Soft Comput (2011) 15:2029–2040 DOI 10.1007/s00500-011-0702-7

ORIGINAL PAPER

Robust intelligent backstepping tracking control for wheeled inverted pendulum Chih-Hui Chiu • Ya-Fu Peng • You-Wei Lin

Published online: 29 March 2011 Ó Springer-Verlag 2011

Abstract In this study, a robust intelligent backstepping tracking control (RIBTC) system combined with adaptive output recurrent cerebellar model articulation controller (AORCMAC) and H? control technique is proposed for wheeled inverted pendulums (WIPs) with unknown system dynamics and external disturbance. The AORCMAC is a nonlinear adaptive system with simple computation, good generalization capability and fast learning property. Therefore, the WIP can stand upright when it moves to a designed position stably. In the proposed control system, an AORCMAC is used to copy an ideal backstepping control, and a robust H? controller is designed to attenuate the effect of the residual approximation errors and external disturbances with desired attenuation level. Moreover, the all adaptation laws of the RIBTC system are derived based on the Lyapunov stability analysis, the Taylor linearization technique and H? control theory, so that the stability of the closed-loop system and H? tracking performance can be guaranteed. The proposed control scheme is practical and efficacious for WIPs by simulation results. Keywords Wheeled inverted pendulum  Backstepping tracking control  H? control  Output recurrent cerebellar model articulation control

C.-H. Chiu (&)  Y.-W. Lin Department of Electrical Engineering, Yuan-Ze University, Chung-Li, Tao-Yuan 320, Taiwan, ROC e-mail: [email protected] Y.-F. Peng Department of Electrical Engineering, Ching-Yun University, Chung-Li, Tao-Yuan 320, Taiwan, ROC

1 Introduction In the past several years, there are many literatures to study the inverted pendulum (Tao et al. 2008; Takimoto et al. 2008; Casavola et al. 2004, 2006; Lin and Mon 2005; El-Hawwary et al. 2006; Er et al. 2002). The main topic of this research is how to keep the pendulum balance at the upright position. This system would become a primary tool for studying balance in active balancing system. It is also to be an important precursor to the field of legged machines and locomotion studies for robotics. An inverted pendulum is a typical unstable complex nonlinear system. It is also a very popular experiment used for educational purposes and control researches. Until now, the rail-cart structure is the most usual type in control experiments. Obviously, the railcart type inverted pendulum is a one-dimensional control structure. Nowadays, many studies of extensions of the onedimensional inverted pendulum control system have been proposed. The most interesting and challenging problem is how to control a mobile wheeled inverted pendulum (WIP) system which the cart is no longer run on a guide rail. In Ha and Yuta (1996), a trajectory tracking algorithm based on a linear state space model was proposed to control mobile robot. Graser et al. (2002) used Newtonian approach and linearization method to get a dynamic model to design a controller for a mobile inverted pendulum. In Pathak et al. (2005), a dynamic model of the WIP was derived with respect to the wheel motor torques as input while taking the nonholonomic no-slip constrains into considerations. Furthermore, two controllers were proposed to stabilize the vehicle’s pitch and position. Jung and Kim (2008) developed a mobile inverted pendulum using the neural network (NN) control combined with proportional-integralderivative (PID) controller. Ren et al. (2008) proposed a

