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Neural network observer-based networked control for a class of nonlinear systems
Neurocomputing 133 (2014) 103–110
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Neural network observer-based networked control for a class of nonlinear systems Changchun Hua, Caixia Yu, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City 066004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2013 Received in revised form 1 November 2013 Accepted 1 November 2013 Communicated by Z. Wang Available online 30 January 2014
A new neural network observer-based networked control structure for a class of nonlinear systems is developed and analyzed. The structure is divided into three parts: local linearized subsystem, communication channels and remote predictive controller. A neural-network-based adaptive observer is presented to approximate the state of the time-delay-free nonlinear system. The neural-network (NN) weights are tuned on-line and no exact knowledge of nonlinearities is required. The time delays considered in the forward and backward communication channels are constant and equal. A modified Smith predictor is proposed to compensate the time delays. The controller is designed based on the developed NN observer and the proposed Smith predictor. By using the Lyapunov theory, rigorous stability proofs for the closed-loop system are presented. Finally, simulations are performed and the results show the effectiveness of the proposed control strategy. & 2014 Elsevier B.V. All rights reserved.
Keywords: Networked control system Time-delay Neural network observer Smith predictor
1. Introduction Networked control system (NCS) is such a system that a remote controller communicates with a plant through two independent communication channels. NCSs have been used in a wide range of areas because of their advantages in practical applications such as reduced system wiring, eased maintenance and diagnosis, and increased flexibility [1]. The insertion of communication network also brings many challenging problems. The key one is the network-induced delay which may make the system unstable or demonstrate undesired performance. The network delay can be modeled as a constant delay, an independent random delay, and a delay with known probability distribution [2]. Many works have been made to solve these delay problems [3,4]. Different mathematical-, heuristic-, and statisticalbased approaches are taken for different delay compensation [5]. A gain scheduler middleware (GSM) was proposed in [6] to alleviate the network delay effect. A new control scheme consisting of a control prediction generator and a network delay compensator was developed in [7]. A time-delay-compensation method based on the concept of network disturbance and communication disturbance observer has been proposed in [8]. Due to its no use of delay-time model, the method can be flexibly applied to many kinds of timedelayed control systems. Some methods consider NCS as a classical control system with slowly changing delay times and adopt wellknown control methods such as the Smith predictor [8,9]. The queuing/buffering method turns the NCS into a time-invariant
E-mail address: [email protected] (C. Hua). 0925-2312/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.11.026
system to alleviate the delays. The constant network delay is considered in this paper. To the best of our knowledge, most of the results about NCSs are based on linear systems [10,11], and only a few literatures focus on the nonlinear ones. Besides, most literatures consider the discrete-time systems [12]. In this paper, we consider a nonlinear continuous-time system which has uncertain nonlinear terms and unknown external disturbance. This type of system is closer to the practical one and owns a far worthier study. In order to deal with the nonlinear terms, the neural network is proposed and widely used due to its versatile features such as learning capability mapping and parallel processing. A slidingmode neural-network (SMNN) control system for the tracking control of robot manipulators to achieve high-precision position control was investigated in [13]. Two neural network-based controllers were designed for the teleoperation system in free motion in [14]. One is a new adaptive controller using the acceleration signal, another without the acceleration signal. In this paper, we employ the NNs to estimate the uncertainties. In most practical situations, the velocity signals are always difficult to measure. Hence, an observer is imperative to estimate the state signals when we design the controller. A dynamic neuralnetwork-based adaptive observer for a class of nonlinear systems was presented in [15], which does not require exact knowledge of nonlinearities. However, the output error equation in [15] is strictly positive real (SPR) which is a restrictive assumption for nonlinear systems. Therefore, paper [16] proposed a new observer without the SPR condition by introducing a vector b0. Besides, more observer design methods have been proposed in [17–20]. In this paper, we consider the networked control design problem for a class of nonlinear systems with unknown nonlinear
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functions. The state variables of system could not be completely obtained. A NN observer is designed to estimate the unmeasurable state. The Smith predictor is employed to deal with the network induced time delay. Based on the constructed observer and the Smith predictor, we design the nonlinear controller to render that the system output follows the given reference signal. By constructing Lyapunov function, the stability of the closed-loop system is proved. Finally, simulations are performed to show the effectiveness of the proposed strategy. This paper is organized as follows. The nonlinear system and the problem we discussed are presented in Section 2. Preliminaries about neural network are introduced in Section 3. A neural network observer and its stability analysis are presented in Section 4. In Section 5, we introduce the whole structure of the control strategy and design a neural network predictive controller. Finally, simulation results are presented in Section 6.
2. Background and problem formulation In this paper, we discuss a class of single-input single-output (SISO) nonlinear systems with networked time delay as follows: 8 x_ 1 ðtÞ ¼ x2 ðtÞ > > > > _ > > < x 2 ðtÞ ¼ x3 ðtÞ ⋮ ð1Þ > > > > x_ n ðtÞ ¼ f ðxðtÞÞ þ gðxðtÞÞuðt TÞ þ dðtÞ > > :y¼x 1
where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ ⋯ xn ðtÞT A Rn , uðtÞ A R, yðtÞ A R are the state, the control, and the output of the plant, respectively; d(t) is the unknown disturbance with a known upper bound bd; f(x) and g(x) are unknown smooth functions; T is the network induced time delay. The above equation can be expressed as ( x_ ¼ As x þ b½f 0 ðxÞ þ gðxÞuðt TÞ þ d ð2Þ y ¼ cT x where 2 6 6 6 As ¼ 6 6 6 4
0
3
1 ⋱
⋱ ⋱
⋱ 0
an
⋯
⋯
a2
7 7 7 7; 7 1 7 5 a1
2 3 0 6⋮7 6 7 6 7 7 b¼6 6 ⋮ 7; 6 7 405 1
2 3 1 607 6 7 6 7 7 c¼6 6⋮7 6 7 4⋮5 0
ð3Þ
and f 0 ðxÞ ¼ f ðxÞ þ∑ni¼ 1 an i þ 1 xi . The parameters ai are suitably chosen such that As is stable. Given a symmetric positive definite matrix Q, there exists a symmetric positive definite matrix Ps satisfying the following Lyapunov equation: ATs P s þ P s As
¼ Q
ð4Þ
In this work, we assume that there is a constant time delay T in the input channel and the feedback channel, and only the output signal y can be measured. Therefore, the first thing we do is to design a nonlinear observer to estimate the state which will be used in the design of the control input u(t). Due to the constant time delay, the control input u(t) can only be applied to the plant after a delay of T. In order to eliminate the influence of the time delay, we employ a modified Smith predictor which confirms the accurate output tracking of the system. The control objective can be described as follows: given a desired state xd(t), design a neural network observer and a control u(t) such that the observer state estimates the actual state exactly and both of them follow the delayed desired trajectory with an acceptable accuracy.
The desired trajectory vector is defined as 1Þ xd ðtÞ ¼ ½yd ðtÞ y_ d ðtÞ … yðn ðtÞT d
ð5Þ
For this purpose, we make some mild assumptions as follows: Assumption 1. The desired trajectory vector xd(t) is continuous and available for measurement, and xd(t) is bounded. Assumption 2. The time delay T is a known constant, and the time delay in the input channel is equal to that in the feedback one. Assumption 3. There exist constant g 0 4 0, and known smooth function gd(x) such that g d ðxÞ ZjgðxÞj Zg 0 . Remark 1. Assumption 3 implies that the smooth function g(x) is strictly either positive or negative. It is reasonable because being away from zero is a controllable condition of system (1), which is made in most control schemes. For a given practical system, the upper bound of g(x) is not difficult to determine by choosing gd(x) large enough. Besides, the low bound g0 is only required for analytical purpose, so its true value is not necessarily known. Definition 1. We say that the solution of a dynamic system is uniformly ultimately bounded (UUB) if for a compact set U of Rn and for all xðt 0 Þ ¼ x0 A U there exists an ɛ 4 0 and a number Tðɛ; x0 Þ such that J xðtÞ J o ɛ for all t Z t 0 þ T. Remark 2. Neural network predictive control strategies for nonlinear dynamic systems and telerobot are presented in [21] and [22], respectively. The systems considered are simpler than that we discuss in this paper. What is more, all the state variables are required to be available in the literatures, which is not the case in most practical systems. In this paper, we discuss the situation that not all states can be measured. In this paper, we denote J J as the Euclidean norm, and ‖ ‖F as the Frobenius norm.
