China

Received 24 January 2014; Accepted 12 May 2014; Published 29 May 2014

Academic Editor: Chengjian Zhang

Copyright © 2014 Jian Ding and Huizhong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper addresses the problem of P-type iterative learning control for a class of multiple-input multiple-output linear discrete-time systems, whose aim is to develop robust monotonically convergent control law design over a finite frequency range. It is shown that the 2 D iterative learning control processes can be taken as 1 D state space model regardless of relative degree. With the generalized Kalman-Yakubovich-Popov lemma applied, it is feasible to describe the monotonically convergent conditions with the help of linear matrix inequality technique and to develop formulas for the control gain matrices design. An extension to robust control law design against systems with structured and polytopic-type uncertainties is also considered. Two numerical examples are provided to validate the feasibility and effectiveness of the proposed method.

1. 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 [26], 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 [79], point-to-point control [10, 11], and positioning control [1215].

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 two-dimensional (2 D) structure of every ILC process, 2 D system theory has been developed to design ILC based on linear repetitive processes [1618], 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 gradient-based 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 can be described as LMIs and formulas for the control law design, and the tracking error can be ensured to converge monotonically in the sense of norm [3134].

However, the aforementioned monotonically convergent ILC works treat control law design over the complete frequency range which is not practical in many cases. In particular, the reference signal and design specification are often given for a certain frequency range of relevance. This viewpoint motivates the present study. In this paper, an integrated ILC framework is developed for multiple-input multiple-output (MIMO), LTI discrete systems with a relative degree, and the frequency design can be specified over a finite range. This benefits from the well-established generalized Kalman-Yakubovich-Popov (KYP) lemma that can tie together a frequency domain inequality (FDI) over finite frequency range and an LMI. It is shown that monotonic convergence conditions can be described in terms of LMIs, as well as formulas obtained for the control law design. Furthermore, this approach is extended to handling the robust issues for the systems with norm-bounded and polytopic-type uncertainties.

Briefly, the paper is organized as follows. Section 2 introduces several useful LMIs. The problem formulation is supplied in Section 3. The convergence performance of the proposed scheme is analyzed in Section 4. Section 5 provides two illustrative examples. Finally, some concluding remarks are given in Section 6.

Throughout this paper, the following notations are employed. denotes the set of nonnegative integers. For a matrix , its transpose, complex conjugate transpose, and orthogonal complement are denoted by , , and , respectively. and are the identity matrix and the zero matrix with appropriate dimensions, respectively. and denote positive definiteness and negative definiteness, respectively. The symbol represents the transposed elements in a symmetric matrix and denotes the spectral radius of its matrix argument. For matrices and , denotes the Kronecker product. is a forward shift operator along the discrete-time axis; that is, .

2. Preliminary Knowledge

Before presenting the main results, the following well-known results are briefly introduced in this section.

Lemma 1 (Schur complement, [35]). Given a symmetric matrix , and are square, and then the following inequalities are equivalent:(1) (2) and (3) and .

Lemma 2 (see [36]). Assume , , and are real matrices with appropriated dimensions. Then for any matrix satisfying , the following inequality: holds if and only if there exists a scalar 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 , the following statements are equivalent:

(1) the frequency domain inequality or holds for all , where is a given real symmetric matrix and where and denotes the frequency ranges specified by as shown in Table 1.

(2) There exists Hermitian matrices , such that and

tab1
Table 1

3. Problem Formulation

3.1. System Description

Consider the following MIMO discrete-time LTI system over : where is the iteration number, which denotes the th repetitive operation of the system. The task interval is finite and discretized in a set that consists of sampled instances . is the state vector, is the control input vector, and is the output vector. are constant matrices of appropriate dimensions. is the initial condition for each iteration. The relative degree of system (9) can be defined by [34](1) if ;(2) if it holds that(a) ;(b) and for all .

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, , . Without loss of generality, it is considered that .

Remark 6. Obviously, the transfer function matrix from to can be expressed as where is usually obtained by discretizing the original continuous-time domain model using a sampling mechanism that consists of a sampler with the sampling interval and a zero-order 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, and , is usually considered in the literature for discrete-time ILC [5].

3.2. Design of ILC

In this section, the ILC law is introduced as follows: where denotes an polynomial gain operator to be designed.

Subtract from and then use (11) and (13) to obtain which leads to where .

4. Convergence Analysis

4.1. Super-Vector Approach

Using the lifting approach, system (9) and ILC law (13) can be described respectively as where , , and , , are the supervectors which are lifted to contain sampled points, and and are two lower triangular block Toeplitz matrices. The elements of are the system Markov parameters (or the pulse response coefficients).

Equation (16) gives where . 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).

4.2. Frequency Domain Approach

The following proposition will be helpful for developing frequency-domain monotonic convergence condition.

Proposition 9 (see [31]). Assume is stable and causal, if and then ; that is, the monotonic convergence of tracking error can be accomplished in norm.