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self-tuning PID control strategy, based on a deduced model, for implementing a motion control system that stabilizes a two-wheeled vehicle (TWV) and follows the desired motion commands. Lin and Tsai (2009) presented an interesting pedagogical tool, a self-balancing human transportation vehicle (HTV), for the teaching of feedback control concepts in undergraduate electrical, mechatronics, and mechanical engineering environments. Many recent studies achieved identification and control of dynamic systems using NNs (Agarwal 1997; Khan and Rahman 2009; Lin and Chou 2009; Mazumdar and Harley 2008; Jemei et al. 2008; Albus 1975a). In addition, many researchers have argued that NNs are powerful building blocks for a wide class of complex nonlinear system control strategies when model information is absent or when a controlled plant is considered a ‘‘black box’’ (Agarwal 1997). In terms of structure, NNs can be classified as feedforward NNs (FNNs) (Khan and Rahman 2009; Lin and Chou 2009) and recurrent NNs (RNNs) (Mazumdar and Harley 2008; Jemei et al. 2008). The ability of NNs to uniformly approximate arbitrary input–output linear or nonlinear mappings on closed subsets is extremely useful. Thus, NN-based controllers have been applied to compensate for effects of nonlinearities and system uncertainties in a control system to improve system stability, convergence and robustness. Moreover, RNNs have such capabilities as dynamic response and ability to store information that make them superior to FNNs (Mazumdar and Harley 2008; Jemei et al. 2008). The internal feedback loop of an RNN captures the dynamic response of a system with external feedback through delays. Thus, an RNN is a dynamic mapping and has good control performance in the presence of unmodeled dynamics. However, regardless of whether an FNN or RNN is used, learning is slow as all weights are updated during each learning cycle. Therefore, the appropriateness of NNs is limited for problems requiring on-line learning. Cerebellar model articulation controller (CMAC) is a type of associative memory neural network inspired by neurophysiologic theory of the cerebellum. It is first introduced by Albus (1975a, b) and has been proposed in the literature. The major advantages of CMAC included simple computation, fast learning property, good generalization capability, and easier hardware implementation. The application of CMAC can be found in many fields such as signal processing, pattern recognition, and control system. The CMAC has been already validated that it can approximate a nonlinear function over a domain of interest to any desired accuracy. The advantages of using CMAC over neural network in many practical applications have been presented in recent literatures (Lu et al. 2009; Yeh 2007; Peng et al. 2004). However, the major drawback of

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existing CMAC is that they belong to static networks. In other words, the application domain of CMAC will be limited to static mapping due to its feedforward network structure (Lee and Teng 2000; Lin et al. 2001; Chow and Fang 1998). The recently developed backstepping control technique (Peng 2009; Kokotovic 1992) is a powerful and systematic design methodology for nonlinear systems, which offers a choice to accommodate the unmodelled nonlinear effects and parameter uncertainties. The key idea of backstepping design is to select recursively some appropriate functions of state variables as fictitious control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new fictitious control design, expressed in terms of the fictitious control design from preceding design stages. The procedure terminates a feedback design for the true control input which achieves the original design objective by virtue of a final Lyapunov function, which is formed by summing the Lyapunov functions associated with each individual design stage. Thus, the backstepping control approach is capable of keeping the robustness properties with respect to the uncertainties (Krstic et al. 1995; Wai et al. 2002; Lin et al. 2005). Since the dynamic characteristics of WIPs are nonlinear and precise models are difficult to obtain, the traditional control approaches are hard to be implemented. To overcome this drawback, a robust intelligent backstepping tracking control (RIBTC) system has been proposed for WIPs. The developed RIBTC system comprises an adaptive output recurrent cerebellar model articulation controller (AORCMAC) and a robust H? controller. The AORCMAC is used to mimic an ideal backstepping control (IBC), and a robust H? controller is designed to attenuate the effect of the residual approximation errors and external desired attenuation level. Here, the output recurrent cerebellar model articulation controller (ORCMAC) architecture is a modified version of the conventional CMAC network such that a small number of receptive fields are used to capture the system dynamics and convert the static CMAC into a dynamic controller. Since the proposed controller captures the dynamic response of controlled system, the ORCMAC will achieve good control performance for a nonlinear system. In addition, the all adaptation laws of the RIBTC system are derived based on the Lyapunov stability analysis, the Taylor linearization technique and H? control theory, so that the stability of the closed-loop system and H? tracking performance can be guaranteed. Finally, the proposed RIBTC system is applied to control a WIP system. The effectiveness of the proposed robust control scheme is verified by simulation results.

