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Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint$ Wei He a, Yiting Dong b, Changyin Sun a,n a b
School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 July 2014 Received in revised form 13 December 2014 Accepted 26 May 2015 This paper was recommended for publication by Dr. Y. Chen
In this paper, we aim to solve the control problem of nonlinear affine systems, under the condition of the input deadzone and output constraint with the external unknown disturbance. To eliminate the effects of the input deadzone, a Radial Basis Function Neural Network (RBFNN) is introduced to compensate for the negative impact of input deadzone. Meanwhile, we design a barrier Lyapunov function to ensure that the output parameters are restricted. In support of the barrier Lyapunov method, we build an adaptive neural network controller based on state feedback and output feedback methods. The stability of the closed-loop system is proven via the Lyapunov method and the performance of the expected effects is verified in simulation. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Unknown nonlinear affine system Input deadzone Output constraint Radial Basis Function Neural Network (RBFNN) Barrier Lyapunov Function
1. Introduction From engineering point view, both input deadzone and output constraint are common static nonlinearities, which require to be tackled practical control systems. Neglecting the deadzone effects may lead to undesired performances such as excessive steady state error, poor transient response and large overshoot [1]. Moreover, performance degradation, hazards or system damage may result from violation of the constraints during operation. Therefore, it is necessary to take both deadzone effect and output constraint effect of the nonlinear systems into account. Recent years have witnessed a rapid development of neural network control techniques and successful applications in tracking control of nonlinear systems with external disturbance [2,3]. Many control approaches, published in various books [4,5], journals [6–10] and conference proceedings [11], show that artificial neural networks play an important role in function approximation and control design to deal with the nonlinearities of complex systems. In spite of having made significant achievements in neural network fields, due to the complexity of nonlinear systems, most works merely focus on one kind of nonlinearities, while a variety of nonlinearities are usually
☆ This work was supported by the National Natural Science Foundation of China under Grant 61203057 and 61125306, the National Basic Research Program of China (973 Program) under Grant 2014CB744206. n Corresponding author. E-mail address: [email protected] (C. Sun).
rarely considered together, especially taking the input deadzone and output constraint into account together. Numerous approaches have been proposed to address the control problems nonlinear affine systems with deadzone or constraint, many techniques have been developed, most of which [12,13] are based on the assumption that the deadzone function can be parameterized into a few values. However, the deadzone function is usually difficult to achieve, which leads these controllers are not practical and utility. Hence, numerous controllers have been proposed to address the control problems of solving unknown deadzone effect. A control method for a special nonlinear system with symmetrical deadzone is proposed in [14], whose deadzone slope and width are unknown. The design based on neural network is recently presented to solve the deadzone effect. In [15], an adaptive neural network control is proposed for single-master–multiple-slaves teleoperation in consideration of time delays and input deadzone uncertainties. To handle asymmetric and nonlinear deadzone function, a scheme based on Multi-layer NNs is proposed in [16] to compensate the deadzone effect, which increases the robustness of the close-loop system. These researches show the superiority and feasibility of applying neural network to compensate the deadzone effect. Meanwhile, some ideas of utilizing the RBFNNs in nonlinear systems have been come up with. A modified adaptive RBFNN for the compensation of deadzone is described in [17]. By constructing a deadzone compensator, a neural design control scheme based on RBFNN is developed by using backstepping design techniques in [18]. However, these proposed control design algorithms are simpler in both designing and applying
http://dx.doi.org/10.1016/j.isatra.2015.05.014 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
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than MNNs. Constructive and destructive parsimonious extreme learning machines are proposed in [19]. Compared with other structures, the RBFNN-based control usually consumes less calculation resources and achieves the desired trajectory faster. [20,21] also have proved that the RBFNN has the learning ability. Nonetheless, the approaches merely focus on approximating the errors of deadzone effect. To address the problem of output constraint, numerous existing works have demonstrated Barrier Lyapunov Function (BLF) that can guarantee the constraints satisfaction. BLF has a better property that can grow to infinity as some limits, which guarantee the boundedness in the closed-loop system. Based on Lyapunov's direct function, the stability of the system can be guaranteed. Barrier Lyapunov function is first proposed to tackle the problem of constraints in Brunovsky form [22] systems. In [23], the authors propose BLF-based control design for nonlinear systems with output constraint. In [24], BLF-based control is investigated in output constrained nonlinear systems with unknown constant control gain function. Meanwhile, BLF is widely used in flexible systems. The authors propose BLF-based adaptive control to suppress the vibration and guarantee the constraint satisfaction of a flexible string system in [25,26]. BLF is also proposed to suppress the undesirable vibration of a flexible crane system in [27] and a flexible beam in [28]. Nevertheless, it should be noted that a large number of researches on output constrained systems have assumed that there exist only constraint and disturbance in the nonlinear systems, and do not take other nonlinearities into account, such as input deadzone, and external disturbance. The NN control for the nonlinear system has received much attention [29–36]. As an effective method in a large number of nonlinear system fields, neural networks require relatively less information about the system dynamics. Numerous researches have proven that artificial neural networks can approximate a wide range of nonlinear functions accurately [37–41]. The authors propose a local network structure for nonlinear system trajectory tracking with uncertainties in [42]. A robust neural network output feedback scheme is developed for the motion control of robot manipulators in [43]. [44] studies the problem of learning from neural network control of a class of nonlinear non-affine systems in uncertain dynamic environments. [45] presents the problems of accurate identification and learning control of ocean surface ship using neural network. For a class of wheeled inverted pendulum models of vehicle systems, the authors present an adaptive NN-based control in [46]. In [47], the authors develop an output feedback adaptive neural network control to achieve stable dynamic balance and track of the desired trajectory. In [48–51], adaptive neural control is proposed for a class of uncertain nonlinear systems. The main aim of this paper is to develop an RBF-based neural network system to identify the nonlinear system parameters with deadzone and constraint and then to design an adaptive controller for deadzone compensation and control the system. To achieve this goal, two RBF-based neural networks are applied in the proposed approach. One is used for identifying parameters of input deadzone. External disturbance is also rejected with another RBF-based proposed control. Once the neural network's center and width factor of the basis functions are determined, the RBF neural network that is similar to a typically single layer neural network is trained based on the gradient decent technique [17]. Meanwhile, the output violation is prevented by utilizing a barrier Lyapunov function. The rest of this paper is organized as follows: we propose the problem formulation and necessary assumptions, use lemmas in Section 2. Section 3 illustrates the direct adaptive NN control design process under the state feedback and the simulation of the state. Output feedback control is investigated in Section 4. Section 5 reveals the validity and performance of proposed method by simulation. The last section introduces our paper.
2. Problem formulation and preliminaries 2.1. Problem formulation The unknown nonlinear affine system [52] is stated as 8 _ > < x i ¼ xi þ 1 ði ¼ 1; 2; …; n 1Þ x_ n ¼ f ðxÞ þ gðxÞu þ dðtÞ > :y¼x 1
ð1Þ
where x ¼ ½x1 ; x2 ; …; xn T A Rn is the state vector, and u A R is the control input to the nonlinear affine system (1) after deadzone and output constraint effect. The external disturbance to the above system is dðtÞ : R þ -R. The unknown function f ðxÞ A R, gðxÞ A R are sufficiently smooth. The deadzone is properly expressed as 8 > < hr ðvÞ; v Z br bl o v o br u ¼ 0; ð2Þ > : h ðvÞ; v r b l l where v is the desired input to deadzone, br and bl are unknown parameters of the deadzone and hr ðÞ and hl ðÞ are unknown functions of the deadzone, which is shown in Fig. 1. The control goal is to develop an adaptive NN controller for system (1) so that the output y is forced to trace the desired trajectory yd asymptotically, the tracking error e1 -0 as t-1, while the output y is guaranteed to be within certain predefined bounds for all time. The control objective will be achieved under the following assumptions. Assumption 1. The desired trajectory yd(t) and its derivatives yð1Þ ðtÞ, yð2Þ ðtÞ, …, ydðnÞ ðtÞ are smooth, bounded and known. d d Assumption 2. For the given system described in (1), the unknown affine term f(x), g(x) is smooth with respect to input u. Assumption 3 (Ge et al. [53]). There exist two constants g0 and g1 such that g 1 Z j gðxÞj Z g 0 4 0 on a compact subset Ω Rn . Assumption 3 implies that smooth function g(x) is strictly either positive or negative. From now onwards, without losing generality, we shall assume gðxÞ 40. Assumption 4 (Ge et al. [53]). The adaptive control problem is studied for a class of nonlinear systems satisfying d½gðxÞ=dxn ¼ 0; 8 x A Ω. Although the assumption d½gðxÞ=dxn ¼ 0 may restrict the range of applicable plants, it brings us a nice property _ gðxÞ ¼
1 d½gðxÞ d½gðxÞ dx nX d½gðxÞ ¼ ¼ x dt dx dt dxi i þ 1 i¼0
ð3Þ
which only depends on states x. Such a property is utilized to design novel adaptive controllers while avoiding the singular problem discussed earlier.
Fig. 1. Model of the nonlinear deadzone constraint.
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
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Assumption 5. The desired trajectory yd is continuously differentiable and available for controller design. Furthermore, yd is upper bounded such that j yj r ydm with ydm being a known positive constant.
exist positive constants r k such that J βn J r ϵr k ; k ¼ 2; 3; …; n, i.e. π kϵþk 1 will asymptotical converge to yðkÞ with a bounded error, and yðkÞ converges to zero with a small constant provided that y and its nth derivatives are bounded.
Assumption 6. For the system external disturbance dðtÞ : R þ -R, there exists unknown positive constant dM such that J dðtÞ J r J dM J .
