51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
NN-based Asymptotic Tracking Control for a Class of Strict-feedback Uncertain Nonlinear Systems with Output Constraints Wenchao Meng, Qinmin Yang, Donghao Pan, Huiqin Zheng, Guizi Wang, and Youxian Sun Abstract— An asymptotic tracking control law is proposed for a class of strict-feedback nonlinear systems with unknown nonlinearities. A Barrier Lyapunov function in combination with backstepping is proposed to guarantee that the output trajectory is contained in a predefined set. A single neural network (NN), whose weights are tuned online, is utilized in our design to approximate the unknown functions in the system dynamics, while the singularity problem of the control gain function is avoided. Meanwhile, in order to compensate for the NN residual reconstruction error and system uncertainties, a robust term is introduced and asymptotic tracking stability is achieved. All the signals in the closed-loop system are proved to be bounded via Lyapunov synthesis and the output converges to the desired trajectory asymptotically without transgressing a given bound. Finally, the merits of the proposed controller are verified in the simulation environment.
I. INTRODUCTION Recent years have witnessed ever increasing interests in tracking control of nonlinear systems with unknown dynamics from control community [1], [2]. In the literatures associated with adaptive control, neural networks are considered to be a promising candidate to estimate the uncertain or poorly known dynamics by exploiting their universal approximation capabilities, and then various NN-based controllers have been developed rather than relying on building a model of the actual plant [3], [4], [5]. However, most works have focused on the stabilization of the interested system, while the output constraints that the plants are usually subjected to are rarely considered. To tackle the problem of output constraints, the idea of Barrier Lyapunov function (BLF) was recently raised and has been extensively utilized to assist the controller design for systems in different forms by exploiting the property of the value of BLF which will escape to infinite when the associated states approach some limits. Barrier Lyapunov function was first proposed to address the issue of state constraints in Brunovsky form [6] systems. In [7], BLF-based control design was proposed for a class of strict feedback nonlinear This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61104008, National Key Basic Research Program of China (973 Program) under Grant 2012CB724404, Qianjiang Program of Zhejiang Province under Grant 2011R10024, Commonweal Technology Research Foundation of Zhejiang Province under Grant 2011C31021, and Research Fund for the Doctoral Program of Higher Education of China under Grant 20110101120063. Wenchao Meng, Qinmin Yang, and Youxian Sun are with the State Key Laboratory of Industrial Control Technology, and the Department of Control Science and Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, China (e-mail:
[email protected]). Donghao Pan, Huiqin Zheng, and Guizi Wang are with Zhejiang Windey Co.,Ltd and the State Key Laboratory of Wind Power System, Hangzhou, Zhejiang 310012, China.
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