Robust intelligent backstepping tracking control

2031

2 Backstepping control system for WIP

aðtÞ ¼ k1 e1 ðtÞ þ xd2 ðtÞ;

Figure 1 shows the WIP system with its three degrees of freedom. It can be able to rotate around the z axis and its vertical axis. Consider the WIP, where the dynamic equation is described as following (Grasser et al. 2002):

where k1 is a positive constant. The first Lyapunov function is chosen as

x_ 1 ðtÞ ¼ x2 ðtÞ

1 V1 ðtÞ ¼ e21 ðtÞ: 2

ð1Þ

x_ 2 ðtÞ ¼ f ðx; tÞ þ gðx; tÞuðtÞ þ dðtÞ; _ T is the state (angle) where x ¼ ½x1 ðtÞ; x2 ðtÞT ¼ ½xðtÞ; xðtÞ vector of the WIP system which is assumed to be available for measurement, f ðx; tÞ 2 R is the nonlinear dynamic function, gðx; tÞ 2 R denotes the control gain of the system and g(x, t) [ 0 for all x and t; uðtÞ 2 R is the control input and dðtÞ 2 R denotes the unknown external disturbance. The control object is to design a suitable control law for the system (Eq. 1) so that the state trajectory vector x can track a desired reference trajectory vector xd ¼ ½xd1 ðtÞ; xd2 ðtÞT ¼ ½xd ðtÞ; x_d ðtÞT despite the presence of unknown system dynamics and external disturbance. Assuming all system dynamics [f(x, t), g(x, t) and d(t)] of the system (Eq. 1) are well known, the design of IBC for the uncertain nonlinear system is described step-by- step as follows: Step 1. Define the tracking error e1 ðtÞ ¼ xd1 ðtÞ  x1 ðtÞ:

ð5Þ

Define e2 ðtÞ ¼ aðtÞ  x_ 1 ðtÞ ¼ aðtÞ  x2 ðtÞ ¼ e_ 1 ðtÞ þ k1 e1 ðtÞ:

ð6Þ

The deductive of V1(t) is V_1 ðtÞ ¼ e1 ðtÞe_ 1 ðtÞ ¼ e1 ðtÞ½e2 ðtÞ  k1 e1 ðtÞ ¼ k1 e21 ðtÞ þ e1 ðtÞe2 ðtÞ

ð7Þ

:

Then, if e2(t) = 0, we would achieve V_1 ðtÞ ¼ k1 e21 ðtÞ  0 with aðtÞ ¼ k1 e1 ðtÞ þ xd2 ðtÞ Step 2: The derivative of e2(t) is expressed as _  x€1 ðtÞ ¼ aðtÞ _  x_ 2 ðtÞ ¼ k1 e_ 1 ðtÞ þ x_ d2 ðtÞ e_ 2 ðtÞ ¼ aðtÞ  f ðx; tÞ  gðx; tÞuðtÞ  dðtÞ:

ð8Þ

The Lyapunov function is chosen as 1 V2 ðtÞ ¼ V1 ðtÞ þ e22 ðtÞ: 2

ð2Þ

Then the deductive of tracking error can be represented as

ð4Þ

ð9Þ

The derivative of V2(t) is

ð3Þ

V_2 ðtÞ ¼ V_1 ðtÞ þ e2 ðtÞe_ 2 ðtÞ ¼ k1 e21 ðtÞ þ e2 ðtÞðk1 e_1 ðtÞ þ x_ d2 ðtÞ  f ðx; tÞ  gðx; tÞuðtÞ  dðtÞÞ: ð10Þ

The x2(t) can be viewed as a virtual control in above equation. Define the following stabilizing function

Step 3: Since the system dynamics and the external disturbance are well known, an IBC can be obtained as

e_ 1 ðtÞ ¼ x_d1 ðtÞ  x_1 ðtÞ ¼ xd2 ðtÞ  x2 ðtÞ:

y

uIBC ¼

yaw

1 ðk1 e_1 ðtÞ þ x_ d2 ðtÞ þ e1 ðtÞ þ k2 e2 ðtÞ gðx; tÞ  f ðx; tÞ  dðtÞÞ;

ð11Þ

where k2 is a positive constant. Substituting Eq. 11 into 10, the following equation can be obtained: V_2 ðEðtÞÞ ¼ k1 e21  k2 e22 ¼ ET KE  0;

ð12Þ

where E ¼ ½e1 ðtÞ; e2 ðtÞT and K ¼ diagðk1 ; k2 Þ. Since V_2 ðEðtÞÞ  0, V_2 ðEðtÞÞ is negative semi-definite ði:e:; V_2 ðEðtÞÞ  V_2 ðEð0ÞÞÞ, which implies e1(t) and e2(t) are bounded. Define the following term:

x

XðtÞ ¼ ET KE ¼ V_2 ðEðtÞÞ Zt

roll

pitch

z Fig. 1 WIP system with its three degrees of freedom

XðsÞds ¼ V_2 ðEð0ÞÞ  V_2 ðEðtÞÞ:

