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Adaptive NN control for discrete-time pure-feedback systems with unknown control direction under amplitude and rate actuator constraints
ISA Transactions 48 (2009) 304–311
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Adaptive NN control for discrete-time pure-feedback systems with unknown control direction under amplitude and rate actuator constraints Weisheng Chen ∗ Department of Applied Mathematics, Xidian University, Xi’an 710071, PR China
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Article history: Received 21 August 2008 Received in revised form 4 February 2009 Accepted 3 April 2009 Available online 28 April 2009 Keywords: Pure-feedback systems Discrete Nussbaum gain Implicit function theorem Neural network
abstract This paper focuses on the problem of adaptive neural network tracking control for a class of discrete-time pure-feedback systems with unknown control direction under amplitude and rate actuator constraints. Two novel state-feedback and output-feedback dynamic control laws are established where the function tanh(·) is employed to solve the saturation constraint problem. Implicit function theorem and mean value theorem are exploited to deal with non-affine variables that are used as actual control. Radial basis function neural networks are used to approximate the desired input function. Discrete Nussbaum gain is used to estimate the unknown sign of control gain. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. A simulation example is provided to illustrate the effectiveness of control schemes proposed in this paper. © 2009 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction Neural networks (NNs) have been proven to be particularly useful for controlling nonlinear systems with nonlinearly parameterized uncertainties owing to their universal approximation property. In general, NNs are used as function approximators to approximate some suitable uncertainties appearing in controllers or systems. Many important theoretical and practical results have been reported in the past two decades [1–10]. In particular, NNs are employed to solve the control problem of nonlinear systems with special lower-triangular forms, including strict-feedback form [11–17], output-feedback form [18–21], and pure-feedback form [22–25]. As far as continuous-time systems are concerned, the control laws of systems and the adaptive laws of NN weights are generally derived by using Lyapunov stability theory. Therefore, the stability of closed-loop systems is easily guaranteed. In the development of adaptive NN control for lower-triangular systems, some important control techniques are successfully proposed. For example, adaptive bounding technique is used to deal with the NN approximation error [12,13]; Integral-type Lyapunov function is exploited to avoid the possible controller singularity [14]; Nussbaum-type functions are utilized to cope with the unknown control gain signs [15]; dynamic surface control is employed to solve the problem of ‘explosion of complexity’ [16]. Then,
∗
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NN control techniques are further extended to output-feedback systems [17–19] and time-delay systems [20,21]. It should be mentioned that the adaptive NN control problem of pure-feedback systems, which are the most general lower-triangular systems, has been widely investigated in past several years [22–25], where mean value theorem [25] or implicit function theorem [22–24] are usually used to deal with non-affine system functions. For discrete-time systems, especially lower-triangular nonlinear discrete-time systems, adaptive NN control is more complex due to some new problems such as the noncausal problem and the coupling problem among subsystems, inputs and outputs. Only a few results are obtained [26–33]. For example, NN control for strict-feedback discrete-time systems is successfully solved in [34– 36], and then the ideas are applied to the multiple-input-multiple output (MIMO) systems [26–28] and the systems with unknown control directions [29,30]. Very recently, adaptive NN control problem for a class of discrete-time systems with non-strict form are addressed [31,32]. However, to the best of our knowledge, no work has been reported on adaptive NN control for the following purefeedback discrete-time systems under amplitude and rate actuator constraints:
(
xi (k + 1) = fi [¯xi (k), xi+1 (k)], xn (k + 1) = fn [¯xn (k), u(k), d(k)] y(k) = x1 (k)
i = 1, . . . , n − 1 (1)
where x¯ i (k) = [x1 (k), x2 (k), . . . , xi (k)]T ∈ Ri , i = 1, . . . , n, y(k) ∈ R and u(k) ∈ R are the state variables, system output and input respectively; d(k) denotes the external disturbance ¯ d¯ > 0; fi : Ri+1 → R are unknown smooth satisfying |d(k)| < d,
0019-0578/$ – see front matter © 2009 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2009.04.002
W. Chen / ISA Transactions 48 (2009) 304–311
305
functions. The control input u(k) is subjected to amplitude and rate limitations defined as follow. • Amplitude constraints: |u(k)| ≤ ua , where ua > 0 denotes the control amplitude bound. • Rate constraint: |1u(k)| = |u(k) − u(k − 1)| ≤ ur , where ur > 0 denotes the rate control bound. If no constraints are imposed on the control input u(k), the control problem of system (1) has been studied in very recent papers [30,33]. However, the physical impossibility of applying unlimited control signals makes actuator saturation a ubiquitous problem in control systems. In particular, it is well known that both amplitude and rate saturations of control input are a source of performance degeneration, limit cycles, different equilibrium points, and even instability. So, the control problem of system (1) under amplitude and rate saturation constraints is more challenging. In this paper, to solve the above problem, and noting that the function tanh(·) possesses the saturation property and its derivative is positive and bounded, so the function tanh(·) is introduced skillfully to design a novel dynamic NN-based control law. In addition, the discrete Nussbaum gain is combined with the adaptive law to deal with the unknown control direction similarly to [30,31]. Finally, a simulation example is provided to illustrate the effectiveness of control schemes proposed in this paper. This paper is organized as follows. Radial basis function (RBF) NN, discrete Nussbaum gain and preliminaries are given in Section 2. In Section 3, we present the problem formulation and the control design procedure. Section 4 gives a simulation example to illustrate the effectiveness of the proposed controllers. In Section 5, we conclude the work of this paper.
properties: (i) if xs (k) increase without bound, then
2. RBF NN, discrete Nussbaum gain and preliminaries
(
1
sup
xs (k)≥δ0 xs (k)
inf
xs (k)≥δ0
1 xs (k)
SN (x(k)) = +∞, SN (x(k)) = −∞;
(ii) if xs (k) ≤ δ1 , then |Sn (x(k0 ))| ≤ δ2 with some positive constants δ1 and δ2 ; where SN (k) is defined as k X
SN (x(k)) =
N (x(k0 ))1x(k0 )
k0 =0
with 1x(k) = x(k + 1) − x(k). In this paper, the discrete Nussbaum gain is chosen as [30] N (x(k)) = xs (k)sN (x(k))
3
If SN (x(k1 )) ≤ xs2 (k1 ),
then go to Step (2)
If SN (x(k1 )) > xs (k1 ),
then go to Step (3)
3 2
2.1. RBF NN
(3)
where the discrete sequence x(k) satisfies x(0) = 0, x(k) > 0, |1x(k)| ≤ δ0 , ∀k. sN (x(k)) is the sign function of the discrete Nussbaum gain, i.e., sN (x(k)) = ±1. The initial value is set as sN (x(0)) = +1. Thereafter, the sign function sN (x(k)) will be chosen by comparing the summation SN (x(k)) with a pair of 3/2 switching curves defined by f (xs (k)) = ±xs (k). The details are as follows. Step (1) At k = k1 , measure the output y(k1 ) and compute 1x(k1 ) and x(k1 + 1) = x(k1 ) + 1x(k1 ) and SN (x(k1 )) = SN (x(k1 − 1)) + N (x(k1 ))1x(k1 ). Case (SN (x(k1 )) = +1):
Case (SN (x(k1 )) = −1):
Many approximators are used to approximate unknown functions, such as RBF NN, multilayer NN, fuzzy system, etc. Owing to the simple structure and good approximation capability, in this paper we will exploit RBF NN [37] to approximate a continuous function h(z ) : Rq → R defined on a compact set Ωz ⊂ Rq , i.e.,
3
(
If SN (x(k1 )) < xs2 (k1 ),
then go to Step (2)
If SN (x(k1 )) ≥ xs (k1 ),
then go to Step (3).
3 2
(2)
Step (2) Set sN (x(k1 + 1)) = 1, go to Step (4). Step (3) Set sN (x(k1 + 1)) = −1, go to Step (4). Step (4) Return to Step 1 and wait for the measurement of output.
where S (z ) = [s1 (z ), s2 (z ), . . . , sl (z )] ∈ R is a known continuous smooth vector-valued function, l is called the NN node number. The components sj (z ), j = 1, 2, . . . , l are commonly chosen as Gaussian function sj (z ) = exp[−kz − µj k2 /η2 ], µj ∈ Ωz is a constant which is called the center of sj (z ), and η > 0 is a real number which is called the width of sj (z ). The optimal weight vector W = [w1 , . . . , wl ]T is defined as
Lemma 1 ([30]). Consider discrete Nussbaum gain defined in (3). (i) Given an arbitrary bounded function g (k) : R → R, and g1 ≤ |g (k)| ≤ g2 , where g1 and g2 are unknown positive constants, then N 0 (x(k)) = g (k)N (x(k)) is also a discrete Nussbaum gain if 1x(k) ≥ 0. (ii) Given an arbitrary function −0 ≤ C (k) ≤ 0 , then N 0 (x(k)) = N (x(k)) + C (k) is still a discrete Nussbaum gain if 1x(k) ≥ 0.