3. Preliminaries In the control engineering, RBF neural network is usually used as a tool for modeling nonlinear functions because of their good capabilities in function approximation. The RBF neural network can be considered as a two-layer network in which the hidden layer performs a fixed nonlinear transformation with no adjustable parameters, i.e., the input space is mapped into a new space. The output layer then combines the outputs in the latter space linearly. Therefore, they belong to a class of linearly parameterized networks. In this paper, the following RBF neural network is used to approximate the continuous function f ðxðtÞÞ : Rq -R: ^ T ΦðxðtÞÞ; f^ ðxðtÞÞ ¼ W
ð6Þ q
where the input vector x A Q x R , and q is the neural network ^ ¼ ½W ^ 1; W ^ 2 ; …; W ^ T A Rl , the NN input dimension. Weight vector W l node number l 4 1, and ΦðxðtÞÞ ¼ ½Φ1 ðxðtÞÞ; …; Φl ðxðtÞÞT , with Φi ðxðtÞÞ chosen as the commonly used Gaussian function, which is in the following form: " # ðxðtÞ μi ÞT ðxðtÞ μi Þ Φi ðxðtÞÞ ¼ exp ; ð7Þ 2η2i where μi ¼ ½μi1 ; μi2 ; …; μiq is the center of the receptive field and ηi is the width of the Gaussian function. It has been proven that the neural network can approximate any continuous function over a compact set Ωx Rq to arbitrary accuracy as f ðxðtÞÞ ¼ W T ΦðxðtÞÞ þ ɛðxÞ;
8 x A Ωx ;
ð8Þ
C. Hua et al. / Neurocomputing 133 (2014) 103–110
where W is the ideal constant weight, ɛ is the approximation error, and both are bounded, i.e. ‖W‖F rW M ; ɛ r ɛM . The ideal weight vector W is an artificial quantity required for ^ that minimizes analytical purposes. W is defined as the value of W J ɛðxÞ J for all xðtÞ A Ωx in a compact region, i.e. T
^ ΦðxðtÞÞjg; W≔arg min fsupjf ðxðtÞÞ W ^ A Rl W
x A Ωx :
ð9Þ
Neural network approximation idea has been applied to the controller design for uncertain nonlinear systems extensively [23]. In [24–26], neural network state feedback controllers were constructed for time delay nonlinear systems. In [27], the deterministic learning mechanism and the neural learning control scheme were presented. For nonlinear MIMO systems, neural network controllers were constructed in [28]. With the control direction unknown, Du et al. [29] designed the neural network controller for nonlinear system with triangular structure. Observer-based output feedback controller was designed in [20]. In [30,31], decentralized state feedback controllers were constructed for interconnected nonlinear systems via adaptive neural network. In this paper, we will use the neural networks to approximate unknown nonlinear functions and construct the predictive controller. The corresponding neural-network-based adaptive observer will be proposed in Section 4.
105
(SPR) condition, we choose a polynomial L(s) so that ~ T ϕ þ φ þ ɛ 1 þ ς þ v1 y~ ¼ HðsÞLðsÞ½W 1 1 1 1 ~ T ϕ þ φ þ ɛ 2 Þu þ ς þ v 2 þ d þ ðW 2 2 2 2
ð15Þ
where HðsÞLðsÞ is SPR. The ‘overbar’ indicates the signal filtered by L 1 ðsÞ. And the terms ς1 and ς2 are defined as ς1 ðtÞ ¼ L 1 ^ , ς ðtÞ ¼ L 1 ðsÞW ^ u with ^ ^ ~ Tϕ ~ T 1 ðsÞϕ ~ Tϕ ~ T 1 ðsÞϕ ðsÞW 2 1 2 1 1 W 1L 2 2u W 2L ~ J ς J r li ‖W ‖F ; i ¼ 1; 2. i
Then the state-space realization of (15) can be expressed as 8 < z~_ ¼ A z~ þ b ½W ~ T ϕ þ φ þ ɛ 1 þ ς þ v 1 þ ðW ~ T ϕ þ φ þ ɛ 2 Þu þ ς þ v 2 þ d c c 1 2 1 2 1 1 2 2 : y~ ¼ cTc z~
ð16Þ 1
with HðsÞLðsÞ ¼ cTc ðsI Ac Þ bc and cc ¼ ½1 0 ⋯ 0T . Since HðsÞLðsÞ is SPR, for Q ¼ Q T 4 0, there exists P ¼ P T 4 0 such that ATc P þ PAc ¼ Q ; Pbc ¼ cc
ð17Þ
Lemma 1. Given Assumptions 1–3, consider the nonlinear system (2) and the NN observer given by (12). Design the NN weight adaptation laws as ^ 1 ^_ 1 ¼ F 1 ϕ ðxÞ ^ y~ K 1 F 1 jyj ~ W W 1
4. Neural network observer without time-delay
^ 2 ^_ 2 ¼ F 2 ϕ ðxÞ ^ yu ~ K 2 F 2 jyj ~ W W 2
ð18Þ
According to the approximation property of NNs, we can assume that the continuous nonlinear functions in the system (2) can be represented by
where Fi are the positive definite matrices, and Ki are the positive parameters, i¼1,2. Let the robustifying terms be
f 0 ðxÞ ¼ W T1 ϕ1 ðxÞ þ ɛ1 ðxÞ
~ ~ v1 ðtÞ ¼ D1 yðtÞ=j yðtÞj;
gðxÞ ¼ W T2 ϕ2 ðxÞ þ ɛ2 ðxÞ
ð10Þ
where Wi is the ideal weight, ϕi ðxÞ is the basis function and ɛ i ðxÞ is the approximation error, i¼ 1,2. So the system (2) without time-delay can be written as ( x_ ¼ As x þb½W T1 ϕ1 ðxÞ þ ɛ 1 ðxÞ þ ðW T2 ϕ2 ðxÞ þ ɛ 2 ðxÞÞu þ d ð11Þ y ¼ cT x
^ T ϕ ðxÞ ^ T ϕ ðxÞu ^ ^ þW ^ v1 ðtÞ v2 ðtÞ þ Kðy cT xÞ x^_ ¼ As x^ þ b½W 1 1 2 2 ^y ¼ cT x^
ð12Þ
^ T ϕ ðxÞ ^ T ϕ ðxÞ ^ and W ^ represent the estimation of f 0 ðxÞ and where W 1 1 2 2 g(x) respectively, K ¼ ½K 1 ; K 2 ; …; K n T is the observer gain vector, chosen so that the characteristic polynomial of As KcT is strictly Hurwitz. And v1 ðtÞ and v2 ðtÞ are robustifying terms that provide robustness in the face of bounded disturbances which will be designed later. Define the state and observer output errors as x~ ¼ x x^ and y~ ¼ y y^ respectively. Subtracting (12) from (11) yields ( ^ ~ Tϕ ~ T^ x~_ ¼ ðAs KcT Þx~ þ b½W 1 1 þ φ1 þ ɛ 1 þ ðW 2 ϕ 2 þ φ2 þ ɛ 2 Þu þ dþ v1 þ v2 y~ ¼ cT x~ ð13Þ ^ and ϕ ^ are the abbreviations of ϕ ðxÞ ~ i ¼ Wi W ^ i, ϕ ^ where W and 1 2 1 ^ ^ ϕ2 ðxÞ. φi ¼ W Ti ½ϕi ðxÞ ϕi ðxÞ, and J φi ðtÞ J r φdi with φdi 40; i ¼ 1; 2. Then the error dynamics may be written as ^ ~ Tϕ ~ T^ y~ ¼ HðsÞ½W 1 1 þ φ1 þɛ 1 þ ðW 2 ϕ 2 þ φ2 þ ɛ 2 Þu þ d þ v1 þ v2
ð19Þ
where D1 Z φd1 þ ɛ M1 is a constant scalar, D2 ðtÞ Zðφd2 þɛ M2 ÞuðtÞ is a ~ function. Then the state estimation error xðtÞ and the neural network ~ 1 ðtÞ and W ~ 2 ðtÞ are uniformly ultimately weight estimation errors W bounded (UUB). Proof. Define the Lyapunov function as
Now, we design the observer as (
~ ~ v2 ðtÞ ¼ D2 ðtÞyðtÞ=j yðtÞj:
ð14Þ
where s denotes the differential operator d=dt, and H(s) is a linear transfer function with stable poles and is realized by ðAs KcT ; b; cÞ. In order to make the system (14) satisfy the strictly positive real
1 T 1 n ~ T 1 ~ o 1 n ~ T 1 ~ o V ¼ z~ P z~ þ tr W 1 F 1 W 1 þ tr W 2 F 2 W 2 : 2 2 2
ð20Þ
The time derivative of V is given by T T ~ T ϕ þ φ þ ɛ 1 þ ς þ v 1 þ ðW ~ Tϕ V_ ¼ 12 z~ ðATc P þ PAc Þz~ þ z~ Pbc ½W 1 1 1 1 2 2
~_ 1 g þ trfW ~_ 2 g: ~ T F 1W ~ T F 1W þ φ 2 þ ɛ 2 Þu þ ς2 þ v 2 þd þ trfW 1 1 2 2 ð21Þ
Using (17), (18) and (19), we obtain 1 T V_ ¼ z~ Q z~ þ y~ T ðɛ 1 þɛ 2 u þd þ ς1 þ ς2 þ φ 1 þ v 1 þ φ 2 u þ v 2 Þ 2 n T o n T o ^ 1 þK 2 jyjtr ^ 2 ~ W ~ W ~ ~ þK 1 jyjtr W W 1
2
1 ~ 1 ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ‖F α2 =2Þ2 ~ þ K 1 ð‖W ~ λ ðQ Þjyj r jyj 2 min ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ
ð22Þ
with lM the maximum singular value of L 1 ðsÞ and αi ¼ W Mi þ li =K i ; i ¼ 1; 2.