systems with output constraints and known dynamics, and extended it to uncertain systems with partially unknown nonlinearities. In [8], the authors investigated its use in output constrained nonlinear systems with unknown constant control gain function. However, it should be noted that a number of works on output constrained systems have assumed that the control gain function was known [7] or unknown constant [8] whereas the controller design for systems with completely unknown control gain function will become much more complex. At the same time, without considering the output constraints, control of uncertain nonlinear systems in the case of unknown control gain have been extensively studied. For example, Ge [4] developed an adaptive controller by using neural network to learn a combined nonlinearities. Park [9] estimated the control gain function directly which may result in singularities. Nevertheless, all these methods can not be applied to the case when the control gain function is completely unknown. Therefore, control of nonlinear output constrained systems with completely unknown nonlinearities needs to be further developed. Meanwhile, due to NN reconstruction errors and other system uncertainties, NN-based control methodologies often guarantee the tracking error to be uniformly ultimately bounded (UUB) [3], i.e. the aforementioned literatures about BLF-based control achieved UUB results when there exist completely unknown nonlinearities. A significant effort has been made in order to obtain asymptotic tracking stability by blending neural networks and various discontinuous feedback strategies, such as sliding-mode control (SMC) [10], [11] and variable structure controllers (VSC) [12], [13]. A main drawback of these discontinuous controllers is the requirement of infinite bandwidth and chattering. To overcome these problems, several researchers [14], [15] have proposed continuous control laws to cancel the steady-state error and achieve asymptotic tracking goal. For example, in [5], the RISE technical combining with two-layer neural networks was used to obtain semi-global asymptotic tracking performance. However, all these literatures have not consider the output constraints problem. Thus, control of outputconstrained systems with asymptotic tracking error is still a challenging problem. By contrast, in this paper, the control of a class of strictfeedback unknown nonlinear systems with output constraints is considered and a novel asymptotic tracking NN controller is developed. The output violation is prevented by utilizing a Barrier Lyapunov function. However, compared with the above-mentioned literatures on BLF-based control design, our proposed approach address the challenging issue where
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the control gain function are assumed to be totally unknown apart from it’s a function of state. Meanwhile, a novel robust strategy which yields continuous control input is developed and asymptotic tracking stability is achieved. Finally, the asymptotical tracking performance and the boundedness of all signals in the closed-loop system are shown by using Lyapunov synthesis. II. PRELIMINARIES
III. CONTROLLER METHODOLOGY
A. Problem formulation Consider a class of high order, single-input-single-output (SISO) continuous-time strict-feedback nonlinear systems of the form x˙ i = fi (xi ) + gi (xi )xi+1 , x˙ n = fn (xn ) + gn (xn )u, y = x1
