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An adaptive neuro-fuzzy tracking control for multi-input nonlinear dynamic systems
Automatica 44 (2008) 1418–1425 www.elsevier.com/locate/automatica
Brief paper
An adaptive neuro-fuzzy tracking control for multi-input nonlinear dynamic systemsI S.P. Moustakidis, G.A. Rovithakis ∗ , J.B. Theocharis Aristotle University of Thessaloniki, Department of Electrical and Computer Engineering, 54124, Thessaloniki, Greece Received 28 September 2006; received in revised form 13 September 2007; accepted 10 October 2007 Available online 10 March 2008
Abstract An adaptive neuro-fuzzy control design is suggested in this paper, for tracking of nonlinear affine in the control dynamic systems with unknown nonlinearities. The plant is described by a Takagi–Sugeno (T–S) fuzzy model, where the local submodels are realized through nonlinear dynamical input–output mappings. Our approach relies upon the effective approximation of certain terms that involve the derivative of the Lyapunov function and the unknown system nonlinearities. The above task is achieved locally, using linear in the weights neural networks. A novel resetting scheme is proposed that assures validity of the control input. Stability analysis provides the control law and the adaptation rules for the network weights, assuring uniform ultimate boundedness of the tracking and the signals appearing in the closed-loop configuration. Illustrative simulations highlight the approach. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Adaptive control; Neuro-fuzzy control; Resetting
1. Introduction Neural network and fuzzy systems-based adaptive control methodologies are receiving considerable attention, emerging as promising approaches for controlling highly uncertain and nonlinear dynamical systems. In the realm of neural networks, many controllers have been suggested (Lewis, Liu, & Yesildirek, 1995; Polycarpou, 1996; Rovithakis, 2001, 2004; Xu & Ioannou, 2003). Although stable tracking is guaranteed, these approaches cannot incorporate linguistic system descriptions within their framework. Recently, several stable adaptive fuzzy control schemes have been introduced based on Takagi–Sugeno (T–S) modeling (Koo, 2001; Lee & Tomizuka, 2000; Ordonez & Passino, 1999; Spooner & Passino, 1996). The reasoning behind this growing research is the capability of adaptive schemes to deal with large parametric uncertainties, while on the other hand, taking advantage of the qualitative knowledge description of the fuzzy systems, in I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ra´ul Ord´on˜ ez under the direction of Editor Miroslav Krstic. ∗ Corresponding author. E-mail address: [email protected] (G.A. Rovithakis).
c 2008 Elsevier Ltd. All rights reserved. 0005-1098/$ - see front matter doi:10.1016/j.automatica.2007.10.019
the form of If/Then linguistics. Unfortunately, they all impose restrictive growing assumptions on systems nonlinearities, as well as they require the boundedness away from zero of a norm of the control input vector field. However, for general multiinput affine in the control nonlinear dynamic systems, such estimates may change signs frequently, or even worse, they may be in a neighbourhood of the origin despite the fact that their corresponding actual values may indeed be far from zero, thus making the estimation model uncontrollable. In this paper, we relax the aforementioned restrictions, developing a stable adaptive neuro-fuzzy controller with robust tracking characteristics, for multi-input nonlinear affine in the control dynamic systems with unknown nonlinearities. Compared to prior research, the distinctions of this work are described as follows: (1) The system dynamics are described by a T–S fuzzy model where the consequent parts of the rules are represented by unknown nonlinear dynamic subsystems. Hence, a better approximation of the actual system dynamics is achieved compared to the conventional T–S models with linear consequent submodels. (2) Instead of modelling the plant dynamics directly, our method relies upon the effective approximation of certain terms that involve the derivative of an unknown Lyapunov function and the unknown system
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nonlinearities, using linear in the weights neural network structures. (3) A novel resetting scheme is proposed to assure the boundedness away from zero of certain signals, thus guaranteeing the validity of the control input. Uniform ultimate boundedness of the tracking error with respect to an arbitrarily small neighbourhood of zero, plus the boundedness of all other signals in the closed loop, is guaranteed. The control design is general, assuming partially known systems. Nevertheless, completely unknown systems can be handled by setting the known parts to zero, throughout the design process. The rest of the paper is organized as follows. In Section 2, we describe the plant dynamics in terms of T–S dynamic fuzzy rules and formulate the problem. In Section 3 the suggested adaptive neuro-fuzzy control is presented while in Section 4, simulation results are provided, to demonstrate the effectiveness of the method. Finally, conclusions are drawn in Section 5.
final state of the T–S fuzzy model is obtained by x˙ =
r X
(k)
(k)
(u)
ςi (x){[ f i (x) + G i (x)u] + [ f i
(u)
(x) + G i (x)u]}
i=1
(4) where ςi (x) ∈ [0, 1] are the normalized firings of the rules: r Q
ςi (x) =
n X
j=1
µ F i (x j ) j
n P
r Q
i=1
j=1
!,
ςi (x(t)) ≥ 0,
µ F i (x j ) j
ςi (x(t)) = 1.
(5)
i=1
Consider affine in the control nonlinear dynamic systems, of the form
The control objective is to devise a control law u(t) such that the system state x(t) is forced to follow a given bounded reference trajectory xr (t). For xr (t), we further assume that x˙r (t) is also bounded, i.e. |x˙r | ≤ dr , where dr is a known upper bound. With the tracking error defined as e(t) = x(t) − xr (t), the error dynamics are described by
x˙ = f (x) + G(x)u
e˙ =
2. Problem formulation
(1)
where x ∈ Rn denotes the system’s state which is assumed to be completely measurable, u ∈ Rm is the control input, f ∈ Rn and G ∈ Rn×m are continuous, locally Lipschitz, vector fields. T–S fuzzy models provide an effective framework for representing complex multi-input multi-output (MIMO) systems (1). Extending the traditional approach of most fuzzy control schemes that use T–S rules with linear consequent models (Tseng, Chen, & Uang, 2001), we consider here a T–S fuzzy dynamic fuzzy model which is composed of generalized If /Then rules where the local models are represented by nonlinear dynamic subsystems of simpler structure: R (i) : IF x1 (t) is F1i AND . . . AND xn (t) is Fni THEN x(t) ˙ = f i (x) + G i (x)u
i = 1, . . . , r
(2)
where r is the number of rules, and F ji , i = 1, . . . , r , j = 1, . . . , n are fuzzy sets associated with the input x j (t) and quantified by a membership function µ F i (x j ) ∈ [0, 1]. j f i (x), G i (x), i = 1, . . . , r are assumed to be unknown and nonlinear. Owing to the nonlinear nature of the consequent functions, (2) approximates the local dynamics of the system more effectively compared to T–S rules with linear functions (x(t) ˙ = Ai x + Bi u, Ai ∈ Rn×n , Bi ∈ Rn×m ). We consider that the consequent functions are partially known, which allows embedding a priori knowledge. (k)
(u)
f i (x) = f i (x) + f i
(x),
(k)
(k)
(u)
G i (x) = G i (x) + G i (x) (3) (k)
(u)
(u)
for i = 1, . . . , r . f i (x), G i (x) and f i (x), G i (x) represent the known and the unknown plant dynamics, respectively. For a bounded state vector x, it is assumed that (k) (k) (u) (u) f i (x), G i (x) and f i (x), G i (x) are also bounded. The
r X
(k)
(k)
ςi (x){[ f i (x) + G i (x)u]
i=1 (u)
+ [ fi
(u)
(x) + G i (x)u]} − x˙r .
(6)
Assumption 1. The solution of (6) can be forced to be uniformly ultimately bounded with respect to an arbitrarily small neighbourhood of the origin, e = 0. Consider a set E, defined as E = {e ∈ Rn : 0 ≤ |e| ≤ e0 }, e0 being an arbitrarily small positive constant. Due to Assumption 1, there exists a radially unbounded Lyapunov function V (e) : Rn → R+ and an appropriate control u 0 (e, x), satisfying V˙ (e) ≤ 0 for all e lying outside E. Usually, the derivation of stabilizing feedback laws is based upon the explicit knowledge of the system’s structure and the Lyapunov function (Sontag, 1989). In our formulation though, we consider a highly uncertain environment, whereby the Lyapunov function, V (e), is assumed to be unknown. To cope with the highly uncertain conditions involved in our framework, we make use of linear in the weights neural networks (LWNNs). Mathematically, LWNNs are represented by y T = (W ∗ )T S(υ), where υ ∈ Rn 2 and y ∈ Rn 1 denote the approximator input and output respectively, W ∗ is a L-dimensional vector of synaptic weights, and S(υ) is a L × n 1 matrix of regressor terms. All LWNNs share the density property: For every continuous function f (υ) : Rn 2 → Rn 1 there exist an integer L and optimal weight values W ∗ such that for every ε > 0, supυ∈Ω | f (υ)T − (W ∗ )T S(υ)| ≤ ε, where Ω ⊂ Rn 2 is a given compact set. Since the Lyapunov function V (e) and the nonlinear terms (u) (u) f i (x), G i (x), i = 1, . . . , r are assumed unknown, we use
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S.P. Moustakidis et al. / Automatica 44 (2008) 1418–1425 ∗ , i = 1, . . . , r , Wc , are estimates of the unknown values Wk,i k = a, b, i = 1, . . . , r , and Wc∗ . Further, consider the Lyapunov function candidate
LWNNs to approximate the expressions: ∂ V (e) (k) ∂ V (e) (u) f i (x) + f (x) ∂e ∂e i (k) (u) = Ai (e, x) + Ai (e, x)
(7)
∂ V (e) (k) ∂ V (e) (u) Bi (e, x) = G i (x) + G i (x) ∂e ∂e (k) (u) = Bi (e, x) + Bi (e, x)
(8)
C(e) = ∂ V (e)/∂e
(9)
Ai (e, x) =
with i = 1, . . . , r , Ai (e, x) ∈ R, Bi (e, x) ∈ R1×m , C(e) ∈ R1×n . More specifically, the term (9) devoted to the estimation of the Lyapunov function derivative is modelled as: C(e) = (Wc∗ )T Sc (e) + ωc (e), while terms related to the known parts are approximated by (k)
(k)
(k)
(k)
(k)
(k)
Ai (e, x) = (Wc∗ )T Sc (e) f i (x) + ωc (e) f i (x) Bi (e, x) = (Wc∗ )T Sc (e)G i (x) + ωc (e)G i (x). (k) f i (x)
L = kV (e) +
r r 1X 1X 1 |W˜ a,i |2 + |W˜ b,i |2 + |W˜ c |2 2 i=1 2 i=1 2
(14)
where V (e) denotes the unknown Lyapunov function, and k > 0 is a design constant. Differentiating L with respect to time, using (6) and the terms substitutions defined in Section 2, we obtain r X (k) (k) L˙ = k ςi (x){Ai (e, x) + Bi (e, x)u} i=1
+k
r X
(u)
(u)
ςi (x){Ai (e, x) + Bi (e, x)u} − kC(e)x˙r
i=1
(10) + (11)
(k) G i (x)
Since and are known expressions, the (k) (k) unknown term ∂ V (e)/∂e in Ai (e, x),Bi (e, x) and C(e) are approximated using a common LWNN. Furthermore, for the completely unknown parts we obtain
r X
{(W˜ a,i )T W˙ a,i + (W˜ b,i )T W˙ b,i } + (W˜ c )T W˙ c
i=1
=k
r X
∗ T ∗ T ςi (x){(Wa,i ) Sa,i (e, x) + (Wb,i ) Sb,i (e, x)u}
i=1 r X
+
{(W˜ a,i )T W˙ a,i + (W˜ b,i )T W˙ b,i } + (W˜ c )T W˙ c
i=1 (u) Ai (e, x)
∗ T (Wa,i ) Sa,i (e, x) + ωa,i (e)
(12)
∗ T Bi (e, x) = (Wb,i ) Sb,i (e, x) + ωb,i (e)
(13)
=
(u)
where Sa,i (e, x) ∈ R L a,i , Sb,i (e, x) ∈ R L b,i ×m , i = 1, . . . , r , and Sc (e) ∈ R L c ×n are suitably selected regressors. Further, ∗ ∈ R L a,i , W ∗ ∈ R L b,i , i = 1, . . . , r and W ∗ ∈ R L c are Wa,i c b,i unknown weight values. Finally, ωa,i (e), ωb,i (e), i = 1, . . . , r , and ωc (e) denote modelling error terms.