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C. Hua et al. / Neurocomputing 133 (2014) 103–110
Thus, V_ is negative outside the region: 9 8 4lM bd þ K 1 α21 þ K 2 α22 > > jyj > > ~ o ; > > > > > > 2λmin ðQ Þ > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > > 2 2 = < 4lM bd þ K 1 α1 þK 2 α2 > 1 ~ ~ ~ ; ‖ W ‖ o α þ ~ W 1; W 2 1 F 1 Ω1 ¼ y; 2 4K 1 > > > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > > > 2 2 > > 4l b þ K α þK α 1 M 1 2 > d 1 2 > ~ 2 ‖F o α2 þ > > ‖ W : > > ; : 2 4K 2
nonlinear system in the first. To locally compensate the nonlinear part of the system, let the local control input be given by uðtÞ ¼ ð23Þ
Therefore, V_ is negative outside the compact set Ω1, which is thus shown to be an attractive set for the system. According to a standard Lyapunov theorem extension, this demonstrates the UUB ~ 1 and W ~ 2. □ ~ W of both y, Remark 3. From (23) we can see that the bound on the approx~ decreases with the value of Ki. On the other hand, imation error jyj ~ 1 ‖F decreases with the increase of K1, the the NN weight error ‖W ~ 1 ‖F is opposite. So the parameters Ki offer a decrease of K2, and ‖W ~ design tradeoff between the relative eventual magnitudes of jyj ~ i ‖F , i¼ 1,2. and ‖W 5. The observer based networked predictive control 5.1. Control structure The whole structure of the networked control system is shown in Fig. 1, which is composed of three parts: remote predictive controller, communication channels and local linearized subsystem. As shown in Fig. 1, the communication channels consist of two time delays. One is in the input channel and the other in the feedback one. In this paper, we use the Smith predictor principle to deal with the time delays. To the best of our knowledge, the Smith predictor is usually based on a linear model, so the linearization of the nonlinear plant is necessary. While the feedback signal to the controller is the observer state, the linearization should aim at the observer. Here, we use a local feedback, called the local nonlinear compensator, to cancel the nonlinear effects. We use the Smith predictor theory in remote predictive controller part to deal with the time delays. If the linear predictor is an accurate model of the local linearized subsystem, then, referring to Fig. 1, ^ TÞ and zðtÞ ¼ xðt ^ þTÞ. This means that the observer zðt 2TÞ ¼ xðt ^ state xðtÞ ahead of time T can be predicted via the predictor's output z(t). Therefore, we compensate the time-delay. 5.2. Smith predictor control
1 ^ T ϕ ðxÞ ^ W 2 2
T
^ ϕ ðxÞ ^ ½τðt TÞ W 1 1
ð24Þ
where τðt TÞ stands for a new control variable from the output of the remote controller with time-delay T in the input channel which will be designed later. Then the local state observer (12) becomes a linear model: ^ þ b½τðt TÞ v1 ðtÞ v2 ðtÞ þ K½yðtÞ yðtÞ ^ x^_ ðtÞ ¼ As xðtÞ
ð25Þ
In the following, we will find a control action τðtÞ such that the ^ state xðtÞ follows the delayed desired trajectory xd ðt TÞ with an acceptable accuracy. Let the linear predictor be z_ ðtÞ ¼ As zðtÞ þ bτðtÞ
ð26Þ
We can use the state of the predictor z(t) as a state feedback to design the remote controller. However, due to the possible pre^ TÞ þ zðtÞ sence of disturbances and modeling errors, we use xðt zðt 2TÞ as the feedback signal to incorporate the Smith predictor loop in Fig. 1. Define an error vector as ^ TÞ zðtÞ þzðt 2TÞ δðtÞ ¼ xd ðtÞ xðt
ð27Þ
Define a filtered error as h T i T rðtÞ ¼ Λ δðtÞ ¼ Λ 1 δðtÞ
ð28Þ
where Λ ¼ ½λ1 λ2 ⋯ λn 1 T is an appropriately chosen coefficient vector such that δ-0 exponentially as r-0, i.e., sn 1 þ λn 1 sn 2 þ ⋯ þ λ1 is Hurwitz. Differentiating r(t) and using (26) and (27), the filtered error dynamics may be written as T T r_ ðtÞ ¼ Λ δ_ ðtÞ ¼ Λ ½x_ d ðtÞ x^_ ðt TÞ z_ ðtÞ þ z_ ðt 2TÞ
^ TÞ þbτðt 2TÞ bv1 ðt TÞ bv2 ðt TÞ ¼ Λ fx_ d ðtÞ fAs xðt ^ TÞg As zðtÞ bτðtÞ þ As zðt 2TÞ þ bτðt 2TÞg þ K½yðt TÞ yðt T
^ TÞ þ zðtÞ zðt 2TÞ þ bv1 ðt TÞ ¼ Λ fx_ d ðtÞ As ½xðt ^ TÞg τðtÞ þ bv2 ðt TÞ K½yðt TÞ yðt T
ð29Þ
Now define controller as ^ TÞ þ zðtÞ zðt 2TÞ τðtÞ ¼ ΛT fx_ d ðtÞ As ½xðt ^ TÞg þ K v rðtÞ K½yðt TÞ yðt
~ TÞ þ K v rðtÞ ¼ Λ fx_ d ðtÞ As xd ðtÞ þ As δðtÞg K yðt T
ð30Þ
where Kv is a positive parameter. Then, (29) becomes
In this section, we adopt the Smith predictor to solve the timedelay problem. As we all know, the Smith predictor is used in the linear system. So, we must cancel the nonlinear terms of the
r_ ðtÞ ¼ K v rðtÞ þ v1 ðt TÞ þ v2 ðt TÞ
Fig. 1. The networked control strategy for nonlinear system.
ð31Þ
C. Hua et al. / Neurocomputing 133 (2014) 103–110
From (27) and (28), we can get
c5 2 c5 kc6 2 kc6 2 r ðtÞ r 2 ðt 2TÞ þ y~ ðtÞ y~ ðt 3TÞ 2 2 2 2 1 ~ 1 ðtÞ‖F ~ ~ λ ðQ ÞjyðtÞj þ K 1 ð‖W jyðtÞj 2 min ~ 2 ðtÞ‖F α2 =2Þ2 α1 =2Þ2 þ K 2 ð‖W 1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þ K 1 α21 =4 þ K 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þ K 1 ð‖W ðlM bd þ K 1 α21 =4 þ K 2 α22 =4Þ
107
þ
J δðtÞ J r jrðtÞj=Λm þ c0
ð32Þ
where Λm ¼ minfλ1 ; λ2 ; …; λn 1 ; 1g, c0 is a positive constant scalar. From (24) and (30), we can get ~ 2TÞj juðtÞj rc1 þ c2 jrðt TÞj þ c3 jyðt
ð33Þ
where c1 ; c2 ; c3 are positive parameters. So we can choose ~ 2TÞjÞ jD2 ðtÞj ¼ ðφd2 þ ɛM2 Þðc1 þ c2 jrðt TÞj þ c3 jyðt ~ 2TÞj ¼ c4 þc5 jrðt TÞj þc6 jyðt
ð34Þ
with c4 ¼ c1 ðφd2 þ ɛ M2 Þ; c5 ¼ c2 ðφd2 þ ɛM2 Þ; c6 ¼ c3 ðφd2 þ ɛ M2 Þ. Theorem 1. Given Assumptions 1–3, consider the nonlinear system (2) with input and feedback time delays. Take the control input as (24) and (30). Let the NN weights tuning be provided by (18) and the robustifying term be provided by (19). Then the filtered error r(t), the estimation error ~ 1 ðtÞ; W ~ 2 ðtÞ are UUB. ~ yðtÞ and the weight tuning error W Proof. Define the Lyapunov function as Z Z 1 c5 t kc6 t V ¼ r 2 ðtÞ þ y~ 2 ðtÞ dt r 2 ðtÞ dt þ 2 2 t 2T 2 t3 T o 1 n T o 1 T 1 n ~ T 1 ~ 1 ~ ~ þ z~ ðtÞP z~ ðtÞ þ tr W 1 ðtÞF 1 W 1 ðtÞ þ tr W 2 ðtÞF 2 W 2 ðtÞ 2 2 2 o 1 T 1 n ~ T 1 ~ 1 ðt TÞ þ z~ ðt TÞP z~ ðt TÞ þ tr W 1 ðt TÞF 1 W 2 2 o 1 n ~ T 1 ~ þ tr W ðt TÞF ð35Þ W 2 ðt TÞ 2 2 2 with k a positive parameter. Using the results of Section 4 and the equality (31), we have ~ TÞ ~ TÞ yðt yðt rðtÞD2 ðt TÞ V_ r K v r 2 ðtÞ rðtÞD1 ~ TÞj ~ TÞj jyðt jyðt c5 2 c5 2 kc6 2 kc6 2 y~ ðtÞ y~ ðt 3TÞ þ r ðtÞ r ðt 2TÞ þ 2 2 2 2 1 ~ 1 ðtÞ‖F α1 =2Þ2 ~ ~ λ ðQ ÞjyðtÞj þK 1 ð‖W jyðtÞj 2 min ~ 2 ðtÞ‖F α2 =2Þ2 þ K 2 ð‖W 1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þ K 1 α21 =4 þK 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þ K 1 ð‖W ðlM bd þ K 1 α21 =4 þK 2 α22 =4Þ ð36Þ
ð37Þ
It is easy to get 1 1 jrðtÞ J rðt 2TÞj r jrðtÞj2 þ jrðt 2TÞj2 2 2 1 k ~ 3TÞj2 ~ jrðtÞ J yðt 3TÞj r jrðtÞj2 þ jyðt 2k 2 So
ð38Þ
i c6 K v c5 jrðtÞj ðD1 þ c4 Þ 2k 1 kc ~ ~ jyðtÞj λmin ðQ Þ 6 jyðtÞj 2 2 ~ 1 ðtÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðtÞ‖F α2 =2Þ2 þK 1 ð‖W
1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þK 1 ð‖W ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ ð39Þ
V_ r jrðtÞj
h
Choose k o λmin ðQ Þ=c6 , and K v 4 c5 þ c6 =2k. Then V_ is negative outside the region: 9 8 D1 þ c 4 > > jrðtÞj o > > ; > > > > K c c =2k > > v 5 6 > > > > > > 2 2 > > > > 4l b þ K α þ K α M d 1 1 2 2 > > > > ~ jyðÞj o ; > > > > > > 2 λ ðQ Þ 2kc 6 min = < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ~ ~ ~ Ω ¼ r; y; W 1 ; W 2 ð40Þ 2 2 4lM bd þ K 1 α1 þ K 2 α2 > 1 ~ > > ;> ‖W 1 ðÞ‖F o α1 þ > > > > > > 2 4K 1 > > > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 2 2 > > > > 4l b þ K α þ K α 1 M 1 2 d > > 1 2 > ‖W ~ > > ðÞ‖ o α þ : > > F 2 2 ; : 2 4K 2
With (34), one has V_ r K v r 2 ðtÞ þ D1 jrðtÞj þ c4 jrðtÞj ~ 3TÞj þ c5 jrðtÞ J rðt 2TÞj þ c6 jrðtÞ J yðt
Since V is a positive definite function, and V_ is negative outside ~ 1 ðtÞ and W ~ 2 ðtÞ are UUB. ~ the region Ω, we conclude that r(t), yðtÞ, W
Fig. 2. Overall networked control structure for nonlinear system with time-delay.
