Robust tracking control of uncertain dynamic nonholonomic systems ...
Neurocomputing 142 (2014) 216–227
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Robust tracking control of uncertain dynamic nonholonomic systems using recurrent neural networks$ Zhiqiang Miao n, Yaonan Wang, Yimin Yang College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 September 2013 Received in revised form 18 December 2013 Accepted 31 March 2014 Communicated by Hongli Dong Available online 2 June 2014
The tracking problem for a class of dynamic nonholonomic systems with uncertainties is considered. First, under the assumption that the dynamics of the nonholonomic systems are exactly known without uncertainties, a simpler model-based controller is proposed by means of cascade design approach, in which the virtual velocity controller is linear, and the actual torque controller is derived by conventional computed-torque law. Then, to deal with uncertainties, a recurrent neural network control system is developed without requiring explicit knowledge of the system dynamics. The closed-loop stability analysis is presented based on a technical lemma developed for nonlinear cascaded systems with vanishing disturbances. Comparing with the existing results, the resulting control system has a simpler structure, and can deal with parametric uncertainties as well as non-parametric uncertainties, yet guarantees asymptotic stability of the tracking error dynamics. Simulation results for a wheeled mobile robot verify the good tracking performance and robustness of the proposed control system. & 2014 Elsevier B.V. All rights reserved.
Keywords: Robust control Nonholonomic systems Recurrent neural networks Tracking control Cascaded systems
1. Introduction In recent years, much attention has been devoted to the problem of controlling nonlinear mechanical systems with nonholonomic constraints [1,2]. Depending on whether the nonholonomic system is presented by a kinematic or dynamic model, the control problem can be classified as either kinematic control or dynamic control. Due to Brockett's theorem [3], it is well known that a nonholonomic system cannot be asymptotically stabilized to an equilibrium point via smooth or even continuous purefeedback laws. However, several approaches have been proposed for tackling the stabilization problem at kinematic level [4–8] or dynamic level [9–11]. The tracking problem has also received a great deal of attention because of its practical importance. Several authors have studied the kinematic tracking problem in which velocity acts as the control input. In [12], a linearization-based tracking control was proposed for nonholonomic systems under the assumption that the linearized system is uniformly completely controllable along the desired trajectory. In [13,14], based on dynamic feedback
☆ This paper was not presented at any IFAC meeting. This work was supported by the National Natural Science Foundation of China (61175075 and 61203207), National High Technology Research and Development Program of China (863 Program: 2012AA111004 and 2012AA112312), and Hunan Provincial Innovation Foundation for Postgraduate (521298960). n Corresponding author. E-mail address: [email protected] (Z. Miao).
http://dx.doi.org/10.1016/j.neucom.2014.03.061 0925-2312/& 2014 Elsevier B.V. All rights reserved.
linearization, controllers with structural singularity were proposed for the tracking problem of mobile robots. However, these controllers only solve the local tracking problem. There are mainly two methods to deal with the global tracking problem: backstepping method [15,16] and cascade design approach [17,18]. The main difference of these two methods lies in the way of dealing with the coupling terms between subsystems. Unlike the backstepping method, the coupling terms are neglected and the control law is chosen without canceling the coupling terms in cascade design to reduce complexity of the controllers. In practice, however, it is more realistic to consider the tracking problem at dynamic level, where the torque and force are taken as the control inputs. The dynamics of the systems usually cannot be neglected if high performance of the control systems is required. With the assumption of known dynamics, model-based controls can be obtained for the dynamic nonholonomic systems using backstepping [19]. However, there often exist uncertainties in the dynamics of the nonholonomic systems inevitably, such as unmolded dynamics, parameter perturbations and load variation. To confront this, some adaptive controls and robust controls have been developed. In [20,21], adaptive control techniques were employed to solve the tracking problem of nonholonomic systems with unknown inertia parameters. However, the major problem of the adaptive control approaches is that certain functions must be assumed “linearity in the parameters”, and tedious preliminary computation of “regression matrices” is needed [22]. In [23], a robust adaptive controller was designed for nonholonomic mechanical systems with model uncertainties. In [11,24], robust
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
adaptive controllers were proposed for dynamic nonholonomic systems with parametric and non-parametric uncertainties, in which adaptive control techniques were used to compensate for the parametric uncertainties and sliding mode control was used to suppress the bounded disturbances. In [25], robust control strategies were presented systematically for both holonomic mechanical systems and a large class of nonholonomic systems in the presence of uncertainties and disturbances. Li et al. [26–28] have studied the motion/force control for a mobile manipulator under both holonomic and nonholonomic constraints, and some adaptive robust control strategies have been proposed. In order to cope with highly uncertain nonlinear systems, as an alternative, the applications of neural networks (NNs) were also studied for the control of nonholonomic robotic systems. A multilayer perception network-based controller was suggested by Fierro and Lewis [29] to deal with parametric or non-parametric uncertainties for a nonholonomic mobile robot without any prior knowledge of the uncertainties. Other feedforword neural networks, like radial basis function (RBF) neural network [30] and wavelet network [31,32], were also adopted for the robust control of mobile manipulators and mobile robots. In the past decade, a great progress has been achieved in the study of using neural networks to control uncertain nonlinear systems. Extensive works demonstrate that adaptive neural control is particularly suitable for controlling highly uncertain, nonlinear, and complex systems [33–35]. In these neuro-adaptive control schemes, the neural network is used to compensate the effects of nonlinearity and system uncertainties, so the stability, convergence and robustness of the control system can be improved [36]. According to the structure, the neural networks can be mainly classified as feedforward neural networks (FNNs) and recurrent neural networks (RNNs). It is well known that FNN is capable of approximating arbitrary continuous function closely. However, FNN is a static mapping and unable to represent a dynamic mapping without the aid of tapped delays [37]. On the other hand, he RNNs, which comprise both feedforward and feedback connections, have superior capabilities than the FNNs. Since the RNN has a feedback loop, it can capture the past information of the network and adapt rapidly to sudden changes of the control environment [38]. The RNNs have the ability to deal with time-varying input or output though their own natural temporal operation [39]. For this ability, the structure of the neural network is simplified. Due to its dynamic characteristic and relative simple structure, the RNN is a useful tool in real-time application [40]. In this paper, the tracking problem is considered for a class of dynamic nonholonomic systems in which the nonholonomic kinematic subsystem is assumed to be capable of being transformed into the chained form and the dynamic subsystem is completely unknown. First, a simpler model-based controller is proposed for dynamic nonholonomic systems without uncertainties, in which the virtual velocity controller is linear, and the actual torque controller is derived by conventional computed-torque law. The stability of the closed-loop systems is presented based on the results on nonlinear cascaded systems with vanishing disturbances. To deal with disturbances and unstructured unmodeled dynamics in the nonholomonic system, a robust control system is then developed based on recurrent neural networks. On-line weights tuning algorithms that do not require off-line learning yet guaranty asymptotic stability of the tracking error dynamics are utilized. Simulation results for a wheeled mobile robot are provided to demonstrate the effectiveness of the proposed control method. The remainder of this paper is organized as follows. Problem is formulated in Section 2. In Section 3, a model-based controller is presented and the closed-loop stability analysis is given based on a
217
technical lemma developed for nonlinear cascaded systems with vanishing disturbances. A RNN-based control system is developed in Section 4. In Section 5, the effectiveness of the proposed approach is verified by simulation on a wheeled mobile robot. Section 6 concludes the paper.