+ kω(e, u) + k(Wc∗ )T Sc (e) ψ (15) P (k) (k) where ψ = ri=1 ζi (x)[ f i (x) + G i (x)u] − x˙r . The term ω(e, u) denotes the generalized modelling error, defined as ω(e, u) =
r X
ςi (x)ωa,i (e) +
i=1
r X
ςi (x)ωb,i (e)u + ωc (e)ψ.
i=1
(16) Pr
Assumption 2. Owing to the density property of the LWNNs, we assume that the modelling errors are bounded in a compact region Ω ⊂ Rn , i.e. |ωa,i (e)| ≤ ω¯ a,i , |ωb,i (e)| ≤ ω¯ b,i , i = 1, . . . , r , and |ωc (e)| ≤ ω¯ c , ω¯ a,i , ω¯ b,i , i = 1, . . . , r , and ω¯ c , being unknown but small positive constants. 3. Adaptive neuro-fuzzy control design In this section, a systematic methodology is presented for the design of stable adaptive neuro-fuzzy controllers following a three-stages approach. At stage 1, the control law and the weight adaptation rules are developed, guaranteeing the uniform ultimate boundedness of the tracking error with respect to an arbitrary small set around the origin. Additionally, the boundedness of all signals involved in the closed loop control configuration is ensured. The above qualities are fulfilled provided that the denominator of the control signal is bounded away zero. At stage 2, a novel resetting scheme is introduced, performing on the weight estimates Wb,i , i = 1, . . . , r and Wc , to guarantee the validity of the control law. Finally, at stage 3, it is demonstrated that e ∈ Ω ⊂ Rn , ∀t ≥ 0. ∗ , Stage 1. Consider the parameter errors W˜ k,i = Wk,i − Wk,i k = a, b, i = 1, . . . , r , W˜ c = Wc − Wc∗ , where Wk,i , k = a, b,
After k i=1 ςi (x)(Wa,i )T Sa,i (e, x), Pr adding and subtracting T k i=1 ςi (x)(Wb,i ) Sb,i (e, x)u and k(Wc )T Sc (e)e˙(k) , (15) is reformulated as r X L˙ = −k ςi (x)(W˜ a,i )T Sa,i (e, x) −k
i=1 r X
ςi (x)(W˜ b,i )T Sb,i (e, x)u − k(W˜ c )T Sc (e)e˙(k)
i=1
+k
r X
ςi (x)(Wa,i )T Sa,i (e, x) − k(Wc )T Sc (e)e˙(k)
i=1
+k
r X
ςi (x)(Wb,i )T Sb,i (e, x)u + (W˜ c )T W˙ c
i=1
+
r X i=1
(W˜ a,i )T W˙ a,i +
r X
(W˜ b,i )T W˙ b,i + kω(e, u).
(17)
i=1
Define the weight update rules. W˙ a,i = Pa,i {−ka,i Wa,i + kςi (x)Sa,i (e, x)}, W˙ b,i = −kb,i Wb,i + kςi (x)Sb,i (e, x)u, Wc = Pc {−kc Wc + k Sc (e)ψ}
i = 1, . . . , r (18)
i = 1, . . . , r
(19) (20)
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S.P. Moustakidis et al. / Automatica 44 (2008) 1418–1425
where ka,i , kb,i > 0, i = 1, . . . , r , and kc > 0 are constants determined by the designer and Pa,i , Pc are projection operators with respect to the convex sets Wa,i = {Wa,i ∈ R L a,i : |Wa,i | ≤ Ma,i }, Wc = {Wc ∈ R L c : |Wc | ≤ Mc }. All Ma,i , i = 1, . . . , r , Mc are chosen to take sufficiently large positive values. Definitions of the projection operators can be found in Ioannou and Sun (1996). Moreover, it is well known that projection modification can only make the derivative of the Lyapunov function more negative provided ∗ , W (0) ∈ M , i = 1, . . . , r and W ∗ , W (0) ∈ that Wa,i a,i a,i c c Ma,i . Hence, L˙ becomes L˙ ≤ ka(e, x, Wa , Wc , x˙r ) + kb(e, x, Wb , Wc )u + kω(e, u) r r X X − ka,i (W˜ a,i )T Wa,i − kb,i (W˜ b,i )T Wb,i i=1
i=1
(21)
where we have defined a(e, x, Wa , Wc , x˙r ) =
ςi (x)(Wa,i )T Sa,i (e, x)
i=1
b(e, x, Wb , Wc ) =
(22)
ςi (x)(Wb,i ) Sb,i (e, x) T
i=1
+ (Wc )T Sc (e)G (k) (x).
(23)
Recalling that for any z, z˜ ∈ Rn , with z˜ = z − z ∗ , we have z˜ T z = 12 |˜z |2 + 12 |z|2 − 12 |z ∗ |2 and after dropping the P P negative terms − ri=1 ka,i /2|Wa,i |2 , − ri=1 kb,i /2|Wb,i |2 , −kc /2|Wc |2 we obtain L˙ ≤ ka(e, x, Wa , Wc , x˙r ) + kb(e, x, Wb , Wc )u + kω(e, u) r r X X ka,i kb,i kc ∗ 2 ∗ 2 |Wa,i |Wb,i + | + | + |Wc∗ |2 2 2 2 i=1 i=1 −
r X ka,i i=1
2
|W˜ a,i |2 −
r X kb,i i=1
2
kc |W˜ b,i |2 − |W˜ c |2 . 2
(24)
Let us now consider the control law, a(e, x, Wa , Wc , x˙r ) + γ (|e|) T u=− b (e, x, Wb , Wc ) (25) |b(e, x, Wb , Wc )|2 where γ (|e|) is a positive, bounded and invertible function, ∀e ∈ Ω ⊂ Rn . We assume for the moment that |b(e, x, Wb , Wc )| is bounded away from zero. Stated otherwise, there is a small bound δb such that |b(e, x, Wb , Wc )| > δb ,
r X ka,i i=1
2
|W˜ a,i |2 −
r X kb,i i=1
2
|W˜ b,i |2
kc ˜ 2 | Wc | + µ 2 ≤ −kγ (|e|) + µ
(27)
−
(28) ka,i i=1 2
kb,i i=1 2
Pr
∗ |2 + ∗ |2 + where µ = k ω¯ + |Wa,i |Wb,i kc ∗ 2 ˙ 2 |Wc | . Hence, since γ (|e|) is invertible, L ≤ 0 provided that
|e| > γ −1
+ (Wc )T Sc (e){ f (k) (x) − x˙r } r X
L˙ ≤ kγ (|e|) −
Pr
− kc (W˜ c )T Wc r X
The above lemma can be verified following similar steps as in Rovithakis (2001), and hence, the proof is omitted. Notice that (19) is in a bounded input bounded output (BIBO) form, with the driving term given by ςi (x)Sb,i (e, x)u. Since |Sb,i (e, x)| ≤ s¯b,i by construction, ςi (x) ∈ [0, 1], and u ∈ L ∞ from Lemma 1, the driving term is bounded, and hence, Wb,i ∈ L ∞ , i = 1, . . . , r . Substituting (25) into (24), L˙ becomes
∀e ∈ Ω ⊂ Rn .
(26)
The aforementioned condition will be fulfilled through a resetting mechanism developed at stage 2. Lemma 1. Provided that (26) holds, the following statements are valid: (1) The control law (25) is uniformly bounded, u ∈ L ∞ and (2) Owing to u ∈ L ∞ , the generalized modelling error (16) is uniformly bounded by an unknown but small bound (i.e. |ω(e, u)| ≤ ω, ¯ ∀e ∈ Ω ).
! r 1 X kc ∗ 2 ∗ 2 ∗ 2 × ω¯ + {ka,i |Wa,i | + kb,i |Wb,i | } + |Wc | . (29) 2k i=1 2k The above inequality suggests that the tracking error e is uniformly ultimately bounded with respect to a set E: ( r 1 X ∗ 2 [ka,i |Wa,i | E = e ∈ Rn : |e| ≤ γ −1 ω¯ + 2k i=1 ) kc ∗ 2 ∗ 2 (30) + kb,i |Wb,i | ] + |Wc | 2k which can be made arbitrarily small by properly selecting the γ function, as well as choosing adequately small values of ka,i , kb,i , i = 1, . . . , r , kc and a large enough k. Theorem 1. For the error dynamics described by (6), the control law (25) for which (26) holds, combined with the weight adaptation scheme (18)–(20), guarantee the uniform ultimate boundedness of the tracking error e with respect to the arbitrarily small set E, plus the uniform boundedness of all other signals in the closed loop. Stage 2. From the aforementioned analysis, it becomes apparent that there is a need to assure that the term |b(e, x, Wb , Wc )| that appears in the denominator of the control law (25) is bounded away from zero, in order to attain the stability attributes of the closed-loop system. In this subsection, a novel resetting scheme is suggested, assuring (26) ∀t ≥ 0. We assume that the system initiates, taking values x(0), e(0), and Wb,i (0), i = 1, . . . , r , Wc (0), outside the set B = {e, x ∈ Rn , Wb,i ∈ R L b,i : |b(e, x, Wb , Wc )| < δb }, where δb > 0 is a design constant. Apparently, as long as |b(e, x, Wb , Wc )| > δb is maintained, there is no need for any modification on Wb,i , i = 1, . . . , r and Wc . The weights are ordinarily updated through (18)–(20) and the control law is derived from (25). Under these circumstances, boundedness of the control signal is achieved, as suggested by Lemma 1. The problem arises whenever |b(e, x, Wb , Wc )| ≤ δb .
