This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS

1

Global Asymptotic Stabilization Using Adaptive Fuzzy PD Control Yongping Pan, Haoyong Yu, Member, IEEE, and Tairen Sun

Abstract—It is well-known that standard adaptive fuzzy control (AFC) can only guarantee uniformly ultimately bounded stability due to inherent fuzzy approximation errors (FAEs). This paper proves that standard AFC with proportional-derivative (PD) control can guarantee global asymptotic stabilization even in the presence of FAEs for a class of uncertain affine nonlinear systems. Variable-gain PD control is designed to globally stabilize the plant. An optimal FAE is shown to be bounded by the norm of the plant state vector multiplied by a globally invertible and nondecreasing function, which provides a pivotal property for stability analysis. Without discontinuous control compensation, the closed-loop system achieves global and partially asymptotic stability in the sense that all plant states converge to zero. Compared with previous adaptive approximation-based global/asymptotic stabilization approaches, the major advantage of our approach is that global stability and asymptotic stabilization are achieved concurrently by a much simpler control law. Illustrative examples have further verified the theoretical results. Index Terms—Adaptive control, asymptotic stabilization, fuzzy approximation, global stability, proportional-derivative (PD) control, uncertain nonlinear system.

I. I NTRODUCTION DAPTIVE approximation-based control (AAC) using fuzzy logic systems (FLSs) or neural networks (NNs) is efficient for tackling nonparametric uncertainties in nonlinear systems [1] and has attracted great concern in the intelligent control community recently. For example, see [2]–[27]. In a general sense, asymptotic tracking and global stability are two major challenges for AAC design. Unlike classic adaptive control which can usually guarantee global asymptotic stability of the closed-loop system, standard AAC usually can only obtain local uniformly ultimately bounded (UUB) stability due to the local approximation property of most function approximators and the existence of inherent approximation errors [1]. At the early stage, discontinuous sliding-mode control [2] and discontinuous supervisory control [3] are widely applied to

A

Manuscript received November 30, 2013; revised March 11, 2014 and June 7, 2014; accepted June 11, 2014. This work was supported in part by the Seed Fund from the Engineering Design and Innovation Center, National University of Singapore under Grant R-261-503-002-133 and in part by the MINDEF-NUS Joint Applied Research and Development Cooperation Programme under Grant MINDEF/NUS/JPP/13/01/01. This paper was recommended by Associate Editor P. Shi. Y. Pan and H. Yu are with the Department of Biomedical Engineering, National University of Singapore, Singapore 117575 (e-mail: biepany@nus. edu.sg; [email protected]). T. Sun is with the School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2331460

achieve asymptotic tracking and global stability in AAC systems, respectively. Nevertheless, those discontinuous control strategies are inapplicable to many practical control problems since they require infinite control bandwidth and cause high-frequency chattering at the control input. Yet interestingly, recent results in [20]–[22] show that partially asymptotic stabilization can be obtained by the variations of standard AAC without discontinuous control compensation. In [20], asymptotic AAC was developed for a special class of affine nonlinear systems, where the plant dynamics are expressed into linear-in-state forms, and each form is estimated by n approximators with n being the plant order. In [21], the approach of [20] was extended to a class of largescale interconnected strict-feedback nonlinear systems, where each lumped uncertainty is estimated by one NN, and the control singularity problem caused by the unknown control gain functions is resolved by integral Lyapunov functions. In [22], global asymptotic AAC was proposed for a general class of affine nonlinear systems, where a global NN is designed by a partition of unity technique in differential topology, an optimal approximation error is proven to be bounded by the norm of the state vector multiplied by a certain constant, and asymptotic stabilization is obtained by the combination of three types of adaptive laws. Compared with the standard AAC approach, the approaches of [20]–[22] have the following drawbacks: 1) the control structures are more complex and 2) permanently positive or negative adaptive laws that make their estimations unbounded are applied to update control parameters. Recently, global stabilization of AAC systems without discontinuous control compensation has also obtained some concern [22]–[27]. As mentioned before, the global asymptotic AAC based on global NN approximation was proposed in [22]. A full-state-feedback proportional-derivative (PD) control term with variable gain was applied to globally stabilize the AAC system in [23]. It was claimed in [24] that simply setting the PD gain to be larger than 1/2 can also obtain the results in [23]. Yet, this result depends on an inappropriate implicit assumption that a lumped uncertainty can be globally approximated by a linear parameterized NN. In [25], a discontinuous ideal control law with PD control was proposed to reduce the number of NN inputs and to relax the bound conditions in [23]. Note that the approaches in [22]–[25] are based on a direct AAC scheme. To achieve global stabilization under an indirect AAC scheme, PD control was applied in [26], and NN feedforward was applied in [27]. It is worth noting that all plant dynamics must be known at the procedure of controller design in [27], which violates the original intention of AAC

c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267  See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON CYBERNETICS