h(z ) = W T S (z ) + ε(z ) T
l
T ˆ W := arg min sup h(z ) − W S (z ) ˆ ∈Rl W
z ∈Ωz
and ε(z ) is the inherent NN approximation error with the minimum upper bound ε¯ > 0. It has been proven that RBF NN satisfies the conditions of the Stone–Weierstrass Theorem and can approximate any continuous function to any desired accuracy over the compact set Ωz . That is, ε¯ can be arbitrarily decreased by increasing the NN node number l. In general, the NN weight W is unknown and needs to be ˆ denotes the estimate of W , estimated. Throughout this paper, W ˜ =W ˆ − W. and the estimation error is defined as W 2.2. The discrete Nussbaum gain Definition 1 ([30]). Consider a discrete nonlinear function N (x(k)) defined on a sequence x(k) with xs (k) = supk0 ≤k {x(k0 )}. N (x(k)) is a discrete Nussbaum gain if and only if it satisfies the following two
Lemma 2 ([30]). Let V (k) be a positive-define function defined ∀k, N (·) be discrete Nussbaum gain, and θ be a nonzero constant. If the following inequality holds, ∀k: V (k) ≤
k X
(c1 + θ N (x(k0 )))1x(k0 ) + c2 x(k) + c3
k 0 =k
1
where c1 , c2 , and c3 are some constants and k1 is a positive integer, Pk 0 0 then V (k), x(k), and k0 =k1 (c1 + θ N (x(k )))1x(k ) + c2 x(k) + c3 must be bounded ∀k. 2.3. Preliminaries Definition 2 ([38]). The trajectory x(k) of the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB), if for any Ω , a compact subset of Rn and all χ (k0 ) ∈ Ω , there exists an ε > 0 and a number N (ε, χ (k0 )) such that kχ (k)k < ε for all k ≥ k0 + N.
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W. Chen / ISA Transactions 48 (2009) 304–311
Lemma 3 ([39]). For some given real scalar sequences s(k), b1 (k), and b2 (k) and vector sequence σ (k), if the following conditions hold: (i) limk→∞ (s2 (k))/(b1 (k)z + b2 (k)σ T (k)σ (k)) = 0; (ii) 0 < b1 (k) < K and 0 ≤ b2 (k) < K with a finite K ; (iii) kσ (k)k ≤ C1 + C2 max0≤k0 ≤k |x(k0 )|, where C1 and C2 are some finite constants; then we have (a) limk→∞ s(k) = 0, and (b) σ (k) is bounded. Lemma 4 (Implicit Function Theorem [38]). Assume that f (x, y): Rn × R → R is continuous differentiable ∀(x, y) ∈ Rn × R, and there exists a positive constant d such that
∂ f (x, y) ∂ y > d > 0,
∀(x, y) ∈ Rn × R
Lemma 5 (Mean Value Theorem [38]). Assume that f (x, y): Rn ×R → R is continuously differentiable at each point of an open set Rn ×(a, b), and assume also that it is continuous at both endpoints y = a and y = b, then there exists a positive constant 0 ≤ θ ≤ 1 such that
Lemma 6 ([34]). Let U be an open subset of Rn and let φ (φ1 , . . . , φn ) : U → Rn be a smooth map. If the Jacobian Matrix
∂φ
1
∂ x1 . = .. dx ∂φ
dφ
n
∂ x1
··· ..
.
···
(6)
where Fi (·) is an unknown function depending on fj (·), j = 1, . . . , n − 1. The above system can be also described as follows x1 (k + n) = F [¯xn (k), u(k), d(k)]
(7)
where F [¯xn (k), u(k), d(k)]
(4)
then there exists a unique continuous (smooth) function y = g (x) such that f (x, g (x)) = 0.
∂ f (x, y) f (x, b) − f (x, a) = (b − a). ∂ y y=θ a+(1−θ)b
x1 (k + n) = F1 [¯xn (k), x2 (k + n − 1)] . .. xn−1 (k + 2) = Fn−1 [¯xn (k), xn (k + 1)] xn (k + 1) = fn [¯xn (k), u(k), d(k)] y(k) = x1 (k)
(5)
=
∂φ1 ∂ xn .. . ∂φ n
∂ xn
is nonsingular at some point p ∈ U, or equivalently, Rank(dφ/dx) = n at some point p ∈ U, then there exists a neighborhood V ⊂ U of p such that φ: V → φ(V ) is a diffeomorphism. 3. Problem formulation and controller design 3.1. Problem formulation and system transformation The control objective of this paper is formulated as follows. For a known and bounded trajectory yd (k) ∈ Ωd , where Ωd is a compact set, design two adaptive control schemes satisfying the amplitude and rate saturation constraints (i.e., state-feedback one and output-feedback one) to make the system output y(k) follow yd (k), while maintaining all closed-loop signals SGUUB. For the control of pure-feedback system (1), define
= F1 [¯xn (k), F2 [¯xn (k), . . . , fn [¯xn (k), u(k), d(k)] · · ·]]. Note that under Assumption 1, we still have
∂ F ∂ F1 ∂ fn = 0 < g ≤ ··· ≤ g¯0 , ∀¯xn ∈ Ωx . 0 ∂u ∂ x2 ∂u
3.2. State-feedback controller design In this subsection, we will design a NN-based state-feedback control for system (7). To do this, we will write the system (7) as follows. x1 (k + n) = F0 [¯xn (k), u(k)] + d0 (k)
(8)
where F0 [¯xn (k), u(k)] = F [¯xn (k), u(k), 0], and d0 (k) = F [¯xn (k), u(k), 0] − F [¯xn (k), u(k), d(k)]. Note that since u(k) is required to ¯ for x¯ n ∈ Ωx , there exists a finite satisfy |u(k)| ≤ ua and |d(k)| ≤ d, constant d¯ 0 such that |d0 (k)| ≤ d¯ 0 . To deal with the problem of amplitude and rate saturation of the control input, we design the following new dynamic control law. u(k) = ua tanh(v(k)),
v(k) = v(k − 1) + ur tanh(α(k)),
(9)
v(−1) = 0
(10)
where α(k) is the new control variable designed later. Obviously, since | tanh(·)| ≤ 1, it is ensured that |u(k)| ≤ ua and |1u(k)| = |v(k) − v(k − 1)| ≤ ur . Substituting (9) and (10) into (8) results into x1 (k + n) = F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
+ d0 (k).
(11)
Define the tracking error e(k) = y(k) − yd (k), we have e(k + n) = F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
− yd (k + n) + d0 (k).
(12)
For v ∈ Ωv and α ∈ Ωα , where Ωv ⊂ R and Ωα ⊂ R are compact sets, there must exist constants g , g > 0 such that d tanh(v)
v
α
d tanh(α)
∂ fi (¯xi , xi+1 ) gi (¯xi , xi+1 ) =: , i = 1, . . . , n − 1, ∂ xi+1 ∂ fi (¯xi , u(k), d(k)) gn (¯xn , u(k), d(k)) =: . ∂ u(k)
0 < g ≤ ≤ 1 and 0 < g α ≤ ≤ 1. Based on dv dα v Assumption 1 and Lemma 4, we conclude that for (¯xn (k), v(k), α(k)) ∈ Ω =: Ωx × Ωv × Ωα , there exists a desired control variable α ∗ (k) = ψ1 [¯xn (k), v(k − 1), yd (k + n)] such that F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α ∗ (k)))] − yd (k + n) = 0. Then, using Lemma 5, we get
Assumption 1. There exist constants 0 < g ≤ g i such that i g ≤ |gi (·)| ≤ g i , ∀¯xn ∈ Ωx ⊂ Rn , where Ωx is a compact set. i However, the signs of gi (·), i = 1, . . . , n − 1 are unknown.
F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
If we consider the original system (1) as a one-step ahead predictor, we can transform it into an equivalent maximum n-step ahead one with the following form, which can predict the future states x1 (k + n), x2 (k + n − 1), . . . , xn (k + 1); then, the causality contradiction is avoided when the controller is constructed based on the maximum n-step ahead prediction [34].
= F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α ∗ (k)))] + gλ (k)[α(k) − α ∗ (k)] where
∂ F0 [¯xn , ua tanh(v + ur tanh(xλ ))] gλ (k) := ∂ xλ xλ =%∗ with %∗ = λα(k) + (1 − λ)α ∗ (k), 0 ≤ λ ≤ 1.
(13)
W. Chen / ISA Transactions 48 (2009) 304–311
Note valid for gλ (k), i.e., 0 < Qnthat Assumption 1 on gi (·) is stillQ n ¯ = ¯i for (¯xn (k), v(k), g g i=1 g i = g ≤ |gλ (k)| ≤ g i=1 g v α α(k)) ∈ Ω . Substituting (13) back into (12) yields e(k + n) = gλ (k)[α(k) − α ∗ (k)] + d0 (k).
