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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 8, AUGUST 2015
Adaptive Neural Control of Nonaffine Systems With Unknown Control Coefficient and Nonsmooth Actuator Nonlinearities Zaiyue Yang, Qinmin Yang, and Youxian Sun
Abstract— This brief considers the asymptotic tracking problem for a class of high-order nonaffine nonlinear dynamical systems with nonsmooth actuator nonlinearities. A novel transformation approach is proposed, which is able to systematically transfer the original nonaffine nonlinear system into an equivalent affine one. Then, to deal with the unknown dynamics and unknown control coefficient contained in the affine system, online approximator and Nussbaum gain techniques are utilized in the controller design. It is proven rigorously that asymptotic convergence of the tracking error and ultimate uniform boundedness of all the other signals can be guaranteed by the proposed control method. The control feasibility is further verified by numerical simulations.
Index Terms— Adaptive neural control, nonaffine systems, nonsmooth nonlinearity, unknown control coefficient. I. I NTRODUCTION Due to both practical needs and theoretical challenges, a lot of efforts have been devoted to adaptive control techniques for nonlinear uncertain systems in recent years [1]. Most of them are concentrated on affine systems in the control input [2]. However, the control signal can be present nonlinearly in the system dynamics in a lot of applications, including air conditioning systems, wind turbines [3], and aircrafts [4]. To tackle the stability problem of nonaffine systems from the perspective of adaptive control, numerous attempts have been made in [5]–[7]. By proving the existence of an implicit desired feedback control law, [5] managed to achieve stability of a class of single-input single-output (SISO) nonaffine systems in normal form via both state and output feedback. To address a more general class of multipleinput multiple-output nonaffine nonlinear systems, a direct adaptive approach is introduced in [6] by using linearly parameterized neural network (NN) structure to approximate a fictional ideal controller, whose existence is assured by the implicit function theorem. The coefficient of control term determines the direction and amplitude of the system motion under a given control signal, which is undoubtedly vital to the stability and control performance of the closed-loop system. Unfortunately, it is clear that for unknown nonaffine systems, the control coefficient is often unknown and probably time-varying and state-dependent. That is, the control design of the unknown nonaffine systems certainly needs to address the unknown control coefficient problem. In recent decades, several methods have been proposed to tackle the unknown control coefficient Manuscript received December 3, 2013; revised August 12, 2014; accepted August 27, 2014. Date of publication September 26, 2014; date of current version July 15, 2015. This work was supported in part by the National High Technology Research and Development Program (863 Program) of China under Grant 2012AA062201 and Grant 2012AA041709 and in part by the National Natural Science Foundation of China under Grant 61004057 and Grant 61104008. (Corresponding author: Qinmin Yang.) The authors are with the State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2354533
problem, among which the Nussbaum gain [8] is widely studied and now becomes a widely used design tool. This method involves an adaptive mechanism to identify the unknown control coefficient. Its major advantage lies in the fact that it can be readily integrated with other control design techniques, such as robust control [9], adaptive control [10], learning control [11], and backstepping design [12]. Meanwhile, in many practical systems, the input–output characteristics of actuators often exhibit unknown nonsmooth nonlinearities, including dead-zone [13], backlash [14], and hysteresis [15]. Undoubtedly, the control design problem for those systems is challenging, due to the the nonsmooth nature of the nonlinearities. Recently, Zheng et al. [16] proposes an unified approach of transforming kinds of nonsmooth nonlinearities into a general form, by approximately modeling nonsmooth nonlinearities as an affine timevarying function. In this brief, we shall focus on a class of SISO nonlinear nonaffine systems with nonsmooth actuator nonlinearities, and a novel control scheme is proposed to achieve asymptotic tracking. By augmenting the original system with a low-pass filter, it can be transformed into an affine strict-feedback system with unknown control coefficient and nonsmooth dynamics. Subsequently, an NN unit and the Nussbaum gain design are employed to compensate for the uncertain dynamics and unknown control coefficient. It is proven that the system output can track a reference signal asymptotically, while all other signals remain bounded via standard Lyapunov analysis. II. P ROBLEM S TATEMENT Consider a class of nonaffine nonlinear systems described by nth-order differential equations of the form x˙i = f i (x¯i , xi+1 ), i = 1, . . . , n − 1 x˙n = f n (x¯n , c(u, t)), n ≥ 2 y = x1
(1)
where x¯i = [x1 , . . . , xi ] ∈ i, i = 1, . . . , n are state vectors; u, y ∈ denote the system input and output, respectively; fi (·),
i = 1, . . . , n are unknown smooth functions, and c(u, t) denotes the input–output characteristics of an actuator that may contain nonsmooth nonlinearities. Assumption 1: The functions ∂ fi (x¯i , xi+1 )/∂ xi+1 , i = 1, . . . , n − 1, and ∂ fn (x¯n , c(u, t))/∂c(u, t) are nonzero ∂ fi (x¯i , xi+1 ) ≥ gi0 > 0 ∂ xi+1 ∂ f n (x¯n , c(u, t)) ≥ gn0 > 0 (2) ∂c(u, t)
with positive constants gi0 , gn0 . The above assumption implies that the considered system is controllable, which can be commonly found in control literature [7], [17]. Remark 1: The assumption on the upper bound of control effectiveness terms (∂ f i (x¯i , xi+1 )/∂ xi+1 , ∂ fn (x¯n , u)/∂u) is eliminated in this brief as opposed to [17]–[19].
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where b2 (x¯3 ) = (∂ f 1 (x1 , x2 )/∂ x1 ) f1 (x1 , x2 ) + (∂ f1 (x1 , x2 )/∂ x2 ) f 2 (x¯2 , x3 ). The following fact holds:
Fig. 1.
∂b2 (x¯3 ) ∂ f 1 (x1 , x2 ) ∂ f2 (x¯2 , x3 ) = . (7) ∂ x3 ∂ x2 ∂ x3 Step 1 (i = 3, . . . , n): In a similar fashion, let si = s˙i−1 = bi−1 (x¯i ), and the time derivative of si is obtained as
Augmented system dynamics.
s˙i = b˙i−1 (x¯i ) =
Assumption 2: The function c(u, t) satisfies the following condition: c(u, t) = κ(·)u + μ(·)
j =1
=
(3)
where κ(·) = 0 is a time-varying bounded function and takes value / I . μ(·) is a timein the closed interval I = [l − , l + ] with 0 ∈ varying bounded function satisfying |μ(·)| ≤ μ∗ for an unknown constant μ∗ . Remark 2: As presented in [16], (3) can represent a large class of nonsmooth actuator nonlinearities, such as, general models of dead-zone, backlash, and hysteresis. Assumption 3: The desired trajectory yd is continuously differen(n) tiable up to the nth order and the vector Yd = [yd , . . . , yd ] is n+1 with d being a compact set. available, where Yd ∈ d ⊂ The primary control goal of this brief is to design an adaptive NN controller for system (1) such that the system output y can track the desired trajectory yd while all the other signals in the closed-loop system remain bounded.
