H [infinity] ILC Design for Discrete Linear Systems with Packet Dropouts and Iteration-Varying Disturbances

Abstract

An [subscript]H∞[/subscript] iterative learning controller is designed for networked systems with intermittent measurements and iteration-varying disturbances. By modeling the measurement dropout as a stochastic variable satisfying the Bernoulli random binary distribution, the design can be transformed into [subscript]H∞[/subscript] control of a 2D stochastic system described by Roesser model. A sufficient condition for mean-square asymptotic stability and [subscript]H∞[/subscript] disturbance attenuation performance for such 2D stochastic system is established by means of linear matrix inequality (LMI) technique, and formulas can be given for the control law design simultaneously. A numerical example is given to illustrate the effectiveness of the proposed results.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Bu Xuhui 1,2 and Zhang Hongwei 1 and Song YunZhong 1 and Yu Fashan 1

Academic Editor:Wenwu Yu

1, School of Electrical Engineering & Automation, Henan Polytechnic University, Jiaozuo 454003, China
2, Henan Provincial Open Lab for Control Engineering Key Discipline, Henan Polytechnic University, Jiaozuo 454003, China

Received 22 February 2014; Accepted 22 May 2014; 9 June 2014

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.

1. Introduction

Iterative learning control (ILC) has been extensively studied with significant progress in theory and widely applied in many fields [1-3]. Most of the reported results are based on an implicit assumption that the communication between the physical plant and controller is perfect; that is, the signals transmitted from the plant will arrive at the controller simultaneously and perfectly. However, in many practical situations, the systems may have intermittent measurements, especially in networked systems, which are becoming more and more popular for the reason that they have several advantages over traditional systems, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability [4-6]. If network is introduced to ILC design, the data packet dropout phenomenon, which appears in a typical network environment, will naturally induce intermittent measurements from the plant to the controller.

There have already been a few results in this issue. In [7-9], some stability conditions for networked-based ILC systems are given. Key conclusions of these works are that the intermittent ILC systems can guarantee convergence even though there may be significant data dropout. In [10, 11], an optimal ILC controller is designed for linear intermittent systems. The proposed ILC schemes can compensate the packet dropout effectively in the iteration domain. In [12], an averaging ILC algorithm is proposed to overcome the random data dropout, and it is shown that such an ILC algorithm can perform well and achieve asymptotic convergence in ensemble average along the iteration axis.

_{H ∞} optimization is a powerful tool that can be used to design a robust controller or filter [13-15], which has been proved to be one of the most important strategies for disturbance attenuation. In [16], an algebraic _H∞ approach is introduced to design higher-order ILC for the plants that are subject to model uncertainties and iteration-varying disturbance. In [17], an _H∞ ILC design approach is proposed for linear systems with iteration-varying disturbances. In [18], an _H∞ ILC design is proposed for linear systems with intermittent measurement. A sufficient condition guaranteeing both exponentially mean-square stability of such ILC process and the desired _H∞ performance in the iteration domain is presented. However, these designs are all based on lifted system representation. It does not address the computational complexity of the lifted ILC design method that might hamper their real-life application [19]. Alternatively, _H∞ ILC design based on 2D system theory is an effective approach for linear systems. Recently, several _H∞ ILC methods have been proposed to cope with parameter uncertainties in ILC systems based on the results of _H∞ control for 2D system or repetitive system [20-25]. However, _H∞ ILC design based on 2D system and linear repetitive process are only considered for systems without intermittent measurements. To the best of our knowledge, the problem of intermittent ILC design has not been investigated in the framework of 2D system or linear repetitive process, which motivates the present study.

In this paper, the 2D design approach is developed to treat the problem of _H∞ ILC design with intermittent measurements and iteration-varying disturbance. For the ILC system to be stochastic instead of a deterministic one by considering intermittent measurement, a 2D stochastic Roesser system is established to describe the entire dynamics. To analyze the tracking performance of the 2D stochastic system, the definition of stochastic mean-square asymptotic stability is introduced. In this case, a sufficient condition can be established by means of LMI technique, and formulas can be given for the control law design simultaneously. Numerical example is also proposed to illustrate the effectiveness of theoretical results.