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C. Hua et al. / Neurocomputing 133 (2014) 103–110
The overall networked control structure for nonlinear system with time-delay is shown in Fig. 2. □ Remark 4. Since, from Theorem 1, r(t) is UUB, then δðtÞ is UUB ^ TÞ zðtÞ þ z by appropriately choosing Λ. Since δðtÞ ¼ xd ðtÞ xðt ^ TÞ þ zðtÞ zðt 2TÞ is UUB. ðt 2TÞ and xd(t) is bounded, then xðt What is more, from Theorem 1, the estimate error y~ is UUB. Therefore, by definition (30) τðtÞ is UUB for a stable As and a bounded x_ d .
7 6 5 4 3 2 1
6. Simulations In this section, a simulation example is given to verify the effectiveness of the main result. We consider a single-link robot whose dynamics is given by
−1 −2
y¼q
ð41Þ −3
where q is the angle, u is the input torque, M, g, m, l are the moment of inertia, the gravity acceleration, the mass and length of the link respectively, d is the external disturbance. Letting x1 ¼ q, _ the state-space description of system is x2 ¼ q, 8 1 0 > < x_ ¼ 0 xþ ½f ðxÞ þgðxÞu þ d ð42Þ 1 2 1 0 > : y ¼ x1 where f 0 ðxÞ ¼ 12mgl sin x1 =M ð 1 2Þx; gðxÞ ¼ 1=M, and the disturbance d ¼ 0:5 sin ð0:5π tÞ. The parameters of robot are M ¼ 0:5, l ¼ 0:4, m ¼ 0:4, g ¼ 9:8: We select the linear filter L 1 ðsÞ ¼ 1=ðs þ 3Þ. In simulation, the initial conditions of the system and the parameters are xð0Þ ¼ ^ xð0Þ ¼ zð0Þ ¼ ½0 0:5T , K ¼ ½400 800T , F 1 ¼ diag½5 104 , F 2 ¼ diag T ½5 103 , K 1 ¼ K 2 ¼ 0:01, K V ¼ 200, Λ ¼ ½0:2 1. The desired traT jectory is xd ¼ ½ sin ðtÞ cos ðtÞ . When the time-delay T ¼1 s, the simulation results are shown from Figs. 3 to 6. Fig. 3 shows the tracking performance of the state x1, from which we can see that the observer state x^ 1 and the actual state x1 are almost completely coincidental, and both of them track with the desired state xd1 ðt 1Þ at about 7 s. Fig. 5 shows the observer estimated error x~ 1 , and it goes to about zero very quickly. Figs. 4 and 6 show the same performance. From these figures, we can see that the observer can accurately estimate the actual
0
5
10
15
20
25
time(sec) Fig. 4. x2 tracking performance with T ¼1 s. (solid, x2 ðtÞ; dashed, x^ 2 ðtÞ; dashdotted, xd2 ðt TÞ). −3
20
x 10
15 error betweem x1 and x1p
M q€ þ 12 mgl sin q ¼ u þ d;
0
10
5
0
−5
0
5
10
15
20
25
20
25
time(sec) Fig. 5. Estimate error x~ 1 ðtÞ with T ¼ 1 s.
2 7
1.5
6 5 error betweem x2 and x2p
1
0.5
0
−0.5
4 3 2 1 0 −1
−1
0
5
10
15
20
25
time(sec) Fig. 3. x1 tracking performance with T ¼1 s. (solid, x1 ðtÞ; dashed, x^ 1 ðtÞ; dashdotted, xd1 ðt TÞ).
−2
0
5
10
15 time(sec)
Fig. 6. Estimate error x~ 2 ðtÞ with T ¼ 1 s.
C. Hua et al. / Neurocomputing 133 (2014) 103–110
109
7
2
6
1.5
error betweem x2 and x2p
5
1
0.5
0
−0.5
4 3 2 1 0
−1
−1.5
−1 −2
0
5
10
15
20
25
0
5
15
20
25
Fig. 10. Estimate error x~ 2 ðtÞ with T ¼ 2 s.
Fig. 7. x1 tracking performance with T ¼ 2 s. (solid, x1 ðtÞ; dashed, x^ 1 ðtÞ; dashdotted, xd1 ðt TÞ).
state, and the proposed control strategy can cancel the time-delay effect in the closed loop system. Figs. 7–10 are simulation results when T ¼2 s. From these figures, we can see that the control performance is also very well. So we can conclude that the proposed control strategy is a timedelay independent method. No matter how large the time-delay is, the observer can estimate the actual state and both of them track the desired xd ðt TÞ and remain stable.
7 6 5 4 3 2
7. Conclusions
1 0 −1 −2
10 time(sec)
time(sec)
0
5
10
15
20
25
time(sec) Fig. 8. x2 tracking performance with T ¼ 2 s. (solid, x2 ðtÞ; dashed, x^ 2 ðtÞ; dashdotted, xd2 ðt TÞ).
In this paper, we have proposed a neural network observer based predictive control strategy for a class of nonlinear systems with time delays caused by communication channels. The strategy solves the problem perfectly that not all state variables could be completely obtained. The Smith predictor could compensate the network induced time delay properly. A rigorous stability analysis has been performed based on the Lyapunov method, which shows that the tracking errors are uniformly ultimately bounded. The numerical simulations have validated the effectiveness of the proposed strategy.
−3
20
x 10
Acknowledgments This work was supported in part by the Natural Science Foundation of Hebei Province under Grants F2011203110 and F2013203186, in part by the Hebei Province Hundred Excellent Innovation Talents Support Program, in part by the Doctoral Fund of Ministry of Education of China under Grant 20121333110008, in part by the Hebei Province Applied Basis Research Project under Grant 13961806D, and in part by the National Natural Science Foundation of China under Grants 61290322, 61273222, and 61322303.
error betweem x1 and x1p
15
10
5
References 0
−5
0
5
10
15 time(sec)
Fig. 9. Estimate error x~ 1 ðtÞ with T ¼2 s.
20
25
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Changchun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a Research Fellow in National University of Singapore from 2006 to 2007. From 2007 to 2009, he worked in Carleton University, Canada, funded by Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2011, he worked in University of Duisburg-Essen, Germany, funded by Alexander von Humboldt Foundation. Now he is a Full Professor in Yanshan University, China. He is the author or coauthor of more than 110 papers in mathematical, technical journals, and conferences. He has been involved in more than 10 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. His research interests are in nonlinear control systems, control systems design over network, teleoperation systems and intelligent control.
Caixia Yu received her B.Sc. degree in electrical engineering from Yan-shan University, Qinhuangdao, China, in 2012. Her research interests are in nonlinear networked system control and cold rolling mill control.
Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from Harbin Institute of Technology, in 1986, 1991, and 1999, respectively. He is with the Department of Automation, Shanghai Jiao Tong University. He is the (co)author of more than 200 papers in mathematical, technical journals, and conferences. As (a)an (co)-investigator, he has finished more than 20 projects supported by National Natural Science Foundation of China (NSFC), the National Education Committee Foundation of China, and other important foundations. He is a Cheung Kong Scholars Programme Special appointment Professor. His current research interests include networked control systems, robust control and intelligent control for complex systems and their applications. Dr. Guan is serving as a Reviewer of Mathematic Review of America, a Member of the Council of Chinese Artificial Intelligence Committee, and Chairman of Automation Society of Hebei Province, China.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Neural network observer-based networked control for a class of nonlinear systems Changchun Hua, Caixia Yu, Xinping Guan Institute of Electrical Engineering, Yanshan University, Qinhuangdao City 066004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 June 2013 Received in revised form 1 November 2013 Accepted 1 November 2013 Communicated by Z. Wang Available online 30 January 2014
A new neural network observer-based networked control structure for a class of nonlinear systems is developed and analyzed. The structure is divided into three parts: local linearized subsystem, communication channels and remote predictive controller. A neural-network-based adaptive observer is presented to approximate the state of the time-delay-free nonlinear system. The neural-network (NN) weights are tuned on-line and no exact knowledge of nonlinearities is required. The time delays considered in the forward and backward communication channels are constant and equal. A modified Smith predictor is proposed to compensate the time delays. The controller is designed based on the developed NN observer and the proposed Smith predictor. By using the Lyapunov theory, rigorous stability proofs for the closed-loop system are presented. Finally, simulations are performed and the results show the effectiveness of the proposed control strategy. & 2014 Elsevier B.V. All rights reserved.