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S.P. Moustakidis et al. / Automatica 44 (2008) 1418–1425
The components of b(e, x, Wb , Wc ) ∈ R1×m are given by b j (e, x, Wb , Wc ) =
r X
T ςi (Wb,i ) j (Sb,i (e, x)) j
(Wb,i )(tr+ ) = (Wb,i )(tr ) + (∆Wb,i )(tr ),
i=1
+ (WcT ) j (Sc (e)G (k) (x)) j T (Wb,i ) j (Sb,i (e, x)) j
=
(Q b,i ) j X
(Wb,i ) j,` (Sb,i (e, x)) j,`
(31)
`=1
(WcT ) j (Sc (e)G (k) (x)) j =
Q c, j X
(Wc )(tr+ ) = T )(tr ) (∆Wb,i
i = 1, . . . , r
(Wc )(tr ) + (∆Wc )(tr ) =
(36) (37)
T )1 (tr ), . . . , [(∆Wb,i T T )m (tr )] (∆Wb,i ) j (tr ), . . . , (∆Wb,i
T (∆Wb,i ) j (tr ) = [0, . . . , (∆Wb,i ) j,` j (tr ), . . . , 0],
` j ∈ [1, . . . , (Q b,i ) j ],
(WcT ) j,` (Sc (e)G (k) (x)) j,`
`=1 T ) , (S (e, x)) (Q b,i ) j and (W T ) , (S (e) with (Wb,i j b,i j ∈ R c c j,` G (k) (x)) j ∈ R(Q c ) j , j = P 1, . . . , m. Assume that at T S (e, x) + t = tr we have |b(tr )| = | ri=1 ςi (x)Wb,i b,i T ) , (W ) , T (k) Wc Sc (e)G (x)| = δb . A single weight from (Wb,i j c j i = 1, . . . , r , j = 1, . . . , m changes its value, provided that it multiplies a nonzero sigmoid term, to guarantee that at t = tr+ |b(tr+ )| > |b(tr )|. We select the ` j th component (Wb,i ) j,` j T ) , ` of (Wb,i j j ∈ [1, . . . , (Q b,i ) j ], and the q j th component (Wc ) j,q j of (Wc ) j , q j ∈ [1, . . . , (Q c ) j ], and apply resetting according to
(Wb,i ) j,` j (tr+ ) = (Wb,i ) j,` j (tr ) + (∆Wb,i ) j,` j (tr ) (∆Wb,i ) j,` j (tr )
(32)
= sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` j }(φb,i ) j,` j (ν) (Wc ) j,q j (tr+ ) = (Wc ) j,q j (tr ) + (∆Wc ) j,q j (tr ) (∆Wc ) j,q j (tr ) = sgn{b j (tr )}sgn{(Sc
Remark 1. The new weights obtained at t = tr+ after applying the resetting changes are
(33) (e)G (k) (x))
j,q j }(φc ) j,q j (ν)
for i = 1, . . . , r , j = 1, . . . , m, where sgn{·} is the sign function. Furthermore, (φb,i ) j,` j (ν) and (φc ) j,q j (ν) are positive, bounded functions, sharing the property that limv→∞ (φb,i ) j,` (v) = 0 and limv→∞ (φc ) j,q (v) = 0, with ν being a positive integer, denoting the number of times the resetting procedure has been activated for a particular weight component. To guarantee that |b(tr+ )| > |b(tr )| it suffices to show that |b j (tr+ )| > |b j (tr )|, ∀ j = 1, . . . , m. Introducing (32)–(33) in (31), we obtain b j (tr+ ) = b j (tr ) + ςi sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` j } × (Sb,i (e, x)) j,` j (φb,i ) j,` j (ν) + sgn{b j (tr )}
(∆WcT )(tr ) = [(∆WcT )1 (tr ), . . . , (∆WcT ) j (tr ), . . . , (∆WcT )m (tr )] (∆WcT ) j (tr ) = [0, . . . , (∆Wc ) j,q j (tr ), . . . , 0], q j ∈ [1, . . . , (Q c ) j ]. Since (Sb,i ) j,` j (e, x) is bounded by construction and (φb,i ) j,` j (ν) is bounded by definition, it is concluded that |(∆Wb,i ) j,` j (tr )| < (∆W b,i ) j , j = 1, . . . , m. Accordingly, we have |(∆Wb,i )(tr )| < (∆W b,i ) = max j=1,...,m {(∆W b,i ) j }. Similarly, owing to the boundedness of Sc (e) and G (k) (x), ∀x ∈ Ω the elements (Sc (e)G (k) (x)) j,q j are bounded. Therefore, we have |(∆Wc ) j,q j (tr )| < (∆W c ) j , j = 1, . . . , m, and |(∆Wc )(tr )| < (∆W c ) = max j=1,...,m {(∆W c ) j }. Hence, the new weights attained immediately after the resetting instant, (Wb,i )(tr+ ) and (Wc )(tr+ ) are bounded. Furthermore, since any bounded and positive function (φc ) j,q j (ν) suffices to guarantee boundedness of (Wc )(tr+ ), we can always select the upper bound of (φc ) j,q j (ν) at each t = tr to further guarantee that (Wc )(tr+ ) ∈ Wc . Remark 2. Owing to the weight changes imposed at t = tr , an instantaneous jump is induced in the Lyapunov function: L(tr+ ) = L(tr ) + ∆L(tr ). Assuming that e(tr+ ) = e(tr ) then ∆L(tr ) =
r 1X {|W˜ b,i (tr+ )|2 − |W˜ b,i (tr )|2 } 2 i=1
1 + {|W˜ c (tr+ )|2 − |W˜ c (tr )|2 } 2 ( ) r 1 X 2 2 ≤ |∆Wb,i (tr )| + |∆Wc (tr )| 2 i=1 ( ) r 1 X 2 2 < (∆W b,i ) + (∆W c ) = ∆L. 2 i=1
× sgn{(Sc (e)G (k) (x)) j,q j }(Sc (e)G (k) (x)) j,q j (φc ) j,q j (ν) = sgn{b j (tr )}[|b j (tr )| + ςi |(Sb,i ) j,` j |(φb,i ) j,` j (ν) + |(Sc (e)G (k) (x)) j,q j |(φc ) j,q j (ν)].
(34)
Since the resetting law assures that sgn{b j (tr+ )} = sgn{b j (tr )}, it is concluded that for j = 1, . . . , m: |b j (tr+ )| = |b j (tr )| + ζi |(Sb,i ) j,` j |(φb,i ) j,` j (ν) + |(Sc (e)G (k) (x)) j,q j |(φc ) j,q j (ν) > |b j (tr )|. (35)
Remark 3. Consider now the time interval [t1 , t2 ] = [tr+ (v), tr (v + 1)] ∈ [0, ∞), starting at tr+ and ending at tr (v + 1), the time instant of the subsequent resetting. Then ∀t ∈ [t1 , t2 ] the weight update laws (18)–(20) can be applied and the control law is provided by (25). Owing to the projection operators and the boundedness of the weights at tr+ (v), the signals a(e, x, Wa , Wc ) and u are bounded, as indicated by Lemma 1. As a result, all stabilizing attributes of Theorem 1 are valid, ∀t ∈ [t1 , t2 ], which ensures that the tracking error e is uniformly ultimately bounded with respect to the arbitrarily
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small set E. The entire course over time of the suggested control approach can be viewed as a success of time intervals [t1 , t2 ], determined by the resetting instants, with instantaneous weight changes at the initiating times and guaranteed operation for the intermediate stages. Under these conditions, the time evolution of the Lyapunov function is piecewise continuous. Remark 4. The only issue left that may become a cause of instability, is that the weights (Wb,i ) should be maintained bounded throughout the whole control process. To guarantee that the proposed resetting scheme may not drive the weights (Wb,i ) to infinity, owing to possibly infinite resettings, we argue that since (Wb,i ) j,l (0) ∈ L ∞ and (Wb,i ) j,` (t) ∈ L ∞ , P∀t ∈ [t1 , t2 ] ⊂ [0, ∞), it suffices to show ∞ that ν=1 sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` }(φb,i ) j,` (v) ∈ L ∞ . However, this is true owing to the boundedness of (φb,i ) j,` (v), ∀t ≥ 0, and the fact that limν→∞ (φb,i ) j,` (v) = 0. Accordingly, the weights Wc , are restricted to the bounded convex set Wc under all operating conditions, a task achieved through the projection operation Pc {·} in (20). Even in the presence of a resetting, we have argued in Remark 4 that we can always select the upper bound of φc (at each t = tr ) to guarantee that (Wc )(tr+ ) ∈ Wc . Additionally, the overall increase of the weights (Wc ) j,q j is also bounded against infinite resettings, P∞ (k) since ∈ ν=1 sgn{b j (tr )}sgn{(Sc (e)G (x)) j,q }(φc ) j,q (v) L ∞ which holds due to the boundedness of (φc ) j,q (v) and its convergence as ν → ∞ to zero. Remark 5. The developed resetting scheme can be visualized as an improved modification of the one first proposed in Rovithakis (2001) since: (i) it guarantees stability even in presence of infinite resettings, (ii) it can be easily combined with other robust adaptive modifications, namely the projection modification, (iii) it has a less complex structure, leading to easier implementation. Stage 3. Based on the definitions of γ (|e|) and V (e), there exists a positive but unknown constant cγ such that γ (|e|) ≥ cγ V (e), ∀e ∈ Ω ⊂ Rn . Defining ca = mini=1,...,r {ka,i } and cb = mini=1,...,r {kb,i } we obtain from (14), (27) that r ca − cγ X |W˜ a,i |2 L˙ ≤ −cγ L + 2 i=1 X r cγ − cb cγ − k c 2 ˜ |Wb,i | + + |W˜ c |2 + µ. 2 2 i=1 ˙ Thus selecting, c = cγ = min{ca , cb , kc } we arrive at L(t) ≤ −cL(t) + µ, and hence µ n µ o −ct 0 ≤ L(t) ≤ + L(0) − e , ∀t ≥ 0. (38) c c The stability properties are valid, assuming that e ∈ Ω , with Ω ⊂ Rn Also recall that due to Theorem 1, e is uniformly ultimately bounded with respect to set E. Hence, starting with e(0) ∈ E, the tracking error will remain consistently within this set, i.e. e(t) ∈ E ⊂ Ω , ∀t ≥ 0. Consider now the case where e(0) ∈ Ω , while satisfying the condition e(0) 6∈ E. Under these circumstances, we distinguish the following cases:
Case 1. L(0) > µ/c. In that case, based on the definition of L(t) and (38), we have kV (e) ≤ L(t) ≤ L(0), ∀t ≥ 0. Hence, it can be easily verified that there exists a gain k such that ( r 1 X E1 = e ∈ Rn : |e| ≤ V −1 V (0) + |W˜ a,i (0)|2 2k i=1 !) r 1 X 1 2 2 |W˜ b,i (0)| + |W˜ c (0)| + ⊂ Ω. (39) 2k i=1 2k Case 2. L(0) ≤ µ/c. In that case, the following condition holds kV (e) ≤ L(t) ≤ µ/c and the tracking error satisfies µ |e| ≤ V −1 { kc }. Hence, there also exists a constant k making µ o n E2 = e ∈ Rn : |e| ≤ V −1 ⊂ Ω. (40) kc Theorem 2. Consider the error system described by (6), where Assumptions 1 and 2 and Lemma 1 are satisfied. The control law (25) and the weight updates (18)–(20), supported by the resetting scheme (32)–(33) ensure the uniform ultimate boundedness of the tracking error e with respect to an arbitrary small set (30), plus the boundedness of all other signals in the closed loop. 4. Simulation results To demonstrate the suggested adaptive neuro-fuzzy control approach, we consider a highly nonlinear MIMO system, namely, a two-link robot manipulator. The plant dynamics are described by q¨ = −M(q)−1 [C(q, q) ˙ q˙ + G(q)] + M(q)−1 τ, q = [q1 , q2 ]T α1 + 2α2 c2 α3 + α2 c2 M(q) = α 3 + α 2 c2 α3 −q˙2 (q˙1 + q˙2 ) C(q, q) ˙ = α2 s2 , q˙1 0
(41)
g G(q) = 1 g2
where α1 = Iz 1 + Iz 2 + m 1 `2c1 + m 2 (`21 + `2c2 ),
α2 = m 2 `1 `c2
α3 = Iz 2 + m 2 `2c2 , `c1 = `1 /2, `c2 = `2 /2 g1 = m 1 g`c1 c1 + m 2 g(`1 c1 + `c2 c12 ), g2 = m 2 g`c2 c12 , q1 , q2 (rad) are the angle positions of the joints 1 and 2 constrained within [−(π/2), (π/2)]. M(q) is the moment of inertia, C(q, q) ˙ includes coriolis and centripetal forces, and G(q) is the gravitational force. Other quantities denote: link masses m 1 , m 2 (kg), link lengths `1 , `2 (m), applied torques τ1 , τ2 (N-m), acceleration due to gravity g = 9.8 m/s2 , and shorthand notations c1 = cos(q1 ), s2 = sin(q2 ), c2 = cos(q2 ), and c12 = cos(q1 + q2 ). The parameter values used in our simulations are: m 1 = m 2 = 0.32, `1 = `2 = 0.3, Iz 1 = Iz 2 = 97.63 × 10−4 . The system states are taken as x1 = q1 , x2 = q2 , x3 = q˙1 , and x4 = q˙2 . The motion dynamics (41) are modelled by fuzzy rules with nonlinear consequents of the form (2). To minimize model and
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Fig. 1. Membership functions along the x1 , x2 premise variables of the T–S fuzzy model.