With Proposition 9, there exist matrices , , , and 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 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 for scalars and . Then if holds, follows immediately, and at the same time (22) equivalently becomes where .

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 of Lemma 4 as

We can get

Obviously, inequality (26) is equivalent to condition (24).

4.2.1. Zero Relative Degree

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 and Assumption 5, and the gain operator matrix is defined by (28). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , and matrices , , , 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 and where , , , , , and 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 and .

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 is required. Moreover, is equivalent to , 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 , , , and give that

pre- and postmultiplying this last inequality by and 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 , , , and represent admissible uncertainties which are assumed to satisfy where , , , and are known matrices of appropriate dimensions, and is an unknown matrix satisfying

In this case, the following robust result can be presented.

Corollary 11. Consider the ILC system (9) and (13) satisfying and Assumption 5. Assume that the plant has uncertain matrices described by (43) and (44), and the gain operator matrix is defined by (28). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , , and matrices , , and , satisfying (23) and the following LMI:

If the LMIs of (23) and (45) are feasible, then the gain matrix is given by (31).

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 satisfying if and only if there exists a scalar 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 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 (9) are known to lie within a convex bounded uncertain domain :

The following result enables robust ILC design when the model matrices of system (9) belong to a polytope-type uncertain domain .

Corollary 12. Consider the ILC system (9) and (13) satisfying and Assumption 5. Assume that the plant matrices have polytope uncertainty described by (51), and the gain operator matrix is defined by (28). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , and matrices , , , satisfying (23) and the following LMIs:

If the LMIs of (23) and (52) are feasible, then the gain matrix is given by (31).

Proof. For all systems of the type of (9) falling within , let , . 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.

4.2.2. Higher-Order Relative Degree

Consider now that system (9) has a higher-order relative degree of , leading to systems of the form (9) with matrices satisfying

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 or repetitively yields that can be expressed as or

Then can be derived as or

Theorem 14. Consider the ILC system (9), (13), and (55) satisfying and Assumption 5, and the gain operator matrix is defined by (54). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , , and matrices , , , 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 , 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 and represent admissible uncertainties that are assumed to satisfy where , , and are known constant matrices, and is an unknown matrix satisfying . Now the effects of norm-bounded uncertainty can be addressed via the following result.

Corollary 15. Consider the ILC system (9), (13), and (55) satisfying and Assumption 5. Assume that the plant has uncertain matrices described by (63) and (64), and the gain operator matrix is defined by (54). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , , and matrices , , , satisfying (23) and the following LMI:

If the LMIs of (23) and (65) are feasible, then the gain matrix is given by (62).

Proof. This proof is omitted since it follows in an identical manner to that of Corollary 11.

Furthermore, the matrices and are known to lie within an uncertainty polytope where

The following result is able to address the problems of polytope-type uncertainty.

Corollary 16. Consider the ILC system (9), (13), and (55) satisfying and Assumption 5. Assume that the plant matrices have polytope uncertainty described by (66), and the gain operator matrix is defined by (54). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , and matrices , , , satisfying (23) and the following LMIs:

If the LMIs of (23) and (67) are feasible, then the gain matrix is given by (62).

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.

Theorem 18. Consider the ILC system (9), (13), and (56) satisfying and Assumption 5, and the gain operator matrix is defined by (54). Then, converges monotonically to zero over the low frequency range when , if there exist scalars , , and matrices , , , and satisfying (23) and the following LMI:

If the LMIs of (23) and (68) are feasible, then the gain matrix is given by (62).

Based on Theorem 18, the robust results with are omitted since it follows identical steps to those of Corollary 15 and Corollary 16.

5. Simulation Examples

Example 1. Consider the following SISO system given by [28]:

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 .

fig1
Figure 1: (a) The reference trajectory and (b) the frequency spectrum for the reference trajectory.

For uncertainties modeled by (64), assume that , , , and , where , , and vary randomly between and . Then apply Corollary 15 and use the LMI solver “feasp” in the Matlab toolbox to obtain .

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.

450241.fig.002
Figure 2: Example 1, norm of tracking error with respect to the iteration number.
450241.fig.003
Figure 3: Example 1, process of the tracking error along both the iteration axis and the time axis.
450241.fig.004
Figure 4: Example 1, the reference and output for and .

Example 2. In this example, the system is considered with matrices given by where and are uncertain parameters in the form of . Here, , , and , and and are uncertain variables lying in the interval .

The simulation condition is performed identically to Example 1. Then apply Corollary 16 to such a system and solve LMIs (23) and (67) to obtain . 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.

450241.fig.005
Figure 5: Example 2, norm of tracking error with respect to the iteration number.
450241.fig.006
Figure 6: Example 2, process of the tracking error along both the iteration axis and the time axis.
450241.fig.007
Figure 7: Example 2, the reference and output for and .

6. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (nos. 61273070 and 61203092).

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