for uncertain nonlinear systems. The major limitation of the global PD control approaches in [23]–[26] is that they only guarantee UUB stability. Based on our preliminary work of [30], this paper focuses on global asymptotic stabilization via standard adaptive fuzzy control (AFC) for a class of uncertain affine nonlinear systems. Because almost all AFCs include PD control terms, applying PD control to globally stabilize the plant is relatively simpler but more effective since it does not complicate the control structure. Another reason for applying PD control is that it is also pivotal for achieving asymptotic stability. Under the direct adaptive scheme, an optimal fuzzy approximation error (FAE) is proven to be bounded by the norm of the state vector multiplied by a globally invertible and nondecreasing function. This result is applied to control design so that global and partially asymptotic stability of the closed-loop system can be obtained without discontinuous control compensation. Compared with the previous AAC-based asymptotic stabilization approaches of [20]–[22] and the previous global PD control approaches of [23]–[26], our approach has some merits as follows: 1) The complexity of control structure and control derivation is reduced sharply compared with [20]–[22]; 2) The permanently positive or negative adaptation applied in [20]–[22] is avoided completely; 3) The global asymptotic stability obtained is better than those global UUB results in [23]–[26]. The rest of this paper is organized as follows. The considered problem is formulated in Section II. The control strategy is developed in Section III. Illustrative examples are given in Section IV. Conclusions are summarized in Section V. Throughout this paper, R, R+ , Rn , and Rn×m denote the spaces of real numbers, real positive numbers, real n-vectors, and real n × m matrixes, respectively, diag(·) denotes the diagonal matrix, L∞ denotes the space of essentially bounded signals, λmin (·) and λmax (·) denote the functions of minimal and maximal eigenvalues, respectively, min{·}, max{·}, and sup{·} denote the functions of minimum, maximum, and supremum, respectively, and C k represents the space of functions whose k-order derivatives all exist and are continuous, where n, m, and k are nonnegative integers. II. P ROBLEM F ORMULATION Consider a class of single input single output (SISO) affine nonlinear systems as follows [1]: ⎧ ⎨ x˙ i = xi+1 (i = 1, 2, · · · , n − 1) x˙ n = f (x) + bu (1) ⎩ y = x1 where x(t) : = [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is the measurable state vector, u(t) ∈ R and y(t) ∈ R are the control input and system output, respectively, f (x):Rn → R satisfying f (0) = 0 [20], [21] is the unknown C 1 nonlinear function, and b ∈ R+ is the unknown control gain. Assumption 1 [23]: There exist a continuous function f (x) and a finite constant b0 ∈ R+ such that the inequalities |f (x)| ≤ f (x) and 0 < b0 ≤ b hold, ∀x ∈ Rn .

Choose a vector k = [k1 , k2 , · · · , kn ]T ∈ Rn such that h(s) : = sn + kn sn−1 + · · · + k2 s + k1 is a Hurwitz polynomial, where s is a complex variable. It follows from the result in [1] that the following ideal control law: u∗ (x) : = (−kT x − f (x))/b

(2)

makes the closed-loop system globally and exponentially stable. However, (2) is not enforceable since f (·) and b are unknown a priori here. Accordingly, the control objective of this paper is to determine a standard AFC-based control strategy for the system in (1) under Assumption 1 such that the closed-loop system achieves global asymptotic stabilization in the sense that limt→∞ x(t) = 0, ∀x(0) ∈ Rn . Remark 1: To simplify discussion, this paper considers the SISO affine nonlinear system in (1) under Assumption 1. Since the design tools in [20] and [21] are standard except the treatment of optimal approximation errors, the results in this paper can be extended to the systems in [20] and [21] by the corresponding derivation steps therein without much difficulty. Please also see [17, Remark 1] for additional discussions. III. G LOBAL A SYMPTOTIC S TABILIZATION S TRATEGY Let Dx : = {x| x ≤ cx } be a domain of fuzzy approximation, Dg ⊂ Dx be a domain of global stabilization, and D¯ g : = Rn −Dg where cx ∈ R+ is a prespecified constant. Since fuzzy approximation is inapplicable outside of the interested domain, the FLS output is zero, ∀x(t) ∈ D¯ g . In this section, the closed-loop dynamics is firstly derived to facilitate control design, and then a two-step design procedure is proposed to achieve global asymptotic stabilization of (1). The outline of this two-step procedure is as follows. Step 1: A PD control with variable gain is applied to ensure x(t) ∈ Dg after a finite time Tη , ∀x(0) ∈ Rn . Step 2: An AFC with PD control law is applied to guarantee limt→∞ x(t) = 0, ∀x(Tη ) ∈ Dg . A. Closed-Loop System Dynamics Since f (·) and b in are unknown a priori, one introduces a class C 1 linearly parameterized FLS [1] T uf (x|θˆ ) = θˆ ξ (x)

(3)

to approximate u∗ (·) in (2), where θˆ : = [θˆ1 , θˆ2 , · · · , θˆM ]T ∈ RM is the vector of adjustable parameters, ξ (x) : = [ξ1 (x), ξ2 (x), · · · , ξM (x)]T ∈ RM satisfying ξ (·) ≤ ψ is the vector of fuzzy basis functions (FBFs), ψ ∈ R+ is a constant, and M is the number of fuzzy rules. Let θ : = θ1 ∩ θ2 , where ˆ θˆ ≤ cθ }, θ : = {θ| ˆ θˆ T ξ(0) = 0}, and cθ ∈ R+ θ1 : = {θ|

2 is a prespecified finite constant. Noting u∗ (0) = 0, define an optimal FAE w as follows: w(x) : = uf (x|θ ∗ ) − u∗ (x) where θ ∗ is a vector of optimal parameters given by1   θ ∗ = arg min sup |uf (x|θˆ ) − u∗ (x)| . ˆ θ θ∈

x∈Dx

(4)

(5)

1 The radial basis function can be applied to construct a proper FLS with guaranteed approximation capacity under θˆ ∈ θ . Please refer to [19, Sec. IV] and [19, Remark 3] for the details.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PAN et al.: GLOBAL ASYMPTOTIC STABILIZATION USING ADAPTIVE FUZZY PD CONTROL

Next, design the control law u as follows: u = uf (x|θˆ ) + uc (x)