i ∂ f j (x¯ j , x j +1 ) ∂b (x¯ ) ∂ fi (x¯i , xi+1 ) ∂bi (x¯i+1 ) = i−1 i = . ∂ xi+1 ∂ xi ∂ xi+1 ∂ x j +1 j =1
where a is a positive constant. Now, the original system (1) becomes
(5)
Remark 3: Notice that the augmented system (5) is increased by one order compared with the original system (1) owing to the utilization of the low-pass filter. In this augmented system, the original control input u can be viewed as one state, and v can be viewed as the new control input as shown in Fig. 1. Therefore, in what follows, our main task is to design v which can drive the augmented system to track a given reference signal yd . For conciseness, we define xn+1 = u. Then, a set of new state variables {s1 , . . . , sn+1 } are introduced to proceed with the transformation task. The entire state transformation is described in the following steps. Step 1: Define new state s1 = y = x1 . Step 2: Let s2 = s˙1 = f 1 (x1 , x2 ), and the time derivative of s2 can be expressed as ∂ f 1 (x1 , x2 ) ∂ f1 (x1 , x2 ) x˙1 + x˙2 b2 (x¯3 ) s˙2 = f˙1 (x1 , x2 ) = ∂ x1 ∂ x2
j =1 n−1
∂bn (x¯n , c(u, t)) f j (x¯ j , x j +1 ) ∂x j
∂bn (x¯n , c(u, t)) f n (x¯n , c(u, t)) ∂ xn ∂bn (x¯n , c(u, t)) + κ(·)u. ˙ ∂c(u, t) +
(10)
Substituting (4) into (10), we have s˙n+1 = f (x¯n , u) + g(x¯n , u)v
(11)
where f (x¯n , u) =
n−1
∂bn (x¯n , c(u, t)) f j (x¯ j , x j +1 ) ∂x j
∂bn (x¯n , c(u, t)) fn (x¯n , c(u, t)) ∂ xn ∂bn (x¯n , c(u, t)) − κ(·)au ∂c(u, t) ∂bn (x¯n , c(u, t)) κ(·). g(x¯n , u) = ∂c(u, t) +
x˙n = f n (x¯n , c(u, t)), n ≥ 2 u˙ = −au + v y = x1 .
(9)
s˙n+1 = b˙n (x¯n , c(u, t)) n ∂bn (x¯n , c(u, t)) ∂bn (x¯n , c(u, t)) x˙ j + = u˙ ∂x j ∂u
j =1
x˙i = f i (x¯i , xi+1 ), i = 1, . . . , n − 1
(8)
Step n + 1: This is the final step. An affine term with respect to the equivalent v will appear. Let sn+1 = s˙n = bn (x¯n , u). Recalling ∂c(u, t)/∂u = κ(·) and taking the time derivative of sn+1 yield
j =1
(4)
∂bi−1 (x¯i ) f j (x¯ j , x j +1 ) bi (x¯i+1 ). ∂x j
Making use of the definition of bi (x¯i+1 ) and (7) yields
III. S YSTEM T RANSFORMATION
u˙ = −au + v
i j =1
= To circumvent the difficulties introduced by the cascade and nonaffine properties, a novel system transformation is developed to transform the nonaffine system (1) into an affine strict-feedback one in normal form. This transformation approach consists of a low-pass filter and a set of newly defined state variables. Let the control input u be generated by a low-pass filter driven by an equivalent control input v
i ∂bi−1 (x¯i ) x˙ j ∂x j
(6)
(12) (13)
From (9) with i = n and Assumption 1, we have the following control gain property: n
|g(x¯n , u)| ≥ gmin > 0
(14)
− + j =1 g j 0 , l = min(|l |, |l |). As a result, the
where gmin = l original nonaffine system (1) is now transformed into the following strict-feedback affine system: s˙i = si+1 , i = 1, . . . , n s˙n+1 = f (x¯n , u) + g(x¯n , u)v y = s1 .
(15)
Apparently, the tracking control of the transformed system is sufficient to guarantee the tracking goal of the original system since y = s1 = x1 .
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Remark 4: In Section IV, the unknown dynamics f (x¯n , u) in (15) will be estimated by a simple NN-based online approximator (OLA), while the unknown control coefficient g(x¯n , u) with respect to v will be addressed by the Nussbaum gain. IV. C ONTROLLER D ESIGN
W ∗ are supposed to be unknown since they are only needed for analytical purposes. Therefore, define nonsmooth function Q(Z) = f (x¯n , u) + m, ˆ can be where Z ≡ [x¯nT , u, m]T . Then, a continuous function Q(Z) ˆ first selected such that the distance between Q(Z) and Q(Z) over an arbitrary compact set Z is bounded ˜ ˆ sup | Q(Z)| = sup |Q(Z) − Q(Z)| ≤ εQ
A. Filtered Tracking Error Let the tracking error e be defined as e = y − yd . Then, a filtered tracking error is given by r = e(n) + dn−1 e(n−1) + · + d1 · · · e + d0 e = d T e
(16)
(n) where d = [d0 , . . . , dn−1 , 1]T , e = [e1 , e˙1 , . . . , e1 ]T , and d j , j = + 0, . . . , n − 1 ∈ are selected appropriately so that the polynomial s n + dn−1 s n−1 + · · · + d1 s + d0 is Hurwitz. According to the linear
differential equation theory [20], the boundedness of filtered tracking error r implies the boundedness of tracking error e provided that the initial error e(0) is finite. Recalling (15), (16), and the definition of e, the dynamics of filtered tracking error r can be written as r˙ = e(n+1) + dn−1 e(n) + · · · + d1 e(2) + d0 e˙ n (n+1) = f (x¯n , u) + g(x¯n , u)v − yd + d j −1 e( j ) .
j =1
˜ ˆ ˜ Q(Z) = Q(Z) + Q(Z) = W ∗T φ(Z) + ε + Q(Z).
(22)
ˆ It has to be noted that the selection of Q(Z) is also for analytical purpose and it is not required to be implemented in the controller. In Section IV-C, the weight W ∗ will be estimated by Wˆ and the following assumption is needed. Assumption 4: The optimal weight matrix W ∗ and reconstruction error ε are upper bounded such that W ∗ ≤ wm , |ε| ≤ εm with wm , εm ∈ + being unknown constants.
Remark 5: It is assumed that the time derivatives of e are available, which indeed can accurately be estimated via well-developed observers, e.g., high-gain observer with arbitrarily small observation error [21], or adaptive high-gain observer with asymptotic observation error [22]. Consequently, r and m are available for later controller design. The observer design is out of the scope of this brief and thus omitted here. B. Online Approximator The OLA is utilized to cope with the unknown dynamics in the desired equivalent controller. In this brief, a two-layer NN is employed as the OLA model. The output of this NN is given by
N(θ) = θ 2 sin θ.
and n 1 and n 3 are the number of nodes in the input and output layers; V ∈ n 1 ×n 2 and Wˆ ∈ n 2 ×n 3 are hidden and output layer weights with n 2 the number of hidden layer nodes; φ(·) is the so-called activation function chosen as the hyperbolic tangent function in this brief. According to the well-known universal approximation property [23], any continuous function F(X) in an arbitrary compact set X can be expressed as (20)
where W ∗ is the optimal weight matrix of output layer, and ε is the corresponding reconstruction error. It has been proved that [24] if the hidden layer weight V is initialized at random and held fixed, while n 2 is sufficiently large, the NN reconstruction error ε can be made arbitrarily small. Since V is fixed during the learning process, we omit it in the following parts for clarity. Note that both ε and
(24)
Clearly, the variation of N(θ) is driven by θ, which is an index measuring the control performance of the closed-loop system governed by following mechanism: θ˙ = r w.