Lemma 3. For any positive constant b A R, the following inequality holds for x A R in the interval jxj o b:
Remark 1. The above-mentioned assumptions are not very strict and they are common assumptions in the control literatures. Also the bounds of g(x) are not required to be known since they are only needed for stability analysis. 2.2. Preliminaries ~ ¼W ^ W n and note in advance that In this paper, we define W ~e i ¼ e^ i ei . A barrier Lyapunov function is introduced to assist our control design which is defined as follows. Definition 1 (Tee et al. [23]). If in an open region D defined with respect to the system x_ ¼ f ðxÞ, a scalar function V(x) which is positive definite and continuously differentiable, has the property that VðxÞ-1 as x approaches the boundary of D, and satisfies bounded by a small positive constant along any state trajectory of system x_ ¼ f ðxÞ with xð0Þ ¼ D, then V(x) is said to a barrier Lyapunov function for this system. In our control design, the following symmetric barrier Lyapunov function is chosen which satisfies definition: 2
1 b VðxÞ ¼ log 2 2 b e2
ð4Þ
where e ¼ x xd , and such a choice for V(x) yields : j ej o b, namely e is bounded by b. Given that xd denotes the desired trajectory, thus there always exists a bound c for any xd, namely j xd j o c. Therefore the output x is bounded: j xj o b þc. Lemma 1 (Ge and Wang [54]). If the state vector xðtÞ ¼ ½x1 ðtÞT ; x2 ðtÞT T ; x1 ðtÞ A R; x2 ðtÞ A R in the continuous system _ ¼ f ðxðtÞÞ xðtÞ
ð5Þ
has an associated Lyapunov function VðxÞ ¼ V 1 ðx1 Þ þ V 2 ðx2 Þ defined on x1 A x1 j j x1 j 〈b; b〉0 ; x2 A R with the following properties: V 1 ðx1 Þ-1 as j x1 j -b;
κ 1 ð J x1 J Þ r V 2 ðx1 Þ rκ 2 ð J x1 J Þ
ð6Þ
2
ln
b 2
b
x2
r
x2
ð11Þ
2
b x2
Proof. For the right-hand side of this inequivalent, we have 0 n 1 x2 1 BX C 2 2 x2 2 2 B C b x2 ð12Þ ¼ lnðex =ðb x Þ Þ ¼ lnB C 2 2 @ A n! b x n¼0 On the basis of the inequivalent above and noting that we can have x2 x2 Z ln 1 þ 2 2 b x2 b x2 2 b □ ¼ ln 2 b x2
x2 Z 0, 2 b x2
ð13Þ
3. Full state feedback control State feedback control schemes are presented in the section. While maintaining system's stability, the control aims to approximate the deadzone effect. The overall control strategy for state feedback control is shown as follows in Fig. 2. In this part, the control design for system (1) is proposed by using the barrier Lyapunov function with the backstepping scheme. For clarity, we are supposed to present the detailed controller design procedure in a constructive way as follows. Step 1: Define e1 ¼ x1 yd , and its derivative with respect to time is e_ 1 ¼ x_ 1 y_ d ¼ x2 y_ d
ð14Þ
According to the backstepping scheme, let x2d ¼ x2 and the ideal virtual controller xn2d can be chosen as 2 xn2d ¼ k1 e1 ðb e21 Þke1 þ y_ d
ð15Þ
where k1 4 0; k 4 0, then e1 is asymptotically stable.
where κ1 and κ2 are K class functions. Thus, x1 ðtÞ remains in the set x1 ðtÞ A x1 j j x1 j o b; b 4 0 provided that the initial value x1 ð0Þ belongs to the set x1 ð0Þ A x1 j j x1 j o b; b 40 and the derivative of V(x) satisfies
Proof. Substituting the ideal controller x2 ¼ xn2d into (14), we 2 obtain e_ 1 ¼ k1 e1 ðb e21 Þke1 . Choose the barrier Lyapunov function as
V_ ðxÞ r ρVðxÞ þ C
1 b V 1 ¼ ln 2 2 b e2
2
ð7Þ
ð16Þ
1
where ρ 4 0; C 40. Lemma 2 (Behtash [55]). Suppose the system output y (t) and its first n derivatives are bounded, so that yðkÞ oY k with positive constant Yk. Consider the following linear system: ϵπ_ i ¼ π i þ 1
ð8Þ
ϵπ_ n ¼ l1 π n l2 π n 1 ⋯ ln 1 π 2 π 1 þ yðtÞ
ð9Þ
where ϵ is a small positive constant and the values of li ; i ¼ 1; 2; …; n 1, are chosen such that the polynomial sn þ l1 sn 1 þ⋯ þ ln 1 s þ 1 is Hurwitz. The following property holds: π βk ¼ k k 1 yðk 1Þ ¼ ϵςðkÞ ; k ¼ 1; 2; …; n ð10Þ ϵ where ς ¼ π n þ l1 π n 1 þ ⋯ þ ln 1 π 1 with ςðkÞ representing there kth derivative of ς. Meanwhile, if π k r π k with positive constant π k , there
Fig. 2. State feedback control strategy.
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
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we have
Making the time derivative of V 1 gives V_ 1 ¼
e1 e_ 1 2
b
_ 2n fe en dðtÞ x_ nd en 1 ge V_ n ¼ V_ n 1 þ n þ ðv ΔuÞen þ g 2 g2 g g
e21
r k1 V 1
ð17Þ
Since k1 4 0 and k 4 0, according to the Lyapunov theorem, this yields limt-1 e1 ¼ 0.□ Step 2: Let e2 ¼ x2 x2d , and its derivative with respect to time is e_ 2 ¼ x_ 2 x_ 2d ¼ x3 x_ 2d
ð18Þ
According to the backstepping scheme, let x3d ¼ x3 and the ideal virtual controller xn3d can be chosen as xn3d ¼ k2 e2 þ x_ 2d
ð19Þ
Proof. Substituting the ideal controller x3 ¼ xn3d into (18), we obtain e_ 2 ¼ k2 e2 . Choose the barrier Lyapunov function as V 2 ¼ V 1 þ 12 e22
ð20Þ
2
b
b e21
k2 e22
¼ κ2 V 2
ð21Þ
where κ 2 ¼ minf2k1 ; 2k2 g
ð22Þ
Step i ði ¼ 3; 4; …; n 1Þ: In a similar way, a virtual controller xði þ 1Þd can be developed to force xid to track the state xi . The detail design procedure is very similar to step 2 and thus omitted here. We only give the main results about the virtual controller design. An ideal controller xnði þ 1Þd is chosen as ð23Þ
ð24Þ
and its time derivative can be obtained as V_ i ¼ k1
e21 2
b e21 r κi V i
k2 e22 ⋯ ki e2i ð25Þ
where κ i ¼ minf2k1 ; 2k2 ; …; 2ki g
ð26Þ
To ensure ρ 4 0, the control gains should be properly chosen to satisfy the following conditions: κi 40. According to the Lyapunov theorem, this yield is stability.□ Step n: Let en ¼ xn xnd , and its derivative with respect to time is e_ n ¼ x_ n x_ nd ¼ f ðxÞ þ gðxÞu x_ nd þ dðtÞ
ð27Þ
Proof. Substituting the v ¼ vn (31) into (30), we have the closedloop system _ n x_ f 1 ge fe en dðtÞ en þ k k e V_ n ¼ V_ n 1 þ n þ en nd n x n g 2 g2 g g g g2 _ 2n en dðtÞ en x_ nd 1 ge þ g 2 g2 g r κn V n ð32Þ where ð33Þ
Consider the vn (31), given that Δu, and the other parameters are all undermined unknown terms, we are supposed to use RBFNNs to approximate these. We give the ideal controller v T
^ SðZÞ þ W ^ SðZ τ Þ v ¼ kx en þ W τ
ð34Þ
The network updating laws are designed as ^ Þ ^_ ¼ ΓðSðZÞen þ σ W W
ð35Þ
^ τÞ ^_ τ ¼ ΓðSðZ τ Þen þ σ τ W W T
^ T SðZ τ Þ ^ SðZÞ and W Since W τ nT ðZ τ Þ; W SðZÞ and W nτ T SðZ τ Þ
ð36Þ nT
approximate to the W SðZÞ and W nτ T S are given by
W nT SðZÞ ¼
ð37Þ
en en x_ nd f 1 g_ en þ kn 2 ε g 2 g2 g g
ð38Þ
which W nT and W τnT are optimal weights and ε and εr are approximation errors of the neural network. ε and εr satisfy maxZ A ΩZ j εj o εn and maxZ τ A ΩZ j εr j o εnr respectively [53]. And Z ¼ ½xT1 ; xT2 ; …; xTi ; en ; Z τ ¼ ½xT1 ; xT2 ; …; xTi ; v. ^ T SðZÞ þ W ^ T SðZ τ Þ into e_ n , we will have Substituting v ¼ kx en þ W τ e_ n ¼ x_ n x_ nd _ n en dðtÞ en ~ T SðZ τ Þ þ W ~ T SðZÞ þ 1 ge 2 ¼g W kn εr ε kx en þ τ 2 2 g g g g ð39Þ We choose the Lyapunov function of the form Vn ¼ Vn1 þ
1 e2n 1 ~ T 1 ~ 1 ~ T 1 ~ þ W Γ W þ Wτ Γ Wτ 2 gðxÞ 2 2
ð40Þ
ð28Þ
ð29Þ
e V_ n ¼ V_ n 1 kn n e2n en ðεr þ ε þ kx en Þ g
Choose the barrier Lyapunov function as 1 e2n Vn ¼ Vn1 þ 2 gðxÞ
where kn 4 0; kx 40 are design parameters, then limt-1 en ¼ 0.
