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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS, 2017 VOL. 23, NO. 6, 595–612 https://doi.org/10.1080/13873954.2016.1278392 Identification of non-uniformly sampled Wiener systems with dead-zone non-linearities a,b b b b Ranran Liu , Tianhong Pan , Shan Chen and Zhengming Li School of Automotive and Traffic Engineering, Jiangsu University of Technology, Changzhou, China; School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, China ABSTRACT ARTICLE HISTORY In multi-rate systems, identifying non-uniformly sampled data (NUSD) Received 6 August 2015 models is a challenge. This study proposes an iteratively recursive least- Accepted 30 December 2016 squares identification algorithm for non-uniformly sampled Wiener sys- KEYWORDS tems with dead-zone non-linearities. First, an extended information vec- Parameter estimation; tor is designed, in which both unknown parameters and inner variables iterative least-squares exist. Then, based on the auxiliary model and iterative method, an algorithm; Wiener systems; auxiliary model-based iteratively recursive least-squares algorithm is non-uniform sampling; developed to estimate the system parameters directly. Furthermore, to dead-zones improve the convergence rate and disturbance rejection, a new modified forgetting factor function is presented. Compared with no or fixed for- getting factor algorithms, the proposed algorithm has a higher conver- gence speed and is more robust to white noise with different variances. The numerical simulation shows the effectiveness of the proposed algo- rithm, and it can be extended to other NUSD non-linear systems. Nomenclature uðkT þ t Þ System input at a specific time φðkTÞ Extended information vector j1 instant yðkT þ TÞ System output at a specific time JðθÞ Cost function instant yðkT þ TÞ Input of the non-linear part I Identity matrix xðkT þ TÞ Output of the non-linear part PðkTÞ Covariance matrix vðkT þ TÞ White noise with zero mean λðkTÞ Varying forgetting factor H Zero-order holder λ The minimum of λðkTÞ τ min τ ; j ¼ 1; 2; .. . ; m Irregularly sampled intervals ρ Gain coefficient P Linear dynamic block ðkTÞ Error of the output between the system and its estimation S Sampler with frame period NINTðÞ The nearest integer T Frame period E½ Mathematical expected value xðtÞ2 R State vector δ Parameter error yðtÞ2 R Output of P σ Noise variance uðtÞ2 R Input of P ðÞ Matrix transpose nn n 1n A 2 R ; B 2 R , C 2 R ; D 2 R Parameter matrices of P Abbreviations c c c z Backward shift operator NUSW Non-uniformly sampled Wiener m ; m Slopes of the corresponding linear AM-IRLS-VFF Auxiliary model-based iteratively 1 2 segment recursive least-squares algorithm with variable forgetting factor r ; r Dead-zone points PEE Parameter estimation error 1 2 (Continued ) CONTACT Tianhong Pan firstname.lastname@example.org © 2017 Informa UK Limited, trading as Taylor & Francis Group 596 R. LIU ET AL. (Continued). hðÞ Switching function AM-IRLS Auxiliary model-based iteratively recursive least-squares YðkTÞ Internal variable AM-IRFLS Auxiliary model-based iteratively recursive forgetting least- squares n ; n Model order of output and input PE Persistent excitation a b θ Parameter vector 1. Introduction A multi-rate non-linear system is a type of non-linear system that has two or more sampling frequencies. Because of hardware limitations, economic constraints and environmental require- ments, many multi-rate systems are employed in industrial processes such as petroleum, chemical, food and medicine [1,2]. If we consider a polymer reactor, for example, the density and other composition measurements are typically obtained after several minutes of analysis, whereas the control signal can update at a relatively fast rate. As a general multi-rate systems, the non- uniformly sampled-data (NUSD) system have attracted much attention in recent years. For example, Ding presented a method to reconstruct continuous-time systems from their NUSD discrete-time systems . Then, a lifted state-space models were derived from their continuous- time systems, and an auxiliary-model-based recursive least-squares algorithm was proposed to estimate system parameters . However, the computational burden is high for calculating inverse matrix. Ding proposed a partially couple stochastic gradient algorithm to improve computational