ð13Þ ð14Þ

0

Because V_2 ðEð0ÞÞ is bounded, and V_2 ðEðtÞÞ is nonincreasing and bounded, the following result can be obtained

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Zt

ð15Þ

XðsÞds\1:

lim

t!1 0

_ is bounded, so by Barbalat’s Lemma (Slotine and Also XðtÞ Li 1991), it can be shown that limt??X(t) = 0. This will imply that e1(t) and e2(t) converge to zero as t ? ?. Therefore, the IBC in Eq. 11 will asymptotically stabilize the system. 3 Development of RIBTC system The proposed RIBTC system is described in this section. The configuration of the proposed system is shown in Fig. 2, xdh is the original reference angle, xdh is the system reference angle and xh is the angle output of the WIP. Moreover, the modified reference angle signal Dxh is specified by a reference model following a position command input xdp and position output of the WIP, xp. Clearly, Dxh is obtained from the position error. It is a virtual angle which is used to move the WIP forward or backward to target position. In this work, a virtual angle can be obtained as Dxh ðNÞ ¼ kx ðxdp ðNÞ  xp ðNÞÞ; where kx is a small positive constant. Obviously, Dxh will be zero when xp(N) equals xdp(N). It means that the WIP moves to the target position already. In practical application, f(x, t) cannot be exactly obtained in general, and the external disturbance d(t) is always unknown. Therefore, the IBC in Eq. 11 should not be precisely obtained. Thus, a RIBTC system is proposed and shown in Fig. 2 to achieve H? tracking performance. The control law is defined to take the following form: u ¼ uAORCMAC þ uR ;

and the robust H? controller is design to recover the residual approximation error and to achieve H? tracking performance with desired attenuation level. The design of the RIBTC system is analyzed as follows. 3.1 Output recurrent cerebellar model articulation controller An output recurrent cerebellar model articulation controller (ORCMAC) is proposed and shown in Fig. 3. This ORCMAC is composed of input space Q, association memory space A, receptive-field space T, weight memory space W and output space Y. The signal propagation and the basic function in each space are introduced as follows. 1. Input space Q: For a given q ¼ ½q1 ; q2 ; . . .; qn T 2 Rn , each input state variable qi must be quantized into discrete regions (called elements) according to given control space. The number of element, nE, is termed as a resolution. 2. Association memory space A: Several elements can be accumulated as a block, the number of blocks nB in a CMAC is usually greater than two. The A denotes an association memory space with nA (nA = n 9 nB) constituents. In this space, each block performs a receptive-field basis function. The Gaussian function is accepted here as the receptive-field basis function, which can expressed as " # ðqri  mik Þ2 /ik ¼ exp ð17Þ ; for k ¼ 1; 2; . . .; nB ; v2ik

ð16Þ

where /ik is the kth block of the ith input qi with the mean mik and variance vik. In addition, the input of this block for discrete time N can be represented as

where uAORCMAC is the AORCMAC and uR is the robust H? controller. The AORCMAC is used to copy the IBC, Fig. 2 Block diagram of RIBTC system

xdp xp

u (t ) wheeled inverted pendulum

uR

Robust H ∞ controller

e2 δ

e2

u ABORCMAC Γˆ , C , H , R

ORCMAC

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Reference model

Adaptation laws e2 e1

e2

xθ

d

dt

Stabilizing e1 α function

xθ

Δxθ xdθ xdθ

Robust intelligent backstepping tracking control

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Fig. 3 Structure of ORCMAC

r1 Association Memory Space A Input Space X

Output Space Y

Weight Memory Space W

φ1k

z −1

x1 bk

wk

y

xn

φnk

z −1

Receptive-Field Space T rn

qri ðNÞ ¼ qi ðNÞ þ ri yðN  1Þ;

