Globally stable adaptive robust tracking control using ... - Springer Link
Neural Comput & Applic (2012) 21:351–363 DOI 10.1007/s00521-010-0455-8
ORIGINAL ARTICLE
Globally stable adaptive robust tracking control using RBF neural networks as feedforward compensators Weisheng Chen • L. C. Jiao • Jianshe Wu
Received: 28 January 2010 / Accepted: 19 September 2010 / Published online: 15 October 2010 Springer-Verlag London Limited 2010
Abstract In previous adaptive neural network control schemes, neural networks are usually used as feedback compensators. So, only semi-globally uniformly ultimate boundedness of closed-loop systems can be guaranteed, and no methods are given to determine the neural network approximation domain. However, in this paper, it is showed that if neural networks are used as feedforward compensators instead of feedback ones, then we can ensure the globally uniformly ultimate boundedness of closedloop systems and determine the neural network approximation domain via the bound of known reference signals. It should be pointed out that this domain is very important for designing the neural network structure, for example, it directly determines the choice of the centers of radial basis function neural networks. Simulation examples are given to illustrate the effectiveness of the proposed control approaches. Keywords Adaptive tracking control Backstepping Determination of approximation domain Feedforward compensators Globally uniformly ultimate boundedness Neural networks
W. Chen (&) Department of Applied Mathematics and the Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, China e-mail: [email protected] L. C. Jiao J. Wu Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Institute of Intelligent Information Processing, Xidian University, Xi’an 710071, China
1 Introduction During the past two decades, neural networks have received considerable attention in the control of uncertain nonlinear systems owing to their universal approximation property. In the control design, in general, neural networks are used as function approximators to model unknown system functions. So far, much work has been done in this area. For example, adaptive neural network control for robotic systems [1–4]; adaptive backstepping neural (or fuzzy) control for lower-triangular-structured systems [4–17]; neural network-based adaptive control for timedelay systems [18–24] or discrete-time systems [25–39], and so on. Among all neural networks, radial basis function (RBF) neural network is one of the most commonly used neural networks due to its linear-parameterized structure. So, it is also called the linear-parameterized-neural-network (LPNN). As pointed out in [40], RBF neural network is a local neural network. Its merit is to possess the simple structure and the easily designed learning algorithm for unknown weights, but its drawbacks are also obvious. Specifically speaking, the choice of RBFs heavily depends on the approximation domain. For example, the centers and widths of the Gaussian-type RBFs are directly related to the approximation domain. Therefore, if the approximation domain cannot be given before control design, then in practice the designers cannot deign RBFs, and theoretically the closed-loop stability cannot be guaranteed. This drawback of RBF neural network makes it more important to determine the approximation domain a priori. Unfortunately, as pointed out in [7] and [15], this is still an open problem at present stage. The main obstacle is that RBF neural network is almost used as feedback compensators to approximate the unknown uncertainties in the existing
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literature, where the system states or outputs are directly used as the neural network inputs. The other drawback of using RBF neural network as feedback compensators is to only guarantee the semiglobally uniformly ultimate boundedness of closed-loop control systems since the approximation ability of RBF neural network holds only over a given compact set. It is worth pointing out that in some existing works (see, e.g. [5]), authors state they obtain the global results under the assumption that the approximation errors are bounded by unknown parameters multiplying known functions. However, this assumption is very strict and not easy to be verified in the engineering applications. The objective of this paper is to establish a global stable neural network tracking control scheme and to give a method to determine the neural network approximation domain. Compared with the existing neural-network-based control schemes [1–40], the main difference of the proposed control scheme is that neural networks are used as the feedforward compensators instead of the feedback ones. The design idea of feedforward compensation is partly inspired by the recent works [41, 42] where the neural networks are used as feedforward compensators to reject the external disturbances. But, in our work, the RBF neural network is used as feedforward compensators in order to solve the aforementioned two open problems. The main contributions of this paper are summarized as follows:
^ Throughout this paper, |||| denotes the two-norm, H denotes the estimate of an unknown parameter H with the ~ :¼ H H, ^ kmax(B) and kmin(B) denote estimation error H the largest and smallest eigenvalues of a positive definite matrix B, respectively.
1.
where x [ R is the system state, and u [ R is the control input; f : R ? R is an unknown smooth function. The control objective is formulated as follows. For a given continuous and bounded reference signal xr(t) with the continuous and bounded derivative x_r ðtÞ, design the control input u(t) such that the tracking error z(t) =: x(t) xr(t) converges to a small neighborhood around the origin, while keeping all closed-loop signals uniformly ultimately bounded. Neural networks have been found to be very useful for controlling nonlinear systems with uncertainties. In particular, RBF neural network is usually used as a tool for modeling nonlinear functions due to its simple structure and good approximation ability [43]. Therefore, in this paper, we employ the RBF neural network to approximate an unknown continuous function .ðvÞ; v 2 X Rl , where X is a compact set, i.e.,
2.
Under the assumption that unknown system functions are bounded by partly known functions, it is shown in this paper that if RBF neural network is used as feedforward compensators, the globally uniformly ultimate boundedness of closed-loop system can be obtained by means of robust adaptive control approaches, and the neural network approximation domain is also easily determined based on the reference signal a priori. To clearly demonstrate the main idea proposed in this brief paper, we compare the feedback compensation control scheme and the feedforward one for a simple first-order system and give the simulation results to illustrate the advantages of feedforward compensation scheme. Then, we extend the design idea to the more general high-order systems with the Brunovsky canonical form using the backstepping technique.
The rest of this paper is organized as follows. Neural network feedforward compensation design idea is demonstrated for a simple first-order system in Sect. 2. Then, in Sect. 3, the design method is applicable to a class of highorder nonlinear systems. In Sect. 4, we conclude the work of this paper.
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2 Neural network feedforward compensation for first-order system In this section, we will demonstrate the difference between feedback and feedforward compensation by a simple firstorder system. For a more general class of high-order systems with the canonical Brunovsky form, we will give the detailed design procedure in the next section. To do so, we first give the following definition. Definition 1 The time-varying vector v(t) is said to be uniformly ultimately bounded (UUB), if for any X, a compact subset of Rn and all v(t0) [ X, there exists an e [ 0 and a number T(e, v(t0)) such that ||v(t)|| \ e for all t C t0 ? T. 2.1 Problem formulation and RBF neural network description Consider a simple first-order system _ ¼ f ðxðtÞÞ þ uðtÞ xðtÞ
.ðvÞ ¼ W T SðvÞ þ d. ðvÞ
ð1Þ
ð2Þ
where SðvÞ : X ! Rm is a vector-valued function and the neural node number m [ 1. The components of S(v), denoted by sk(v), 1 B k B m, are called the basis functions that are commonly chosen as Gaussian functions with the form
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# ðv lk ÞT ðv lk Þ sk ðvÞ ¼ exp ; g2k
353
"
k ¼ 1; 2; . . .; m
where lk [ X is a constant vector which is called the center of sk(v), and gk [ 0 is a real number which is called the width of sk(v). The optimal weight vector W = [w1, ..., wm]T is defined as ( ) T ^ SðvÞ W :¼ arg min sup .ðvÞ W ^ l W2R
v2X
and d.(v) is the inherent neural network approximation error, which can be decreased by increasing the neural node number m. Remark 1 As pointed out in [40], the RBF neural network is a local neural network, and its structure including the centers lk, the width gk and the neural node number m, depends heavily on the approximation domain. So, the determination of this domain a priori plays a crucial role in the RBF-neural-network-based control schemes. In general, if the designer can know this domain a priori, then the centers lk, k = 1, ..., m, are usually designed to cover this domain evenly. 2.2 RBF neural network as a feedback compensator In this subsection, we will show if the RBF neural network is used as a feedback compensator to directly approximate the unknown function f(x) in system (1), then some open problems will arise. In the following, we will demonstrate these problems. Based on the existing feedback compensation design idea, the unknown function f(x) is directly approximated as follows f ðxÞ ¼ W T SðxÞ þ df ðxÞ:
ð3Þ
According to the universal approximation property of RBF neural network, the above expression holds provided that the state x(t) always remains over a compact set X R; df(x) denotes the approximation error, which satisfies jdf ðxÞj df with df being the minimal upper bound of df(x). Then, it is easy to derive the dynamic of the tracking error as follows z_ ¼ u þ W T SðxÞ þ df ðxÞ x_ r :
~ T C1 W, ~ whose Define a Lyapunov function V ¼ 12 z2 þ 12 W derivative is expressed as 1 2 _ ^ ~ T W: V ¼ cþ ð8Þ z þ df ðxÞz þ rW 2i Using the inequalities df ðxÞz ~ TW ~ þ 1 W T W, we have 12 W 2
ð6Þ
^ denotes the estimate of W. c [ 0 is called the where W control gain whose size will affect the convergence peed of
1 2 2i z
~ TW ^ þ 2i d2f and W
r ~T ~ r T i V_ cz2 W W þ W W þ d2f 2 2 2 ‘V þ 1 where ‘ ¼ minf2c; k
r
min ðC
1
Þ
ð9Þ
g and 1 ¼ r2 W T W þ 2i d2f . Equation
(9) implies 1 1 VðtÞ Vð0Þ e‘t þ ‘ ‘
ð10Þ
which shows that V(t) is uniformly ultimately bounded. From the boundedness of V(t), it is easy to conclude the control objective can be achieved. For the more detailed derivations, the reader may be referred to [4–13]. However, the following three problems need be further investigated. 1.
ð4Þ
Based on (4), the control law and the adaptive law are usually designed as follows [4, 6, 12, 13] 1 ^ T SðxÞ þ x_ r u¼ cþ ð5Þ zW 2i _^ ¼ CðSðxÞz rWÞ ^ W
closed-loop errors. i [ 0 is used to encounter the effect of the neural network approximation error df, whose size also has effect on the ultimate system errors. r [ 0 is a small design parameter called the r-modification efficient, which is introduced to avoid the parameter drift [4, 7]. C [ 0 is the adaptive gain matrix, which will affect the convergence speed of estimated parameter. Then, the dynamic of closed-loop error system is given by 8 > ~ T SðxÞ þ df ðxÞ < z_ ¼ c þ 1 z þ W 2i ð7Þ > : _~ ^ W ¼ CðSðxÞz rWÞ
2.
