Adaptive fuzzy PD control with stable H∞ tracking guarantee

Author’s Accepted Manuscript Adaptive fuzzy PD control with stable H∞ tracking guarantee Yongping Pan, Meng Joo Er, Tairen Sun, Bin Xu, Haoyong Yu www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)30987-0 http://dx.doi.org/10.1016/j.neucom.2016.08.091 NEUCOM17525

To appear in: Neurocomputing Received date: 11 March 2016 Revised date: 8 August 2016 Accepted date: 28 August 2016 Cite this article as: Yongping Pan, Meng Joo Er, Tairen Sun, Bin Xu and Haoyong Yu, Adaptive fuzzy PD control with stable H∞ tracking guarantee, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.091 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive fuzzy PD control with stable H ∞ tracking guarantee  Yongping Pan a, Meng Joo Er b, Tairen Sun c , Bin Xu d, Haoyong Yu a a Department b School c School

of Biomedical Engineering, National University of Singapore, Singapore 117583, Singapore

of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore

of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China

d School

of Automation, Northwestern Polytechnical University, Xi’an 710072, China

Abstract For indirect adaptive fuzzy H ∞ tracking control (AFHC) of perturbed uncertain nonlinear systems, sliding-mode control (SMC) compensation usually has to be applied to ensure stability and H ∞ robustness of the closed-loop system. We prove that indirect AFHC without SMC compensation is sufficient to guarantee stable H ∞ tracking under given initial conditions and parameter constraints. The control structure only includes an indirect adaptive fuzzy control term and a proportional derivative (PD) control term. A certainty equivalent control law is slightly modified such that both a lumped perturbation and adaptive laws are independent of the PD control term. This modification is significant since it not only plays a key role in stability analysis, but also alleviates some drawbacks of existing AFHC approaches for practical applications. An illustrative example has been provided to verify correctness of the developed theoretical result. Key words: Adaptive control, disturbance, neuro-fuzzy system, H ∞ tracking, nonlinear system, uncertainty.

 Corresponding author: H. Yu. Email addresses: [email protected] (Yongping Pan), [email protected] (Meng Joo Er), [email protected] (Tairen Sun), [email protected] (Bin Xu), [email protected] (Haoyong Yu). Preprint submitted to Elsevier Science

29 August 2016

1

Introduction

Adaptive approximation-based control (AAC) using fuzzy systems or neural networks is effective for handling functional uncertainties in nonlinear systems [1] and has kept great attraction in recent years, where some latest results can be referred to [2–20]. This study is focused on the control of perturbed uncertain nonlinear systems. For simplifying illustration, consider a class of single-input single-output (SISO) perturbed affine nonlinear systems in the Brunovsky canonical form as follows [21]: ⎧ ⎪ ⎪ ⎪ x˙ i ⎪ ⎪ ⎨

x˙ n ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= xi+1 (i = 1, 2, · · · , n − 1) = f (x) + g(x)u + d(t)

,

(1)

y = x1

in which x(t) := [x1 (t), x2 (t), · · · , xn (t)]T ∈ Rn is a state vector, u(t) ∈ R and y(t) ∈ R are a control input and a system output, respectively, f (x), g(x) : Rn → R are unknown functions, and d(t) ∈ R is an unknown external disturbance. Typically, AAC includes two different schemes, namely direct and indirect schemes. In the direct scheme, a single approximator is applied as a controller to approximate an ideal control law, whereas in the indirect scheme, two approximators are applied as estimation models to approximate the functions f and g, respectively. In practice, AAC systems can usually be influenced by various perturbations such as approximation errors, external disturbances and unmodelled dynamics, which cannot be handled by function approximation and can degrade control performances or even destroy system stability [31]. Proportional derivative (PD) control is a popular method for suppressing the aforementioned perturbations [21–40]. It is shown in [21] that adaptive fuzzy H ∞ tracking control (AFHC), which is composed of an indirect adaptive fuzzy control (AFC) term and a PD control term, guarantees that the L2 -gain from a lumped perturbation wo (including optimal approximation errors) to tracking errors is not larger than a prescribed attenuation level. Usually, g must be known in the direct AFHC [21–23]. This limitation was relaxed by applying some constants to replace g during control synthesis in [24–26]. Yet, such a treatment greatly increases the gain of wo . The indirect AFHC does not have any special requirement on g [21]. However, the classical indirect AFHC [21] and some of its variations such as those in [27–30] are subjected to several fundamental limitations as follows [39,40]: 1) Standard H ∞ robustness cannot be guaranteed since wo relies on u; 2) there is a critical stability problem because a stability condition may not be satisfied even if optimal approximation errors are small. Sliding-mode control (SMC) compensation with known certain plant bounds or adaptive bounding techniques is an effective method for tackling 2

these limitations [31–38]. However, this method results in some side effects as follows: 1) More priori knowledge of the plant has to be known; 2) there is a trade-off between chattering at u and tracking accuracy; 3) implementation cost can be increased due to the additional SMC term. To our knowledge, how to tackle the drawbacks of the classical indirect AFHC without SMC compensation is still an open question.