Keywords: Networked control system Time-delay Neural network observer Smith predictor
1. Introduction Networked control system (NCS) is such a system that a remote controller communicates with a plant through two independent communication channels. NCSs have been used in a wide range of areas because of their advantages in practical applications such as reduced system wiring, eased maintenance and diagnosis, and increased flexibility [1]. The insertion of communication network also brings many challenging problems. The key one is the network-induced delay which may make the system unstable or demonstrate undesired performance. The network delay can be modeled as a constant delay, an independent random delay, and a delay with known probability distribution [2]. Many works have been made to solve these delay problems [3,4]. Different mathematical-, heuristic-, and statisticalbased approaches are taken for different delay compensation [5]. A gain scheduler middleware (GSM) was proposed in [6] to alleviate the network delay effect. A new control scheme consisting of a control prediction generator and a network delay compensator was developed in [7]. A time-delay-compensation method based on the concept of network disturbance and communication disturbance observer has been proposed in [8]. Due to its no use of delay-time model, the method can be flexibly applied to many kinds of timedelayed control systems. Some methods consider NCS as a classical control system with slowly changing delay times and adopt wellknown control methods such as the Smith predictor [8,9]. The queuing/buffering method turns the NCS into a time-invariant
E-mail address: [email protected] (C. Hua). 0925-2312/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.11.026
system to alleviate the delays. The constant network delay is considered in this paper. To the best of our knowledge, most of the results about NCSs are based on linear systems [10,11], and only a few literatures focus on the nonlinear ones. Besides, most literatures consider the discrete-time systems [12]. In this paper, we consider a nonlinear continuous-time system which has uncertain nonlinear terms and unknown external disturbance. This type of system is closer to the practical one and owns a far worthier study. In order to deal with the nonlinear terms, the neural network is proposed and widely used due to its versatile features such as learning capability mapping and parallel processing. A slidingmode neural-network (SMNN) control system for the tracking control of robot manipulators to achieve high-precision position control was investigated in [13]. Two neural network-based controllers were designed for the teleoperation system in free motion in [14]. One is a new adaptive controller using the acceleration signal, another without the acceleration signal. In this paper, we employ the NNs to estimate the uncertainties. In most practical situations, the velocity signals are always difficult to measure. Hence, an observer is imperative to estimate the state signals when we design the controller. A dynamic neuralnetwork-based adaptive observer for a class of nonlinear systems was presented in [15], which does not require exact knowledge of nonlinearities. However, the output error equation in [15] is strictly positive real (SPR) which is a restrictive assumption for nonlinear systems. Therefore, paper [16] proposed a new observer without the SPR condition by introducing a vector b0. Besides, more observer design methods have been proposed in [17–20]. In this paper, we consider the networked control design problem for a class of nonlinear systems with unknown nonlinear
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functions. The state variables of system could not be completely obtained. A NN observer is designed to estimate the unmeasurable state. The Smith predictor is employed to deal with the network induced time delay. Based on the constructed observer and the Smith predictor, we design the nonlinear controller to render that the system output follows the given reference signal. By constructing Lyapunov function, the stability of the closed-loop system is proved. Finally, simulations are performed to show the effectiveness of the proposed strategy. This paper is organized as follows. The nonlinear system and the problem we discussed are presented in Section 2. Preliminaries about neural network are introduced in Section 3. A neural network observer and its stability analysis are presented in Section 4. In Section 5, we introduce the whole structure of the control strategy and design a neural network predictive controller. Finally, simulation results are presented in Section 6.
2. Background and problem formulation In this paper, we discuss a class of single-input single-output (SISO) nonlinear systems with networked time delay as follows: 8 x_ 1 ðtÞ ¼ x2 ðtÞ > > > > _ > > < x 2 ðtÞ ¼ x3 ðtÞ ⋮ ð1Þ > > > > x_ n ðtÞ ¼ f ðxðtÞÞ þ gðxðtÞÞuðt TÞ þ dðtÞ > > :y¼x 1
where xðtÞ ¼ ½x1 ðtÞ x2 ðtÞ ⋯ xn ðtÞT A Rn , uðtÞ A R, yðtÞ A R are the state, the control, and the output of the plant, respectively; d(t) is the unknown disturbance with a known upper bound bd; f(x) and g(x) are unknown smooth functions; T is the network induced time delay. The above equation can be expressed as ( x_ ¼ As x þ b½f 0 ðxÞ þ gðxÞuðt TÞ þ d ð2Þ y ¼ cT x where 2 6 6 6 As ¼ 6 6 6 4
0
3
1 ⋱
⋱ ⋱
⋱ 0
an
⋯
⋯
a2
7 7 7 7; 7 1 7 5 a1
2 3 0 6⋮7 6 7 6 7 7 b¼6 6 ⋮ 7; 6 7 405 1
2 3 1 607 6 7 6 7 7 c¼6 6⋮7 6 7 4⋮5 0
ð3Þ
and f 0 ðxÞ ¼ f ðxÞ þ∑ni¼ 1 an i þ 1 xi . The parameters ai are suitably chosen such that As is stable. Given a symmetric positive definite matrix Q, there exists a symmetric positive definite matrix Ps satisfying the following Lyapunov equation: ATs P s þ P s As
¼ Q
ð4Þ
In this work, we assume that there is a constant time delay T in the input channel and the feedback channel, and only the output signal y can be measured. Therefore, the first thing we do is to design a nonlinear observer to estimate the state which will be used in the design of the control input u(t). Due to the constant time delay, the control input u(t) can only be applied to the plant after a delay of T. In order to eliminate the influence of the time delay, we employ a modified Smith predictor which confirms the accurate output tracking of the system. The control objective can be described as follows: given a desired state xd(t), design a neural network observer and a control u(t) such that the observer state estimates the actual state exactly and both of them follow the delayed desired trajectory with an acceptable accuracy.
The desired trajectory vector is defined as 1Þ xd ðtÞ ¼ ½yd ðtÞ y_ d ðtÞ … yðn ðtÞT d
ð5Þ
For this purpose, we make some mild assumptions as follows: Assumption 1. The desired trajectory vector xd(t) is continuous and available for measurement, and xd(t) is bounded. Assumption 2. The time delay T is a known constant, and the time delay in the input channel is equal to that in the feedback one. Assumption 3. There exist constant g 0 4 0, and known smooth function gd(x) such that g d ðxÞ ZjgðxÞj Zg 0 . Remark 1. Assumption 3 implies that the smooth function g(x) is strictly either positive or negative. It is reasonable because being away from zero is a controllable condition of system (1), which is made in most control schemes. For a given practical system, the upper bound of g(x) is not difficult to determine by choosing gd(x) large enough. Besides, the low bound g0 is only required for analytical purpose, so its true value is not necessarily known. Definition 1. We say that the solution of a dynamic system is uniformly ultimately bounded (UUB) if for a compact set U of Rn and for all xðt 0 Þ ¼ x0 A U there exists an ɛ 4 0 and a number Tðɛ; x0 Þ such that J xðtÞ J o ɛ for all t Z t 0 þ T. Remark 2. Neural network predictive control strategies for nonlinear dynamic systems and telerobot are presented in [21] and [22], respectively. The systems considered are simpler than that we discuss in this paper. What is more, all the state variables are required to be available in the literatures, which is not the case in most practical systems. In this paper, we discuss the situation that not all states can be measured. In this paper, we denote J J as the Euclidean norm, and ‖ ‖F as the Frobenius norm.