Since the RBF neural network approximation ability holds only over a compact set, theoretically, only semiglobally uniformly ultimate boundedness of closedloop system can be obtained. That is, the system state x (i.e., the neural network inputs) must remain over the compact X for all t C 0. This a theoretical drawback. As pointed out in [7] and [15], how to guarantee the global stability is an open problem in neural network control community. Another open problem is to how to determine the neural network approximation domain X a priori. It must be emphasized that practically, this domain is very important for designing the RBF neural network structure. For example, as stated above, it directly determines the choice of centers and widths of Gaussian basis functions.
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3.
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Using RBF neural network to approximate the unknown function is equivalent to making a strong assumption on the nonlinear function, for example, from (3), it follows that jf ðxÞ f ðxr Þj jW T ðSðxÞ Sðxr ÞÞj þ jdðxÞ dðxr Þj: ð11Þ
Since the basis functions si(x) meet the Lipschitz condition and the approximation error d(x) and d(xr) are bounded, so there must exist constants a and b such that jf ðxÞ f ðxr Þj ajx xr j þ b:
ð12Þ
That is, the nonlinear function f(x) must satisfies linear growth condition which, however, is not easily satisfied in engineering practice. 2.3 RBF neural network as a feedforward compensator In the above subsection, we have shown the disadvantages of RBF neural network used as the feedback compensators. In this subsection, we will show if the RBF neural network is used as the feedforward compensator instead of the feedback one, the aforementioned three open problems can be solved very well. First, we make an assumption on the unknown function f(x) which is used to ensure the globally uniformly ultimate boundedness of closed-loop system. Assumption 1 There exist two unknown parameters a, and a known smooth function w(x, xr), such that the unknown function f(x) satisfies jf ðxÞ f ðxr Þj awðx; xr Þjx xr j;
8ðx; xr Þ 2 R2 :
ð13Þ
Remark 2 Compared with (12), Assumption 1 further relaxes the linear bound assumption on f(x) to the nonlinear bound case. Obviously, the inequality (12) is a special case of inequality (13). That is, when w(x, xr) = 1, the inequality (13) become the inequality (12) with b = 0. In fact, according to the well-known Mean Value Theorem, we have f(x) - f(xr) = f0 (xr ? e(x - xr))(x - xr), where 0 \ e \ 1. So, if |f0 (xr ? e(x - xr))| B aw(x, xr), then Assumption 1 holds naturally. In order to use the RBF neural network as a feedforward compensator, we rewrite the unknown function f(x) as f ðxÞ ¼ f ðxr Þ þ f ðxÞ f ðxr Þ ¼ f ðxr Þ þ Kðx; xr Þ
ð14Þ
where K(x, xr) = f(x) - f(xr) is called the replacement error, which, according to Assumption 1, is bounded by jKðx; xr Þj awðx; xr Þjx xr j;
ð15Þ
and the unknown function f(xr) in (14) will be approximated by the RBF neural network as follows
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f ðxr Þ ¼ W T Sðxr Þ þ df ðxr Þ
ð16Þ
where xr [ X and jdf ðxr Þj df ; 8xr 2 X. Then, differentiating the tracking error yields z_ ¼ u þ W T Sðxr Þ þ Kðx; xr Þ þ df ðxr Þ x_ r :
ð17Þ
Based on (17), we design the following control law and adaptive laws 1 ^ T Sðxr Þ þ x_r u¼ cþ ð18Þ z a^wðx; xr Þz W 2i _^ ¼ CðSðx Þz rWÞ ^ W r
ð19Þ
aÞ a^_ ¼ cðwðx; xr Þz2 r^
ð20Þ
where c; i; r [ 0 are the design parameters, C ¼ CT [ 0 is the adaptive gain matrix. Then, the dynamic of closed-loop error system is expressed as 8 1 > > ~ T Sðxr Þ þ Kðx;xr Þ þ df ðxÞ z _ ¼ c þ z a^wðx;xr Þz þ W > > < 2i _~ ¼ CðSðx Þz rWÞ ^ > W r > > > : a~_ ¼ Cðwðx;xr Þz2 r^ aÞ ð21Þ ~ T C1 W ~ Consider the Lyapunov function V ¼ 12 z2 þ 12 W 1 2 þ 2c a~ , whose derivative is easily computed as follows 1 2 _ ~ T Sðxr Þz V ¼ cþ z a^wðx; xr Þz2 þ W 2i þ Kðx; xr Þz þ df ðxÞz ~ T ðSðxr Þz rWÞ ^ a~ðwðx; xr Þz2 r^ W aÞ 1 2 ¼ cþ z awðx; xr Þz2 þ Kðx; xr Þz 2i ~ TW ^ þ a~a^Þ: þ df ðxÞz þ rðW
ð22Þ
Based on Assumption 1, we have the following inequalities Kðx; xr Þz þ df ðxÞz awðx; xr Þz2 þ ðdf Þjzj 1 2 i 2 z þ ð df Þ 2i 2 1 1 1 ~ TW ~ TW ^ þ a~a^ W ~ a~2 þ W T W þ 1 a2 : W 2 2 2 2 awðx; xr Þz2 þ
ð23Þ ð24Þ
Substituting (23) and (24) back into (22) yields r ~T ~ r 2 r T i df Þ 2 W a~ þ ðW W þ a2 Þ þ ð V_ cz2 W 2 2 2 2 ‘V þ 1 ð25Þ where ‘ ¼ minf2c; K
r
min ðC
1
Þ
; rcg and 1 ¼ r2 ðW T W þ a2 Þþ
i 2 2 ð df Þ .
It follows from (25) that 1 VðtÞ Vð0Þe‘t þ ‘
ð26Þ
Neural Comput & Applic (2012) 21:351–363
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which shows the boundedness of V(t). Similarly, from (26), we can further conclude the tracking error z(t) converges to a small neighborhood around the origin by adjusting the design parameters and the closed-loop signals are uniformly ultimately bounded. It can be seen from (26) that the neural network approximation error df and the design parameters c; i; r; c; C both affect the ultimate bound of tracking error. Moreover, in order to illustrate the design idea, the schematic diagram of the control scheme with neural network as feedforward compensator is shown in Fig. 1. Now, by comparing two control schemes, we easily find the feedforward compensation scheme is superior to the feedback compensation scheme in the following aspects: 1.
2.
3.
Since the RBF neural network input xr is always bounded for all t C 0 and Assumption 1 holds globally, the obtained uniformly ultimate boundedness of closed-loop system is global. That is, it is not required that the state x(t) always remains over a given compact set. This point is obviously superior to the feedback compensation scheme where only the semi-globally uniformly ultimate boundedness is obtained. Since the RBF neural network input is the reference signal xr instead of the system state x, and xr is provided a priori, the designer can determine the neural network approximation domain via the scale of the reference signal and then design the RBF neural network structure, e.g., the center lk, width gk and neural node number m. However, in the existing feedback compensation schemes, no guidelines are provided and the designer only determine this domain based on their experience. By comparing with (12) and (13), we can find that the linear bound assumption on the nonlinear functions is relaxed to the nonlinear bound one.
Fig. 1 Schematic diagram of the control scheme with neural network as feedforward compensator
2.4 Simulation comparison In this subsection, we compare two control schemes by a simulation example. Consider the following first-order system x_ ¼ 1:1x cosðxÞ þ u
ð27Þ
where it is assumed that f(x) = 1.1cos(x)x is unknown, and the control objective is to let the system state x follow the reference signal xr = sin(t). On the one hand, we employ the RBF neural network as the feedforward compensator. It is easily verified that f 0 ðxr þ ðx xr ÞÞ ¼ 1:1sinðxr þ ðx xr ÞÞðxr þ ðx xr ÞÞþ 1:1cosðxr þ ðx xr ÞÞ which implies that jf ðxÞ f ðxr Þj 1:1ðjxr j þ jðx xr Þj þ 1Þjx xr j ¼: awðx; xr Þjx xr j: where we assume that w(x,xr) = |xr| ? |(x - xr)| ? 1 is known, but a is unknown. The control scheme is given by (18–20). Since the reference signal xr(t) [ [-1,1], we determine the approximation domain X = [-1,1], and then design the basis functions as follows ! ðxr lk Þ2 sk ðxr Þ ¼ exp ; k ¼ 1; 2; . . .; 9 ð28Þ g2 where the centers lk are chosen as ± 1, ± 0.75, ± 0.5, pffiffiffiffiffiffiffi ± 0.25, 0 and the width g as 0:4. The control parameters are specified as c ¼ 0:1; i ¼ 5; r ¼ 0:001; C ¼ diagf1g and c = 0.2. On the other hand, we employ the RBF neural network as the feedback compensator, the control law and the adaptive law are given by (5) and (6). It is worth pointing out that in this case, no method can be provided to determine the approximation domain since the designer cannot know the scale of state x(t). For fair comparison purpose, we still employ the approximation domain X = [-1,1] and
Error feedback compensator:
(c 1 /(2 ))( x
+
xr )
NN feedforward compensator:
xr
+
Wˆ T S ( x r ) Robust adaptive compensator:
aˆ ( x, x r )( x
xr
System:
u
x
x
f ( x) u
+
xr ) +
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the basis functions (28). The control parameters are still chosen as c ¼ 0:1; i ¼ 5; r ¼ 0:001; and C ¼ diagf1g. All simulations are run by the Matlab ‘‘ode45’’ method and the max step size is set to be 0.1. In order to illustrate the effect of the approximation domain on the stability of closed-loop system, we give the simulation results in three cases. Figure 2 shows the case when x(0) = - 0.8 [ X, but Figs. 2 and 3 present the cases when x(0) = 5 and x(0) = - 5.5 which are outsider the approximation domain X. Figure 2 (a1, b1, c1, d1) show the simulation results using the RBF neural network as the feedback compensator with x(0) = - 0.8, from which we can see that the system state x(t) can track the reference signal xr(t) to a desired accuracy degree (see Fig. 1a1), and the RBF neural net^ T SðxÞ can match the unknown function f(x) very work W ^ and well (see Fig. 2 (c1)), and the closed-loop signals jjWjj u are uniformly ultimately bounded (see Fig. 1b1, d1). The simulation results using the RBF neural network as the feedforward compensator with x(0) = - 0.8 are shown in
(a1) r
x
x
(a2) 2
r
x, xr, x−xr
2
x, xr, x−x
Fig. 2 Simulation of system (27) with x(0) = -0.8 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
Fig. 2 (a2, b2, c2, d2), from which it can be seen that the same control performance is also obtained. By comparing two simulation results, we can find that both control schemes can achieve the desired control performance. This is because the initial state x(0) belongs to the approximation domain X. So, theoretically, both control schemes can guarantee the stability of closed-loop system and the desired tracking performance. Figure 3 (a1, b1, c1, d1) show the simulation results using the RBF neural network as the feedback compensator with x(0) = 5 which is far away from X. It can be seen that the closed-loop stability is destroyed and the state x(t) cannot follow the desired reference signal xr(t) (see Fig. 3a1). In fact, in this case, the RBF neural network loses its approximation ability (see Fig. 3c1) since x(t) is far away from X. However, the simulation results using the RBF neural network as the feedforward compensator with x(0) = 5 are shown in Fig. 3 (a2, b2, c2, d2), from which we can see that the state x(t) still can track xr(t) with very
0
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(a2)
x−xr
x
4
x
r
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xr
r
0
x
−2 0
x−x
r
r
8 6 4 2 0
x, x , x−x
r
(a1) x, x , x−x
Fig. 3 Simulation of system (27) with x(0) = 5 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
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−2 −10 −3
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small error (see Fig. 2a2), and the RBF neural network ^ T Sðxr Þ also can approximate f(xr) very well (see Fig. 3 W c2). The similar simulation results with x(0) = - 5.5 are shown in Fig. 4. The detailed discussion is omitted. In fact, for this example, we find when -5 B x(0) B 4.7, both control schemes can achieve the desired control objective. However, when x(0) \ -5 or x(0) [ 4.7, the feedback compensation scheme will lose the closed-loop stability, but the feedforward compensation scheme still works very well, which accords with the theoretical analysis given in two subsections above.