(25)
Intuitively, θ keeps varying if the control objective, i.e., asymptotic tracking of yd and r converging to 0, has not been achieved yet. Consequently, N(θ) will keep adjusting, until it matches the sign and magnitude of g(x¯n , u), so that an appropriate feedback control law has been built and the control objective can be achieved. Furthermore, w is given as w = kr + Wˆ T φ(Z) +
(19)
where X ∈ n 1 and O ∈ n 3 denote the NNs input and output,
F(X) = W ∗T φ(V T X) + ε
(23)
where N(θ) is the Nussbaum gain, which is introduced to deal with the unknown control coefficient g(x¯n , u). In this brief, the following type of Nussbaum gain is adopted:
d j −1 e( j ) .
O(X) = Wˆ T φ(V T X)
v = N(θ)w
(18)
where +
where ε Q is an unknown positive constant. ˆ Subsequently, the continuous function Q(Z) can be approximated by an NN with arbitrary accuracy ε, and thus Q(Z) can be written as
The adaptive control law is
r˙ = f (x¯n , u) + g(x¯n , u)v + m
(n+1)
(21)
C. Control Law
Then, it can be further rephrased as
m = −yd
Z ∈ Z
(17)
j =1
n
Z ∈ Z
hˆ 2r ˆh|r | + exp(−t)
(26)
where kr is a feedback term with k > 0, and the second term Wˆ T φ(Z) is an estimate of the optimal NN W ∗T φ(Z). The weight Wˆ is updated adaptively as W˙ˆ = r φ(Z) − σ |r |Wˆ , Wˆ (0) = 0
(27)
where σ is a positive learning rate. The corresponding estimation error is defined as W˜ = Wˆ − W ∗ .
(28)
ˆ | + exp(−t)) can be regarded as a modified The third term (hˆ 2r /h|r sign function, which is always continuous and can approximate the ˆ because of the following property: sign function sgn(r )h, r
hˆ 2r ˆ | + exp(−t) h|r
hˆ 2r 2
ˆ | − exp(−t) > h|r ˆ | + exp(−t) h|r = r sgn(r )hˆ − exp(−t).
=
(29)
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In (29), hˆ is a nonnegative variable, which is adaptively updated according to ˆ h˙ˆ = |r |, h(0) =0
(30)
which is used to estimate the unknown constant h given as h = εm + ε Q +
2σ wm . 4
(31)
The estimation error is defined as h˜ = hˆ − h.
(32)
V. C ONVERGENCE AND B OUNDEDNESS The main theoretical result of the asymptotic convergence of tracking error and the uniform boundedness of the other signals is presented in this section. Theorem 1: Consider the nonaffine plant (1) under Assumptions 1–3, the control law generated by (4) with equivalent control v given by (23), and the update law (25). Then, for bounded initial conditions, the tracking error e converges asymptotically to 0, and all the other signals are uniformly ultimately bounded. Proof: Consider the following Lyapunov candidate: 1 1 1 V ≡ r 2 + W˜ T W˜ + h˜ 2 , V0 = V (0). 2 2 2 Its time derivative is
(33)
˜ |. V˙ = r ( f (x¯n , u) + g(x¯n , u)v + m) + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r Recalling (22) and (23), we have ˜ V˙ = r W ∗T φ(Z) + ε + Q(Z) + r g(x¯n , u)N(θ)w
(34)
Then, inserting (26) into above yields ˜ V˙ = r W ∗T φ(Z) + ε + Q(Z) + r (g(x¯n , u)N(θ) + 1)w ˆ 2r h T − r kr + Wˆ φ(Z) + ˆ | + exp(−t) h|r ˜ | + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r ∗T T T ˜ = r W − Wˆ + W˜ φ(Z) + r (ε + Q(Z))
Then, taking integration yields t t t ˙ + e−τ dτ +V0 0 ≤ V ≤ −k |r |2 ds + (g(x¯n , u)N(θ) +1)θdτ 0 0 0 t θt t 2 ≤ −k |r | ds + g(x¯n , u)N(θ)dθ + θt + e−τ dτ + V0 0
0
0
(38) where θt = θ(t). It further gives θt g(x¯n , u)N(θ)dθ + θt + 1 + V0 0≤V ≤
(39)
0
or equivalently −(θt + 1 + V0 ) ≤
θt 0
g(x¯n , u)N(θ)dθ.
(40)
Note that since (14) holds, the Nussbaum function N(θ) has the following properties: 1
g(x¯n , u)N(θ)dθ = +∞ (41) lim sup
0
→∞ 1
lim inf g(x¯n , u)N(θ)dθ = −∞. (42)
0
→∞ Then, we can show that θt is bounded by the method of contradiction. Suppose that θt becomes divergent, then there are two possibilities.
θ 1) If θt → +∞, (40) gives limθt →+∞ (1/θt ) 0 t g(x¯n , u)N ≤ (θ) dθ ≥ −1, which contradicts with (42).
2) If θt → −∞, (40) gives limθt →−∞ (1/θt ) 0θt g(x¯n , u)N(θ) dθ ≤ −1, which contradicts with (41).
θ As a result, θt is bounded, and 0 t g(x¯n , u)N(θ)dθ is also bounded; hence, we can always find a positive constant C, such that θt g(x¯n , u)N(θ)dθ + θt + 1 + V0 ≤ C. (43) Then, (39) implies
(35)
Using (28) and (29), we have the following inequality: ˆ | V˙ ≤ (g(x¯n , u)N(θ) + 1)θ˙ + (εm + ε Q )|r | − kr 2 − h|r T ˜ | + exp(−t) − σ |r |W˜ (W˜ + W ) + h|r ˆ | ≤ (g(x¯n , u)N(θ) + 1)θ˙ + (εm + ε Q )|r | − kr 2 − h|r ˜ | + exp(−t) − σ |r | W˜ 2 + σ |r | W˜ wm + h|r 2 ˆ | ˙ = (g(x¯n , u)N(θ) + 1)θ + (εm + ε Q )|r | − kr − h|r
2σ wm 2 wm ˜ | + |r | + h|r + exp(−t) − σ |r | W˜ − 2 4
wm 2 = (g(x¯n , u)N(θ) + 1)θ˙ − σ |r | W˜ − 2
wm σ + |r | εm + ε Q − hˆ + + h˜ − kr 2 + exp(−t). 4
˜ = (g(x¯n , u)N(θ) + 1)θ˙ + |r |(h − hˆ + h)
2 w m 2 − kr − σ |r | W˜ − + exp(−t) 2
wm 2 = (g(x¯n , u)N(θ) + 1)θ˙ − kr 2 − σ |r | W˜ − + exp(−t) 2 2 ≤ (g(x¯n , u)N(θ) + 1)θ˙ − kr + exp(−t). (37)
0
hˆ 2 r 2
+ (g(x¯n , u)N(θ) + 1)θ˙ ˆ | + exp(−t) h|r ˜ |. − kr 2 − σ |r |W˜ T Wˆ + h|r −
By recalling the definition of h, one has
and
˜ | + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r ∗T ˜ = r W φ(Z) + ε + Q(Z) + r (g(x¯n , u)N(θ) + 1)w ˜ |. − r w + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r
1825
(36)
0 ≤ V ≤ −k
t 0
|r |2 dτ + C.