In this paper, a novel methodology of indirect AFHC is proposed to guarantee stable H ∞ tracking without SMC compensation. The design procedure of the proposed approach is as follows: Firstly, the certainty equivalent control law is slightly modified such that both a lumped perturbation and adaptive laws are independent of the PD control term; secondly, a domain of fuzzy approximation is determined under given initial conditions; thirdly, parameter constraints that guarantee system stability are obtained by the Lyapunov synthesis. It is proven that the closed-loop system achieves stable H ∞ tracking in the sense that all involved signals are uniformly bounded and all tracking errors converge to a small neighbourhood of zero dominated by control parameters. Compared with existing studies of indirect AFHC, the contributions of this study include: 1) A slightly modified certainty equivalent control law is proposed to facilitate control analysis; 2) it is proven that the proposed indirect AFHC is sufficient to guarantee stable H ∞ tracking without SMC compensation.

The rest of this paper is organized as follows: The problem considered is formulated in Section 2; the stable AFHC is designed in Section 3; an illustrative example is given in Section 4; conclusions are summarized in Section 5. Throughout this paper, R, R+ , Rn and Rn×m denote the spaces of real numbers, positive real numbers, real n-vectors and real n×m-matrices, respectively, · and ·∞ denote the Euclidean-norm and ∞-norm, respectively, L2 and L∞ denote the spaces of square-integrable signals and bounded signals, respectively, λmin (·) and λmax (·) are the functions of minimal and maximal eigenvalues, respectively, min(·), max(·) and sup(·) are the functions of minimum, maximum and supremum, respectively, col(α, β) := [αT , β T ]T , and C k represents the space of functions whose k-order derivatives all exist and are continuous, where α, β ∈ Rn , and n, m and k are positive integers. 3

2

Problem formulation

2.1 System description Revisit the system (1) with f and g being of C 1 , and rewrite it in the following compact form: ⎧ ⎪ ⎨x ˙

= Ax + b (f (x) + g(x)u + d(t))

⎪ ⎩y

= cT x

(2)

where A, b and c are given by ⎛

⎜0 ⎜. ⎜. ⎜.

1 ··· .. . . . .

⎜0 ⎜ ⎝

0 ···

A=⎜ ⎜

⎞

⎛ ⎞

0⎟ .. ⎟ .⎟ ⎟

⎟,b ⎟ 1⎟ ⎟ ⎠

0 0 ··· 0

⎛ ⎞

⎜0⎟ ⎜.⎟ ⎜.⎟ ⎜.⎟

⎟ =⎜ ⎜ ⎟,c = ⎜0⎟ ⎜ ⎟ ⎝ ⎠

1

⎜1⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜ ⎟. ⎜.⎟ ⎜ .. ⎟ ⎜ ⎟ ⎝ ⎠

0

Let yd (t) ∈ R denote a desired output and e1 (t) := y(t) − yd (t) denote an out(n−1) put tracking error. Define yd (t) := [yd (t), y˙ d(t), · · · , yd (t)]T ∈ Rn , yde (t) := (n) col(yd (t), yd (t)) ∈ Rn+1 and e(t) := x(t) − yd (t) = [e1 (t), e2 (t), · · · , en (t)]T . Then, the following assumptions are made accordingly [37]. Assumption 1 : There exist constants g0 , d¯ ∈ R+ to satisfy |g(x)| > g0 and ¯ Without loss of generality, it is assumed that g(x) > g0 . |d(t)| ≤ d. (i)

Assumption 2 : It has yd (t) ∈ L∞ for i = 0, 1, · · · , n. Select k = [k1 , k2 , · · · , kn ]T ∈ Rn such that Ac := (A − bkT ) ∈ Rn×n is strictly Hurwitz. For any given Q ∈ Rn×n satisfying Q = QT > 0, a unique solution P ∈ Rn×n satisfying P = P T > 0 exists for the following Riccati-like equation: ATc P



2 1 − 2 bT P = 0 + P Ac + Q − P b r ρ

(3)

with r, ρ ∈ R+ if and only if 2ρ2 ≥ r [21]. Making 2ρ2 = r, one also obtains the following Lyapunov equation: ATc P + P Ac = −Q.