3 Globally stable neural network control for a class of high-order nonlinear systems 3.1 Problem formulation and control design Consider a class of nonlinear systems with the Brunovsky canonical form
20
time (s)
time (s)
8 > < x_ i ¼ xiþ1 ; 1 i n 1 . x_ n ¼ f ðxÞ þ wðtÞ þ u > : y ¼ x1
ð29Þ
where x ¼ ½x1 ; . . .; xn T 2 Rn is the system state vector, u [ R is the control input and y [ R is the system output; f:Rn?R is an unknown smooth function; w(t) denotes the external disturbance which is bounded by an unknown constant, i.e., jwðtÞj -; 8t 2 R. The control objective is formulated as follows. For a given bounded reference signal yr(t) with the bounded and continuous derivatives up to order n , using the RBF neural network as the feedforward compensator, design a globally stable adaptive control scheme such that the tracking error y(t) - yr(t) converges to a small neighborhood around the origin, while maintaining all the closed-loop signals globally uniformly ultimately bounded. To achieve the above globally stable control objective, similar to Assumption 1, the following additional assumption on the unknown function f(x) is needed.
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Fig. 4 Simulation of system (27) with x(0) = -5.5 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
(a2) x
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The unknown function f(x) satisfies n X Þj aj wj ðx; yðn1Þ Þjxj yðj1Þ j þ b ð30Þ jf ðxÞ f ð yðn1Þ r r r Assumption 2
j¼1 ðn1Þ
where yr
constants,
ðn1Þ T
¼ ½yr ; y_r ; . . .; yr
ðn1Þ and wj ðx; yr Þ
, aj and b are unknown
are known continuous functions.
time (s)
; i ¼ 2; . . .; n: xi ¼ zi þ ai1 þ yði1Þ r
ð33Þ
Step 1. From (33) for i = 2, we get x2 ¼ z2 þ a1 þ y_ r , and then the time derivative of z1 is z_1 ¼ x2 y_ r ¼ z2 þ a1
ð34Þ
from which, we design the first virtual control function as
Remark 3 As pointed out in Remark 2, compared with the control approaches in which the RBF neural networks are used as the feedback compensators, Assumption 2 further relaxes the linear bound requirement on the unknown function f(x) to the nonlinear bound case.
a1 ¼ c1;1 z1
We employ the backstepping technique to design the control law, define a change of coordinates
z_2 ¼ x_2 a_ 1 y€r ¼ x3 þ c1;1 z_1 y€r :
ð35Þ
where c1,1 =: c1 [ 0 is a design parameter. Then, we have z_1 ¼ c1 z1 þ z2 :
ð36Þ
Step 2. The time derivative of z2 is given by ð37Þ
z1 ¼ y yr
ð31Þ
From (33) for i = 3, it follows that x3 ¼ z3 þ a2 þ y€r . Noticing (36), we further obtain
zi ¼ xi ai1 yði1Þ ; i ¼ 2; . . .; n r
ð32Þ
z_2 ¼ z3 þ a2 c1;1 c1 z1 þ c1;1 z2
where ai-1, i = 2,…, n, are virtual control functions designed later. From (32), we have
123
ð38Þ
from which, we design the following virtual control function
Neural Comput & Applic (2012) 21:351–363
359
a2 ¼ c2;1 z1 c2;2 z2
ð39Þ
where c2,1 = -c1,1c1 ? 1 and c2,2 = c1,1 ? c2 with c2 [ 0 being a design parameter. Then, substituting (39) back into (38) results in z_2 ¼ z1 c2 z2 þ z3 :
V_z ¼
n X
ci z2i þ zn ðf ðxÞ þ xðtÞ þ unn þ ua1 þ ua2 Þ
i¼1
¼
n X
ci z2i þ zn ðf ð yðn1Þ Þ þ Kðx; yðn1Þ Þ r r
i¼1
ð40Þ
þ xðtÞ þ unn þ ua1 þ ua2 Þ
ð49Þ
Step i (i = 3, ..., n - 1). Using the similar derivations above, the time derivative of zi is given as
where the unknown function f ð yr
z_i ¼ x_ i a_ i1 yðiÞ r
f ð yðn1Þ Þ ¼ W T Sð yðn1Þ Þ þ df ð yðn1Þ Þ r r r
¼ ziþ1 þ ai þ
i1 X
ðn1Þ
and the replacement error Kðx; yr dealt with as follows
ci1;j ðzj1 cj zj þ zjþ1 Þ
j¼1
¼ ziþ1 þ ai þ
i X
ðn1Þ
ð41Þ
di;j zj
j¼1
jzn Kðx; yðn1Þ Þj r
n X
Þ is approximated by ð50Þ ðn1Þ
Þ ¼ f ðxÞ f ð yr
aj jzn jwj ðx; yðn1Þ Þjxj yðj1Þ j þ jzn jb r r
j¼1
with di;j ¼ ci1;j1 ci1;j cj þ ci1;jþ1
ð42Þ
where for convenience of denotation, we define z0 = 0 and ci,j = 0 if j = 0 or j C i. From (41), we design the virtual control function ð43Þ ai ¼ ci;1 z1 ci;2 z2 ci;i zi where ci,j = di,j, j B i - 2, ci,i-1 = di,i-1 ? 1 and ci,i = di,j ? ci with ci [ 0 being a design parameter. Then, we have z_i ¼ zi1 ci zi þ ziþ1 :
¼
j¼1
ð45Þ
n X
aj jzn jwj ðx; yðn1Þ Þjzj þ aj1 j þ jzn jb r
j¼1
j1 X ¼ aj jzn jwj ðx; yðn1Þ Þzj þ cj1;k zk þ jzn jb r j¼1 k¼1 n X
max faj ; aj jcj1;k jg
1 k\j n
j n X X
jzn wj ðx; yðn1Þ Þzk j þ jzn jb r
j¼1 k¼1
max faj ; aj jcj1;k jg
1 k\j n
ð44Þ
Step n. Similarly, the time derivative of zn is given by n X ¼ f ðxÞ þ xðtÞ þ u þ dn;j zj yðnÞ z_n ¼ x_n a_ n1 yðnÞ r r
j n X X 1 2 2 z fw2j ðx; yðn1Þ Þz þ þ jzn jb r n 4f k j¼1 k¼1 max faj ; aj jcj1;k jg
n X
1 k\j n
þ
j¼1
n X nkþ1 k¼1
jfw2j ðx; yðn1Þ Þz2n r
4f
z2k þ jzn jb n X niþ1
where dn,j is defined as in (42) for i = n. From (45), we design the control law containing five parts
Þz2n þ Hbðx; yðn1Þ r
u ¼ an þ unn þ ua1 þ ua2 þ yðnÞ r
where f [ 0 is an adjustable design parameter, and
ð46Þ
where an is defined as (43) for i = n, unn is the RBF neural network feedforward compensation term, ua,1 is used to ðn1Þ
compensate for the replacement error f ðxÞ f ð yr Þ, ua,2 is used to compensate for the bounded approximation error and disturbance. Then, we have z_n ¼ zn1 cn zn þ f ðxÞ þ xðtÞ þ unn þ ua1 þ ua2 Consider the Lyapunov function n 1X Vz ¼ z2 : 2 i¼1 i
Þ is
ð47Þ
i¼1
From (36), (40), (44) and (47), the derivative of Vz is given by
z2i þ jzn jb:
ð51Þ
H¼
max faj ; aj jcj1;k jg 1 k\j n n X Þ ¼ jfw2j ðx; yðn1Þ Þ: bðx; yðnÞ r r j¼1 Substituting (50) and (51) back into (49) yields V_z
ð48Þ
4f
n X i¼1
niþ1 2 ci zi 4f
þ zn Hbðx; yðn1Þ Þzn þ W T Sð yðn1Þ Þ þ unn þ ua1 þ ua2 r r þ jzn jðb þ df þ -Þ
ð52Þ
123
360
Neural Comput & Applic (2012) 21:351–363
where df denotes the upper bound of d(t). By using the inequality [4] jvj vtanhðv=iÞ þ ji with j = 0.2785 and i[ 0, we have the following inequality zn ð53Þ jzn jðb þ df þ -Þ zn q tanh þ qji i with q ¼: b þ df þ -. Substituting (53) into (52) yields n X niþ1 2 _ Vz ci yðn1Þ Þ þ u zi þ zn W T Sð nn r 4f i¼1 þ zn Hbðx; yðn1Þ Þzn þ ua1 r zn þ zn q tanh þ ua2 þ qji: ð54Þ i Then, we design the other parts of control law as follows 8 ^ T Sð > yðn1Þ Þ unn ¼ W > r > < ðn1Þ ^ Þzn ua1 ¼ Hbðx; yr ð55Þ > > z n > u ¼ ^ : q tanh a2 i with the adaptive laws h i 8 _^ ¼ C Sð ^ > W yðn1Þ Þzn rW > r > > < h i ^_ ¼ c bðx; yðn1Þ Þz2 rH ^ H H r n > > h i > zn > : q ^_ ¼ cq zn tanh r^ q i
ð56Þ
where C; cH ; cq [ 0 are the adaptive gains. Substituting (55) into (54) yields n X niþ1 2 ~ T Sð ci yðn1Þ Þzn V_z zi þ W r 4f i¼1 zn ~ ~zn tanh þ qji: ð57Þ þ Hbðx; yðn1Þ Þz2n þ q r i If we choose the design parameters such that ci [
niþ1 4f ,
0 or equivalently, ci niþ1 4f ¼: ci [ 0, then we have the following globally stable results.
