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Optimal tracking performance for SIMO systems with packet dropouts and control energy constraints

2018; Institution of Engineering and Technology; Volume: 12; Issue: 12 Linguagem: Inglês

10.1049/iet-cta.2017.1164

1751-8652

Xiaowei Jiang, Chaoyang Chen, Qingsheng Yang, Xiu‐Jun Qu, Huaicheng Yan,

Adaptive Control of Nonlinear Systems

IET Control Theory & ApplicationsVolume 12, Issue 12 p. 1714-1721 Research ArticleFree Access Optimal tracking performance for SIMO systems with packet dropouts and control energy constraints Xiao-Wei Jiang, Corresponding Author Xiao-Wei Jiang jxw07045136@126.com School of Automation, China University of Geosciences, Wuhan, 430074 People's Republic of China College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of ChinaSearch for more papers by this authorChao-Yang Chen, Chao-Yang Chen School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, 411201 People's Republic of ChinaSearch for more papers by this authorQing-Sheng Yang, Qing-Sheng Yang College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of ChinaSearch for more papers by this authorXiu-Jun Qu, Xiu-Jun Qu Department of Military, Tongji University, Shanghai, 200092 People's Republic of ChinaSearch for more papers by this authorHuai-Cheng Yan, Huai-Cheng Yan College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of China Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and Technology, Shanghai, 200237 People's Republic of ChinaSearch for more papers by this author Xiao-Wei Jiang, Corresponding Author Xiao-Wei Jiang jxw07045136@126.com School of Automation, China University of Geosciences, Wuhan, 430074 People's Republic of China College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of ChinaSearch for more papers by this authorChao-Yang Chen, Chao-Yang Chen School of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan, 411201 People's Republic of ChinaSearch for more papers by this authorQing-Sheng Yang, Qing-Sheng Yang College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of ChinaSearch for more papers by this authorXiu-Jun Qu, Xiu-Jun Qu Department of Military, Tongji University, Shanghai, 200092 People's Republic of ChinaSearch for more papers by this authorHuai-Cheng Yan, Huai-Cheng Yan College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi, 435002 People's Republic of China Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and Technology, Shanghai, 200237 People's Republic of ChinaSearch for more papers by this author First published: 01 August 2018 https://doi.org/10.1049/iet-cta.2017.1164Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the optimal tracking performance for single-input multiple-output (SIMO) systems under control energy constraints is investigated. The packet dropouts in the upstream channel and down channel are adequately considered. The minimal tracking error is obtained by searching through all stablising controllers. The main results of this study show that the optimal tracking performance in these kinds of system is closely dependent on the structure of the given plant and statistic characteristics of the reference signal. Furthermore, the packet dropouts in the communication channel will degenerate the tracking performance. Finally, a typical example is given to illustrate the theoretical results. 1 Introduction In recent years, the growing attention has been paid to the study of performance of control systems [1, 2], especially for the tracking and regulation problems [3, 4]. In traditional control systems, some researchers focus on revealing the relationship between tracking performance and intrinsic characteristics of the plant. It is well known that the limitations of non-minimum phase zeros, unstable poles and time delay in the plant will affect the fundamental tracking performance of control systems [5, 6], in which tracking performance is evaluated by tracking error between output signal and reference input signal. With the rapid development of internet and communication technology, networked control systems (NCSs) arise and have important applications in many areas [7, 8], such as vehicle suspension systems [9], aerospace [10] and smart grid [11, 12]. However, it is worth noting that the results of tracking performance of traditional control systems are obtained from a basic assumption. That is, the data information transmitted among the plant, the sensor, the controller and actuator are in an ideal manner. For example, for unit output feedback control system, the output signal of the plant can be real-time and non-destructive transmitted to the controller, and the control signals can also be transmitted in an ideal way to the plant. Thus, the study of NCSs has become a hot topic and challenge problem recently. Many works have been devoted to dealing with the stabilisation problem of the control system over communication channel. For instance, in [13], the packet dropouts channel is modelled as an independent and identically distributed (i.i.d.) process. The minimum data rate is explicitly given in terms of unstable eigenvalues of the open-loop matrix and packet dropouts rate from time-domain angle. In [14], stabilisation of NCSs with multirate sampling has been studied, in which input channels are modelled in two different ways: an ideal transmission system together with an additive norm bounded uncertainty, and a discrete zero-order hold. In [15], the problem of finding a stabilising feedback controller for a linear time invariant (LTI) plant subjects to a constraint on the signal-to-noise ratio (SNR) of a channel is investigated. Other related works can also be found in [16–19]. The previous results [20–22] provide some useful information about the relationship