~_ ¼ W ^_ . _ n ¼ 0 and W ~ ¼W ^ W n and W n is a real value, W Since W Consider the derivative of the Lyapunov function and substitute e_ n into it, we will have
Let Δu ¼ v u; Δu is the error. Substituting Δu into e_ n e_ n ¼ f ðxÞ þ gðxÞðv ΔuÞ x_ nd þ dðtÞ
ð31Þ
W τnT SðZ τ Þ ¼ Δu εr
Choose the barrier Lyapunov function as V i ¼ V i 1 þ 12 e2i
_ n x_ nd f 1 ge en dðtÞ en þ 2 kn kx en þ Δu g 2 g2 g g g
T
Since κ 2 4 0, according to the Lyapunov theorem, this yield is stable.□
xnði þ 1Þd ¼ ki ei þ x_ id
vn ¼
To ensure κn 4 0, the control gains should be properly chosen to satisfy the following conditions: ki 4 0 ði ¼ 1; 2; …n 1Þ; kn þ 1 4 0, according to the Lyapunov theorem, this yield is stability.□
V_ 2 ¼ V_ 1 þ e2 e_ 2 2
Consider system (1) satisfying Assumptions 1–6. If the input v is chosen as
κn ¼ minf2k1 ; 2k2 ; …; 2ðkn þ 1Þg
Making the time derivative of V 2 gives
r k1 ln
ð30Þ
2
Substituting e_ n into the derivative of the Lyapunov function (29),
2
~ TW ^ στ W ~ TW ^ τ þ en dðtÞ en σW τ g g2
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
ð41Þ
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~ TW ~ Þ ¼ W ~ TW ~ T ^ ¼ W ~ T ðW n þ W ~ W ~ T W n and W Since W ~ TW ~ TW ~ þ W nT W n Þ, we have W ^ r 1W ~ TW ~ þ 1W nT W n . W n r 1 ðW 2
2
Meanwhile, en ε en εr r e2n þ 12ε2 þ 12ε2r r e2n þ en dðtÞ g
e2n g2
2
1 n 2 2Jε J þ
2 1 2 J εr J ;
r J dðtÞ J 2 r J dM J 2 . We substitute these into the V_ n
V_ n r k1
e21 2
b e21
k2 e22 ⋯ kn
e2n g
1 2 C e rV n ð0Þ þ 2 i κ
1 ~ T ~ 1 ~ T ~ σW W Wτ Wτ 2 2
(
2σ τ 2σ κ ¼ min 2k1 ; 2k2 ; …; 2kn ; ; λmax ðΓ τ 1 Þ λmax ðΓ 1 Þ
)
C ¼ 12 W nT W n þ 12 W nτ T W nτ þ 12 J εn J 2 þ 12 J εr J 2 þ J dM J 2
ð54Þ
For ei , we have
1 1 þ σW nT W n þ σ τ W nτ T W nτ e2n en ðεr þε þ kx en Þ þ J dM J 2 2 2 1 1 1 1 r κV n þ σW nT W n þ σ τ W nτ T W nτ þ J εn J 2 þ J εr J 2 þ J dM J 2 2 2 2 2 ¼ κV n þ C ð42Þ where
Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b ðe2D 1Þ J e1 J r e2D
5
ð43Þ
ð44Þ
To sure κ 4 0 and C 4 0, the design parameters kx Z 1; ki 4 0 ði ¼ 1; 2; …; nÞ. There, we have the following theorem.
ð55Þ
Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C J ei J r 2 V n ð0Þ þ κ ~ and W ~ τ , we also can obtain As for W vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u C u uV n ð0Þ þ t κ ~ JW J r λmin ðΓ 1 Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u C u uV n ð0Þ þ t κ□ ~ JW τ J r λmin ðΓ τ 1 Þ
ð56Þ
ð57Þ
ð58Þ
4. Output feedback control Theorem 1. Considering the nonlinear affine system (1) with unknown disturbance, input deadzone and output constraint. The proposed state feedback RBFNN control law (34), neural network updating laws (35), (36) with bounded initial conditions, the closedloop system signals ei ; W and others are semi-globally bounded. ~ and W ~ τ will remain Furthermore, the closed-loop error signals ei ; W within compact sets Ωei ; ΩW~ ; ΩW~ τ respectively, defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 2 < b ðe2D 1Þ= n ð45Þ Ωe1 : ¼ e1 A R j J e1 J r : ; e2D n
o pffiffiffiffiffiffiffi Ωei : ¼ ei A Rn j J ei J r 2D ði ¼ 2; …; n 1Þ Ωen
n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio : ¼ en A R j J en J r 2gðxÞD n
( ~ A Rn j J W ~ Jr W
ΩW~ : ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ 1 Þ
( ΩW~ τ : ¼
~ τ A Rn j J W ~ τJr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ τ 1 Þ
ð46Þ ð47Þ
ð48Þ
ð49Þ
In this section, the high-gain observer is introduced to complete the output feedback control. According to Lemma 3, we know that πkϵþk 1 converges to yðkÞ under bounded errors so that π kϵþk 1 will be a suitable observer to estimate the unmeasurable state xk þ 1 . Define the estimate of unmeasurable states as h π π π n iT 2 3 x^ ¼ x1 ; ; 2 ; ⋯ n 1 ð59Þ ϵ ϵ ϵ Considering the following linear system: ϵπ_ 1 ¼ π 2 ϵπ_ n ¼ l1 π n l2 π n 1 ⋯ ln 1 π 2 π 1 þ yðtÞ According to Lemma 2, we have πn yðn 1Þ ¼ ϵςðnÞ ϵn 1
ð60Þ
ð61Þ
where ϵ is any small constant, and there exist positive constants t n and r n such that 8 t 4 t n , we have J βn J r ϵr n . So we can use ϵnπn 1 to estimate xn , then xn ; en can be estimated as follows: πn ð62Þ x^ n ¼ n 1 ϵ e^ n ¼ x^ n xnd
ð63Þ
where D ¼ V n ð0Þ þ Cκ with C and κ given in (43) and (44) where both are positive definite.
e~ n ¼ e^ n en ¼
Proof. Multiplying (42) by eκt yields
Revisit the full state control, RBFNN updating law with e^ n , we have
d ðV n eκt Þ r Ceκt dt Integrating the above inequality, we obtain C C C r V n ð0Þ þ V n r V n ð0Þ e κt þ κ κ κ Then, we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C J en J r 2ngðxÞn V n ð0Þ þ κ
T
ð50Þ
ð51Þ
ð52Þ
2
1
^ SðZ^ Þ þ W ^ τ SðZ^ τ Þ v ¼ kx e^ n þ W
ð64Þ
ð65Þ
The network updating laws are designed as
For e1 , we have 1 b C ln r V n ð0Þ þ 2 b2 e2 κ
πn πn xnd ðxn xnd Þ ¼ n 1 yðn 1Þ ¼ βn ϵn 1 ϵ
ð53Þ
^_ ¼ ΓðSðZ^ Þe^ n þ σ W ^ Þ W
ð66Þ
^ τÞ ^_ τ ¼ ΓðSðZ^ τ Þe^ n þσ τ W W
ð67Þ
The overall control strategy for output feedback control is given in Fig. 3. ^ T SðZ^ Þ þ W ^ τ SðZ^ τ Þ into the e_ n , we will Substituting v ¼ kx e^ n þ W have ^ T SðZ^ Þ þ W ^ τ SðZ^ τ Þ W nT SðZÞ W nT SðZ τ Þ e_ n ¼ g kx e^ n þ W τ _ n 1 ge en dðtÞ en 2 εr ε kn þ ð68Þ þ 2 g2 g g g
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
1 1 V_ n r V_ n 1 þ J γ S J 2 J W n J 2 þ J γ τ Sτ J 2 J W nτ J 2 2 2 σ ~ 2 σ τ ~ 2 l 2 lτ 2 þ J W J þ J W τ J þ e~ n þ e~ n 4 σ 4 στ 1 e2 þ ð1 kx Þe2n kn n en ðεr þεÞ 2 g T 1 2 ~ W ^ στ W ~ TW ^ τ þ J dM J 2 þ kx e~ n σ W τ 2
ð81Þ
Since en ðεr þεÞ r e2n þ 12ε2r þ 12ε2 re2n þ 12 J εnr J 2 þ 12 J εn J 2 , meanwhile, ~ TW ~ Þ ¼ W ~ TW ~ T Wn r ^ ¼ W ~ T ðW n þ W ~ W ~ T W n and W W T T T 1 ~ ~ W ~ þ W nT W n Þ, we have W ^ r 1W ~ W ~ þ 1W nT W n . SubðW W 2
stituting these inequalities into V_ n
Fig. 3. Output feedback control strategy.
ð69Þ
V_ n r V_ n 1 þen ðW nT ðSðZ^ Þ SðZÞÞ þ W τnT ðSðZ^ τ Þ sðZ τ ÞÞÞ ~ T SðZ^ Þ e~ n W ~ T SðZ^ τ Þ e~ n W τ e2n ^ στ W ~ TW ^ τ kx en e^ n þ J dM J 2 ~ TW en ðεr þ εÞ σ W τ g
ð70Þ
Meanwhile, we have SðZ^ Þ SðZÞ ¼ γ S
ð71Þ
SðZ^ τ Þ SðZ τ Þ ¼ γ τ Sτ
ð72Þ
where S and Sτ are bounded vector functions with satisfying j Sj o Sn and j Sτ j o Snτ and γ 4 0. Consider V_ n , by applying en e~ n r 12 e2n þ 12 e~ 2n
J γ τ Snτ J 2 þ σ τ l lτ J W nτ J 2 þ e~ 2n þ e~ 2n σ 2 στ 1 1 1 þ kx e~ 2n þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2 2 þ
Let estimation error e~ n ¼ e^ n en . Consider the derivative of the Lyapunov function and substitute e_ n into it, we have
kn
2
e2 σ ~ 2 σ τ ~ 2 J γ Sn J þ σ J JW τ J þ J Wn J2 V_ n r V_ n 1 kn n J W 2 g 4 4
Choose the barrier Lyapunov function as 1 e2n 1 ~ T 1 ~ 1 ~ T 1 ~ þ W Γ W þ WτΓ Wτ Vn ¼ Vn1 þ 2 gðxÞ 2 2
2
ð73Þ
ð82Þ
Substituting (64) into the above equation, we have e2 σ ~ 2 σ τ ~ 2 J γ Sn J 2 þσ J JW τ J þ JWn J2 V_ n r V_ n 1 kn n J W 2 g 4 4 J γ Sn J 2 þ σ τ l lτ 1 þ þ kx ðβn Þ2 J W nτ J 2 þ þ τ τ σ στ 2 2 1 1 þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2 ¼ κV n þ C where
ð83Þ
(
) στ σ ; λmax ðΓ τ 1 Þ λmax ðΓ 1 Þ n 2 JγS J þσ J γ Sn J 2 þ σ τ l lτ 1 JWn J2 þ τ τ þ þ kx ϵ2 r 2n J W nτ J 2 þ C¼ 2 σ στ 2 2
κ ¼ min 2k1 ; 2k2 ; …; 2kn ;
ð84Þ
we have 1 1 þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2
~ T SðZ^ Þ e~ n W ~ T SðZ^ τ Þ V_ n r V_ n 1 þen W n γ S þ en W nτ γ τ Sτ e~ n W τ e2n
~ TW ^ στ W ~ TW ^ τ þ 1kx e~ 2 1kx e2 en ðεr þ εÞ σ W τ n 2 2 n
ð85Þ
ð74Þ
To sure κ 4 0; C 4 0, the design parameters kx 4 4, ki 4 0. There, we have the following theorem.