2. Problem formulation In general, a mechanical system with nonholonomic constraints can be described as JðqÞq_ ¼ 0
ð1Þ
_ q_ þ GðqÞ þdðtÞ ¼ J T ðqÞλ þBðqÞτ MðqÞq€ þ Cðq; qÞ n
ð2Þ nn
where q A R denotes the generalized coordinates, MðqÞ A R is _ A Rnn is the the symmetric, positive definite inertia matrix, Cðq; qÞ centripetal and coriolis matrix, GðqÞ A Rn is the gravitational vector, dðtÞ A Rn denotes unknown disturbances including unstructured unmodeled dynamics, JðqÞ A Rmn is the constrained matrix, λ A Rm is the associated Lagrange multipliers which expresses the contact force, τ A Rr is the vector of control input, BðqÞ A Rnr is the input transformation matrix, and is assumed to be known because it is a function of fixed geometry of the system. The dynamic system (2) has the following properties. _ GðqÞ and d(t) are bounded. Property 1. MðqÞ; Cðq; qÞ; _ 2C is a skew-symmetric matrix, i.e. xT ðM _ 2CÞ Property 2. M x ¼ 0; 8 x a 0. Let SðqÞ A Rnðn mÞ be a full rank matrix formed by a set of smooth and linearly independent vector fields spanning the null space of J(q), i.e. ST ðqÞJ T ðqÞ ¼ 0
ð3Þ
According to (2) and (3), there always exists an auxiliary vector of independent generalized velocities v A Rn m , that make the system (1) and (2) be transformed in to a more appropriate representation for control purposes: q_ ¼ SðqÞv
ð4Þ
_ þ G1 ðqÞ þ d1 ðtÞ ¼ B1 ðqÞτ M 1 ðqÞv_ þ C 1 ðq; qÞv
ð5Þ
_ ¼ ST ðMðqÞS_ þ Cðq; qÞSÞ, _ where M 1 ðqÞ ¼ S MðqÞS, C 1 ðq; qÞ G1 ðqÞ ¼ ST GðqÞ, B1 ðqÞ ¼ ST BðqÞ, d1 ðtÞ ¼ ST dðtÞ. It should be noted that the reduced system consists of a reduced state dynamics (5) and the so-called kinematic model (4) of nonholonomic systems in the literature. It is assumed that B1 ðqÞ A Rðn mÞr is a full rank matrix and r Z n m to completely actuate the nonholonomic system. For ease of controller design, the existing results for the control of nonholonomic canonical forms in the literature are exploited. We assume that there exists a coordinate transformation x ¼ T 1 ðqÞ and a state feedback v ¼ T 2 ðqÞu, so that the kinematic model of the nonholonomic system given in Eq. (4) can be converted to the chained form. Since most of non-holonomic robotic systems (such as wheeled mobile robots) often include two pseudo-velocities, we only consider two independent generalized velocities ðn m ¼ 2Þ case for the sake of simplicity; however, the results can be extended to more general case. That is, the nonholonomic chained system considered in this paper is the 2-input, single-chain, single-generator chained form T
x_ 1 ¼ u1 x_ i ¼ u1 xi þ 1 x_ n ¼ u2
ð2 r ir n 1Þ ð6Þ
The necessary and sufficient condition for the existence of the transformation can be founded in [41–43]. Based on the above
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transformation, the dynamic model is converted as _ þ G2 ðxÞ þ d2 ðtÞ ¼ B2 ðxÞτ M 2 ðxÞu_ þ C 2 ðx; xÞu where
ð7Þ
_ ¼ T T2 ðqÞðM 1 ðqÞ M 2 ðxÞ ¼ T T2 ðqÞM 1 ðqÞT 2 ðqÞjq ¼ T 1 ðxÞ ; C 2 ðx; xÞ 1
To facilitate the subsequent control development, the new error variables are defined as follows: y1 ¼ x1e xj1e i xjd j ¼ i þ 1 ðj iÞ! n
_ 2 ðqÞÞjq ¼ T 1 ðxÞ , G2 ðxÞ ¼ T T2 ðqÞG1 ðqÞjq ¼ T 1 ðxÞ , B2 ðxÞ ¼ T_ 2 ðqÞ þ C 1 ðq; qÞT
yi ¼ xie
T T2 ðqÞB1 ðqÞjq ¼ T 1 ðxÞ , d2 ðtÞ ¼ T T2 ðqÞd1 ðtÞjq ¼ T 1 ðxÞ . The following properties of the dynamic model (7) can be easily proved.
ð2 r i r n 1Þ yn ¼ xne
Property 3. Matrix M 2 ðxÞ is symmetric positive definite; M 2 ðxÞ; _ G2 ðxÞ and d2 ðtÞ are bounded. C 2 ðx; xÞ;
Then, the following error model for tracking system can be attained:
_ 2 2C 2 is a skew-symmetric matrix. Property 4. M
y_ 1 ¼ u1 u1d
Given a differential desired trajectory qd(t) satisfies the constraint equation (1), it can be easily proved that there exists a time-varying reference control vd(t) satisfying
z_ ¼ u1d Az þ ðu1 u1d ÞAz þ Bðu2 u2d Þ ΔBðy1 ; u2d Þy1
1
1
1
1
q_ d ¼ Sðqd Þvd
ð8Þ
then there must exist a coordinate transformation xd ¼ T 1 ðqd Þ and a state feedback vd ¼ T 2 ðqd Þud , so that x_ 1d ¼ u1d x_ id ¼ u1d xði þ 1Þd
ð2 r ir n 1Þ
x_ nd ¼ u2d
ð9Þ
Then according to the above transformations, the tracking problem considering in this paper is equivalent to design the control law τ for the system (6) and (7) such that lim ðxðtÞ xd ðtÞÞ ¼ 0:
ð10Þ
t-1
∑
ð12Þ
ð13Þ
ð14Þ
where z ¼ ½z1 ; z2 ; …; zn 1 ¼ ½y2 ; y3 ; …; yn , ΔBðy1 ; u2d Þ ¼ ðn 2Þ!; yn1 4 =ðn 3Þ!; …; 1; 0T u2d , and (A, B) is in the controllable canonical form, that is 2 3 2 3 0 1 0 … 0 0 60 0 1 … 07 607 6 7 6 7 6 7 6 7 7 6 7 A¼6 6 ⋮ ⋮ ⋮ ⋱ ⋮ 7; B ¼ 6 ⋮ 7 6 7 6 7 0 0 … … 1 4 5 405 0 0 … … 0 1 T
T
Hereafter, we consider system (13) and (14) as our starting point. For subsystem (14), we define α ¼ ½α1 ; α2 ; …; αn 1 T with α1 ¼ 0; α2 ¼ 0 and for 2 r ir n 2 ∂α i zj þ 1 ki 1 ðzi 1 αi 1 Þ j ¼ 1 ∂zj i1
αi þ 1 ¼ ∑
3. Model-based control In this section, under the assumption that the dynamic model of the nonholonomic system is exactly known without considering disturbances, a model-based control system shown in Fig. 1 is developed. The control design procedure can be divided into two steps: first, a kinematic controller for subsystem (6) is designed, and then the conventional computed-torque control method is adopted to develop a toque law using backstepping approach.
½yn1 3 =
ð15Þ
where ki ð1 r i r n 3Þ are positive constants. From the definition of α, it can be easily checked that αi ð1 r i r n 1Þ are linear functions with respect to z1 ; z2 ; …; zi 1 , that is α ¼ Ψ z, where Ψ is a constant strictly lower triangular matrix. Let z ¼ z α, we have z_ 1 ¼ u1d z 2 þ ðu1 u1d Þz 2 u2d yn1 2 =ðn 2Þ! z_ ¼ u ð k z þz Þ u yn i 1 =ðn i 1Þ! i
i1 i1
1d
iþ1
2d 1
∂α i u2d y1n j 1 =ðn j 1Þ! þ ∑ j ¼ 1 ∂zj i1
∂α i zj þ 1 j ¼ 1 ∂zj i1
!
þðu1 u1d Þ zi þ 1 ∑
3.1. Kinematic Control As a precursor to the full dynamic system study, let us first neglect the dynamics of system (6) and (7) and consider the kinematic tracking problem only. Now u is assumed to be the virtual control, the object is to design the control input u¼ η such that (10) holds. Define xe ¼ x xd , it can be directly checked that the xe dynamics satisfy the following differential equations:
ð2 r i r n 2Þ n 2 ∂α n1 z_ n 1 ¼ u2 u2d u1d ∑ zj þ 1 j ¼ 1 ∂zj n2 ∂
þ ∑
j¼1
αn 1 ∂zj
u2d yn1 j 1 =ðn j 1Þ!
n2 ∂
ðu1 u1d Þ ∑
j¼1
x_ 1e ¼ u1 u1d x_ ie ¼ u1d xði þ 1Þe þ xi þ 1 ðu1 u1d Þ
αn 1 ∂zj
ð16Þ
zj þ 1
With the aid of z, we have the following theorem.