Accordingly, it can be shown that (6)

where uc is the PD control term given by uc (x) = −k(t)xT pn

(7)

in which k(t) ∈ R+ is a PD control gain that will be defined for global stabilization, and pn ∈ Rn is the nth column of a real matrix P ∈ Rn×n . Applying (2) and (6) to (1) and making some transformations yields   (8) x˙ = Ax + b uf (x|θˆ ) − u∗ (x) + uc (x) in which

⎡ ⎤ ⎤ 0 0 1 ··· 0 ⎢ .. ⎥ ⎢ .. .. . . .. ⎥ ⎢ ⎥ ⎢ . . ⎥ . A=⎢ . ⎥,b = ⎢ . ⎥. ⎣0⎦ ⎣ 0 0 ··· 1 ⎦ b −k1 −k2 · · · −kn ⎡

From the selection of k(t) in (12), one gets V˙ s (t) ≤ −xT Qx/2 + ϕ 2 ≤ −λ1 (Vs (t) − ϕ 2 /λ1 ) where λ1 : = λmin (Q)/λmax (P). Involving the Comparison Lemma [23], one deduces Vs (t) ≤ (Vs (0) − ϕ 2 /λ1 ) exp(−λ1 t) + ϕ 2 /λ1 .

x(t) 2 ≤ (λ2 x(0) 2 − λ3 ϕ 2 ) exp(−λ1 t) + λ3 ϕ 2

(9)

Let θ˜ : = θˆ − θ ∗ . Noting (4), it is easy to obtain the closedloop system dynamics as follows:  T  x˙ = Ax + b θ˜ ξ (x) + w(x) + uc (x) . (10) B. Step 1: Global Robust Stabilization To make the fuzzy approximation in Section III-A applicable, one should ensure x(t) ∈ Dx as in (5). Choose a Lyapunov function candidate for (8) as follows: Vs (x) = xT Px/2.

V˙ s ≤ −xT Qx/2 + (f¯ (x) + |kT x|)2 /(4k(t)b0 )  2 − k(t)b0 |xT pn | − (f¯ (x) + |kT x|)/(2k(t)b0 ) ≤ −xT Qx/2 + (f¯ (x) + |kT x|)2 /(4k(t)b0 ). (13)

(14)

Applying (11) to the above inequality, one immediately gets

From the selection of k, A is a stable matrix. Hence, for any given positive definite symmetric matrix Q ∈ Rn×n , there must exist a unique positive definite symmetric matrix solution P for the Lyapunov equation AT P + PA = −Q.

3

(11)

Then, we provide the following theorem to solve this issue. Theorem 1: For the system in (1) with x(0) ∈ Rn satisfying Assumption 1 with known f (x) and b0 , choose u = uc (x) in (7) with k(t) = κ(x(t)) as the control law, where     (12) 4b0 ϕ 2 κ(x(t)) = 1 + (f¯ (x(t)) + |kT x(t)|)2 and ϕ ∈ R+ is a prespecified control parameter. Then the closed-loop system is globally UUB stable in the sense that x is bounded by the transient bound in (16) and exponentially converges to Dg in (18) after a finite time Tη in (17). Proof: Differentiating Vs with respect to time t and using (2), (8), and (9), one obtains   V˙ s = −xT Qx/2 + xT Pb uc (x) − u∗ (x)   = −xT Qx/2 + xT Pb uc (x) + f (x) + kT x . Applying (7), Pb = pn b and Assumption 1 to the above equality, one obtains  V˙ s ≤ −xT Qx/2 − k(t)b0 |xT pn |2  − |xT pn |(f¯ (x) + |kT x|)/(k(t)b0 ) .

(15)

with λ2 : = λmax (P)/λmin (P) and λ3 : = 2/(λ1 λmin (P)), which implies that x is bounded by  

x(t) ≤ λ2 x(0) exp(−λ1 t/2) + λ3 ϕ. (16) Let η2 : = (λ3 + 1)ϕ 2 . From the time solution of (15), there exists a finite time [23]    (17) Tη = max 0, (2/λ1 )ln( λ2 x(0) /ϕ) such that x converges to the compact set  Dg = {x(t)| x(t) ≤ λ3 + 1ϕ}, ∀t ≥ Tη .

(18)

From the definition of UUB [31], √ (18) implies that x is UUB with the uniform bound η = λ3 + 1ϕ. From (7) and u = uc (x), one immediately gets u is uniformly bounded. Since Vs (x) is radially unbounded, i.e., Vs (x) → ∞ as x → ∞, and the above analysis is applicable ∀x(0) ∈ Rn , the UUB stability result is global in this sense. Corollary 1: For the system in (1) with x(0) ∈ Sg satisfying Assumption 1 with unknown f (x) and b0 , choose u = uc (x) in (7) with k(t) = κ(x(t)) as the control law, where Sg ⊂ Rn is a domain of semiglobal attraction, and κ(x(t)) = (1 + x(t) 2 )/ϕ 2 .