en W n γ S r 12 e2n þ 12 J γ S J 2 J W n J 2
ð75Þ
en W n γ τ Sτ r 12 e2n þ 12 J γ τ Sτ J 2 J W nτ J 2
ð76Þ
Theorem 2. Considering system consisting of (1) under the assumptions and the bounded changing rate of the external disturbance, the output feedback control (65), the NN weights updating laws (66), (67). For bounded initial conditions, all signals of the closed-loop system are bounded. Furthermore, the closed-loop error signals ~ and W ~ τ will remain within compact sets Ωei ; Ω ~ ; Ω ~ respecei ; W W Wτ tively, defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 2 < b ðe2D 1Þ= ð86Þ Ωe1 ¼ e1 A Rn j J e1 J r : ; e2D
kn
g
Since
and ~ J 2 þ 1 J SðZ^ Þ J 2 e~ 2 ~ T SðZ^ Þ r σ J W e~ n W n 4 σ ~ SðZ^ τ Þ r e~ n W τ T
στ ~ 2 1 J W τ J þ J SðZ^ τ Þ J 2 e~ 2n στ 4
ð77Þ ð78Þ
Substituting J SðZ^ Þ J 2 r l and J SðZ^ τ Þ J 2 r lτ , we have ~ J 2 þ 2l e~ 2 ~ SðZ^ Þ r σ J W e~ n W 4 2σ n σ ~ 2 2l 1 2 J þ e~ ¼ JW 4 σ2 n T
~ T SðZ^ τ Þ r σ τ J W ~ τ J2 þ e~ n W τ 4 σ τ ~ 2 2lτ 1 2 e~ ¼ JW τ J þ 4 στ 2 n
ð79Þ
ð87Þ
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio Ωen ¼ en A Rn j J en J r 2gðxÞD
ð88Þ
(
2lτ 2 e~ 2σ τ n
Substituting (79) and (80) into (74), we have
n o pffiffiffiffiffiffiffi Ωei ¼ ei A Rn j J ei J r 2D ði ¼ 2; …; n 1Þ
ΩW~ ¼
~ A Rn j J W ~ Jr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ 1 Þ
(
ð80Þ ΩW~ τ ¼
~ τ A Rn j J W ~ τJr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ τ 1 Þ
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
ð89Þ
ð90Þ
W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Proof. The proof of theorem is straightforward using the method of proof in Theorem 1, and hence, is omitted.□ 5. Simulation study In this section, we will make use of the proposed adaptive neural control (31), (34), (65) and to the nonlinear affine system with deadzone, output constraint and disturbance. The system is described as follows: 8 x_ 1 ¼ x2 > > > > < x_ 2 ¼ x3 ð91Þ x_ 3 ¼ x1 x2 3x3 þ ð1 þ 0:2x1 Þu þ dðtÞ > > > > : y ¼ x1 The deadzone is set to be br ¼ 2:5 and bl ¼ 4:5. The output constraint is set to be b ¼ 0:05. The external disturbance is set as dðtÞ ¼ 3ð1 expð 0:01tÞÞ sin ð0:5πtÞ
ð92Þ
previously defined in Case 2, the new parameters are added to the simulation: kp ¼ 20; kd ¼ 0:5. The tracking performance of the closed-loop system is given in Fig. 5. From the figure, we can state that all the four kinds of control ((31), (34), (65) and the PD control) can successfully track the desired trajectory, where the errors of the closed-loop system are converging to a small value close to zero. However, from Fig. 6,
5
ð94Þ
In order to verify the methods for improving the robustness of the system, we have carried out four cases for simulation. Case 1: Model based control (31). When the deadzone function, the external disturbance and the dynamics are fully known, we examine the control performance for the model based control. The control gains are given as k1 ¼ 20; k2 ¼ 30, k3 ¼ 40; k ¼ 20 and kx ¼ 50. The model based control needs to know the external disturbance dðtÞ, the system dynamics f ðxÞ; gðxÞ, which are difficult to obtain in practice. Case 2: State feedback control (34). The new parameters involved in the state feedback control are added to the simulation. Unknown deadzone is defined to have br ¼ 2:5 and bl ¼ 4:5. Given the situation that we have no information about the deadzone function, system dynamics and disturbance, the proposed control (34) can be implied to track the desired trajectory. We use two RBFNNs to estimate the unknown functions, which can approximate the deadzone error, system dynamics and external disturbance, respectively. Furthermore, the duration of simulation t f ¼ 40. For the two RBFNNs with Z ¼ ½x1 ; x2 ; x3 ; e 3 and Z τ ¼ ½x1 ; x2 ; x3 ; v, numbers of nodes l1 for S is 24 and l2 for Sτ is 24 . Variances for both neural networks are set to be σ ¼ 0:001; σ τ ¼ 0:001; Γ ¼ 50I 1616 , Γ τ ¼ 50I 1616 . RBFNN centers for S are evenly distributed in the domain of ½ 1; 1 ½ 1; 1 ½ 1; 1 ½ 1; 1 and centers for Sτ are evenly distributed in domain of ½ 1; 1 ½ 1; 1 ½ 1; 1 ½ 1; 1. Initial weights of the neural network are all zero. The control gains are given as k1 ¼ 20, k2 ¼ 30; k3 ¼ 30; k ¼ 30 and kx ¼ 30. Case 3: Output feedback control (65). With all parameters being the same as previously defined in Case 2, the new parameters are added to the simulation: ϵ ¼ 0:005; l1 ¼ 1 and initial terms π 1 ¼ 0; π 2 ¼ _ 0. 0; π 3 ¼ 0; π_ 1 ¼ 0; π_ 2 ¼ 0; π 3 ¼ Case 4: PD control. The PD control is designed as follows: v ¼ kp e3 kd …e3 . With all parameters being the same as
2
0
5
10
15
20
25
30
35
40
t [s] Fig. 4. Disturbance.
The tracking performance 2 the PD control the control based model the state feedback control the output feedback control the desired trajectory the upper bound of the output constraint the lower bound of the output constraint
1 0.95
1.5
0.9 3.5 4 4.5
1 position [rad]
y_ d ¼ 0:5 ð0:5 cos ð0:5 tÞ 1:5 sin ð1:5 tÞÞ
3
0
0.5
0
−0.5
−1 0
5
10
15
20
25
30
35
40
t [s]
Fig. 5. System output and desired trajectory.
The error of the tracking performance 0.02 the error of the PD control the error of the control based model the error of the state feedback control the error of the output feedback control
0.015
0.01 position [rad]
ð93Þ
4
1
Take note that the external disturbance vibrates with a high frequency, which indicates a big value for the time derivative. In general cases, the time derivative usually consider to be a small value with none or very low vibration frequency. In our case, shown in Fig. 4, which is the external disturbance, the high frequency disturbance is utilized to test the feasibility of proposed nonlinear disturbance observer. Initial positions of the system are y1 ð0Þ ¼ 0:54 and y2 ð0Þ ¼ 0. And desired trajectory as follows: yd ¼ 0:5 ð sin ð0:5 tÞ þ cos ð1:5 tÞÞ
External disturbance
6
external disturbance [rad]
where D ¼ V n ð0Þ þ Cκ with C and κ given in (84) and (85) where both are positive definite.
7
0.005
0
−0.005
−0.01 0
5
10
15
20
25
30
35
t [s]
Fig. 6. The tracking error.
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
40
W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6. Conclusion In this paper, adaptive control schemes have been studied for a class of nonlinear affine systems with both deadzone and constraint by using RBFNN. An adaptive controller is firstly developed for the systems with the assumptions holding on a given compact set. The stability of the closed-loop adaptive neural system is guaranteed via Lyapunov stability theorem. In addition, with the The error of the tracking performance 40 the control input of the control based model the control input of the state feedback control the control input of the output feedback control
30
control input [Nm]
20
30 Norms for output feedback control norm W
we can see that the errors of the PD control are larger than others, because of the neural network learning ability. Meanwhile, it is satisfying that the tracking trajectory of the NN control is showing a good performance with small errors, which indicates that the constraint satisfaction can be ensured. However, in order to achieve the control objective, the model-based control needs to require the fully known system dynamics, which is difficult to obtain. Furthermore, the contribution of neural networks is shown clearly by comparing the tracking performances and errors of different controllers, which can state that the neural networks can also approximate the unknown system dynamics and ensure the stability of the system. Fig. 7 shows the control inputs. It can be seen that, within the time interval ½0 2, the control v will establish a large control action to drive the error to a small value closed to zero. That is the reason why there exist some spikes in this time interval. The boundedness of NN weights is also depicted in Figs. 8 and 9.
20 10 0
0
5
10
15
20
25
30
35
40
t [s] 80 Norms for output feedback control norm Wtau
8
60 40 20 0
0
5
10
15
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25
30
35
40
t [s]
Fig. 9. The norms of output feedback control NN weights.
aim of compensating for the input deadzone and the output constraint effects, the direct neural network control is built. In order to guarantee the closed-loop stability, Lyapunov's direct method has been applied in the control design process. Furthermore, simulation study has demonstrated that the proposed direct neural network control method could successfully drive the system to track desired curve under the effects of the input deadzone and the output constraint. Overall, the control objective has been reached, all signals in the closed-loop systems have been guaranteed to be semi-globally uniformly ultimately bounded, and eventually converge to the corresponding small compact sets. This article employs the RBF neural network to solve the input deadzone and output constraint of the nonlinear affine systems, and the simulation results show that the proposed control is effective.
10
Acknowledgment
0
The authors would like to thank the Editor-In-Chief, the Associate Editor and the anonymous reviewers for their constructive comments which helped improve the quality and presentation of this paper.
−10 −20 −30
References
−40 0
5
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15
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25
30
35
40
t [s]
Fig. 7. The control input.
30
norm W
Norms for state feedback control
20
10
0
0
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35
40
t [s] 0.4
norm W
tau
Norms for state feedback control
0.3 0.2 0.1 0 0
5
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25
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t [s]
Fig. 8. The norms of state feedback control NN weights.