ð2 r ir n 1Þ x_ ne ¼ u2 u2d
ð11Þ
ud
Theorem 1. Consider the kinematic tracking problem of subsystem (6). Assume that u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0;
d dt
xe
xd
Kinematic Controller (17)
u
Torque Controller (19)
τ
Dynamic Model (7)
u
x Fig. 1. Structure of model-based control system.
Kinematic Model (6)
x
x
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Consider the system
then the virtual control law u ¼ η ¼ ζ þ ud k0 y1
2
ζ¼6 4 k
ξ_ ¼ f ðt; ξÞ þ gðt; ξÞ
3
n 2 u1d z n 2 kn 1 z n 1 þ u1d
n 2∂
∑
αn 1
j¼1
makes (10) holds, where k0 ; kn 2 ; kn 1 parameters.
∂zj
7 zj þ 1 5
ð17Þ
are positive control
3.2. Dynamic control Now the control law for τ is designed to make the outputs of the dynamic subsystem u tend to the auxiliary signal η. To design the torque input, denote: u~ ¼ u η, the subsystem (7) can be rewritten as _ u~ ¼ B2 ðxÞτ M 2 ðxÞu~_ þC 2 ðx; xÞ _ η þ G2 ðxÞ þ d2 ðtÞÞ ðM 2 ðxÞη_ þ C 2 ðx; xÞ
ð18Þ
Assume that the dynamics of the system is exactly known and dðtÞ ¼ 0. If the control law is chosen as _ η þG2 ðxÞ τ ¼ B2þ ðxÞ½ K p u~ þ ðM2 ðxÞη_ þC 2 ðx; xÞ
ð19Þ
B2þ ðxÞ
is the right pseudo inverse of B2 ðxÞ, Kp is a positive where definite matrix, and η is defined in Eq. (17). Then substituting (19) into (18), which yields _ u~ K p u~ M 2 ðxÞu~_ ¼ C 2 ðx; xÞ
ð20Þ
Substituting (17) into (13) and (16) and rewriting them in a more compact form: y_1 ¼ k0 y1 þ u~ 1
ð21Þ
z_ ¼ Aðu1d Þz þ ðu~ 1 k0 y1 ÞðI Ψ ÞAðI Ψ Þ 1 z þ Bu~ 2 ðI Ψ ÞΔBðy1 ; u2d Þy1
0
6 k u 1 1d 6 6 0 Aðu1d Þ ¼ 6 6 6 ⋮ 4 0
where ξ A R , f ðt; ξÞ and gðt; ξÞ are piecewise continuous in t and locally Lipschitz in ξ. We think of this system as a perturbation of the nominal system:
ξ_ ¼ f ðt; ξÞ
ð24Þ
The perturbation term gðt; ξÞ satisfies
Remark 2. Since z ¼ ðI Ψ Þz, it follows that the new z dynamics can be obtained by a linear coordinate transformation of the zcoordinates. This idea was used and the explicit expression of Ψ was given in [20,11]. In our method, Ψ does not occur in the controller expression, however, it will be used in the following stability analysis for ease of notation.
2
ð23Þ
n
Remark 1. The proof of this theorem can be easily validated in the following procedure of proving Theorem 2. The new tracking error used in our paper is first introduced in [44]. Another different tracking error is given in [45] via the use of Lie group operation. However, it can be easily checked that the definitions of tracking error in [44,45] are equivalent. It should be mentioned that the persistency of excitation (PE) assumption on u1d ðtÞ made in our method is also needed in most of the results on tracking control of nonholonomic systems. Roughly speaking, the fulfillment of this condition implies that the desired reference trajectory is a moving trajectory, instead of fixed set point.
where
219
ð22Þ
u1d
0
…
0
0
u1d
…
0
k2 u1d
0
⋱
⋮
⋮
⋮
⋱
u1d
0
…
kn 2 u1d
kn 1
3 7 7 7 7 7 7 5
Based on the above analysis, the closed-loop system of (6), (7), (17) and (19) consists of three subsystems (20)–(22). Before giving the completely stability analysis, we present a technical lemma on the stability of nonlinear cascaded systems with vanishing disturbances that will be used in the proof of our main results.
J gðt; ξÞ J r γ ðtÞ J ξ J þ δðtÞ
ð25Þ
where γ ðtÞ A R; δðtÞ A R are nonnegative, differentiable, and bounded for all t Z 0. For the perturbed system (23), we have the following. Lemma 1. Let ξ ¼0 be a globally exponentially stable equilibrium point of the nominal system (24), and suppose gðt; ξÞ is vanishing, that is, it satisfies (25) with lim γ ðtÞ ¼ 0;
t-1
lim δðtÞ ¼ 0
ð26Þ
t-1
Then the origin of perturbed system (23) is globally asymptotically stable. Proof. Since ξ_ ¼ f ðt; ξÞ is globally exponentially stable, using converse Lyapunov theory [46], there exists a Lyapunov function Vðt; ξÞ that satisfies c1 ξ2 r Vðt; ξÞ r c2 ξ2
∂V ∂V þ f ðt; ξÞ r c3 ξ2 ∂t ∂ξ ∂V J J r c4 J ξ J ∂ξ
ð27Þ
for all ðt; ξÞ A ½0; 1Þ Rn and some positive constants c1 ; c2 ; c3 and c4. The derivative of Vðt; ξÞ along the trajectories of (23) is given by ∂V ∂V ∂V þ f ðt; ξÞ þ gðt; ξÞ V_ ðt; ξÞ ¼ ∂t ∂ξ ∂ξ ∂V 2 r c3 ‖ξ‖ þ J J J gðt; ξÞ J ∂ξ r ð c3 c4 γ ðtÞÞ‖ξ‖2 þ c4 δðtÞ J ξ J Using (27), we can find an upper bound on V_ as sffiffiffiffiffi c3 c4 V V þc4 δðtÞ V_ r γ ðtÞ c1 c2 c1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By defining WðtÞ ¼ Vðt; ξðtÞ, we have c4 _ ðtÞ r 1 c3 γ ðtÞc4 WðtÞ þ p ffiffiffiffiffiδðtÞ W 2 c2 c1 2 c1
ð28Þ
ð29Þ
ð30Þ
Since γ ðtÞ tends to zero as t-1, there exists a time moment T 0 4t 0 and a positive constant c5 such that γ ðtÞ oðc1 c3 c5 Þ=c2 c4 for all t Z T 0 . Which implies c4 _ ðtÞ r c5 WðtÞ þ p W ffiffiffiffiffiδðtÞ; 2c1 c2 2 c1
t ZT 0
ð31Þ
By the comparison theorem, WðtÞ r UðtÞðt Z T 0 Þ, where c5 c4 U_ ðtÞ ¼ UðtÞ þ pffiffiffiffiffiδðtÞ; 2c1 c2 2 c1
Uðt 0 Þ ¼ Wðt 0 Þ
ð32Þ
On the other hand, by assumption, for an arbitrary constant ε1 4 0, pffiffiffiffiffi there exists T 1 ðε1 Þ 4 0 such that δðtÞ r ðc5 =2 c1 c2 c4 Þε1 for all t Z T 0 þ T 1 . Solving the scalar differential equation (32) yields c5 UðtÞ ¼ UðT 0 þT 1 Þ exp ðt T 0 T 1 Þ 2c1 c2 Z t c4 c5 pffiffiffiffiffiδðsÞ exp ðt sÞ ds þ 2c1 c2 T þ T 2 c1 0 1 c5 ε1 ð33Þ r exp ðt T 0 T 1 Þ UðT 0 þ T 1 Þ þ 2c1 c2 2
220
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
δðtÞ ¼ J B J ju~ 2 ðtÞj þ J I Ψ J J ΔBðy1 ; u2d Þ J jy1 ðtÞj
It follows that UðtÞ r ε1 for all t Z ð2c1 c2 =c5 Þ ln½2UðT 0 þ T 1 Þ=ε1 þ T 0 þ T 1 . Observing that ε1 can beffi arbitrarily small, then U(t) tends pffiffiffiffiffiffiffiffi to zero as t-1, hence VðtÞ and J ξðtÞ J are asymptotically convergent to zero. □
Since Ψ , A and B are bounded, ΔBðy1 ; u2d Þ is bounded as y1 ðtÞ and ~ u2d are bounded, moreover, uðtÞ and y1 ðtÞ tend to zero as t-1, it follows that γ ðtÞ and δðtÞ satisfy the vanishing condition (26). Thus, if the origin of system
We now view our closed-loop system as a form of (23) in ~ Lemma 1, with uðtÞ and y1 ðtÞ acting as vanishing perturbations of the dynamics of zðtÞ. Then we have the following theorem.