(14)
The desired control variable α ∗ (k) can be expressed by using a RBF NN as follows
α ∗ (k) = W T S (z (k)) + ε(z (k)) (15) where the NN input z (k) = [¯xTn (k), v(k − 1), yd (k + n)]T ∈ Ωx × Ωv × Ωd =: Ω0 . The NN approximation error |ε(z (k))| ≤ ε¯ , and the NN node number l > 0. Based on (14) and (15), the new control variable α(k) is designed as ˆ T (k)S (z (k)). α(k) = W
(16)
Substituting (15) and (16) back into (14) leads to
˜ T (k)S (z (k)) + d∗ (k). e1 (k + n) = gλ (k)W
(17)
where d (k) = −g (k)ε1 (z (k)) + d0 (k) and it is trivial to show that |d∗ (k)| ≤ g¯ ε¯ + d¯ 0 := d∗0 . Now, the adaptive law of W is given as ∗
ˆ (k) = W ˆ (k − n) − γ N (x(k))S (k − n) W
(k)
D(k) where N (x(k)) is the discrete Nussbaum gain given by (3) and γ e(k) (k) = , G(k) = 1 + |N (x(k))|, G(k)
(18)
a(k)G(k) 2 (k) D(k)
x(0) = 0, a(k) =
1, 0,
ˆ (j) = 0, W
,
and a2 (k) = a(k). Then, by the same derivation to [30, Theorem 2], we have V (k) ≤ −2
k X
N (x(k))
1
−
a(k)G(k) 2 k
lim 1x(k) = lim
k→∞
D(k)
k→∞
= 0.
(22)
Let us define a time interval as Z1 = [k|a(k) = 1] and suppose that Z1 is an infinite set. Then based on Lemma 3, we have limk→∞,k∈Z1 (k) = limk→∞Z1 a(k)(k) = 0, which conflicts with a(k) = 1, k ∈ Z1 , because |(k)| ≥ ν , when a(k) = 1. Therefore, Z1 must be a finite set and then, we have lim sup |(k)| ≤ ν
k→∞
Cν
γ
.
Then, following the same procedure as in [30, Theorem 2], the SGUUB of other close-loop signals can be concluded. This completes the proof.
j = −n − 1, . . . , 0.
gλ (k − n)
(21)
where c1 = γ 2 + |2d∗0 /g ν|, and N 0 (x(k)) = g (k−n) which is still a discrete Nassbaum gain based on Lemma 1. Then, according to ˆ (k), G(k), x(k), and Lemma 2, we conclude the boundedness of W N (x(k)), and further, from the boundedness of x(k), we have
lim sup |e(k)| ≤
if |(k)| > ν others
˜ T (k − n)S (k − n) = γW
N 0 (x(k0 ))1x(k0 ) + c1 x(k) + c1
k0 =0
k→∞
Proof. The proof closely follows [30]. First, we assume the NN is constructed to cover a large enough compact set Ω0 such that the NN input z ∈ Ω0 . From (17), we have G(k)(k)
1 gλ (k − n)
γ d∗ (k − n).
(19)
Choose a positive-definite function V (k) as
˜ T (k − n + j)W ˜ (k − n + j) W
3.3. Output-feedback NN controller design In the above subsection, we assume the system states x¯ n (k) are measurable. However, the control scheme becomes infeasible when some system states are unmeasurable. In this subsection, we assume that only the system output is measurable, and an outputfeedback NN control scheme will be designed. To do so, define the new system states ξ (k) = [ξ1 (k), ξ2 (k), . . . ξn (k)]T for the system (6) as
ξ1 (k) = x1 (k), ξ2 (k) = x1 (k + 1) = f1 (¯x1 (k), x2 (k)) = hn (¯x2 (k)) ξ3 (k) = x1 (k + 2) = f1 (¯x1 (k + 1), x2 (k + 1)) = f1 (¯x2 (k), f2 (¯x2 (k), x3 (k))) = hn−1 (¯x3 (k)), .. .
(23)
ξn (k) = x1 (k + n − 1) = f1 [¯xn−1 , f2 [¯xn−1 , . . . , fn−1 [¯xn−1 , xn ] · · ·]] = h2 (¯xn (k)) which can be written as ξ (k) = T (¯xn (k)), where T (¯xn (k)) is a nonlinear coordinate transformation of the form T (¯xn (k)) = [x1 (k), hn (¯x2 (k)), . . . , h2 (¯x2 (k))]T .
(24)
The following lemma guarantees that the coordinate transformation mapping T is diffeomorphism.
n
j =1
∗
which implies that N (x(k)) will converge to a constant ultimately. By denoting the limit of G(k) as C , it can be derived from the definition of (k) that
Theorem 1. Under Assumption 1, consider the closed-loop adaptive system consisting of the plant (1), the control law (9), (10) and (16) with the adaptive laws (18). All the signals in the closed-loop system are SGUUB, and the discrete Nussbaum gain N (x(k)) will converge to a constant ultimately. Denote C = limk→∞ G(k), then the tracking error satisfies limk→∞ sup |e(k)| < C ν/γ , where the tuning rate γ > 0 and the threshold value ν > 0 can be arbitrary constants to be specified by the designer.
X
2d∗ 0 N (x(k))d (k − n)(k) ≤ a(k) G(k) 2 (k) gν g (k − n) 2a(k)
lim a(k) = 0,
Remark 1. The adaptive law (18) is motivated by [30]. Although the adaptive law is same as that of [30] in the form, the new dynamic control law consisting of (9), (10) and (16) is very different from that of [30], where no dynamic equations (9) and (10) are introduced due to no constraints being imposed on the control input. Moreover, the introduction of (9) and (10) makes the above analysis and derivations more difficult than [30]. It is emphasized d tanh(·) that the good property of tanh(·) (i.e., 0 < ≤ 1) plays a d· crucial role in the above derivations.
V (k) =
and note that
k→∞
D(k) = 1 + kS (t − n)k2 + |N (x(k))| + 2 (k),
1x(k) = x(k + 1) − x(k) =
307
(20)
Lemma 7. For system (1) (or system (6)), the nonlinear coordinate transformation T (¯xn (k)) in (24), is a smooth map and a
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W. Chen / ISA Transactions 48 (2009) 304–311
diffeomorphism, namely, there exists a unique transformation function T −1 (ξ (k)) such that x¯ n = T −1 (ξ ), and both T and T −1 are oneto-one.
ˆ (k) = W ˆ (k − n) − γ N (x(k))S (k − n) W
Proof. From (24), it is easily derived that
1
∗ dT (¯xn (k)) = dx(k) ∗ ∗
0
∂ f1 ∂ x2 .. . ∗
···
0
···
0
.. . ···
Based on the above analysis, we can obtain the cascade system description, which is equivalent to the original system (1), as follows:
(25)
where ξ (k) = [ξ1 (k), . . . , ξn (k)]T ∈ Rn , and f (ξ (k), u(k), d(k)) = F (¯xn , u(k), d(k))
= F (T −1 (ξ (k)), u(k), d(k)) is an unknown smooth function. Since ξ (k) is unmeasured, similarly to [34, Eqs. (31)–(35)], we write ξ (k) as y(k + 1) = ξ2 (k) = f (y(k), u(k − n + 1), d(k − n + 1))
=: ϕ2 (z (k), d(k)) y(k + 2) = ξ3 (k) = f (y(k + 1), u(k − n + 2), d(k − n + 2))
=: ϕ3 (z (k), d(k)) .. .
(26)
y(k + n − 1) = ξn (k) = f (y(k + n − 2), u(k − 1), d(k − 1))
=: ϕn (z (k), d(k)) y(k + n) = ξn (k + 1) = f (y(k + n − 1), u(k), d(k))
=: f0 (z (k), u(k), d(k)) where y(k) = [y(k − n + 1), . . . , y(k − 1), y(k)]T , uk−1 (k) =
[u(k − 1), . . . , u(k − n + 1)]T , z (k) = [yT (k), uTk−1 (k)]T , d(k) = [d(k), . . . , d(k − n + 1)]T and f0 (z (k), u(k), d(k)) = f (x1 (k), ϕ2 (z (k)), . . . , ϕn (z (k)), u(k), d(k)). We rewrite the system (26) as y(k + n) = f¯0 (z (k), u(k)) + d∗ (k)
(27)
where f¯0 (z (k), u(k)) = f0 (z (k), u(k), 0), and which satisfies |d∗ (k)| ≤ d¯ for z (k) ∈ Ωz1 , where Ωz1 is a compact set. Note that f¯0 (z (k), u(k)) still satisfies Assumption 1. Since the system (27) has the same form as the system (7), so the control design procedure is similar. Here we only give the following control law and the detailed procedure and derivation are omitted.
ˆ T (k)S (¯z (k)) α(k) = W
(28)
v(−1) = 0
γ e(k) , G(k) = 1 + |N (x(k))|, G(k) D(k) = 1 + kS (t − n)k2 + |N (x(k))| + 2 (k), a(k)G(k) 2 (k) 1x(k) = x(k + 1) − x(k) = , D(k) x(0) = 0, 1, if |(k)| > ν a(k) = 0, others ˆ (j) = 0, W
j = −n − 1, . . . , 0.