ð18Þ

where ri is the recurrent weight of the recurrent unit. It is clear that the input of this block contains the memory terms y(N - 1), which store the past information of the network. This is the apparent difference between the proposed ORCMAC and the conventional CMAC. 3. Receptive-field space T: Areas formed by blocks are called receptive-fields. The number of receptive field, nR, is equal to nB in this study. The multidimensional receptive-field function is define as " # n n Y X ðqrik  mik Þ2 bk ¼ /ik ðqi Þ ¼ exp ; v2ik ð19Þ i¼1 i¼1 k ¼ 1; 2; . . .; na ; where bk is associated with the kth receptive field. The multidimensional receptive-field function can be expressed in a vector form as Cðq; m; v; rÞ ¼ ½b1 ; b2 ; . . .; bk ; . . .; bnR T ;

ð20Þ

where m ¼ ½m11 ; m21 ; . . .; mn1 ; . . .; m1k ; . . .; mnk T 2 RnnR , v ¼ ½v11 ; v21 ; . . .; vn1 ; . . .; v1k ; . . .; vnk T 2 RnnR and r ¼ ½rT1 ; . . .; rTk ; . . .; rTnR T 2 RnnR : 4. Weight memory space W: Every location of T to a particular adjustable value in the weight memory space can be expressed as w ¼ ½w1 ; w2 ; . . .; wk ; . . .; wnR T ;

ð21Þ

where wk denotes the connecting weight value of the output associated with the kth receptive-field. The weight wk is initialized from zero and is automatically adjusted during on-line operation.

5.

Output space Y: The output of ORCMAC is the algebraic sum of the activated weights in the weight memory, and is expressed as yo ¼ wCðq; m; v; r Þ ¼

nR X

ð22Þ

wk bk :

k¼1

3.2 Robust intelligent backstepping tracking control The design of RIBTC system for the uncertain nonlinear WIP system is described step-by-step as follows: Step 1. Define the tracking error e1(t) as in Eq. 2, the stabilizing functions a(t) as in Eq. 4, and e2(t) as in Eq. 6, respectively. Step 2. Since the ORCMAC is utilized to estimate the IBC presented in Eq. 11, so that uAORCMAC can be written as follows uAORCMAC ðq; m; v; rÞ ¼ y ¼ WCðq; m; v; rÞ:

ð23Þ

Assume there exists an optimal uAORCMAC to approach the uIBC such that uIBC ¼ uAORCMAC ðq; w ; m ; v ; r  Þ þ e ¼ w C þ e;

ð24Þ

where e is a minimum approximation error, w ; m ; v ; C and r  are optimal parameters of w; m; v; C and r, respectively. However, the optimal uAORCMAC cannot be obtained, so that the on-line estimation uABORCMAC is used to approach the uIBC . From Eq. 23, the control law (Eq. 16) can be rewritten as follows: ^ þ uR ; ^ m; ^ v^; r^Þ þ uR ¼ w ^C u ¼ uAORCMAC ðq; w;

ð25Þ

^ and r^ are some estimates of the optimal ^ m; ^ v^; C where w;  parameters w ; m ; v ; C and r  , respectively. Subtracting Eq. 25 from 24, an approximation error u~ is defined as

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^  uR ^C u~ ¼ uIBC  u ¼ W  C þ e  W ; ~ þ e  uR ~ þW ^C ¼ WC

ð26Þ

~ ¼ C  C. ^ Moreover, the ^ and C ~ ¼ w  w where w linearization technique is employed to transform the multidimensional receptive-field basis functions into ~ in Taylor partially linear form so that the expansion of C series can be obtained as (Leu et al. 1999) 2 

ob1 om

T 3

 7 3 6  7 6 b~1 7 6 7 7 6 6 .. 7 6 .7 6 . 7 6 .. 7 6 7 6   7 6 7 6 6 T 7 7 6 obk 7 ~ 6 C¼ 7 6 b~1 7 ¼ 6 7 7 6 6 6 . 7 6 om 7 7 6 .7 6 .. 7 6 .7 6  5 6 4 .7 7 6 ~  7 6 bnR 4 obn T 5 m¼ m ^ R om 2  T 3 ob1  7 6 ov 7 6 7 6  6 .. 7  6 .7 7 6 6   7 6 T 7 6 obk 7 ^ 6 ðm  mÞþ 7 7 6 ov 7 6 7 6 .. 7 6 7 6 . 7 6 7 6  T 5 v¼ v^ 4 obn 2