3. Preliminaries In the control engineering, RBF neural network is usually used as a tool for modeling nonlinear functions because of their good capabilities in function approximation. The RBF neural network can be considered as a two-layer network in which the hidden layer performs a fixed nonlinear transformation with no adjustable parameters, i.e., the input space is mapped into a new space. The output layer then combines the outputs in the latter space linearly. Therefore, they belong to a class of linearly parameterized networks. In this paper, the following RBF neural network is used to approximate the continuous function f ðxðtÞÞ : Rq -R: ^ T ΦðxðtÞÞ; f^ ðxðtÞÞ ¼ W
ð6Þ q
where the input vector x A Q x R , and q is the neural network ^ ¼ ½W ^ 1; W ^ 2 ; …; W ^ T A Rl , the NN input dimension. Weight vector W l node number l 4 1, and ΦðxðtÞÞ ¼ ½Φ1 ðxðtÞÞ; …; Φl ðxðtÞÞT , with Φi ðxðtÞÞ chosen as the commonly used Gaussian function, which is in the following form: " # ðxðtÞ μi ÞT ðxðtÞ μi Þ Φi ðxðtÞÞ ¼ exp ; ð7Þ 2η2i where μi ¼ ½μi1 ; μi2 ; …; μiq is the center of the receptive field and ηi is the width of the Gaussian function. It has been proven that the neural network can approximate any continuous function over a compact set Ωx Rq to arbitrary accuracy as f ðxðtÞÞ ¼ W T ΦðxðtÞÞ þ ɛðxÞ;
8 x A Ωx ;
ð8Þ
C. Hua et al. / Neurocomputing 133 (2014) 103–110
where W is the ideal constant weight, ɛ is the approximation error, and both are bounded, i.e. ‖W‖F rW M ; ɛ r ɛM . The ideal weight vector W is an artificial quantity required for ^ that minimizes analytical purposes. W is defined as the value of W J ɛðxÞ J for all xðtÞ A Ωx in a compact region, i.e. T
^ ΦðxðtÞÞjg; W≔arg min fsupjf ðxðtÞÞ W ^ A Rl W
x A Ωx :
ð9Þ
Neural network approximation idea has been applied to the controller design for uncertain nonlinear systems extensively [23]. In [24–26], neural network state feedback controllers were constructed for time delay nonlinear systems. In [27], the deterministic learning mechanism and the neural learning control scheme were presented. For nonlinear MIMO systems, neural network controllers were constructed in [28]. With the control direction unknown, Du et al. [29] designed the neural network controller for nonlinear system with triangular structure. Observer-based output feedback controller was designed in [20]. In [30,31], decentralized state feedback controllers were constructed for interconnected nonlinear systems via adaptive neural network. In this paper, we will use the neural networks to approximate unknown nonlinear functions and construct the predictive controller. The corresponding neural-network-based adaptive observer will be proposed in Section 4.
105
(SPR) condition, we choose a polynomial L(s) so that ~ T ϕ þ φ þ ɛ 1 þ ς þ v1 y~ ¼ HðsÞLðsÞ½W 1 1 1 1 ~ T ϕ þ φ þ ɛ 2 Þu þ ς þ v 2 þ d þ ðW 2 2 2 2
ð15Þ
where HðsÞLðsÞ is SPR. The ‘overbar’ indicates the signal filtered by L 1 ðsÞ. And the terms ς1 and ς2 are defined as ς1 ðtÞ ¼ L 1 ^ , ς ðtÞ ¼ L 1 ðsÞW ^ u with ^ ^ ~ Tϕ ~ T 1 ðsÞϕ ~ Tϕ ~ T 1 ðsÞϕ ðsÞW 2 1 2 1 1 W 1L 2 2u W 2L ~ J ς J r li ‖W ‖F ; i ¼ 1; 2. i
Then the state-space realization of (15) can be expressed as 8 < z~_ ¼ A z~ þ b ½W ~ T ϕ þ φ þ ɛ 1 þ ς þ v 1 þ ðW ~ T ϕ þ φ þ ɛ 2 Þu þ ς þ v 2 þ d c c 1 2 1 2 1 1 2 2 : y~ ¼ cTc z~
ð16Þ 1
with HðsÞLðsÞ ¼ cTc ðsI Ac Þ bc and cc ¼ ½1 0 ⋯ 0T . Since HðsÞLðsÞ is SPR, for Q ¼ Q T 4 0, there exists P ¼ P T 4 0 such that ATc P þ PAc ¼ Q ; Pbc ¼ cc
ð17Þ
Lemma 1. Given Assumptions 1–3, consider the nonlinear system (2) and the NN observer given by (12). Design the NN weight adaptation laws as ^ 1 ^_ 1 ¼ F 1 ϕ ðxÞ ^ y~ K 1 F 1 jyj ~ W W 1
4. Neural network observer without time-delay
^ 2 ^_ 2 ¼ F 2 ϕ ðxÞ ^ yu ~ K 2 F 2 jyj ~ W W 2
ð18Þ
According to the approximation property of NNs, we can assume that the continuous nonlinear functions in the system (2) can be represented by
where Fi are the positive definite matrices, and Ki are the positive parameters, i¼1,2. Let the robustifying terms be
f 0 ðxÞ ¼ W T1 ϕ1 ðxÞ þ ɛ1 ðxÞ
~ ~ v1 ðtÞ ¼ D1 yðtÞ=j yðtÞj;
gðxÞ ¼ W T2 ϕ2 ðxÞ þ ɛ2 ðxÞ
ð10Þ
where Wi is the ideal weight, ϕi ðxÞ is the basis function and ɛ i ðxÞ is the approximation error, i¼ 1,2. So the system (2) without time-delay can be written as ( x_ ¼ As x þb½W T1 ϕ1 ðxÞ þ ɛ 1 ðxÞ þ ðW T2 ϕ2 ðxÞ þ ɛ 2 ðxÞÞu þ d ð11Þ y ¼ cT x
^ T ϕ ðxÞ ^ T ϕ ðxÞu ^ ^ þW ^ v1 ðtÞ v2 ðtÞ þ Kðy cT xÞ x^_ ¼ As x^ þ b½W 1 1 2 2 ^y ¼ cT x^
ð12Þ
^ T ϕ ðxÞ ^ T ϕ ðxÞ ^ and W ^ represent the estimation of f 0 ðxÞ and where W 1 1 2 2 g(x) respectively, K ¼ ½K 1 ; K 2 ; …; K n T is the observer gain vector, chosen so that the characteristic polynomial of As KcT is strictly Hurwitz. And v1 ðtÞ and v2 ðtÞ are robustifying terms that provide robustness in the face of bounded disturbances which will be designed later. Define the state and observer output errors as x~ ¼ x x^ and y~ ¼ y y^ respectively. Subtracting (12) from (11) yields ( ^ ~ Tϕ ~ T^ x~_ ¼ ðAs KcT Þx~ þ b½W 1 1 þ φ1 þ ɛ 1 þ ðW 2 ϕ 2 þ φ2 þ ɛ 2 Þu þ dþ v1 þ v2 y~ ¼ cT x~ ð13Þ ^ and ϕ ^ are the abbreviations of ϕ ðxÞ ~ i ¼ Wi W ^ i, ϕ ^ where W and 1 2 1 ^ ^ ϕ2 ðxÞ. φi ¼ W Ti ½ϕi ðxÞ ϕi ðxÞ, and J φi ðtÞ J r φdi with φdi 40; i ¼ 1; 2. Then the error dynamics may be written as ^ ~ Tϕ ~ T^ y~ ¼ HðsÞ½W 1 1 þ φ1 þɛ 1 þ ðW 2 ϕ 2 þ φ2 þ ɛ 2 Þu þ d þ v1 þ v2
ð19Þ
where D1 Z φd1 þ ɛ M1 is a constant scalar, D2 ðtÞ Zðφd2 þɛ M2 ÞuðtÞ is a ~ function. Then the state estimation error xðtÞ and the neural network ~ 1 ðtÞ and W ~ 2 ðtÞ are uniformly ultimately weight estimation errors W bounded (UUB). Proof. Define the Lyapunov function as
Now, we design the observer as (
~ ~ v2 ðtÞ ¼ D2 ðtÞyðtÞ=j yðtÞj:
ð14Þ
where s denotes the differential operator d=dt, and H(s) is a linear transfer function with stable poles and is realized by ðAs KcT ; b; cÞ. In order to make the system (14) satisfy the strictly positive real
1 T 1 n ~ T 1 ~ o 1 n ~ T 1 ~ o V ¼ z~ P z~ þ tr W 1 F 1 W 1 þ tr W 2 F 2 W 2 : 2 2 2
ð20Þ
The time derivative of V is given by T T ~ T ϕ þ φ þ ɛ 1 þ ς þ v 1 þ ðW ~ Tϕ V_ ¼ 12 z~ ðATc P þ PAc Þz~ þ z~ Pbc ½W 1 1 1 1 2 2
~_ 1 g þ trfW ~_ 2 g: ~ T F 1W ~ T F 1W þ φ 2 þ ɛ 2 Þu þ ς2 þ v 2 þd þ trfW 1 1 2 2 ð21Þ
Using (17), (18) and (19), we obtain 1 T V_ ¼ z~ Q z~ þ y~ T ðɛ 1 þɛ 2 u þd þ ς1 þ ς2 þ φ 1 þ v 1 þ φ 2 u þ v 2 Þ 2 n T o n T o ^ 1 þK 2 jyjtr ^ 2 ~ W ~ W ~ ~ þK 1 jyjtr W W 1
2
1 ~ 1 ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ‖F α2 =2Þ2 ~ þ K 1 ð‖W ~ λ ðQ Þjyj r jyj 2 min ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ
ð22Þ
with lM the maximum singular value of L 1 ðsÞ and αi ¼ W Mi þ li =K i ; i ¼ 1; 2.