Theorem 2. Under Assumption 1, consider the closed-loop adaptive system consisting of the plant (1), the control law (28)–(30) with the adaptive laws (31). All the signals in the closed-loop system are SGUUB, and the discrete Nussbaum gain N (x(k)) will converge to a constant ultimately. Denote C = limk→∞ G(k), then the tracking error satisfies limk→∞ sup |e(k)| < C ν/γ , where the tuning rate γ > 0 and the threshold value ν > 0 can be arbitrary constants to be specified by the designer. Proof. The proof can be easily completed by following the same derivation of Theorem 1. The detailed procedures are omitted. 4. Simulation example In this section, a numerical simulation example is provided to illustrate the effectiveness of the proposed control approaches. Consider the following second-order system
2 x1 (k + 1) = x1 (k) cos(x1 (k)) + 0.5 sin[x2 (k)e−x1 (k) ] + x2 (k) x1 (k) + x2 (k) x2 (k + 1) = (32) 1 + [x1 (k) + x2 (k)]2 3 +ρ( u ( k ) + 0 . 1u ( k )) + d ( k ) y(k) = x1 (k) where d(k) = 0.001 sin(0.1k), and ρ is a nonzero constant whose sign denotes the control direction. Suppose that there is no a priori knowledge of the system nonlinearities. It is easily checked that the system (32) satisfies Assumption 1. The control input is limited by the amplitude and rate saturation constraints: |u(k)| ≤ 1.2 and |1u(k)| ≤ 0.8. The control objective is to make the output y(k) follow a desired reference signal yd (k) = 0.5 sin(kπ /200) + 0.5 sin(kπ /100). 4.1. State feedback NN control Based on the control approach developed in Section 3.2, the control law u(k) is given as
d∗ (k) = f0 (z (k), u(k), d(k)) − f0 (z (k), u(k), 0)
u(k) = ua tanh(v(k)),
(31)
(k) =
Based on Assumption 1, Trace (dT (¯xn (k)/dx(k))) 6= 0, therefore T is a diffeomorphism according to Lemma 6. Equivalently, T −1 is also a diffeomorphism.
ξ1 (k + 1) = ξ2 (k), ξ2 (k + 1) = ξ3 (k), .. . ξn (k + 1) = f (ξ (k), u(k), d(k))
(k) D(k)
where N (x(k)) is the discrete Nussbaum gain given by (3) and
. .. . ∂ f1 ∂ f n −1 ··· ∂ x2 ∂ xn
v(k) = v(k − 1) + ur tanh(α(k)),
where z¯ = [z T , v(k − 1), yd (k + n)]T ∈ Ω1 , and the adaptive law is given by
(29) (30)
u(k) = 1.2 tanh(v(k)), v(k) = v(k − 1) + 0.8 tanh(α(k)), ˆ T (k)S (z (k)). α(k) = W
v(−1) = 0
(33)
where z (k) = [x1 (k), x2 (k), v(k − 1), yd (k + 2)]T , and the adaptive ˆ is given by (18) with n = 2. law of W In simulation, the initial conditions for system states are set to ˆ (0) = 0. The number be x1 (0) = x2 (0) = 0, and those for NN are W of NN nodes is chosen as l = 64 . The centers of RBFs evenly cove the
W. Chen / ISA Transactions 48 (2009) 304–311
309
Fig. 1. Simulation results of state-feedback control for system (32) with ρ = 1.
Fig. 2. Simulation results of state-feedback control for system (32) with ρ = −1.
compact set [−1, 1]4 , and the widths of RBFs are set to be η = 0.6. The design parameters are specified as γ = 0.51 and ν = 0.005. The simulation results of ρ ± 1 are shown in Figs. 1 and 2. Fig. 1 shows the case of the control gain ρ = 1. Fig. 1(a) shows the output y(k), the reference signal yd (k), and the tracking
error e(k). Fig. 1(b) gives the response curves of the N (z (k)) and z (k). Fig. 1(c) displays the boundedness of the norm of NN weight ˆ (k)k. Fig. 1(d) gives the response curves of u(k) and 1u(k). kW It can be seen from Fig. 1 that the tracking error converge to a desired neighborhood around the origin, and the control input
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W. Chen / ISA Transactions 48 (2009) 304–311
Fig. 3. Simulation results of output-feedback control for system (32) with ρ = 1.
Fig. 4. Simulation results of output-feedback control for system (32) with ρ = −1.
satisfies the saturation constrain condition, and the closed-loop ˆ (k)k and u(k) are uniformly bounded. The signals x1 (k), x2 (k), kW case of ρ = 1 is shown in Fig. 2.
4.2. Output feedback NN control When the system states are not available, based on the outputfeedback control approach developed in Section 3.3, the control
W. Chen / ISA Transactions 48 (2009) 304–311
u(k) is given as
u(k) = 1.2 tanh(v(k)), v(k) = v(k − 1) + 1.2 tanh(α(k)), ˆ T (k)S (¯z (k)) α(k) = W
v(−1) = 0
(34)
where z (k) = [y(k), y(k − 1), u(k − 1)(k), v(k − 1), yd (k + 2)]T , ˆ (k) is given by (31) with n = 2. and the adaptive law of W In simulation, the initial conditions for system states are set to ˆ (0) = 0. The number be x1 (0) = x2 (0) = 0, and those for NN are W of NN nodes is chosen as l = 55 . The centers of RBFs evenly cove the compact set [−1.15, 1.15]5 , and the widths of RBFs are set to be η = 0.9. The design parameters are specified as γ = 0.78 and ν = 0.005. The simulation results of ρ = ±1 are shown in Figs. 3 and 4. For two cases, it can be seen from Figs. 3 and 4 that the tracking errors both converge to a desired neighborhood around the origin, and the control input satisfies the saturation constraints, ˆ (k)k and u(k) are still while the closed-loop signals x1 (k), x2 (k), kW uniformly bounded. 5. Conclusions This paper considers the adaptive NN tracking control problem of discrete-time pure-feedback system with unknown control direction under the amplitude and rate saturation constraints of control input. This is a further extension of the existing results [30, 33]. By employing the good property of the function tanh(·), we successfully design two kinds of novel dynamic NN-based control schemes to deal with the problem of amplitude and rate saturation of control input. Though we only consider the single-input-singleoutput (SISO) systems in this paper, the design idea is also applied to MIMO cases [26–28]. Acknowledgments The author would like to thank the anonymous reviewers for their helpful comments and advice which contributed much to the improvement of this paper. The work was supported by the National Natural Science Funds of PR China under grant 60804021. References [1] Chen FC, Khalil HK. Adaptive control of nonlinear systems using neural networks. International Journal of Control 1992;55(6):1299–317. [2] Chen FC, Liu CC. Adaptively controlling nonlinear continuous-time systems using multilayer neural networks. IEEE Transactions on Automatic Control 1994;39(10):1306–10. [3] Chen FC, Khalil HK. Adaptive control of a class of nonlinear discrete-time systems using neural networks. IEEE Transactions on Automatic Control 1995; 40(5):791–801. [4] Lewis FL, Yesildirek A, Liu K. Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Transactions on Neural Network 1996; 7(2):388–98. [5] Lewis FL, Jagannathan S, Yesildirek A. Neural network control of robot manipulators. Singapore: World Scientific; 1998. [6] Jagannathan S, Lewis FL. Discrete-time neural net controller for a class of nonlinear dynamical systems. IEEE Transactions on Automatic Control 1996; 41(10):1693–9. [7] Jagannathan S. Control of a class of nonlinear discrete-time system using multilayer neural networks. IEEE Transactions on Neural Networks 2008; 12(5):1113–20. [8] Savran A, Tasaltin R, Becerikli Y. Intelligent adaptive nonlinear flight control for a high performance aircraft with neural networks. ISA Transactions 2006; 45(2):225–47. [9] Kolla SR, Altman SD. Artificial neural network based fault identification scheme implementation for a three-phase induction motor. ISA Transactions 2007;46(2):261–6.