design complexity only the states x1 and x3 are considered in the antecedent part. Each premise variable is partitioned using three fuzzy sets (Fig. 1), leading to a fuzzy model with r = 9 rules. We assume that no a priori knowledge is available, i.e. (k) (k) f i (x) = G i (x) ≡ 0, i = 1, . . . , 9. The objective is to design a stable control law for τ1 (t) and τ2 (t), such that q1 and q2 are forced to follow consistently a desired trajectory: xr,1 (t) = 2π/9 + π/9 cos(π t), xr,2 (t) = −π/9 cos(π t). (u) (u) To accomplish our goal, the terms Ai (e, x), Bi (e, x) and C(e) are approximated via LWNNs. We develop common vectors Sa,i (e, x) ∈ R Q a,i and Sb,i (e, x) ∈ R Q b,i for all rules, i = 1, . . . , 9, comprising Q a,i = 16 and Q b,i = 12, respectively. Let diagq [z]denote a q × q diagonal matrix with q identical entries equal to z. The regressor vectors are given by T Sa,i (e, x) = S1T (e)Sa (x)
S1T (e) = [se (e1 ), se (e2 ), se (e3 ), se (e4 )] Sa (x) = [diag4 (s1 (x1 )), diag4 (s2 (x2 )), diag4 (s3 (x3 )), diag4 (s3 (x4 ))] T Sb,i (e, x) = S1T (e) · Sb (x) + S2T (e) · [diag4 (s2 (x2 ))]
Sb (x) = [diag4 (s2 (x2 )), diag4 (s22 (x2 ))] S2T (e) = [se2 (e1 ), se2 (e2 ), se2 (e3 ), se2 (e4 )].
Fig. 2. The trajectories of the state variables (solid lines) and the reference signals (dotted lines): (x1 , xr,1 ) (a), (x2 , xr,2 ) (b), (x3 , x˙r,1 ) (c), and (x4 , x˙r,2 ) (b).
Further, Sc (e) ∈ R Q c ×4 contains a total of 20 terms (Q c = 5), described as: Sc (e) = [c1 , c2 , c3 , c4 ], where c1T = [se2 (e1 ), se (e1 ), se (e1 )se (e3 ), se2 (e2 )se (e4 ), se2 (e1 )se (e2 )] c2T = [se2 (e2 ), se (e2 ), se (e2 )se (e4 ), se2 (e4 )se (e2 ), se2 (e2 )se (e1 )] c3T = [se2 (e3 ), se (e3 ), se2 (e1 )se (e3 ), se2 (e1 )se (e2 ), se2 (e3 )se (e4 )] c4T = [se2 (e4 ), se (e4 ), se2 (e3 )se (e1 ), se (e3 )se (e4 ), se2 (e4 )se (e2 )]. The sigmoid functions are defined as follows: s1 (x) = s2 (x) =
2π − (π − 0.5), 1 + e−x
2π/3 − π/3, s3 (x3 ) = 1 + e−x3 2π/3 − π/3. se (e) = 1 + e−(e−0.001) The control law is obtained by applying (25), with γ (|e|) = (1 − e−0.1 |e| ). The weight values are updated using (18)–(20), and ψ = −x˙r . The learning rates are chosen as ka,i = ka = 0.001, kb,i = kb = 0.001, i = 1, . . . , 9, kc = 0.001, while k = 0.5. The system states are initialized at (x1 (0), x2 (0), x3 (0), x4 (0))T = (1.1363, 1.1363, 0.4, 0.4)T . Further, the initial weight values are Wa,i = row16 [−5] ∈ R16 , Wb,i = row12 [−1] ∈ R12 , i = 1, . . . , 9, Wc = row5 [−2] ∈ R5 , where
Fig. 3. Evolution of |b(e, x, Wb , Wc )|.
rowq [z] denotes a row vector with q identical entries equal to z. The aforementioned LWNNs used as well as the determination of the design constants were selected according to a trial and error procedure. Notice though that Theorem 1 provides guidelines for a qualitative selection. The performance of the suggested neuro-fuzzy control scheme is demonstrated in Figs. 2 and 3. In Fig. 2, the state trajectories are plotted along with their respective reference signals xr and x˙r . Apparently, the tracking error diminishes almost exponentially fast. Fig. 3 depicts the evolution of the signal b(e, x, Wb , Wc ) which illustrates that resetting
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guarantees |b(e, x, Wb )| > 0, ∀t ≥ 0. In our simulations, 1 we have selected δb = 0.008 and ϕ(v) = v − 4 . Whenever |b(e, x, Wb , Wc )| − δb = 0, the resetting procedure is activated, forcing |b(e, x, Wb , Wc )| to admit values greater that δb . 5. Conclusions A novel control design is proposed for tracking affine in the control dynamic systems, with unknown nonlinearities. The plant is described by a T–S fuzzy model with nonlinear consequent parts. A rigorous Lyapunov-based stability analysis is presented, providing the network weights’ update laws and the derivation of the control law. A novel resetting scheme is also developed, ensuring the validity of the control input and uniform ultimate boundedness of the tracking error. References Ioannou, P. A., & Sun, J. (1996). Robust adaptive control. Upper Saddle Press NJ, USA: Prentice Hall. Koo, T. J. (2001). Stable model reference adaptive fuzzy control of a class of nonlinear systems. IEEE Transactions on Fuzzy Systems, 9(4), 624–636. Lee, H., & Tomizuka, M. (2000). Robust adaptive control using a universal approximator for SISO nonlinear systems. IEEE Transactions on Fuzzy Systems, 8(1), 95–106. Lewis, F. L, Liu, K., & Yesildirek, A. (1995). Neural net robot controller with guaranteed tracking performance. IEEE Transactions on Neural Networks, 6(3), 703–715. Ordonez, R., & Passino, K. (1999). Stable multi-input multi-output adaptive fuzzy/neural control. IEEE Transactions on Fuzzy Systems, 7(3), 345–353. Polycarpou, M. M. (1996). Stable adaptive neural control scheme for nonlinear systems. IEEE Transactions on Automatic Control, 41(3), 447–451. Rovithakis, G. A. (2001). Stable adaptive neuro-control design via Lyapunov function derivative estimation. Automatica, 37(8), 1213–1221. Rovithakis, G. A. (2004). Robust redesign of a neural network controller in the presence of unmodeled dynamics. IEEE Transactions on Neural Networks, 15(6), 1482–1490. Sontag, E. D. (1989). A universal construction of Artstein’s theorem on nonlinear stabilization. Systems Control Letters, 13, 117–123. Spooner, J. T., & Passino, M. (1996). Stable adaptive control using fuzzy systems and neural networks. IEEE Transactions on Fuzzy Systems, 4(3), 339–359. Tseng, C. S., Chen, B.-S., & Uang, H.-. J (2001). Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model. IEEE Transactions on Fuzzy Systems, 9(3), 381–392.
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Xu, H., & Ioannou, P. A. (2003). Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds. IEEE Transactions on Automatic Control, 48(5), 728–742. S.P. Moustakidis was born in Thessaloniki, Greece, in 1981. He received a degree in electrical engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2004. He is currently working toward a Ph.D. degree at the Department of Electrical and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. His research interests include fuzzy logic systems, support vector machines, wavelet analysis, control with neural networks and genetic algorithms. G.A. Rovithakis was born in Chania, Crete, Greece in 1967. He received the diploma in Electrical Engineering from the Aristotle University of Thessaloniki, Greece in 1990 and the M.S. and Ph.D degrees in Electronic and Computer Engineering both from the Technical University of Crete, Greece in 1994 and 1995 respectively. After holding a visiting Assistant Professor position with the Department of Electronic and Computer Engineering, Technical University of Crete from 1995 to 2002, he joints the Aristotle University of Thessaloniki where he is currently an Assistant Professor with the Department of Electrical and Computer Engineering. His research interests include nonlinear systems, neural network systems, robust adaptive control, identification-control of unknown systems using neural networks, control of computer networks, QoS control, production control, intelligent control, fault detection, isolation and accommodation in nonlinear dynamical systems, automated inspection systems where he has authored or co-authored over 90 publications in scientific journals, referred conference proceedings and book chapters. He has also co-authored the book Adaptive Control with Recurrent High-Order Neural Networks (London UK: Springer-Verlag, 2000). Dr Rovithakis serves as a reviewer for various journals and conferences and has served as session chairman or co-chairman in international conferences. He is a Senior Member of IEEE, Member of the IEEE Control Systems Society Conference Editorial Board and a Member of the Technical Chamber of Greece. J.B. Theocharis (M’90) received a degree in electrical engineering and the Ph.D. degree from Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1980 and 1985, respectively. He is currently an Associate Professor in the Department of Electronic and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. His research activities include fuzzy systems, neural networks, neuron-fuzzy modelling, time-series prediction, recurrent fuzzy neural networks, evolutionary algorithms and adaptive control.