efficiency . Furthermore, an auxiliary model based multi-innovation generalized extended stochastic gradient algorithm was presented for NUSD systems with coloured noise . However, the dimension of the estimated parameter is high. Based on the hierarchical identifica- tion principle, Liu proposed a novel hierarchical least squares algorithm for NUSD systems . The complex system was decomposed into sub-models with lower dimensional parameter vectors. Then, under the constraints of causality, system parameters were estimated. Ding presented a subspace identification model to deal with the causality constraints . The above-mentioned methods are for NUSD systems with irregular inputs. Based on a time-varying backward shift operator, Xie proposed a novel model of NUSD systems with asynchronous input-output data . Although there are many works involving the identification of NUSD linear systems, they cannot be used to the actual industry with non-linear characteristics - especially the dead-zone characteristics in manufacturing equipment, hydraulic servo valves and sensors that are insensi- tive to small signals . Because these characteristics reflect actual industrial processes, the identification of these systems has already attracted widespread attention of researchers. In previous studies, many types of identification algorithms have been developed for use in non- linear or non-uniformly sampled systems. Most studies have focused on single-rate systems such as the Hammerstein, Wiener and Hammerstein-Wiener systems [11–14]. An overview of identification methods is given in Table 1. Hagenblad et al. presented a maximum likelihood identification approach for Wiener models with general disturbances . Ding et al. proposed several estimation schemes to identify Hammerstein systems. These include the gradient strategy, projection method and Newton method based on recursive/iterative algorithms . However, these algorithms assume that the non-linear characteristics are polynomials with known orders, and the parameters are estimated using the over-parameterization method [17,18]. This means that the non-linear system is reparametrized to linearize the output in the unknown parameter space. Therefore, the identifica- tion approaches of a linear system (e.g. the recursive least-squares algorithm, the stochastic gradient algorithm) can be used to estimate the parameters of the transformed model directly. However, the dimension of the estimated parameter vector is huger than system parameters vector, and the system parameters must be separated from this vector. Nevertheless, this increases MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 597 Table 1. Identification algorithms for non-linear systems. No Algorithms Applicability Pros Cons Ref 1 Maximum likelihood method, gradient Single-rate Wiener/Hammerstein The traditional algorithm for linear The computational burden increases [15,16] strategy, projection methods and Newton models written in polynomial form systems can be used method based on recursive/iterative algorithm 2 Hierarchical multi-innovation stochastic Single-rate Hammerstein models The computational burden is reduced The system state cannot be estimated  gradient algorithm written in polynomial form 3 Filtering-based multi-innovation stochastic Single-rate Hammerstein models The system parameters and states can The algorithm cannot be used for  gradient algorithm written in polynomial form with be estimated hard non-linear systems coloured noise 4 Separable least-squares algorithm and Single-rate Hammerstein models with The identification is equivalent to a Only an unknown constant in non-  correlation analysis method hard non-linearity one-dimensional minimization linearity exists problem 5 Iterative identification method with internal Single-rate Wiener/Hammerstein The model parameters are estimated The method has a priori knowledge of [22,23] variable systems with multi-segment simultaneously the limits for domain partition and piecewise-linear non-linearities cannot be used for dead-zone non- linearities 6 Auxiliary model-based recursive generalized Single-rate Hammerstein OEAR Parameters of system model, noise The algorithm cannot be used for  least squares algorithm systems model and internal variable are dead-zone non-linear systems in estimated simultaneously which the slops are zero 7 Gradient-based