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IEEE Transactions on Systems, Man, and Cybernetics-Part B 2004;34(1):499–516. [16] Wang D, Huang J. Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Transactions on Neural Networks 2005;16(1):195–202. [17] Ho DWC, Li J, Gang Y. Adaptive neural control for a class of nonlinear parametric time delay systems. IEEE Transactions on Neural Networks 2005; 16(3):625–35. [18] Choi JY, Farrell JA. Adaptive observer backstepping control using neural networks. IEEE Transactions on Neural Networks 2001;12(5):1103–12. [19] Stoev J, Choi JY, Farrell J. Adaptive control for output feedback nonlinear systems in the presence of modeling errors. Automatica 2002;38(10): 1761–1767. [20] Chen W, Li J. Decentralized output-feedback neural control for systems with unknown interconnections. IEEE Transactions on Systems, Man, and Cybernetics-Part B 2008;38(1):258–66. [21] Hua C, Guan X, Shi P. Robust output feedback tracking control for timedelay nonlinear systems using neural network. IEEE Transactions on Neural Networks 2007;18(2):495–505. [22] Wang D, Huang J. Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica 2002;38(8): 1365–1372. [23] Ge SS, Wang C. Adaptive NN control of uncertain nonlinear pure-feedback systems. Automatica 2002;38(4):671–82. [24] Wang C, Hill DJ, Ge SS, Chen G. An ISS-modular approach for adaptive neural control of pure-feedback systems. Automatica 2006;42(5):723–31. [25] Du H, Shao H, Yao P. Adaptive neural network control for a class of low-triangular-structured nonlinear systems. IEEE Transactions on Neural Networks 2006;17(2):509–14. [26] Ge SS, Zhang J, Lee TH. Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time. IEEE Transactions on Systems, Man, and Cybernetics-Part B 2004;34(4):1630–45. [27] Zhang J, Ge SS, Lee TH. Output feedback control of a class of discrete MIMO nonlinear systems with triangular form inputs. IEEE Transactions on Neural Networks 2005;16(6):1491–503. [28] Alanis AY, Sanchez EN, Loukianov AG. Discrete-time adaptive backstepping nonlinear control via high-order neural networks. IEEE Transactions on Neural Networks 2007;18(4):1185–95. [29] Ge SS, Yang C, Lee TH. Adaptive robust control of a class of nonlinear strictfeedback discrete-time systems with unknown control directions. Systems & Control Letters 2008;57(11):888–95. [30] Yang C, Ge SS, Xiang C, Chai TY, Lee TH. Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach. IEEE Transactions on Neural Networks 2008;19(11):1873–86. [31] Jagannathan S, He P. Neural-network-based state-feedback control of a nonlinear discrete-time system in nonstrict feedback form. IEEE Transactions on Neural Networks 2008;19(12):2073–87. [32] Vance J, Jagannathan S. 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Adaptive NN control for discrete-time pure-feedback systems with unknown control direction under amplitude and rate actuator constraints Weisheng Chen ∗ Department of Applied Mathematics, Xidian University, Xi’an 710071, PR China
article
info
Article history: Received 21 August 2008 Received in revised form 4 February 2009 Accepted 3 April 2009 Available online 28 April 2009 Keywords: Pure-feedback systems Discrete Nussbaum gain Implicit function theorem Neural network
abstract This paper focuses on the problem of adaptive neural network tracking control for a class of discrete-time pure-feedback systems with unknown control direction under amplitude and rate actuator constraints. Two novel state-feedback and output-feedback dynamic control laws are established where the function tanh(·) is employed to solve the saturation constraint problem. Implicit function theorem and mean value theorem are exploited to deal with non-affine variables that are used as actual control. Radial basis function neural networks are used to approximate the desired input function. Discrete Nussbaum gain is used to estimate the unknown sign of control gain. The uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. A simulation example is provided to illustrate the effectiveness of control schemes proposed in this paper. © 2009 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction Neural networks (NNs) have been proven to be particularly useful for controlling nonlinear systems with nonlinearly parameterized uncertainties owing to their universal approximation property. In general, NNs are used as function approximators to approximate some suitable uncertainties appearing in controllers or systems. Many important theoretical and practical results have been reported in the past two decades [1–10]. In particular, NNs are employed to solve the control problem of nonlinear systems with special lower-triangular forms, including strict-feedback form [11–17], output-feedback form [18–21], and pure-feedback form [22–25]. As far as continuous-time systems are concerned, the control laws of systems and the adaptive laws of NN weights are generally derived by using Lyapunov stability theory. Therefore, the stability of closed-loop systems is easily guaranteed. In the development of adaptive NN control for lower-triangular systems, some important control techniques are successfully proposed. For example, adaptive bounding technique is used to deal with the NN approximation error [12,13]; Integral-type Lyapunov function is exploited to avoid the possible controller singularity [14]; Nussbaum-type functions are utilized to cope with the unknown control gain signs [15]; dynamic surface control is employed to solve the problem of ‘explosion of complexity’ [16]. Then,
∗
Tel.: +86 029 88202860; fax: +86 029 88202861. E-mail address: [email protected].
NN control techniques are further extended to output-feedback systems [17–19] and time-delay systems [20,21]. It should be mentioned that the adaptive NN control problem of pure-feedback systems, which are the most general lower-triangular systems, has been widely investigated in past several years [22–25], where mean value theorem [25] or implicit function theorem [22–24] are usually used to deal with non-affine system functions. For discrete-time systems, especially lower-triangular nonlinear discrete-time systems, adaptive NN control is more complex due to some new problems such as the noncausal problem and the coupling problem among subsystems, inputs and outputs. Only a few results are obtained [26–33]. For example, NN control for strict-feedback discrete-time systems is successfully solved in [34– 36], and then the ideas are applied to the multiple-input-multiple output (MIMO) systems [26–28] and the systems with unknown control directions [29,30]. Very recently, adaptive NN control problem for a class of discrete-time systems with non-strict form are addressed [31,32]. However, to the best of our knowledge, no work has been reported on adaptive NN control for the following purefeedback discrete-time systems under amplitude and rate actuator constraints:
(
xi (k + 1) = fi [¯xi (k), xi+1 (k)], xn (k + 1) = fn [¯xn (k), u(k), d(k)] y(k) = x1 (k)
i = 1, . . . , n − 1 (1)
where x¯ i (k) = [x1 (k), x2 (k), . . . , xi (k)]T ∈ Ri , i = 1, . . . , n, y(k) ∈ R and u(k) ∈ R are the state variables, system output and input respectively; d(k) denotes the external disturbance ¯ d¯ > 0; fi : Ri+1 → R are unknown smooth satisfying |d(k)| < d,
0019-0578/$ – see front matter © 2009 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2009.04.002
W. Chen / ISA Transactions 48 (2009) 304–311
305
functions. The control input u(k) is subjected to amplitude and rate limitations defined as follow. • Amplitude constraints: |u(k)| ≤ ua , where ua > 0 denotes the control amplitude bound. • Rate constraint: |1u(k)| = |u(k) − u(k − 1)| ≤ ur , where ur > 0 denotes the rate control bound. If no constraints are imposed on the control input u(k), the control problem of system (1) has been studied in very recent papers [30,33]. However, the physical impossibility of applying unlimited control signals makes actuator saturation a ubiquitous problem in control systems. In particular, it is well known that both amplitude and rate saturations of control input are a source of performance degeneration, limit cycles, different equilibrium points, and even instability. So, the control problem of system (1) under amplitude and rate saturation constraints is more challenging. In this paper, to solve the above problem, and noting that the function tanh(·) possesses the saturation property and its derivative is positive and bounded, so the function tanh(·) is introduced skillfully to design a novel dynamic NN-based control law. In addition, the discrete Nussbaum gain is combined with the adaptive law to deal with the unknown control direction similarly to [30,31]. Finally, a simulation example is provided to illustrate the effectiveness of control schemes proposed in this paper. This paper is organized as follows. Radial basis function (RBF) NN, discrete Nussbaum gain and preliminaries are given in Section 2. In Section 3, we present the problem formulation and the control design procedure. Section 4 gives a simulation example to illustrate the effectiveness of the proposed controllers. In Section 5, we conclude the work of this paper.
properties: (i) if xs (k) increase without bound, then
2. RBF NN, discrete Nussbaum gain and preliminaries
(
1
sup
xs (k)≥δ0 xs (k)
inf
xs (k)≥δ0
1 xs (k)
SN (x(k)) = +∞, SN (x(k)) = −∞;
(ii) if xs (k) ≤ δ1 , then |Sn (x(k0 ))| ≤ δ2 with some positive constants δ1 and δ2 ; where SN (k) is defined as k X
SN (x(k)) =
N (x(k0 ))1x(k0 )
k0 =0
with 1x(k) = x(k + 1) − x(k). In this paper, the discrete Nussbaum gain is chosen as [30] N (x(k)) = xs (k)sN (x(k))
3
If SN (x(k1 )) ≤ xs2 (k1 ),
then go to Step (2)
If SN (x(k1 )) > xs (k1 ),
then go to Step (3)
3 2
2.1. RBF NN
(3)
where the discrete sequence x(k) satisfies x(0) = 0, x(k) > 0, |1x(k)| ≤ δ0 , ∀k. sN (x(k)) is the sign function of the discrete Nussbaum gain, i.e., sN (x(k)) = ±1. The initial value is set as sN (x(0)) = +1. Thereafter, the sign function sN (x(k)) will be chosen by comparing the summation SN (x(k)) with a pair of 3/2 switching curves defined by f (xs (k)) = ±xs (k). The details are as follows. Step (1) At k = k1 , measure the output y(k1 ) and compute 1x(k1 ) and x(k1 + 1) = x(k1 ) + 1x(k1 ) and SN (x(k1 )) = SN (x(k1 − 1)) + N (x(k1 ))1x(k1 ). Case (SN (x(k1 )) = +1):
Case (SN (x(k1 )) = −1):
Many approximators are used to approximate unknown functions, such as RBF NN, multilayer NN, fuzzy system, etc. Owing to the simple structure and good approximation capability, in this paper we will exploit RBF NN [37] to approximate a continuous function h(z ) : Rq → R defined on a compact set Ωz ⊂ Rq , i.e.,
3
(
If SN (x(k1 )) < xs2 (k1 ),
then go to Step (2)
If SN (x(k1 )) ≥ xs (k1 ),
then go to Step (3).