Brief paper
An adaptive neuro-fuzzy tracking control for multi-input nonlinear dynamic systemsI S.P. Moustakidis, G.A. Rovithakis ∗ , J.B. Theocharis Aristotle University of Thessaloniki, Department of Electrical and Computer Engineering, 54124, Thessaloniki, Greece Received 28 September 2006; received in revised form 13 September 2007; accepted 10 October 2007 Available online 10 March 2008
Abstract An adaptive neuro-fuzzy control design is suggested in this paper, for tracking of nonlinear affine in the control dynamic systems with unknown nonlinearities. The plant is described by a Takagi–Sugeno (T–S) fuzzy model, where the local submodels are realized through nonlinear dynamical input–output mappings. Our approach relies upon the effective approximation of certain terms that involve the derivative of the Lyapunov function and the unknown system nonlinearities. The above task is achieved locally, using linear in the weights neural networks. A novel resetting scheme is proposed that assures validity of the control input. Stability analysis provides the control law and the adaptation rules for the network weights, assuring uniform ultimate boundedness of the tracking and the signals appearing in the closed-loop configuration. Illustrative simulations highlight the approach. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Adaptive control; Neuro-fuzzy control; Resetting
1. Introduction Neural network and fuzzy systems-based adaptive control methodologies are receiving considerable attention, emerging as promising approaches for controlling highly uncertain and nonlinear dynamical systems. In the realm of neural networks, many controllers have been suggested (Lewis, Liu, & Yesildirek, 1995; Polycarpou, 1996; Rovithakis, 2001, 2004; Xu & Ioannou, 2003). Although stable tracking is guaranteed, these approaches cannot incorporate linguistic system descriptions within their framework. Recently, several stable adaptive fuzzy control schemes have been introduced based on Takagi–Sugeno (T–S) modeling (Koo, 2001; Lee & Tomizuka, 2000; Ordonez & Passino, 1999; Spooner & Passino, 1996). The reasoning behind this growing research is the capability of adaptive schemes to deal with large parametric uncertainties, while on the other hand, taking advantage of the qualitative knowledge description of the fuzzy systems, in I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ra´ul Ord´on˜ ez under the direction of Editor Miroslav Krstic. ∗ Corresponding author. E-mail address: [email protected] (G.A. Rovithakis).
c 2008 Elsevier Ltd. All rights reserved. 0005-1098/$ - see front matter doi:10.1016/j.automatica.2007.10.019
the form of If/Then linguistics. Unfortunately, they all impose restrictive growing assumptions on systems nonlinearities, as well as they require the boundedness away from zero of a norm of the control input vector field. However, for general multiinput affine in the control nonlinear dynamic systems, such estimates may change signs frequently, or even worse, they may be in a neighbourhood of the origin despite the fact that their corresponding actual values may indeed be far from zero, thus making the estimation model uncontrollable. In this paper, we relax the aforementioned restrictions, developing a stable adaptive neuro-fuzzy controller with robust tracking characteristics, for multi-input nonlinear affine in the control dynamic systems with unknown nonlinearities. Compared to prior research, the distinctions of this work are described as follows: (1) The system dynamics are described by a T–S fuzzy model where the consequent parts of the rules are represented by unknown nonlinear dynamic subsystems. Hence, a better approximation of the actual system dynamics is achieved compared to the conventional T–S models with linear consequent submodels. (2) Instead of modelling the plant dynamics directly, our method relies upon the effective approximation of certain terms that involve the derivative of an unknown Lyapunov function and the unknown system
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nonlinearities, using linear in the weights neural network structures. (3) A novel resetting scheme is proposed to assure the boundedness away from zero of certain signals, thus guaranteeing the validity of the control input. Uniform ultimate boundedness of the tracking error with respect to an arbitrarily small neighbourhood of zero, plus the boundedness of all other signals in the closed loop, is guaranteed. The control design is general, assuming partially known systems. Nevertheless, completely unknown systems can be handled by setting the known parts to zero, throughout the design process. The rest of the paper is organized as follows. In Section 2, we describe the plant dynamics in terms of T–S dynamic fuzzy rules and formulate the problem. In Section 3 the suggested adaptive neuro-fuzzy control is presented while in Section 4, simulation results are provided, to demonstrate the effectiveness of the method. Finally, conclusions are drawn in Section 5.
final state of the T–S fuzzy model is obtained by x˙ =
r X
(k)
(k)
(u)
ςi (x){[ f i (x) + G i (x)u] + [ f i
(u)
(x) + G i (x)u]}
i=1
(4) where ςi (x) ∈ [0, 1] are the normalized firings of the rules: r Q
ςi (x) =
n X
j=1
µ F i (x j ) j
n P
r Q
i=1
j=1
!,
ςi (x(t)) ≥ 0,
µ F i (x j ) j
ςi (x(t)) = 1.
(5)
i=1
Consider affine in the control nonlinear dynamic systems, of the form
The control objective is to devise a control law u(t) such that the system state x(t) is forced to follow a given bounded reference trajectory xr (t). For xr (t), we further assume that x˙r (t) is also bounded, i.e. |x˙r | ≤ dr , where dr is a known upper bound. With the tracking error defined as e(t) = x(t) − xr (t), the error dynamics are described by
x˙ = f (x) + G(x)u
e˙ =
2. Problem formulation
(1)
where x ∈ Rn denotes the system’s state which is assumed to be completely measurable, u ∈ Rm is the control input, f ∈ Rn and G ∈ Rn×m are continuous, locally Lipschitz, vector fields. T–S fuzzy models provide an effective framework for representing complex multi-input multi-output (MIMO) systems (1). Extending the traditional approach of most fuzzy control schemes that use T–S rules with linear consequent models (Tseng, Chen, & Uang, 2001), we consider here a T–S fuzzy dynamic fuzzy model which is composed of generalized If /Then rules where the local models are represented by nonlinear dynamic subsystems of simpler structure: R (i) : IF x1 (t) is F1i AND . . . AND xn (t) is Fni THEN x(t) ˙ = f i (x) + G i (x)u
i = 1, . . . , r
(2)
where r is the number of rules, and F ji , i = 1, . . . , r , j = 1, . . . , n are fuzzy sets associated with the input x j (t) and quantified by a membership function µ F i (x j ) ∈ [0, 1]. j f i (x), G i (x), i = 1, . . . , r are assumed to be unknown and nonlinear. Owing to the nonlinear nature of the consequent functions, (2) approximates the local dynamics of the system more effectively compared to T–S rules with linear functions (x(t) ˙ = Ai x + Bi u, Ai ∈ Rn×n , Bi ∈ Rn×m ). We consider that the consequent functions are partially known, which allows embedding a priori knowledge. (k)
(u)
f i (x) = f i (x) + f i
(x),
(k)
(k)
(u)
G i (x) = G i (x) + G i (x) (3) (k)
(u)
(u)
for i = 1, . . . , r . f i (x), G i (x) and f i (x), G i (x) represent the known and the unknown plant dynamics, respectively. For a bounded state vector x, it is assumed that (k) (k) (u) (u) f i (x), G i (x) and f i (x), G i (x) are also bounded. The
r X
(k)
(k)
ςi (x){[ f i (x) + G i (x)u]
i=1 (u)
+ [ fi
(u)
(x) + G i (x)u]} − x˙r .
(6)
Assumption 1. The solution of (6) can be forced to be uniformly ultimately bounded with respect to an arbitrarily small neighbourhood of the origin, e = 0. Consider a set E, defined as E = {e ∈ Rn : 0 ≤ |e| ≤ e0 }, e0 being an arbitrarily small positive constant. Due to Assumption 1, there exists a radially unbounded Lyapunov function V (e) : Rn → R+ and an appropriate control u 0 (e, x), satisfying V˙ (e) ≤ 0 for all e lying outside E. Usually, the derivation of stabilizing feedback laws is based upon the explicit knowledge of the system’s structure and the Lyapunov function (Sontag, 1989). In our formulation though, we consider a highly uncertain environment, whereby the Lyapunov function, V (e), is assumed to be unknown. To cope with the highly uncertain conditions involved in our framework, we make use of linear in the weights neural networks (LWNNs). Mathematically, LWNNs are represented by y T = (W ∗ )T S(υ), where υ ∈ Rn 2 and y ∈ Rn 1 denote the approximator input and output respectively, W ∗ is a L-dimensional vector of synaptic weights, and S(υ) is a L × n 1 matrix of regressor terms. All LWNNs share the density property: For every continuous function f (υ) : Rn 2 → Rn 1 there exist an integer L and optimal weight values W ∗ such that for every ε > 0, supυ∈Ω | f (υ)T − (W ∗ )T S(υ)| ≤ ε, where Ω ⊂ Rn 2 is a given compact set. Since the Lyapunov function V (e) and the nonlinear terms (u) (u) f i (x), G i (x), i = 1, . . . , r are assumed unknown, we use
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S.P. Moustakidis et al. / Automatica 44 (2008) 1418–1425 ∗ , i = 1, . . . , r , Wc , are estimates of the unknown values Wk,i k = a, b, i = 1, . . . , r , and Wc∗ . Further, consider the Lyapunov function candidate
LWNNs to approximate the expressions: ∂ V (e) (k) ∂ V (e) (u) f i (x) + f (x) ∂e ∂e i (k) (u) = Ai (e, x) + Ai (e, x)
(7)
∂ V (e) (k) ∂ V (e) (u) Bi (e, x) = G i (x) + G i (x) ∂e ∂e (k) (u) = Bi (e, x) + Bi (e, x)
(8)
C(e) = ∂ V (e)/∂e
(9)
Ai (e, x) =
with i = 1, . . . , r , Ai (e, x) ∈ R, Bi (e, x) ∈ R1×m , C(e) ∈ R1×n . More specifically, the term (9) devoted to the estimation of the Lyapunov function derivative is modelled as: C(e) = (Wc∗ )T Sc (e) + ωc (e), while terms related to the known parts are approximated by (k)
(k)
(k)
(k)
(k)
(k)
Ai (e, x) = (Wc∗ )T Sc (e) f i (x) + ωc (e) f i (x) Bi (e, x) = (Wc∗ )T Sc (e)G i (x) + ωc (e)G i (x). (k) f i (x)
L = kV (e) +
r r 1X 1X 1 |W˜ a,i |2 + |W˜ b,i |2 + |W˜ c |2 2 i=1 2 i=1 2
(14)
where V (e) denotes the unknown Lyapunov function, and k > 0 is a design constant. Differentiating L with respect to time, using (6) and the terms substitutions defined in Section 2, we obtain r X (k) (k) L˙ = k ςi (x){Ai (e, x) + Bi (e, x)u} i=1
+k
r X
(u)
(u)
ςi (x){Ai (e, x) + Bi (e, x)u} − kC(e)x˙r
i=1
(10) + (11)
(k) G i (x)
Since and are known expressions, the (k) (k) unknown term ∂ V (e)/∂e in Ai (e, x),Bi (e, x) and C(e) are approximated using a common LWNN. Furthermore, for the completely unknown parts we obtain
r X
{(W˜ a,i )T W˙ a,i + (W˜ b,i )T W˙ b,i } + (W˜ c )T W˙ c
i=1
=k
r X
∗ T ∗ T ςi (x){(Wa,i ) Sa,i (e, x) + (Wb,i ) Sb,i (e, x)u}
i=1 r X
+
{(W˜ a,i )T W˙ a,i + (W˜ b,i )T W˙ b,i } + (W˜ c )T W˙ c
i=1 (u) Ai (e, x)
∗ T (Wa,i ) Sa,i (e, x) + ωa,i (e)
(12)
∗ T Bi (e, x) = (Wb,i ) Sb,i (e, x) + ωb,i (e)
(13)
=
(u)
where Sa,i (e, x) ∈ R L a,i , Sb,i (e, x) ∈ R L b,i ×m , i = 1, . . . , r , and Sc (e) ∈ R L c ×n are suitably selected regressors. Further, ∗ ∈ R L a,i , W ∗ ∈ R L b,i , i = 1, . . . , r and W ∗ ∈ R L c are Wa,i c b,i unknown weight values. Finally, ωa,i (e), ωb,i (e), i = 1, . . . , r , and ωc (e) denote modelling error terms.