This paper is organized as follows. In Section 2, the mathematical description and design objectives of networked-based ILC system are presented, together with its transformation into an equivalent 2D stochastic Roesser system. In Section 3, a mean-square asymptotic stability condition for such 2D stochastic systems is derived, and an _H∞ ILC design approach can be given by means of LMI technique. The effectiveness of the proposed method is illustrated by a numerical example in Section 4. Finally, the conclusions are given in Section 5.

2. Problem Formulation

Consider the following linear discrete time system: [figure omitted; refer to PDF] where the subscript k denotes iteration and t denotes discrete time. x(t,k) , u(t,k) , y(t,k) , and w(t,k) are state, input, output variables, and iteration-varying disturbances. A , B , C , _B1 are the system matrices with appropriate dimension. x(0,k)=_x0k stands for the initial condition of the process in the k th cycle. The system is operated repeatedly in the iteration domain with a desired output _yd (t) , t∈[0,T] .

In this paper, the ILC law is given as [figure omitted; refer to PDF] where e(t,k)=_yd (t)-y(t,k) is the tracking error and K is gain matrix to be designed.

Assume the ILC scheme (2) is implemented via a networked control system, where the data e(t+1,k) is transferred from the remote plant to the ILC controller. In this process, the data e(t+1,k) may be missed due to the network transmission failure. In this case, ILC law (2) can be described as [figure omitted; refer to PDF] where stochastic parameter α is a random Bernoulli variable taking the values of 0 and 1 with [figure omitted; refer to PDF] in which α- satisfying 0...4;α-...4;1 is a known constant.

The design objective of this paper can be described as follows. For an initial condition _x0k and packet dropout satisfying (4), design an ILC law (3) such that the ILC system is stable, and the influence of the iteration-varying disturbances should be as small as possible.

The ILC systems (1) and (3) are essentially a 2D system with evolution along two independent axes: time t and iteration k . We can use the 2D analysis approach to ILC to derive an expression for the tracking error and the state error. Using (1) and (3), we can obtain [figure omitted; refer to PDF] where η(t,k)=x(t-1,k+1)-x(t-1,k) , w~(t,k)=w(t-1,k+1)-w(t-1,k) .

Next, from (1) and (3), the following can also be obtained: [figure omitted; refer to PDF] Equations (5) and (6) can be rewritten as follows: [figure omitted; refer to PDF]

Denoting η(t,k)=^xh (t,k) , e(t,k)=^xv (t,k) ; that is, [figure omitted; refer to PDF]

We know that system (8) is a typical 2D Roesser system. Hence, the synthetic for ILC system under the control law (3) is equivalent to synthetic of Roesser's system in (8). Notice that the introduction of the stochastic variable α renders the 2D system to be stochastic instead of a deterministic one. Before proceeding further, we need to introduce the following definition of stochastic stability for the 2D Roesser system (8), which will be essential for our derivation.

Definition 1 (stochastic stability [26]).

The 2D stochastic system (8) is said to be mean-square asymptotically stable if for every bounded initial condition ^xh (i,0) , ^xv (0,i) , the following is satisfied: [figure omitted; refer to PDF]

Definition 2 (_H∞ performance).

Given a scalar γ>0 , the 2D stochastic system (8) is said to be mean-square asymptotically stable with an _H∞ disturbance attenuation level γ , if it is mean-square asymptotically stable and under zero boundary conditions, _||x||E <γ_||w~||2 , for all w~∈[0,∞) , where [figure omitted; refer to PDF]

To this end, the problem to be addressed in this paper can be transformed as follows. Consider the system in (1) with packet dropouts described in (4). Given a real number γ>0 , design a controller in the form of (3) such that the 2D stochastic system (8) is mean-square asymptotically stable with an _H∞ disturbance attenuation level γ .