z_ ¼ Aðu1d Þz
V 3 ¼ 12 z T Kz
V_ 3 ¼ kn 1 z 2n 1
Along the trajectories of (20), the time derivative of V1 is given as ð35Þ
Remark 3. Unlike the standard backstepping method, in order to reduce complexity of the controllers, the control law is chosen without canceling the coupling terms. Based on cascaded systems theory, and exploiting the inherent cascade interconnect structure of the nonholonomic system, we show that it is possible to design controllers for subsystems independently. Comparing with the existing results [20,21,23], our controller is simpler, since the virtual velocity controller is linear static state feedback law, and the actual torque controller is a conventional computed-torque law.
where λmin ðK p Þ is the minimum eigenvalue of matrix Kp. The above ~ inequality implies that uðtÞ globally asymptotically converges to zero. For y_ 1 ¼ k0 y1 þ u~ 1 , define Lyapunov function ð36Þ
The time derivative of V2 along the trajectory of y1 is V_ 2 ¼ k0 y21 þ u~ 1 y1 r 2k0 V 2 þju~ 1 j
pffiffiffiffiffiffiffiffiffi 2V 2
ð37Þ
Then the following inequality holds: pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dð V 2 ðtÞÞ=dt r k0 V 2 ðtÞ þ ju~ 1 ðtÞj= 2
ð38Þ
By the comparison theorem, we have pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ðtÞ r V 2 ðt 0 Þ expð k0 ðt t 0 ÞÞ Z t pffiffiffi ju~ 1 ðsÞj expð k0 ðt sÞÞ ds= 2 þ
ð39Þ
4. RNN-based control In practice, however, uncertainties and external disturbances, such as mechanical parameter variation, external load disturbance, and unstructured uncertainty do exist in the dynamics of the nonholonomic systems. Thus, the model-based controller cannot be precisely obtained. To solve this problem, a RNN-based control system (shown in Fig. 2) is developed for the nonholonomic systems without requiring explicit knowledge of the system dynamics, and can deal with bounded disturbances and unstructured unmodeled dynamics. The description of the recurrent neural network and the development of the control system are described as follows.
t0
pffiffiffiffiffiffiffiffiffiffiffi Since u~ 1 ðtÞ tends to zero as t-1, it follows that V 2 ðtÞ as well as y1 ðtÞ are asymptotically convergent. Step 2: For (22), we view it as a form of (23), that is z_ ¼ Aðu1d Þz þ gðt; zÞ
ð44Þ
It is well known [47] that the origin of the system (42) is globally exponentially stable if the pair ðAðu1d Þ; CÞ with C ¼ ½0; 0; …; kn 1 is uniformly completely observable (UCO). It follows immediately from [16, Corollary 2.3.4] that the pair ðAðu1d Þ; CÞ is UCO, which completes the proof. □
ð34Þ
V 2 ¼ 12 y21
ð43Þ
where K ¼ diag½ðk1 k2 ⋯kn 2 Þ; ðk2 k3 ⋯kn 2 Þ; …; kn 2 ; 1. From the skew symmetric property of KAðu1d Þ, we have
Proof. The whole proof process is divided into two steps: first we ~ prove that uðtÞ and y1 ðtÞ tend to zero as t-1, then by means of Lemma 1, we prove zðtÞ-0 as t-1. Step 1: For (20), consider the following Lyapunov function candidate:
~ 2 V_ 1 ¼ u~ T K p u~ r λmin ðK p Þ‖u‖
ð42Þ
is globally exponentially stable, then by means of Lemma 1, Theorem 2 can be proved. Now, for system (42), consider the following Lyapunov function candidate:
Theorem 2. Consider the dynamic tracking problem of system (6) and (7). Assume that the dynamics of the system is exactly known and dðtÞ ¼ 0, u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0; then the closed-loop system (20)–(22) is globally asymptotically stable, hence, the control law defined by (19) makes (10) hold.
V 1 ¼ 12 u~ T M 2 u~
ð41Þ
ð40Þ
where
4.1. Recurrent neural network
y1 ðtÞ. It can be easily checked that gðt; zÞ satisfies (25) with
A three-layer RNN, as shown in Fig. 3, which has n1 neurons in the input layer, n2 neurons in the hidden layer and n3 neurons in the output layer, is adopted to implement the proposed control
gðt; zÞ ¼ ðu~ 1 ðtÞ k0 y1 ðtÞÞðI Ψ ÞAðI Ψ Þ 1 z þBu~ 2 ðtÞ ðI Ψ ÞΔBðy1 ; u2d Þ
γ ðtÞ ¼ J I Ψ J J A J J ðI Ψ Þ 1 J ðju~ 1 ðtÞj þ jk0 y1 ðtÞjÞ
ud xd
RNN
_
x
xe
Kinematic Controller (17)
η_
u~
fˆ
Kp
_+
τ
Dynamic Model (7)
u
K s sgn( )
Fig. 2. Structure of RNN-based control system.
Kinematic Model (6)
x
∫
x
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
221
network approximation theory, a general function Hðχ Þ : Rn1 -Rn3 can be written as
z−1
^ χ jθn Þ þ ε Hðχ Þ ¼ Hð z χ1
ð53Þ
^ χ jθ Þ ¼ Υ ðχ jθ Þ, ε is the functional reconstruction error; where Hð n the ideal weights θ can be defined as ( ) n ^ θ ¼ arg min sup J Hðχ Þ Hðχ jθÞ J ð54Þ n
−1
1
θ
n
χ
However, the ideal weights are difficult to determine. If we denote
χn
1
z−1
^ χ jθn Þ is θ^ as the estimation of θn , the Taylor expansion of Hð
n3
^ ^ ^ χ jθn Þ ¼ Hð ^ χ jθ^ Þ þ ∂Hðχ jθ Þ θ~ þ Δ Hð ^ ∂θ n where θ~ ¼ θ θ^ , Δ is a vector of higher order terms;
ð55Þ
^ χ jθ^ Þ ∂Hð ∂H^ 1 ðχ jθ^ Þ ∂H^ 2 ðχ jθ^ Þ ∂H^ n3 ðχ jθ^ Þ T ¼ð ; ; …; Þ ^ ^ ^ ∂θ ∂θ ∂θ ∂θ^
z−1 Fig. 3. Three-layer RNN structure.