(4)

Remark 1 : In this study, only the class of SISO affine nonlinear systems in (1) or (2) under Assumptions 1 and 2 is considered for the simplification of illustration. In Assumption 1, |g(x)| > g0 and |d(t)| ≤ d¯ imply the controllability of the system (1) and the boundedness of d(t), respectively. In Assumption 2, 4

(n)

yd , y˙ d , · · · , yd being of L∞ ensures the boundedness of yde and the existence of an ideal control law for the system (1). Similar theoretical results of this study are passible to be derived for a more general class of multi-input multioutput nonaffine nonlinear systems [37] and a class of time-delayed nonlinear systems [41] under certain conditions.

2.2 Function approximation If f and g are known a priori and d(t) = 0 in the system (1), then the following ideal control law: u∗ = (−f (x) + v)/g(x) (5) (n)

with v := yd − kT e makes the closed-loop dynamics e˙ = Ac e globally and exponentially stable [1]. As f and g are unknown and d = 0 in (1) in this study, the following C 1 linearly parameterized fuzzy systems [1]: ⎧ ⎪ ˆ θˆf ) ⎨ f(x|

 ˆ = θˆfT ξ(x) = M j=1 θf j ξj (x)

⎪ ⎩g ˆ(x|θˆ

ˆT g ) = θg ξ(x) =

M

ˆ j=1 θgj ξj (x)

(6)

are applied to approximate f and g in (5), respectively, where ξ : Rn → RM satisfying ξ ≤ φ is a vector of fuzzy basis functions, θˆf = [θˆf 1 , θˆf 2 , · · · , θˆf M ]T ∈ RM and θˆg = [θˆg1 , θˆg2 , · · · , θˆgM ]T ∈ RM are vectors of adjustable parameters, φ ∈ R+ is a finite constant, and M is the number of fuzzy rules. For facilitating presentation, define compact sets 1 Ωe := {e|eT P e/2 ≤ ce }, Ωd := {yd |yd  ≤ md }, (n)

Ωde := {yde |yd ∈ Ωd , |yd | ≤ mde }, Dx := {x| x = e + yd , yd ∈ Ωd , e ∈ Ωe }, Ωf := {θˆf | − mf ≤ θˆf j ≤ mf , j = 1, · · · , M}, Ωg := {θˆg | 0 < mg ≤ θˆgj ≤ mg , j = 1, · · · , M} where Dx = Ωd × Ωe is a domain of fuzzy approximation, and ce , md , mde , mf , mf , mg , mg ∈ R+ are prespecified parameters, Define optimal approximation errors wf and wg as follows: ⎧ ⎪ ⎨ wf (x)

:= f (x) − fˆ(x|θf∗ )

⎪ ⎩ wg (x)

:= g(x) − gˆ(x|θg∗ )

1

(7)

In mathematics, a compact set is a closed and bounded subset of an Euclidean space Rk for any k = 1, 2, 3 ....

5

where θf∗ and θg∗ are vectors of optimal parameters given by θf∗



:= arg min

θˆf ∈Ωf

θg∗ := arg min

sup



θˆg ∈Ωg

x∈Dx

    ˆ ˆ f (x) − f (x|θf ) , 



  sup g(x) − gˆ(x|θˆg ) .

x∈Dx

2.3 Revisit of previous results A certainty equivalent control law with respect to (5) is presented as follows [21]: u = (−fˆ(x|θˆf ) + v)/ˆ g(x|θˆg ) −eT P b/(rˆ g(x|θˆg )) 





uf





(8)

uc

where uf denotes an indirect AFC part, and uc denotes a PD control part that is essential for achieving the H ∞ tracking performance as shown in [21]. Applying (5)-(8) to (2), one get a closed-loop tracking error dynamics as follows: 

e˙ = Ac e + b θ˜fT ξ(x) + θ˜gT ξ(x)u + wo − eT P b/r



(9)

with θ˜f := θf∗ − θˆf and θ˜g := θg∗ − θˆg , in which wo is a lumped perturbation in [21] given by wo (x, u, t) := wf (x) + wg (x)u + d(t). (10) Adaptive laws of θˆf and θˆg in [21] are given as follows: ⎧ ⎪ ˆ˙ f ⎨θ

= proj(γf ξ(x)eT P b)