35
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Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint$ Wei He a, Yiting Dong b, Changyin Sun a,n a b
School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 July 2014 Received in revised form 13 December 2014 Accepted 26 May 2015 This paper was recommended for publication by Dr. Y. Chen
In this paper, we aim to solve the control problem of nonlinear affine systems, under the condition of the input deadzone and output constraint with the external unknown disturbance. To eliminate the effects of the input deadzone, a Radial Basis Function Neural Network (RBFNN) is introduced to compensate for the negative impact of input deadzone. Meanwhile, we design a barrier Lyapunov function to ensure that the output parameters are restricted. In support of the barrier Lyapunov method, we build an adaptive neural network controller based on state feedback and output feedback methods. The stability of the closed-loop system is proven via the Lyapunov method and the performance of the expected effects is verified in simulation. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Unknown nonlinear affine system Input deadzone Output constraint Radial Basis Function Neural Network (RBFNN) Barrier Lyapunov Function
1. Introduction From engineering point view, both input deadzone and output constraint are common static nonlinearities, which require to be tackled practical control systems. Neglecting the deadzone effects may lead to undesired performances such as excessive steady state error, poor transient response and large overshoot [1]. Moreover, performance degradation, hazards or system damage may result from violation of the constraints during operation. Therefore, it is necessary to take both deadzone effect and output constraint effect of the nonlinear systems into account. Recent years have witnessed a rapid development of neural network control techniques and successful applications in tracking control of nonlinear systems with external disturbance [2,3]. Many control approaches, published in various books [4,5], journals [6–10] and conference proceedings [11], show that artificial neural networks play an important role in function approximation and control design to deal with the nonlinearities of complex systems. In spite of having made significant achievements in neural network fields, due to the complexity of nonlinear systems, most works merely focus on one kind of nonlinearities, while a variety of nonlinearities are usually
☆ This work was supported by the National Natural Science Foundation of China under Grant 61203057 and 61125306, the National Basic Research Program of China (973 Program) under Grant 2014CB744206. n Corresponding author. E-mail address: [email protected] (C. Sun).
rarely considered together, especially taking the input deadzone and output constraint into account together. Numerous approaches have been proposed to address the control problems nonlinear affine systems with deadzone or constraint, many techniques have been developed, most of which [12,13] are based on the assumption that the deadzone function can be parameterized into a few values. However, the deadzone function is usually difficult to achieve, which leads these controllers are not practical and utility. Hence, numerous controllers have been proposed to address the control problems of solving unknown deadzone effect. A control method for a special nonlinear system with symmetrical deadzone is proposed in [14], whose deadzone slope and width are unknown. The design based on neural network is recently presented to solve the deadzone effect. In [15], an adaptive neural network control is proposed for single-master–multiple-slaves teleoperation in consideration of time delays and input deadzone uncertainties. To handle asymmetric and nonlinear deadzone function, a scheme based on Multi-layer NNs is proposed in [16] to compensate the deadzone effect, which increases the robustness of the close-loop system. These researches show the superiority and feasibility of applying neural network to compensate the deadzone effect. Meanwhile, some ideas of utilizing the RBFNNs in nonlinear systems have been come up with. A modified adaptive RBFNN for the compensation of deadzone is described in [17]. By constructing a deadzone compensator, a neural design control scheme based on RBFNN is developed by using backstepping design techniques in [18]. However, these proposed control design algorithms are simpler in both designing and applying
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Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
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than MNNs. Constructive and destructive parsimonious extreme learning machines are proposed in [19]. Compared with other structures, the RBFNN-based control usually consumes less calculation resources and achieves the desired trajectory faster. [20,21] also have proved that the RBFNN has the learning ability. Nonetheless, the approaches merely focus on approximating the errors of deadzone effect. To address the problem of output constraint, numerous existing works have demonstrated Barrier Lyapunov Function (BLF) that can guarantee the constraints satisfaction. BLF has a better property that can grow to infinity as some limits, which guarantee the boundedness in the closed-loop system. Based on Lyapunov's direct function, the stability of the system can be guaranteed. Barrier Lyapunov function is first proposed to tackle the problem of constraints in Brunovsky form [22] systems. In [23], the authors propose BLF-based control design for nonlinear systems with output constraint. In [24], BLF-based control is investigated in output constrained nonlinear systems with unknown constant control gain function. Meanwhile, BLF is widely used in flexible systems. The authors propose BLF-based adaptive control to suppress the vibration and guarantee the constraint satisfaction of a flexible string system in [25,26]. BLF is also proposed to suppress the undesirable vibration of a flexible crane system in [27] and a flexible beam in [28]. Nevertheless, it should be noted that a large number of researches on output constrained systems have assumed that there exist only constraint and disturbance in the nonlinear systems, and do not take other nonlinearities into account, such as input deadzone, and external disturbance. The NN control for the nonlinear system has received much attention [29–36]. As an effective method in a large number of nonlinear system fields, neural networks require relatively less information about the system dynamics. Numerous researches have proven that artificial neural networks can approximate a wide range of nonlinear functions accurately [37–41]. The authors propose a local network structure for nonlinear system trajectory tracking with uncertainties in [42]. A robust neural network output feedback scheme is developed for the motion control of robot manipulators in [43]. [44] studies the problem of learning from neural network control of a class of nonlinear non-affine systems in uncertain dynamic environments. [45] presents the problems of accurate identification and learning control of ocean surface ship using neural network. For a class of wheeled inverted pendulum models of vehicle systems, the authors present an adaptive NN-based control in [46]. In [47], the authors develop an output feedback adaptive neural network control to achieve stable dynamic balance and track of the desired trajectory. In [48–51], adaptive neural control is proposed for a class of uncertain nonlinear systems. The main aim of this paper is to develop an RBF-based neural network system to identify the nonlinear system parameters with deadzone and constraint and then to design an adaptive controller for deadzone compensation and control the system. To achieve this goal, two RBF-based neural networks are applied in the proposed approach. One is used for identifying parameters of input deadzone. External disturbance is also rejected with another RBF-based proposed control. Once the neural network's center and width factor of the basis functions are determined, the RBF neural network that is similar to a typically single layer neural network is trained based on the gradient decent technique [17]. Meanwhile, the output violation is prevented by utilizing a barrier Lyapunov function. The rest of this paper is organized as follows: we propose the problem formulation and necessary assumptions, use lemmas in Section 2. Section 3 illustrates the direct adaptive NN control design process under the state feedback and the simulation of the state. Output feedback control is investigated in Section 4. Section 5 reveals the validity and performance of proposed method by simulation. The last section introduces our paper.
2. Problem formulation and preliminaries 2.1. Problem formulation The unknown nonlinear affine system [52] is stated as 8 _ > < x i ¼ xi þ 1 ði ¼ 1; 2; …; n 1Þ x_ n ¼ f ðxÞ þ gðxÞu þ dðtÞ > :y¼x 1
ð1Þ
where x ¼ ½x1 ; x2 ; …; xn T A Rn is the state vector, and u A R is the control input to the nonlinear affine system (1) after deadzone and output constraint effect. The external disturbance to the above system is dðtÞ : R þ -R. The unknown function f ðxÞ A R, gðxÞ A R are sufficiently smooth. The deadzone is properly expressed as 8 > < hr ðvÞ; v Z br bl o v o br u ¼ 0; ð2Þ > : h ðvÞ; v r b l l where v is the desired input to deadzone, br and bl are unknown parameters of the deadzone and hr ðÞ and hl ðÞ are unknown functions of the deadzone, which is shown in Fig. 1. The control goal is to develop an adaptive NN controller for system (1) so that the output y is forced to trace the desired trajectory yd asymptotically, the tracking error e1 -0 as t-1, while the output y is guaranteed to be within certain predefined bounds for all time. The control objective will be achieved under the following assumptions. Assumption 1. The desired trajectory yd(t) and its derivatives yð1Þ ðtÞ, yð2Þ ðtÞ, …, ydðnÞ ðtÞ are smooth, bounded and known. d d Assumption 2. For the given system described in (1), the unknown affine term f(x), g(x) is smooth with respect to input u. Assumption 3 (Ge et al. [53]). There exist two constants g0 and g1 such that g 1 Z j gðxÞj Z g 0 4 0 on a compact subset Ω Rn . Assumption 3 implies that smooth function g(x) is strictly either positive or negative. From now onwards, without losing generality, we shall assume gðxÞ 40. Assumption 4 (Ge et al. [53]). The adaptive control problem is studied for a class of nonlinear systems satisfying d½gðxÞ=dxn ¼ 0; 8 x A Ω. Although the assumption d½gðxÞ=dxn ¼ 0 may restrict the range of applicable plants, it brings us a nice property _ gðxÞ ¼
1 d½gðxÞ d½gðxÞ dx nX d½gðxÞ ¼ ¼ x dt dx dt dxi i þ 1 i¼0
ð3Þ
which only depends on states x. Such a property is utilized to design novel adaptive controllers while avoiding the singular problem discussed earlier.
Fig. 1. Model of the nonlinear deadzone constraint.
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Assumption 5. The desired trajectory yd is continuously differentiable and available for controller design. Furthermore, yd is upper bounded such that j yj r ydm with ydm being a known positive constant.
exist positive constants r k such that J βn J r ϵr k ; k ¼ 2; 3; …; n, i.e. π kϵþk 1 will asymptotical converge to yðkÞ with a bounded error, and yðkÞ converges to zero with a small constant provided that y and its nth derivatives are bounded.
Assumption 6. For the system external disturbance dðtÞ : R þ -R, there exists unknown positive constant dM such that J dðtÞ J r J dM J .