Theorem 1 Consider the closed-loop adaptive system consisting of the system (29), the control law (46), (55), and the adaptive law (56), then for any initial conditions, all closed-loop signals are globally uniformly ultimately bounded and the tracking error can converge to a neighborhood around the origin, whose size can be tuned arbitrarily small by choosing the suitable design parameters. Proof
Consider the following Lyapunov function
1 ~ T 1 ~ 1 ~2 1 2 ~ : V ¼ Vz þ W C Wþ q H þ 2 2cH 2cq
ð58Þ
From (56) and (57), the derivative of V is easily computed as follows
123
V_
n X
~H ^ þq ^ þH ~ TW ~q ^Þ þ qji: c0i z2i þ rðW
ð59Þ
i¼1
By completing square, we have the following inequality r ~T ~ ~ 2 ~H ^ þq ~ TW ^ þH ~q ^ Þ ðW ~2 Þ rðW W þH þq 2 r þ ðW T W þ H2 þ q2 Þ: ð60Þ 2 Substituting (60) into (59) yields n X r ~T ~ ~ 2 ~2 Þ W þH þq V_ c0i z2i ðW 2 i¼1 r T þ ðW W þ H2 þ q2 Þ þ qji ‘V þ t 2 where ‘ ¼ min1 i n f2c0i ; k r T 2 ðW W
2
r min ðC
1
Þ
ð61Þ
; rcH ; rcq g and t ¼
2
þ H þ q Þ þ qji. It follows from (25) that t ð62Þ VðtÞ Vð0Þe‘t þ ‘ ^ which shows the boundedness of V(t), which implies zi, W, ^ and q ^ are GUUB. From (35), (39) and (43), we can H further obtain the boundedness of ai. Then, it can be seen from (33) that xi are GUUB. Finally, we can easily get the boundedness of u from the boundedness of the above signals. Moreover, from (62), we have rffiffiffi pffiffiffiffiffiffiffiffiffiffiffi t lim jy yr j lim 2VðtÞ : ð63Þ t!þ1 t!þ1 ‘ By increasing C; cH ; cq ; c0i , and reducing r; i, we can make pffi the value of t‘ arbitrarily small. This completes the whole proof. 3.2 Simulation example Consider the Duffing’s equation described by ( x_1 ¼ x2 x_2 ¼ p1 x1 p2 x31 px2 þ q cosðdtÞ þ u
ð64Þ
where p1, p2, p, q and d are real constants. Depending on the choice of these constants, the solution of system (64) with u = 0 may display complex phenomena. Obviously, system (64) has the form (1), where f ðxÞ ¼ p1 x1 p2 x31 px2 and xðtÞ ¼ q cosðdtÞ. It is easily verified that f(x) satisfies jf ðx1 ; x2 Þ f ðyr ; y_r Þj jx1 yr ja1 w1 ðx1 ; yr Þ þ jx2 y_r ja2 w2 ðx2 ; yr Þ
ð65Þ
where a1 ¼ maxfjp1 j; jp2 jg, w1 ¼ 1 þ jx21 þ y2r þ x1 yr j, a2 = |p|, and w2 = 1. Based on the proposed control scheme, we design the control law as follows
Neural Comput & Applic (2012) 21:351–363
361
u ¼ a2 þ unn þ u1 þ u2 þ y€r
ð66Þ
with a2 ¼ ðc21 þ 1Þz1 ðc1 þ c2 Þz2 ^ T Sðyr ; y_ r Þ unn ¼ W ^ yr ; y_r Þz2 u1 ¼ Hbðx; z2 u2 ¼ ^ q tanh i where z1 = y - yr, z2 ¼ x2 a1 y_ r , a1 = - c1z1 and ^ ^ and w h; H bðx; yr ; y_ r Þ ¼ fw21 þ 2fw22 . The adaptive laws of ^ are given by (56). For simulation purpose, it is assumed that p1 = -1.1, p2 = 1, p = 0.4, q = 2.1 and d = 1.8. In simulation, the reference signal is specified as yr ðtÞ ¼ sinðtÞ. So, the RBF neural network approximation domain is X = [-1, 1] 9 [-1, 1]. Then, this neural node number is appropriately chosen as 36 (If this number is too large, then
the more computation loads are needed. If this number is too small, then the neural network approximation error cannot satisfy the performance requirement). Moreover, to ensure the centers of RFBs evenly cover X, we set the centers ni = (ai, bi) with ai and bi chosen arbitrarily from {-1, -0.6, -0.2, 0.2, 0.6, 1}, and the width is set to be gi = 1/(25 ln 2) so that the sum of two neighboring RFB values is one at the midpoint between two centers. The control parameters are set to be f = 2, c1 = 0.6, c2 = 0.3 22þ1 (c1 [ 21þ1 2f , c2 [ 2f ), i ¼ 0:01; r ¼ 0:0001; C ¼ 2I; ^ ^ cH ¼ 0:02; cq ¼ 0:5. and the initial values of hð0Þ; Hð0Þ ^ and wð0Þ are zeros. The simulation results are shown in Fig. 5. Figure 5 (a1, b1, c1, d1) present the case when xð0Þ ¼ ½0:8; 0T 2 X, from which we can see that the control performance is satisfactory. To further verify the advantages of our feedforward compensation method, we present the
(a1)
2
(a2)
6
y
y
r
y, y
y, yr
4 0
2 0
y −2
r
0
10
20
30
40
−2
50
yr 0
10
20
time (s)
50
40
50
40
50
r
f(x )
f(xr)
40
0 −2
RBFNN 0
10
20
30
40
50
RBFNN 0
10
20
time (s)
(c1)
4
ρ
||W|| 2 0 −2
x2 0
10
Θ 20
30
40
(c2)
4 2 0
x2
−2
50
0
10
ρ 20
30
time (s)
(d1)
(d2)
u
200
0
−5
Θ
||W||
time (s) 5
30
time (s)
x2, ||W||, Θ, ρ
x2, ||W||, Θ, ρ
50
f(xr)
2
0 −2
u
40
(b2)
(b1) f(xr)
2
30
time (s)
100
0 0
10
20
30
time (s)
40
50
0
10
20
30
time (s)
Fig. 5 Simulation of system (64) using feedforward compensation design a1–d1: x1(0) = 0.8; a2–d2: x1(0) = 6
123
362
simulation results of x(0) = [6, 0]T [ X in Fig. 5 (a2, b2, c2, d2). In this case, theoretically, the traditional feedback compensation method cannot ensure the stability of closedloop system since x(0) is far away from X. However, the simulation results show the feedforward compensation scheme still can guarantee the stability of closed-loop system. This further verifies the advantages of the feedforward compensation method proposed in this paper.
4 Conclusions In this paper, we discuss the two open problems in neural network control community, i.e., the globally uniformly ultimate boundedness and the approximation domain determination. It is shown that the two problems can be solved well under the assumption that the unknown system functions are bounded by partly known functions. The main design idea is to use the RBF neural network as feedforward compensators instead of feedback ones. In the authors’ opinion, since the universal approximation property of neural networks holds only provided that the neural inputs must remain on a compact set, it is more valid and practical to use the neural network as a feedforward compensator than a feedback one. Acknowledgments The authors would like to thank the anonymous reviewers for their comments that improve the quality of the paper. This work was supported by the National Natural Science Foundation of P. R. China (60804021, 61072106, 61072139, 61001202), the Fundamental Research Funds for the Central Universities (JY10000970001), and the China Postdoctoral Science Foundation funded project (20090461282).