ð30Þ

Step 4. Define the Lyapunov function as V3 ðtÞ ¼ V2 ðtÞ þ

1 ~ ~T 1 1 T 1 T ~m ~T þ WW þ m v~v~ þ r~r~ ; 2g1 2g2 2g3 2g4 ð31Þ

where g1, g2, g3 and g4 are positive constants. Taking the derivative of the Lyapunov function (Eq. 31) and using Eq. 30, it is concluded that ð27Þ

R

where b~k ¼ bk  b^k ; bk is the optimal parameter of bk; b^k is ~ ¼ m  m; ^ v~ ¼ v  v^; Ot 2 RnR is a an estimate of bk ; m h i obnR 1 vector of higher-order terms; C ¼ ob    :2 om om j m¼m ^   h i h i obnR  obnR  1 1 RnnR nR ; H ¼ ob 2 RnnR nR ; R ¼ ob ov  ov  or  or  v¼^ v r¼^ r

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^ þ CT m ^ Tm ~ C ~ þ H T v~ þ RT r~ þ Ot Þ þ WðC ~ þ H T v~ u~ ¼ Wð ^ þ WðC ~C ^ Tm ~ þ H T v~ þ RT r~ þ Ot Þ þ e  uR ¼ W   ~ T ðC T m ~ þ H T v~ þ RT r~Þ þ W  Ot þ e  uR þ RT r~Þ þ W   ^ þW ~C ^ CT m ~ þ H T v~ þ RT r~ þ n  uR ; ð29Þ ¼W

 e1 ðtÞ  k2 e2 ðtÞ:

or T T ~ C mþH v~þRT r~þOt ;

:

Substituting Eqs. 27 and 28 into 26, yields

¼ gðx; tÞ~ u  e1 ðtÞ  k2 e2 ðtÞ     ^ þW ~C ^ CT m ~ þ H T v~ þ RT r~ þ n  uR ¼ gðx; tÞ W

R

2R

ð28Þ

~ T m þ H T v~ þ RT r~ þ W  Ot þ e where n ¼ W½C Step 3. In order to develop the robust H? controller, the dynamic equation 8 can be expressed via Eqs. 11 and 29 as   e_ 2 ðtÞ ¼ gðx; tÞ uIBC  u1  e1 ðtÞ  k2 e2 ðtÞ

ov 2  T 3 ob1  7 6 or 7 6 7 6  6 .. 7  6 .7 7 6 6   7 6 T 7 6 obk 7 ðv  v^Þþ 6 ðr  ^ r ÞþOt 7 7 6 or 7 6 7 6 .. 7 6 7 6 . 7 6 7 6  4 obn T 5 r ¼ r^

nnR

Rewriting Eq. 27, it can be obtained that ^ þ CT m ~ þ H T v~ þ RT r~ þ Ot : C ¼C 

1 ~ _~ T 1 _ T 1 1 ~ m ~ þ v~_ T v~þ r~_ T r~ V_3 ðtÞ ¼ V_2 ðtÞ þ W W þ m g1 g2 g3 g4 1 ~ _^ T 2 ¼ k1 e1 ðtÞ þ e2 ðtÞ½e1 ðtÞ þ e_2 ðtÞ  W W g1 1 _T 1 _T 1 _T ~  v^ v~ r^ r~ ¼ ET KE þ e2 ðtÞgðx;tÞ ^ m  m g2 g3 g4    T  ^ ~ ^ ~ þ H T v~þ RT r~ þ n  uR  WC þ W C m 1 ~ _^ T 1 _ T 1 1 ~  v^_ T v~ r^_ T r~ ^ m WW  m g1 g2 g3 g4

1 _^ T T ^ ~ ¼ E KE þ W e2 ðtÞgðx;tÞC  W g

1 1 ^ T m ^_ T m ~ þ e2 ðtÞgðx;tÞWC g2

^ T  1 v^_ T v~ þ e2 ðtÞgðx;tÞWH g3

1 _T T ^ þ e2 ðtÞgðx;tÞWR  r^ þ e2 ðtÞgðx;tÞðn  uR Þ g4 

ð32Þ Step 5. The RIBTC system is designed as in Eq. 16. The adaptive laws of the AORCMAC are chosen as ^T ^_ ¼ g1 e2 ðtÞgðx; tÞC w