Lemma 3. For any positive constant b A R, the following inequality holds for x A R in the interval jxj o b:
Remark 1. The above-mentioned assumptions are not very strict and they are common assumptions in the control literatures. Also the bounds of g(x) are not required to be known since they are only needed for stability analysis. 2.2. Preliminaries ~ ¼W ^ W n and note in advance that In this paper, we define W ~e i ¼ e^ i ei . A barrier Lyapunov function is introduced to assist our control design which is defined as follows. Definition 1 (Tee et al. [23]). If in an open region D defined with respect to the system x_ ¼ f ðxÞ, a scalar function V(x) which is positive definite and continuously differentiable, has the property that VðxÞ-1 as x approaches the boundary of D, and satisfies bounded by a small positive constant along any state trajectory of system x_ ¼ f ðxÞ with xð0Þ ¼ D, then V(x) is said to a barrier Lyapunov function for this system. In our control design, the following symmetric barrier Lyapunov function is chosen which satisfies definition: 2
1 b VðxÞ ¼ log 2 2 b e2
ð4Þ
where e ¼ x xd , and such a choice for V(x) yields : j ej o b, namely e is bounded by b. Given that xd denotes the desired trajectory, thus there always exists a bound c for any xd, namely j xd j o c. Therefore the output x is bounded: j xj o b þc. Lemma 1 (Ge and Wang [54]). If the state vector xðtÞ ¼ ½x1 ðtÞT ; x2 ðtÞT T ; x1 ðtÞ A R; x2 ðtÞ A R in the continuous system _ ¼ f ðxðtÞÞ xðtÞ
ð5Þ
has an associated Lyapunov function VðxÞ ¼ V 1 ðx1 Þ þ V 2 ðx2 Þ defined on x1 A x1 j j x1 j 〈b; b〉0 ; x2 A R with the following properties: V 1 ðx1 Þ-1 as j x1 j -b;
κ 1 ð J x1 J Þ r V 2 ðx1 Þ rκ 2 ð J x1 J Þ
ð6Þ
2
ln
b 2
b
x2
r
x2
ð11Þ
2
b x2
Proof. For the right-hand side of this inequivalent, we have 0 n 1 x2 1 BX C 2 2 x2 2 2 B C b x2 ð12Þ ¼ lnðex =ðb x Þ Þ ¼ lnB C 2 2 @ A n! b x n¼0 On the basis of the inequivalent above and noting that we can have x2 x2 Z ln 1 þ 2 2 b x2 b x2 2 b □ ¼ ln 2 b x2
x2 Z 0, 2 b x2
ð13Þ
3. Full state feedback control State feedback control schemes are presented in the section. While maintaining system's stability, the control aims to approximate the deadzone effect. The overall control strategy for state feedback control is shown as follows in Fig. 2. In this part, the control design for system (1) is proposed by using the barrier Lyapunov function with the backstepping scheme. For clarity, we are supposed to present the detailed controller design procedure in a constructive way as follows. Step 1: Define e1 ¼ x1 yd , and its derivative with respect to time is e_ 1 ¼ x_ 1 y_ d ¼ x2 y_ d
ð14Þ
According to the backstepping scheme, let x2d ¼ x2 and the ideal virtual controller xn2d can be chosen as 2 xn2d ¼ k1 e1 ðb e21 Þke1 þ y_ d
ð15Þ
where k1 4 0; k 4 0, then e1 is asymptotically stable.
where κ1 and κ2 are K class functions. Thus, x1 ðtÞ remains in the set x1 ðtÞ A x1 j j x1 j o b; b 4 0 provided that the initial value x1 ð0Þ belongs to the set x1 ð0Þ A x1 j j x1 j o b; b 40 and the derivative of V(x) satisfies
Proof. Substituting the ideal controller x2 ¼ xn2d into (14), we 2 obtain e_ 1 ¼ k1 e1 ðb e21 Þke1 . Choose the barrier Lyapunov function as
V_ ðxÞ r ρVðxÞ þ C
1 b V 1 ¼ ln 2 2 b e2
2
ð7Þ
ð16Þ
1
where ρ 4 0; C 40. Lemma 2 (Behtash [55]). Suppose the system output y (t) and its first n derivatives are bounded, so that yðkÞ oY k with positive constant Yk. Consider the following linear system: ϵπ_ i ¼ π i þ 1
ð8Þ
ϵπ_ n ¼ l1 π n l2 π n 1 ⋯ ln 1 π 2 π 1 þ yðtÞ
ð9Þ
where ϵ is a small positive constant and the values of li ; i ¼ 1; 2; …; n 1, are chosen such that the polynomial sn þ l1 sn 1 þ⋯ þ ln 1 s þ 1 is Hurwitz. The following property holds: π βk ¼ k k 1 yðk 1Þ ¼ ϵςðkÞ ; k ¼ 1; 2; …; n ð10Þ ϵ where ς ¼ π n þ l1 π n 1 þ ⋯ þ ln 1 π 1 with ςðkÞ representing there kth derivative of ς. Meanwhile, if π k r π k with positive constant π k , there
Fig. 2. State feedback control strategy.
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we have
Making the time derivative of V 1 gives V_ 1 ¼
e1 e_ 1 2
b
_ 2n fe en dðtÞ x_ nd en 1 ge V_ n ¼ V_ n 1 þ n þ ðv ΔuÞen þ g 2 g2 g g
e21
r k1 V 1
ð17Þ
Since k1 4 0 and k 4 0, according to the Lyapunov theorem, this yields limt-1 e1 ¼ 0.□ Step 2: Let e2 ¼ x2 x2d , and its derivative with respect to time is e_ 2 ¼ x_ 2 x_ 2d ¼ x3 x_ 2d
ð18Þ
According to the backstepping scheme, let x3d ¼ x3 and the ideal virtual controller xn3d can be chosen as xn3d ¼ k2 e2 þ x_ 2d
ð19Þ
Proof. Substituting the ideal controller x3 ¼ xn3d into (18), we obtain e_ 2 ¼ k2 e2 . Choose the barrier Lyapunov function as V 2 ¼ V 1 þ 12 e22
ð20Þ
2
b
b e21
k2 e22
¼ κ2 V 2
ð21Þ
where κ 2 ¼ minf2k1 ; 2k2 g
ð22Þ
Step i ði ¼ 3; 4; …; n 1Þ: In a similar way, a virtual controller xði þ 1Þd can be developed to force xid to track the state xi . The detail design procedure is very similar to step 2 and thus omitted here. We only give the main results about the virtual controller design. An ideal controller xnði þ 1Þd is chosen as ð23Þ
ð24Þ
and its time derivative can be obtained as V_ i ¼ k1
e21 2
b e21 r κi V i
k2 e22 ⋯ ki e2i ð25Þ
where κ i ¼ minf2k1 ; 2k2 ; …; 2ki g
ð26Þ
To ensure ρ 4 0, the control gains should be properly chosen to satisfy the following conditions: κi 40. According to the Lyapunov theorem, this yield is stability.□ Step n: Let en ¼ xn xnd , and its derivative with respect to time is e_ n ¼ x_ n x_ nd ¼ f ðxÞ þ gðxÞu x_ nd þ dðtÞ
ð27Þ
Proof. Substituting the v ¼ vn (31) into (30), we have the closedloop system _ n x_ f 1 ge fe en dðtÞ en þ k k e V_ n ¼ V_ n 1 þ n þ en nd n x n g 2 g2 g g g g2 _ 2n en dðtÞ en x_ nd 1 ge þ g 2 g2 g r κn V n ð32Þ where ð33Þ
Consider the vn (31), given that Δu, and the other parameters are all undermined unknown terms, we are supposed to use RBFNNs to approximate these. We give the ideal controller v T
^ SðZÞ þ W ^ SðZ τ Þ v ¼ kx en þ W τ
ð34Þ
The network updating laws are designed as ^ Þ ^_ ¼ ΓðSðZÞen þ σ W W
ð35Þ
^ τÞ ^_ τ ¼ ΓðSðZ τ Þen þ σ τ W W T
^ T SðZ τ Þ ^ SðZÞ and W Since W τ nT ðZ τ Þ; W SðZÞ and W nτ T SðZ τ Þ
ð36Þ nT
approximate to the W SðZÞ and W nτ T S are given by
W nT SðZÞ ¼
ð37Þ
en en x_ nd f 1 g_ en þ kn 2 ε g 2 g2 g g
ð38Þ
which W nT and W τnT are optimal weights and ε and εr are approximation errors of the neural network. ε and εr satisfy maxZ A ΩZ j εj o εn and maxZ τ A ΩZ j εr j o εnr respectively [53]. And Z ¼ ½xT1 ; xT2 ; …; xTi ; en ; Z τ ¼ ½xT1 ; xT2 ; …; xTi ; v. ^ T SðZÞ þ W ^ T SðZ τ Þ into e_ n , we will have Substituting v ¼ kx en þ W τ e_ n ¼ x_ n x_ nd _ n en dðtÞ en ~ T SðZ τ Þ þ W ~ T SðZÞ þ 1 ge 2 ¼g W kn εr ε kx en þ τ 2 2 g g g g ð39Þ We choose the Lyapunov function of the form Vn ¼ Vn1 þ
1 e2n 1 ~ T 1 ~ 1 ~ T 1 ~ þ W Γ W þ Wτ Γ Wτ 2 gðxÞ 2 2
ð40Þ
ð28Þ
ð29Þ
e V_ n ¼ V_ n 1 kn n e2n en ðεr þ ε þ kx en Þ g
Choose the barrier Lyapunov function as 1 e2n Vn ¼ Vn1 þ 2 gðxÞ
where kn 4 0; kx 40 are design parameters, then limt-1 en ¼ 0.
