Identification and adaptive multi‐dimensional Taylor network control of single‐input single‐output non‐linear uncertain time‐varying systems with noise disturbances

IET Control Theory & ApplicationsVolume 13, Issue 6 p. 841-853 Research ArticleFree Access Identification and adaptive multi-dimensional Taylor network control of single-input single-output non-linear uncertain time-varying systems with noise disturbances Chao Zhang, Chao Zhang Key Laboratory of Measurement and Control of Complex Systems of Engineering, School of Automation, Southeast University, Ministry of Education, Nanjing, 210096 People's Republic of China Department of Computer Science and Technology, Henan Institute of Technology, Xinxiang, 453003 People's Republic of China School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorHong-Sen Yan, Corresponding Author Hong-Sen Yan hsyan@seu.edu.cn orcid.org/0000-0002-9305-5427 Key Laboratory of Measurement and Control of Complex Systems of Engineering, School of Automation, Southeast University, Ministry of Education, Nanjing, 210096 People's Republic of China School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this author Chao Zhang, Chao Zhang Key Laboratory of Measurement and Control of Complex Systems of Engineering, School of Automation, Southeast University, Ministry of Education, Nanjing, 210096 People's Republic of China Department of Computer Science and Technology, Henan Institute of Technology, Xinxiang, 453003 People's Republic of China School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorHong-Sen Yan, Corresponding Author Hong-Sen Yan hsyan@seu.edu.cn orcid.org/0000-0002-9305-5427 Key Laboratory of Measurement and Control of Complex Systems of Engineering, School of Automation, Southeast University, Ministry of Education, Nanjing, 210096 People's Republic of China School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this author First published: 27 March 2019 https://doi.org/10.1049/iet-cta.2018.5542Citations: 10AboutSectionsPDF 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 paper, an adaptive control approach based on the multi-dimensional Taylor network (MTN) is proposed to control the non-linear uncertain time-varying systems with noise disturbances. MTNs are introduced to formulate adaptive filtering, non-linear identification and optimal control. First, an MTN filter is developed to eliminate the control interference and measurement noise, so that the model output without stochastic disturbance can be obtained. Then, an MTN identifier (MTNI) is so designed as to be capable of dynamic mapping and require fewer weights than traditional neural networks. On the basis of the above, the MTN controller (MTNC) is developed to realise the precise tracking control of the system. The non-linear uncertain time-varying system is identified by MTNI, which then provides sensitivity information of the plant to MTNC to make it adaptive. Furthermore, the skeletonisation algorithm is adopted to remove redundant inputs and redundant regression items from MTNI and MTNC for concise MTNs. Successful convergence and faster learning are guaranteed using the Lyapunov theorem, and the optimal learning rates are identified. Simulation results demonstrate that the proposed approach features its accurate identification, excellent tracking and better anti-interference capability for the adaptive real-time control of uncertain, stochastic and time-varying non-linear systems. 1 Introduction As Wen et al. [1] pointed out, the development in the non-linear control domain has triggered the need to handle increasingly complex systems, especially the non-linear time-varying ones with noise disturbance, and the need to meet various requirements, such as for real-time operation, with less precise advanced knowledge of the plant and its environment. Though there are good results achieved on the control of non-linear systems, they fail to involve the overall consideration of the uncertainty, time-varying characteristics, measurement noise or other stochastic factors. Generally speaking, the control algorithms and structures adopted should be simple enough to be implemented. The control schemes should be designed to address the issues of non-linearity, robustness, flexibility and learning ability. If not addressed, the measurement noises and stochastic disturbances may turn up in the controller's inputs without inhibition, as a result of which the stability of adaptive process and the control accuracy will be seriously affected. It is generally accepted that the object in the industrial process is a complex, non-linear and time-varying system, where the system noise and observation noise are also often considered to be time varying, and may lead to unsmooth output and even the drastic change of the performance of the control system. Thus, how to minimise the influences of randomness, time variation and uncertain non-linearity and how to improve the system's real-time performance are of great significance. However, the fundamental shortcomings of current adaptive control methods [2–4], such as their rising complexity with the increasing number of unknown parameters, their failure to consider measurement noises and their general unsuitability for real-time applications, have obliged designers to seek other solutions. In addition, to deal with noises and disturbances, a fixed low-pass filter is commonly used to eliminate the high-frequency disturbance from measurement and other parts of the system [5, 6]. However, due to that the filter's parameters remain unchanged in the whole de-noising process, the noise elimination algorithm is unsuitable for the time-varying system, especially