and scheme. The signal propagation and the activation function in each layer are introduced as follows: Layer 1: Input layer
∂H^ k ðχ jθ^ Þ ð1 rk rn3 Þ ∂θ^
net 1i ðNÞ ¼ χ 1i ðNÞ;
are defined as " # ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ¼ ; ; ^ ∂w ∂θ^ ∂ϑ^ ∂β^
1 O1i ðNÞ ¼ f i ðnet 1i ðNÞÞ
¼
1 ; 1 þexpð net 1i ðNÞÞ
i ¼ 1; 2; …; n1 :
ð45Þ
where χ1i represents the ith input to the node of input layer; N denotes the number of iterations; f1i is the activation function, which is a sigmoid function. Layer 2: Hidden layer net 2j ðNÞ ¼
β
ϑχ
2 2 j Oj ðN 1Þ þ ∑ ij i ðNÞ; i
1 ; 1 þexpð net 2j ðNÞÞ
j ¼ 1; 2; …; n2 :
ð46Þ
where βj are the weight of the self-feedback loop in the hidden layer; ϑij are the connective weights between the input layer and the hidden layer; and f2j is the activation function, which is also a sigmoid function. Layer 3: Output layer net 3k ðNÞ ¼ ∑wjk χ 3j ðNÞ; j
3
O3k ðNÞ ¼ f k ðnet 3k ðNÞÞ ¼ net 3k ðNÞ;
ðk 1Þn2
ð57Þ
ðn3 kÞn2
2 3 ∂H^ k ðχ jθ^ Þ 6 ∂H^ k ∂H^ k 7 ¼ 4 0…0 … 0…0 5 |ffl{zffl} ∂ϑ^ ∂ϑ^ n1 k |ffl{zffl} ∂ϑ^ 1k ðk 1Þn1
2 O2j ðNÞ ¼ f j ðnet 2j ðNÞÞ
¼
2 3 ∂H^ k ðχ jθ^ Þ 6 ∂H^ k ∂H^ k 7 … 0…0 5 ¼ 4 0…0 |ffl{zffl} ∂w ^ ^ 1k ∂w ^ n2 k |ffl{zffl} ∂w
ð56Þ
ð58Þ
ðn2 kÞn1
" # ∂H^ k ðχ jθ^ Þ ∂H^ k ∂H^ k ¼ … ∂β^ 1 ∂β^ n2 ∂β^
ð59Þ
Based on the above analysis, for a arbitrarily function Hðχ Þ, we can transform it into the following form: ^ ^ ^ χ jθ^ Þ þ ∂Hðχ jθ Þ θ~ þ ðΔ þ εÞ ð60Þ Hðχ Þ ¼ Hð ^ ∂θ ^ χ jθ^ Þ can be represented by a RNN output; ½∂Hð ^ χ jθ^ Þ=∂θ^ θ~ where Hð is a parametric linear term respect to θ~ , and ðΔ þ εÞ is treated as an bounded disturbance.
k ¼ 1; 2; …; n3 :
ð47Þ
where wjk are the connective weights between the hidden layer and the output layer; f3k is the activation function, which is set to be unit; and O3k is the output of the RNN. Moreover, we denote
4.2. Control design
w ¼ ðw11 w21 …wn2 1 ; …; w1n3 w2n3 …wn2 n3 ÞT
ð48Þ
_ η þ G2 ðxÞ þ d2 ðtÞ f ¼ M 2 ðxÞη_ þ C 2 ðx; xÞ
ϑ ¼ ðϑ11 ϑ21 …ϑn1 1 ; …; ϑ1n2 ϑ2n2 …ϑn1 n2 ÞT
ð49Þ
β ¼ ðβ1 β2 …βn2 ÞT
ð50Þ
θ ¼ ðwT ; ϑT ; βT ÞT
ð51Þ
The output of the RNN can be rewritten as follows: O ¼ Υ ðχ ; w; ϑ; βÞ ¼ Υ ðχ jθÞ 3
ð52Þ
where χ ¼ ðχ 1 ; χ 2 ; …; χ n1 Þ are the inputs of RNN, Υ ¼ ðγ 1 ; γ 2 ; …; γ n3 ÞT are the outputs of RNN. According to the neural T
From Eqs. (18) and (19), we know that the important nonlinear function ð61Þ
is required to be known to derive the control law τ. However, function f contains all the dynamic quantities of the nonholonomic system that often imperfectly known and difficult to determine, therefore, a RNN is adopted to adaptively learn this nonlinear function, that is ∂f^ ðχ jθ^ Þ ~ f ðχ Þ ¼ f^ ðχ jθ^ Þ þ θ þ ðΔ þ εÞ ∂θ^
ð62Þ
where the vector required to compute f is defined as χ ¼ ½xT ; ηT T , and we assume that ðΔ þ εÞ is bounded, i.e., J Δ þ ε J r K s , Ks is a positive constant. We have the following result:
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Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Theorem 3. Consider the robust dynamic tracking problem of system (6) and (7). Assume that u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0; then the control law ~ þ f^ ðχ jθ^ Þ τ ¼ B2þ ðqÞ½ K p u~ K s sgnðuÞ
ð63Þ
" #T ∂f^ ðχ jθ^ Þ _ ^ u~ θ ¼ Γ ∂θ^
ð64Þ
makes (10) holds, where Γ is a positive definite matrix. Proof. The closed-loop system of (6), (7), (17), (63) and (64) can be written as _ u~ K p u~ M 2 ðqÞu~_ ¼ C 2 ðq; qÞ
∂f^ ðχ jθ^ Þ ~ θ ∂θ^
~ ðΔ þ εÞ K s sgnðuÞ _
"
θ^ ¼ Γ
∂f^ ðχ jθ^ Þ ∂θ^
u~
ð66Þ ð67Þ
þ ðu~ 1 k0 y1 ÞðI Ψ ÞAðI Ψ Þ 1 z
ð68Þ
~ We will prove that for subsystem (65) and (66), uðtÞ still tends to zero as t-1, and the rest is the same as in Theorem 2. For (65) and (66), consider Lyapunov function candidate V 1 1
θ~ ¼ 12 u~ T M2 u~ þ 12 θ~ Γ 1 θ~ T
ð69Þ
the time derivative of V 1 is
! ∂f^ ðχ jθ^ Þ ~ _ T 1 V_ 1 ¼ u~ T K p u~ þ θ~ Γ u~ T θ ∂θ^ ~ þ ðΔ þ εÞÞ u~ T ðK s sgnðuÞ r u~ T K p u~
From the analysis in Sections 3 and 4, a design procedure for the proposed RNN-based control system is outlined as follows: Step 1: Compute the auxiliary tracking error (12). Step 2: Compute the virtual control law η by (17). Step 3: Compute the hybrid torque control law as (63), where the RNN weight parameters are updated as (64). Step 4: Return to Step 1.
ð65Þ
z_ ¼ Aðu1d Þz þ Bu~ 2 ðI Ψ ÞΔBðy1 ; u2d Þy1
T
Remark 7. The controller parameter Ks has a significant effect on the control performance. If Ks is selected too large, the sign function of the robust controller will result in serious chattering phenomena. On the other hand, if Ks is selected too small, the stability may not be guaranteed. Therefore, when we implement the controller, a suitable Ks is needed.
#T
y_1 ¼ k0 y1 þ u~ 1
V 1 ¼ V 1 þ 12 θ~ Γ
weights between input and hidden layer. This training scheme will increase the learning capability of the RNN.
5. Simulation results A model of wheeled mobile robot moving on a horizontal plane, as shown in Fig. 4, is used to verify the validity of the proposed control scheme. Let the midpoint between two wheels Q be the reference point, then the complete dynamics of the system _ ¼ 0, is given by q ¼ ½x; y; θT ; MðqÞ ¼ diag½m0 ; m0 ; I 0 , GðqÞ ¼ 0, Cðq; qÞ BðqÞ ¼ ½colf cos θ; sin θ; Lg, colf cos θ; sin θ; Lg=R, and JðqÞ ¼ ½ sin θ; cos θ; 0, where m0 is the mass of the mobile robot, I0 being its inertia moment around the vertical axis at point Q, R is the radius of the wheels, 2L is the length of axis of the front wheels. Let SðqÞ ¼ ½colf cos θ; sin θ; 0g; colf0; 0; 1g, and the coordinate transformation x ¼ T 1 ðqÞ and a state feedback u ¼ T 2 1 ðqÞv is 2 3 2 32 3 x1 0 0 1 x 6 x 7 6 sin θ cos θ 0 76 y 7 4 25¼4 54 5 "
ð70Þ
The inequality implies that V 1 is bounded. Recalling that _ ½∂f^ ðχ jθ^ Þ=∂θ^ and ðΔ þ εÞ are bounded, then it can M 2 ðqÞ; C 2 ðq; qÞ; be easily checked that V€ 1 ðtÞ is also uniformly bounded, thus by ~ Barbalat's lemma, V_ 1 ðtÞ as well as uðtÞ are asymptotically convergent. Along the same lines in the proof of Theorem 2, Theorem 3 is proved. □ Remark 4. The RNN-based control law (63) consists of two ~ is the sliding components. The first component ½ K p u~ K s sgnðuÞ mode controller which is used to guarantee the stability of the closed-loop system, and the second component ½f^ ðχ jθ^ Þ is the outputs of a RNN to compensate the uncertainties. Hence, the RNN-based law is a hybrid robust adaptive controller. Remark 5. Most of the conventional adaptive control schemes consider the uncertain dynamics of nonholonomic systems as a product of a regressor matrix and the inertia parameters. It can only deal with parameter uncertain systems in which the unknown parameters must be assumed to be constants or varying slowly. However, comparing with the existing results [20,21,23], our RNN-based controller can deal with parametric uncertainties as well as non-parametric uncertainties such as disturbances or unstructured unmodeled dynamics. Remark 6. For parameter tuning, an online parameter training methodology is derived in the sense of the Lyapunov function, so that the stability of the system can be guaranteed. Not only the connecting weights between hidden and output layer are adjusted online but also the self-feedback weights and the connecting
x3 u1 u2
#
" ¼
cos θ
v2 v1 v2 x2
sin θ #
0
θ
From the analysis in Sections 3 and 4, the virtual control law can be easily obtained as " # k0 y1 þ u1d η¼ k1 u1d z 1 k2 z 2 þu2d
η
and the adaptive law (64) is defined by Eqs. (56)–(59) after ^ Specifically, ∂f k ; ∂f k ; ∂f k are defined by the substituting f^ for H. ∂wjk ∂ϑ ∂β ij
j
Y
2R Q y
2L
x Fig. 4. Wheeled mobile robot configuration.