ð33Þ

^ ^_ ¼ g2 e2 ðtÞgðx; tÞC w m

T

ð34Þ

Robust intelligent backstepping tracking control

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^T v^_ ¼ g3 e2 ðtÞgðx; tÞH w

ð35Þ

^ r^_ ¼ g4 e2 ðtÞgðx; tÞRw

ð36Þ

and the robust H? controller is chosen as 2

uR ¼

ðd þ 1Þ e2 ðtÞ 2d2

ð37Þ

Assume n 2 L2 ½0; T; 8T 2 ½0; 1Þ. Integrating above equation from t = 0 to t = T, yields V3 ðTÞ  V3 ð0Þ  ZT ZT 1 1 2 2  e2 ðtÞgðx; tÞdt þ d gðx; tÞn2 ðtÞdt: 2 2 0

where d is a positive constant. From Eqs. 33 to 37, Eq. 32 can be rewritten as V_3 ðtÞ ¼ ET KE þ e2 ðtÞgðx; tÞðn  uR Þ ¼ ET KE þ e2 ðtÞgðx; tÞn  e2 ðtÞgðx; tÞuR e2 ðtÞgðx; tÞðd2 þ 1Þ ¼ ET KE þ e2 ðtÞgðx; tÞn  2 2 2d

2 1 1 e2 ðtÞ  dn ¼ ET KE  gðx; tÞe22 ðtÞ  gðx; tÞ 2 2 d 1 1 1 2 2 2 þ gðx; tÞd n   gðx; tÞe2 ðtÞ þ gðx; tÞd2 n2 : 2 2 2

ð38Þ

the

ð39Þ

0

Since V(T) C 0, the above inequality implies the following inequality 1 2

ZT 0

e22 ðtÞgðx; tÞdt  V3 ð0Þ

1 þ d2 2

ZT

gðx; tÞn2 ðtÞdt:

ð40Þ

0

Using Eq. 31, the above inequality is equivalent to the following

Fig. 4 Simulation results, RIBTC with xdh = 0°, xdp = 1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10s: a, b for d = 0.5; c, d for d=1

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ZT

e22 ðtÞdt  ET ð0ÞEð0Þ

0

C.-H. Chiu et al.

1 1 T ~ ~ w ~T ð0Þ þ m ~ ð0Þmð0Þ þ wð0Þ g1 g2

1 1 vð0Þ þ r~T ð0Þ~ r ð0Þ þ d2 þ v~T ð0Þ~ g3 g4

ZT

n2 ðtÞdt:

ð41Þ

0

4 Simulation results

If the system starts with initial condition E(0) = 0, ~ ~ wð0Þ ¼ 0, mð0Þ ¼ 0, v~ð0Þ ¼ 0, r~ð0Þ ¼ 0, the H? tracking performance in Eq. 41 can be rewritten as ke2 k d n2L2 ½0;T knk sup

ke2 k and knk. If d = ?, this is the case of minimum error tracking control without disturbance attenuation (Chen and Lee 1996). Then, the desired robust tracking performance in (41) can be achieved for a prescribed attenuation level d.

ð42Þ

RT RT where ke2 k2 ¼ 0 e22 ðtÞg1 ðx; tÞdt and kn2 k2 ¼ 0 n2 ðtÞ g1 ðx; tÞdt. The attenuation constant d can be specified by the designer to achieve desired attenuation ratio between

In this portion, the AORCMAC used in this simulation is characterized by q = 4, nE = 5, nB = 8 and nR = 2 9 4. The receptive-field basis functions are chosen as /ik ¼ exp½ðqi  mik Þ2 =v2ik : The initial values of the parameter for the receptive-field basis functions are chosen as [mi1, mi2, mi3, mi4, mi5, mi6, mi7, mi8] = [-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5], rik = 0.5 for all i and k and r1 = r2 = 0.01. The control objective is to let the state

Fig. 5 Simulation results, RIBTC control with xdh = 0°, xdp = -1.5 m, an external disturbance (x1(10) = -0.5 rad) at 10 s and an extra mass (5 kg) added on WIP: a, b for d = 0.5; c, d for d = 1

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Fig. 6 Simulation results, PID control with xdh = 0°, xdp = 1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10 s