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ORIGINAL ARTICLE
Globally stable adaptive robust tracking control using RBF neural networks as feedforward compensators Weisheng Chen • L. C. Jiao • Jianshe Wu
Received: 28 January 2010 / Accepted: 19 September 2010 / Published online: 15 October 2010 Springer-Verlag London Limited 2010
Abstract In previous adaptive neural network control schemes, neural networks are usually used as feedback compensators. So, only semi-globally uniformly ultimate boundedness of closed-loop systems can be guaranteed, and no methods are given to determine the neural network approximation domain. However, in this paper, it is showed that if neural networks are used as feedforward compensators instead of feedback ones, then we can ensure the globally uniformly ultimate boundedness of closedloop systems and determine the neural network approximation domain via the bound of known reference signals. It should be pointed out that this domain is very important for designing the neural network structure, for example, it directly determines the choice of the centers of radial basis function neural networks. Simulation examples are given to illustrate the effectiveness of the proposed control approaches. Keywords Adaptive tracking control Backstepping Determination of approximation domain Feedforward compensators Globally uniformly ultimate boundedness Neural networks
W. Chen (&) Department of Applied Mathematics and the Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, Xi’an 710071, China e-mail: [email protected] L. C. Jiao J. Wu Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Institute of Intelligent Information Processing, Xidian University, Xi’an 710071, China
1 Introduction During the past two decades, neural networks have received considerable attention in the control of uncertain nonlinear systems owing to their universal approximation property. In the control design, in general, neural networks are used as function approximators to model unknown system functions. So far, much work has been done in this area. For example, adaptive neural network control for robotic systems [1–4]; adaptive backstepping neural (or fuzzy) control for lower-triangular-structured systems [4–17]; neural network-based adaptive control for timedelay systems [18–24] or discrete-time systems [25–39], and so on. Among all neural networks, radial basis function (RBF) neural network is one of the most commonly used neural networks due to its linear-parameterized structure. So, it is also called the linear-parameterized-neural-network (LPNN). As pointed out in [40], RBF neural network is a local neural network. Its merit is to possess the simple structure and the easily designed learning algorithm for unknown weights, but its drawbacks are also obvious. Specifically speaking, the choice of RBFs heavily depends on the approximation domain. For example, the centers and widths of the Gaussian-type RBFs are directly related to the approximation domain. Therefore, if the approximation domain cannot be given before control design, then in practice the designers cannot deign RBFs, and theoretically the closed-loop stability cannot be guaranteed. This drawback of RBF neural network makes it more important to determine the approximation domain a priori. Unfortunately, as pointed out in [7] and [15], this is still an open problem at present stage. The main obstacle is that RBF neural network is almost used as feedback compensators to approximate the unknown uncertainties in the existing
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Neural Comput & Applic (2012) 21:351–363
literature, where the system states or outputs are directly used as the neural network inputs. The other drawback of using RBF neural network as feedback compensators is to only guarantee the semiglobally uniformly ultimate boundedness of closed-loop control systems since the approximation ability of RBF neural network holds only over a given compact set. It is worth pointing out that in some existing works (see, e.g. [5]), authors state they obtain the global results under the assumption that the approximation errors are bounded by unknown parameters multiplying known functions. However, this assumption is very strict and not easy to be verified in the engineering applications. The objective of this paper is to establish a global stable neural network tracking control scheme and to give a method to determine the neural network approximation domain. Compared with the existing neural-network-based control schemes [1–40], the main difference of the proposed control scheme is that neural networks are used as the feedforward compensators instead of the feedback ones. The design idea of feedforward compensation is partly inspired by the recent works [41, 42] where the neural networks are used as feedforward compensators to reject the external disturbances. But, in our work, the RBF neural network is used as feedforward compensators in order to solve the aforementioned two open problems. The main contributions of this paper are summarized as follows:
^ Throughout this paper, |||| denotes the two-norm, H denotes the estimate of an unknown parameter H with the ~ :¼ H H, ^ kmax(B) and kmin(B) denote estimation error H the largest and smallest eigenvalues of a positive definite matrix B, respectively.
1.
where x [ R is the system state, and u [ R is the control input; f : R ? R is an unknown smooth function. The control objective is formulated as follows. For a given continuous and bounded reference signal xr(t) with the continuous and bounded derivative x_r ðtÞ, design the control input u(t) such that the tracking error z(t) =: x(t) xr(t) converges to a small neighborhood around the origin, while keeping all closed-loop signals uniformly ultimately bounded. Neural networks have been found to be very useful for controlling nonlinear systems with uncertainties. In particular, RBF neural network is usually used as a tool for modeling nonlinear functions due to its simple structure and good approximation ability [43]. Therefore, in this paper, we employ the RBF neural network to approximate an unknown continuous function .ðvÞ; v 2 X Rl , where X is a compact set, i.e.,
2.
Under the assumption that unknown system functions are bounded by partly known functions, it is shown in this paper that if RBF neural network is used as feedforward compensators, the globally uniformly ultimate boundedness of closed-loop system can be obtained by means of robust adaptive control approaches, and the neural network approximation domain is also easily determined based on the reference signal a priori. To clearly demonstrate the main idea proposed in this brief paper, we compare the feedback compensation control scheme and the feedforward one for a simple first-order system and give the simulation results to illustrate the advantages of feedforward compensation scheme. Then, we extend the design idea to the more general high-order systems with the Brunovsky canonical form using the backstepping technique.
The rest of this paper is organized as follows. Neural network feedforward compensation design idea is demonstrated for a simple first-order system in Sect. 2. Then, in Sect. 3, the design method is applicable to a class of highorder nonlinear systems. In Sect. 4, we conclude the work of this paper.
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2 Neural network feedforward compensation for first-order system In this section, we will demonstrate the difference between feedback and feedforward compensation by a simple firstorder system. For a more general class of high-order systems with the canonical Brunovsky form, we will give the detailed design procedure in the next section. To do so, we first give the following definition. Definition 1 The time-varying vector v(t) is said to be uniformly ultimately bounded (UUB), if for any X, a compact subset of Rn and all v(t0) [ X, there exists an e [ 0 and a number T(e, v(t0)) such that ||v(t)|| \ e for all t C t0 ? T. 2.1 Problem formulation and RBF neural network description Consider a simple first-order system _ ¼ f ðxðtÞÞ þ uðtÞ xðtÞ
.ðvÞ ¼ W T SðvÞ þ d. ðvÞ
ð1Þ
ð2Þ
where SðvÞ : X ! Rm is a vector-valued function and the neural node number m [ 1. The components of S(v), denoted by sk(v), 1 B k B m, are called the basis functions that are commonly chosen as Gaussian functions with the form
Neural Comput & Applic (2012) 21:351–363
# ðv lk ÞT ðv lk Þ sk ðvÞ ¼ exp ; g2k
353
"
k ¼ 1; 2; . . .; m
where lk [ X is a constant vector which is called the center of sk(v), and gk [ 0 is a real number which is called the width of sk(v). The optimal weight vector W = [w1, ..., wm]T is defined as ( ) T ^ SðvÞ W :¼ arg min sup .ðvÞ W ^ l W2R
v2X
and d.(v) is the inherent neural network approximation error, which can be decreased by increasing the neural node number m. Remark 1 As pointed out in [40], the RBF neural network is a local neural network, and its structure including the centers lk, the width gk and the neural node number m, depends heavily on the approximation domain. So, the determination of this domain a priori plays a crucial role in the RBF-neural-network-based control schemes. In general, if the designer can know this domain a priori, then the centers lk, k = 1, ..., m, are usually designed to cover this domain evenly. 2.2 RBF neural network as a feedback compensator In this subsection, we will show if the RBF neural network is used as a feedback compensator to directly approximate the unknown function f(x) in system (1), then some open problems will arise. In the following, we will demonstrate these problems. Based on the existing feedback compensation design idea, the unknown function f(x) is directly approximated as follows f ðxÞ ¼ W T SðxÞ þ df ðxÞ:
ð3Þ
According to the universal approximation property of RBF neural network, the above expression holds provided that the state x(t) always remains over a compact set X R; df(x) denotes the approximation error, which satisfies jdf ðxÞj df with df being the minimal upper bound of df(x). Then, it is easy to derive the dynamic of the tracking error as follows z_ ¼ u þ W T SðxÞ þ df ðxÞ x_ r :
~ T C1 W, ~ whose Define a Lyapunov function V ¼ 12 z2 þ 12 W derivative is expressed as 1 2 _ ^ ~ T W: V ¼ cþ ð8Þ z þ df ðxÞz þ rW 2i Using the inequalities df ðxÞz ~ TW ~ þ 1 W T W, we have 12 W 2
ð6Þ
^ denotes the estimate of W. c [ 0 is called the where W control gain whose size will affect the convergence peed of
1 2 2i z
~ TW ^ þ 2i d2f and W
r ~T ~ r T i V_ cz2 W W þ W W þ d2f 2 2 2 ‘V þ 1 where ‘ ¼ minf2c; k
r
min ðC
1
Þ
ð9Þ
g and 1 ¼ r2 W T W þ 2i d2f . Equation
(9) implies 1 1 VðtÞ Vð0Þ e‘t þ ‘ ‘
ð10Þ
which shows that V(t) is uniformly ultimately bounded. From the boundedness of V(t), it is easy to conclude the control objective can be achieved. For the more detailed derivations, the reader may be referred to [4–13]. However, the following three problems need be further investigated. 1.
ð4Þ
Based on (4), the control law and the adaptive law are usually designed as follows [4, 6, 12, 13] 1 ^ T SðxÞ þ x_ r u¼ cþ ð5Þ zW 2i _^ ¼ CðSðxÞz rWÞ ^ W
closed-loop errors. i [ 0 is used to encounter the effect of the neural network approximation error df, whose size also has effect on the ultimate system errors. r [ 0 is a small design parameter called the r-modification efficient, which is introduced to avoid the parameter drift [4, 7]. C [ 0 is the adaptive gain matrix, which will affect the convergence speed of estimated parameter. Then, the dynamic of closed-loop error system is given by 8 > ~ T SðxÞ þ df ðxÞ < z_ ¼ c þ 1 z þ W 2i ð7Þ > : _~ ^ W ¼ CðSðxÞz rWÞ
2.
Since the RBF neural network approximation ability holds only over a compact set, theoretically, only semiglobally uniformly ultimate boundedness of closedloop system can be obtained. That is, the system state x (i.e., the neural network inputs) must remain over the compact X for all t C 0. This a theoretical drawback. As pointed out in [7] and [15], how to guarantee the global stability is an open problem in neural network control community. Another open problem is to how to determine the neural network approximation domain X a priori. It must be emphasized that practically, this domain is very important for designing the RBF neural network structure. For example, as stated above, it directly determines the choice of centers and widths of Gaussian basis functions.
123
354
3.