+ kω(e, u) + k(Wc∗ )T Sc (e) ψ (15) P (k) (k) where ψ = ri=1 ζi (x)[ f i (x) + G i (x)u] − x˙r . The term ω(e, u) denotes the generalized modelling error, defined as ω(e, u) =
r X
ςi (x)ωa,i (e) +
i=1
r X
ςi (x)ωb,i (e)u + ωc (e)ψ.
i=1
(16) Pr
Assumption 2. Owing to the density property of the LWNNs, we assume that the modelling errors are bounded in a compact region Ω ⊂ Rn , i.e. |ωa,i (e)| ≤ ω¯ a,i , |ωb,i (e)| ≤ ω¯ b,i , i = 1, . . . , r , and |ωc (e)| ≤ ω¯ c , ω¯ a,i , ω¯ b,i , i = 1, . . . , r , and ω¯ c , being unknown but small positive constants. 3. Adaptive neuro-fuzzy control design In this section, a systematic methodology is presented for the design of stable adaptive neuro-fuzzy controllers following a three-stages approach. At stage 1, the control law and the weight adaptation rules are developed, guaranteeing the uniform ultimate boundedness of the tracking error with respect to an arbitrary small set around the origin. Additionally, the boundedness of all signals involved in the closed loop control configuration is ensured. The above qualities are fulfilled provided that the denominator of the control signal is bounded away zero. At stage 2, a novel resetting scheme is introduced, performing on the weight estimates Wb,i , i = 1, . . . , r and Wc , to guarantee the validity of the control law. Finally, at stage 3, it is demonstrated that e ∈ Ω ⊂ Rn , ∀t ≥ 0. ∗ , Stage 1. Consider the parameter errors W˜ k,i = Wk,i − Wk,i k = a, b, i = 1, . . . , r , W˜ c = Wc − Wc∗ , where Wk,i , k = a, b,
After k i=1 ςi (x)(Wa,i )T Sa,i (e, x), Pr adding and subtracting T k i=1 ςi (x)(Wb,i ) Sb,i (e, x)u and k(Wc )T Sc (e)e˙(k) , (15) is reformulated as r X L˙ = −k ςi (x)(W˜ a,i )T Sa,i (e, x) −k
i=1 r X
ςi (x)(W˜ b,i )T Sb,i (e, x)u − k(W˜ c )T Sc (e)e˙(k)
i=1
+k
r X
ςi (x)(Wa,i )T Sa,i (e, x) − k(Wc )T Sc (e)e˙(k)
i=1
+k
r X
ςi (x)(Wb,i )T Sb,i (e, x)u + (W˜ c )T W˙ c
i=1
+
r X i=1
(W˜ a,i )T W˙ a,i +
r X
(W˜ b,i )T W˙ b,i + kω(e, u).
(17)
i=1
Define the weight update rules. W˙ a,i = Pa,i {−ka,i Wa,i + kςi (x)Sa,i (e, x)}, W˙ b,i = −kb,i Wb,i + kςi (x)Sb,i (e, x)u, Wc = Pc {−kc Wc + k Sc (e)ψ}
i = 1, . . . , r (18)
i = 1, . . . , r
(19) (20)
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where ka,i , kb,i > 0, i = 1, . . . , r , and kc > 0 are constants determined by the designer and Pa,i , Pc are projection operators with respect to the convex sets Wa,i = {Wa,i ∈ R L a,i : |Wa,i | ≤ Ma,i }, Wc = {Wc ∈ R L c : |Wc | ≤ Mc }. All Ma,i , i = 1, . . . , r , Mc are chosen to take sufficiently large positive values. Definitions of the projection operators can be found in Ioannou and Sun (1996). Moreover, it is well known that projection modification can only make the derivative of the Lyapunov function more negative provided ∗ , W (0) ∈ M , i = 1, . . . , r and W ∗ , W (0) ∈ that Wa,i a,i a,i c c Ma,i . Hence, L˙ becomes L˙ ≤ ka(e, x, Wa , Wc , x˙r ) + kb(e, x, Wb , Wc )u + kω(e, u) r r X X − ka,i (W˜ a,i )T Wa,i − kb,i (W˜ b,i )T Wb,i i=1
i=1
(21)
where we have defined a(e, x, Wa , Wc , x˙r ) =
ςi (x)(Wa,i )T Sa,i (e, x)
i=1
b(e, x, Wb , Wc ) =
(22)
ςi (x)(Wb,i ) Sb,i (e, x) T
i=1
+ (Wc )T Sc (e)G (k) (x).
(23)
Recalling that for any z, z˜ ∈ Rn , with z˜ = z − z ∗ , we have z˜ T z = 12 |˜z |2 + 12 |z|2 − 12 |z ∗ |2 and after dropping the P P negative terms − ri=1 ka,i /2|Wa,i |2 , − ri=1 kb,i /2|Wb,i |2 , −kc /2|Wc |2 we obtain L˙ ≤ ka(e, x, Wa , Wc , x˙r ) + kb(e, x, Wb , Wc )u + kω(e, u) r r X X ka,i kb,i kc ∗ 2 ∗ 2 |Wa,i |Wb,i + | + | + |Wc∗ |2 2 2 2 i=1 i=1 −
r X ka,i i=1
2
|W˜ a,i |2 −
r X kb,i i=1
2
kc |W˜ b,i |2 − |W˜ c |2 . 2
(24)
Let us now consider the control law, a(e, x, Wa , Wc , x˙r ) + γ (|e|) T u=− b (e, x, Wb , Wc ) (25) |b(e, x, Wb , Wc )|2 where γ (|e|) is a positive, bounded and invertible function, ∀e ∈ Ω ⊂ Rn . We assume for the moment that |b(e, x, Wb , Wc )| is bounded away from zero. Stated otherwise, there is a small bound δb such that |b(e, x, Wb , Wc )| > δb ,
r X ka,i i=1
2
|W˜ a,i |2 −
r X kb,i i=1
2
|W˜ b,i |2
kc ˜ 2 | Wc | + µ 2 ≤ −kγ (|e|) + µ
(27)
−
(28) ka,i i=1 2
kb,i i=1 2
Pr
∗ |2 + ∗ |2 + where µ = k ω¯ + |Wa,i |Wb,i kc ∗ 2 ˙ 2 |Wc | . Hence, since γ (|e|) is invertible, L ≤ 0 provided that
|e| > γ −1
+ (Wc )T Sc (e){ f (k) (x) − x˙r } r X
L˙ ≤ kγ (|e|) −
Pr
− kc (W˜ c )T Wc r X
The above lemma can be verified following similar steps as in Rovithakis (2001), and hence, the proof is omitted. Notice that (19) is in a bounded input bounded output (BIBO) form, with the driving term given by ςi (x)Sb,i (e, x)u. Since |Sb,i (e, x)| ≤ s¯b,i by construction, ςi (x) ∈ [0, 1], and u ∈ L ∞ from Lemma 1, the driving term is bounded, and hence, Wb,i ∈ L ∞ , i = 1, . . . , r . Substituting (25) into (24), L˙ becomes
∀e ∈ Ω ⊂ Rn .
(26)
The aforementioned condition will be fulfilled through a resetting mechanism developed at stage 2. Lemma 1. Provided that (26) holds, the following statements are valid: (1) The control law (25) is uniformly bounded, u ∈ L ∞ and (2) Owing to u ∈ L ∞ , the generalized modelling error (16) is uniformly bounded by an unknown but small bound (i.e. |ω(e, u)| ≤ ω, ¯ ∀e ∈ Ω ).
! r 1 X kc ∗ 2 ∗ 2 ∗ 2 × ω¯ + {ka,i |Wa,i | + kb,i |Wb,i | } + |Wc | . (29) 2k i=1 2k The above inequality suggests that the tracking error e is uniformly ultimately bounded with respect to a set E: ( r 1 X ∗ 2 [ka,i |Wa,i | E = e ∈ Rn : |e| ≤ γ −1 ω¯ + 2k i=1 ) kc ∗ 2 ∗ 2 (30) + kb,i |Wb,i | ] + |Wc | 2k which can be made arbitrarily small by properly selecting the γ function, as well as choosing adequately small values of ka,i , kb,i , i = 1, . . . , r , kc and a large enough k. Theorem 1. For the error dynamics described by (6), the control law (25) for which (26) holds, combined with the weight adaptation scheme (18)–(20), guarantee the uniform ultimate boundedness of the tracking error e with respect to the arbitrarily small set E, plus the uniform boundedness of all other signals in the closed loop. Stage 2. From the aforementioned analysis, it becomes apparent that there is a need to assure that the term |b(e, x, Wb , Wc )| that appears in the denominator of the control law (25) is bounded away from zero, in order to attain the stability attributes of the closed-loop system. In this subsection, a novel resetting scheme is suggested, assuring (26) ∀t ≥ 0. We assume that the system initiates, taking values x(0), e(0), and Wb,i (0), i = 1, . . . , r , Wc (0), outside the set B = {e, x ∈ Rn , Wb,i ∈ R L b,i : |b(e, x, Wb , Wc )| < δb }, where δb > 0 is a design constant. Apparently, as long as |b(e, x, Wb , Wc )| > δb is maintained, there is no need for any modification on Wb,i , i = 1, . . . , r and Wc . The weights are ordinarily updated through (18)–(20) and the control law is derived from (25). Under these circumstances, boundedness of the control signal is achieved, as suggested by Lemma 1. The problem arises whenever |b(e, x, Wb , Wc )| ≤ δb .