where is the approximation error which can be made arbitrarily small by increasing the number of hidden layer nodes. Since the hidden layer weight matrix V is fixed during the learning process, we omit it in the following parts for clarity. Assumption 3: The approximation error is upper bounded such that || ≤ m , with m being a known constant.
1≤i≤n−1 n≥2 (1)
where u, y ∈ denote the input and output, respectively; xi = [x1 , x2 , · · · , xi ]T ∈ Ωxi ⊂ i , i = 1, . . . , n are state variables; fi (·) and gi (·), i = 1, . . . , n are totally unknown smooth functions. Assumption 1: The functions gi (¯ xi ), i = 1, . . . , n are positive for ∀xi ∈ Ωxi ⊂ i , and there exist constants so that gi0 ≤ gi (¯ xi ) ≤ gim with gi0 , gim > 0 being its lower and upper bounds. Assumption 2: The desired trajectory yd is continuously differentiable and available for controller design. Furthermore, yd is upper bounded such that |y| ≤ ydm with ydm being a known positive constant. 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, i.e. the tracking error e1 → 0 as t → ∞, while the output y is guaranteed to be within certain predefined bounds for all time. Remark 1: The above-mentioned assumptions are not very strict and they are common assumptions in the control literatures [1]. Also the bounds of gi (¯ xi ) are not required to be known since they are only need for stability analysis. B. Two-layer neural networks A class of two-layer neural networks is utilized for our controller design. The NN output is defined as O(Z) = W T φ(V T Z)
(2)
where Z ∈ n1 and O ∈ n3 are the NN input and output with n1 , n3 the number of input and output nodes. W ∈ n2 ×n3 represents the output layer weight, and V ∈ n1 ×n2 denotes the hidden layer weight which is initialized to random values and held fixed in learning phase. φ(·) is the so-called activation function determined as the hyperbolic tangent function in our study. It have been proven that for any smooth function f (Z) over a compact set ΩZ there exist an optimal weight W ∗ such that [16] f (Z) = W ∗T φ(V T Z) +
(3)
A. Barrier Lyapunov Function In this section, a Barrier Lyapunov function is introduced to assist our controller design which is defined as follows. Definition 1: If, in an open region BR , a scalar function V (x) is positive definite and continuously differentiable , and V (x) → ∞ as x approaches the boundary of BR , bounded by a small positive constant along any state trajectory of system x˙ = f (x) with x(0) ∈ BR . then V (x) is said to a Barrier Lyapunov function for this system [7]. In our controller design, the following symmetric Barrier Lyapunov function [6] is chosen which satisfies definition 1. 1 b2 (4) log 2 2 b − x2 where x denotes the state to be constrained, and b denotes the bound of x. T Lemma 1: If the state vector x(t) = x1 (t)T , x2 (t)T x1 (t) ∈ , x2 (t) ∈ n in the continuous system V (x) =
x(t) ˙ = f (x(t))
(5)
has an associated Lyapunov function V (x) = V1 (x1 ) + V2 (x2 ) defined on x1 ∈ {x1 | |x1 | < b, b > 0} , x2 ∈ n with the following properties. V1 (x1 ) → ∞ as |x1 | → b, κ1 ( x1 ) ≤ V2 (x1 ) ≤ κ2 ( x1 ) (6) where κ1 and κ2 are K class functions. Thus, x1 (t) remain in the set x1 (t) ∈ {x1 | |x1 | < b, b > 0} provided that the initial value x1 (0) belongs to the set x1 (0) ∈ {x1 | |x1 | < b, b > 0} and the derivative of V (x) satisfies 2 V˙ (x) ≤ −λ x + ε