X
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2 x/m
1 0 −1 −2 −3 0
5
10 t/s
15
2
y/m
0 −0.5 −1 −1.5 −2 5
10 t/s
15
30
15 10 5 0
10 t/s
15
20
Model−based RNN−based
0
Error of θ /rad
θ / rad
20
5
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based Desired
25
Model−based RNN−based
0
Error of y /m
1 0.5
0
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based Desired
1.5
223
5
10 t/s
15
20
Model−based RNN−based
0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
−5 0
5
10 t/s
15
1.5
20
Norm of weight vectors
y/m
0.5 0 −0.5 −1 −1.5 −3
Start point −2
−1
0 x/m
1
2
10 t/s
15
Norm of w Norm of ϑ Norm of β
5 4 3 2 1 0
5
10 t/s
15
1
Model−based RNN−based
0.5
0
20
0
3
Control τ2 /(N.m)
Control τ1 /(N.m)
1
5
6
Model−based RNN−based Desired
1
0
20
Model−based RNN−based
0.5
0
−0.5
−0.5 0
5
10 t/s
15
20
0
5
10 t/s
15
20
Fig. 5. Simulation results of Case 1. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
224
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2
x/m
1 0 −1 −2 −3 0
5
10
15
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based
0
5
t/s
2
Model−based RNN−based Desired
Error of y /m
1
y/m
0.5 0 −0.5 −1 −1.5 5
10
15
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0
20
5
10
15
20
t/s
30
0.3
Model−based RNN−based Desired
25
Model−based RNN−based
0.2
Error of θ /rad
20
θ / rad
20
Model−based RNN−based
t/s
15 10 5
0.1 0 −0.1 −0.2
0 −5
−0.3 0
5
10
15
20
0
5
t/s
10
15
20
t/s
1.5
6
Norm of weight vectors
Model−based RNN−based Desired
1 0.5
y/m
15
t/s
1.5
−2 0
10
0 −0.5 −1
Norm of w Norm of ϑ Norm of β
5 4 3 2 1
Start point
−1.5 −3
0 −2
−1
0
1
2
3
0
5
10
15
20
t/s
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3
1
Model−based RNN−based
Control τ2 / (N.m)
Control τ1 / (N.m)
x/m
Model−based RNN−based
0.5
0
−0.5 0
5
10
t/s
15
20
0
5
10
15
20
t/s
Fig. 6. Simulation results of Case 2. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2
x/m
1 0 −1 −2 −3 0
5
10
15
225
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based
0
5
t/s
2
Model−based RNN−based Desired
1.5
Error of y /m
y/m
1 0.5 0 −0.5 −1 −1.5 0
5
10
15
20
Model−based RNN−based
0
Model−based RNN−based Desired
25
Error of θ / rad
θ / rad
20 15 10 5 0 −5
5
10
15
20
20
Model−based RNN−based
0
5
10
15
20
6
Norm of weight vectors
Model−based RNN−based Desired
1 0.5
y/m
15
t/s
1.5
0 −0.5 −1 Start point
Norm of Norm of Norm of
5
w ϑ β
4 3 2 1 0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
0
5
x/m
10
15
20
t/s
1
1
Model−based RNN−based
Model−based RNN−based
Controlτ2 / (N.m)
Controlτ1 / (N.m)
10
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
t/s
−1.5 −2.5
20
t/s
30
5
15
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
t/s
0
10
t/s
0.5
0
0.5
0
−0.5
−0.5 0
5
10
t/s
15
20
0
5
10
15
20
t/s
Fig. 7. Simulation results of Case 3. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
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Z. Miao et al. / Neurocomputing 142 (2014) 216–227
following equations: ∂f k 2 1 ¼ f j ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞ ∂wjk i ∂f k 2 1 1 ¼ wjk ðf j Þ0 ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞnf i ðχ 1i ðNÞÞ ∂ϑij i ∂f k 2 1 ¼ wjk ðf j Þ0 ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞnO2j ðN 1Þ ∂β j i where ðf j Þ0 is the derivative of ðf j Þ; χ1i is the ith element of vector ½x1 ; x2 ; x3 ; η1 ; η2 T ; and i ¼ 1; 2; ‥; 5; j ¼ 1; 2; …; 10; k ¼ 1; 2. In the simulation, the parameters of the systems are assumed to be L ¼ 1; R ¼ 0:1; m 0 ¼ 1; I 0 ¼ 1, where the overbar symbol represents the system parameter in the nominal condition. To investigate the effectiveness of the proposed controllers, three simulation cases including parametric variation (parametric uncertainties) and external disturbance (non-parametric uncertainties) are considered in the following: 2
2
Case 1: m0 ¼ m 0 ; dðtÞ ¼ ½0; 0; 0T . Case 2: m0 ¼ 1:5m 0 ; dðtÞ ¼ ½0; 0; 0T . Case 3: m0 ¼ m 0 ; dðtÞ ¼ ½0:5 sin 3t; 0:5 cos 2t; cos 3tT . Given a desired trajectory qd ðtÞ ¼ ½xd ðtÞ; yd ðtÞ; θd ðtÞT , with xd ðtÞ ¼ 2 sin t; yd ðtÞ ¼ cos t, and θd ðtÞ is determined by the nonholonomic constraint x_ d sin θd y_ d cos θd ¼ 0. The initial _ condition is qð0Þ ¼ ½0; 1:2; 0T ; qð0Þ ¼ ½0; 0; 0T , and the control parameters used in the controller are set as k0 ¼ 3; k1 ¼ 5; k2 ¼ 5; K p ¼ diag½5; 5; K s ¼ 2; Γ ¼ I. In addition, the structure of RNN is 5-input, 10-hidden, and 2-output, all the weights except the self-feedback loop are initialize randomly in the range of [ 1, 1], the self-feedback units are chosen as 0, that is, there are no feedback units initially. A comparison study of the model-based controller in Theorem 2 and RNN-based controller in Theorem 3 is presented to show the effectiveness of the proposed algorithms. The simulation results are shown in Figs. 5–7. It can be seen from the results in Case 1 that both model-based controller and RNN-based controller can achieve good tracking performance when no uncertainties and disturbances exist in the dynamics of the mobile robot. However, as shown in Fig. 6 for parametric variation and Fig. 7 for external disturbance, the modelbased controller fails to cope with these cases while the RNN-based controller still works well. In addition, from the respective tracking errors shown in Fig. 7(b), (d) and (f), we can see that the tracking errors of the RBF-based controller are small enough and eventually tend to zero, thus the asymptotical stability of the tracking error dynamics is guaranteed. Thus, the effectiveness and robustness of the RNN-based control system is verified.