(19)

Then the closed-loop system is semiglobally UUB stable in the sense that x is bounded by the transient bound in (16) and exponentially converges to Dg in (18) after a finite time Tη in (17), and Sg can be arbitrarily enlarged by the increase of ϕ, where λ1 and λ2 are the same as those in Theorem 1, and λ3 is defined in the following proof. Proof: Since f (x) is of class C 1 and f (0) = 0, one gets f (x) ≤ |f (x) − f (0)| ≤ λx x , ∀x ∈ Sg , where λx ∈ R+ is a finite constant. Applying f (x) ≤ λx x to (13) leads to V˙ s ≤ −xT Qx/2 + (λx x + k

x )2 /(4k(t)b0 ) ≤ −xT Qx/2 + x 2 δ 2 /(k(t)b0 ) where δ : = max{λx , k }/2. Applying the selection of k(t) in (19) to the above result, one obtains V˙ s (t) ≤ −xT Qx/2 + (δϕ)2 /b0 ≤ −λ1 (Vs (t) − (δϕ)2 /(b0 λ1 ))

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4

IEEE TRANSACTIONS ON CYBERNETICS

with λ1 being the same as in Theorem 1. Involving the Comparison Lemma [23], one deduces Vs (t) ≤ (Vs (0) − (δϕ)2 /(b0 λ1 )) exp(−λ1 t) + (δϕ)2 /(b0 λ1 ) which is similar to (14). Let λ3 : = 2δ 2 /(b0 λ1 λmin (P)) and λ2 be the same as in Theorem 1. Using the same derivation in the proof of Theorem 1, one immediately gets that x is UUB with the transient bound in (16) and exponentially converges to Dg in (18) after the time Tη in (17), ∀x(0) ∈ Sg . Since Sg can be arbitrarily enlarged by the decrease of ϕ, the UUB stability result is semiglobal in this sense [28]. C. Step 2: Regional Asymptotic Stabilization Consider the SISO version of a pretreatment method in [20]. By the Mean Value Theorem, f (·) can be rewritten as f (x) = (x)x

(20)

in which (x) : Rn → Rn is a vector of unknown functions. Then, n approximators are applied to estimate (·) such that the resulting optimal approximation errors lie in a small gain-type norm bounded conic sector. This pretreatment plays a key role in ensuring closed-loop asymptotic stabilization in [20]. Yet, it also greatly complicates the control structure since n approximators, rather than one, should be applied to indirectly estimate the scale function f (·). A similar method to that of [20] was proposed in [21], where the major difference is that only one approximator is needed in [21] to estimate each lumped uncertainty. However, this pretreatment still makes the control law derivation complex. In fact, the pretreatment in [20] and [21] is not needed and the asymptotic stabilization result can still be guaranteed without increasing the complexity of both control structure and control derivation. To show this, we first establish the following lemma. Lemma 1: The optimal FAE w in (4) can be bounded by w(x) ≤ ρ( x ) x , ∀x ∈ Dx

(21)

in which ρ( x ):R+ → R+ is a certain globally invertible and nondecreasing function. Proof: Since u∗ in (2) and uf in (3) are of class C1 , w in (4) is of class C1 . In addition, θ ∗ is constant from its definition in (5). Thus, the Mean Value Theorem can be applied to the expression of w in (4) such that w(x) − w(0) = w (x)x, ∀x ∈ Dx

w(0) = uf (0, θ ∗ ) − u∗ (0) = 0.

Combining (23) with (24), one immediately gets (21). Choose a Lyapunov function candidate T ˜ V(z) = xT Px/2 + bθ˜ θ/(2γ )

(25)

for the system in (10), where z : = [xT , θ˜ ]T and γ ∈ R+ is a learning rate, and design the adaptive law of θˆ to be θ˙ˆ = proj(−γ xT pn ξ (x))

(26)

where proj(·) is a projection operator given by [1] ⎧ ⎪ •, if θˆ < cθ ⎪ ⎪ ⎪ ⎨ or θˆ = c & θˆ T · • ≤ 0 θ proj(•) = . T ˆ ˆ ⎪ • − θ θ · •/ θˆ 2 ⎪ ⎪ ⎪ T ⎩ if θˆ = cθ & θˆ · • > 0 The system in (1) is assumed to be started from t = Tη . Now, we establish the main result of this paper. Theorem 2: Consider the system in (1) with Assumption 1 driven by the control law in (6) with (3), (7), (26), and k(t) = κ0 : = maxx∈Dg {κ(x(t))} with κ(x(t)) in (12). The schematic diagram of this approach is depicted in Fig. 1. Then, for all x(Tη ) ∈ Dg , θˆ (Tη ) ∈ θ1 and k ∈ Rn , there exist suitable parameters ϕ and Q such that the closed-loop system achieves partially asymptotic stability in the sense that all involved signals are ultimately bounded and limt→∞ x(t) = 0. Proof: First, since x(Tη ) ∈ Dg ⊂ Dx , the fuzzy approximation in Section III-A and the closed-loop system dynamics in (10) are applicable. Differentiating (25) along (10) with respect to time t and using (9) leads to   V˙ = −xT Qx/2 + xT pn b w(x) + uc (x)  T + bθ˜ (xT p )ξ (x) + θ˙ˆ /γ . From the projection operator results in [1, Sec. 25], the adaptive law in (26) guarantees that: i) θˆ (t) ∈ θ1 , ∀t ≥ Tη if ˙ˆ ) ≤ 0. Applying these ˆ η ) ∈ θ ; ii) θ˜ T ((xT pn )ξ (x) + θ/γ θ(T 1 results to the above expression, one obtains   V˙ = −xT Qx/2 + xT pn w(x) + uc (x) . Substituting (7) with k(t) = κ0 and (21) to the above equality, it is easy to obtain

Applying the above expression to (22) leads to (23)

According to [29, Remark 3], there exists a globally invertible and nondecreasing function ρ( x ) such that w (x)x ≤ ρ( x ) x , ∀x ∈ Dx .