(44)
Therefore, according to Barbalat Lemma, (44) implies r → 0 when t → ∞, which further gives e → 0 when t → ∞. Then, we can conclude the asymptotic convergence of the tracking error, which directly implies the boundedness of new state variables {s1 , . . . , sn+1 }. Furthermore, the boundedness of other signals also can be readily obtained. Equation (44) shows the uniform boundedness of V since the right-hand side term is bounded. Therefore, W˜ and h˜ are bounded according to (33), which also implies the boundedness of Wˆ and hˆ by definition. In addition, since the activation function φ(Z) is bounded, the new control signal v is bounded by (23) and (26). As a result, the original control input u is also bounded which can be viewed as the output of a stable linear time invariant system driven by v. Furthermore, the state x1 is bounded by definition. Thus, x2 is also bounded based on the fact that (∂ f1 (x1 , x2 )/∂ x2 ) > 0 and
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Fig. 4.
Actual and desired system output and states.
ˆ Norm of NN weights and adaptive term h.
of NN neurons is 10, and hyperbolic tangent function is adopted as the activation function. The other controller parameters are selected as k = 7, σ = 0.05. Figs. 2–4 show the simulation results. It can be readily found that satisfactory tracking performance is obtained from Fig. 2, where the system output y converges fast to a small neighborhood of the desired output. Fig. 3 shows the generated control signal u. The spikes is due to the fact that the controller does not know the control direction initially, but it soon converges and stays within a reasonable bound. The boundedness of NN weights Wˆ and adaptive term hˆ is also shown in Fig. 4. VII. C ONCLUSION Fig. 3.
Trajectory of control input signal u.
s2 = f1 (x1 , x2 ) is bounded. Similarly, the boundedness of original states xi , i = 3, . . . , n can be achieved. VI. S IMULATION In order to verify the feasibility of the proposed theoretic design, it is applied to the following nonlinear nonaffine system: x˙1 =
1 − e x1 + x2 + 0.5 sin(x2 ) 1 + e x1
x˙2 = 0.2e−x1 x2 − (0.9 + 0.05e−x1 )c(u) − 0.1 sin(c(u)) 2
y = x1
(45)
where c(u) represents a nonsymmetric dead-zone nonlinearity modeled by [25] ⎧ u ≥ 2.5 ⎨ (1 − 0.3 sin(u))(u − 2.5) c(u) = 0 −1.5 < u < 2.5 (46) ⎩ (0.8 − 0.2 cos(u))(u + 1.5) u ≤ −1.5 c(u) can be rewritten into the form given in (3) as 1 − 0.3 sin(u) u≥0 κ(·) = 0.8 − 0.2 cos(u) u < 0 ⎧ ⎪ ⎪ −2.5(1 − 0.3 sin(u)) u ≥ 2.5 ⎨ −(1 − 0.3 sin(u))u 0 ≤ u < 2.5 μ(·) = −(0.8 − 0.2 cos(u))u −1.5 < u < 0 ⎪ ⎪ ⎩ 1.5(0.8 − 0.2 cos(u)) u ≤ −1.5.
(47)
Clearly, |μ(·)| ≤ μ∗ = max(2.5 × 1.3, 1 × 1.5) = 3.25. The desired trajectory to be tracked is yd (t) = −cos(t). The filtered tracking error is then defined as r = e˙ + 3e. The number
In this brief, we concentrate on a class of nth-order SISO nonaffine nonlinear dynamical systems with nonsmooth actuator nonlinearities, and study its asymptotic tracking problem. In order to avoid the difficulty of control design caused by the nonaffine structure, we first propose a novel transformation approach to transfer systematically the original nonaffine system into an equivalent affine one. Then, an adaptive controller is devised for the affine system, which consists of two parts: 1) an NN-based OLA is utilized to approximate the unknown dynamics and 2) the Nussbaum gain technique is integrated to tackle the unknown and varying control coefficient. Finally, it is proven rigorously via standard Lyapunov analysis that asymptotic tracking of a given smooth enough reference signal, as well as the ultimate uniform boundedness of all the other signals can be guaranteed by the proposed control method. Numerical simulations demonstrate the effectiveness of the proposed method. R EFERENCES [1] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [2] D. Han and L. Shi, “Guaranteed cost control of affine nonlinear systems via partition of unity method,” Automatica, vol. 49, no. 2, pp. 660–666, 2013. [3] W. Meng, Q. Yang, Y. Ying, Y. Sun, and Z. Yang, “Adaptive power capture control of variable-speed wind energy conversion systems with guaranteed transient and steady-state performance,” IEEE Trans. Energy Convers., vol. 28, no. 3, pp. 716–725, Sep. 2013. [4] L. R. Hunt and G. Meyer, “Stable inversion for nonlinear systems,” Automatica, vol. 33, no. 8, pp. 1549–1554, 1997. [5] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900–918, Jul. 2003. [6] N. Hovakimyan, A. J. Calise, and N. Kim, “Adaptive output feedback control of a class of multi-input multi-output systems using neural networks,” Int. J. Control, vol. 77, no. 15, pp. 1318–1329, 2004.
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[7] B.-J. Yang and A. J. Calise, “Adaptive control of a class of nonaffine systems using neural networks,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1149–1159, Jul. 2007. [8] N. D. Nussbaum, “Some remarks on a conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, no. 5, pp. 243–246, 1983. [9] L. Liu and J. Huang, “Global robust stabilization of cascade-connected systems with dynamic uncertainties without knowing the control direction,” IEEE Trans. Autom. Control, vol. 51, no. 10, pp. 1693–1699, Oct. 2006. [10] S. S. Ge and J. Wang, “Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients,” IEEE Trans. Autom. Control, vol. 48, no. 8, pp. 1463–1469, Aug. 2003. [11] Y.-J. Liu, C. L. Chen, G.-X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Trans. Neural Netw., vol. 22, no. 7, pp. 1162–1167, Jul. 2011. [12] Y. Zhang, C. Wen, and Y. C. Soh, “Adaptive backstepping control design for systems with unknown high-frequency gain,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2350–2354, Dec. 2000. [13] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown dead-zones,” IEEE Trans. Autom. Control, vol. 39, no. 1, pp. 59–68, Jan. 1994. [14] G. Tao and P. V. Kokotovic, “Continuous-time adaptive control of systems with unknown backlash,” IEEE Trans. Autom. Control, vol. 40, no. 6, pp. 1083–1087, Jun. 1995. [15] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown hystereses,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [16] Y. Zheng, C. Wen, and Z. Li, “Robust adaptive asymptotic tracking control of uncertain nonlinear systems subject to nonsmooth actuator nonlinearities,” Int. J. Adapt. Control Signal Process., vol. 27, no. 1, pp. 108–121, 2013.