for time-varying noises and disturbances, and is prone to over-adjustment. Furthermore, with the rapid development of the intelligent control techniques to solve the uncertainty and complexity of the control object, the massive parallelism, natural fault tolerance and implicit programming of neural network (NN) computing architectures suggest that they may be good candidates for implementing adaptive controllers and filters. In the past two decades, several NN models and neural training schemes were applied to system controller design, and many promising results were presented [7, 8]. However, the NN has the shortcomings of slow learning speed, weak generalisation ability and unsatisfied robust performance [9, 10]. In short, the existing methods cannot fully solve the control problem of the non-linear uncertain time-varying system with time-varying noise disturbance. Representing a dynamic system by mapping with the aid of tapped delays, the multi-dimensional Taylor network (MTN, see Fig. 1) and the idea of MTN optimal control were proposed by Hong-Sen Yan in 2010 [11]. MTN for single-input single-output (SISO) systems is, in essence, a polynomial type of non-linear auto-regressive moving average model. Its polynomial function is composed of a number of linear and non-linear terms which can trend to infinity, which enables it to well represent the state dynamics in general sense, especially for unstable systems, serving as the explicit description of the system behaviour [12]. Moreover, MTN is of simple structure and fast computing speed for feasible real-time control, as illustrated in Table 1. The proportional–integral–derivative (PID) controller is found to be a special case of the MTN controller (MTNC), and its parameters can be taken as the initial ones of the latter, as illustrated in Section 5. This model is commonly applied for model prediction [13], system identification [14], disaster prediction [15], motor control [16] and non-linear control [17–22]. However, Yan et al. [18] provided only a basic idea of the MTNC. Many problems, such as the overall consideration of the uncertainty, time-varying characteristics and measurement noise have not been investigated. The initial stability of adaptive controller has not been designed. Zhang and Yan [21] have not involved the adaptive filtering problem. More importantly, the convergence and faster learning have not been guaranteed, the optimal learning rates have not been identified and the computational complexity has not been considered. In [22], the MTNs were used to approximate the non-linearities, and then, an adaptive MTNC is constructed via backstepping technique. Wherein, only the approximation of MTN was discussed. The basic properties of MTN which affect the control performance have not been analysed. Table 1. Comparison of computational complexity between MTN and BPNN MTN NN weight addition multiplication Fig. 1Open in figure viewerPowerPoint MTN model Ge and Li [23] pointed out that one should not expect a major increase in the performance of a perceptron in every situation, just by 'throwing in' feedback. In most cases, the best network structure will probably turn out to have feedback only in a smaller group of units. Taking the above into consideration, our work is carried out in the following line. For system identification, the identifier is chosen to be of a small number of parameters [24], i.e. fewer weights for our MTN model. This is because the more parameters, the greater random influence on the model. As real-time implementation is the key to control applications, the MTN identifier (MTNI) needs to be so designed as to be capable of converging with a relatively small number units, which means more generalisation ability. That is, if several networks fit equally well for the training sample, the simplest structure's network has a more generalisation ability. To this end, the skeletonisation algorithm [25] is adopted to remove its redundant inputs and regression items. As the plant is unknown in general, the desirable sensitivity of controller required for training is usually undeterminable. Most applications simply ignore it and use the direct or simple control method of sign changes to set plant response as sensitivity. However, in this study, a MTNI is utilised to identify the sensitivity to facilitate the weight adjustment of the MTNC. In addition, the prediction principle [26] of signal processing utilised in the MTN filter (MTNF) is introduced to eliminate control interference and measurement noise for desirable control stability and accuracy. However, the convergence of MTN-based control scheme turns out to be a real challenge. With regard to MTNI and MTNC, a high training rate makes the system unstable while a low one may slow down the learning process. Thus, an approach to find the bounds on training rates using the Lyapunov function is developed here. The use of adaptive learning rates guarantees convergence, and the optimal learning rates are found. In addition, for ensuring the stability of MTNF, a strong condition imposed on the step size is deduced. Three MTNs are proposed to formulate MTNI, MTNC and MTNF, respectively. MTNI and MTNC are designed and trained first off-line for non-linear uncertain time-varying systems with noise disturbance on the basis of the open-loop input–output data. Once the training process is completed, their initial weights and structures can be determined simultaneously. For precise control, MTNF whose weights are adjusted by set-membership normalised least mean square (LMS) algorithm is designed to eliminate stochastic disturbance. Adaptive control is applied to the non-linear