iterative algorithm Single-rate Hammerstein/Wiener By introducing a switching function, Under the constraints of causality, the [25,26] systems with saturation and dead- the traditional algorithm for linear algorithm cannot estimate system zone non-linearities systems can be used parameters directly 8 Least-squares-based iterative algorithm Non-uniformly sampled Hammerstein A linear regressive model is derived The method has a huge computation  systems written in polynomial form 9 An auxiliary model-based recursive least- Non-uniformly sampled Hammerstein The computational burden is reduced The system parameters are not  squares algorithm systems written in polynomial form estimated directly 598 R. LIU ET AL. the computational burden and reduces accuracy. By decomposing the identification model into two sub-models, Ding proposed a hierarchical stochastic gradient algorithm. The dimension of the estimated parameter vector and computation is reduced . Furthermore, a filtering-based multi-innovation stochastic gradient algorithm was developed to estimate system parameters and states . Hard non-linear characteristics include preload, dead-zone, saturation, saturation with dead- zone, piecewise-linear and their composition . These types of non-linearities cannot be well described through a polynomial form model. Therefore, the above-mentioned methods cannot be used directly to estimate parameters. Bai proposed two identification approaches used in Hammerstein models with hard non-linearity: a separable least-squares algorithm and a corre- lation analysis method . However, in this study, only a single unknown constant exists in hard non-linearity. Iterative identification [22,23] and an auxiliary model-based recursively generalized least-squares parameter estimation  have been proposed for multi-segment piecewise-linear Hammerstein/Wiener systems with different linear segment slopes and parti- tion points. These methods can be used for systems with noninvertible characteristics except dead-zones. Chen used an appropriate switching function to derive a gradient-based iterative algorithm for Hammerstein and Wiener systems with saturation and dead-zone parts [25,26], and the algorithms were numerically validated. By introducing switching function, the non- linear system is reparameterized to a linear systems, namely transformed model. Then, the proposed algorithm can be used to estimate the parameters of the transformed model. However, this algorithm was used to estimate the parameters of the transformed model rather than those of the original model. Under the constraints of causality, the system parameters should be separated from the estimated parameters. Alonso identified a friction model that includes dead- zone effects and successfully applied it to the modelling of an inertia wheel pendulum bench- mark . Most studies focus on single-rate systems and ignore general non-uniformly sampled characteristics. Li proposed a least-squares-based iterative algorithm for non-uniformly sampled Hammerstein non-linear systems. Unknown variables are estimated by using an over-parame- terization technique . However, the parameter estimates of the identification model include the product terms of the parameters of the original systems. And some parameters are repeat estimated. Then,under theconstraintofcausality,theaveragemethod isusedtoobtain system parameters. Thus, the algorithm requires considerable computational effort. Combining the auxiliary model identification idea and the key-term separation principle, Li presented a discrete identification model for a non-uniformly sampled Hammerstein system without the products of parameters. Thus, computational burden of the auxiliary model-based recursive least-squares algorithm is reduced . However, the methodcannotbeuseddirectlytoidentify hard non-linear systems. The direct identification method is investigated by extending an information vector. And the product terms of the original system parameters are then divided into an independent system parameter vector and parts of the extended information vector. Based on the auxiliary model and iterative method, parameters are estimated directly. Then, an online iteratively recursive algorithm is proposed. First, a NUSW model with hard non-linearity is transformed into an analytic form by using a switching function. An iteratively recursive