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[17] M. Wang, X. Liu, and P. Shi, “Adaptive neural control of pure-feedback nonlinear time-delay systems via dynamic surface technique,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 41, no. 6, pp. 1681–1692, Dec. 2011. [18] T.-P. Zhang, H. Wen, and Q. Zhu, “Adaptive fuzzy control of nonlinear systems in pure feedback form based on input-to-state stability,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 80–93, Feb. 2010. [19] W.-Y. Wang, Y.-H. Chien, Y.-G. Leu, and T.-T. Lee, “Adaptive T–S fuzzy-neural modeling and control for general MIMO unknown nonaffine nonlinear systems using projection update laws,” Automatica, vol. 46, no. 5, pp. 852–863, 2010. [20] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. New York, NY, USA: McGraw-Hill, 1955. [21] H. K. Khalil and A. N. Atassi, “A separation principle for the stabilization of a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1672–1687, Sep. 1999. [22] L. Praly, “Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate,” IEEE Trans. Autom. Control, vol. 48, no. 6, pp. 1103–1108, Jun. 2003. [23] D. A. White and D. A. Sofge, Handbook of Intelligent Control: Neural, Fuzzy, and Adaptive Approaches. New York, NY, USA: Van Nostrand, 1993. [24] B. Igelnik and Y.-H. Pao, “Stochastic choice of basis functions in adaptive function approximation and the functional-link net,” IEEE Trans. Neural Netw., vol. 6, no. 6, pp. 1320–1329, Nov. 1995. [25] T. P. Zhang and S. S. Ge, “Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form,” Automatica, vol. 44, no. 7, pp. 1895–1903, 2008.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 8, AUGUST 2015
Adaptive Neural Control of Nonaffine Systems With Unknown Control Coefficient and Nonsmooth Actuator Nonlinearities Zaiyue Yang, Qinmin Yang, and Youxian Sun
Abstract— This brief considers the asymptotic tracking problem for a class of high-order nonaffine nonlinear dynamical systems with nonsmooth actuator nonlinearities. A novel transformation approach is proposed, which is able to systematically transfer the original nonaffine nonlinear system into an equivalent affine one. Then, to deal with the unknown dynamics and unknown control coefficient contained in the affine system, online approximator and Nussbaum gain techniques are utilized in the controller design. It is proven rigorously that asymptotic convergence of the tracking error and ultimate uniform boundedness of all the other signals can be guaranteed by the proposed control method. The control feasibility is further verified by numerical simulations.
Index Terms— Adaptive neural control, nonaffine systems, nonsmooth nonlinearity, unknown control coefficient. I. I NTRODUCTION Due to both practical needs and theoretical challenges, a lot of efforts have been devoted to adaptive control techniques for nonlinear uncertain systems in recent years [1]. Most of them are concentrated on affine systems in the control input [2]. However, the control signal can be present nonlinearly in the system dynamics in a lot of applications, including air conditioning systems, wind turbines [3], and aircrafts [4]. To tackle the stability problem of nonaffine systems from the perspective of adaptive control, numerous attempts have been made in [5]–[7]. By proving the existence of an implicit desired feedback control law, [5] managed to achieve stability of a class of single-input single-output (SISO) nonaffine systems in normal form via both state and output feedback. To address a more general class of multipleinput multiple-output nonaffine nonlinear systems, a direct adaptive approach is introduced in [6] by using linearly parameterized neural network (NN) structure to approximate a fictional ideal controller, whose existence is assured by the implicit function theorem. The coefficient of control term determines the direction and amplitude of the system motion under a given control signal, which is undoubtedly vital to the stability and control performance of the closed-loop system. Unfortunately, it is clear that for unknown nonaffine systems, the control coefficient is often unknown and probably time-varying and state-dependent. That is, the control design of the unknown nonaffine systems certainly needs to address the unknown control coefficient problem. In recent decades, several methods have been proposed to tackle the unknown control coefficient Manuscript received December 3, 2013; revised August 12, 2014; accepted August 27, 2014. Date of publication September 26, 2014; date of current version July 15, 2015. This work was supported in part by the National High Technology Research and Development Program (863 Program) of China under Grant 2012AA062201 and Grant 2012AA041709 and in part by the National Natural Science Foundation of China under Grant 61004057 and Grant 61104008. (Corresponding author: Qinmin Yang.) The authors are with the State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2354533
problem, among which the Nussbaum gain [8] is widely studied and now becomes a widely used design tool. This method involves an adaptive mechanism to identify the unknown control coefficient. Its major advantage lies in the fact that it can be readily integrated with other control design techniques, such as robust control [9], adaptive control [10], learning control [11], and backstepping design [12]. Meanwhile, in many practical systems, the input–output characteristics of actuators often exhibit unknown nonsmooth nonlinearities, including dead-zone [13], backlash [14], and hysteresis [15]. Undoubtedly, the control design problem for those systems is challenging, due to the the nonsmooth nature of the nonlinearities. Recently, Zheng et al. [16] proposes an unified approach of transforming kinds of nonsmooth nonlinearities into a general form, by approximately modeling nonsmooth nonlinearities as an affine timevarying function. In this brief, we shall focus on a class of SISO nonlinear nonaffine systems with nonsmooth actuator nonlinearities, and a novel control scheme is proposed to achieve asymptotic tracking. By augmenting the original system with a low-pass filter, it can be transformed into an affine strict-feedback system with unknown control coefficient and nonsmooth dynamics. Subsequently, an NN unit and the Nussbaum gain design are employed to compensate for the uncertain dynamics and unknown control coefficient. It is proven that the system output can track a reference signal asymptotically, while all other signals remain bounded via standard Lyapunov analysis. II. P ROBLEM S TATEMENT Consider a class of nonaffine nonlinear systems described by nth-order differential equations of the form x˙i = f i (x¯i , xi+1 ), i = 1, . . . , n − 1 x˙n = f n (x¯n , c(u, t)), n ≥ 2 y = x1
(1)
where x¯i = [x1 , . . . , xi ] ∈ i, i = 1, . . . , n are state vectors; u, y ∈ denote the system input and output, respectively; fi (·),
i = 1, . . . , n are unknown smooth functions, and c(u, t) denotes the input–output characteristics of an actuator that may contain nonsmooth nonlinearities. Assumption 1: The functions ∂ fi (x¯i , xi+1 )/∂ xi+1 , i = 1, . . . , n − 1, and ∂ fn (x¯n , c(u, t))/∂c(u, t) are nonzero ∂ fi (x¯i , xi+1 ) ≥ gi0 > 0 ∂ xi+1 ∂ f n (x¯n , c(u, t)) ≥ gn0 > 0 (2) ∂c(u, t)
with positive constants gi0 , gn0 . The above assumption implies that the considered system is controllable, which can be commonly found in control literature [7], [17]. Remark 1: The assumption on the upper bound of control effectiveness terms (∂ f i (x¯i , xi+1 )/∂ xi+1 , ∂ fn (x¯n , u)/∂u) is eliminated in this brief as opposed to [17]–[19].
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where b2 (x¯3 ) = (∂ f 1 (x1 , x2 )/∂ x1 ) f1 (x1 , x2 ) + (∂ f1 (x1 , x2 )/∂ x2 ) f 2 (x¯2 , x3 ). The following fact holds:
Fig. 1.
∂b2 (x¯3 ) ∂ f 1 (x1 , x2 ) ∂ f2 (x¯2 , x3 ) = . (7) ∂ x3 ∂ x2 ∂ x3 Step 1 (i = 3, . . . , n): In a similar fashion, let si = s˙i−1 = bi−1 (x¯i ), and the time derivative of si is obtained as
Augmented system dynamics.
s˙i = b˙i−1 (x¯i ) =
Assumption 2: The function c(u, t) satisfies the following condition: c(u, t) = κ(·)u + μ(·)
j =1
=
(3)
where κ(·) = 0 is a time-varying bounded function and takes value / I . μ(·) is a timein the closed interval I = [l − , l + ] with 0 ∈ varying bounded function satisfying |μ(·)| ≤ μ∗ for an unknown constant μ∗ . Remark 2: As presented in [16], (3) can represent a large class of nonsmooth actuator nonlinearities, such as, general models of dead-zone, backlash, and hysteresis. Assumption 3: The desired trajectory yd is continuously differen(n) tiable up to the nth order and the vector Yd = [yd , . . . , yd ] is n+1 with d being a compact set. available, where Yd ∈ d ⊂ The primary control goal of this brief is to design an adaptive NN controller for system (1) such that the system output y can track the desired trajectory yd while all the other signals in the closed-loop system remain bounded.
i ∂ f j (x¯ j , x j +1 ) ∂b (x¯ ) ∂ fi (x¯i , xi+1 ) ∂bi (x¯i+1 ) = i−1 i = . ∂ xi+1 ∂ xi ∂ xi+1 ∂ x j +1 j =1
where a is a positive constant. Now, the original system (1) becomes
(5)
Remark 3: Notice that the augmented system (5) is increased by one order compared with the original system (1) owing to the utilization of the low-pass filter. In this augmented system, the original control input u can be viewed as one state, and v can be viewed as the new control input as shown in Fig. 1. Therefore, in what follows, our main task is to design v which can drive the augmented system to track a given reference signal yd . For conciseness, we define xn+1 = u. Then, a set of new state variables {s1 , . . . , sn+1 } are introduced to proceed with the transformation task. The entire state transformation is described in the following steps. Step 1: Define new state s1 = y = x1 . Step 2: Let s2 = s˙1 = f 1 (x1 , x2 ), and the time derivative of s2 can be expressed as ∂ f 1 (x1 , x2 ) ∂ f1 (x1 , x2 ) x˙1 + x˙2 b2 (x¯3 ) s˙2 = f˙1 (x1 , x2 ) = ∂ x1 ∂ x2
j =1 n−1
∂bn (x¯n , c(u, t)) f j (x¯ j , x j +1 ) ∂x j
∂bn (x¯n , c(u, t)) f n (x¯n , c(u, t)) ∂ xn ∂bn (x¯n , c(u, t)) + κ(·)u. ˙ ∂c(u, t) +
(10)
Substituting (4) into (10), we have s˙n+1 = f (x¯n , u) + g(x¯n , u)v
(11)
where f (x¯n , u) =
n−1
∂bn (x¯n , c(u, t)) f j (x¯ j , x j +1 ) ∂x j
∂bn (x¯n , c(u, t)) fn (x¯n , c(u, t)) ∂ xn ∂bn (x¯n , c(u, t)) − κ(·)au ∂c(u, t) ∂bn (x¯n , c(u, t)) κ(·). g(x¯n , u) = ∂c(u, t) +
x˙n = f n (x¯n , c(u, t)), n ≥ 2 u˙ = −au + v y = x1 .
(9)
s˙n+1 = b˙n (x¯n , c(u, t)) n ∂bn (x¯n , c(u, t)) ∂bn (x¯n , c(u, t)) x˙ j + = u˙ ∂x j ∂u
j =1
x˙i = f i (x¯i , xi+1 ), i = 1, . . . , n − 1
(8)
Step n + 1: This is the final step. An affine term with respect to the equivalent v will appear. Let sn+1 = s˙n = bn (x¯n , u). Recalling ∂c(u, t)/∂u = κ(·) and taking the time derivative of sn+1 yield
j =1
(4)
∂bi−1 (x¯i ) f j (x¯ j , x j +1 ) bi (x¯i+1 ). ∂x j
Making use of the definition of bi (x¯i+1 ) and (7) yields
III. S YSTEM T RANSFORMATION
u˙ = −au + v
i j =1
= To circumvent the difficulties introduced by the cascade and nonaffine properties, a novel system transformation is developed to transform the nonaffine system (1) into an affine strict-feedback one in normal form. This transformation approach consists of a low-pass filter and a set of newly defined state variables. Let the control input u be generated by a low-pass filter driven by an equivalent control input v
i ∂bi−1 (x¯i ) x˙ j ∂x j
(6)
(12) (13)
From (9) with i = n and Assumption 1, we have the following control gain property: n
|g(x¯n , u)| ≥ gmin > 0
(14)
− + j =1 g j 0 , l = min(|l |, |l |). As a result, the
where gmin = l original nonaffine system (1) is now transformed into the following strict-feedback affine system: s˙i = si+1 , i = 1, . . . , n s˙n+1 = f (x¯n , u) + g(x¯n , u)v y = s1 .
(15)
Apparently, the tracking control of the transformed system is sufficient to guarantee the tracking goal of the original system since y = s1 = x1 .
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Remark 4: In Section IV, the unknown dynamics f (x¯n , u) in (15) will be estimated by a simple NN-based online approximator (OLA), while the unknown control coefficient g(x¯n , u) with respect to v will be addressed by the Nussbaum gain. IV. C ONTROLLER D ESIGN
W ∗ are supposed to be unknown since they are only needed for analytical purposes. Therefore, define nonsmooth function Q(Z) = f (x¯n , u) + m, ˆ can be where Z ≡ [x¯nT , u, m]T . Then, a continuous function Q(Z) ˆ first selected such that the distance between Q(Z) and Q(Z) over an arbitrary compact set Z is bounded ˜ ˆ sup | Q(Z)| = sup |Q(Z) − Q(Z)| ≤ εQ
A. Filtered Tracking Error Let the tracking error e be defined as e = y − yd . Then, a filtered tracking error is given by r = e(n) + dn−1 e(n−1) + · + d1 · · · e + d0 e = d T e
(16)
(n) where d = [d0 , . . . , dn−1 , 1]T , e = [e1 , e˙1 , . . . , e1 ]T , and d j , j = + 0, . . . , n − 1 ∈ are selected appropriately so that the polynomial s n + dn−1 s n−1 + · · · + d1 s + d0 is Hurwitz. According to the linear
differential equation theory [20], the boundedness of filtered tracking error r implies the boundedness of tracking error e provided that the initial error e(0) is finite. Recalling (15), (16), and the definition of e, the dynamics of filtered tracking error r can be written as r˙ = e(n+1) + dn−1 e(n) + · · · + d1 e(2) + d0 e˙ n (n+1) = f (x¯n , u) + g(x¯n , u)v − yd + d j −1 e( j ) .
j =1
˜ ˆ ˜ Q(Z) = Q(Z) + Q(Z) = W ∗T φ(Z) + ε + Q(Z).