6. Conclusion In this paper, a robust RNN-based control system for tracking control of uncertain dynamic nonholonomic systems is developed based on the universal approximation property of recurrent neural networks. The control scheme can learn the full dynamics of nonholonomic system on-line, and can deal with parametric as well as non-parametric uncertainties, yet guarantees tracking errors asymptotically converge to zero. The application to a simplified mobile robot is described, and the simulation results verify the effectiveness of the proposed control system. References [1] I. Kolmanovsky, N.H. McClamroch, Developments in nonholonomic control problems, IEEE Control Syst. Mag. 15 (6) (1995) 20–36.
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Xu, Tracking control of uncertain dynamic nonholonomic system and its application to wheeled mobile robots, IEEE Trans. Robot. Autom. 16 (6) (2000) 870–874. [21] W.J. Dong, W.L. Xu, Adaptive tracking control of uncertain nonholonomic dynamic system, IEEE Trans. Autom. Control 46 (3) (2001) 450–454. [22] Z.G. Hou, A.M. Zou, L. Cheng, M. Tan, Adaptive control of an electrically driven nonholonomic mobile robot via backstepping and fuzzy approach, IEEE Trans. Control Syst. Technol. 17 (4) (2009) 803–815. [23] M. Oya, C.Y. Su, R. Katoh, Robust adaptive motion/force tracking control of uncertain nonholonomic mechanical systems, IEEE Trans. Robot. Autom. 19 (1) (2003) 175–181. [24] K. Shojaei, A.M. Shahri, Adaptive robust time varying control of uncertain nonholonomic robotic systems, IET Control Theory Appl. 6 (1) (2012) 90–102. [25] Z. Wang, S.S. Ge, T.H. Lee, Robust motion/force control of uncertain holonomic/ nonholonomic mechanical systems, IEEE/ASME Trans. 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[31] C. de Sousa Jr., E.M. Hemerly, R.K.H. Galvao, Adaptive control for mobile robot using wavelet networks, IEEE Trans. Syst., Man, Cybern., B, Cybern. 32 (4) (2002) 493–504. [32] B.S. Park, S.J. Yoo, Y.H. Choi, Adaptive neural sliding mode control of nonholonomic wheeled mobile robots with model uncertainty, IEEE Trans. Control Syst. Technol. 17 (1) (2009) 207–214. [33] S.S. Ge, C. Wang, Adaptive neural control of uncertain MIMO nonlinear systems, IEEE Trans. Neural Netw. 15 (3) (2004) 674–692. [34] C.C. Hua, Y. Yang, X. Guan, Neural network-based adaptive position tracking control for bilateral teleoperation under constant time delay, Neurocomputing 113 (2013) 204–212.
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[35] T. Das, I.N. Kar, S. Chaudhury, Simple neuron-based adaptive controller for a nonholonomic mobile robot including actuator dynamics, Neurocomputing 69 (2006) 2140–2151. [36] C.M. Lin, C.F. Hsu, Recurrent neural network based adaptive backstepping control for induction servo motors, IEEE Trans. Ind. Electron. 52 (6) (2005) 1677–1684. [37] F.J. Lin, R.J. Wai, Robust recurrent fuzzy neural network control for linear synchronous motor drive system, Neurocomputing 50 (2003) 365–390. [38] F.J. Lin, R.J. Wai, W.D. Chou, S.P. Hsu, Adaptive backstepping control using recurrent neural network for linear induction motor drive, IEEE Trans. Ind. Electron. 49 (1) (2002) 134–146. [39] F.J. Lin, H.J. Shieh, P.H. Shieh, P.H. Shen, An adaptive recurrent-neural-network motion controller for X Y table in CNC machine, IEEE Trans. Syst., Man, Cybern., B, Cybern. 36 (2) (2006) 286–299. [40] C.F. Hsu, K.H. Cheng, Recurrent fuzzy-neural approach for nonlinear control using dynamic structure learning scheme, Neurocomputing 71 (2008) 3447–3459. [41] R.M. Murray, S.S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Autom. Control 38 (5) (1993) 700–716. [42] L. Bushnell, D. Tilbury, S.S. Sastry, Steering three-input chained form nonholonomic systems using sinusoids: the firetruck example, in: Proceedings of the European Control Conference 1993, pp. 1432–1437. [43] G.C. Walsh, L.G. Bushnell, Stabilization of multiple input chained form control systems, Syst. Control Lett. 25 (3) (1995) 227–234. [44] T.C. Lee, Z.P. Jiang, A generalization of Krasovskii–LaSalle theorem for nonlinear time-varying systems: converse results and applications, IEEE Trans. Autom. Control 50 (8) (2005) 1147–1163. [45] Z.H. Qu, J. Wang, C.E. Plaisted, R.A. Hull, Global-stabilizing near-optimal control design for nonholonomic chained systems, IEEE Trans. Autom. Control 51 (9) (2006) 1440–1456. [46] H.K. Khalil, Nonlinear Systems, Prentice-Hall, New Jersey, 2002, pp. 162–165. [47] W.J. Rugh, Linear Systems Theory, Prentice-Hall, New Jersey, 1996.
Zhiqiang Miao received the B.S. degree in Electrical and Information Engineering from Hunan University, Changsha, China, in 2010, where he is currently working toward the Ph.D. degree with the college of Electrical and Information Engineering. His research interests include neural networks, robot control, robust control and nonlinear systems.
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Yaonan Wang received the B.S. degree in Computer Engineering from East China Science and Technology University (ECSTU), Fuzhou, China, in 1981 and the M.S. and Ph.D. degrees in Electrical Engineering from Hunan University, Changsha, China, in 1990 and 1994, respectively. From 1994 to 1995, he was a postdoctoral Research Fellow with the National University of Defence Technology. From 1981 to 1994, he worked with ECSTU. From 1998 to 2000, he was a senior Humboldt Fellow in Germany, and from 2001 to 2004, he was a visiting professor with the University of Bremen, Bremen, Germany. He has been a professor at Hunan University since 1995. His research interests include intelligent control and information processing, robot control, industrial process control, and image processing.
Yimin Yang received the B.S. and M.S. degrees in 2005 and 2009, respectively, from Xiangtan University and Hunan University, Hunan, China. Now, He is currently working toward the Ph.D. degree in Hunan University. His research interests include artificial neural networks, robot control and navigation, optimization algorithm.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Robust tracking control of uncertain dynamic nonholonomic systems using recurrent neural networks$ Zhiqiang Miao n, Yaonan Wang, Yimin Yang College of Electrical and Information Engineering, Hunan University, Changsha 410082, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 September 2013 Received in revised form 18 December 2013 Accepted 31 March 2014 Communicated by Hongli Dong Available online 2 June 2014
The tracking problem for a class of dynamic nonholonomic systems with uncertainties is considered. First, under the assumption that the dynamics of the nonholonomic systems are exactly known without uncertainties, a simpler model-based controller is proposed by means of cascade design approach, in which the virtual velocity controller is linear, and the actual torque controller is derived by conventional computed-torque law. Then, to deal with uncertainties, a recurrent neural network control system is developed without requiring explicit knowledge of the system dynamics. The closed-loop stability analysis is presented based on a technical lemma developed for nonlinear cascaded systems with vanishing disturbances. Comparing with the existing results, the resulting control system has a simpler structure, and can deal with parametric uncertainties as well as non-parametric uncertainties, yet guarantees asymptotic stability of the tracking error dynamics. Simulation results for a wheeled mobile robot verify the good tracking performance and robustness of the proposed control system. & 2014 Elsevier B.V. All rights reserved.