∀x ∈ m+1
(7)
where λ > 0 and ε → 0 as t → ∞. Proof: The proof is similar to [8] and thus omitted . Remark 2: In lemma 2, the system states are divided into x1 and x2 , where only x1 is required to be constrained and x2 is not, Therefore, only x1 requires the Barrier Lyapunov function. B. Controller design The controller design for system (1) is proposed in this section by using the Barrier Lyapunov function in combination with the beckstepping scheme. For clarity, we shall present the detailed controller design procedure in a constructive way as follows. Step 1: Let e1 = x1 − yd , and its derivative with respect to time is e˙ 1 = f1 (x1 ) + g1 (x1 )x2 − y˙ d (8)
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According to the backstepping scheme, let x2d x2 and the ideal virtual controller x∗2d can be chosen according to the following lemma. Lemma 2: For the system (8), if the ideal virtual controller is chosen as x∗2d = −k1 e1 (b2 − e1 2 ) −
f1 (x1 ) − y˙ d g1 (x1 )
(9)
with k1 > 0, then e1 is asymptotically stable. Proof: Substituting the ideal controller x2 = x∗2d into (8), we obtain e˙ 1 = −k1 g1 (x1 )e1 (b2 − e21 ). Choose the Barrier Lyapunov function as V1 =
1 b2 log 2 2 b − e21
ˆ 1 > φm1 /σ1 . V˙ w will become negative as long as W ˆ ˆ Therefore, W1 ∈ Ωw1 if W1 (0) ∈ Ωw1 for t ≥ 0 Define e2 = x2 − x2d and substitute (14) into (8), we have ˆ 1T φ1 (X1 ) + f1 (x1 ) − y˙ d e˙ 1 =g1 (x1 )[−k1 e1 (b2 − e21 ) − W g1 (x1 ) + e2 + ur1 + N1 − N1 ] ˜ T φ1 (X1 ) + e2 + ur1 + 1 ] =g1 (x1 )[−k1 e1 (b2 − e21 ) − W 1 (18) ˆ 1 − W ∗ . In the following parts, we all use ˜1 = W where W 1 the definition that (˜·) = (ˆ·) − (·)∗ . Choose the Barrier Lyapunov function
(10) V1 =
Making the time derivative of V1 gives e1 e˙ 1 V˙ 1 = 2 = −k1 g1 (x1 )e21 b − e21
(11)
(12)
where N1 (f1 (x1 ) − y˙ d )/g1 (x1 ). Note that this ideal controller is not implementable owing to the unknown N1 . However, N1 is a smooth function in a compact set Ωx1 , thus it can be approximated by the two layer neural networks as N1 = W1∗T φ1 (X1 ) + 1
e1 e˙ 1 = −g1 (x1 )k1 e21 V˙ 1 = 2 b − e21 g1 (x1 )e1 ˜ T −W1 φ1 (X1 ) + 1 + ur1 + e2 + 2 2 b − e1
ˆ 1T φ1 (X1 ) + ur1 x2d = −k1 e1 (b2 − e1 2 ) − W
(14)
where k1 > 0 is a design constant, ur1 is the robust term to improve the robustness with respect to the system ˆ 1 is chosen as uncertainties. The adaptive law for W ˆ 1) ˆ˙ 1 = Γ1 (φ1 (X1 )e1 − σ1 |e1 |W W
(15)
where σ1 , Γ1 = ΓT1 > 0, the following lemma shows the ˆ 1 [2] boundedness of W Lemma 3: For adaptive law (15), there exists a compact set φm1 ˆ ˆ Ωw1 = W1 | W1 ≤ (16) σ1 ˆ 1 (t) ∈ where φ1 (X1 ) ≤ φm1 with φm1 > 0, such that W ˆ Ωw1 , ∀t ≥ 0 provided that W1 (0) ∈ Ωw1 . ˆ 1 , its time derivative ˆ T Γ−1 W Proof: Let Vw1 = 1/2W 1 1 is ˆ T (φ1 (X1 )e1 − σ1 |e1 |W ˆ 1) V˙ w = W 1 ˆ 1 (σ1 W ˆ 1 − φm1 ) ≤ −|e1 | W
(20)
ˆ 1 is upper bounded as stated in Since g1 (x1 ) > 0 and W ˜ 1 ≤ w1 with w1 being a Lemma 3, which implies that W positive constant, it is also noted that φ1 (X1 ) ≤ φm1 and |1 | < m1 . Therefore, the derivative of V˙ 1 can be rewritten as g1 (x1 )e1 e2 V˙ 1 ≤ −g1 (x1 )k1 e21 + + Λ1 b2 − e21
(13)
where W1∗ is the optimal weight vector of N1 and |1 | < m1 with m1 > 0. In practice, the optimal weight W1∗ is difficult to obtain. ˆ 1 , we propose Thus, by replacing it with its estimated value W the following implementable virtual controller.