(22)
ˆ It has to be noted that the selection of Q(Z) is also for analytical purpose and it is not required to be implemented in the controller. In Section IV-C, the weight W ∗ will be estimated by Wˆ and the following assumption is needed. Assumption 4: The optimal weight matrix W ∗ and reconstruction error ε are upper bounded such that W ∗ ≤ wm , |ε| ≤ εm with wm , εm ∈ + being unknown constants.
Remark 5: It is assumed that the time derivatives of e are available, which indeed can accurately be estimated via well-developed observers, e.g., high-gain observer with arbitrarily small observation error [21], or adaptive high-gain observer with asymptotic observation error [22]. Consequently, r and m are available for later controller design. The observer design is out of the scope of this brief and thus omitted here. B. Online Approximator The OLA is utilized to cope with the unknown dynamics in the desired equivalent controller. In this brief, a two-layer NN is employed as the OLA model. The output of this NN is given by
N(θ) = θ 2 sin θ.
and n 1 and n 3 are the number of nodes in the input and output layers; V ∈ n 1 ×n 2 and Wˆ ∈ n 2 ×n 3 are hidden and output layer weights with n 2 the number of hidden layer nodes; φ(·) is the so-called activation function chosen as the hyperbolic tangent function in this brief. According to the well-known universal approximation property [23], any continuous function F(X) in an arbitrary compact set X can be expressed as (20)
where W ∗ is the optimal weight matrix of output layer, and ε is the corresponding reconstruction error. It has been proved that [24] if the hidden layer weight V is initialized at random and held fixed, while n 2 is sufficiently large, the NN reconstruction error ε can be made arbitrarily small. Since V is fixed during the learning process, we omit it in the following parts for clarity. Note that both ε and
(24)
Clearly, the variation of N(θ) is driven by θ, which is an index measuring the control performance of the closed-loop system governed by following mechanism: θ˙ = r w.
(25)
Intuitively, θ keeps varying if the control objective, i.e., asymptotic tracking of yd and r converging to 0, has not been achieved yet. Consequently, N(θ) will keep adjusting, until it matches the sign and magnitude of g(x¯n , u), so that an appropriate feedback control law has been built and the control objective can be achieved. Furthermore, w is given as w = kr + Wˆ T φ(Z) +
(19)
where X ∈ n 1 and O ∈ n 3 denote the NNs input and output,
F(X) = W ∗T φ(V T X) + ε
(23)
where N(θ) is the Nussbaum gain, which is introduced to deal with the unknown control coefficient g(x¯n , u). In this brief, the following type of Nussbaum gain is adopted:
d j −1 e( j ) .
O(X) = Wˆ T φ(V T X)
v = N(θ)w
(18)
where +
where ε Q is an unknown positive constant. ˆ Subsequently, the continuous function Q(Z) can be approximated by an NN with arbitrary accuracy ε, and thus Q(Z) can be written as
The adaptive control law is
r˙ = f (x¯n , u) + g(x¯n , u)v + m
(n+1)
(21)
C. Control Law
Then, it can be further rephrased as
m = −yd
Z ∈ Z
(17)
j =1
n
Z ∈ Z
hˆ 2r ˆh|r | + exp(−t)
(26)
where kr is a feedback term with k > 0, and the second term Wˆ T φ(Z) is an estimate of the optimal NN W ∗T φ(Z). The weight Wˆ is updated adaptively as W˙ˆ = r φ(Z) − σ |r |Wˆ , Wˆ (0) = 0
(27)
where σ is a positive learning rate. The corresponding estimation error is defined as W˜ = Wˆ − W ∗ .
(28)
ˆ | + exp(−t)) can be regarded as a modified The third term (hˆ 2r /h|r sign function, which is always continuous and can approximate the ˆ because of the following property: sign function sgn(r )h, r
hˆ 2r ˆ | + exp(−t) h|r
hˆ 2r 2
ˆ | − exp(−t) > h|r ˆ | + exp(−t) h|r = r sgn(r )hˆ − exp(−t).
=
(29)
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In (29), hˆ is a nonnegative variable, which is adaptively updated according to ˆ h˙ˆ = |r |, h(0) =0
(30)
which is used to estimate the unknown constant h given as h = εm + ε Q +
2σ wm . 4
(31)
The estimation error is defined as h˜ = hˆ − h.
(32)
V. C ONVERGENCE AND B OUNDEDNESS The main theoretical result of the asymptotic convergence of tracking error and the uniform boundedness of the other signals is presented in this section. Theorem 1: Consider the nonaffine plant (1) under Assumptions 1–3, the control law generated by (4) with equivalent control v given by (23), and the update law (25). Then, for bounded initial conditions, the tracking error e converges asymptotically to 0, and all the other signals are uniformly ultimately bounded. Proof: Consider the following Lyapunov candidate: 1 1 1 V ≡ r 2 + W˜ T W˜ + h˜ 2 , V0 = V (0). 2 2 2 Its time derivative is
(33)
˜ |. V˙ = r ( f (x¯n , u) + g(x¯n , u)v + m) + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r Recalling (22) and (23), we have ˜ V˙ = r W ∗T φ(Z) + ε + Q(Z) + r g(x¯n , u)N(θ)w
(34)
Then, inserting (26) into above yields ˜ V˙ = r W ∗T φ(Z) + ε + Q(Z) + r (g(x¯n , u)N(θ) + 1)w ˆ 2r h T − r kr + Wˆ φ(Z) + ˆ | + exp(−t) h|r ˜ | + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r ∗T T T ˜ = r W − Wˆ + W˜ φ(Z) + r (ε + Q(Z))
Then, taking integration yields t t t ˙ + e−τ dτ +V0 0 ≤ V ≤ −k |r |2 ds + (g(x¯n , u)N(θ) +1)θdτ 0 0 0 t θt t 2 ≤ −k |r | ds + g(x¯n , u)N(θ)dθ + θt + e−τ dτ + V0 0
0
0
(38) where θt = θ(t). It further gives θt g(x¯n , u)N(θ)dθ + θt + 1 + V0 0≤V ≤
(39)
0
or equivalently −(θt + 1 + V0 ) ≤
θt 0
g(x¯n , u)N(θ)dθ.
(40)
Note that since (14) holds, the Nussbaum function N(θ) has the following properties: 1
g(x¯n , u)N(θ)dθ = +∞ (41) lim sup
0
→∞ 1
lim inf g(x¯n , u)N(θ)dθ = −∞. (42)
0
→∞ Then, we can show that θt is bounded by the method of contradiction. Suppose that θt becomes divergent, then there are two possibilities.
θ 1) If θt → +∞, (40) gives limθt →+∞ (1/θt ) 0 t g(x¯n , u)N ≤ (θ) dθ ≥ −1, which contradicts with (42).
2) If θt → −∞, (40) gives limθt →−∞ (1/θt ) 0θt g(x¯n , u)N(θ) dθ ≤ −1, which contradicts with (41).