⎪ ⎩θ ˆ˙

= proj(γg ξ(x)eT P bu)

g

(11)

in which γf , γg ∈ R+ are learning rates, and proj(•) = [proj(•1 ), · · · , proj(•M )]T is a projection operator given by [1]

proj(•j ) :=

⎧ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨

0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ •j

if θˆj = m and •j < 0 if θˆj = m and •j > 0 otherwise

with θˆj = θˆf j or θˆgj , m = −mf or mg and m = mf or mg . Combining the definitions of Ωf and Ωg with the above projection operator, one has that if θˆf and θˆg are in the interiors of Ωf and Ωg , respectively or on the boundaries but moving toward the insides of Ωf and Ωg , respectively, then the projection 6

is not effective. Otherwise, if θˆf and θˆg are on the boundaries but moving toward the outsides of Ωf and Ωg , respectively, then θˆf and θˆg are projected to the boundaries of Ωf and Ωg , respectively. Choose a Lyapunov function candidate 1 ˜T ˜ 1 ˜T ˜ 1 θf θf + θ θg V (z) = eT P e + 2 2γf 2γg g

(12)

with z := col(e, θ˜f , θ˜g ) for the closed-loop system constituted by (8)-(11). The following lemma shows the performance of the previous indirect AFHC in [21] for the system (1). Lemma 1 [21]: Consider the system (1) with Assumptions 1 and 2 driven by the control law comprised of (8) and (11), where P in (8) is obtained by solving (3). If θˆf (0) ∈ Ωf and θˆg (0) ∈ Ωg , then the closed-loop system achieves the following H ∞ tracking performance:  T 0

eT (t)Qe(t)dt ≤ 2V (0) + ρ2

 T 0

wo2 (t)dt

(13)

∀T ∈ [0, ∞) and ∀wo ∈ L2 [0, T ], which implies the L2 -gain from w0 to e is not larger than the prescribed attenuation level ρ while V (0) = 0. Remark 2 : The fundamental limitations of the indirect AFHC in [21] and some of its variations such as those in [27–30] are shown as follows: • The performance (13) cannot guarantee standard H ∞ robustness as a small ρ also leads to a large wo (using (8), (10) and 2ρ2 ≥ r) which may cancel out the effect of ρ [39]; • The closed-loop system has a critical stability problem as a stability condition may not be guaranteed even if both wf and wg are small 2 . The SMC compensation with known certain plant bounds or adaptive bounding estimates [32–38] is the only effective method for tackling these limitations so far. Nevertheless, this method results in some side effects as discussed in Section I.

2

Please refer to the stability condition (26) and Remark 5 in the subsequent contents for better understanding this stability problem. Specifically, if w ¯o , an upper ¯n in (26), then because decreasing ρ also bound of wo in (10), is applied to replace w increases w ¯o , the last term of (26) cannot be diminished by the decrease of ρ such that the inequality (26) may not be guaranteed.

7

3

Stable adaptive fuzzy control

3.1 Closed-loop system dynamics This section shows that the drawbacks of the indirect AFHC manifested in Remark 2 can be avoided by a slight modification on the certainty equivalent control law (8) as follows: u = (−fˆ(x|θˆf ) + v)/ˆ g (x|θˆg ) −eT P b/r 





uf



(14)



uc

with r = 2ρ2 , where the PD control term uc is independent of gˆ. From the uf part in (14) and the definition of v, one obtains (n) yd = kT e + fˆ(x|θˆf ) + gˆ(x|θˆg )uf .

Subtracting the above equality from the second line of (1) and noting e = x − yd and u = uf + uc , one obtains a new closed-loop tracking error dynamics as follows: 

e˙ = Ac e + b θ˜fT ξ(x) + θ˜gT ξ(x)uf + wn − g(x)eT P b/r



(15)

where wn is a new lumped perturbation given by wn (x, uf , t) := wf (x) + wg (x)uf + d(t).

(16)

It is worth noting that differing from the lumped perturbation wo in (10), the lumped perturbation wn in (16) is independent of the PD control term uc . From (14) and the definitions of v and e, one gets uf = uf (e, yde , θˆf , θˆg ). Let a possible upper bound of wn be w¯n := max |wn (x, uf , t)| = w¯f + w¯g u¯f + d¯