(19)
and invoking (18), we obtain the derivative of V1 as
Since k1 > 0 and g1 (x1 ) > 0, according to the Lyapunov theorem, this yields lim e1 = 0. t→∞ The ideal controller (9) can be rewritten as x∗2d = −k1 e1 (b2 − e1 2 ) − N1
1 b2 log 2 2 b − e21
e1 g1 (x1 )e1 ur1 E1 with Λ1 = + g1 (x1 ) 2 b2 − e21 b − e21 Δ
(21)
(22)
Δ
where E1 = w1 φm1 +m1 . To attain the asymptotic tracking control goal, the robust term ur1 must be designed properly to cancel the effect of E1 in (21). Then, we determine the robust term ur1 as follows ur1 = −
ρE12 |ρ|E1 + exp(−β1 t)
(23)
Δ
where ρ = e1 /(b2 − e21 ), and β1 > 0 is a design parameter, exp(·) denotes the exponential function. Substituting ur1 into (22), we will have the following inequality
ρ2 E12 Λ1 =g1 (x1 ) |ρ|E1 − |ρ|E1 + exp(−β1 t) ≤g1 (x1 ) exp(−β1 t) (24) Then, combining (24) with (21), V˙ 1 becomes g1 (x1 )e1 e2 V˙ 1 ≤ −g1 (x1 )k1 e21 + 2 + g1 (x1 ) exp(−β1 t) (25) b − e21 Step 2: We attempt to force the virtual controller x2d to track the state x2 . The time derivative of e2 is obtained as
(17) 5412
e˙ 2 = f2 (¯ x2 ) + g2 (¯ x2 )x3 − x˙ 2d
(26)
Similarly, x3 can be seen as the virtual control input for the e2 -subsystem, i.e., if we choose x∗3d = x3 . An ideal controller can be chosen as x2 ) − x˙ 2d f2 (¯ g1 (x1 )e1 x∗3d = −k2 e2 − − (27) g2 (¯ x2 ) g2 (¯ x2 )(b2 − e21 ) with k2 > 0 being a design constant. Note that the last term in (27) is introduced to cancel the coupling term in derivative of V1 . The ideal controller x∗3d can be rewritten as x∗3d = −k2 e2 − N2
(28)
Δ
x2 ) − x˙ 2d )/g2 (¯ x2 ) + (g1 (x1 )e1 )/(g2 (¯ x2 ) where N2 = (f2 (¯ (b2 − e21 )). Since the nonlinearities f2 (¯ x2 ) and g2 (¯ x2 ) are unknown, the ideal controller x∗3d can not be implemented in practice. By using the neural network approximation property, N2 can be expressed as N2 = W2∗T φ2 (X2 ) + 2
(29)
where |2 | < m2 with m2 > 0. Similarly, Choose the virtual controller x3d as ˆ 2T φ2 (X2 ) + ur2 x3d = −k2 e2 − W
(30)
and substituting it into (26), we will have ˆ 2T φ2 (X2 ) + ur2 + e3 e˙ 2 =g2 (¯ x2 )[−k2 e2 − W f2 (¯ x2 ) − x˙ 2d + N2 − N2 ] g2 (¯ x2 ) ˜ 2T φ2 (X2 ) + ur2 + e3 + 2 ] x2 )[−k2 e2 − W =g2 (¯ g1 (x1 )e1 + 2 (31) (b − e2 ) where e3 is defined as e3 = x3 − x3d . The adaptive law for ˆ 2 is chosen similar to W ˆ 1 as W ˆ˙ 2 = Γ2 (φ2 (X2 )e2 − σ2 |e2 |W ˆ 2) W
(32)
ˆ2 where σ2 , Γ2 > 0. For the same reason as Lemma 1, the W ˆ is also upper bounded such that W2 ≤ φm2 /σ2 provided that W2 (0) ≤ φm2 /σ2 with σ2 being a design parameter. We choose the Lyapunov function of the form 1 (33) V2 = V1 + e22 2 By utilizing (31), the derivative of V2 can be rewritten as ˜ T φ2 (X2 ) + 2 + ur2 + e3 V˙ 2 =V˙ 1 + g2 (¯ x2 )e2 −W 2 − k2 g2 (¯ x2 )e22 +
g1 (x1 )e1 e2 (b2 − e2 )
(34)
V˙ 2 ≤−k1 g1 (x1 )e21 −k2 g2 (¯ x2 )e22 + g1 (x1 ) exp(−β1 t) x2 )e2 e3 + Λ2 (35) + g2 (¯ Δ
ur2 = −
e2 E22 |e2 |E2 + exp(−β2 t)
(37)
with β2 > 0 a design parameter. Substituting ur2 into (36), we will have
e22 E22 Λ2 =g2 (¯ x2 ) |e2 |E2 − |e2 |E2 + exp(−β2 t) x2 ) exp(−β2 t) (38) ≤g2 (¯ Invoking (38) and (35), one has V˙ 2 ≤
2
−kl gl (¯ xl )e2l +
l=1
2
gl (¯ xl ) exp(−βl t) + g2 (¯ x2 )e2 e3
l=1