3 2
(2)
Step (2) Set sN (x(k1 + 1)) = 1, go to Step (4). Step (3) Set sN (x(k1 + 1)) = −1, go to Step (4). Step (4) Return to Step 1 and wait for the measurement of output.
where S (z ) = [s1 (z ), s2 (z ), . . . , sl (z )] ∈ R is a known continuous smooth vector-valued function, l is called the NN node number. The components sj (z ), j = 1, 2, . . . , l are commonly chosen as Gaussian function sj (z ) = exp[−kz − µj k2 /η2 ], µj ∈ Ωz is a constant which is called the center of sj (z ), and η > 0 is a real number which is called the width of sj (z ). The optimal weight vector W = [w1 , . . . , wl ]T is defined as
Lemma 1 ([30]). Consider discrete Nussbaum gain defined in (3). (i) Given an arbitrary bounded function g (k) : R → R, and g1 ≤ |g (k)| ≤ g2 , where g1 and g2 are unknown positive constants, then N 0 (x(k)) = g (k)N (x(k)) is also a discrete Nussbaum gain if 1x(k) ≥ 0. (ii) Given an arbitrary function −0 ≤ C (k) ≤ 0 , then N 0 (x(k)) = N (x(k)) + C (k) is still a discrete Nussbaum gain if 1x(k) ≥ 0.
h(z ) = W T S (z ) + ε(z ) T
l
T ˆ W := arg min sup h(z ) − W S (z ) ˆ ∈Rl W
z ∈Ωz
and ε(z ) is the inherent NN approximation error with the minimum upper bound ε¯ > 0. It has been proven that RBF NN satisfies the conditions of the Stone–Weierstrass Theorem and can approximate any continuous function to any desired accuracy over the compact set Ωz . That is, ε¯ can be arbitrarily decreased by increasing the NN node number l. In general, the NN weight W is unknown and needs to be ˆ denotes the estimate of W , estimated. Throughout this paper, W ˜ =W ˆ − W. and the estimation error is defined as W 2.2. The discrete Nussbaum gain Definition 1 ([30]). Consider a discrete nonlinear function N (x(k)) defined on a sequence x(k) with xs (k) = supk0 ≤k {x(k0 )}. N (x(k)) is a discrete Nussbaum gain if and only if it satisfies the following two
Lemma 2 ([30]). Let V (k) be a positive-define function defined ∀k, N (·) be discrete Nussbaum gain, and θ be a nonzero constant. If the following inequality holds, ∀k: V (k) ≤
k X
(c1 + θ N (x(k0 )))1x(k0 ) + c2 x(k) + c3
k 0 =k
1
where c1 , c2 , and c3 are some constants and k1 is a positive integer, Pk 0 0 then V (k), x(k), and k0 =k1 (c1 + θ N (x(k )))1x(k ) + c2 x(k) + c3 must be bounded ∀k. 2.3. Preliminaries Definition 2 ([38]). The trajectory x(k) of the closed-loop system is semi-globally uniformly ultimately bounded (SGUUB), if for any Ω , a compact subset of Rn and all χ (k0 ) ∈ Ω , there exists an ε > 0 and a number N (ε, χ (k0 )) such that kχ (k)k < ε for all k ≥ k0 + N.
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W. Chen / ISA Transactions 48 (2009) 304–311
Lemma 3 ([39]). For some given real scalar sequences s(k), b1 (k), and b2 (k) and vector sequence σ (k), if the following conditions hold: (i) limk→∞ (s2 (k))/(b1 (k)z + b2 (k)σ T (k)σ (k)) = 0; (ii) 0 < b1 (k) < K and 0 ≤ b2 (k) < K with a finite K ; (iii) kσ (k)k ≤ C1 + C2 max0≤k0 ≤k |x(k0 )|, where C1 and C2 are some finite constants; then we have (a) limk→∞ s(k) = 0, and (b) σ (k) is bounded. Lemma 4 (Implicit Function Theorem [38]). Assume that f (x, y): Rn × R → R is continuous differentiable ∀(x, y) ∈ Rn × R, and there exists a positive constant d such that
∂ f (x, y) ∂ y > d > 0,
∀(x, y) ∈ Rn × R
Lemma 5 (Mean Value Theorem [38]). Assume that f (x, y): Rn ×R → R is continuously differentiable at each point of an open set Rn ×(a, b), and assume also that it is continuous at both endpoints y = a and y = b, then there exists a positive constant 0 ≤ θ ≤ 1 such that
Lemma 6 ([34]). Let U be an open subset of Rn and let φ (φ1 , . . . , φn ) : U → Rn be a smooth map. If the Jacobian Matrix
∂φ
1
∂ x1 . = .. dx ∂φ
dφ
n
∂ x1
··· ..
.
···
(6)
where Fi (·) is an unknown function depending on fj (·), j = 1, . . . , n − 1. The above system can be also described as follows x1 (k + n) = F [¯xn (k), u(k), d(k)]
(7)
where F [¯xn (k), u(k), d(k)]
(4)
then there exists a unique continuous (smooth) function y = g (x) such that f (x, g (x)) = 0.
∂ f (x, y) f (x, b) − f (x, a) = (b − a). ∂ y y=θ a+(1−θ)b
x1 (k + n) = F1 [¯xn (k), x2 (k + n − 1)] . .. xn−1 (k + 2) = Fn−1 [¯xn (k), xn (k + 1)] xn (k + 1) = fn [¯xn (k), u(k), d(k)] y(k) = x1 (k)
(5)
=
∂φ1 ∂ xn .. . ∂φ n
∂ xn
is nonsingular at some point p ∈ U, or equivalently, Rank(dφ/dx) = n at some point p ∈ U, then there exists a neighborhood V ⊂ U of p such that φ: V → φ(V ) is a diffeomorphism. 3. Problem formulation and controller design 3.1. Problem formulation and system transformation The control objective of this paper is formulated as follows. For a known and bounded trajectory yd (k) ∈ Ωd , where Ωd is a compact set, design two adaptive control schemes satisfying the amplitude and rate saturation constraints (i.e., state-feedback one and output-feedback one) to make the system output y(k) follow yd (k), while maintaining all closed-loop signals SGUUB. For the control of pure-feedback system (1), define
= F1 [¯xn (k), F2 [¯xn (k), . . . , fn [¯xn (k), u(k), d(k)] · · ·]]. Note that under Assumption 1, we still have
∂ F ∂ F1 ∂ fn = 0 < g ≤ ··· ≤ g¯0 , ∀¯xn ∈ Ωx . 0 ∂u ∂ x2 ∂u
3.2. State-feedback controller design In this subsection, we will design a NN-based state-feedback control for system (7). To do this, we will write the system (7) as follows. x1 (k + n) = F0 [¯xn (k), u(k)] + d0 (k)
(8)
where F0 [¯xn (k), u(k)] = F [¯xn (k), u(k), 0], and d0 (k) = F [¯xn (k), u(k), 0] − F [¯xn (k), u(k), d(k)]. Note that since u(k) is required to ¯ for x¯ n ∈ Ωx , there exists a finite satisfy |u(k)| ≤ ua and |d(k)| ≤ d, constant d¯ 0 such that |d0 (k)| ≤ d¯ 0 . To deal with the problem of amplitude and rate saturation of the control input, we design the following new dynamic control law. u(k) = ua tanh(v(k)),
v(k) = v(k − 1) + ur tanh(α(k)),
(9)
v(−1) = 0
(10)
where α(k) is the new control variable designed later. Obviously, since | tanh(·)| ≤ 1, it is ensured that |u(k)| ≤ ua and |1u(k)| = |v(k) − v(k − 1)| ≤ ur . Substituting (9) and (10) into (8) results into x1 (k + n) = F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
+ d0 (k).
(11)
Define the tracking error e(k) = y(k) − yd (k), we have e(k + n) = F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
− yd (k + n) + d0 (k).
(12)
For v ∈ Ωv and α ∈ Ωα , where Ωv ⊂ R and Ωα ⊂ R are compact sets, there must exist constants g , g > 0 such that d tanh(v)
v
α
d tanh(α)
∂ fi (¯xi , xi+1 ) gi (¯xi , xi+1 ) =: , i = 1, . . . , n − 1, ∂ xi+1 ∂ fi (¯xi , u(k), d(k)) gn (¯xn , u(k), d(k)) =: . ∂ u(k)
0 < g ≤ ≤ 1 and 0 < g α ≤ ≤ 1. Based on dv dα v Assumption 1 and Lemma 4, we conclude that for (¯xn (k), v(k), α(k)) ∈ Ω =: Ωx × Ωv × Ωα , there exists a desired control variable α ∗ (k) = ψ1 [¯xn (k), v(k − 1), yd (k + n)] such that F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α ∗ (k)))] − yd (k + n) = 0. Then, using Lemma 5, we get
Assumption 1. There exist constants 0 < g ≤ g i such that i g ≤ |gi (·)| ≤ g i , ∀¯xn ∈ Ωx ⊂ Rn , where Ωx is a compact set. i However, the signs of gi (·), i = 1, . . . , n − 1 are unknown.
F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α(k)))]
If we consider the original system (1) as a one-step ahead predictor, we can transform it into an equivalent maximum n-step ahead one with the following form, which can predict the future states x1 (k + n), x2 (k + n − 1), . . . , xn (k + 1); then, the causality contradiction is avoided when the controller is constructed based on the maximum n-step ahead prediction [34].