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Using RBF neural network to approximate the unknown function is equivalent to making a strong assumption on the nonlinear function, for example, from (3), it follows that jf ðxÞ f ðxr Þj jW T ðSðxÞ Sðxr ÞÞj þ jdðxÞ dðxr Þj: ð11Þ
Since the basis functions si(x) meet the Lipschitz condition and the approximation error d(x) and d(xr) are bounded, so there must exist constants a and b such that jf ðxÞ f ðxr Þj ajx xr j þ b:
ð12Þ
That is, the nonlinear function f(x) must satisfies linear growth condition which, however, is not easily satisfied in engineering practice. 2.3 RBF neural network as a feedforward compensator In the above subsection, we have shown the disadvantages of RBF neural network used as the feedback compensators. In this subsection, we will show if the RBF neural network is used as the feedforward compensator instead of the feedback one, the aforementioned three open problems can be solved very well. First, we make an assumption on the unknown function f(x) which is used to ensure the globally uniformly ultimate boundedness of closed-loop system. Assumption 1 There exist two unknown parameters a, and a known smooth function w(x, xr), such that the unknown function f(x) satisfies jf ðxÞ f ðxr Þj awðx; xr Þjx xr j;
8ðx; xr Þ 2 R2 :
ð13Þ
Remark 2 Compared with (12), Assumption 1 further relaxes the linear bound assumption on f(x) to the nonlinear bound case. Obviously, the inequality (12) is a special case of inequality (13). That is, when w(x, xr) = 1, the inequality (13) become the inequality (12) with b = 0. In fact, according to the well-known Mean Value Theorem, we have f(x) - f(xr) = f0 (xr ? e(x - xr))(x - xr), where 0 \ e \ 1. So, if |f0 (xr ? e(x - xr))| B aw(x, xr), then Assumption 1 holds naturally. In order to use the RBF neural network as a feedforward compensator, we rewrite the unknown function f(x) as f ðxÞ ¼ f ðxr Þ þ f ðxÞ f ðxr Þ ¼ f ðxr Þ þ Kðx; xr Þ
ð14Þ
where K(x, xr) = f(x) - f(xr) is called the replacement error, which, according to Assumption 1, is bounded by jKðx; xr Þj awðx; xr Þjx xr j;
ð15Þ
and the unknown function f(xr) in (14) will be approximated by the RBF neural network as follows
123
f ðxr Þ ¼ W T Sðxr Þ þ df ðxr Þ
ð16Þ
where xr [ X and jdf ðxr Þj df ; 8xr 2 X. Then, differentiating the tracking error yields z_ ¼ u þ W T Sðxr Þ þ Kðx; xr Þ þ df ðxr Þ x_ r :
ð17Þ
Based on (17), we design the following control law and adaptive laws 1 ^ T Sðxr Þ þ x_r u¼ cþ ð18Þ z a^wðx; xr Þz W 2i _^ ¼ CðSðx Þz rWÞ ^ W r
ð19Þ
aÞ a^_ ¼ cðwðx; xr Þz2 r^
ð20Þ
where c; i; r [ 0 are the design parameters, C ¼ CT [ 0 is the adaptive gain matrix. Then, the dynamic of closed-loop error system is expressed as 8 1 > > ~ T Sðxr Þ þ Kðx;xr Þ þ df ðxÞ z _ ¼ c þ z a^wðx;xr Þz þ W > > < 2i _~ ¼ CðSðx Þz rWÞ ^ > W r > > > : a~_ ¼ Cðwðx;xr Þz2 r^ aÞ ð21Þ ~ T C1 W ~ Consider the Lyapunov function V ¼ 12 z2 þ 12 W 1 2 þ 2c a~ , whose derivative is easily computed as follows 1 2 _ ~ T Sðxr Þz V ¼ cþ z a^wðx; xr Þz2 þ W 2i þ Kðx; xr Þz þ df ðxÞz ~ T ðSðxr Þz rWÞ ^ a~ðwðx; xr Þz2 r^ W aÞ 1 2 ¼ cþ z awðx; xr Þz2 þ Kðx; xr Þz 2i ~ TW ^ þ a~a^Þ: þ df ðxÞz þ rðW
ð22Þ
Based on Assumption 1, we have the following inequalities Kðx; xr Þz þ df ðxÞz awðx; xr Þz2 þ ðdf Þjzj 1 2 i 2 z þ ð df Þ 2i 2 1 1 1 ~ TW ~ TW ^ þ a~a^ W ~ a~2 þ W T W þ 1 a2 : W 2 2 2 2 awðx; xr Þz2 þ
ð23Þ ð24Þ
Substituting (23) and (24) back into (22) yields r ~T ~ r 2 r T i df Þ 2 W a~ þ ðW W þ a2 Þ þ ð V_ cz2 W 2 2 2 2 ‘V þ 1 ð25Þ where ‘ ¼ minf2c; K
r
min ðC
1
Þ
; rcg and 1 ¼ r2 ðW T W þ a2 Þþ
i 2 2 ð df Þ .
It follows from (25) that 1 VðtÞ Vð0Þe‘t þ ‘
ð26Þ
Neural Comput & Applic (2012) 21:351–363
355
which shows the boundedness of V(t). Similarly, from (26), we can further conclude the tracking error z(t) converges to a small neighborhood around the origin by adjusting the design parameters and the closed-loop signals are uniformly ultimately bounded. It can be seen from (26) that the neural network approximation error df and the design parameters c; i; r; c; C both affect the ultimate bound of tracking error. Moreover, in order to illustrate the design idea, the schematic diagram of the control scheme with neural network as feedforward compensator is shown in Fig. 1. Now, by comparing two control schemes, we easily find the feedforward compensation scheme is superior to the feedback compensation scheme in the following aspects: 1.
2.
3.
Since the RBF neural network input xr is always bounded for all t C 0 and Assumption 1 holds globally, the obtained uniformly ultimate boundedness of closed-loop system is global. That is, it is not required that the state x(t) always remains over a given compact set. This point is obviously superior to the feedback compensation scheme where only the semi-globally uniformly ultimate boundedness is obtained. Since the RBF neural network input is the reference signal xr instead of the system state x, and xr is provided a priori, the designer can determine the neural network approximation domain via the scale of the reference signal and then design the RBF neural network structure, e.g., the center lk, width gk and neural node number m. However, in the existing feedback compensation schemes, no guidelines are provided and the designer only determine this domain based on their experience. By comparing with (12) and (13), we can find that the linear bound assumption on the nonlinear functions is relaxed to the nonlinear bound one.
Fig. 1 Schematic diagram of the control scheme with neural network as feedforward compensator
2.4 Simulation comparison In this subsection, we compare two control schemes by a simulation example. Consider the following first-order system x_ ¼ 1:1x cosðxÞ þ u
ð27Þ
where it is assumed that f(x) = 1.1cos(x)x is unknown, and the control objective is to let the system state x follow the reference signal xr = sin(t). On the one hand, we employ the RBF neural network as the feedforward compensator. It is easily verified that f 0 ðxr þ ðx xr ÞÞ ¼ 1:1sinðxr þ ðx xr ÞÞðxr þ ðx xr ÞÞþ 1:1cosðxr þ ðx xr ÞÞ which implies that jf ðxÞ f ðxr Þj 1:1ðjxr j þ jðx xr Þj þ 1Þjx xr j ¼: awðx; xr Þjx xr j: where we assume that w(x,xr) = |xr| ? |(x - xr)| ? 1 is known, but a is unknown. The control scheme is given by (18–20). Since the reference signal xr(t) [ [-1,1], we determine the approximation domain X = [-1,1], and then design the basis functions as follows ! ðxr lk Þ2 sk ðxr Þ ¼ exp ; k ¼ 1; 2; . . .; 9 ð28Þ g2 where the centers lk are chosen as ± 1, ± 0.75, ± 0.5, pffiffiffiffiffiffiffi ± 0.25, 0 and the width g as 0:4. The control parameters are specified as c ¼ 0:1; i ¼ 5; r ¼ 0:001; C ¼ diagf1g and c = 0.2. On the other hand, we employ the RBF neural network as the feedback compensator, the control law and the adaptive law are given by (5) and (6). It is worth pointing out that in this case, no method can be provided to determine the approximation domain since the designer cannot know the scale of state x(t). For fair comparison purpose, we still employ the approximation domain X = [-1,1] and
Error feedback compensator:
(c 1 /(2 ))( x
+
xr )
NN feedforward compensator:
xr
+
Wˆ T S ( x r ) Robust adaptive compensator:
aˆ ( x, x r )( x
xr
System:
u
x
x
f ( x) u
+
xr ) +
123
356
Neural Comput & Applic (2012) 21:351–363
the basis functions (28). The control parameters are still chosen as c ¼ 0:1; i ¼ 5; r ¼ 0:001; and C ¼ diagf1g. All simulations are run by the Matlab ‘‘ode45’’ method and the max step size is set to be 0.1. In order to illustrate the effect of the approximation domain on the stability of closed-loop system, we give the simulation results in three cases. Figure 2 shows the case when x(0) = - 0.8 [ X, but Figs. 2 and 3 present the cases when x(0) = 5 and x(0) = - 5.5 which are outsider the approximation domain X. Figure 2 (a1, b1, c1, d1) show the simulation results using the RBF neural network as the feedback compensator with x(0) = - 0.8, from which we can see that the system state x(t) can track the reference signal xr(t) to a desired accuracy degree (see Fig. 1a1), and the RBF neural net^ T SðxÞ can match the unknown function f(x) very work W ^ and well (see Fig. 2 (c1)), and the closed-loop signals jjWjj u are uniformly ultimately bounded (see Fig. 1b1, d1). The simulation results using the RBF neural network as the feedforward compensator with x(0) = - 0.8 are shown in
(a1) r
x
x
(a2) 2
r
x, xr, x−xr
2
x, xr, x−x
Fig. 2 Simulation of system (27) with x(0) = -0.8 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
Fig. 2 (a2, b2, c2, d2), from which it can be seen that the same control performance is also obtained. By comparing two simulation results, we can find that both control schemes can achieve the desired control performance. This is because the initial state x(0) belongs to the approximation domain X. So, theoretically, both control schemes can guarantee the stability of closed-loop system and the desired tracking performance. Figure 3 (a1, b1, c1, d1) show the simulation results using the RBF neural network as the feedback compensator with x(0) = 5 which is far away from X. It can be seen that the closed-loop stability is destroyed and the state x(t) cannot follow the desired reference signal xr(t) (see Fig. 3a1). In fact, in this case, the RBF neural network loses its approximation ability (see Fig. 3c1) since x(t) is far away from X. However, the simulation results using the RBF neural network as the feedforward compensator with x(0) = 5 are shown in Fig. 3 (a2, b2, c2, d2), from which we can see that the state x(t) still can track xr(t) with very
0
x−x −2
r
0
20
40
x
r
0
x−x −2
60
x
r
0
20
(b1) 1.5 1 0.5 0
0
20
40
60
time (s)
(b2) estimate of ||W|| 1 0.5 0
estimate of a 0
20
f(xr)
f(x)
60
(c2)
0
RBFNN 0
20
0 −1
f(x) 40
60
RBFNN 0
f(xr)
20
time (s)
time (s)
(d1)
(d2)
40
60
40
60
2
u
2
u
40
time (s)
1
−1
0
0
20
40
time (s)
123
60
1.5
(c1) 1
−2
40
time (s) estimates of ||W||, a
estimate of ||W||
time (s)
60
0 −2
0
20
time (s)
Neural Comput & Applic (2012) 21:351–363
r
(a2)
x−xr
x
4
x
r
2
xr
r
0
x
−2 0
x−x
r
r
8 6 4 2 0
x, x , x−x
r
(a1) x, x , x−x
Fig. 3 Simulation of system (27) with x(0) = 5 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
357
20
40
60
0
20
(b1)
1
0
−1
0
20
40
60
time (s)
4
0
0
r
2 0
RBFNN 0
20
0
20
40
60
40
60
40
60
(c2) RBFNN
2 0 −2
60
40
time (s)
f(x)
f(x )
f(x)
estimate of a estimate of ||W||
2
4
4
−2
60
(b2)
6
(c1) 6
40
time (s) estimates of ||W||, a
estimate of ||W||
time (s)
f(xr) 0
20
time (s)
time (s)
(d1)
(d2) 0
u
u
−1 −5
−2 −10 −3
0
20
40
60
0
small error (see Fig. 2a2), and the RBF neural network ^ T Sðxr Þ also can approximate f(xr) very well (see Fig. 3 W c2). The similar simulation results with x(0) = - 5.5 are shown in Fig. 4. The detailed discussion is omitted. In fact, for this example, we find when -5 B x(0) B 4.7, both control schemes can achieve the desired control objective. However, when x(0) \ -5 or x(0) [ 4.7, the feedback compensation scheme will lose the closed-loop stability, but the feedforward compensation scheme still works very well, which accords with the theoretical analysis given in two subsections above.