Fig. 7 Simulation results, ACMAC control with xdh = 0°, xdp = 1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10 s

trajectory x and x_ track the desired reference trajectory xd and x_ d . The RIBTC system parameters are chosen as g1 = 0.04, g2 = g3 = 0.07, g4 = 0.01, k1 = 4 and kx = 0.2. Moreover, the parameters of the WIP are given as follows: external disturbance is 0.0873, acceleration of gravity is 9.8 m/s2, gravitational constant is 6.673 9 10-11, total weight of the WIP is 7 kg, wheel weight is 2 kg, moment of inertia of the chassis with respect to the horizontal axis and vertical axis are 1.8229 kg m2 and 0.6490 kg m2, lateral distance between wheel and the center of chassis is 0.15 m and height of the WIP from the chassis is 0.2 m. The following cases, including the extra mass added on WIP and the alteration in the attenuation constant d, are used to examine the adaptive and robust control

performance of the proposed method. The simulation results are depicted in Figs. 4 and 5. Figure 4 shows the control response of the WIP system with the target value of angle and position 0° and -1.5 m, respectively, and an external disturbance (x1(10) = 0.5 rad) at 10 s. The tracking responses of angle and position are plotted in Fig. 4a, b for d = 0.5, and Fig. 4c, d for d = 1. Figure 5 shows the control response of the WIP system with the target value of angle and position 0°, respectively, and -1.5 m and an external disturbance (x1(10) = -0.5 rad) at 10 s. Moreover, an extra mass (5 kg) is added on the WIP. The tracking responses of angle and position are plotted in Fig. 5a, b for d = 0.5; and Fig. 5c, d for d = 1. For comparison, several existing methods, including the PID control, the adaptive cerebellar model articulation

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Fig. 8 Simulation results, AORCMAC control with xdh = 0°, xdp = 1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10 s

Fig. 9 Simulation results, NNPID control with xdh = 0°, xdp = 1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10 s

controller (ACMAC) in Peng et al. (2004), the AORCMAC in Chiu (2010) and a neural network controller with PID controllers (NNPID) in Jung and Kim (2007) are used to control the WIP. The parameters of PID controller are selected as KP = 3, KI = 2 and KD = 5. Figure 6 shows the simulation results for the PID control. The tracking responses of angle and position are plotted in Fig. 6a, b, respectively. The system parameters of the ACMAC and the AORCMAC are the same as those for the RIBTC system. Figures 7 and 8 show the simulation results for the ACMAC and AORCMAC, respectively. The parameters of NNPID controller are the same as those for the NNPID system shown in Jung and Kim (2007). Figure 9 shows the simulation results obtained using the NNPID. Figure 9a is

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the system responses for the angle and velocity of the angle. The position and velocity of the position of the WIP are plotted in Fig. 9b. The four simulations above used the same reference conditions, which are xdh = 0 , xdp = -1.5 m and an external disturbance (x1(10) = 0.5 rad) at 10 s. Comparing with the simulation results shown in Figs. 4, 6, 7, 8 and 9, the tracking responses shown in Fig. 4 are better than the others. It shows that the tracking error of the RIBTC converges faster than the others. Since the proposed output recurrent structure captures the dynamic response of controlled system, the RIBTC will achieve good control performance for the WIP system. Moreover, the backstepping control technique proposes a powerful ability to accommodate the unmodelled

Robust intelligent backstepping tracking control

disturbance and uncertainty effects, Furthermore, the better tracking performance can be achieved as the attenuation constant d is chosen smaller. The effectiveness of the proposed robust control scheme is verified.

5 Conclusions In this paper, a RIBTC system using an AORCMAC is proposed for WIPs. The RIBTC comprises an AORCMAC and a robust H? controller. The AORCMAC is used to mimic an IBC, and the robust H? controller is designed to recover the residual approximation error and to achieve H? tracking performance with desired attenuation level. The all adaptation laws of the RIBTC system are derived based on Lyapunov stability analysis, the Taylor linearization technique and H? control theory, so that the stability of the closed-loop system and H? tracking performance can be guaranteed. Finally, the proposed RIBTC system is applied to control the WIP system. The simulation results demonstrate the effectiveness of the proposed robust scheme for the WIP system.

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