between the stability and the characteristics of the plant. However, it should be noted that this paper will investigate the tracking performance of NCSs by using frequency-domain method, which is the main difference from [20–22]. Thus, it is interesting and meaningful to study the relationship between performance of NCSs and communication parameter, such as bandwidth [23], time-delay [24], packet dropouts [25], quantisation [26] and so on. On the other hand, for tracking control systems, if we only seek the minimum of tracking error, the optimal controller will not be proper. Thus, the amplitude of the control signal will become infinite. In order to get the optimal proper controller, the balance between the power of tracking error and control energy should be fully taken into account. The contributions of this paper include: (1) The new performance index is proposed for single-input multiple-output (SIMO) systems. Then the relationship between optimal tracking performance and intrinsic characteristics of the plant will be revealed quantitatively. (2) By using a binary stochastic process to model the packet dropouts in communication channel, we will demonstrate that optimal tracking performance will be badly degraded with the increase of packet dropouts probability. (3) In order to obtain the optimal tracing performance, that is, the minimum tracking error value, some parameters of designed controllers are determined from the Youla parameterisation. The rest of this paper is organised as follows. In Section 2, some notations, assumptions and problem descriptions are formulated. Optimal estimation of tracking error of NCSs with packet dropouts and control energy constraints are derived in Section 3. Some corollaries and discussion are also given. The results are illustrated by the numerical examples in Section 4. Finally, some conclusions are drawn in Section 5. 2 Preliminaries and problem formulations The notations used throughout this paper are described as follows. denotes the conjugate of a complex number z. The transpose and conjugate transpose of a vector u and a matrix A are denoted by and , respectively. Let the open unit disc be denoted by , the closed unit disc by , the unit circle by and the complement of by . Moreover, let denote the Euclidean vector norm and the Frobenius norm, . denote the probability of some event. Define Hilbert space as with the inner product . It is well known that admits an orthogonal decomposition into the subspaces and , where and It follows that for any and , we have . It is worth pointing out that the same notation will be used to denote these norms, as the meaning of each of these norms will be clear from the context. Let denote the set of all stable, proper, rational transfer function matrices. We define by the Z-transform of sequence . Given two vector zero mean wide sense stationary (w.s.s.) processes, x and y, we define their covariance as , and the associated cross-spectrum as , provided the sum converges. Given x as above, we define its variance as , and its power spectral density (PSD) via It is noted that if x has an always positive rational spectrum, then we can always find a stable, minimum phase and biproper transfer function matrix such that For the rational transfer function matrix , where denotes the packet dropouts probability in the communication channel and , let its right and left coprime factorisations be given by (1)where and satisfy the double Bezout identity [26] (2)for some . It is well known that to stabilise P, each two-degree-of-freedom compensator can be characterised by the Youla parametrisation [10] (3)where . A transfer function , not necessarily square, is called an inner if is in and for all , where . A transfer function is called an outer if is in and has a right inverse which is analytic in . For (4)where is an inner and is an outer. Equation (4) is thus called an inner-outer factorisation of . On the other hand, if has poles , and V is a real diagonal matrix, it is also possible to factorise , yielding (5)where is minimum phase, and is all-pass and collects all the poles of in . Specifically, can be constructed as (6)with being a unitary vector , which can be computed analogously from the pole direction vectors of and being a matrix such that The problem considered in the paper is described in the following Fig. 1. Fig. 1Open in figure viewerPowerPoint Control structure of tracking systems As depicted in Fig. 1, P represents the plant model and , where , is the scalar transfer function. is denoted as the two-parameter compensator. r is the unit step signal, which can be expressed as (7)where . For the channel i, we denote the spectral density of by and define . The amplification of the noisy channel with input y and output is depicted in Fig. 2, where and n are described in the following assumptions. Fig. 2Open in figure viewerPowerPoint Simplified model of the communication channel with packet dropouts and noise Assumption 1.