3 Globally stable neural network control for a class of high-order nonlinear systems 3.1 Problem formulation and control design Consider a class of nonlinear systems with the Brunovsky canonical form
20
time (s)
time (s)
8 > < x_ i ¼ xiþ1 ; 1 i n 1 . x_ n ¼ f ðxÞ þ wðtÞ þ u > : y ¼ x1
ð29Þ
where x ¼ ½x1 ; . . .; xn T 2 Rn is the system state vector, u [ R is the control input and y [ R is the system output; f:Rn?R is an unknown smooth function; w(t) denotes the external disturbance which is bounded by an unknown constant, i.e., jwðtÞj -; 8t 2 R. The control objective is formulated as follows. For a given bounded reference signal yr(t) with the bounded and continuous derivatives up to order n , using the RBF neural network as the feedforward compensator, design a globally stable adaptive control scheme such that the tracking error y(t) - yr(t) converges to a small neighborhood around the origin, while maintaining all the closed-loop signals globally uniformly ultimately bounded. To achieve the above globally stable control objective, similar to Assumption 1, the following additional assumption on the unknown function f(x) is needed.
123
358
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Fig. 4 Simulation of system (27) with x(0) = -5.5 a1–d1: feedback compensation design; a2–d2: feedforward compensation design
(a2) x
2
r
0 −2 −4 −6 −8
x−xr
0
r
x
r
x
x, x , x−x
x, xr, x−xr
(a1)
20
40
0 −2
60
x−xr
xr
−4 0
20
(b1)
1
0
−1
0
20
40
60
time (s)
(c1)
2
f(xr)
f(x)
−2 −4 −6 20
60
40
60
40
60
estimate of ||W||
2 0
0
40
time (s)
(c2) f(xr)
0
−4
60
20
RBFNN 0
20
time (s)
time (s)
(d1)
3
40
estimate of a
4
−2
f(x) 0
60
(b2)
6
2
RBFNN
0
40
time (s) estimates of ||W||, a
estimate of ||W||
time (s)
(d2)
20
u
u
2
10
1 0 0
0
20
40
60
0
20
time (s)
The unknown function f(x) satisfies n X Þj aj wj ðx; yðn1Þ Þjxj yðj1Þ j þ b ð30Þ jf ðxÞ f ð yðn1Þ r r r Assumption 2
j¼1 ðn1Þ
where yr
constants,
ðn1Þ T
¼ ½yr ; y_r ; . . .; yr
ðn1Þ and wj ðx; yr Þ
, aj and b are unknown
are known continuous functions.
time (s)
; i ¼ 2; . . .; n: xi ¼ zi þ ai1 þ yði1Þ r
ð33Þ
Step 1. From (33) for i = 2, we get x2 ¼ z2 þ a1 þ y_ r , and then the time derivative of z1 is z_1 ¼ x2 y_ r ¼ z2 þ a1
ð34Þ
from which, we design the first virtual control function as
Remark 3 As pointed out in Remark 2, compared with the control approaches in which the RBF neural networks are used as the feedback compensators, Assumption 2 further relaxes the linear bound requirement on the unknown function f(x) to the nonlinear bound case.
a1 ¼ c1;1 z1
We employ the backstepping technique to design the control law, define a change of coordinates
z_2 ¼ x_2 a_ 1 y€r ¼ x3 þ c1;1 z_1 y€r :
ð35Þ
where c1,1 =: c1 [ 0 is a design parameter. Then, we have z_1 ¼ c1 z1 þ z2 :
ð36Þ
Step 2. The time derivative of z2 is given by ð37Þ
z1 ¼ y yr
ð31Þ
From (33) for i = 3, it follows that x3 ¼ z3 þ a2 þ y€r . Noticing (36), we further obtain
zi ¼ xi ai1 yði1Þ ; i ¼ 2; . . .; n r
ð32Þ
z_2 ¼ z3 þ a2 c1;1 c1 z1 þ c1;1 z2
where ai-1, i = 2,…, n, are virtual control functions designed later. From (32), we have
123
ð38Þ
from which, we design the following virtual control function
Neural Comput & Applic (2012) 21:351–363
359
a2 ¼ c2;1 z1 c2;2 z2
ð39Þ
where c2,1 = -c1,1c1 ? 1 and c2,2 = c1,1 ? c2 with c2 [ 0 being a design parameter. Then, substituting (39) back into (38) results in z_2 ¼ z1 c2 z2 þ z3 :
V_z ¼
n X
ci z2i þ zn ðf ðxÞ þ xðtÞ þ unn þ ua1 þ ua2 Þ
i¼1
¼
n X
ci z2i þ zn ðf ð yðn1Þ Þ þ Kðx; yðn1Þ Þ r r
i¼1
ð40Þ
þ xðtÞ þ unn þ ua1 þ ua2 Þ
ð49Þ
Step i (i = 3, ..., n - 1). Using the similar derivations above, the time derivative of zi is given as
where the unknown function f ð yr
z_i ¼ x_ i a_ i1 yðiÞ r
f ð yðn1Þ Þ ¼ W T Sð yðn1Þ Þ þ df ð yðn1Þ Þ r r r
¼ ziþ1 þ ai þ
i1 X
ðn1Þ
and the replacement error Kðx; yr dealt with as follows
ci1;j ðzj1 cj zj þ zjþ1 Þ
j¼1
¼ ziþ1 þ ai þ
i X
ðn1Þ
ð41Þ
di;j zj
j¼1
jzn Kðx; yðn1Þ Þj r
n X
Þ is approximated by ð50Þ ðn1Þ
Þ ¼ f ðxÞ f ð yr
aj jzn jwj ðx; yðn1Þ Þjxj yðj1Þ j þ jzn jb r r
j¼1
with di;j ¼ ci1;j1 ci1;j cj þ ci1;jþ1
ð42Þ
where for convenience of denotation, we define z0 = 0 and ci,j = 0 if j = 0 or j C i. From (41), we design the virtual control function ð43Þ ai ¼ ci;1 z1 ci;2 z2 ci;i zi where ci,j = di,j, j B i - 2, ci,i-1 = di,i-1 ? 1 and ci,i = di,j ? ci with ci [ 0 being a design parameter. Then, we have z_i ¼ zi1 ci zi þ ziþ1 :
¼
j¼1
ð45Þ
n X
aj jzn jwj ðx; yðn1Þ Þjzj þ aj1 j þ jzn jb r
j¼1
j1 X ¼ aj jzn jwj ðx; yðn1Þ Þzj þ cj1;k zk þ jzn jb r j¼1 k¼1 n X
max faj ; aj jcj1;k jg
1 k\j n
j n X X
jzn wj ðx; yðn1Þ Þzk j þ jzn jb r
j¼1 k¼1
max faj ; aj jcj1;k jg
1 k\j n
ð44Þ
Step n. Similarly, the time derivative of zn is given by n X ¼ f ðxÞ þ xðtÞ þ u þ dn;j zj yðnÞ z_n ¼ x_n a_ n1 yðnÞ r r
j n X X 1 2 2 z fw2j ðx; yðn1Þ Þz þ þ jzn jb r n 4f k j¼1 k¼1 max faj ; aj jcj1;k jg
n X
1 k\j n
þ
j¼1
n X nkþ1 k¼1
jfw2j ðx; yðn1Þ Þz2n r
4f
z2k þ jzn jb n X niþ1
where dn,j is defined as in (42) for i = n. From (45), we design the control law containing five parts
Þz2n þ Hbðx; yðn1Þ r
u ¼ an þ unn þ ua1 þ ua2 þ yðnÞ r
where f [ 0 is an adjustable design parameter, and
ð46Þ
where an is defined as (43) for i = n, unn is the RBF neural network feedforward compensation term, ua,1 is used to ðn1Þ
compensate for the replacement error f ðxÞ f ð yr Þ, ua,2 is used to compensate for the bounded approximation error and disturbance. Then, we have z_n ¼ zn1 cn zn þ f ðxÞ þ xðtÞ þ unn þ ua1 þ ua2 Consider the Lyapunov function n 1X Vz ¼ z2 : 2 i¼1 i
Þ is
ð47Þ
i¼1
From (36), (40), (44) and (47), the derivative of Vz is given by
z2i þ jzn jb:
ð51Þ
H¼
max faj ; aj jcj1;k jg 1 k\j n n X Þ ¼ jfw2j ðx; yðn1Þ Þ: bðx; yðnÞ r r j¼1 Substituting (50) and (51) back into (49) yields V_z
ð48Þ
4f
n X i¼1
niþ1 2 ci zi 4f
þ zn Hbðx; yðn1Þ Þzn þ W T Sð yðn1Þ Þ þ unn þ ua1 þ ua2 r r þ jzn jðb þ df þ -Þ
ð52Þ
123
360
Neural Comput & Applic (2012) 21:351–363
where df denotes the upper bound of d(t). By using the inequality [4] jvj vtanhðv=iÞ þ ji with j = 0.2785 and i[ 0, we have the following inequality zn ð53Þ jzn jðb þ df þ -Þ zn q tanh þ qji i with q ¼: b þ df þ -. Substituting (53) into (52) yields n X niþ1 2 _ Vz ci yðn1Þ Þ þ u zi þ zn W T Sð nn r 4f i¼1 þ zn Hbðx; yðn1Þ Þzn þ ua1 r zn þ zn q tanh þ ua2 þ qji: ð54Þ i Then, we design the other parts of control law as follows 8 ^ T Sð > yðn1Þ Þ unn ¼ W > r > < ðn1Þ ^ Þzn ua1 ¼ Hbðx; yr ð55Þ > > z n > u ¼ ^ : q tanh a2 i with the adaptive laws h i 8 _^ ¼ C Sð ^ > W yðn1Þ Þzn rW > r > > < h i ^_ ¼ c bðx; yðn1Þ Þz2 rH ^ H H r n > > h i > zn > : q ^_ ¼ cq zn tanh r^ q i
ð56Þ
where C; cH ; cq [ 0 are the adaptive gains. Substituting (55) into (54) yields n X niþ1 2 ~ T Sð ci yðn1Þ Þzn V_z zi þ W r 4f i¼1 zn ~ ~zn tanh þ qji: ð57Þ þ Hbðx; yðn1Þ Þz2n þ q r i If we choose the design parameters such that ci [
niþ1 4f ,
0 or equivalently, ci niþ1 4f ¼: ci [ 0, then we have the following globally stable results.