Schematic diagram of AFC plus PD control.

n

(22)

where w (x) : = ∂w(x)/∂xT |x=δ , δ : = [δ1 , δ2 , · · · , δn ]T , δi ∈ [0, xi ] (if xi ≥ 0) or [xi , 0] (if xi < 0), and i = 1, 2, · · · , n. From (2), one obtains u∗ (0) = 0. From the definition of θ ∗ in (5), one obtains uf (0, θ ∗ ) = 0. Applying u∗ (0) = 0 and uf (0, θ ∗ ) = 0 to (4), one immediately gets

w(x) = w (x)x, ∀x ∈ Dx .

Fig. 1.

(24)

V˙ ≤ −λmin (Q) x 2 /2 − κ0 |xT pn |2 + |xT pn |ρ( x ) x

= −λmin (Q) x 2 /2 + ρ 2 ( x ) x 2 /(4κ0 )  2 − κ0 |xT pn | − ρ( x ) x /(2κ0 )   ≤ − λmin (Q) − ρ 2 ( x )/(2κ0 ) x 2 /2.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PAN et al.: GLOBAL ASYMPTOTIC STABILIZATION USING ADAPTIVE FUZZY PD CONTROL

Accordingly, it can be stated that V˙ ≤ −λ4 x 2 /2, ∀λmin (Q) > ρ 2 ( x )/(2κ0 )

(27)

with λ4 : = λmin (Q) − ρ 2 ( x )/(2κ0 ) ∈ R+ . Second, let continuous positive definite functions U1 and U2 , and a continuous positive semi-definite function U be ⎧ ⎨ U1 (z) : = λ5 z 2 U (z) : = λ6 z 2 ⎩ 2 U(z) : = λ4 x 2 /2 in which λ5 : = min{λmin (P)/2, b/2γ }, and λ6 : = max {λmax (P)/2, b/(2γ )}. Then, V(·) in (25) satisfies  U1 (z) ≤ V(z) ≤ U2 (z) (28) ˙ V(z) ≤ −U(z) ∀ t ≥ Tη and ∀z ∈ Dz , where Dz is given by    Dz : = z ∈ Rn+1 | z < ρ −1 2κ0 λmin (Q)

(29)

with ρ −1 (·) being the inverse function of ρ(·). Eq. (28) and (29) imply V(z) ∈ L∞ in Dz . Hence, x, θˆ ∈ L∞ in Dz . From (6) with (3), (7) and k(t) = κ0 , one gets u ∈ L∞ in Dz . Since x, θˆ , ξ (x), w ∈ L∞ in Dz , (10) implies x˙ ∈ L∞ in Dz . From the definition of U(z) and x, x˙ ∈ L∞ in Dz , ˙ one gets U(z) ∈ L∞ in Dz , which is a sufficient condition for U(z) being uniformly continuous in Dz . Third, according to the above analysis, the Invariance-Like Theorem (see [31, Th. 8.4] or [29, Lemma 2]) can be invoked to conclude that lim U(z) = 0, ∀ z(Tη ) ∈ Sl

t→∞

where Sl is a domain of local attraction given by    Sl : = z ∈ Dz |U2 (z) < λ5 ρ −1 2κ0 λmin (Q) .

(30)

Consequently, one also obtains lim x(t) = 0, ∀ z(Tη ) ∈ Sl .

t→∞

From (12), (30) and the definition of κ0 , increasing λmin (Q) and/or 1/ϕ enlarges Sl [29]. From (18) and the definition of λ3 , for a given k, increasing λmin (Q) and/or 1/ϕ lessens Dg . Combining these results with Dg ⊂ Dx , there exist suitably large λmin (Q) and 1/ϕ such that Dg × θ0 ⊆ Sl with ˆ θ ∗ ∈ θ }. Thus, it can be concluded that θ0 : = { θ˜ |θ, 1 lim x(t) = 0, ∀ x(Tη ) ∈ Dg

t→∞

and all involved signals of the closed-loop system are uniformly bounded ∀x(Tη ) ∈ Dg . Remark 2: From the results obtained, while x(0) ∈ D¯ g , the control law in Theorem 1 (or Corollary 1) can be applied to get x(t) ∈ Dg ⊂ Dx , ∀t ≥ Tη . As long as x(Tη ) ∈ Dg , the control law in Theorem 2 can be applied to get limt→∞ x(t) = 0. Thus, global (or semiglobal) asymptotic stabilization of (1) can be guaranteed by the overall control law given by  −k(t)xT pn , if x ∈ D¯ g u= (31) T −k(t)xT pn + θˆ ξ (x), if x ∈ Dg

5

with the adaptive law of θˆ in (26), P in (9), Pb = pn b, and  κ(x(t)) in (12) or (19), if x ∈ D¯ g k(t) = . κ0 = maxx∈Dg {κ(x(t))}, if x ∈ Dg Remark 3: To clearly demonstrate the relationship between the design parameters k, P, Q, γ , and ϕ and the stability and performance improvement, we look into the overall control law in (31) and the stability results in (27) and (30). Since all parts of the ideal control law in (2), including the linear feedback part kT x and the nonlinear feedback part f (x), are estimated together by the FLS in (3), k does not make direct contribution to the stability and performance improvement. This conclusion can be verified by the combination of (9), (26), (27), (30), and (31) that for a given Q, enlarging k decreases pn