least-squares algorithm is then proposed to estimate the parameters of the system directly. To improve identification accuracy and enhance the anti-jamming ability  of the algorithm, a varying forgetting factor is introduced. As a recursive form with iteration, the proposed algorithm is more accurate, more robust and has a faster convergence than the previous algorithms. The remainder of this study is organized as follows. Section 2 describes the characteristics of NUSW systems with dead-zone non-linearities and Section 3 gives a derivation of the proposed algorithm. Section 4 analyses the performance of the algorithm and Section 5 provides an illustrative example. Section 6 compares the proposed algorithm to previous algorithms and Section 7 offers a conclusion. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 599 2. Problem formulation Consider an NUSW system with a dead-zone nonlinearity, as shown in Figure 1.Here, uðkT þ t Þ and yðkT þ TÞ are the input and output of the system, respectively; ykðÞ T þ T j1 and xkðÞ T þ T are the input and output of the non-linear part, respectively and vðkT þ TÞ is the measurement noise with zero mean. In addition, H is a zero-order holder with irregularly sampled intervalsfg τ ; τ ; .. . ; τ , and the input updating period is set as 1 2 m kT þ t , j ¼ 1; 2; :::; m (t ¼ 0, t :¼ τ þ τ þ ::: þ τ ). Finally, P is a linear dynamic block j1 0 j 1 2 j c and S is a sampler with the frame period T :¼ τ þ τ þ ::: þ τ ¼ t . T 1 2 m m By using the lifting technique , H is formulated as follows: uðkTÞ; kT t < kT þ t > 1 uðkT þ t Þ; kT þ t t < kT þ t 1 1 2 uðtÞ (1) . . . . . . uðkT þ t Þ; kT þ t t < kT þ T m1 m1 P is assumed as the state-space model such that: x_ðtÞ¼ A xðtÞþ B uðtÞ c c P :¼ (2) yðtÞ¼ CxðtÞþ DuðtÞ n n where xðtÞ2 R is the state vector, and yðtÞ2 R and uðtÞ2 R are the output and input of P , nn n 1n respectively. A 2 R ,B 2 R , C 2 R and D 2 R are parameter matrices. Once Equation (2) c c is discretized with the frame period T , the linear part can be written as follows: B ðz Þ j¼1 1 1 yðkTÞ¼ uðkT þ t Þ¼½1 Aðz ÞyðkTÞþ B ðz ÞuðkT þ t Þ (3) j1 j j1 Aðz Þ j¼1 with 1 1 2 n Aðz Þ¼ 1 þ a z þ a z þ ... þ a z 1 2 n 1 1 n B ðz Þ¼ b þ b z þ .. . þ b z 1 10 11 1n 1 1 2 n B ðz Þ¼ b z þ b z þ ... þ b z ; j ¼ 2; 3; ; m j j1 j2 jn 1 1 where z is the backward shift operator, that is, z xðkTÞ¼ xðkT TÞ. The hard non-linear part xðkTÞ, as shown in Figure 1, is formulated as follows: m ðyðkTÞ r Þ; r yðkTÞ 1 1 1 xðkTÞ¼ 0 ; r yðkTÞ r (4) 2 1 m ðyðkTÞ r Þ; yðkTÞ r 2 2 2 Figure 1. Structure of an NUSW system with dead-zone nonlinearity and measurement noise interference. 600 R. LIU ET AL. where m and m are slopes of the corresponding linear segment, and r >0 and r <0 are the 1 2 1 2 constants at the dead-zone points. Because the non-linear parameters are unknown, a switching function hð:Þ is introduced  as follows: 0; yðtÞ > 0 hðyðtÞÞ :¼ (5) 1; yðtÞ 0 Based on the switching function, Equation (4) is rewritten as follows: xðkTÞ¼ m hðr yðkTÞÞ½ yðkTÞ rþ m hðyðkTÞ r Þ½ yðkTÞ r (6) 1 1 1 2 2 2 Similarly, yðkTÞ becomes yðkTÞ¼ hðr yðkTÞÞyðkTÞþ hðyðkTÞ r ÞyðkTÞþ hðyðkTÞ r Þhðr yðkTÞÞyðkTÞ (7) 1 2 1 2 Combining Equations (6) and (7) yields xðkTÞ¼ yðkTÞ hðr yðkTÞÞyðkTÞ hðyðkTÞ r ÞyðkTÞ hðyðkT Þ r Þhðr yðkTÞÞyðkTÞ 1 2 1 2 þ m hðr yðkTÞÞ½ yðkTÞ rþ m hðyðkTÞ r Þ½ yðkTÞ r 1 1 1 2 2 2 (8) The system output is formulated as follows: yðkTÞ¼ xðkTÞþ vðkTÞ ¼ yðkTÞ hðr yðkTÞÞyðkTÞ hðyðkTÞ r ÞyðkTÞ hðyðkTÞ r Þhðr yðkTÞÞyðkTÞ 1 2 1 2 þ m hðr yðkTÞÞ½ yðkTÞ rþ m hðyðkTÞ r Þ½ yðkTÞ rþ vðkTÞ 1 1 1 2 2 2 (9) Define an internal variable YðkTÞ as YðkTÞ :¼ yðkTÞþ hðr yðkTÞÞyðkTÞþ hðyðkTÞ r ÞyðkTÞþ hðyðkTÞ r Þhðr yðkTÞÞyðkTÞ 1 2 1 2 (10) When Equations (3) and (9) are combined with Equation (10), YðkTÞ can be reformulated as follows: 1 1 YðkTÞ¼½1 Aðz ÞyðkTÞþ B ðz ÞuðkT þ t Þ j j1 j¼1 (11) þ m hðr yðkTÞÞyðkTÞþ m hðyðkTÞ r ÞyðkTÞ 1 1 2 2 m r hðr yðkTÞÞ m r hðyðkTÞ r Þþ vðkTÞ 1 1 1 2 2 2 Suppose that the model order n ; n is known and that uðkT þ t Þ¼ 0, yðkTÞ¼ 0, vðkTÞ¼ 0 a b j1 and j ¼ 1; 2; .. . ; m when k 0. Define the parameter vector and extended information vector φðÞ kT as follows: n þmn þ5 a b θ :¼½a ; a ; .. . ; a ; b ; b ; ... ; b ; b ; .. . ; b ; .. . ; b ; .. . ; b ; m ; m ; r r 2 R 1 2 n 10 11 1n 21 2n m1 mn 1 2 1; 2 b b b ϕðkTÞ :¼½yðkT TÞ; yðkT 2TÞ; .. . ; yðkT n TÞ; uðkTÞ; uðkT TÞ; ; uðkT n TÞ; a b uðkT T þ t Þ; .. . ; uðkT n T þ t Þ; .. . ; uðkT T þ t Þ; .. . ; uðkT n T þ t Þ; 1 b 1 m1 b m1 hðr yðkTÞÞyðkTÞ; hðyðkTÞ r ÞyðkTÞ; m hðr yðkTÞÞ; m hðyðkTÞ r Þ 1 2 1 1 2 2 n þmn þ5 a b 2 R where the superscript ‘ðÞ ’ denotes the matrix transpose, and φðkTÞ includes the input and output information as well as the unknown parameters. Equation (11) can then be rewritten in a concise form as follows: MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 601 YðkTÞ¼ φ ðkTÞθ þ vðkTÞ (12) With the use of Equation (12), the parameter θ of the linear and non-linear parts can be estimated simultaneously. The challenge here is that φðkTÞ contains both the unknown inner variables yðkT iTÞ; i ¼ 0; 1; .. . ; n and unknown parameters m ; m ; r ; r . Therefore, the standard a 1 2 1 2 recursive least squares  cannot be used to estimate θ directly. 3. Auxiliary model-based iteratively recursive least-squares algorithm with variable forgetting factor 3.1. Algorithm derivation By using the auxiliary model , we can propose an effective solution as shown in Figure 2. The auxiliary model is defined as follows: yðkTÞ¼ φ ^ ðkTÞθ (13) ^ ^ ^ ϕ ðkTÞ :¼½yðkT TÞ; yðkT 2TÞ; ... ; yðkT n TÞ; uðkTÞ; uðkT TÞ; ; uðkT n TÞ; a b uðkT T þ t Þ; .. . ; uðkT n T þ t Þ; .. . ; uðkT T þ t Þ; .. . ; 1 b 1 m1 n þmn þ1 a b uðkT n T þ t Þ 2 R b m1 (14) n þmn þ1 ^ ^ ^ ^ ^ ^ ^ ^ a b θ :¼½^a ; ^a ; ... ; ^a ; b ; b ; .. . ; b ; b ; ... ; b ; ... ; b ; ... ; b 2 R (15) a 1 2 n 10 11 1n 21 2n m1 mn a b b b where φ ^ ðkTÞ and θ are the estimated values of information vector and parameter vector, respectively, in the auxiliary model. At time kT, the unknown variables yðkT iTÞ and m ; m ; r ; r in φðkTÞ are replaced with the output yðkT iTÞ of the auxiliary model and the 1 2 1 2 k1 k1 k1 k1 estimates m ; m ; r ; r at time kT T. The parameter vector is then renamed as φ ^ðkTÞ. 1 2 1 2 ^ ^ Let θðkTÞ and θ ðkTÞ be the estimate of θ and θ at time kT. The following cost function is a a minimized as follows: T 2 JðθÞ¼ ½YðkTÞ φ ^ ðkTÞθ i¼1 The recursive least-squares algorithm can be obtained . To improve the tracking capability and accelerate the response to dynamic characteristics, a varying forgetting factor λðkTÞ is Figure 2. NUSW system with an auxiliary model. 602 R. LIU ET AL. introduced. Then, an auxiliary model-based iteratively recursive least-squares algorithm with varying forgetting factor (AM-IRLS-VFF) is derived : ^ ^ ^ θðkTÞ¼ θðkT TÞþ LðkTÞ½YðkTÞ φ ^ ðkTÞθðkT TÞ (16) k1 k1 ^ ^ ^ ^ YðkTÞ¼ yðkTÞþ hðr yðkTÞÞyðkTÞþ hðyðkTÞ r ÞyðkTÞ 1 2 (17) k1 k1 ^ ^ ^ þ hðyðkTÞ r Þhðr yðkTÞÞyðkTÞ 1 2 LðkTÞ¼ PðkT TÞφ ^ðkTÞ½λðkTÞþ φ ^ ðkTÞPðkT TÞφ ^ðkTÞ (18) PðkTÞ¼ ½I LðkTÞφ ^ ðkTÞPðkT TÞ (19) λðkTÞ ^ ^ ^ ϕðkTÞ :¼½yðkT TÞ; yðkT 2TÞ; .. . ; yðkT n TÞ; uðkTÞ; uðkT TÞ; ; uðkT n TÞ; a b uðkT T þ t Þ; .. . ; uðkT n T þ t Þ; .. . ; uðkT T þ t Þ; .. . ; uðkT n T þ t Þ; 1 b 1 m1 b m1 k1 k1 k1 k1 ^ ^ ^ ^ ^ hðr yðkTÞÞyðkTÞ; hðyðkTÞ r ÞyðkTÞ; m hðr yðkTÞÞ; 1 2 1 1 k1 k1 m hðyðkTÞ r Þ 2 2 (20) where I is an identity matrix and PðÞ kT is the covariance matrix. As the time increases, λðkTÞ reduces the weight of old data. The data saturation is then overcome to some extent. When λðkTÞ is set to a low value, the algorithm has a strong tracking capability and a small parameter estimation error (PEE). Simultaneously, the AM-IRLS-VFF is more sensitive to noise, which causes a large variation in the parameter estimation. By contrast, a high value of λðkTÞ causes the estimation process to stabilize, but produces less sensitivity and a slow convergence rate . To increase the convergence rate, improve the anti-jamming performance and reduce the PEE, a new modified form of the forgetting factor is proposed: RðkTÞ λðkTÞ¼ λ þð1 λ Þ min min (21) RðkTÞ¼ NINTðρjðkTÞjÞ s:t: ðkTÞ¼ YðkTÞ φ ^ ðkTÞθðkT TÞ where λðkTÞ is the forgetting factor at time kT; λ is the minimum of λðkTÞ;0 λ 1; ρ is a min min gain coefficient, which makes λðkTÞ tend to 1; ðkTÞ¼ YðkTÞ φ ^ ðkTÞθðkT TÞ is the error of the output between the system and its estimation; and NINTðÞ is the nearest integer. Note that the magnitude of λðkTÞ is determined by the prediction error. When the prediction error changes, implementing the data discounting mechanism becomes necessary. The higher the value of ðkTÞ, the higher that of RðkTÞ, the lower that of λðkTÞ and the faster the tracking velocity of the algorithm. Otherwise, the algorithm is not only insensitive to