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C. Hua et al. / Neurocomputing 133 (2014) 103–110
Thus, V_ is negative outside the region: 9 8 4lM bd þ K 1 α21 þ K 2 α22 > > jyj > > ~ o ; > > > > > > 2λmin ðQ Þ > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > > 2 2 = < 4lM bd þ K 1 α1 þK 2 α2 > 1 ~ ~ ~ ; ‖ W ‖ o α þ ~ W 1; W 2 1 F 1 Ω1 ¼ y; 2 4K 1 > > > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > > > 2 2 > > 4l b þ K α þK α 1 M 1 2 > d 1 2 > ~ 2 ‖F o α2 þ > > ‖ W : > > ; : 2 4K 2
nonlinear system in the first. To locally compensate the nonlinear part of the system, let the local control input be given by uðtÞ ¼ ð23Þ
Therefore, V_ is negative outside the compact set Ω1, which is thus shown to be an attractive set for the system. According to a standard Lyapunov theorem extension, this demonstrates the UUB ~ 1 and W ~ 2. □ ~ W of both y, Remark 3. From (23) we can see that the bound on the approx~ decreases with the value of Ki. On the other hand, imation error jyj ~ 1 ‖F decreases with the increase of K1, the the NN weight error ‖W ~ 1 ‖F is opposite. So the parameters Ki offer a decrease of K2, and ‖W ~ design tradeoff between the relative eventual magnitudes of jyj ~ i ‖F , i¼ 1,2. and ‖W 5. The observer based networked predictive control 5.1. Control structure The whole structure of the networked control system is shown in Fig. 1, which is composed of three parts: remote predictive controller, communication channels and local linearized subsystem. As shown in Fig. 1, the communication channels consist of two time delays. One is in the input channel and the other in the feedback one. In this paper, we use the Smith predictor principle to deal with the time delays. To the best of our knowledge, the Smith predictor is usually based on a linear model, so the linearization of the nonlinear plant is necessary. While the feedback signal to the controller is the observer state, the linearization should aim at the observer. Here, we use a local feedback, called the local nonlinear compensator, to cancel the nonlinear effects. We use the Smith predictor theory in remote predictive controller part to deal with the time delays. If the linear predictor is an accurate model of the local linearized subsystem, then, referring to Fig. 1, ^ TÞ and zðtÞ ¼ xðt ^ þTÞ. This means that the observer zðt 2TÞ ¼ xðt ^ state xðtÞ ahead of time T can be predicted via the predictor's output z(t). Therefore, we compensate the time-delay. 5.2. Smith predictor control
1 ^ T ϕ ðxÞ ^ W 2 2
T
^ ϕ ðxÞ ^ ½τðt TÞ W 1 1
ð24Þ
where τðt TÞ stands for a new control variable from the output of the remote controller with time-delay T in the input channel which will be designed later. Then the local state observer (12) becomes a linear model: ^ þ b½τðt TÞ v1 ðtÞ v2 ðtÞ þ K½yðtÞ yðtÞ ^ x^_ ðtÞ ¼ As xðtÞ
ð25Þ
In the following, we will find a control action τðtÞ such that the ^ state xðtÞ follows the delayed desired trajectory xd ðt TÞ with an acceptable accuracy. Let the linear predictor be z_ ðtÞ ¼ As zðtÞ þ bτðtÞ
ð26Þ
We can use the state of the predictor z(t) as a state feedback to design the remote controller. However, due to the possible pre^ TÞ þ zðtÞ sence of disturbances and modeling errors, we use xðt zðt 2TÞ as the feedback signal to incorporate the Smith predictor loop in Fig. 1. Define an error vector as ^ TÞ zðtÞ þzðt 2TÞ δðtÞ ¼ xd ðtÞ xðt
ð27Þ
Define a filtered error as h T i T rðtÞ ¼ Λ δðtÞ ¼ Λ 1 δðtÞ
ð28Þ
where Λ ¼ ½λ1 λ2 ⋯ λn 1 T is an appropriately chosen coefficient vector such that δ-0 exponentially as r-0, i.e., sn 1 þ λn 1 sn 2 þ ⋯ þ λ1 is Hurwitz. Differentiating r(t) and using (26) and (27), the filtered error dynamics may be written as T T r_ ðtÞ ¼ Λ δ_ ðtÞ ¼ Λ ½x_ d ðtÞ x^_ ðt TÞ z_ ðtÞ þ z_ ðt 2TÞ
^ TÞ þbτðt 2TÞ bv1 ðt TÞ bv2 ðt TÞ ¼ Λ fx_ d ðtÞ fAs xðt ^ TÞg As zðtÞ bτðtÞ þ As zðt 2TÞ þ bτðt 2TÞg þ K½yðt TÞ yðt T
^ TÞ þ zðtÞ zðt 2TÞ þ bv1 ðt TÞ ¼ Λ fx_ d ðtÞ As ½xðt ^ TÞg τðtÞ þ bv2 ðt TÞ K½yðt TÞ yðt T
ð29Þ
Now define controller as ^ TÞ þ zðtÞ zðt 2TÞ τðtÞ ¼ ΛT fx_ d ðtÞ As ½xðt ^ TÞg þ K v rðtÞ K½yðt TÞ yðt
~ TÞ þ K v rðtÞ ¼ Λ fx_ d ðtÞ As xd ðtÞ þ As δðtÞg K yðt T
ð30Þ
where Kv is a positive parameter. Then, (29) becomes
In this section, we adopt the Smith predictor to solve the timedelay problem. As we all know, the Smith predictor is used in the linear system. So, we must cancel the nonlinear terms of the
r_ ðtÞ ¼ K v rðtÞ þ v1 ðt TÞ þ v2 ðt TÞ
Fig. 1. The networked control strategy for nonlinear system.
ð31Þ
C. Hua et al. / Neurocomputing 133 (2014) 103–110
From (27) and (28), we can get
c5 2 c5 kc6 2 kc6 2 r ðtÞ r 2 ðt 2TÞ þ y~ ðtÞ y~ ðt 3TÞ 2 2 2 2 1 ~ 1 ðtÞ‖F ~ ~ λ ðQ ÞjyðtÞj þ K 1 ð‖W jyðtÞj 2 min ~ 2 ðtÞ‖F α2 =2Þ2 α1 =2Þ2 þ K 2 ð‖W 1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þ K 1 α21 =4 þ K 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þ K 1 ð‖W ðlM bd þ K 1 α21 =4 þ K 2 α22 =4Þ
107
þ
J δðtÞ J r jrðtÞj=Λm þ c0
ð32Þ
where Λm ¼ minfλ1 ; λ2 ; …; λn 1 ; 1g, c0 is a positive constant scalar. From (24) and (30), we can get ~ 2TÞj juðtÞj rc1 þ c2 jrðt TÞj þ c3 jyðt
ð33Þ
where c1 ; c2 ; c3 are positive parameters. So we can choose ~ 2TÞjÞ jD2 ðtÞj ¼ ðφd2 þ ɛM2 Þðc1 þ c2 jrðt TÞj þ c3 jyðt ~ 2TÞj ¼ c4 þc5 jrðt TÞj þc6 jyðt
ð34Þ
with c4 ¼ c1 ðφd2 þ ɛ M2 Þ; c5 ¼ c2 ðφd2 þ ɛM2 Þ; c6 ¼ c3 ðφd2 þ ɛ M2 Þ. Theorem 1. Given Assumptions 1–3, consider the nonlinear system (2) with input and feedback time delays. Take the control input as (24) and (30). Let the NN weights tuning be provided by (18) and the robustifying term be provided by (19). Then the filtered error r(t), the estimation error ~ 1 ðtÞ; W ~ 2 ðtÞ are UUB. ~ yðtÞ and the weight tuning error W Proof. Define the Lyapunov function as Z Z 1 c5 t kc6 t V ¼ r 2 ðtÞ þ y~ 2 ðtÞ dt r 2 ðtÞ dt þ 2 2 t 2T 2 t3 T o 1 n T o 1 T 1 n ~ T 1 ~ 1 ~ ~ þ z~ ðtÞP z~ ðtÞ þ tr W 1 ðtÞF 1 W 1 ðtÞ þ tr W 2 ðtÞF 2 W 2 ðtÞ 2 2 2 o 1 T 1 n ~ T 1 ~ 1 ðt TÞ þ z~ ðt TÞP z~ ðt TÞ þ tr W 1 ðt TÞF 1 W 2 2 o 1 n ~ T 1 ~ þ tr W ðt TÞF ð35Þ W 2 ðt TÞ 2 2 2 with k a positive parameter. Using the results of Section 4 and the equality (31), we have ~ TÞ ~ TÞ yðt yðt rðtÞD2 ðt TÞ V_ r K v r 2 ðtÞ rðtÞD1 ~ TÞj ~ TÞj jyðt jyðt c5 2 c5 2 kc6 2 kc6 2 y~ ðtÞ y~ ðt 3TÞ þ r ðtÞ r ðt 2TÞ þ 2 2 2 2 1 ~ 1 ðtÞ‖F α1 =2Þ2 ~ ~ λ ðQ ÞjyðtÞj þK 1 ð‖W jyðtÞj 2 min ~ 2 ðtÞ‖F α2 =2Þ2 þ K 2 ð‖W 1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þ K 1 α21 =4 þK 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þ K 1 ð‖W ðlM bd þ K 1 α21 =4 þK 2 α22 =4Þ ð36Þ
ð37Þ
It is easy to get 1 1 jrðtÞ J rðt 2TÞj r jrðtÞj2 þ jrðt 2TÞj2 2 2 1 k ~ 3TÞj2 ~ jrðtÞ J yðt 3TÞj r jrðtÞj2 þ jyðt 2k 2 So
ð38Þ
i c6 K v c5 jrðtÞj ðD1 þ c4 Þ 2k 1 kc ~ ~ jyðtÞj λmin ðQ Þ 6 jyðtÞj 2 2 ~ 1 ðtÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðtÞ‖F α2 =2Þ2 þK 1 ð‖W
1 ~ TÞj λmin ðQ Þjyðt ~ TÞj ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ jyðt 2 ~ 1 ðt TÞ‖F α1 =2Þ2 þ K 2 ð‖W ~ 2 ðt TÞ‖F α2 =2Þ2 þK 1 ð‖W ðlM bd þK 1 α21 =4 þ K 2 α22 =4Þ ð39Þ
V_ r jrðtÞj
h
Choose k o λmin ðQ Þ=c6 , and K v 4 c5 þ c6 =2k. Then V_ is negative outside the region: 9 8 D1 þ c 4 > > jrðtÞj o > > ; > > > > K c c =2k > > v 5 6 > > > > > > 2 2 > > > > 4l b þ K α þ K α M d 1 1 2 2 > > > > ~ jyðÞj o ; > > > > > > 2 λ ðQ Þ 2kc 6 min = < ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s ~ ~ ~ Ω ¼ r; y; W 1 ; W 2 ð40Þ 2 2 4lM bd þ K 1 α1 þ K 2 α2 > 1 ~ > > ;> ‖W 1 ðÞ‖F o α1 þ > > > > > > 2 4K 1 > > > > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > 2 2 > > > > 4l b þ K α þ K α 1 M 1 2 d > > 1 2 > ‖W ~ > > ðÞ‖ o α þ : > > F 2 2 ; : 2 4K 2
With (34), one has V_ r K v r 2 ðtÞ þ D1 jrðtÞj þ c4 jrðtÞj ~ 3TÞj þ c5 jrðtÞ J rðt 2TÞj þ c6 jrðtÞ J yðt
Since V is a positive definite function, and V_ is negative outside ~ 1 ðtÞ and W ~ 2 ðtÞ are UUB. ~ the region Ω, we conclude that r(t), yðtÞ, W
Fig. 2. Overall networked control structure for nonlinear system with time-delay.