= F0 [¯xn (k), ua tanh(v(k − 1) + ur tanh(α ∗ (k)))] + gλ (k)[α(k) − α ∗ (k)] where
∂ F0 [¯xn , ua tanh(v + ur tanh(xλ ))] gλ (k) := ∂ xλ xλ =%∗ with %∗ = λα(k) + (1 − λ)α ∗ (k), 0 ≤ λ ≤ 1.
(13)
W. Chen / ISA Transactions 48 (2009) 304–311
Note valid for gλ (k), i.e., 0 < Qnthat Assumption 1 on gi (·) is stillQ n ¯ = ¯i for (¯xn (k), v(k), g g i=1 g i = g ≤ |gλ (k)| ≤ g i=1 g v α α(k)) ∈ Ω . Substituting (13) back into (12) yields e(k + n) = gλ (k)[α(k) − α ∗ (k)] + d0 (k).
(14)
The desired control variable α ∗ (k) can be expressed by using a RBF NN as follows
α ∗ (k) = W T S (z (k)) + ε(z (k)) (15) where the NN input z (k) = [¯xTn (k), v(k − 1), yd (k + n)]T ∈ Ωx × Ωv × Ωd =: Ω0 . The NN approximation error |ε(z (k))| ≤ ε¯ , and the NN node number l > 0. Based on (14) and (15), the new control variable α(k) is designed as ˆ T (k)S (z (k)). α(k) = W
(16)
Substituting (15) and (16) back into (14) leads to
˜ T (k)S (z (k)) + d∗ (k). e1 (k + n) = gλ (k)W
(17)
where d (k) = −g (k)ε1 (z (k)) + d0 (k) and it is trivial to show that |d∗ (k)| ≤ g¯ ε¯ + d¯ 0 := d∗0 . Now, the adaptive law of W is given as ∗
ˆ (k) = W ˆ (k − n) − γ N (x(k))S (k − n) W
(k)
D(k) where N (x(k)) is the discrete Nussbaum gain given by (3) and γ e(k) (k) = , G(k) = 1 + |N (x(k))|, G(k)
(18)
a(k)G(k) 2 (k) D(k)
x(0) = 0, a(k) =
1, 0,
ˆ (j) = 0, W
,
and a2 (k) = a(k). Then, by the same derivation to [30, Theorem 2], we have V (k) ≤ −2
k X
N (x(k))
1
−
a(k)G(k) 2 k
lim 1x(k) = lim
k→∞
D(k)
k→∞
= 0.
(22)
Let us define a time interval as Z1 = [k|a(k) = 1] and suppose that Z1 is an infinite set. Then based on Lemma 3, we have limk→∞,k∈Z1 (k) = limk→∞Z1 a(k)(k) = 0, which conflicts with a(k) = 1, k ∈ Z1 , because |(k)| ≥ ν , when a(k) = 1. Therefore, Z1 must be a finite set and then, we have lim sup |(k)| ≤ ν
k→∞
Cν
γ
.
Then, following the same procedure as in [30, Theorem 2], the SGUUB of other close-loop signals can be concluded. This completes the proof.
j = −n − 1, . . . , 0.
gλ (k − n)
(21)
where c1 = γ 2 + |2d∗0 /g ν|, and N 0 (x(k)) = g (k−n) which is still a discrete Nassbaum gain based on Lemma 1. Then, according to ˆ (k), G(k), x(k), and Lemma 2, we conclude the boundedness of W N (x(k)), and further, from the boundedness of x(k), we have
lim sup |e(k)| ≤
if |(k)| > ν others
˜ T (k − n)S (k − n) = γW
N 0 (x(k0 ))1x(k0 ) + c1 x(k) + c1
k0 =0
k→∞
Proof. The proof closely follows [30]. First, we assume the NN is constructed to cover a large enough compact set Ω0 such that the NN input z ∈ Ω0 . From (17), we have G(k)(k)
1 gλ (k − n)
γ d∗ (k − n).
(19)
Choose a positive-definite function V (k) as
˜ T (k − n + j)W ˜ (k − n + j) W
3.3. Output-feedback NN controller design In the above subsection, we assume the system states x¯ n (k) are measurable. However, the control scheme becomes infeasible when some system states are unmeasurable. In this subsection, we assume that only the system output is measurable, and an outputfeedback NN control scheme will be designed. To do so, define the new system states ξ (k) = [ξ1 (k), ξ2 (k), . . . ξn (k)]T for the system (6) as
ξ1 (k) = x1 (k), ξ2 (k) = x1 (k + 1) = f1 (¯x1 (k), x2 (k)) = hn (¯x2 (k)) ξ3 (k) = x1 (k + 2) = f1 (¯x1 (k + 1), x2 (k + 1)) = f1 (¯x2 (k), f2 (¯x2 (k), x3 (k))) = hn−1 (¯x3 (k)), .. .
(23)
ξn (k) = x1 (k + n − 1) = f1 [¯xn−1 , f2 [¯xn−1 , . . . , fn−1 [¯xn−1 , xn ] · · ·]] = h2 (¯xn (k)) which can be written as ξ (k) = T (¯xn (k)), where T (¯xn (k)) is a nonlinear coordinate transformation of the form T (¯xn (k)) = [x1 (k), hn (¯x2 (k)), . . . , h2 (¯x2 (k))]T .
(24)
The following lemma guarantees that the coordinate transformation mapping T is diffeomorphism.
n
j =1
∗
which implies that N (x(k)) will converge to a constant ultimately. By denoting the limit of G(k) as C , it can be derived from the definition of (k) that
Theorem 1. Under Assumption 1, consider the closed-loop adaptive system consisting of the plant (1), the control law (9), (10) and (16) with the adaptive laws (18). All the signals in the closed-loop system are SGUUB, and the discrete Nussbaum gain N (x(k)) will converge to a constant ultimately. Denote C = limk→∞ G(k), then the tracking error satisfies limk→∞ sup |e(k)| < C ν/γ , where the tuning rate γ > 0 and the threshold value ν > 0 can be arbitrary constants to be specified by the designer.
X
2d∗ 0 N (x(k))d (k − n)(k) ≤ a(k) G(k) 2 (k) gν g (k − n) 2a(k)
lim a(k) = 0,
Remark 1. The adaptive law (18) is motivated by [30]. Although the adaptive law is same as that of [30] in the form, the new dynamic control law consisting of (9), (10) and (16) is very different from that of [30], where no dynamic equations (9) and (10) are introduced due to no constraints being imposed on the control input. Moreover, the introduction of (9) and (10) makes the above analysis and derivations more difficult than [30]. It is emphasized d tanh(·) that the good property of tanh(·) (i.e., 0 < ≤ 1) plays a d· crucial role in the above derivations.
V (k) =
and note that
k→∞
D(k) = 1 + kS (t − n)k2 + |N (x(k))| + 2 (k),
1x(k) = x(k + 1) − x(k) =
307
(20)
Lemma 7. For system (1) (or system (6)), the nonlinear coordinate transformation T (¯xn (k)) in (24), is a smooth map and a
308
W. Chen / ISA Transactions 48 (2009) 304–311
diffeomorphism, namely, there exists a unique transformation function T −1 (ξ (k)) such that x¯ n = T −1 (ξ ), and both T and T −1 are oneto-one.
ˆ (k) = W ˆ (k − n) − γ N (x(k))S (k − n) W
Proof. From (24), it is easily derived that
1
∗ dT (¯xn (k)) = dx(k) ∗ ∗
0
∂ f1 ∂ x2 .. . ∗
···
0
···
0
.. . ···
Based on the above analysis, we can obtain the cascade system description, which is equivalent to the original system (1), as follows:
(25)
where ξ (k) = [ξ1 (k), . . . , ξn (k)]T ∈ Rn , and f (ξ (k), u(k), d(k)) = F (¯xn , u(k), d(k))
= F (T −1 (ξ (k)), u(k), d(k)) is an unknown smooth function. Since ξ (k) is unmeasured, similarly to [34, Eqs. (31)–(35)], we write ξ (k) as y(k + 1) = ξ2 (k) = f (y(k), u(k − n + 1), d(k − n + 1))
=: ϕ2 (z (k), d(k)) y(k + 2) = ξ3 (k) = f (y(k + 1), u(k − n + 2), d(k − n + 2))
=: ϕ3 (z (k), d(k)) .. .
(26)
y(k + n − 1) = ξn (k) = f (y(k + n − 2), u(k − 1), d(k − 1))
=: ϕn (z (k), d(k)) y(k + n) = ξn (k + 1) = f (y(k + n − 1), u(k), d(k))
=: f0 (z (k), u(k), d(k)) where y(k) = [y(k − n + 1), . . . , y(k − 1), y(k)]T , uk−1 (k) =
[u(k − 1), . . . , u(k − n + 1)]T , z (k) = [yT (k), uTk−1 (k)]T , d(k) = [d(k), . . . , d(k − n + 1)]T and f0 (z (k), u(k), d(k)) = f (x1 (k), ϕ2 (z (k)), . . . , ϕn (z (k)), u(k), d(k)). We rewrite the system (26) as y(k + n) = f¯0 (z (k), u(k)) + d∗ (k)
(27)
where f¯0 (z (k), u(k)) = f0 (z (k), u(k), 0), and which satisfies |d∗ (k)| ≤ d¯ for z (k) ∈ Ωz1 , where Ωz1 is a compact set. Note that f¯0 (z (k), u(k)) still satisfies Assumption 1. Since the system (27) has the same form as the system (7), so the control design procedure is similar. Here we only give the following control law and the detailed procedure and derivation are omitted.