Remark 3.

Since w~(t,k)=w(t-1,k+1)-w(t-1,k) , we can obtain that _||w~||2 ...4;2_||w||2 , and as a consequence, it follows that [figure omitted; refer to PDF] Therefore, the _H∞ objective of _||x||E <γ_||w||2 can be guaranteed by ensuring that the _H∞ performance of 2D system (8) fulfills _||x||E <(γ/2)_||w~||2 .

3. Main Results

In this section, the stability analysis problem is first concerned. More specifically, we assume that the system matrices A , B , C , _B1 in (8) are known, and we study the condition under which the 2D system in (8) is mean-square asymptotically stable with a guaranteed _H∞ performance. Then, a feasible ILC controller gain matrix can be given based on the condition.

Define α~=α-α- ; it is obvious that [figure omitted; refer to PDF] then the 2D system (8) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Theorem 4.

Consider the 2D system (13) and suppose the matrices _A-1 , _A-2 , B- are known. Then the system is mean-square asymptotically stable with a given _H∞ disturbance attenuation level γ , if there exists positive definite matrices _P1 , _P2 satisfying [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

We first prove the stochastic stability of 2D system (13) with zero disturbance w~(t,k)=0 . In this case, the system (13) becomes [figure omitted; refer to PDF] and condition (15) is [figure omitted; refer to PDF]

Define [figure omitted; refer to PDF] where x~=[^xh (t,k)^xv (t,k)] .

Consider the following index: [figure omitted; refer to PDF]

Substituting (17) into the index, we can obtain [figure omitted; refer to PDF] where Ψ=^A-1T P_A-1 +^θ2A-2T P_A-2 -P .

Since Ψ<0 , it means that for all x~...0;0 we have [figure omitted; refer to PDF] where δ=1-_λmin... (-Ψ)/_λmax... (P) .

Notice that _λmin... (-Ψ)/_λmax... (P)>0 ; we have δ<1 . From (22), it is also easy to see that δ...5;_W1 /_W2 >0 . Hence, δ∈(0,1) and it is independent of x~ . Thus, we obtain _W1 ...4;δ_W2 , and taking expectation of both sides yields [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Adding both sides of the inequality system (24) yields [figure omitted; refer to PDF] Using this relationship iteratively, we can obtain [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Now, denote _χi ...=^{∑j=0i ||x(i-j,j)||2} ; then upon inequality (27) we have [figure omitted; refer to PDF] Adding both sides of the inequalities yields [figure omitted; refer to PDF] Since x(0,k)=_x0k is satisfied for all k in system (1), then [figure omitted; refer to PDF] and ^xv (i,0)=e(i,0)=_yd (i)-Cx(i,0) is bounded; hence, the right side of inequality (30) is bounded, which means _{lim...i[arrow right]∞} E{_χi }=0 ; that is, _{lim...t+k[arrow right]∞} E{^||x(t,k)||2 }=0 . Then the 2D stochastic system (13) with w~(t,k)=0 is mean-square asymptotically stable.

Now, the _H∞ performance for the 2D stochastic system (13) will be established. Assume zero initial and boundary conditions; that is, ^xh (0,i)=0 , ^xv (i,0)=0 for all i . In this case, the index J becomes [figure omitted; refer to PDF]

Define ξ=^{[x~Tw~T ]T} ; another index is introduced as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] From condition (15), for any ξ...0;0 , we have Π<0 ; that is, [figure omitted; refer to PDF] Taking the expectation of both sides yields [figure omitted; refer to PDF] Due to the relationship (36), it can be established that [figure omitted; refer to PDF]

Adding both sides of the inequality system, we have [figure omitted; refer to PDF] Summing up both sides of these inequalities from i=0 to i=N , we have [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Considering the zero boundary conditions, (40) means [figure omitted; refer to PDF]

This completes the proof.