(39) Step i(3 ≤ i ≤ n−1): In a similar way, a virtual controller x(i+1)d 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 virtual controller design. An ideal controller x∗(i+1)d is chosen as x∗(i+1)d = − ki ei − Ni xi ) − x˙ id xi )ei−1 gi−1 (¯ Δ fi (¯ Ni = − gi (¯ xi ) gi (¯ xi )
(40) (41)
and the following virtual control input is determined. ˆ iT φi (x) + uri x(i+1)d = −ki ei − W with uri = −
ei Ei2 |ei |Ei + exp(−βi t)
(42)
(43)
Choose the Lyapunov function of the form 1 Vi = Vi+1 + e2i 2
(44)
and its time derivative can be obtained as V˙ i ≤
i
−kl gl (¯ xl )e2l +
l=1
i
gl (xl ) exp(−βl t) + gi (¯ x)ei ei+1
l=1
(45) Step n: Compared with previous steps, the actual control input u will appear in this step. An ideal controller u∗ is given by u∗ = −kn en − Nn (46) Δ
ˆ 2 is upper bounded, it follows that Since g1 (x2 ) > 0 and W ˜
W2 ≤ w2 with w2 being a constant, and φ1 (X2 ) ≤ φm2 . Thus, the derivative of V˙ 2 can be rewritten as
x2 )e2 ur2 + g2 (¯ x2 ) |e2 | E2 with Λ2 = g2 (¯
Δ
and E2 = w2 φm2 + m2 . In order to cancel the influence of E2 , the robust term ur2 is designed as
xn ) − x˙ nd )/gn (¯ xn ) + (gn−1 (xn−1 )en−1 )/ with Nn = (fn (¯ gn (¯ xn ), by using the two layer neural network to approximate Nn , an implementable controller is determined as ˆ nT φn (Xn ) + urn u = −kn en − W
(47)
with kn > 0 being a design parameter, Similarly, the adaptive ˆ n is chosen as law for W
(36) 5413
ˆ˙ n = Γn (φn (Xn )en − σn |en |W ˆ n) W
(48)
with σn , Γn > 0 being design parameters, and urn the robust term described as en En2 (49) urn = − |en |En + exp(−βn t) Δ ˜ n ≤ wn , with En = wn φmn + mn such that W
φn (Xn ) ≤ φmn and |n | ≤ mn . Differentiating the error en = xn − xnd , we will have
˜ nT φn (Xn ) + urn + n ] xn )[−k2 en − W e˙ n =gn (¯ xn−1 )en−1 (50) + gn−1 (¯ We choose the Lyapunov function of the form 1 Vn = Vn−1 + e2n (51) 2 By utilizing (50), the derivative of Vn can be rewritten as ˜ nT φn (Xn ) + n + urn V˙ n =V˙ n + gn (¯ xn )en −W − kn gn (¯ xn )e2n + gn−1 (xn−1 )en−1 en ≤
n
−kl gl (¯ xl )e2l +
l=1
n−1
Notice that Φ → 0 as t → ∞. According to a standard Lyapunov theorem extension [17], this implies that |ei | → 0, i = 1, 2, . . . , n as t → ∞. 2) x1 is bounded owing to the boundedness of e1 and yd . Since ei = xi − xid and the definitions of virtual controller xid (14), (30) and (42), we can conclude that xid is bounded, and thus xi is bounded. From the definitions of control input (47), we have that the control u is also bounded. Finally, the ˆ i , i = 1, 2, . . . , n are bounded according NN update laws W to lemma 3. 3) According to Lemma 1 and the global Lyapunov function (57), we can conclude that the tracking error e1 resides in the following set Ωe1 = {e1 | |e1 | < b}
for all the time if e0 ∈ Ωe1 , i.e. e1 (t) ∈ Ωe1 , ∀t > 0 From Assumption 2, It is known that |yd | ≤ ydm , it follows that |y| ≤ |e1 | + |yd | = b + ydm = ym
IV. SIMULATION
Δ
with Λn = gn (¯ xn )en urn + gn (¯ xn ) |en | En
(53)
Invoking (49) and (53), we will have
e2n En2 xn ) |en |En − Λn =gn (¯ |en |En + exp(−βn t) ≤gn (¯ xn ) exp(−βn t) (54) Thus, the derivative of Vn becomes n n V˙ n ≤ −kl gl (¯ xl )e2l + gl (¯ xl ) exp(−βl t)
(55)
l=1
Theorem 1: Consider the plant characterized by (1) satisfying Assumptions 1-2, the controller (47) and the adaptive laws (15), (32) and (48). Then we can obtain the following facts. 1) the tracking error e1 = y − yd converges to zero asymptotically, i.e, e1 (t) → 0 as t → ∞. 2) all the other signals in the closed-looped are bounded. 3) the output y remains in the set Ωy = {y ∈ | |y| ≤ ym } with ym = ydm + b Proof: 1) Following (55), we can obtain V˙ n ≤ −