uncertain time-varying system with noise disturbance, where MTNI, MTNC and MTNF are used as identifier, controller and filter, respectively. This endows the overall MTN control scheme with the capabilities of learning and adaptiveness. The non-linear uncertain time-varying system is identified by MTNI, which provides model information online to update MTNC. As the proposed approach is designed for general non-linear uncertain time-varying systems with noise disturbance rather than for the special ones, it can well be used to identify and control any real-time systems, electrical or mechanical. Rest of the paper is organised as follows. Section 2 gives the problem statement. Presented in Section 3 is the procedure of MTN model design as well as the comparison of MTN and back propagation neural network (BPNN) in terms of their computational complexity. In Section 4, the use of MTNI for identification and its training algorithm are studied, and the skeletonisation algorithm for concise MTNs is proposed. Design of MTNC and its training algorithm are presented in Section 5. In Section 6, the design of MTNF for filtering is discussed for improvement of the stability and control accuracy of the adaptive process. The convergence analyses of MTNI, MTNC and MTNF are provided in Section 7. Section 8 gives the implementation procedure of the proposed MTN integrated control strategy. Simulation results are analysed in Section 9, and conclusions are made in Section 10. 2 Problem statement Consider the SISO non-linear uncertain time-varying discrete-time system described by the following input–output difference equation: (1)where is a non-linear scalar function, is the system output, is the system input and and are the corresponding maximum delays, . If there is the noise (stochastic disturbance) in the environment of the control system, that is, the control system is affected or polluted by the noise, then the corresponding stochastic item should be added to (1). Thus, we can have (2)where is the noise, which can be either model noise or measurement noise, i.e. the time-varying noise addressed here. To facilitate the presentation, the time-varying noise is replaced by noise. That is to say, the non-linear uncertain time-varying system with noise disturbance is a controlled dynamic time-varying system (as described by (2)), whose disturbance can be expressed stochastically. Our target is to automatically generate the controllers that ensure the output of the system (2) tracks the given reference signal as closely as possible in real time. 3 Multi-dimensional Taylor network A general MTN model is shown in Fig. 1, featuring its simple structure and faster computing speed. MTN, in essence, is a polynomial type of non-linear auto-regressive moving average model. Based on the principle of multivariate Taylor formula, if a function in the neighbourhood of a certain point has order derivatives everywhere, then the expansion of this function at this point is not more than m power in the form of variable power series. Hence, for a SISO MTN model, the general dynamic equation of a n -dimensional system can be written as (3)where represents the non-linear component of the MTN model, whose basic idea is to use simple functions to approximate complex ones, is the t th weight before each item, is the total number of the expansion and is the power of in the t th product term. The expressions of and can be derived by recursive methods. Let denote the number of items, each of which has the power sum m for an n -dimensional system; then can be expressed as follows: (4)Each time when m increases by 1, the combinations of variables from to rise, and the corresponding new items are formed. Thus, we have (5)It is apparent that when . Substituting (5) into (4) yields (6)Let be the starting serial number of the r th of subsets of items with the power sum h. So we have (7)where { } and { }. It can be deduced that (8)where { } and { }. The initial value is (9)where { }, { } and . As can be seen from Fig. 1, the MTN model adopts the single middle layer forward structure composed of input layer, middle layer and output layer. As long as is large enough, it can be approximated with sufficient accuracy to any model. Despite its non-linear function of the system's input and output, its parameters are of linear relationship. Dynamics is incorporated via the tapped delay lines at the input into the network for creation of a dynamic MTN network controller. Table 1 gives the comparative results of the computational complexity of classical BPNN [27] and MTN model in one iteration, highlighting the latter's excellent real-time performance guaranteed by the simplicity of its structure. Here, represents the number of nodes in the l th layer of NN , where . and denote the number of nodes in the input and output layers of NN and MTN, respectively, and the NN activation function . 