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The components of b(e, x, Wb , Wc ) ∈ R1×m are given by b j (e, x, Wb , Wc ) =
r X
T ςi (Wb,i ) j (Sb,i (e, x)) j
(Wb,i )(tr+ ) = (Wb,i )(tr ) + (∆Wb,i )(tr ),
i=1
+ (WcT ) j (Sc (e)G (k) (x)) j T (Wb,i ) j (Sb,i (e, x)) j
=
(Q b,i ) j X
(Wb,i ) j,` (Sb,i (e, x)) j,`
(31)
`=1
(WcT ) j (Sc (e)G (k) (x)) j =
Q c, j X
(Wc )(tr+ ) = T )(tr ) (∆Wb,i
i = 1, . . . , r
(Wc )(tr ) + (∆Wc )(tr ) =
(36) (37)
T )1 (tr ), . . . , [(∆Wb,i T T )m (tr )] (∆Wb,i ) j (tr ), . . . , (∆Wb,i
T (∆Wb,i ) j (tr ) = [0, . . . , (∆Wb,i ) j,` j (tr ), . . . , 0],
` j ∈ [1, . . . , (Q b,i ) j ],
(WcT ) j,` (Sc (e)G (k) (x)) j,`
`=1 T ) , (S (e, x)) (Q b,i ) j and (W T ) , (S (e) with (Wb,i j b,i j ∈ R c c j,` G (k) (x)) j ∈ R(Q c ) j , j = P 1, . . . , m. Assume that at T S (e, x) + t = tr we have |b(tr )| = | ri=1 ςi (x)Wb,i b,i T ) , (W ) , T (k) Wc Sc (e)G (x)| = δb . A single weight from (Wb,i j c j i = 1, . . . , r , j = 1, . . . , m changes its value, provided that it multiplies a nonzero sigmoid term, to guarantee that at t = tr+ |b(tr+ )| > |b(tr )|. We select the ` j th component (Wb,i ) j,` j T ) , ` of (Wb,i j j ∈ [1, . . . , (Q b,i ) j ], and the q j th component (Wc ) j,q j of (Wc ) j , q j ∈ [1, . . . , (Q c ) j ], and apply resetting according to
(Wb,i ) j,` j (tr+ ) = (Wb,i ) j,` j (tr ) + (∆Wb,i ) j,` j (tr ) (∆Wb,i ) j,` j (tr )
(32)
= sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` j }(φb,i ) j,` j (ν) (Wc ) j,q j (tr+ ) = (Wc ) j,q j (tr ) + (∆Wc ) j,q j (tr ) (∆Wc ) j,q j (tr ) = sgn{b j (tr )}sgn{(Sc
Remark 1. The new weights obtained at t = tr+ after applying the resetting changes are
(33) (e)G (k) (x))
j,q j }(φc ) j,q j (ν)
for i = 1, . . . , r , j = 1, . . . , m, where sgn{·} is the sign function. Furthermore, (φb,i ) j,` j (ν) and (φc ) j,q j (ν) are positive, bounded functions, sharing the property that limv→∞ (φb,i ) j,` (v) = 0 and limv→∞ (φc ) j,q (v) = 0, with ν being a positive integer, denoting the number of times the resetting procedure has been activated for a particular weight component. To guarantee that |b(tr+ )| > |b(tr )| it suffices to show that |b j (tr+ )| > |b j (tr )|, ∀ j = 1, . . . , m. Introducing (32)–(33) in (31), we obtain b j (tr+ ) = b j (tr ) + ςi sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` j } × (Sb,i (e, x)) j,` j (φb,i ) j,` j (ν) + sgn{b j (tr )}
(∆WcT )(tr ) = [(∆WcT )1 (tr ), . . . , (∆WcT ) j (tr ), . . . , (∆WcT )m (tr )] (∆WcT ) j (tr ) = [0, . . . , (∆Wc ) j,q j (tr ), . . . , 0], q j ∈ [1, . . . , (Q c ) j ]. Since (Sb,i ) j,` j (e, x) is bounded by construction and (φb,i ) j,` j (ν) is bounded by definition, it is concluded that |(∆Wb,i ) j,` j (tr )| < (∆W b,i ) j , j = 1, . . . , m. Accordingly, we have |(∆Wb,i )(tr )| < (∆W b,i ) = max j=1,...,m {(∆W b,i ) j }. Similarly, owing to the boundedness of Sc (e) and G (k) (x), ∀x ∈ Ω the elements (Sc (e)G (k) (x)) j,q j are bounded. Therefore, we have |(∆Wc ) j,q j (tr )| < (∆W c ) j , j = 1, . . . , m, and |(∆Wc )(tr )| < (∆W c ) = max j=1,...,m {(∆W c ) j }. Hence, the new weights attained immediately after the resetting instant, (Wb,i )(tr+ ) and (Wc )(tr+ ) are bounded. Furthermore, since any bounded and positive function (φc ) j,q j (ν) suffices to guarantee boundedness of (Wc )(tr+ ), we can always select the upper bound of (φc ) j,q j (ν) at each t = tr to further guarantee that (Wc )(tr+ ) ∈ Wc . Remark 2. Owing to the weight changes imposed at t = tr , an instantaneous jump is induced in the Lyapunov function: L(tr+ ) = L(tr ) + ∆L(tr ). Assuming that e(tr+ ) = e(tr ) then ∆L(tr ) =
r 1X {|W˜ b,i (tr+ )|2 − |W˜ b,i (tr )|2 } 2 i=1
1 + {|W˜ c (tr+ )|2 − |W˜ c (tr )|2 } 2 ( ) r 1 X 2 2 ≤ |∆Wb,i (tr )| + |∆Wc (tr )| 2 i=1 ( ) r 1 X 2 2 < (∆W b,i ) + (∆W c ) = ∆L. 2 i=1
× sgn{(Sc (e)G (k) (x)) j,q j }(Sc (e)G (k) (x)) j,q j (φc ) j,q j (ν) = sgn{b j (tr )}[|b j (tr )| + ςi |(Sb,i ) j,` j |(φb,i ) j,` j (ν) + |(Sc (e)G (k) (x)) j,q j |(φc ) j,q j (ν)].
(34)
Since the resetting law assures that sgn{b j (tr+ )} = sgn{b j (tr )}, it is concluded that for j = 1, . . . , m: |b j (tr+ )| = |b j (tr )| + ζi |(Sb,i ) j,` j |(φb,i ) j,` j (ν) + |(Sc (e)G (k) (x)) j,q j |(φc ) j,q j (ν) > |b j (tr )|. (35)
Remark 3. Consider now the time interval [t1 , t2 ] = [tr+ (v), tr (v + 1)] ∈ [0, ∞), starting at tr+ and ending at tr (v + 1), the time instant of the subsequent resetting. Then ∀t ∈ [t1 , t2 ] the weight update laws (18)–(20) can be applied and the control law is provided by (25). Owing to the projection operators and the boundedness of the weights at tr+ (v), the signals a(e, x, Wa , Wc ) and u are bounded, as indicated by Lemma 1. As a result, all stabilizing attributes of Theorem 1 are valid, ∀t ∈ [t1 , t2 ], which ensures that the tracking error e is uniformly ultimately bounded with respect to the arbitrarily
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small set E. The entire course over time of the suggested control approach can be viewed as a success of time intervals [t1 , t2 ], determined by the resetting instants, with instantaneous weight changes at the initiating times and guaranteed operation for the intermediate stages. Under these conditions, the time evolution of the Lyapunov function is piecewise continuous. Remark 4. The only issue left that may become a cause of instability, is that the weights (Wb,i ) should be maintained bounded throughout the whole control process. To guarantee that the proposed resetting scheme may not drive the weights (Wb,i ) to infinity, owing to possibly infinite resettings, we argue that since (Wb,i ) j,l (0) ∈ L ∞ and (Wb,i ) j,` (t) ∈ L ∞ , P∀t ∈ [t1 , t2 ] ⊂ [0, ∞), it suffices to show ∞ that ν=1 sgn{b j (tr )}sgn{(Sb,i (e, x)) j,` }(φb,i ) j,` (v) ∈ L ∞ . However, this is true owing to the boundedness of (φb,i ) j,` (v), ∀t ≥ 0, and the fact that limν→∞ (φb,i ) j,` (v) = 0. Accordingly, the weights Wc , are restricted to the bounded convex set Wc under all operating conditions, a task achieved through the projection operation Pc {·} in (20). Even in the presence of a resetting, we have argued in Remark 4 that we can always select the upper bound of φc (at each t = tr ) to guarantee that (Wc )(tr+ ) ∈ Wc . Additionally, the overall increase of the weights (Wc ) j,q j is also bounded against infinite resettings, P∞ (k) since ∈ ν=1 sgn{b j (tr )}sgn{(Sc (e)G (x)) j,q }(φc ) j,q (v) L ∞ which holds due to the boundedness of (φc ) j,q (v) and its convergence as ν → ∞ to zero. Remark 5. The developed resetting scheme can be visualized as an improved modification of the one first proposed in Rovithakis (2001) since: (i) it guarantees stability even in presence of infinite resettings, (ii) it can be easily combined with other robust adaptive modifications, namely the projection modification, (iii) it has a less complex structure, leading to easier implementation. Stage 3. Based on the definitions of γ (|e|) and V (e), there exists a positive but unknown constant cγ such that γ (|e|) ≥ cγ V (e), ∀e ∈ Ω ⊂ Rn . Defining ca = mini=1,...,r {ka,i } and cb = mini=1,...,r {kb,i } we obtain from (14), (27) that r ca − cγ X |W˜ a,i |2 L˙ ≤ −cγ L + 2 i=1 X r cγ − cb cγ − k c 2 ˜ |Wb,i | + + |W˜ c |2 + µ. 2 2 i=1 ˙ Thus selecting, c = cγ = min{ca , cb , kc } we arrive at L(t) ≤ −cL(t) + µ, and hence µ n µ o −ct 0 ≤ L(t) ≤ + L(0) − e , ∀t ≥ 0. (38) c c The stability properties are valid, assuming that e ∈ Ω , with Ω ⊂ Rn Also recall that due to Theorem 1, e is uniformly ultimately bounded with respect to set E. Hence, starting with e(0) ∈ E, the tracking error will remain consistently within this set, i.e. e(t) ∈ E ⊂ Ω , ∀t ≥ 0. Consider now the case where e(0) ∈ Ω , while satisfying the condition e(0) 6∈ E. Under these circumstances, we distinguish the following cases:
Case 1. L(0) > µ/c. In that case, based on the definition of L(t) and (38), we have kV (e) ≤ L(t) ≤ L(0), ∀t ≥ 0. Hence, it can be easily verified that there exists a gain k such that ( r 1 X E1 = e ∈ Rn : |e| ≤ V −1 V (0) + |W˜ a,i (0)|2 2k i=1 !) r 1 X 1 2 2 |W˜ b,i (0)| + |W˜ c (0)| + ⊂ Ω. (39) 2k i=1 2k Case 2. L(0) ≤ µ/c. In that case, the following condition holds kV (e) ≤ L(t) ≤ µ/c and the tracking error satisfies µ |e| ≤ V −1 { kc }. Hence, there also exists a constant k making µ o n E2 = e ∈ Rn : |e| ≤ V −1 ⊂ Ω. (40) kc Theorem 2. Consider the error system described by (6), where Assumptions 1 and 2 and Lemma 1 are satisfied. The control law (25) and the weight updates (18)–(20), supported by the resetting scheme (32)–(33) ensure the uniform ultimate boundedness of the tracking error e with respect to an arbitrary small set (30), plus the boundedness of all other signals in the closed loop. 4. Simulation results To demonstrate the suggested adaptive neuro-fuzzy control approach, we consider a highly nonlinear MIMO system, namely, a two-link robot manipulator. The plant dynamics are described by q¨ = −M(q)−1 [C(q, q) ˙ q˙ + G(q)] + M(q)−1 τ, q = [q1 , q2 ]T α1 + 2α2 c2 α3 + α2 c2 M(q) = α 3 + α 2 c2 α3 −q˙2 (q˙1 + q˙2 ) C(q, q) ˙ = α2 s2 , q˙1 0
(41)
g G(q) = 1 g2
where α1 = Iz 1 + Iz 2 + m 1 `2c1 + m 2 (`21 + `2c2 ),
α2 = m 2 `1 `c2
α3 = Iz 2 + m 2 `2c2 , `c1 = `1 /2, `c2 = `2 /2 g1 = m 1 g`c1 c1 + m 2 g(`1 c1 + `c2 c12 ), g2 = m 2 g`c2 c12 , q1 , q2 (rad) are the angle positions of the joints 1 and 2 constrained within [−(π/2), (π/2)]. M(q) is the moment of inertia, C(q, q) ˙ includes coriolis and centripetal forces, and G(q) is the gravitational force. Other quantities denote: link masses m 1 , m 2 (kg), link lengths `1 , `2 (m), applied torques τ1 , τ2 (N-m), acceleration due to gravity g = 9.8 m/s2 , and shorthand notations c1 = cos(q1 ), s2 = sin(q2 ), c2 = cos(q2 ), and c12 = cos(q1 + q2 ). The parameter values used in our simulations are: m 1 = m 2 = 0.32, `1 = `2 = 0.3, Iz 1 = Iz 2 = 97.63 × 10−4 . The system states are taken as x1 = q1 , x2 = q2 , x3 = q˙1 , and x4 = q˙2 . The motion dynamics (41) are modelled by fuzzy rules with nonlinear consequents of the form (2). To minimize model and
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Fig. 1. Membership functions along the x1 , x2 premise variables of the T–S fuzzy model.