[from (9)], which does not improve system stability [from (27) and (30)] but degrades control capability [from (31) with (26)]. Yet, for a given k, enlarging λmin (Q) increases pn [from (9)], which enhances control capability [from (31) with (26)] so that the system stability and performance can be improved [from (27) and (30)]. Therefore, it can be concluded from the above analysis that for a given k, enlarging γ increases adaptive speed [from (26)], and enlarging λmin (Q) and/or ϕ improves the system stability and performance. Remark 4: In practice, due to the conservative control design in Theorem 1, the control parameter ϕ in (12) does not need to be set as small as that in (18). Thus, to implement the overall control law in (31), one can comply with the following steps: Firstly, determine a proper k to ensure stability of the linear part x˙ = Ax in (10); secondly, set the initial value of θˆ so that ˆ θ(0) ∈ θ1 ; thirdly, decrease ϕ in (12) or (19) to guarantee x(t) ∈ Dg , ∀t ≥ Tη ; finally, activate the FLS uf in (3) and increase γ in (26) to achieve limt→∞ x(t) = 0, ∀ x(Tη ) ∈ Dg under desired control performance. Remark 5: Compared with the previous global PD control approaches in [23]–[26], the proposed approach has the following novelties: 1) global asymptotic stabilization rather than global UUB stabilization is achieved under known plant bounds and 2) semiglobal stabilization is obtained under unknown plant bounds. Compared with the previous AACbased asymptotic stabilization approaches in [20]–[22], the proposed approach has the following virtues: 1) the complexity of both control structure and control derivation is reduced sharply and 2) the problem of permanently positive or negative adaptation is avoided completely. Compared with the recently developed semiglobal asymptotic stabilization approach in [19], the proposed approach is different in the following aspects: 1) the system considered is different so that the use of integral Lyapunov functions is avoided; 2) the proof of global stability is more compact; and 3) the design of the PD gain is improved so that it is more reasonable. IV. I LLUSTRATIVE E XAMPLES A. Example 1: Controlled Li´enard System Consider a controlled Li´enard system that can be represented by the form of (1) with n = 2 [20], where  f (x) = −2(x14 − 1)x2 − (x1 + tanh(x1 )) b=3

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6

Fig. 2.

IEEE TRANSACTIONS ON CYBERNETICS

Simulation trajectories of Li´enard system stabilization. (a) Convergence of x1 . (b) Convergence of x2 . (c) Adaptation of θˆ . (d) Control input u.

which satisfies Assumption 1 with f (0) = 0. For simulation studies, give five different initial state values x(0) as [2, 1.5]T (Case 1), [1, 1]T (Case 2), [0, −3]T (Case 3), [ − 1, −2]T (Case 4), and [ − 2, 0]T (Case 5). For the simplification of control design, choose k(t) in (19) as the PD control gain. The control objective is to make x converge to zero. The procedure of controller design is as follows: Firstly, select cx = [1, 1] ≈ 1.4142 to determine Dx and choose fuzzy membership functions ⎧   π(xi −clii )  ⎨1 i 1 + cos , if |xi − clii | ≤ σi l σi μi (xi ) = 2 ⎩ 0, if |x − cli | > σ i

i

i

with σi = 0.5, clii = (li − 3)/2, li = 1, 2, · · · , 5 and i = 1, 2 to construct the FBFs in (3) with M = 52 = 25; secondly, choose k = [1, 2]T and Q = diag(10, 10), and solve (9) to obtain pn = [5, 5]T ; thirdly, decrease ϕ in (19) so that x converges to Dg = {x| x ≤ 1.4} ⊂ Dx after a finite time Tμ ; finally, set cθ = 50 and θˆ (0) = [0, 0, · · · , 0]T , and increase γ in (26) to achieve desired control performance. For simulation comparisons, give four settings of control parameters as γ = 0 with ϕ = 2.5 (Setting 1), γ = 300 with ϕ = 2.5 (Setting 2), γ = 300 with ϕ = 1 (Setting 3), and γ = 3000 with ϕ = 1 (Setting 4), and choose the tracking indexes

J(ITAE) and J(IAE), and the control energy Ec as performance indexes [6]. Simulation trajectories under Setting 2 are shown in Fig. 2. One observes that x1 and x2 quickly converge to zero [see Fig. 2(a) and (b)] under rapid parameter adaptation [see Fig. 2(c)] and smooth control inputs [see Fig. 2(d)]. Note that the adaptive law of θˆ in (26) is applied from t = 0 while the FLS in (3) is activated when x(t) ∈ Dg after t = Tμ , where Tμ = 0.75, 0, 0.032, 0.24, and 0.73 s for the cases 1, 2, 3, 4, and 5, respectively. Performance comparisons under various cases and settings are given in Table I. It is shown that applying FLS greatly improves control performance under slightly increased control efforts (i.e., control energy or control cost) [see Settings 1 and 2 of Table I], and increasing γ and/or 1/ϕ further improves control performance [see Settings 3 and 4 of Table I] as claimed in Remark 3. Yet, decreasing ϕ also distinctly increases control efforts [see Setting 3 of Table I], and increasing γ does not improves control performance whileγ is large enough [see Setting 4 of Table I]. B. Example 2: Flexible-Joint Robot Arm Consider a flexible-joint robot arm that can be described by the form of (1) with n = 4 [32], where

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PAN et al.: GLOBAL ASYMPTOTIC STABILIZATION USING ADAPTIVE FUZZY PD CONTROL

Fig. 3.

7

ˆ (d) PD gain k(t). Simulation trajectories of robot arm stabilization. (a) Convergence of all states. (b) Control input u. (c) Adaptation of θ.