(17)

with w¯f := max |wf (x)|, w¯g := max |wg (x)| and u¯f := max |uf (e, yde , θˆf , θˆg )|, in which the maximization is done over all e ∈ Ωe , yde ∈ Ωde , θˆf ∈ Ωf and θˆg ∈ Ωg . Remark 3 : It should be mentioned that the possible bound of wn in (17) obtained under e ∈ Ωe is just used for establishing the stability condition in Theorem 1 of Section 3.2 rather than acts as a stability condition directly. In fact, using e ∈ Ωe as a stability condition is improper since e ∈ Ωe implies the boundedness of the system (1). The approach of this study related to the specification of both initial conditions and control parameters is similar to that of the classical work [42], where the proof of Theorem 1 is inspired by the concepts of local attraction and positively invariant sets therein. 8

3.2 Control synthesis and analysis Adaptive laws of θˆf and θˆg are redesigned as follows 3 : ⎧ ⎪ ˆ˙ f ⎨θ

= proj(γf ξ(x)eT P b)

⎪ ⎩θ ˆ˙

= proj(γg ξ(x)eT P buf )

g

(18)

which are independent of the PD control term uc . To facilitate the proof of the following theorem, define two constants cf and cg as follows: ⎧ ⎪ ⎨ cf ⎪ ⎩c

g

:= maxθˆf ,θ∗ ∈Ωf {θ˜fT θ˜f /(2γf )} f

:= maxθˆg ,θg∗ ∈Ωg {θ˜gT θ˜g /(2γg )}

(19)

Then, let a compact set that includes all possible e(0) be Ωe0 := {e|eT P e/2 ≤ ce0 } with ce0 < ce and Dx0 = Ωd × Ωe0 . Choose the Lyapunov function candidate in (12) for the closed-loop system composed of (14), (15) and (18). The following theorem shows the main results of this study. Theorem 1 : Consider the system (1) under Assumptions 1 and 2 driven by the control law (14) with (18), where P in (14) is obtained by solving (4). For any given x(0) ∈ Dx0 , θˆf (0) ∈ Ωf , θˆg (0) ∈ Ωg , yde (t) ∈ Ωde and an approximation domain Dx subjected to (20) ce ≥ ce0 + cf + cg there exist suitably large PD control gain 1/r in (14), learning rates γf and γg in (18) and λmin (Q) in (4) to satisfy (26) and (28) such that the closed-loop system achieves: (1) Stability in the sense that x(t) ∈ Dx , θˆf (t) ∈ Ωf , θˆg (t) ∈ Ωg and u(t) ∈ Ωu in (27), ∀t ≥ 0; (2) The following H∞ tracking performance:  T 0

eT (t)Qe(t)dt ≤ 2V (0) + ρ2

 T 0

w 2 (t)dt

(21)

√ with w := wn / g0 , ∀T ∈ [0, ∞) and ∀w ∈ L2 [0, T ], which implies that the L2 -gain from w to e is not larger than the prescribed attenuation level ρ while V (0) = 0; 3

For application consideration, a dead-zone modification in [1, Th. 7.2.3] can be applied to the adaptive laws (18) to avoid parameter drift. However, this important technique is tangential to the topic of this study.

9

(3) The convergence of e(t) to a small neighbourhood of zero. Proof : Firstly, differentiating (12) along (15) with respect to time t and applying (4) to the resulting expression, one obtains V˙ = −eT Qe/2 + eT P bwn − g(x)(eT P b)2/r   ˆ˙ f /γf − θ˜T θ ˆ˙ + eT P b θ˜fT ξ(x) + θ˜gT ξ(x)uf − θ˜fT θ g g /γg . Noting Assumption 1, the above result leads to V˙ ≤ −eT Qe/2 + eT P bwn − g0 (eT P b)2 /r   ˆ˙ f /γf + θ˜fT eT P bξ(x) − θ   ˆ˙ g /γg . + θ˜gT eT P bξ(x)uf − θ

(22)

From the result in [1, Sec. 4.6.1], proj(·) in (18) guarantees θˆf (t) ∈ Ωf and θˆg (t) ∈ Ωg with ∀t ≥ 0, ∀θˆf (0) ∈ Ωf and ∀θˆg (0) ∈ Ωg , and ⎧   ⎪ ˆ˙ f /γf ≤ 0 ⎨ θ˜T eT P bξ(x) − θ f   ⎪ ⎩ θ˜T eT P bξ(x)u − θ ˆ˙ /γ ≤ f

g

g

g

0

.

Applying the above inequalities to (22) yields V˙ ≤ −eT Qe/2 − g0 (eT P b)2/r + eT P bwn . By using r = 2ρ2 , the above expression becomes 



V˙ ≤ −eT Qe/2 − g0 (eT P b)2 /(2ρ2 ) − eT P bwn /g0 . √ Applying the triangle inequality to the above result and noting w = wn / g0 , one immediately gets V˙ (t) ≤ −eT (t)Qe(t)/2 + ρ2 w 2 (t)/2.