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C. Hua et al. / Neurocomputing 133 (2014) 103–110
The overall networked control structure for nonlinear system with time-delay is shown in Fig. 2. □ Remark 4. Since, from Theorem 1, r(t) is UUB, then δðtÞ is UUB ^ TÞ zðtÞ þ z by appropriately choosing Λ. Since δðtÞ ¼ xd ðtÞ xðt ^ TÞ þ zðtÞ zðt 2TÞ is UUB. ðt 2TÞ and xd(t) is bounded, then xðt What is more, from Theorem 1, the estimate error y~ is UUB. Therefore, by definition (30) τðtÞ is UUB for a stable As and a bounded x_ d .
7 6 5 4 3 2 1
6. Simulations In this section, a simulation example is given to verify the effectiveness of the main result. We consider a single-link robot whose dynamics is given by
−1 −2
y¼q
ð41Þ −3
where q is the angle, u is the input torque, M, g, m, l are the moment of inertia, the gravity acceleration, the mass and length of the link respectively, d is the external disturbance. Letting x1 ¼ q, _ the state-space description of system is x2 ¼ q, 8 1 0 > < x_ ¼ 0 xþ ½f ðxÞ þgðxÞu þ d ð42Þ 1 2 1 0 > : y ¼ x1 where f 0 ðxÞ ¼ 12mgl sin x1 =M ð 1 2Þx; gðxÞ ¼ 1=M, and the disturbance d ¼ 0:5 sin ð0:5π tÞ. The parameters of robot are M ¼ 0:5, l ¼ 0:4, m ¼ 0:4, g ¼ 9:8: We select the linear filter L 1 ðsÞ ¼ 1=ðs þ 3Þ. In simulation, the initial conditions of the system and the parameters are xð0Þ ¼ ^ xð0Þ ¼ zð0Þ ¼ ½0 0:5T , K ¼ ½400 800T , F 1 ¼ diag½5 104 , F 2 ¼ diag T ½5 103 , K 1 ¼ K 2 ¼ 0:01, K V ¼ 200, Λ ¼ ½0:2 1. The desired traT jectory is xd ¼ ½ sin ðtÞ cos ðtÞ . When the time-delay T ¼1 s, the simulation results are shown from Figs. 3 to 6. Fig. 3 shows the tracking performance of the state x1, from which we can see that the observer state x^ 1 and the actual state x1 are almost completely coincidental, and both of them track with the desired state xd1 ðt 1Þ at about 7 s. Fig. 5 shows the observer estimated error x~ 1 , and it goes to about zero very quickly. Figs. 4 and 6 show the same performance. From these figures, we can see that the observer can accurately estimate the actual
0
5
10
15
20
25
time(sec) Fig. 4. x2 tracking performance with T ¼1 s. (solid, x2 ðtÞ; dashed, x^ 2 ðtÞ; dashdotted, xd2 ðt TÞ). −3
20
x 10
15 error betweem x1 and x1p
M q€ þ 12 mgl sin q ¼ u þ d;
0
10
5
0
−5
0
5
10
15
20
25
20
25
time(sec) Fig. 5. Estimate error x~ 1 ðtÞ with T ¼ 1 s.
2 7
1.5
6 5 error betweem x2 and x2p
1
0.5
0
−0.5
4 3 2 1 0 −1
−1
0
5
10
15
20
25
time(sec) Fig. 3. x1 tracking performance with T ¼1 s. (solid, x1 ðtÞ; dashed, x^ 1 ðtÞ; dashdotted, xd1 ðt TÞ).
−2
0
5
10
15 time(sec)
Fig. 6. Estimate error x~ 2 ðtÞ with T ¼ 1 s.
C. Hua et al. / Neurocomputing 133 (2014) 103–110
109
7
2
6
1.5
error betweem x2 and x2p
5
1
0.5
0
−0.5
4 3 2 1 0
−1
−1.5
−1 −2
0
5
10
15
20
25
0
5
15
20
25
Fig. 10. Estimate error x~ 2 ðtÞ with T ¼ 2 s.
Fig. 7. x1 tracking performance with T ¼ 2 s. (solid, x1 ðtÞ; dashed, x^ 1 ðtÞ; dashdotted, xd1 ðt TÞ).
state, and the proposed control strategy can cancel the time-delay effect in the closed loop system. Figs. 7–10 are simulation results when T ¼2 s. From these figures, we can see that the control performance is also very well. So we can conclude that the proposed control strategy is a timedelay independent method. No matter how large the time-delay is, the observer can estimate the actual state and both of them track the desired xd ðt TÞ and remain stable.
7 6 5 4 3 2
7. Conclusions
1 0 −1 −2
10 time(sec)
time(sec)
0
5
10
15
20
25
time(sec) Fig. 8. x2 tracking performance with T ¼ 2 s. (solid, x2 ðtÞ; dashed, x^ 2 ðtÞ; dashdotted, xd2 ðt TÞ).
In this paper, we have proposed a neural network observer based predictive control strategy for a class of nonlinear systems with time delays caused by communication channels. The strategy solves the problem perfectly that not all state variables could be completely obtained. The Smith predictor could compensate the network induced time delay properly. A rigorous stability analysis has been performed based on the Lyapunov method, which shows that the tracking errors are uniformly ultimately bounded. The numerical simulations have validated the effectiveness of the proposed strategy.
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Acknowledgments This work was supported in part by the Natural Science Foundation of Hebei Province under Grants F2011203110 and F2013203186, in part by the Hebei Province Hundred Excellent Innovation Talents Support Program, in part by the Doctoral Fund of Ministry of Education of China under Grant 20121333110008, in part by the Hebei Province Applied Basis Research Project under Grant 13961806D, and in part by the National Natural Science Foundation of China under Grants 61290322, 61273222, and 61322303.
error betweem x1 and x1p
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Fig. 9. Estimate error x~ 1 ðtÞ with T ¼2 s.
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Changchun Hua received the Ph.D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 2005. He was a Research Fellow in National University of Singapore from 2006 to 2007. From 2007 to 2009, he worked in Carleton University, Canada, funded by Province of Ontario Ministry of Research and Innovation Program. From 2009 to 2011, he worked in University of Duisburg-Essen, Germany, funded by Alexander von Humboldt Foundation. Now he is a Full Professor in Yanshan University, China. He is the author or coauthor of more than 110 papers in mathematical, technical journals, and conferences. He has been involved in more than 10 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. His research interests are in nonlinear control systems, control systems design over network, teleoperation systems and intelligent control.
Caixia Yu received her B.Sc. degree in electrical engineering from Yan-shan University, Qinhuangdao, China, in 2012. Her research interests are in nonlinear networked system control and cold rolling mill control.
Xinping Guan received the B.S. degree in mathematics from Harbin Normal University, Harbin, China, and the M.S. degree in applied mathematics and the Ph.D. degree in electrical engineering, both from Harbin Institute of Technology, in 1986, 1991, and 1999, respectively. He is with the Department of Automation, Shanghai Jiao Tong University. He is the (co)author of more than 200 papers in mathematical, technical journals, and conferences. As (a)an (co)-investigator, he has finished more than 20 projects supported by National Natural Science Foundation of China (NSFC), the National Education Committee Foundation of China, and other important foundations. He is a Cheung Kong Scholars Programme Special appointment Professor. His current research interests include networked control systems, robust control and intelligent control for complex systems and their applications. Dr. Guan is serving as a Reviewer of Mathematic Review of America, a Member of the Council of Chinese Artificial Intelligence Committee, and Chairman of Automation Society of Hebei Province, China.