4 Adaptive MTN identification 4.1 Learning algorithm Lemma 1.Any continuous function defined in a closed interval can be accurately approximated by an arbitrary polynomial function [28]. Lemma 2.A continuous function defined in a closed interval can be approximated by , where is the total number of product terms of the expansion, is the weight of the t th product term of the expansion and is the power of the variable in the t th product term [13]. Using the network learning strategy, the MTN weights are updated by gradient descent learning algorithm. The error function is given by (10)where ; and are the system output and the MTN output, respectively, and the adjustable parameter represents the weights. Applying the gradient descent learning algorithm, the optimisation aims to minimise the error function (10) with respect to the adjustable parameter of the network (11)The general update rule at this point is (12)where is the learning rate of MTNI. Furthermore, for the successful identification of time-varying systems and the fast convergence, the algorithms with the optimal step size should be introduced [29]. The incremental gradient method is supposed to well meet the requirements, being good at determining the optimal step size for each step of learning. Since (13)we have (14) , whose weights have been modified, can be expanded into the first-order Taylor series (15)To obtain the optimal step size , whose weights have been modified and which should be close to , we get (16)Substituting (14) (the correction formula of ) into (16) yields (17)Then the optimal step size is (18) 4.2 Skeletonisation technique A more concise MTNI or MTNC will greatly reduce the iterative learning time and simplify the structure, contributing a lot to engineering application and hardware implementation. As verified by [30], approximating any complex non-linear function requires about ten middle-layer components. So streamlining network size is a must. Skeletonisation technique for the best generalisation (minimum structure) is one of the pruning algorithms. In the case that the controlled non-linear object is unknown in advance, the general conservative strategy adopts the top-down design method, i.e. a large enough network should be constructed first. That makes dimension explosion appear due to that the MTN's may be too large. Upon network training, the contribution of nodes (input nodes and middle nodes) to network error can be calculated by the skeletonisation technique, so as to allow us to remove those that contribute the least. The method can not only cut off redundant middle nodes, but also manage to approximate the sensitivity estimation of each input to remove unimportant input node. To improve the performance of MTN, the following is proposed: (19)where E is the error of the MTN network on the learning set. Expression (19) is a straightforward measure of the relevance [25]. However, when an input or middle node is removed, the entire network needs to be retrained. In view of this, a quick way should be found to calculate . For this end, an additional bit of notation is incorporated into (19). Suppose that associated with each input node and each middle node is a coefficient which represents the attentional strength of the node. This coefficient can be thought of as gating the flow of activity from the node i, i.e. (20)where is the activity of middle node i, is the activity of output node j and is the connection strength to j from i. If , node i has no influence on the rest of the MTN network, that is to say, it can be removed. If , node i is a conventional node, and its relevance can then be rewritten as (21) can be approximated using the derivative of the error with respect to as follows: (22)Assuming this equality holds for , that is (23)Then we have (24)This derivative can be computed via an error propagation procedure similar to that for adjusting the weights. Calculation of can be carried out simultaneously with the weights adjustment, making full use of the intermediate results of the latter. The relevance is calculated based on a linear error , where p is an index over patterns, is the target output and is the actual output, which can provide a better estimate of relevance (compared with the usual quadratic error function ) if the output pattern is close to the target. Thus, while is used as the error metric in training the weights, is measured using . This involves separate phases for computing the weight updates and the relevance measures. In practice, is found to fluctuate drastically in time. Taking this into consideration, the following measures are taken to secure a more stable estimate: Step 1: By the exponentially-decaying time average of the derivative, we have (25) Step 2: Using the calculation method of the normalised sensitivity, let us define , where is the sum of the absolute values representing the sensitivity of each node in the same layer (input or middle layer). Then, we get the normalised sensitivity as follows: (26)After normalisation, the sensitivity values of the nodes go between −1 and 1. Step 3: Using the sensitivity value at steady state, the mean value of the normalised sensitivity of a node in a period of time and standard deviation can be calculated as (27) When both and are small, that is, when , this node can be removed. Here, L represents the window length, and the preset threshold. Additionally, as is assumed to be 1, it never needs to be changed. That is to say, is not a real parameter of MTN, but just a bit of notational convenience for estimating relevance, exerting no influence on the weight adjustment of network. 