design complexity only the states x1 and x3 are considered in the antecedent part. Each premise variable is partitioned using three fuzzy sets (Fig. 1), leading to a fuzzy model with r = 9 rules. We assume that no a priori knowledge is available, i.e. (k) (k) f i (x) = G i (x) ≡ 0, i = 1, . . . , 9. The objective is to design a stable control law for τ1 (t) and τ2 (t), such that q1 and q2 are forced to follow consistently a desired trajectory: xr,1 (t) = 2π/9 + π/9 cos(π t), xr,2 (t) = −π/9 cos(π t). (u) (u) To accomplish our goal, the terms Ai (e, x), Bi (e, x) and C(e) are approximated via LWNNs. We develop common vectors Sa,i (e, x) ∈ R Q a,i and Sb,i (e, x) ∈ R Q b,i for all rules, i = 1, . . . , 9, comprising Q a,i = 16 and Q b,i = 12, respectively. Let diagq [z]denote a q × q diagonal matrix with q identical entries equal to z. The regressor vectors are given by T Sa,i (e, x) = S1T (e)Sa (x)
S1T (e) = [se (e1 ), se (e2 ), se (e3 ), se (e4 )] Sa (x) = [diag4 (s1 (x1 )), diag4 (s2 (x2 )), diag4 (s3 (x3 )), diag4 (s3 (x4 ))] T Sb,i (e, x) = S1T (e) · Sb (x) + S2T (e) · [diag4 (s2 (x2 ))]
Sb (x) = [diag4 (s2 (x2 )), diag4 (s22 (x2 ))] S2T (e) = [se2 (e1 ), se2 (e2 ), se2 (e3 ), se2 (e4 )].
Fig. 2. The trajectories of the state variables (solid lines) and the reference signals (dotted lines): (x1 , xr,1 ) (a), (x2 , xr,2 ) (b), (x3 , x˙r,1 ) (c), and (x4 , x˙r,2 ) (b).
Further, Sc (e) ∈ R Q c ×4 contains a total of 20 terms (Q c = 5), described as: Sc (e) = [c1 , c2 , c3 , c4 ], where c1T = [se2 (e1 ), se (e1 ), se (e1 )se (e3 ), se2 (e2 )se (e4 ), se2 (e1 )se (e2 )] c2T = [se2 (e2 ), se (e2 ), se (e2 )se (e4 ), se2 (e4 )se (e2 ), se2 (e2 )se (e1 )] c3T = [se2 (e3 ), se (e3 ), se2 (e1 )se (e3 ), se2 (e1 )se (e2 ), se2 (e3 )se (e4 )] c4T = [se2 (e4 ), se (e4 ), se2 (e3 )se (e1 ), se (e3 )se (e4 ), se2 (e4 )se (e2 )]. The sigmoid functions are defined as follows: s1 (x) = s2 (x) =
2π − (π − 0.5), 1 + e−x
2π/3 − π/3, s3 (x3 ) = 1 + e−x3 2π/3 − π/3. se (e) = 1 + e−(e−0.001) The control law is obtained by applying (25), with γ (|e|) = (1 − e−0.1 |e| ). The weight values are updated using (18)–(20), and ψ = −x˙r . The learning rates are chosen as ka,i = ka = 0.001, kb,i = kb = 0.001, i = 1, . . . , 9, kc = 0.001, while k = 0.5. The system states are initialized at (x1 (0), x2 (0), x3 (0), x4 (0))T = (1.1363, 1.1363, 0.4, 0.4)T . Further, the initial weight values are Wa,i = row16 [−5] ∈ R16 , Wb,i = row12 [−1] ∈ R12 , i = 1, . . . , 9, Wc = row5 [−2] ∈ R5 , where
Fig. 3. Evolution of |b(e, x, Wb , Wc )|.
rowq [z] denotes a row vector with q identical entries equal to z. The aforementioned LWNNs used as well as the determination of the design constants were selected according to a trial and error procedure. Notice though that Theorem 1 provides guidelines for a qualitative selection. The performance of the suggested neuro-fuzzy control scheme is demonstrated in Figs. 2 and 3. In Fig. 2, the state trajectories are plotted along with their respective reference signals xr and x˙r . Apparently, the tracking error diminishes almost exponentially fast. Fig. 3 depicts the evolution of the signal b(e, x, Wb , Wc ) which illustrates that resetting
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guarantees |b(e, x, Wb )| > 0, ∀t ≥ 0. In our simulations, 1 we have selected δb = 0.008 and ϕ(v) = v − 4 . Whenever |b(e, x, Wb , Wc )| − δb = 0, the resetting procedure is activated, forcing |b(e, x, Wb , Wc )| to admit values greater that δb . 5. Conclusions A novel control design is proposed for tracking affine in the control dynamic systems, with unknown nonlinearities. The plant is described by a T–S fuzzy model with nonlinear consequent parts. A rigorous Lyapunov-based stability analysis is presented, providing the network weights’ update laws and the derivation of the control law. A novel resetting scheme is also developed, ensuring the validity of the control input and uniform ultimate boundedness of the tracking error. References Ioannou, P. A., & Sun, J. (1996). Robust adaptive control. Upper Saddle Press NJ, USA: Prentice Hall. Koo, T. J. (2001). Stable model reference adaptive fuzzy control of a class of nonlinear systems. IEEE Transactions on Fuzzy Systems, 9(4), 624–636. Lee, H., & Tomizuka, M. (2000). Robust adaptive control using a universal approximator for SISO nonlinear systems. IEEE Transactions on Fuzzy Systems, 8(1), 95–106. Lewis, F. L, Liu, K., & Yesildirek, A. (1995). Neural net robot controller with guaranteed tracking performance. IEEE Transactions on Neural Networks, 6(3), 703–715. Ordonez, R., & Passino, K. (1999). Stable multi-input multi-output adaptive fuzzy/neural control. IEEE Transactions on Fuzzy Systems, 7(3), 345–353. Polycarpou, M. M. (1996). Stable adaptive neural control scheme for nonlinear systems. IEEE Transactions on Automatic Control, 41(3), 447–451. Rovithakis, G. A. (2001). Stable adaptive neuro-control design via Lyapunov function derivative estimation. Automatica, 37(8), 1213–1221. Rovithakis, G. A. (2004). Robust redesign of a neural network controller in the presence of unmodeled dynamics. IEEE Transactions on Neural Networks, 15(6), 1482–1490. Sontag, E. D. (1989). A universal construction of Artstein’s theorem on nonlinear stabilization. Systems Control Letters, 13, 117–123. Spooner, J. T., & Passino, M. (1996). Stable adaptive control using fuzzy systems and neural networks. IEEE Transactions on Fuzzy Systems, 4(3), 339–359. Tseng, C. S., Chen, B.-S., & Uang, H.-. J (2001). Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model. IEEE Transactions on Fuzzy Systems, 9(3), 381–392.
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Xu, H., & Ioannou, P. A. (2003). Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds. IEEE Transactions on Automatic Control, 48(5), 728–742. S.P. Moustakidis was born in Thessaloniki, Greece, in 1981. He received a degree in electrical engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2004. He is currently working toward a Ph.D. degree at the Department of Electrical and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. His research interests include fuzzy logic systems, support vector machines, wavelet analysis, control with neural networks and genetic algorithms. G.A. Rovithakis was born in Chania, Crete, Greece in 1967. He received the diploma in Electrical Engineering from the Aristotle University of Thessaloniki, Greece in 1990 and the M.S. and Ph.D degrees in Electronic and Computer Engineering both from the Technical University of Crete, Greece in 1994 and 1995 respectively. After holding a visiting Assistant Professor position with the Department of Electronic and Computer Engineering, Technical University of Crete from 1995 to 2002, he joints the Aristotle University of Thessaloniki where he is currently an Assistant Professor with the Department of Electrical and Computer Engineering. His research interests include nonlinear systems, neural network systems, robust adaptive control, identification-control of unknown systems using neural networks, control of computer networks, QoS control, production control, intelligent control, fault detection, isolation and accommodation in nonlinear dynamical systems, automated inspection systems where he has authored or co-authored over 90 publications in scientific journals, referred conference proceedings and book chapters. He has also co-authored the book Adaptive Control with Recurrent High-Order Neural Networks (London UK: Springer-Verlag, 2000). Dr Rovithakis serves as a reviewer for various journals and conferences and has served as session chairman or co-chairman in international conferences. He is a Senior Member of IEEE, Member of the IEEE Control Systems Society Conference Editorial Board and a Member of the Technical Chamber of Greece. J.B. Theocharis (M’90) received a degree in electrical engineering and the Ph.D. degree from Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1980 and 1985, respectively. He is currently an Associate Professor in the Department of Electronic and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. His research activities include fuzzy systems, neural networks, neuron-fuzzy modelling, time-series prediction, recurrent fuzzy neural networks, evolutionary algorithms and adaptive control.