Remark 5.

Theorem 4 provides a sufficient condition for the mean-square asymptotic stability of 2D discrete stochastic systems with intermittent measurement. If the communication links existing between the plant and the controller are perfect, that is, there is no data dropout during their transmission, then α-=1 and θ=0 . In this case, the condition in Theorem 4 becomes [figure omitted; refer to PDF] which also can be easily obtained by the results in [25] for 2D discrete deterministic system. From this point of view, Theorem 4 can be seen as an extension of 2D discrete deterministic systems with intermittent measurement.

Theorem 4 only gives a mean-square asymptotic stability condition for system in (13) where the matrixes A , B , C , _B1 and parameters α- are all known. However, our eventual purpose is to determine the controller matrix K . In the following, we will give an approach to solve the controller design problem for 2D systems with stochastic intermittent measurement.

The following well-known lemma is needed in the proof of our main result.

Lemma 6 (Schur complement [27]).

Assume W , L , V are given matrices with appropriate dimension, where W and V are positive definite symmetric matrices. Then [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

Based on the above lemma, we can give our main result.

Theorem 7.

The 2D discrete stochastic closed system in (13) is mean-square asymptotically stable with a given _H∞ performance γ , if there exist positive definite matrices _Q1 , _Q2 , and M such that the following LMI holds: [figure omitted; refer to PDF] Also, if this condition holds, a suitable gain matrix of ILC law (3) is defined by [figure omitted; refer to PDF]

Proof.

The condition in Theorem 4 can be rewritten as [figure omitted; refer to PDF] By applying Lemma 6, condition (48) is equivalent to the following LMI condition: [figure omitted; refer to PDF] Substituting matrices _A-1 , _A-2 , B- into the LMI condition, we obtain [figure omitted; refer to PDF] Defining _Q1 =^P1-1 , _Q2 =^P2-1 , pro-, and postmultiplying diag...(I,I,_Q1 ,_Q2 ,_Q1 ,_Q2 ,_Q1 ,_Q2 ,I) for LMI condition (50) gives [figure omitted; refer to PDF] Set K_Q2 =M to obtain the LMI of (46) and the proof is complete.

Remark 8.

Theorem 7 provides an LMI condition for the mean-square asymptotic stability and _H∞ performance of 2D stochastic system in (13) which can be solved by LMI Toolbox. Then by (47), we also can give a suitable ILC law. The feasibility of the proposed method will be illustrated by the example given in Section 4.

Remark 9.

For a fixed γ , the feasibility of (47) is a suboptimal _H∞ ILC. When γ is not fixed, the minimization of γ that satisfies (47) can be searched. That is, the optimal performance index can be achieved. Thus, the optimal _H∞ ILC design problem is equivalent to the following convex programming problem: [figure omitted; refer to PDF]

Remark 10.

Even though we consider the ILC law (3) only uses the error information from the previous iteration in networked systems framework, the general tools used can be extended to other ILC laws such as containing state signal in [23-25]. All of these problems end up with the same 2D system formulation as in (13).

4. Numerical Example

In this section, an example is given to illustrate the proposed results. Consider the following SISO linear system: [figure omitted; refer to PDF] where w(k,t) is an iteration-varying disturbance with |w(k,t)|<0.02 . The desired repetitive reference trajectory is given as [figure omitted; refer to PDF]

For the initial state, it is assumed that _x1 (0,k)=_x2 (0,k)=0 for all k . To perform the simulation, we consider the intermittent measurements α-=0.8 . Obviously, it means that there is 20 percent missing measurements. By applying Theorem 7 and solving the optimization problem (52), the minimum _H∞ disturbance attenuation level for the ILC system is based on feasibility of the corresponding LMIs. Meanwhile, we obtain _γopt =27.5 and K=0.72 . Simulation results are shown in Figures 1, 2, 3, and 4, where the tracking error on iteration domain is plotted in Figure 1, and system outputs at 10th, 20th, and 50th iteration are plotted in Figures 2-4, respectively. It is observed that the tracking is worse and significant tracking errors exist in the start iteration due to the effect of significant measurement dropout. However, the tracking error can converge to zero after some iteration and the perfect tracking can be obtained. The ILC system is insensitive to intermittent measurement and iteration-varying disturbance.