(23)

Second, it follows from (17), (19) and (23) that V˙ ≤ −λq V + λq (cf + cg ) + (ρw¯n )2 /(2g0) 

= −λq V − cf − cg − (ρw¯n )2 /(2g0λq )



with λq := λmin (Q)/λmax (P ) ∈ R+ . Thus, one has V˙ (t) ≤ 0, ∀V (t) ≥ cv

(24)

with cv := cf + cg + (ρw¯n )2 /(2g0λq ) ∈ R+ . It follows from x(0) ∈ Dx0 and Dx0 = Ωd × Ωe0 that e(0) ∈ Ωe0 . Using e(0) ∈ Ωe0 , the definition of Ωe0 , (12), 10

(19) and (20), one obtains V (0) ≤ ce0 + cf + cg ≤ ce .

(25)

Using (24) and (25), one gets that the set Ωz := {z|V ≤ ce , θˆf ∈ Ωf , θˆg ∈ Ωg } is positively invariant, ∀ce ≥ cv . Since wn in (16) is independent of uc , one has that w¯n in (17) is independent of ρ. Thus, it follows from the definition of cv that cv can be diminished by the decrease of ρ, cf and/or cg . From r = 2ρ2 and (19), increasing 1/r, γf and γg decrease ρ, cf and cg , respectively. Thus, for any given designed and fixed ce in (20), there exist suitably large 1/r, γf and γg to satisfy (26) ce ≥ cf + cg + (ρw¯n )2 /(2g0λq ) resulting in ce ≥ cv . Therefore, one obtains V (t) ≤ ce , t ≥ 0, which implies that the closed-loop system is stable in the sense that e(t) ∈ Ωe , x(t) ∈ Dx , and u(t) ∈ Ωu with Ωu :={u|e ∈ Ωe , yde ∈ Ωde , θˆf ∈ Ωf , θˆg ∈ Ωg }.

(27)

Integrating both sides of (23) at t = [0, T ) yields V (T ) − V (0) ≤

1 T T 1 T 2 e (t)Qe(t)dt + ρ2 w (t)dt 2 0 2 0

with T ∈ [0, ∞). After simple manipulation on the above inequality, one gets the H ∞ tracking result (21). Finally, it follows from (23), (16) and (17) that ¯ 2 /(2g0 ). V˙ ≤ −λmin (Q)e2 /2 + ρ2 (w¯f + w¯g |uf | + d) Combining with the expression of uf in (14), one gets V˙ ≤ −λmin (Q)e2 /2 + ρ2 (δ + ηe)2 /(2g0) with δ := w¯f +w¯g (max |fˆ(·)|+mde )/ min |ˆ g (·)|+d¯ ∈ R+ and η := w¯g k/ min |ˆ g (·)| ∈ + ˆf R , where the maximization and minimization are done over all x ∈ Dx , W ˆ g ∈ Ωg . Accordingly, it follows from the above result that ∈ Ωf and W V˙ ≤ −λmin (Q)e2 /2 + ρ2 (δ 2 + 2δηe + (ηe)2 )/(2g0) ≤ −λmin (Q)e2 /2 + ρ2 (1 + η 2 )(δ 2 + e2 )/(2g0). If the selection of Q is restricted to λmin (Q) > ρ2 (1 + η 2 )/g0 11

(28)

then one immediately obtains V˙ (t) ≤ −λa e(t)2 /2 + ϕ with λa := λmin (Q) − ρ2 (1 + η 2 )/g0 ∈ R+ and ϕ := (ρδ)2 (1 + η 2 )/(2g0 ) ∈ R+ . Thus, it is straightforward to get V˙ (t) ≤ −λb V (t) + λb (cf + cg ) + ϕ

(29)