noise, but also it has low PEE. 3.2. Algorithm summary ^ ^ To initialize the algorithm, θð0Þ is set as a small real vector; for example, θð0Þ¼ 1=p and Pð0Þ¼ p 1 with p normally a large positive numeric (e.g.,p ¼ 10 ), where 1 is a column vector with an 0 0 0 appropriate dimension whose entries are all 1. Then, the proposed AM-IRLS-VFF algorithm is summarized as follows: Step 1: Let k ¼ 1, and set θð0Þ, yð0Þ, and Pð0Þ to: θð0Þ¼ 1=p ; yð0Þ¼ 1=p ; Pð0Þ¼ p 1 0 0 0 MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 603 Step 2:Collect uðkT þ t Þ; yðkTÞ : j ¼ 1; 2; .. . ; m ,construct φ ^ ðkTÞ using Equation (14) jþ1 k1 k1 k1 k1 and calculate yðkTÞ using Equation (13). Then, separate the parameters m ; m ; r ; r 1 2 1 2 from θðkT TÞ and form φ ^ðkTÞ according to Equation (20). Step 3: Calculate YðkTÞ, λðkTÞ, LðkTÞ and PðkTÞ separately using Equations (17), (18), (19) and (21). Step 4: Update θðkTÞ based on Equation (16). Step 5: Let k ¼ k þ 1, and return to Step 3. If λ ¼ 1, the algorithm is simplified to the auxiliary model-based iteratively recursive least- min squares (AM-IRLS) algorithm. If λðkTÞ is constant, the algorithm is deformalized to the auxiliary model-based iteratively recursive forgetting least-squares (AM-IRFLS) algorithm. 4. Performance analysis Proposition 1: Given the AM-IRLS-VFF algorithm in Equations (16)–(21) and the system in Equation (12), if φ ^ðkTÞ is a persistent excitation (PE) (constants 0<α β < 1 and a positive integer N n exist), the following PE condition holds as follows: N1 αI φ ^ðkT þ iTÞφ ^ ðkT þ iTÞ βI a:s:; k > 0 (A1) i¼0 Given 0 < λ < 1 using Equations (19),(21), and (A1), PðkTÞ satisfies min N1 λ α 1 min k 1 P ðkTÞ αI þ λ P ð0Þ I min 1 λ 1 λ min min Let P ð0Þ satisfy αI 1 λ min P ð0Þ or p (22) 1 λ α min Then, for k N, Equation (23) is obtained as follows: N1 λ α 1 λ min 1 min P ðkTÞ I or PðkTÞ I; 0 < λ < 1 (23) min N1 1 λ min λ α min Proof: Provided by Ding in . Proposition 1 proves that PðkTÞ in the AM-IRLS-VFF algorithm has finite upper bounds under the PE condition. Proposition 2: For the system in Equation (12),fg vðkTÞ is independent white noise with zero mean and variance σ under Equation (A1). In other words, vðkTÞ satisfies E½vðkTÞ ¼ 0 (A2) 2 2 E½v ðkTÞ σ < 1 (A3) v 604 R. LIU ET AL. 2 2 ~ ^ ^ ~ Let E½jjθð0Þjj ¼ E½jjθð0Þ θjj ¼ δ <1. Here, θð0Þ is independent of vðkTÞ, and θðkTÞ¼ θðkTÞ θðkTÞ is the parameter error vector. Therefore, for k N, the AM-IRLS-VFF algorithm in Equations (16)–(21) has the following upper bounds: nð1 λ Þ 2ðN1Þ min 2 2 2 2 2 2 E½jjθðkTÞjj α p λ ðjTÞλ ð1 λ Þ δ þ σ ¼: f ðλ ; kTÞ min 0 u min 0 min v N1 αλ j¼1 min Proof: Provided by Ding in . 1λ min Furthermore, if p is set as p ¼ , f ðλ ; kTÞ can be rewritten as follows: 0 0 u min nð1 λ Þ 2 2ðN1Þ min f ðλ ; kTÞ¼ λ ðjTÞλ δ þ σ u min 0 min N1 v αλ min j¼1 The measurements fuðkT gT þ t Þg; g ¼ 0; 1; :::; n ; j ¼ 1; 2; :::; m; k ¼ 1; 2; :::; L are j1 b collected from physical systems or experiments, and the output of the auxiliary model fyðkT hTÞg; h ¼ 1; 2; :::; n is obtained from the auxiliary model. Thus, φ ^ðkTÞ is known and α; β is calculated according to Equation (A1). Therefore, as k !1, the error upper bound f ðλ ; kTÞ approximates a finite constant, that is, u min nð1 λ Þ min f ðλ ; kTÞ! σ u min N1 αλ min where n ¼ n þ mn þ 5 is the dimension of θðkTÞ that is known, and σ cannot be determined a b 2 2 in practice. Therefore, σ is replaced with its estimation σ ^ : v v 2 T 2 σ ^ ¼ ½YðiTÞ φ ^ ðiTÞθðLTÞ i¼1 The upper bounds of AM-IRLS-VFF are then obtained. 5. Numerical example Many actuators are used in an industrial chemical process, and the performance of an actuator can be described as an NUSW system , as shown in Figure 1. The linear dynamics block is expressed as follows: B ðz Þ j¼1 1 1 yðkTÞ¼ uðkT þ t Þ¼ ½1 Aðz ÞyðkTÞþ B ðz ÞuðkT þ t Þ (24) j1 j j1 Aðz Þ j¼1 where 1 1 2 Aðz Þ¼ 1 1:529z þ 0:7408z 1 1 2 B ðz Þ¼ 0:1234 þ 0:06899z þ 0:01538z 1 1 2 B ðz Þ¼ 0:0421z þ 0:08506z and the parameters of the dead-zone non-linearities are m ¼ 1:2; m ¼ 1:1; r ¼ 0:8; r ¼0:9. 