TABLE I P ERFORMANCE C OMPARISONS OF L Ie´ NARD S YSTEM S TABILIZATION

⎧ mgL sin x1 2 (x2 + mgL cosI x1 +K ) ⎨ f (x) = I mgL sin x1 K x1 −(x3 + )( I + KJ + mgL cos ) I I ⎩ b = K/IJ in which x1 denotes the position of the link, u denotes the generalized torque of the motor, m is the mass of the link,

g is the gravitational acceleration, I and J are the link and the rotor inertia moments, respectively, K is the elastic constant of the joint, and L is the distance from the axis of rotation to the center of mass of the link. For simulation, set x(0) = [4, 0, 0, 0]T , m = 5 kg, g = 9.8 m/s2 , I = 0.45 kg·m2 , J = 0.15 kg·m2 , L = 0.2 m, and K = 5 N·m/rad. For simplifying control design, choose k(t) in (19) as the PD gain. The control objective is to make x converge to zero. The selection of control parameters is the same as that in Section IV-A except that cx = [1, 1, 1, 1] = 2, k = [16, 32, 24, 8]T , Q = diag(500, 10, 10, 10), and μlii has i = 1, · · · , 4 and li = 1, 2, 3 so that M = 54 = 625. Simulation trajectories under γ = 100 and ϕ = 10 are depicted in Fig. 3, where all system states quickly converge to zero [see Fig. 3(a)] under smooth control input u [see Fig. 3(b)], rapid adaptation of θˆ [see Fig. 3(c)], and variable PD gain k(t) [see Fig. 3(d)].

V. C ONCLUSION The major contribution of this paper is to prove that standard AFC plus PD control can guarantee global asymptotic stabilization rather than UUB stability even in the presence of FAEs for a class of uncertain affine nonlinear systems.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8

IEEE TRANSACTIONS ON CYBERNETICS

Compared with previous AAC-based global/asymptotic stabilization approaches, the merit of our approach is that both global stability and asymptotic stabilization are concurrently achieved by a much simpler control structure. Two illustrative examples have been provided to verify the developed theoretical results. Nevertheless, it is not straightforward to extend the proposed approach to the tracking problem. Further work would focus on the standard AAC strategy with global asymptotic tracking performance. R EFERENCES [1] L. X. Wang, A Course in Fuzzy Systems and Fuzzy Control. Englewood Cliffs, NJ, USA: Prentice-Hall, 1997. [2] J. Wang, A. B. Rad, and P. Chan, “Indirect adaptive fuzzy sliding mode control—Part I: Fuzzy switching,” Fuzzy Sets Syst., vol. 122, no. 1, pp. 21–30, Aug. 2001. [3] P. Chan, A. B. Rad, and J. Wang, “Indirect adaptive fuzzy sliding mode control—Part II: Parameter projection and supervisory control,” Fuzzy Sets Syst., vol. 122, no. 1, pp. 31–43, Aug. 2001. [4] S. Barkat, A. Tlemcani, and H. Nouri, “Noninteracting adaptive control of PMSM using interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 925–936, Oct. 2011. [5] T. R. Sun, H. L. Pei, Y. P. Pan, and C. H. Zhang, “Robust wavelet network control for a class of autonomous vehicles to track environmental contour line,” Neurocomputing, vol. 74, no. 17, pp. 2886–2892, Oct. 2011. [6] Y. P. Pan, M. J. Er, D. P. Huang, and Q. R. Wang, “Adaptive fuzzy control with guaranteed convergence of optimal approximation error,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 807–818, Oct. 2011. [7] A. Boulkroune and M. M’Saad, “On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign,” Fuzzy Sets Syst., vol. 201, pp. 71–85, Aug. 2012. [8] A. Boulkroune, M. M’Saad, and M. Farza, “Fuzzy approximation-based indirect adaptive controller for multi-input multi-output non-affine systems with unknown control direction,” IET Control Theory Appl., vol. 6, no. 17, pp. 2619–2629, Nov. 2012. [9] A. Boulkroune, M. M’Saad, and M. Farza, “Adaptive fuzzy tracking control for a class of MIMO nonaffine uncertain systems,” Neurocomputing, vol. 93, pp. 48–55, Sep. 2012. [10] H. Chaoui and P. Sicard, “Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1123–1133, Feb. 2012. [11] Y. P. Pan, M. J. Er, and T. R. Sun, “Composite adaptive fuzzy control for synchronizing generalized Lorenz systems,” Chaos, vol. 22, no. 2, p. 023144, Jun. 2012. [12] Y. P. Pan, M. J. Er, D. P. Huang, and T. R. Sun, “Practical adaptive fuzzy H∞ tracking control of uncertain nonlinear systems,” Int. J. Fuzzy Syst., vol. 14, no. 4, pp. 463–473, Dec. 2012. [13] E. Kayacan, E. Kayacan, H. Ramon, and W. Saeys, “Adaptive neurofuzzy control of a spherical rolling robot using sliding-mode-controltheory-based online learning algorithm,” IEEE Trans. Cybern., vol. 43, no. 1, pp. 170–179, Feb. 2013. [14] M. Chen and S. Z. S. Ge, “Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer,” IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213–1225, Aug. 2013. [15] T. R. Sun, H. L. Pei, Y. P. Pan, and C. H. Zhang, “Robust adaptive neural network control for environmental boundary tracking by mobile robots,” Int. J. Robust Nonlinear Control, vol. 23, no. 2, pp. 123–136, Jan. 2013. [16] Y. P. Pan, Y. Zhou, T. R. Sun, and M. J. Er, “Composite adaptive fuzzy H∞ tracking control of uncertain nonlinear systems,” Neurocomputing, vol. 99, no. 1, pp. 15–24, Jan. 2013. [17] Y. P. Pan and M. J. Er, “Enhanced adaptive fuzzy control with optimal approximation error convergence,” IEEE Trans. Fuzzy Syst., vol. 21, no. 6, pp. 1123–1132, Dec. 2013. [18] B. Chen et al., “Approximation-based adaptive neural control design for a class of nonlinear systems,” IEEE Trans. Cybern., vol. 44, no. 5, pp. 610–619, May 2014. [19] Y. P. Pan, H. Y. Yu, and M. J. Er, “Adaptive neural PD control with semiglobal asymptotic stabilization guarantee,” IEEE Trans. Neural Netw. Learn. Syst., to be published.