(description of )The signal is a binary stochastic process that models the packet dropouts of communication channel, namely with a probability distribution Assumption 2 (independent of signals).The reference input signal r is denoted by , which is uncorrelated with noise n and packet dropouts . Consider the tracking problem under the constraint of control input energy. It follows from Figs. 1 and 2 that and The tracking error is defined by (8)Then, the performance index is given by (9)where is a positive constant and . When , the objective is to find the minimal tracking error without consideration of input energy, which is usually called a cheap control problem. If , it is reduced to an optimal energy regulation problem. So the performance index could give a trade-off between them. If only considering the minimal tracking error performance, the control energy may become very large. So it is necessary to take the control energy constraint into account. This performance index has also been investigated in [27]. The main problem of this paper is to obtain the tracking performance limitation, that is, the minimum tracking error, which can be calculated by using frequency-domain methods. The proper two parameter controllers can be found by searching through all the stabilising controllers, which is a set characterised by the Youla parametrisation where K is given in (3) and denote the set of all the stabilising controllers. It should be noted that the form of performance index in this paper is similar to LQR control problem. However, it should be noted that the problem investigated in this paper is different from LQR control. The purpose of this paper is to calculate the minimum value of tracking error under communication constraints. Before that, we need to know the parametrisation of the controller, which including two parameters to be determined. For the different plant and communication constraint, we have a different choice. On the other hand, the system under consideration in this paper is LTI. There have been no results for non-linear systems, which is the limitation of frequency-domain method by comparing with LQR control. 3 Optimal estimation of tracking error of NCSs Before presenting the main results of this paper, we first give some definitions and lemmas which will be used in derivation. Consider the class of functions in The above class consists of functions with restricted behaviour at infinity. By this, we intend to deal with integration over a contour that becomes arbitrarily long. Generally speaking, if f is analytic and bounded magnitude in , then f is of class . Lemma 1.Let and analytic in . Denote that . Suppose that is conjugate symmetric, that is, . Then Lemma 2.Assume that and are zero mean w.s.s stochastic processes, that is a sequence of i.i.d. Bernoulli random variables with parameter , and that is the transfer function matrix of a proper LTI system. Then (1) If , then . (2) If . (3) If , then To obtain the minimum tracking error variance, we will first derive an expression for the tracking error PSD, . Therefore, from Lemma 2 given in [26] and [25] we have (10)where . To evaluate , we proceed as follows: from (8) and Lemma 2 given in [26] it is clear that (11)From Assumption 2 and (11) we have (12)Expression for can be derived using a similar reasoning, yielding (13)Substitution of (12) and (13) into (10), we have (14)Next we derive the expression for control energy, which can be expressed by the norm of control input signal. From Fig. 1 and the same method used in the derivation of , it follows that (15)and the tracking performance (9) can be transformed into (16) Remark 1.The main results are obtained from the calculation of (16). It should be noted that and have closed relationship, which means that (16) may not be regarded as multi-objective function. is a scalar trade-off factor without the specific physical meaning. Theorem 1.Assume that the plant has unstable poles with direction . Then, the minimal tracking error of SIMO systems with packet dropouts and control energy constraints is given by where is the packet dropouts probability and will be presented in the following part. Proof.According to (14)–(16), we may rewrite Note from (7) and (3), the problem under consideration reduces to Denote Then, we will calculate , respectively. From (4) and for the transfer function matrix it has an inner-outer factorisation (17)where is an inner factor and satisfied . is represented as an outer factor. It is easy to verify that is an inner matrix. Hence (18)Note that we may pick such that and Then it follows from (18) that (19)As is outer, we have A direct calculation of (19) yields Defining . By invoking Lemma 1 given in [24], we thus obtain (20)It follows from (5) and (6) that (21)Based on partial fraction expansion (22)where , and . Substituting (22) into (21), we have where It is clear that Thus, we have (23)In particular, a straightforward calculation of the first term of (23) yields As is an outer factor, an appropriate can be selected such that Thus, from (23) we have (24)By invoking Cauchy's theorem, we can obtain On the other hand, according to the inner-outer factorisation of , we have It then follows from (24) that (25)According to (20) and (25), the proof is completed. □ We next discuss a number of special cases of interest, in the hope of gaining more conceptual insights into this result. Corollary 1.Suppose that the plant is single-input and single-output (SISO) and has non-minimum phase zeros . Without considering packet dropouts and control energy constraints, then one has Note that the result is consistent with [5]. Corollary 2.The result of Theorem 1, that is the optimal value of tracking performance mainly results from the optimal design of two-parameter controllers. Note that Q can be obtained from the following equalities: and That is Then two-parameter controllers can be designed based on the Youla parameterisation (3). However, it should be noted that the controller in this paper is not designed but chosen. The 'optimal' means the proper controller corresponding to the minimum tracking error. In addition, the noise considered in this paper is an additive white Gaussian noise (AWGN), which is uncorrelated with reference signal and packet dropouts. Moreover, we consider the spectral density as statistical characteristics of AWGN. Thus, the systems is LTI. On the other hand, the unstable poles and non-minimum phase zeros have some negative effects on the tracking performance, no matter how the controller is designed [5, 6]. Remark 