~_ ¼ W ^_ . _ n ¼ 0 and W ~ ¼W ^ W n and W n is a real value, W Since W Consider the derivative of the Lyapunov function and substitute e_ n into it, we will have
Let Δu ¼ v u; Δu is the error. Substituting Δu into e_ n e_ n ¼ f ðxÞ þ gðxÞðv ΔuÞ x_ nd þ dðtÞ
ð31Þ
W τnT SðZ τ Þ ¼ Δu εr
Choose the barrier Lyapunov function as V i ¼ V i 1 þ 12 e2i
_ n x_ nd f 1 ge en dðtÞ en þ 2 kn kx en þ Δu g 2 g2 g g g
T
Since κ 2 4 0, according to the Lyapunov theorem, this yield is stable.□
xnði þ 1Þd ¼ ki ei þ x_ id
vn ¼
To ensure κn 4 0, the control gains should be properly chosen to satisfy the following conditions: ki 4 0 ði ¼ 1; 2; …n 1Þ; kn þ 1 4 0, according to the Lyapunov theorem, this yield is stability.□
V_ 2 ¼ V_ 1 þ e2 e_ 2 2
Consider system (1) satisfying Assumptions 1–6. If the input v is chosen as
κn ¼ minf2k1 ; 2k2 ; …; 2ðkn þ 1Þg
Making the time derivative of V 2 gives
r k1 ln
ð30Þ
2
Substituting e_ n into the derivative of the Lyapunov function (29),
2
~ TW ^ στ W ~ TW ^ τ þ en dðtÞ en σW τ g g2
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
ð41Þ
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~ TW ~ Þ ¼ W ~ TW ~ T ^ ¼ W ~ T ðW n þ W ~ W ~ T W n and W Since W ~ TW ~ TW ~ þ W nT W n Þ, we have W ^ r 1W ~ TW ~ þ 1W nT W n . W n r 1 ðW 2
2
Meanwhile, en ε en εr r e2n þ 12ε2 þ 12ε2r r e2n þ en dðtÞ g
e2n g2
2
1 n 2 2Jε J þ
2 1 2 J εr J ;
r J dðtÞ J 2 r J dM J 2 . We substitute these into the V_ n
V_ n r k1
e21 2
b e21
k2 e22 ⋯ kn
e2n g
1 2 C e rV n ð0Þ þ 2 i κ
1 ~ T ~ 1 ~ T ~ σW W Wτ Wτ 2 2
(
2σ τ 2σ κ ¼ min 2k1 ; 2k2 ; …; 2kn ; ; λmax ðΓ τ 1 Þ λmax ðΓ 1 Þ
)
C ¼ 12 W nT W n þ 12 W nτ T W nτ þ 12 J εn J 2 þ 12 J εr J 2 þ J dM J 2
ð54Þ
For ei , we have
1 1 þ σW nT W n þ σ τ W nτ T W nτ e2n en ðεr þε þ kx en Þ þ J dM J 2 2 2 1 1 1 1 r κV n þ σW nT W n þ σ τ W nτ T W nτ þ J εn J 2 þ J εr J 2 þ J dM J 2 2 2 2 2 ¼ κV n þ C ð42Þ where
Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b ðe2D 1Þ J e1 J r e2D
5
ð43Þ
ð44Þ
To sure κ 4 0 and C 4 0, the design parameters kx Z 1; ki 4 0 ði ¼ 1; 2; …; nÞ. There, we have the following theorem.
ð55Þ
Then, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C J ei J r 2 V n ð0Þ þ κ ~ and W ~ τ , we also can obtain As for W vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u C u uV n ð0Þ þ t κ ~ JW J r λmin ðΓ 1 Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u C u uV n ð0Þ þ t κ□ ~ JW τ J r λmin ðΓ τ 1 Þ
ð56Þ
ð57Þ
ð58Þ
4. Output feedback control Theorem 1. Considering the nonlinear affine system (1) with unknown disturbance, input deadzone and output constraint. The proposed state feedback RBFNN control law (34), neural network updating laws (35), (36) with bounded initial conditions, the closedloop system signals ei ; W and others are semi-globally bounded. ~ and W ~ τ will remain Furthermore, the closed-loop error signals ei ; W within compact sets Ωei ; ΩW~ ; ΩW~ τ respectively, defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 2 < b ðe2D 1Þ= n ð45Þ Ωe1 : ¼ e1 A R j J e1 J r : ; e2D n
o pffiffiffiffiffiffiffi Ωei : ¼ ei A Rn j J ei J r 2D ði ¼ 2; …; n 1Þ Ωen
n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio : ¼ en A R j J en J r 2gðxÞD n
( ~ A Rn j J W ~ Jr W
ΩW~ : ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ 1 Þ
( ΩW~ τ : ¼
~ τ A Rn j J W ~ τJr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ τ 1 Þ
ð46Þ ð47Þ
ð48Þ
ð49Þ
In this section, the high-gain observer is introduced to complete the output feedback control. According to Lemma 3, we know that πkϵþk 1 converges to yðkÞ under bounded errors so that π kϵþk 1 will be a suitable observer to estimate the unmeasurable state xk þ 1 . Define the estimate of unmeasurable states as h π π π n iT 2 3 x^ ¼ x1 ; ; 2 ; ⋯ n 1 ð59Þ ϵ ϵ ϵ Considering the following linear system: ϵπ_ 1 ¼ π 2 ϵπ_ n ¼ l1 π n l2 π n 1 ⋯ ln 1 π 2 π 1 þ yðtÞ According to Lemma 2, we have πn yðn 1Þ ¼ ϵςðnÞ ϵn 1
ð60Þ
ð61Þ
where ϵ is any small constant, and there exist positive constants t n and r n such that 8 t 4 t n , we have J βn J r ϵr n . So we can use ϵnπn 1 to estimate xn , then xn ; en can be estimated as follows: πn ð62Þ x^ n ¼ n 1 ϵ e^ n ¼ x^ n xnd
ð63Þ
where D ¼ V n ð0Þ þ Cκ with C and κ given in (43) and (44) where both are positive definite.
e~ n ¼ e^ n en ¼
Proof. Multiplying (42) by eκt yields
Revisit the full state control, RBFNN updating law with e^ n , we have
d ðV n eκt Þ r Ceκt dt Integrating the above inequality, we obtain C C C r V n ð0Þ þ V n r V n ð0Þ e κt þ κ κ κ Then, we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C J en J r 2ngðxÞn V n ð0Þ þ κ
T
ð50Þ
ð51Þ
ð52Þ
2
1
^ SðZ^ Þ þ W ^ τ SðZ^ τ Þ v ¼ kx e^ n þ W
ð64Þ
ð65Þ
The network updating laws are designed as
For e1 , we have 1 b C ln r V n ð0Þ þ 2 b2 e2 κ
πn πn xnd ðxn xnd Þ ¼ n 1 yðn 1Þ ¼ βn ϵn 1 ϵ
ð53Þ
^_ ¼ ΓðSðZ^ Þe^ n þ σ W ^ Þ W
ð66Þ
^ τÞ ^_ τ ¼ ΓðSðZ^ τ Þe^ n þσ τ W W
ð67Þ
The overall control strategy for output feedback control is given in Fig. 3. ^ T SðZ^ Þ þ W ^ τ SðZ^ τ Þ into the e_ n , we will Substituting v ¼ kx e^ n þ W have ^ T SðZ^ Þ þ W ^ τ SðZ^ τ Þ W nT SðZÞ W nT SðZ τ Þ e_ n ¼ g kx e^ n þ W τ _ n 1 ge en dðtÞ en 2 εr ε kn þ ð68Þ þ 2 g2 g g g
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W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
1 1 V_ n r V_ n 1 þ J γ S J 2 J W n J 2 þ J γ τ Sτ J 2 J W nτ J 2 2 2 σ ~ 2 σ τ ~ 2 l 2 lτ 2 þ J W J þ J W τ J þ e~ n þ e~ n 4 σ 4 στ 1 e2 þ ð1 kx Þe2n kn n en ðεr þεÞ 2 g T 1 2 ~ W ^ στ W ~ TW ^ τ þ J dM J 2 þ kx e~ n σ W τ 2
ð81Þ
Since en ðεr þεÞ r e2n þ 12ε2r þ 12ε2 re2n þ 12 J εnr J 2 þ 12 J εn J 2 , meanwhile, ~ TW ~ Þ ¼ W ~ TW ~ T Wn r ^ ¼ W ~ T ðW n þ W ~ W ~ T W n and W W T T T 1 ~ ~ W ~ þ W nT W n Þ, we have W ^ r 1W ~ W ~ þ 1W nT W n . SubðW W 2
stituting these inequalities into V_ n
Fig. 3. Output feedback control strategy.
ð69Þ
V_ n r V_ n 1 þen ðW nT ðSðZ^ Þ SðZÞÞ þ W τnT ðSðZ^ τ Þ sðZ τ ÞÞÞ ~ T SðZ^ Þ e~ n W ~ T SðZ^ τ Þ e~ n W τ e2n ^ στ W ~ TW ^ τ kx en e^ n þ J dM J 2 ~ TW en ðεr þ εÞ σ W τ g
ð70Þ
Meanwhile, we have SðZ^ Þ SðZÞ ¼ γ S
ð71Þ
SðZ^ τ Þ SðZ τ Þ ¼ γ τ Sτ
ð72Þ
where S and Sτ are bounded vector functions with satisfying j Sj o Sn and j Sτ j o Snτ and γ 4 0. Consider V_ n , by applying en e~ n r 12 e2n þ 12 e~ 2n
J γ τ Snτ J 2 þ σ τ l lτ J W nτ J 2 þ e~ 2n þ e~ 2n σ 2 στ 1 1 1 þ kx e~ 2n þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2 2 þ
Let estimation error e~ n ¼ e^ n en . Consider the derivative of the Lyapunov function and substitute e_ n into it, we have
kn
2
e2 σ ~ 2 σ τ ~ 2 J γ Sn J þ σ J JW τ J þ J Wn J2 V_ n r V_ n 1 kn n J W 2 g 4 4
Choose the barrier Lyapunov function as 1 e2n 1 ~ T 1 ~ 1 ~ T 1 ~ þ W Γ W þ WτΓ Wτ Vn ¼ Vn1 þ 2 gðxÞ 2 2
2
ð73Þ
ð82Þ
Substituting (64) into the above equation, we have e2 σ ~ 2 σ τ ~ 2 J γ Sn J 2 þσ J JW τ J þ JWn J2 V_ n r V_ n 1 kn n J W 2 g 4 4 J γ Sn J 2 þ σ τ l lτ 1 þ þ kx ðβn Þ2 J W nτ J 2 þ þ τ τ σ στ 2 2 1 1 þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2 ¼ κV n þ C where
ð83Þ
(
) στ σ ; λmax ðΓ τ 1 Þ λmax ðΓ 1 Þ n 2 JγS J þσ J γ Sn J 2 þ σ τ l lτ 1 JWn J2 þ τ τ þ þ kx ϵ2 r 2n J W nτ J 2 þ C¼ 2 σ στ 2 2
κ ¼ min 2k1 ; 2k2 ; …; 2kn ;
ð84Þ
we have 1 1 þ J εnr J 2 þ J εn J 2 þ J dM J 2 2 2
~ T SðZ^ Þ e~ n W ~ T SðZ^ τ Þ V_ n r V_ n 1 þen W n γ S þ en W nτ γ τ Sτ e~ n W τ e2n
~ TW ^ στ W ~ TW ^ τ þ 1kx e~ 2 1kx e2 en ðεr þ εÞ σ W τ n 2 2 n
ð85Þ
ð74Þ
To sure κ 4 0; C 4 0, the design parameters kx 4 4, ki 4 0. There, we have the following theorem.