[20] T. Hayakawa, W. M. Haddad, and N. Hovakimyan, “Neural network adaptive control for a class of nonlinear uncertain dynamical systems with asymptotic stability guarantees,” IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 80–89, Jan. 2008. [21] S. Mehraeen, S. Jagannathan, and M. L. Crow, “Decentralized dynamic surface control of large-scale interconnected systems in strict-feedback form using neural networks with asymptotic stabilization,” IEEE Trans. Neural Netw., vol. 22, no. 11, pp. 1709–1722, Nov. 2011. [22] P. Chen, H. Qin, M. Sun, and X. Fang, “Global adaptive neural network control for a class of uncertain non-linear systems,” IET Control Theory Appl., vol. 5, no. 5, pp. 655–662, Mar. 2011. [23] T. Zhang, S. S. Ge, and C. C. Hang, “Stable adaptive control for a class of nonlinear systems using a modified Lyapunov function,” IEEE Trans. Autom. Control, vol. 45, no. 1, pp. 129–132, Jan. 2000. [24] S. N. Huang, K. K. Tan, and T. H. Lee, “Further results on adaptive control for a class of nonlinear systems using neural networks,” IEEE Trans. Neural Netw., vol. 14, no. 3, pp. 719–722, May 2003. [25] S. N. Huang, K. K. Tan, and T. H. Lee, “An improvement on stable adaptive control for a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. 49, no. 8, pp. 1398–1403, Aug. 2004. [26] Y. Lee and S. H. Zak, “Uniformly ultimately bounded fuzzy adaptive tracking controllers for uncertain systems,” IEEE Trans. Fuzzy Syst., vol. 12, no. 6, pp. 797–811, Dec. 2004. [27] W. Chen and Z. Zhang, “Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign,” Fuzzy Sets Syst., vol. 161, no. 6, pp. 821–836, 2010. [28] A. Chaillet and A. Loría, “Uniform semiglobal practical asymptotic stability for non-autonomous cascaded systems and applications,” Automatica, vol. 44, no. 2, pp. 337–347, Feb. 2008. [29] B. Xian, D. M. Dawson, M. S. Queiroz, and J. Chen, “A continuous asymptotic tracking control strategy for uncertain nonlinear systems,” IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1206–1211, Apr. 2004. [30] Y. P. Pan, R. J. Chen, H. Z. Tan, and M. J. Er, “Asymptotic stabilization via adaptive fuzzy control,” in Proc. IEEE Int. Conf. Fuzzy Syst., Hyderabad, India, 2013, pp. 1–5. [31] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ, USA: Prentice Hall, 2002. [32] C. S. Chen, “Robust self-organizing neural-fuzzy control with uncertainty observer for MIMO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 694–706, Aug. 2011.

Yongping Pan received the B.Eng. degree in automation, the M.Eng. degree in control theory and control engineering from the Guangdong University of Technology, Guangzhou, China, in 2004 and 2007, respectively, and the Ph.D. degree in control theory and control engineering from the South China University of Technology, Guangzhou, in 2011. From 2007 to 2008, he was a Control Engineer with the Santak Electronic Corporation, Ltd., Eaton Group, Shenzhen, China, and the Light Engineering Corporation, Ltd., Guangzhou. From 2011 to 2013, he was a Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He is currently a Research Fellow with the Department of Biomedical Engineering and the Singapore Institute for Neurotechnology, National University of Singapore, Singapore. He has authored over 40 peer-reviewed papers in journals and conferences. His current research interests include adaptive control, computational intelligence, intelligent robotics, and embedded systems. Dr. Pan received the Rockwell Automation Master Scholarship and the Graduate Students Academic Award from the university in 2006, and the Innovation Fund of Excellent Doctoral Dissertations and the Excellent Graduate Student Award from the university in 2010. He was invited as an Associate Editor for two international journals. He also serves as a Reviewer for some flagship journals.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PAN et al.: GLOBAL ASYMPTOTIC STABILIZATION USING ADAPTIVE FUZZY PD CONTROL

Haoyong Yu (M’10) received the B.S. and M.S. degrees in mechanical engineering from the Shanghai Jiao Tong University, Shanghai, China, in 1988 and 1991, respectively, and the Ph.D. degree in mechanical engineering from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2002. He was a Principal Member of Technical Staff with the DSO National Laboratories, Singapore, until 2002. He is currently an Assistant Professor with the Department of Biomedical Engineering and a Principal Investigator with the Singapore Institute of Neurotechnology, National University of Singapore, Singapore. His current research interests include medical robotics, rehabilitation engineering and assistive technologies, and system dynamics and control. Dr. Yu received the Outstanding Poster Award at the IEEE Life Sciences Grand Challenges Conference in 2013 and also served on a number of IEEE Conference Committees.

9

Tairen Sun received the M.S. degree in operations research and cybernetics from the Sun Yat-sen University, Guangzhou, China, and the Ph.D. degree in control theory and control engineering from the South China University of Technology, Guangzhou, in 2008 and 2011, respectively. He is currently a Lecturer with the School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, China. His current research interests include intelligent control, multiagent cooperative control, and robot control.