1 2 1 2 The parameter vector is then formulated as follows: MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 605 θ ¼½a ; a ; b ; b ; b ; b ; b ; m ; m ; r ; r 1 2 10 11 12 21 22 1 2 1 2 (25) ¼½1:529; 0:7408; 0:1234; 0:06899; 0:01538; 0:0421; 0:08506; 1:2; 1:1; 0:8; 0:9 Set m ¼ 2; τ ¼ 1s; τ ¼ 1:5s;then, t ¼ 0s; t ¼ τ ¼ 1s and t ¼ τ þ τ ¼ 2:5s ¼ T.Inthis 1 2 0 1 1 2 1 2 case, uðkTÞ and uðkT þ t Þ are taken as the PE signal sequence with zero mean and unit variance. When noise variance σ ¼ 0:1, the AM-IRLS-VFF (ρ ¼ 2and λ ¼ 0:95) algorithm min is used to estimate the system parameters (Equation (24)). The parameter estimation and their errors δ ¼ θðkTÞ θjj jj = θ 100% are shown in Table 2.Wecan seethatthePEEdecreases with an increase in the quantity of data and tends to stabilize after 1000 periods. Taking θ as different vectors, similar results are obtained. Estimates of θ ¼½0:86; 0:3417; 0:6; 0:1547; 0:02301; 0:1297; 0:1744; 1:4; 0:8; 0:7; 0:8 and errors with σ ¼ 0:1; λ ¼ 0:98; ρ ¼4are min shown in Table 3.Estimates of θ ¼½0:5803; 0:829; 0:5; 0:7273; 0:06812; 0:3504; 0:2392; 0:86; 1:23; 1:7; 0:521 and errors with σ ¼ 0:1; λ ¼ 0:997; ρ ¼ 6are shown in Table 4. min 6. Comparison To demonstrate the performance of the present algorithm, other existing methods are taken for comparison (shown in Figure 3). We can see that the Gradient-based iterative (GI) algorithm (red dot line, with iterative times k ¼ 500 and date length L ¼ 200) in  had a lower accuracy (PEE was nearly 59%). Moreover, the higher computation burden of the GI algorithm generated a lower executive efficiency (the elapsed time of the GI algorithm was nearly 34 s, whereas the AM-IRLS, AM-IRLS-PVFF algorithm and the algorithm in  abbreviated as AM-IRLS-PVFF was less than 1.6 s). Based on the study in , the AM-IRLS algorithm was used to estimate parameters (black dash-dot line). Without the forgetting factor, the method was stable during the estimating process, and the accuracy was better (PEE was ~9%). When the variable forgetting factor was introduced in , the algorithm is shortened to AM-IRLS-PVFF. We can see that the convergence rate improved. As a result, the estimated precision improved as well (purple dash line), and the PEE reduced to ~3%. By incorporating Equation (21), the AM-IRLS-VFF algorithm was more sensitive in adjusting λðkTÞ. Consequently, it showed a faster convergence and greater accuracy (blue solid line; PEE was ~2%). To validate the robustness of AM-IRLS-VFF, we compared it to AM-IRLS-PVFF and AM- IRFLS (σ ¼ 0:1 and 0:2). Figure 4 shows the results. The estimated parameters of the AM-IRFLS and AM-IRLS-PVFF algorithms proved to be unstable with a high σ. By comparison, the AM- IRLS-VFF algorithm could eliminate interferences and showed good accuracy. The influence of λ and ρ in the AM-IRLS-VFF algorithm is shown in Figure 5. min Figure 5 showed that with an increase in ρ, the convergence rate becomes much more sensitive, and an evident wave exists at the early state of the estimation (blue dot line). Simultaneously, based on Equation (21), the influence of λ and ρ can be summarized as follows: min The higher the value of ρ, the slower is the rate of λðkTÞ (tending to 1), and relatively lower is λðkTÞ. The tracking velocity is then found to be faster and more sensitive to noise, and a lower PEE and more evident wave may occur (blue dot line) The higher the value of λ , the relatively higher is that of λðkTÞ. The algorithm then proves to be min stable during the estimating process, is not sensitive to noise and has a higher PEE (black dash-dot line). 7. Summary In this study, an AM-IRLS-VFF algorithm was proposed to estimate the parameters of NUSW systems with dead-zone nonlinearities. A switching function was used to describe the output, and 606 R. LIU ET AL. Table 2. The VFF-AM-IRLS estimates of θ and errors with σ ¼ 0:1. ka a b b b b b m m r r δð%Þ 1 2 10 11 12 21 22 1 2 1 2 100 −1.21378 0.48726 0.05866 −0.01754 0.09026 0.01073 0.13136 0.75316 0.81880 0.57820 0.08136 45.85271 200 −1.24413 0.52605 0.13211 0.12587 0.09825 0.02086 0.16383 0.81348 0.89518 0.84796 −0.55251 25.56411 500 −1.49611 0.70172 0.12628 0.08638 0.00070 0.03497 0.10563 1.14586 1.08917 0.77022 −0.86628 3.51762 1000 −1.50554 0.71649 0.12494 0.06998 −0.00571 0.03579 0.09253 1.22548 1.12283 0.80268 −0.88968 2.05729 2000 −1.54760 0.75210 0.12301 0.06579 0.00179 0.03919 0.09067 1.20082 1.14096 0.77708 −0.90779 2.05862 3000 −1.53978 0.74718 0.12239 0.06819 0.00647 0.04044 0.08808 1.20274 1.12416 0.78057 −0.89092 1.36170 4000 −1.53572 0.74498 0.12164 0.06775 0.00804 0.04055 0.08707 1.21126 1.09914 0.79090 −0.86527 1.48340 0 −1.52900 0.74080 0.12340 0.06899 0.01538 0.04210 0.08506 1.20000 1.10000 0.80000 −0.90000 0.00000 MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 607 Table 3. The estimates of θ and errors with σ ¼ 0:1; λ ¼ 0:98; ρ ¼ 4. min ka a b b b b b m m r r δð%Þ 1 2 10 11 12 21 22 1 2 1 2 100 0.56135 0.30839 0.53952 −0.04823 0.09681 0.08415 0.13937 1.18697 0.59178 0.62571 −0.41580 27.72588 200 0.65773 0.28474 0.56157 0.01006 0.03944 0.09357 0.14251 1.21342 0.58883 0.64721 −0.47416 22.65312 500 0.81806 0.30627 0.58407 0.11485 −0.00102 0.11295 0.14741 1.29383 0.62458 0.70212 −0.65236 11.83529 1000 0.89138 0.32496 0.59657 0.16376 −0.01297 0.12117 0.16159 1.30225 0.67989 0.69902 −0.75046 7.64474 2000 0.90576 0.33658 0.59424 0.17816 −0.00547 0.12493 0.16962 1.32733 0.71332 0.69596 −0.77233 5.84400 3000 0.89061 0.34829 0.59440 0.17128 0.01046 0.12630 0.16838 1.36136 0.73958 0.71076 −0.80197 3.66910 4000 0.87635 0.34357 0.59423 0.16210 0.01373 0.12630 0.16784 1.37304 0.76580 0.71129 −0.82892 2.58745 0 0.86 0.3417 0.6 0.1547 0.02301 0.1297 0.1744 1.4 0.8 0.7 −0.8 0 608 R. LIU ET AL. Table 4. The estimates of θ and errors with σ ¼ 0:1; λ ¼ 0:997; ρ ¼ 6. min ka a b b b b b m m r r δð%Þ 1 2 10 11 12 21 22 1 2 1 2 100 −0.56483 0.84001 −0.48134 0.59854 −0.03198 0.30836 0.22662 0.46996 1.26621 0.52631 −0.44037 45.88010 200 −0.54568 0.82598 −0.47687 0.65303 −0.02230 0.33279 0.24451 0.59993 1.26879 1.32867 −0.49043 17.14838 500 −0.57838 0.82628 −0.48927 0.70463 −0.04316 0.34649 0.22673 0.77612 1.24208 1.93812 −0.50074 9.42359 1000 −0.58062 0.83113 −0.49055 0.71341 −0.05003 0.34896 0.23232 0.76604 1.23222 1.70157 −0.50811 3.61354 2000 −0.57718 0.82894 −0.49512 0.71843 −0.05132 0.34947 0.23763 0.80212 1.23138 1.74222 −0.51066 2.76023 3000 −0.57756 0.82665 −0.49686 0.72190 −0.05499 0.34912 0.23588 0.81692 1.23089 1.73682 −0.51026 2.19456 4000 −0.57847 0.82828 −0.49823 0.72261 −0.05870 0.34925 0.23575 0.82265 1.23133 1.72880 −0.51363 1.80499 0 −0.58030 0.82900 −0.50000 0.72730 −0.06812 0.35040 0.23920 0.86000 1.23000 1.70000 −0.52100 0.00000 MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 609 0 200 400 600 800 1000 AM−IRLS 90 90 AM−IRLS−PVFF AM−IRLS−VFF k=1000,L=200 GI 50 50 AM−IRLS(λmin=1,ρ=2) AM−IRLS−PVFF(λmin=0.9,ρ=2) AM−IRLS−VFF(λmin=0.9,ρ=2) 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 3. Parameter estimation errors δ versus k. AM−IRLS−VFF AM−IRLS−PVFF AM−IRFLS 0 500 1000 1500 2000 2500 3000 3500 4000 (a) σ =0.1 AM−IRLS−VFF AM−IRLS−PVFF AM−IRFLS 0 500 1000 1500 2000 2500 3000 3500 4000 (b) σ =0.2 Figure 4. Parameter estimation errors δ versus k of AM-IRLS-VFF, AM-IRLS-PVFF and AM-IRFLS algorithms with different noise levels. δ (%) δ (%) δ (%) δ (%) 610 R. LIU ET AL. λmin=0.9,ρ=2 λmin=0.9,ρ=3 λmin=1,ρ=2 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 5. AM-IRLS-VFF algorithm with different λ and ρ. min then the identification model was derived. By defining a new extended information vector, some coupled parameters were separated. In addition, system parameters were estimated directly. Furthermore, a new varying forgetting factor function was also embedded in the algorithm to improve the convergence velocity and enhance the anti-jamming ability. The simulation results demonstrate the effectiveness of the proposed algorithm. This method can be extended to other hard non-linear Hammerstein, Wiener or Hammerstein-Wiener systems using different switching functions, as well as other non-uniformly sampled systems (e.g. the outputs non-uniformly sampled systems and inputs-outputs non-uniformly sampled systems), and multivariable systems. However, the formulation of λðkTÞ needs to be further investigated. Acknowledgments The authors would like to thank the financial support provided by National Natural Science Foundation of China under Grant 61273142, Foundation for Six Talents by Jiangsu Province (2012-DZXX-045), Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Jiangsu Policy Guidance (Industry University Research) Project [Grant no. BY2016030-16] and Jiangsu Planned Projects for Postdoctoral Research Funds [Grant no. 1601138B]. Disclosure statement No potential conflict of interest was reported by the authors. 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Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Nov 2, 2017
Keywords: Parameter estimation; iterative least-squares algorithm; Wiener systems; non-uniform sampling; dead-zones
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