5 Adaptive MTN control As shown in Fig. 2, MTNI is designed here to model the non-linear uncertain time-varying system with noise disturbance for generation of an adaptive MTNC. In the adaptive control phase, the control objective is to adjust the parameters of MTNC to make the system output track the desired trajectory , as well as to alter the parameters of MTNI to make its output model the behaviour in real time. In this way, MTNI can provide updated model information to MTNC to make it adaptive. Fig. 2Open in figure viewerPowerPoint Block diagram of MTN-based control system 5.1 Design of initial MTNC Design of the off-line MTNC (off-line inverse modelling) is the leading stage of online adaptive MTNC design. Off-line experiments require a large number of training samples so as to enable the adaptive MTNC to cover the entire work range. Design of the off-line MTNC is implemented using the direct method [31] as shown in Fig. 3, that is, the inverse model of the object is established directly by using its input–output data. To be specific, the output of the object is taken as the input of MTN so as to make the output of MTN approximate the object input. Its principle is very simple, but its training is not under the goal direction, so this structure can only be carried out off-line. Fig. 3Open in figure viewerPowerPoint Schema of off-line inverse modelling For desirable real-time performance, the skeletonisation algorithm is introduced in the training process to ensure that MTNC is of the minimum network structure controller and the best generalisation ability. The weights can be trained by minimising the following instantaneous cost function: (28)where is the value of modelling signal at the sampling time k. The adjustment formula of in the off-line learning process of MTNC is (29)where , can be obtained by the standard gradient algorithm; the criteria (24)–(27) of the skeletonisation algorithm can be used to cut off the redundant inputs and weights. As a result of the above, a streamlined MTNC can be generated, and its initial parameters are obtained. 5.2 Learning algorithm Generally speaking, the typical discrete-time PID controller [32] can be expressed as (30)where is the control effort at time k; , and are proportional, integral and derivative gains, respectively; and represent the integral and derivative constants, respectively; T denotes the sampling time. Since (30) adopts the full output, the is accumulated upon computing. If the sensor fails, may see a significant change. To avoid this, (30) can be rewritten in the incremental form as (31)Note that (31) is often used to write the control algorithm in practice. As shown in Fig. 1, MTN can act as a controller. If and , the input of MTNC is as follows: Its output can be expressed as (32)Comparing (31) and (32) reveals that the PID controller is just a special case of MTNC if and . Once the controller parameters (, and ) are optimised, its satisfactory tracking performance can be secured. For the successful optimisation, the weights of MTNC corresponding to the PID controller's parameters are updated through online learning. As indicated by our theoretical analysis [17, 19] and simulation verification, the higher-power terms of MTNC are found to be of significant effects on the dynamic performance of the system. The learning algorithm can be formulated to make MTNC adaptive. For the adaptive MTNC, its weights are updated along the negative gradient of a given error function as follows: (33)In general, , and the correction of the weight is proportional to the negative gradient, i.e. (34)where denotes the sensitivity of the plant output with respect to its input. As the precise mathematical model is usually unknown, the sensitivity cannot be calculated directly from the output of the non-linear system. Commonly, , thus the system sensitivity can be calculated as (35)Using the gradient descent learning algorithm, the weights adaptation law is acquired as (36)From the relationship between the input and output of MTNI and MTNC, we get (37)The following recursive forms can be derived as (38)where represents the s th delay of the input of MTNI and MTNC; and are the MTNI's input and middle-layer output; and are the sets of its input variables and , respectively; and denote MTNC's input and middle-layer output; represents a set of its control variable . Any programming language supporting matrix mathematics can easily implement the above control strategy. Addressing the MTN particularity and the time-varying characteristics of the controlled object, in-depth research on the learning algorithm is highly wanted. When the traditional gradient descent method is used to solve the problem, it is likely to result in two main issues: slow convergence and local minimum. Thus, in the adaptive control phase, the use of the variable step size algorithm [33] and competitive algorithm [34] is introduced. (1) Variable step size algorithm Set an initial step size . The step size is multiplied by if the error increases after one iteration, and the next iteration point is recalculated in the original direction. The step size is multiplied by if the error decreases after one iteration. Then we have (39) (2) Competition algorithm Based on the study of biological NN s [35], the competitive learning in the middle layer of network is introduced to avoid falling into local minimum for acquisition of the global optimal solution. The process goes as follows: after calculating the error signal of each unit in the middle layer and comparing these values, the weight vector of the middle node with the maximum value is normally modified, while those of the other middle nodes are modified in the direction opposite to the maximum unit. Substituting the following error signal for the error signal of each unit in the middle layer, we get (40)Thus, the weight adjustment formula is (41)The proposed approach can be applied for identifying and controlling the electrical and mechanical systems with any invertible, unknown and non-linear functions. 6 Adaptive MTN filtering If not addressed, object noise and stochastic disturbance will appear in the controller's input without inhibition. This may greatly reduce the stability of the adaptive process, and seriously affect the control accuracy. Thus, in the adaptive noise disturbance elimination phase, the improved normalised LMS (NLMS) MTN algorithm is introduced to achieve precise control. That is, the error , which contains the stochastic disturban