2.It should be noted that packet dropouts occur frequently in the communication channel. Thus the systems under consideration is discrete and the packet dropouts' effects on the performance of systems have been reveal quantitatively, which is the main difference between [19]. On the other hand, in [19], the system is SISO. However, the plant considered in this paper is SIMO. Remark 3.Compared with [28, 29], the controlled plant discussed in this paper is SIMO, which means that the plant is not right invertible. Thus, the calculation of performance limitation value is more complicated. On the other hand, optimal tracking performance is influenced by the communication factors in the communication channel, such as delay, packet dropouts, coding and decoding, and so on. It is more difficult to discuss these factors at the same time. Thus, we have investigated packet dropouts and control energy constraints in this paper, which is another difference with [28, 29]. Remark 4.Note that . Thus, the product cannot be negative. Moreover, if the plant contains more unstable poles, the tracking error will become larger, the tracking performance is certainly become worse. 4 Illustrative example In this section, an illustrative example is given to show the effectiveness of the obtained theoretic results. For leader–follower multi-agent systems [30], the position, direction and velocity information of leader are considered as the reference signal, and the controller is designed to achieve the minimal tracking error between the leader and the follower. However, owing to the structural characteristics of follower and the communication constraint between the leader and the follower, the minimal tracking error could not be zero. Thus, we focus on the study of the relationship among the tracking performance, structural characteristics of followers and communication parameters (packet dropouts and noise in this paper). Consider the model of follower with transfer function matrix given in [31] where . It is obvious that has non-minimum phase zeros at , and one unstable pole at . We can easily see that coprime factors of can be written as Note that has non-minimum phase zeros , then the inner factor in (3.9) can, without loss of generality, be fixed as We choose . Fixed , Fig. 3 depicts the optimal tracking performance for with respect to packet-dropout rate . Fig. 3Open in figure viewerPowerPoint Relationship between tracking performance limitation and feedback channel noise Figs. 3, 4–5 depict the relationship between optimal tracking performance (minimum tracking error) and feedback channel noise , respectively. We can observe that tracking performance will become worse with the increase of packet dropouts' probability, which means that the data packet loss occurs frequently. On the other hand, feedback channel noise can also lead to the sharp deterioration of tracking performance. Fig. 4Open in figure viewerPowerPoint Optimal tracking performance over a noisy channel subject to packet dropouts Fig. 5Open in figure viewerPowerPoint Optimal tracking performance over a noisy channel subject to packet dropouts Figs. 6 and 7 depict the relationship between optimal tracking performance with . denotes the trade-off between tracking performance and control energy. The increasing of indicates the more control effort used to track the reference input signal, which can improve the tracking performance (the smaller tracking error), to a certain extent. However, the optimal trade-off factors are not found in this paper. Fig. 6Open in figure viewerPowerPoint Optimal tracking performance with packet dropouts and trade-off factor Fig. 7Open in figure viewerPowerPoint Optimal tracking performance with trade-off factor In addition, the results obtained in this paper can also be used to the inverted pendulum system [31, 32]. In [32], the design of an active queue management ensuring the stability of the congestion phenomenon at a router is proposed. 5 Conclusion In this paper, we have studied the optimal estimation of tracking error for SIMO NCSs with packet dropouts and control energy constraints. Based on the integral square criterion, the error signal between the out of the plant and the input signal have been used as the tracking performance index. Due to the plant is not right invertible, by giving the parameterisation of two-parameter controllers, the tracking performance limitations with packet dropouts and noise in the feedback channel have been investigated. The obtained results have shown that the optimal tracking performances in these kinds of systems are closely related to the structure of the given plant and statistic characteristics of the noise signal. Finally, a typical example has been given to illustrate the correctness of the obtained result. In this paper, we have only discussed the trade-off between tracking performance and control energy, which is reflected in the proposed performance index function. An optimal design of trade-off factors may be a difficult and interesting problem, and will be studied in the future. On the other hand, we have only used the packet dropouts and noise to describe the feedback communication channel. However, it is interesting and meaningful to investigate the communication channel with induced parameters (such as network-induced delay, quantisation etc.) and basic parameters (such as communication bandwidth, SNR, channel capacity etc.), revealing that how the tracking performance is influenced by these factors. Moreover, we considered an AWGN in this paper, which is uncorrelated with reference signal and packet dropouts. The key point of the systems is LTI. Thus, the robustness of the designed controller could be guaranteed. 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