Figure 1: Max tracking error on iteration domain.

[figure omitted; refer to PDF]

Figure 2: System output at 10th iteration.

[figure omitted; refer to PDF]

Figure 3: System output at 20th iteration.

[figure omitted; refer to PDF]

Figure 4: System output at 50th iteration.

[figure omitted; refer to PDF]

5. Conclusions

In this paper, the problem of _H∞ ILC design for linear networked systems with intermittent measurements and iteration-varying disturbances has been investigated. A stochastic variable satisfying the Bernoulli random binary distribution is utilized to characterize the data missing phenomenon, and then the design of ILC law has been transformed into _H∞ control problem of a 2D stochastic system. A sufficient condition of mean-square asymptotic stability is established by means of LMI technique. An example is given to demonstrate the effectiveness and feasibility of the proposed design methods. This paper gives a systematic _H∞ design approach for stochastic ILC based on 2D system.

Acknowledgments

This work is supported by the Program of NSFC (no. 61203065 and no. 61340041), the program of Natural Science of Henan Provincial Education Department (12A510013), the program of Open Laboratory Foundation of Control Engineering Key Discipline of Henan Provincial High Education (KG 2011-10), the program of Key Young Teacher of Henan Polytechnic University, and the Doctoral Fund Program of Henan Polytechnic University (B2012-003).

Conflict of Interests

There is no conflict of interests regarding the publication of the paper.

References

[1] S. Arimoto, S. Kawamura, F. Miyazaki, "Bettering operation of robots by learning," Journal of Robotic Systems , vol. 1, no. 2, pp. 123-140, 1984.

[2] D. A. Bristow, M. Tharayil, A. G. Alleyne, "A survey of iterative learning control: a learning-based method for high-performance tracking control," IEEE Control Systems Magazine , vol. 26, no. 3, pp. 96-114, 2006.

[3] H.-S. Ahn, Y. Chen, K. L. Moore, "Iterative learning control: brief survey and categorization," IEEE Transactions on Systems, Man and Cybernetics C: Applications and Reviews , vol. 37, no. 6, pp. 1099-1121, 2007.

[4] T. C. Yang, "Networked control system: a brief survey," IEE Proceedings: Control Theory and Applications , vol. 153, no. 4, pp. 403-412, 2006.

[5] J. P. Hespanha, P. Naghshtabrizi, Y. G. Xu, "A survey of recent results in networked control systems," Proceedings of the IEEE , vol. 95, no. 1, pp. 138-162, 2007.

[6] W. Zhang, M. S. Branicky, S. M. Phillips, "Stability of networked control systems," IEEE Control Systems Magazine , vol. 21, no. 1, pp. 85-99, 2001.

[7] X. H. Bu, Z. S. Hou, F. S. Yu, "Stability of first and high order iterative learning control with data dropouts," International Journal of Control, Automation and Systems , vol. 9, no. 5, pp. 843-849, 2011.

[8] X. H. Bu, F. S. Yu, Z. S. Hou, F. Z. Wang, "Iterative learning control for a class of nonlinear systems with random packet losses," Nonlinear Analysis: Real World Applications , vol. 14, no. 1, pp. 567-580, 2013.

[9] X.-H. Bu, Z.-S. Hou, "Stability of iterative learning control with data dropouts via asynchronous dynamical system," International Journal of Automation and Computing , vol. 8, no. 1, pp. 29-36, 2011.

[10] H.-S. Ahn, Y. Chen, K. L. Moored, "Intermittent iterative learning control," in Proceedings of the IEEE International Symposium on Intelligent Control (ISIC '06), pp. 832-837, Munich, Germany, October 2006.