en W n γ S r 12 e2n þ 12 J γ S J 2 J W n J 2
ð75Þ
en W n γ τ Sτ r 12 e2n þ 12 J γ τ Sτ J 2 J W nτ J 2
ð76Þ
Theorem 2. Considering system consisting of (1) under the assumptions and the bounded changing rate of the external disturbance, the output feedback control (65), the NN weights updating laws (66), (67). For bounded initial conditions, all signals of the closed-loop system are bounded. Furthermore, the closed-loop error signals ~ and W ~ τ will remain within compact sets Ωei ; Ω ~ ; Ω ~ respecei ; W W Wτ tively, defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 8 2 < b ðe2D 1Þ= ð86Þ Ωe1 ¼ e1 A Rn j J e1 J r : ; e2D
kn
g
Since
and ~ J 2 þ 1 J SðZ^ Þ J 2 e~ 2 ~ T SðZ^ Þ r σ J W e~ n W n 4 σ ~ SðZ^ τ Þ r e~ n W τ T
στ ~ 2 1 J W τ J þ J SðZ^ τ Þ J 2 e~ 2n στ 4
ð77Þ ð78Þ
Substituting J SðZ^ Þ J 2 r l and J SðZ^ τ Þ J 2 r lτ , we have ~ J 2 þ 2l e~ 2 ~ SðZ^ Þ r σ J W e~ n W 4 2σ n σ ~ 2 2l 1 2 J þ e~ ¼ JW 4 σ2 n T
~ T SðZ^ τ Þ r σ τ J W ~ τ J2 þ e~ n W τ 4 σ τ ~ 2 2lτ 1 2 e~ ¼ JW τ J þ 4 στ 2 n
ð79Þ
ð87Þ
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio Ωen ¼ en A Rn j J en J r 2gðxÞD
ð88Þ
(
2lτ 2 e~ 2σ τ n
Substituting (79) and (80) into (74), we have
n o pffiffiffiffiffiffiffi Ωei ¼ ei A Rn j J ei J r 2D ði ¼ 2; …; n 1Þ
ΩW~ ¼
~ A Rn j J W ~ Jr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ 1 Þ
(
ð80Þ ΩW~ τ ¼
~ τ A Rn j J W ~ τJr W
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) D λmin ðΓ τ 1 Þ
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
ð89Þ
ð90Þ
W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Proof. The proof of theorem is straightforward using the method of proof in Theorem 1, and hence, is omitted.□ 5. Simulation study In this section, we will make use of the proposed adaptive neural control (31), (34), (65) and to the nonlinear affine system with deadzone, output constraint and disturbance. The system is described as follows: 8 x_ 1 ¼ x2 > > > > < x_ 2 ¼ x3 ð91Þ x_ 3 ¼ x1 x2 3x3 þ ð1 þ 0:2x1 Þu þ dðtÞ > > > > : y ¼ x1 The deadzone is set to be br ¼ 2:5 and bl ¼ 4:5. The output constraint is set to be b ¼ 0:05. The external disturbance is set as dðtÞ ¼ 3ð1 expð 0:01tÞÞ sin ð0:5πtÞ
ð92Þ
previously defined in Case 2, the new parameters are added to the simulation: kp ¼ 20; kd ¼ 0:5. The tracking performance of the closed-loop system is given in Fig. 5. From the figure, we can state that all the four kinds of control ((31), (34), (65) and the PD control) can successfully track the desired trajectory, where the errors of the closed-loop system are converging to a small value close to zero. However, from Fig. 6,
5
ð94Þ
In order to verify the methods for improving the robustness of the system, we have carried out four cases for simulation. Case 1: Model based control (31). When the deadzone function, the external disturbance and the dynamics are fully known, we examine the control performance for the model based control. The control gains are given as k1 ¼ 20; k2 ¼ 30, k3 ¼ 40; k ¼ 20 and kx ¼ 50. The model based control needs to know the external disturbance dðtÞ, the system dynamics f ðxÞ; gðxÞ, which are difficult to obtain in practice. Case 2: State feedback control (34). The new parameters involved in the state feedback control are added to the simulation. Unknown deadzone is defined to have br ¼ 2:5 and bl ¼ 4:5. Given the situation that we have no information about the deadzone function, system dynamics and disturbance, the proposed control (34) can be implied to track the desired trajectory. We use two RBFNNs to estimate the unknown functions, which can approximate the deadzone error, system dynamics and external disturbance, respectively. Furthermore, the duration of simulation t f ¼ 40. For the two RBFNNs with Z ¼ ½x1 ; x2 ; x3 ; e 3 and Z τ ¼ ½x1 ; x2 ; x3 ; v, numbers of nodes l1 for S is 24 and l2 for Sτ is 24 . Variances for both neural networks are set to be σ ¼ 0:001; σ τ ¼ 0:001; Γ ¼ 50I 1616 , Γ τ ¼ 50I 1616 . RBFNN centers for S are evenly distributed in the domain of ½ 1; 1 ½ 1; 1 ½ 1; 1 ½ 1; 1 and centers for Sτ are evenly distributed in domain of ½ 1; 1 ½ 1; 1 ½ 1; 1 ½ 1; 1. Initial weights of the neural network are all zero. The control gains are given as k1 ¼ 20, k2 ¼ 30; k3 ¼ 30; k ¼ 30 and kx ¼ 30. Case 3: Output feedback control (65). With all parameters being the same as previously defined in Case 2, the new parameters are added to the simulation: ϵ ¼ 0:005; l1 ¼ 1 and initial terms π 1 ¼ 0; π 2 ¼ _ 0. 0; π 3 ¼ 0; π_ 1 ¼ 0; π_ 2 ¼ 0; π 3 ¼ Case 4: PD control. The PD control is designed as follows: v ¼ kp e3 kd …e3 . With all parameters being the same as
2
0
5
10
15
20
25
30
35
40
t [s] Fig. 4. Disturbance.
The tracking performance 2 the PD control the control based model the state feedback control the output feedback control the desired trajectory the upper bound of the output constraint the lower bound of the output constraint
1 0.95
1.5
0.9 3.5 4 4.5
1 position [rad]
y_ d ¼ 0:5 ð0:5 cos ð0:5 tÞ 1:5 sin ð1:5 tÞÞ
3
0
0.5
0
−0.5
−1 0
5
10
15
20
25
30
35
40
t [s]
Fig. 5. System output and desired trajectory.
The error of the tracking performance 0.02 the error of the PD control the error of the control based model the error of the state feedback control the error of the output feedback control
0.015
0.01 position [rad]
ð93Þ
4
1
Take note that the external disturbance vibrates with a high frequency, which indicates a big value for the time derivative. In general cases, the time derivative usually consider to be a small value with none or very low vibration frequency. In our case, shown in Fig. 4, which is the external disturbance, the high frequency disturbance is utilized to test the feasibility of proposed nonlinear disturbance observer. Initial positions of the system are y1 ð0Þ ¼ 0:54 and y2 ð0Þ ¼ 0. And desired trajectory as follows: yd ¼ 0:5 ð sin ð0:5 tÞ þ cos ð1:5 tÞÞ
External disturbance
6
external disturbance [rad]
where D ¼ V n ð0Þ þ Cκ with C and κ given in (84) and (85) where both are positive definite.
7
0.005
0
−0.005
−0.01 0
5
10
15
20
25
30
35
t [s]
Fig. 6. The tracking error.
Please cite this article as: He W, et al. Adaptive neural network control of unknown nonlinear affine systems with input deadzone and output constraint. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.014i
40
W. He et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6. Conclusion In this paper, adaptive control schemes have been studied for a class of nonlinear affine systems with both deadzone and constraint by using RBFNN. An adaptive controller is firstly developed for the systems with the assumptions holding on a given compact set. The stability of the closed-loop adaptive neural system is guaranteed via Lyapunov stability theorem. In addition, with the The error of the tracking performance 40 the control input of the control based model the control input of the state feedback control the control input of the output feedback control
30
control input [Nm]
20
30 Norms for output feedback control norm W
we can see that the errors of the PD control are larger than others, because of the neural network learning ability. Meanwhile, it is satisfying that the tracking trajectory of the NN control is showing a good performance with small errors, which indicates that the constraint satisfaction can be ensured. However, in order to achieve the control objective, the model-based control needs to require the fully known system dynamics, which is difficult to obtain. Furthermore, the contribution of neural networks is shown clearly by comparing the tracking performances and errors of different controllers, which can state that the neural networks can also approximate the unknown system dynamics and ensure the stability of the system. Fig. 7 shows the control inputs. It can be seen that, within the time interval ½0 2, the control v will establish a large control action to drive the error to a small value closed to zero. That is the reason why there exist some spikes in this time interval. The boundedness of NN weights is also depicted in Figs. 8 and 9.
20 10 0
0
5
10
15
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25
30
35
40
t [s] 80 Norms for output feedback control norm Wtau
8
60 40 20 0
0
5
10
15
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25
30
35
40
t [s]
Fig. 9. The norms of output feedback control NN weights.
aim of compensating for the input deadzone and the output constraint effects, the direct neural network control is built. In order to guarantee the closed-loop stability, Lyapunov's direct method has been applied in the control design process. Furthermore, simulation study has demonstrated that the proposed direct neural network control method could successfully drive the system to track desired curve under the effects of the input deadzone and the output constraint. Overall, the control objective has been reached, all signals in the closed-loop systems have been guaranteed to be semi-globally uniformly ultimately bounded, and eventually converge to the corresponding small compact sets. This article employs the RBF neural network to solve the input deadzone and output constraint of the nonlinear affine systems, and the simulation results show that the proposed control is effective.
10
Acknowledgment
0
The authors would like to thank the Editor-In-Chief, the Associate Editor and the anonymous reviewers for their constructive comments which helped improve the quality and presentation of this paper.
−10 −20 −30
References
−40 0
5
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35
40
t [s]
Fig. 7. The control input.
30
norm W
Norms for state feedback control
20
10
0
0
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40
t [s] 0.4
norm W
tau
Norms for state feedback control
0.3 0.2 0.1 0 0
5
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15
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25
30
t [s]
Fig. 8. The norms of state feedback control NN weights.
35
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