ˆ T (k)S (¯z (k)) α(k) = W
(28)
v(−1) = 0
γ e(k) , G(k) = 1 + |N (x(k))|, G(k) D(k) = 1 + kS (t − n)k2 + |N (x(k))| + 2 (k), a(k)G(k) 2 (k) 1x(k) = x(k + 1) − x(k) = , D(k) x(0) = 0, 1, if |(k)| > ν a(k) = 0, others ˆ (j) = 0, W
j = −n − 1, . . . , 0.
Theorem 2. Under Assumption 1, consider the closed-loop adaptive system consisting of the plant (1), the control law (28)–(30) with the adaptive laws (31). All the signals in the closed-loop system are SGUUB, and the discrete Nussbaum gain N (x(k)) will converge to a constant ultimately. Denote C = limk→∞ G(k), then the tracking error satisfies limk→∞ sup |e(k)| < C ν/γ , where the tuning rate γ > 0 and the threshold value ν > 0 can be arbitrary constants to be specified by the designer. Proof. The proof can be easily completed by following the same derivation of Theorem 1. The detailed procedures are omitted. 4. Simulation example In this section, a numerical simulation example is provided to illustrate the effectiveness of the proposed control approaches. Consider the following second-order system
2 x1 (k + 1) = x1 (k) cos(x1 (k)) + 0.5 sin[x2 (k)e−x1 (k) ] + x2 (k) x1 (k) + x2 (k) x2 (k + 1) = (32) 1 + [x1 (k) + x2 (k)]2 3 +ρ( u ( k ) + 0 . 1u ( k )) + d ( k ) y(k) = x1 (k) where d(k) = 0.001 sin(0.1k), and ρ is a nonzero constant whose sign denotes the control direction. Suppose that there is no a priori knowledge of the system nonlinearities. It is easily checked that the system (32) satisfies Assumption 1. The control input is limited by the amplitude and rate saturation constraints: |u(k)| ≤ 1.2 and |1u(k)| ≤ 0.8. The control objective is to make the output y(k) follow a desired reference signal yd (k) = 0.5 sin(kπ /200) + 0.5 sin(kπ /100). 4.1. State feedback NN control Based on the control approach developed in Section 3.2, the control law u(k) is given as
d∗ (k) = f0 (z (k), u(k), d(k)) − f0 (z (k), u(k), 0)
u(k) = ua tanh(v(k)),
(31)
(k) =
Based on Assumption 1, Trace (dT (¯xn (k)/dx(k))) 6= 0, therefore T is a diffeomorphism according to Lemma 6. Equivalently, T −1 is also a diffeomorphism.
ξ1 (k + 1) = ξ2 (k), ξ2 (k + 1) = ξ3 (k), .. . ξn (k + 1) = f (ξ (k), u(k), d(k))
(k) D(k)
where N (x(k)) is the discrete Nussbaum gain given by (3) and
. .. . ∂ f1 ∂ f n −1 ··· ∂ x2 ∂ xn
v(k) = v(k − 1) + ur tanh(α(k)),
where z¯ = [z T , v(k − 1), yd (k + n)]T ∈ Ω1 , and the adaptive law is given by
(29) (30)
u(k) = 1.2 tanh(v(k)), v(k) = v(k − 1) + 0.8 tanh(α(k)), ˆ T (k)S (z (k)). α(k) = W
v(−1) = 0
(33)
where z (k) = [x1 (k), x2 (k), v(k − 1), yd (k + 2)]T , and the adaptive ˆ is given by (18) with n = 2. law of W In simulation, the initial conditions for system states are set to ˆ (0) = 0. The number be x1 (0) = x2 (0) = 0, and those for NN are W of NN nodes is chosen as l = 64 . The centers of RBFs evenly cove the
W. Chen / ISA Transactions 48 (2009) 304–311
309
Fig. 1. Simulation results of state-feedback control for system (32) with ρ = 1.
Fig. 2. Simulation results of state-feedback control for system (32) with ρ = −1.
compact set [−1, 1]4 , and the widths of RBFs are set to be η = 0.6. The design parameters are specified as γ = 0.51 and ν = 0.005. The simulation results of ρ ± 1 are shown in Figs. 1 and 2. Fig. 1 shows the case of the control gain ρ = 1. Fig. 1(a) shows the output y(k), the reference signal yd (k), and the tracking
error e(k). Fig. 1(b) gives the response curves of the N (z (k)) and z (k). Fig. 1(c) displays the boundedness of the norm of NN weight ˆ (k)k. Fig. 1(d) gives the response curves of u(k) and 1u(k). kW It can be seen from Fig. 1 that the tracking error converge to a desired neighborhood around the origin, and the control input
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W. Chen / ISA Transactions 48 (2009) 304–311
Fig. 3. Simulation results of output-feedback control for system (32) with ρ = 1.
Fig. 4. Simulation results of output-feedback control for system (32) with ρ = −1.
satisfies the saturation constrain condition, and the closed-loop ˆ (k)k and u(k) are uniformly bounded. The signals x1 (k), x2 (k), kW case of ρ = 1 is shown in Fig. 2.
4.2. Output feedback NN control When the system states are not available, based on the outputfeedback control approach developed in Section 3.3, the control
W. Chen / ISA Transactions 48 (2009) 304–311
u(k) is given as
u(k) = 1.2 tanh(v(k)), v(k) = v(k − 1) + 1.2 tanh(α(k)), ˆ T (k)S (¯z (k)) α(k) = W
v(−1) = 0
(34)
where z (k) = [y(k), y(k − 1), u(k − 1)(k), v(k − 1), yd (k + 2)]T , ˆ (k) is given by (31) with n = 2. and the adaptive law of W In simulation, the initial conditions for system states are set to ˆ (0) = 0. The number be x1 (0) = x2 (0) = 0, and those for NN are W of NN nodes is chosen as l = 55 . The centers of RBFs evenly cove the compact set [−1.15, 1.15]5 , and the widths of RBFs are set to be η = 0.9. The design parameters are specified as γ = 0.78 and ν = 0.005. The simulation results of ρ = ±1 are shown in Figs. 3 and 4. For two cases, it can be seen from Figs. 3 and 4 that the tracking errors both converge to a desired neighborhood around the origin, and the control input satisfies the saturation constraints, ˆ (k)k and u(k) are still while the closed-loop signals x1 (k), x2 (k), kW uniformly bounded. 5. Conclusions This paper considers the adaptive NN tracking control problem of discrete-time pure-feedback system with unknown control direction under the amplitude and rate saturation constraints of control input. This is a further extension of the existing results [30, 33]. By employing the good property of the function tanh(·), we successfully design two kinds of novel dynamic NN-based control schemes to deal with the problem of amplitude and rate saturation of control input. Though we only consider the single-input-singleoutput (SISO) systems in this paper, the design idea is also applied to MIMO cases [26–28]. Acknowledgments The author would like to thank the anonymous reviewers for their helpful comments and advice which contributed much to the improvement of this paper. The work was supported by the National Natural Science Funds of PR China under grant 60804021. References [1] Chen FC, Khalil HK. Adaptive control of nonlinear systems using neural networks. International Journal of Control 1992;55(6):1299–317. [2] Chen FC, Liu CC. Adaptively controlling nonlinear continuous-time systems using multilayer neural networks. IEEE Transactions on Automatic Control 1994;39(10):1306–10. [3] Chen FC, Khalil HK. Adaptive control of a class of nonlinear discrete-time systems using neural networks. IEEE Transactions on Automatic Control 1995; 40(5):791–801. [4] Lewis FL, Yesildirek A, Liu K. Multilayer neural-net robot controller with guaranteed tracking performance. IEEE Transactions on Neural Network 1996; 7(2):388–98. [5] Lewis FL, Jagannathan S, Yesildirek A. Neural network control of robot manipulators. Singapore: World Scientific; 1998. [6] Jagannathan S, Lewis FL. Discrete-time neural net controller for a class of nonlinear dynamical systems. IEEE Transactions on Automatic Control 1996; 41(10):1693–9. [7] Jagannathan S. Control of a class of nonlinear discrete-time system using multilayer neural networks. IEEE Transactions on Neural Networks 2008; 12(5):1113–20. [8] Savran A, Tasaltin R, Becerikli Y. Intelligent adaptive nonlinear flight control for a high performance aircraft with neural networks. ISA Transactions 2006; 45(2):225–47. [9] Kolla SR, Altman SD. Artificial neural network based fault identification scheme implementation for a three-phase induction motor. ISA Transactions 2007;46(2):261–6.
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