[11] H.-S. Ahn, Y. Q. Chen, K. L. Moore, "Discrete-time intermittent iterative learning controller with independent data dropouts," in Proceedings of the 17th World Congress, International Federation of Automatic Control (IFAC '08), pp. 12442-12447, Seoul, South Korea, July 2008.

[12] C. P. Liu, J. X. Xu, J. Wu, "Iterative learning control for network systems with communication delay or data dropout," in Proceedings of the 48th IEEE Conference on Decision and Control, pp. 4858-4863, Shanghai, China, December 2009.

[13] L. H. Xie, "Output feedback _H∞ control of systems with parameter uncertainty," International Journal of Control , vol. 63, no. 4, pp. 741-750, 1996.

[14] G. Pascal, A. Pierre, "A linear matrix inequality approach to _H∞ control," International Journal of Robust and Nonlinear Control , vol. 4, no. 4, pp. 421-448, 1994.

[15] Y. Zhao, Z. S. Duan, G. H. Wen, G. R. Chen, "Distributed _H∞ consensus of multi-agent systems: a performance region-based approach," International Journal of Control , vol. 85, no. 3, pp. 332-341, 2012.

[16] K. L. Moore, H.-S. Ahn, Y. Chen, "Iteration domain _H∞ -optimal iterative learning controller design," International Journal of Robust and Nonlinear Control , vol. 18, no. 10, pp. 1001-1017, 2008.

[17] D. Y. Meng, Y. M. Jia, J. P. Du, F. Yu, " _{H ∞} -based design approach to discrete-time learning control systems with iteration-varying disturbances," in Proceedings of the 48th IEEE Conference on Decision and Control, pp. 4882-4887, Shanghai, China, December 2009.

[18] X. H. Bu, Z. S. Hou, F. S. Yu, F. Z. Wang, " _{H ∞} iterative learning controller design for a class of discrete-time systems with data dropouts," International Journal of Systems Science , 2013.

[19] A. Haber, R. Fraanje, M. Verhaegen, "Linear computational complexity robust ILC for lifted systems," Automatica , vol. 48, no. 6, pp. 1102-1110, 2012.

[20] C. Du, L. Xie, C. Zhang, " _{H ∞} control and robust stabilization of two-dimensional systems in Roesser models," Automatica , vol. 37, no. 2, pp. 205-211, 2001.

[21] E. Rogers, K. Galkowski, D. H. Owens Control Systems Theory and Applications for Linear Repetitive Processes , vol. 349, of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2001.

[22] E. Rogers, K. Galkowski, A. Gramacki, J. Gramacki, D. H. Owens, "Stability and controllability of a class of 2-D linear systems with dynamic boundary conditions," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications , vol. 49, no. 2, pp. 181-195, 2002.

[23] L. Hladowski, K. Galkowski, Z. Cai, E. Rogers, C. T. Freeman, P. L. Lewin, "Experimentally supported 2D systems based iterative learning control law design for error convergence and performance," Control Engineering Practice , vol. 18, no. 4, pp. 339-348, 2010.

[24] L. Hladowski, K. Galkowski, Z. Cai, E. Rogers, C. T. Freeman, P. L. Lewin, "A 2D systems approach to iterative learning control for discrete linear processes with zero Markov parameters," International Journal of Control , vol. 84, no. 7, pp. 1246-1262, 2011.

[25] J. Shi, F. Gao, T.-J. Wu, "Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system," Journal of Process Control , vol. 15, no. 8, pp. 907-924, 2005.

[26] H. Gao, J. Lam, S. Xu, C. Wang, "Stability and stabilization of uncertain 2-D discrete systems with stochastic perturbation," Multidimensional Systems and Signal Processing , vol. 16, no. 1, pp. 85-106, 2005.

[27] S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan Linear Matrix Inequalities in System and Control Theory , vol. 15, of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.