where λb := λa /λmax (P ) ∈ R+ . Solving the above inequality using [1, Lemma A.3.2], one obtains V (t) ≤V (0) exp (−λb t) + (cf + cg + ϕ/λb ). The above result implies that e(t) converges to a set Ωe∞ = {e|eT P e ≤ cf + cg + ϕ/λb } that can be contracted to a small neighborhood of zero by the increase of 1/r, λmin (Q), γf and γg . 2 Remark 4 : The results in Theorem 1 depend on the constraints in (20) and (26) related to two constants cf and cg in (19). Since the fuzzy approximation in (7) is done over x ∈ Dx , the compact sets Ωf and Ωg to which the optimal parameters θf∗ and θg∗ belong themselves depend on the compact set Dx . Hence, Dx must be defined prior, and consequently, is independent of Ωf and Ωg , which results in the requirement of large learning rates γf and γg to satisfy the inequalities (20) and (26) [42]. Noting θˆf , θf∗ ∈ Ωf and θˆg , θg∗ ∈ Ωg , one gets that θ˜f and θ˜g are finite. Therefore, cf and cg in (19) can be diminished by the increase of γf and γg , respectively. If a persistent-exciting condition is satisfied such that θ˜f → 0 and θ˜g → 0, the requirement on large γf and γg can be relaxed to a certain extent. Remark 5 : Compared with previous AFHC laws, the major differences of the proposed control law (14) with (18) are that the PD control term uc in (14) is independent of gˆ and the adaptive laws in (18) are independent of uc . Compared with the previous H ∞ tracking result (13), the significance of the deduced H ∞ tracking result (21) is that the term w in (21) is independent of uc . The proof of Theorem 1 has successfully shown that the slight modification in the proposed control law (14) leads to several salient features as follows: • Standard H ∞ robustness can be guaranteed because decreasing ρ in uc √ enhances the H ∞ tracking result (21) without enlarging w = wn / g0 , where wn is given by (16); • Since w¯n given by (17) is independent of ρ, closed-loop stability can be guaranteed under a suitable choice of the approximation domain Dx restricted 12

by (20) and the control parameters ρ, Q, γf and γg restricted by (26) and (28). Consequently, the proposed approach completely avoids the drawbacks of the classical indirect AFHC in [21] and some of its variations such as those in [27– 30] without SMC compensation. The tracking performance can be enhanced by the increase of 1/r, λmin (Q), γf and γg , and the terms w¯n and g0 in (26) do not need to be known in practice as the constraint (26) can be easily satisfied by trail-and-error. The cost of applying the proposed approach is that the global stability obtained by the indirect AFHC with SMC compensation is degraded to be local.

4

An Illustrative Example

To verify the theoretical result in Theorem 1, consider an inverted pendulum that can be modeled by (1) with [37] ⎧ ⎪ ⎨ f (x)

=

⎪ ⎩ g(x)

=

gv sin x1 −mp lp x22 cos x1 sin x1 /(mc +mp ) 4lp /3−lp mp cos2 x1 /(mc +mp ) cos x1 /(mc +mp ) 4lp /3−(lp mp cos2 x1 )/(mc +mp )

in which x1 (rad) and x2 (rad/s) are the angular position and velocity of the pendulum, respectively, gv (m/s2 ) is the gravitational acceleration, mc (kg) is the mass of the cart, mp (kg) is the mass of the pendulum, and lp (m) is the half-length of the pendulum. For simulations, let mc = 1, mp = 0.1, gv = 9.8, lp = 0.5, d(t) = 5 sin(t), x(0) = [π/6, 0]T and yd (t) = (π/6) sin(t). According to the above control information, let Ωd = [−π/6, π/6]×[−π/6, π/6], Dx0 = [−π/6, π/6] × [−π/6, π/6] such that Ωe0 = [−π/3, π/3] × [−π/3, π/3], and design a sufficiently large Dx = [−π/4, π/4] × [−π/4, π/4] such that its corresponding ce can satisfy ce ≥ ce0 + cf + cg in (20). The construction of the proposed control law (14) with (18) is given as follows: First, to construct the fuzzy system (6), select fuzzy membership functions as μlii = exp(−(xi − (π/8)(li − 3))2 /2/(0.12)2) with i = 1, 2 and li = 1 to 5 to cover Dx ; second, to construct the control law (14), select k = [1, 2]T , ρ ∈ [2−7 , 2−2 ], Q = diag(q, q) with q ∈ [10 × 50 , 10 × 55 ] and obtain P by solving (4); finally, to determine the adaptive law (18), set mf = mf = 20, mg = 1, mg = 5, γf ∈ [51 , 56 ], γg = 1, θˆf (0) = 0 and θˆg (0) = [1.2, 1.2, · · · , 1.2]T . Let ρ = 1/4, q = 10 and γf = 5 be initial values of the control parameters for all simulation cases. Simulations are carried out on MATLAB software running in Windows 7, where the step size is set to be 0.001 s, and the other settings are kept default. To show the merits of the proposed approach, the indirect AFHC law (8) with (11) given by [21] is selected as a baseline controller. 13