n
ki gi0 e2i + Φ
(56)
(57)
i=1 Δ
with Φ =
n
(61)
gl (¯ xl ) exp(−βl t) + Λn (52)
l=1
l=1
(60)
gim exp(−βi t)
(58)
i=1
where gi0 and gim are the gi (¯ xi )’s lower and upper bounds, respectively. Hence, V˙ n will become negative as long as Φ , i = 1, 2, . . . , n (59) |ei | ≥ ki gi0
To show the effectiveness and merits of our proposed controller, simulation studies are done on the system described by x˙ 1 = 0.2x1 + (4 + 0.2x41 )x2
x˙ 2 = x1 x2 + [3 + sin2 (x1 )]u y = x1
(62)
The desired signal which is followed by the output y is given as yd = sin(t), and the output y is required to reside in the set y ∈ {y||y| ≤ ym }, ∀t ≥ 0 with ym = 3. Apparently, there exists a constant ydm = 1 such that |yd | ≤ ydm . From (56), we can determine the tracking error bound b = ym − ydm = 2. The adaptive gain matrices Γ1 = Γ2 = diag{3.5} with proper dimensions. The initial ˆ 1 (0) = conditions x0 = [x10 , x20 ]T = [1.8, 2]T and W ˆ W2 (0) = 0. Other parameters utilized in this study is shown in Table I. The tracking performance using the proposed controller is shown in Fig. 1. It can be observed that the system output can still track the desired trajectory with asymptotical performance even in the presence of NN reconstruction errors. Moreover, the output y resides in the set Ωy = {y ∈ | |y| ≤ ym = 3} in the simulation time interval. Note that the initial tracking error e1 (0) = x1 (0) − yd = 1.8 belongs to the set {e0 | |e0 | < b = 2}. Fig.2 shows the trajectories of tracking error e1 = y −yd and control input u. In can be seen that, within the time interval [0, 3], the control u(t) will establish a large control action to drive the e1 (t) from the bound b when the tracking error e1 (t) approaches the bound. That is the reason why there exist some spikes in this time interval. We can also see that e1 (t) remains in the set |e1 (t)| < b. In the remain time interval [3, 10], Since the tracking error e1 is very small, the spikes in control u(t)
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TABLE I
NN weights 5
SUMMARY OF PARAMETERS USED IN SIMULATION Parameter k1 k2 β1 β2 E1 E2 σ1 σ2 Value 4 5 10 10 0.1 0.2 0.1 0.1
norm of W1 norm of W2 4
disappear. The boundedness of NN weights is also depicted in Fig. 3.
3
2 Desired/Actual Trajectory 3 actual output y desired trajectory yd
2.5
1
2 0 1.5
0
2
4
6
8
10
time/sec
1
Fig. 3.
L2 norm of NN weights
0.5 0 −0.5 −1 −1.5
0
2
4
6
8
10
time/sec
Fig. 1.
Actual and desired system trajectory
tracking error 4 tracking error e1 tracking error bound b 3
2
1
0
−1
0
2
4
6
8
10
time/sec control signal 100 control input u
60
20
−20
−60
0
2
4
6
8
10
time/sec
Fig. 2.
Top: Tracking error e1 Bottom: The NN based controller
V. CONCLUSIONS In this paper, a novel continuous control law was proposed for uncertain strict-feedback nonlinear systems with an output constraints. by incorporating with the Barrier Lyapunov function and a special robust compensator, The asymptotic tracking goal is attained with the output residing in a predefined set for all time. Furthermore, the developed control approach avoids the controller singularity problem completely.
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