Theorem 1 Consider the closed-loop adaptive system consisting of the system (29), the control law (46), (55), and the adaptive law (56), then for any initial conditions, all closed-loop signals are globally uniformly ultimately bounded and the tracking error can converge to a neighborhood around the origin, whose size can be tuned arbitrarily small by choosing the suitable design parameters. Proof
Consider the following Lyapunov function
1 ~ T 1 ~ 1 ~2 1 2 ~ : V ¼ Vz þ W C Wþ q H þ 2 2cH 2cq
ð58Þ
From (56) and (57), the derivative of V is easily computed as follows
123
V_
n X
~H ^ þq ^ þH ~ TW ~q ^Þ þ qji: c0i z2i þ rðW
ð59Þ
i¼1
By completing square, we have the following inequality r ~T ~ ~ 2 ~H ^ þq ~ TW ^ þH ~q ^ Þ ðW ~2 Þ rðW W þH þq 2 r þ ðW T W þ H2 þ q2 Þ: ð60Þ 2 Substituting (60) into (59) yields n X r ~T ~ ~ 2 ~2 Þ W þH þq V_ c0i z2i ðW 2 i¼1 r T þ ðW W þ H2 þ q2 Þ þ qji ‘V þ t 2 where ‘ ¼ min1 i n f2c0i ; k r T 2 ðW W
2
r min ðC
1
Þ
ð61Þ
; rcH ; rcq g and t ¼
2
þ H þ q Þ þ qji. It follows from (25) that t ð62Þ VðtÞ Vð0Þe‘t þ ‘ ^ which shows the boundedness of V(t), which implies zi, W, ^ and q ^ are GUUB. From (35), (39) and (43), we can H further obtain the boundedness of ai. Then, it can be seen from (33) that xi are GUUB. Finally, we can easily get the boundedness of u from the boundedness of the above signals. Moreover, from (62), we have rffiffiffi pffiffiffiffiffiffiffiffiffiffiffi t lim jy yr j lim 2VðtÞ : ð63Þ t!þ1 t!þ1 ‘ By increasing C; cH ; cq ; c0i , and reducing r; i, we can make pffi the value of t‘ arbitrarily small. This completes the whole proof. 3.2 Simulation example Consider the Duffing’s equation described by ( x_1 ¼ x2 x_2 ¼ p1 x1 p2 x31 px2 þ q cosðdtÞ þ u
ð64Þ
where p1, p2, p, q and d are real constants. Depending on the choice of these constants, the solution of system (64) with u = 0 may display complex phenomena. Obviously, system (64) has the form (1), where f ðxÞ ¼ p1 x1 p2 x31 px2 and xðtÞ ¼ q cosðdtÞ. It is easily verified that f(x) satisfies jf ðx1 ; x2 Þ f ðyr ; y_r Þj jx1 yr ja1 w1 ðx1 ; yr Þ þ jx2 y_r ja2 w2 ðx2 ; yr Þ
ð65Þ
where a1 ¼ maxfjp1 j; jp2 jg, w1 ¼ 1 þ jx21 þ y2r þ x1 yr j, a2 = |p|, and w2 = 1. Based on the proposed control scheme, we design the control law as follows
Neural Comput & Applic (2012) 21:351–363
361
u ¼ a2 þ unn þ u1 þ u2 þ y€r
ð66Þ
with a2 ¼ ðc21 þ 1Þz1 ðc1 þ c2 Þz2 ^ T Sðyr ; y_ r Þ unn ¼ W ^ yr ; y_r Þz2 u1 ¼ Hbðx; z2 u2 ¼ ^ q tanh i where z1 = y - yr, z2 ¼ x2 a1 y_ r , a1 = - c1z1 and ^ ^ and w h; H bðx; yr ; y_ r Þ ¼ fw21 þ 2fw22 . The adaptive laws of ^ are given by (56). For simulation purpose, it is assumed that p1 = -1.1, p2 = 1, p = 0.4, q = 2.1 and d = 1.8. In simulation, the reference signal is specified as yr ðtÞ ¼ sinðtÞ. So, the RBF neural network approximation domain is X = [-1, 1] 9 [-1, 1]. Then, this neural node number is appropriately chosen as 36 (If this number is too large, then
the more computation loads are needed. If this number is too small, then the neural network approximation error cannot satisfy the performance requirement). Moreover, to ensure the centers of RFBs evenly cover X, we set the centers ni = (ai, bi) with ai and bi chosen arbitrarily from {-1, -0.6, -0.2, 0.2, 0.6, 1}, and the width is set to be gi = 1/(25 ln 2) so that the sum of two neighboring RFB values is one at the midpoint between two centers. The control parameters are set to be f = 2, c1 = 0.6, c2 = 0.3 22þ1 (c1 [ 21þ1 2f , c2 [ 2f ), i ¼ 0:01; r ¼ 0:0001; C ¼ 2I; ^ ^ cH ¼ 0:02; cq ¼ 0:5. and the initial values of hð0Þ; Hð0Þ ^ and wð0Þ are zeros. The simulation results are shown in Fig. 5. Figure 5 (a1, b1, c1, d1) present the case when xð0Þ ¼ ½0:8; 0T 2 X, from which we can see that the control performance is satisfactory. To further verify the advantages of our feedforward compensation method, we present the
(a1)
2
(a2)
6
y
y
r
y, y
y, yr
4 0
2 0
y −2
r
0
10
20
30
40
−2
50
yr 0
10
20
time (s)
50
40
50
40
50
r
f(x )
f(xr)
40
0 −2
RBFNN 0
10
20
30
40
50
RBFNN 0
10
20
time (s)
(c1)
4
ρ
||W|| 2 0 −2
x2 0
10
Θ 20
30
40
(c2)
4 2 0
x2
−2
50
0
10
ρ 20
30
time (s)
(d1)
(d2)
u
200
0
−5
Θ
||W||
time (s) 5
30
time (s)
x2, ||W||, Θ, ρ
x2, ||W||, Θ, ρ
50
f(xr)
2
0 −2
u
40
(b2)
(b1) f(xr)
2
30
time (s)
100
0 0
10
20
30
time (s)
40
50
0
10
20
30
time (s)
Fig. 5 Simulation of system (64) using feedforward compensation design a1–d1: x1(0) = 0.8; a2–d2: x1(0) = 6
123
362
simulation results of x(0) = [6, 0]T [ X in Fig. 5 (a2, b2, c2, d2). In this case, theoretically, the traditional feedback compensation method cannot ensure the stability of closedloop system since x(0) is far away from X. However, the simulation results show the feedforward compensation scheme still can guarantee the stability of closed-loop system. This further verifies the advantages of the feedforward compensation method proposed in this paper.
4 Conclusions In this paper, we discuss the two open problems in neural network control community, i.e., the globally uniformly ultimate boundedness and the approximation domain determination. It is shown that the two problems can be solved well under the assumption that the unknown system functions are bounded by partly known functions. The main design idea is to use the RBF neural network as feedforward compensators instead of feedback ones. In the authors’ opinion, since the universal approximation property of neural networks holds only provided that the neural inputs must remain on a compact set, it is more valid and practical to use the neural network as a feedforward compensator than a feedback one. Acknowledgments The authors would like to thank the anonymous reviewers for their comments that improve the quality of the paper. This work was supported by the National Natural Science Foundation of P. R. China (60804021, 61072106, 61072139, 61001202), the Fundamental Research Funds for the Central Universities (JY10000970001), and the China Postdoctoral Science Foundation funded project (20090461282).
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