A comparison of tracking performances between the two controllers under various values of ρ is shown in Fig. 1, where an integral absolute error (IAE) with respect to e and a logarithmic transformation N = log2 (1/ρ) − 1 are applied to the vertical and horizontal axes, respectively for clear illustration. It is observed that these two approaches have very similar tracking performances. Nevertheless, the control law (8) has two drawbacks that can only be observed during simulation processes rather than from simulation results as follows. Firstly, since 1/ˆ g (x|θˆg ) can be regarded as a variable gain of the PD control term uc in (8), the performance of uc in (8) can be affected by the settings of both the initial adaptive parameter θˆg (0) and the design parameters of 3.5 by the AFHC law (8) by the AFHC law (14) 3

IAE

2.5 2 1.5 1 0.5 1

2

3

4

5

6

N

Fig. 1. Relationships between ρ and IAE by both the approaches. 3.5

by varying ρ by varying Q by varying γ

3

f

IAE

2.5 2 1.5 1 0.5 1

2

3

4

5

6

N

Fig. 2. Relationships among ρ, Q and γf and IAE by the proposed approach. 0.8

ρ = 1/4 ρ = 1/8 ρ = 1/16

0.7 0.6

0.04 0.03 0.02 0.01

||e||

0.5

0

0.4

−0.01

20

18

16

0.3 0.2 0.1 0 −0.1 0

2

4

6

8

10 time(s)

12

14

16

18

20

Fig. 3. Tracking error results by the proposed approach with various ρ.

14

the adaptive laws in (11), which is undesirable since the dependence of uc on gˆ(x|θˆg ) makes H ∞ tracking blurry. Specifically, the tracking performance would be degraded while θˆg (0) and/or mg are too high but the learning rate γg is too small. Secondly, the adaptation of θˆg in (11) can be diverged easily during simulations while the change rate of u (related to uc and ρ) and γg are too large. A smaller step size can be adopted to solve this problem at the cost of increasing computational burden. Therefore, these two drawbacks are detrimental for practical applications. The merits of the applied control law (14) are that it not only makes stable H∞ tracking without SMC compensation possible, but also completely avoids the aforementioned two drawbacks. A comparison of tracking performances by the proposed approach with various values of ρ, Q and γf is given in Fig. 2, where logarithmic transformations N = log2 (1/ρ) − 1, N = log5 (q/10) + 1 and N = log5 (γf ) are applied to ρ, q and γf , respectively to uniform horizontal axes. It is observed during simulations that the step size must be decreased to ensure simulation stability if ρ, Q and/or γf are increased, which results in increased computational burden. It is worth noting that the closed-loop system is unstable as long as q < 1 and/or γf < 1, which implies (26) and (28) are unsatisfied. Finally, Figs. 3-5 are also provided to further show how the control parameters ρ, Q and γf affect tracking accuracy in the proposed approach. 0.8

0.04

Q = 10 Q = 50 Q = 250

0.7 0.6

0.03 0.02 0.01

||e||

0.5

0

0.4

−0.01

16

18

20

0.3 0.2 0.1 0 −0.1 0

2

4

6

8

10 time(s)

12

14

16

18

20

Fig. 4. Tracking error results by the proposed approach with various Q. 0.8

0.04

γ =5 f

0.7

3

0.03

γf = 56

0.01

γ =5 f

0.6

0.02

||e||

0.5

0

0.4

−0.01 15

17

16

18

19

20

0.3 0.2 0.1 0 −0.1 0

2

4

6

8

10 time(s)

12

14

16

18

20

Fig. 5. Tracking error results by the proposed approach with various γf .

15

5

Conclusions

This study has successfully proven that an indirect AFHC approach without SMC compensation can guarantee stable H ∞ tracking under given initial conditions and parameter constraints. A slight modification on the certainty equivalent control law is significant since it not only makes stable H∞ tracking without SMC compensation possible, but also alleviates some drawbacks of existing AFHC approaches for practical applications. Simulation results has verified correctness of the developed theoretical result. As the proposed method is based on the general class of affine nonlinear systems (1), the dissipativity and L2 -L∞ techniques in [43–45] can be naturally incorporated into this method to enhance its robustness against perturbations. This work would be done in further studies.

Acknowledgment The authors would like to thank the reviewers for their valuable comments that have greatly improved the quality of this article. This work was supported in part by the Biomedical Engineering Programme, Agency for Science, Technology and Research (A*STAR), Singapore under Grant 1421480015, in part by the Ministry of Education, Singapore (Tier 1 AcRF, RG29/15), in part by the Natural Science Foundation of Jiangsu Province, China under Grant BK20130536, and in part by the International Science and Technology Cooperation Program of China under Grant 2014DFA11580.

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