θ As a result, θt is bounded, and 0 t g(x¯n , u)N(θ)dθ is also bounded; hence, we can always find a positive constant C, such that θt g(x¯n , u)N(θ)dθ + θt + 1 + V0 ≤ C. (43) Then, (39) implies
(35)
Using (28) and (29), we have the following inequality: ˆ | V˙ ≤ (g(x¯n , u)N(θ) + 1)θ˙ + (εm + ε Q )|r | − kr 2 − h|r T ˜ | + exp(−t) − σ |r |W˜ (W˜ + W ) + h|r ˆ | ≤ (g(x¯n , u)N(θ) + 1)θ˙ + (εm + ε Q )|r | − kr 2 − h|r ˜ | + exp(−t) − σ |r | W˜ 2 + σ |r | W˜ wm + h|r 2 ˆ | ˙ = (g(x¯n , u)N(θ) + 1)θ + (εm + ε Q )|r | − kr − h|r
2σ wm 2 wm ˜ | + |r | + h|r + exp(−t) − σ |r | W˜ − 2 4
wm 2 = (g(x¯n , u)N(θ) + 1)θ˙ − σ |r | W˜ − 2
wm σ + |r | εm + ε Q − hˆ + + h˜ − kr 2 + exp(−t). 4
˜ = (g(x¯n , u)N(θ) + 1)θ˙ + |r |(h − hˆ + h)
2 w m 2 − kr − σ |r | W˜ − + exp(−t) 2
wm 2 = (g(x¯n , u)N(θ) + 1)θ˙ − kr 2 − σ |r | W˜ − + exp(−t) 2 2 ≤ (g(x¯n , u)N(θ) + 1)θ˙ − kr + exp(−t). (37)
0
hˆ 2 r 2
+ (g(x¯n , u)N(θ) + 1)θ˙ ˆ | + exp(−t) h|r ˜ |. − kr 2 − σ |r |W˜ T Wˆ + h|r −
By recalling the definition of h, one has
and
˜ | + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r ∗T ˜ = r W φ(Z) + ε + Q(Z) + r (g(x¯n , u)N(θ) + 1)w ˜ |. − r w + r W˜ T φ(Z) − σ |r |W˜ T Wˆ + h|r
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(36)
0 ≤ V ≤ −k
t 0
|r |2 dτ + C.
(44)
Therefore, according to Barbalat Lemma, (44) implies r → 0 when t → ∞, which further gives e → 0 when t → ∞. Then, we can conclude the asymptotic convergence of the tracking error, which directly implies the boundedness of new state variables {s1 , . . . , sn+1 }. Furthermore, the boundedness of other signals also can be readily obtained. Equation (44) shows the uniform boundedness of V since the right-hand side term is bounded. Therefore, W˜ and h˜ are bounded according to (33), which also implies the boundedness of Wˆ and hˆ by definition. In addition, since the activation function φ(Z) is bounded, the new control signal v is bounded by (23) and (26). As a result, the original control input u is also bounded which can be viewed as the output of a stable linear time invariant system driven by v. Furthermore, the state x1 is bounded by definition. Thus, x2 is also bounded based on the fact that (∂ f1 (x1 , x2 )/∂ x2 ) > 0 and
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Fig. 2.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 26, NO. 8, AUGUST 2015
Fig. 4.
Actual and desired system output and states.
ˆ Norm of NN weights and adaptive term h.
of NN neurons is 10, and hyperbolic tangent function is adopted as the activation function. The other controller parameters are selected as k = 7, σ = 0.05. Figs. 2–4 show the simulation results. It can be readily found that satisfactory tracking performance is obtained from Fig. 2, where the system output y converges fast to a small neighborhood of the desired output. Fig. 3 shows the generated control signal u. The spikes is due to the fact that the controller does not know the control direction initially, but it soon converges and stays within a reasonable bound. The boundedness of NN weights Wˆ and adaptive term hˆ is also shown in Fig. 4. VII. C ONCLUSION Fig. 3.
Trajectory of control input signal u.
s2 = f1 (x1 , x2 ) is bounded. Similarly, the boundedness of original states xi , i = 3, . . . , n can be achieved. VI. S IMULATION In order to verify the feasibility of the proposed theoretic design, it is applied to the following nonlinear nonaffine system: x˙1 =
1 − e x1 + x2 + 0.5 sin(x2 ) 1 + e x1
x˙2 = 0.2e−x1 x2 − (0.9 + 0.05e−x1 )c(u) − 0.1 sin(c(u)) 2
y = x1
(45)
where c(u) represents a nonsymmetric dead-zone nonlinearity modeled by [25] ⎧ u ≥ 2.5 ⎨ (1 − 0.3 sin(u))(u − 2.5) c(u) = 0 −1.5 < u < 2.5 (46) ⎩ (0.8 − 0.2 cos(u))(u + 1.5) u ≤ −1.5 c(u) can be rewritten into the form given in (3) as 1 − 0.3 sin(u) u≥0 κ(·) = 0.8 − 0.2 cos(u) u < 0 ⎧ ⎪ ⎪ −2.5(1 − 0.3 sin(u)) u ≥ 2.5 ⎨ −(1 − 0.3 sin(u))u 0 ≤ u < 2.5 μ(·) = −(0.8 − 0.2 cos(u))u −1.5 < u < 0 ⎪ ⎪ ⎩ 1.5(0.8 − 0.2 cos(u)) u ≤ −1.5.
(47)
Clearly, |μ(·)| ≤ μ∗ = max(2.5 × 1.3, 1 × 1.5) = 3.25. The desired trajectory to be tracked is yd (t) = −cos(t). The filtered tracking error is then defined as r = e˙ + 3e. The number
In this brief, we concentrate on a class of nth-order SISO nonaffine nonlinear dynamical systems with nonsmooth actuator nonlinearities, and study its asymptotic tracking problem. In order to avoid the difficulty of control design caused by the nonaffine structure, we first propose a novel transformation approach to transfer systematically the original nonaffine system into an equivalent affine one. Then, an adaptive controller is devised for the affine system, which consists of two parts: 1) an NN-based OLA is utilized to approximate the unknown dynamics and 2) the Nussbaum gain technique is integrated to tackle the unknown and varying control coefficient. Finally, it is proven rigorously via standard Lyapunov analysis that asymptotic tracking of a given smooth enough reference signal, as well as the ultimate uniform boundedness of all the other signals can be guaranteed by the proposed control method. Numerical simulations demonstrate the effectiveness of the proposed method. R EFERENCES [1] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY, USA: Wiley, 1995. [2] D. Han and L. Shi, “Guaranteed cost control of affine nonlinear systems via partition of unity method,” Automatica, vol. 49, no. 2, pp. 660–666, 2013. [3] W. Meng, Q. Yang, Y. Ying, Y. Sun, and Z. Yang, “Adaptive power capture control of variable-speed wind energy conversion systems with guaranteed transient and steady-state performance,” IEEE Trans. Energy Convers., vol. 28, no. 3, pp. 716–725, Sep. 2013. [4] L. R. Hunt and G. Meyer, “Stable inversion for nonlinear systems,” Automatica, vol. 33, no. 8, pp. 1549–1554, 1997. [5] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900–918, Jul. 2003. [6] N. Hovakimyan, A. J. Calise, and N. Kim, “Adaptive output feedback control of a class of multi-input multi-output systems using neural networks,” Int. J. Control, vol. 77, no. 15, pp. 1318–1329, 2004.
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