Keywords: Robust control Nonholonomic systems Recurrent neural networks Tracking control Cascaded systems
1. Introduction In recent years, much attention has been devoted to the problem of controlling nonlinear mechanical systems with nonholonomic constraints [1,2]. Depending on whether the nonholonomic system is presented by a kinematic or dynamic model, the control problem can be classified as either kinematic control or dynamic control. Due to Brockett's theorem [3], it is well known that a nonholonomic system cannot be asymptotically stabilized to an equilibrium point via smooth or even continuous purefeedback laws. However, several approaches have been proposed for tackling the stabilization problem at kinematic level [4–8] or dynamic level [9–11]. The tracking problem has also received a great deal of attention because of its practical importance. Several authors have studied the kinematic tracking problem in which velocity acts as the control input. In [12], a linearization-based tracking control was proposed for nonholonomic systems under the assumption that the linearized system is uniformly completely controllable along the desired trajectory. In [13,14], based on dynamic feedback
☆ This paper was not presented at any IFAC meeting. This work was supported by the National Natural Science Foundation of China (61175075 and 61203207), National High Technology Research and Development Program of China (863 Program: 2012AA111004 and 2012AA112312), and Hunan Provincial Innovation Foundation for Postgraduate (521298960). n Corresponding author. E-mail address: [email protected] (Z. Miao).
http://dx.doi.org/10.1016/j.neucom.2014.03.061 0925-2312/& 2014 Elsevier B.V. All rights reserved.
linearization, controllers with structural singularity were proposed for the tracking problem of mobile robots. However, these controllers only solve the local tracking problem. There are mainly two methods to deal with the global tracking problem: backstepping method [15,16] and cascade design approach [17,18]. The main difference of these two methods lies in the way of dealing with the coupling terms between subsystems. Unlike the backstepping method, the coupling terms are neglected and the control law is chosen without canceling the coupling terms in cascade design to reduce complexity of the controllers. In practice, however, it is more realistic to consider the tracking problem at dynamic level, where the torque and force are taken as the control inputs. The dynamics of the systems usually cannot be neglected if high performance of the control systems is required. With the assumption of known dynamics, model-based controls can be obtained for the dynamic nonholonomic systems using backstepping [19]. However, there often exist uncertainties in the dynamics of the nonholonomic systems inevitably, such as unmolded dynamics, parameter perturbations and load variation. To confront this, some adaptive controls and robust controls have been developed. In [20,21], adaptive control techniques were employed to solve the tracking problem of nonholonomic systems with unknown inertia parameters. However, the major problem of the adaptive control approaches is that certain functions must be assumed “linearity in the parameters”, and tedious preliminary computation of “regression matrices” is needed [22]. In [23], a robust adaptive controller was designed for nonholonomic mechanical systems with model uncertainties. In [11,24], robust
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
adaptive controllers were proposed for dynamic nonholonomic systems with parametric and non-parametric uncertainties, in which adaptive control techniques were used to compensate for the parametric uncertainties and sliding mode control was used to suppress the bounded disturbances. In [25], robust control strategies were presented systematically for both holonomic mechanical systems and a large class of nonholonomic systems in the presence of uncertainties and disturbances. Li et al. [26–28] have studied the motion/force control for a mobile manipulator under both holonomic and nonholonomic constraints, and some adaptive robust control strategies have been proposed. In order to cope with highly uncertain nonlinear systems, as an alternative, the applications of neural networks (NNs) were also studied for the control of nonholonomic robotic systems. A multilayer perception network-based controller was suggested by Fierro and Lewis [29] to deal with parametric or non-parametric uncertainties for a nonholonomic mobile robot without any prior knowledge of the uncertainties. Other feedforword neural networks, like radial basis function (RBF) neural network [30] and wavelet network [31,32], were also adopted for the robust control of mobile manipulators and mobile robots. In the past decade, a great progress has been achieved in the study of using neural networks to control uncertain nonlinear systems. Extensive works demonstrate that adaptive neural control is particularly suitable for controlling highly uncertain, nonlinear, and complex systems [33–35]. In these neuro-adaptive control schemes, the neural network is used to compensate the effects of nonlinearity and system uncertainties, so the stability, convergence and robustness of the control system can be improved [36]. According to the structure, the neural networks can be mainly classified as feedforward neural networks (FNNs) and recurrent neural networks (RNNs). It is well known that FNN is capable of approximating arbitrary continuous function closely. However, FNN is a static mapping and unable to represent a dynamic mapping without the aid of tapped delays [37]. On the other hand, he RNNs, which comprise both feedforward and feedback connections, have superior capabilities than the FNNs. Since the RNN has a feedback loop, it can capture the past information of the network and adapt rapidly to sudden changes of the control environment [38]. The RNNs have the ability to deal with time-varying input or output though their own natural temporal operation [39]. For this ability, the structure of the neural network is simplified. Due to its dynamic characteristic and relative simple structure, the RNN is a useful tool in real-time application [40]. In this paper, the tracking problem is considered for a class of dynamic nonholonomic systems in which the nonholonomic kinematic subsystem is assumed to be capable of being transformed into the chained form and the dynamic subsystem is completely unknown. First, a simpler model-based controller is proposed for dynamic nonholonomic systems without uncertainties, in which the virtual velocity controller is linear, and the actual torque controller is derived by conventional computed-torque law. The stability of the closed-loop systems is presented based on the results on nonlinear cascaded systems with vanishing disturbances. To deal with disturbances and unstructured unmodeled dynamics in the nonholomonic system, a robust control system is then developed based on recurrent neural networks. On-line weights tuning algorithms that do not require off-line learning yet guaranty asymptotic stability of the tracking error dynamics are utilized. Simulation results for a wheeled mobile robot are provided to demonstrate the effectiveness of the proposed control method. The remainder of this paper is organized as follows. Problem is formulated in Section 2. In Section 3, a model-based controller is presented and the closed-loop stability analysis is given based on a
217
technical lemma developed for nonlinear cascaded systems with vanishing disturbances. A RNN-based control system is developed in Section 4. In Section 5, the effectiveness of the proposed approach is verified by simulation on a wheeled mobile robot. Section 6 concludes the paper.
2. Problem formulation In general, a mechanical system with nonholonomic constraints can be described as JðqÞq_ ¼ 0
ð1Þ
_ q_ þ GðqÞ þdðtÞ ¼ J T ðqÞλ þBðqÞτ MðqÞq€ þ Cðq; qÞ n
ð2Þ nn
where q A R denotes the generalized coordinates, MðqÞ A R is _ A Rnn is the the symmetric, positive definite inertia matrix, Cðq; qÞ centripetal and coriolis matrix, GðqÞ A Rn is the gravitational vector, dðtÞ A Rn denotes unknown disturbances including unstructured unmodeled dynamics, JðqÞ A Rmn is the constrained matrix, λ A Rm is the associated Lagrange multipliers which expresses the contact force, τ A Rr is the vector of control input, BðqÞ A Rnr is the input transformation matrix, and is assumed to be known because it is a function of fixed geometry of the system. The dynamic system (2) has the following properties. _ GðqÞ and d(t) are bounded. Property 1. MðqÞ; Cðq; qÞ; _ 2C is a skew-symmetric matrix, i.e. xT ðM _ 2CÞ Property 2. M x ¼ 0; 8 x a 0. Let SðqÞ A Rnðn mÞ be a full rank matrix formed by a set of smooth and linearly independent vector fields spanning the null space of J(q), i.e. ST ðqÞJ T ðqÞ ¼ 0
ð3Þ
According to (2) and (3), there always exists an auxiliary vector of independent generalized velocities v A Rn m , that make the system (1) and (2) be transformed in to a more appropriate representation for control purposes: q_ ¼ SðqÞv
ð4Þ
_ þ G1 ðqÞ þ d1 ðtÞ ¼ B1 ðqÞτ M 1 ðqÞv_ þ C 1 ðq; qÞv
ð5Þ
_ ¼ ST ðMðqÞS_ þ Cðq; qÞSÞ, _ where M 1 ðqÞ ¼ S MðqÞS, C 1 ðq; qÞ G1 ðqÞ ¼ ST GðqÞ, B1 ðqÞ ¼ ST BðqÞ, d1 ðtÞ ¼ ST dðtÞ. It should be noted that the reduced system consists of a reduced state dynamics (5) and the so-called kinematic model (4) of nonholonomic systems in the literature. It is assumed that B1 ðqÞ A Rðn mÞr is a full rank matrix and r Z n m to completely actuate the nonholonomic system. For ease of controller design, the existing results for the control of nonholonomic canonical forms in the literature are exploited. We assume that there exists a coordinate transformation x ¼ T 1 ðqÞ and a state feedback v ¼ T 2 ðqÞu, so that the kinematic model of the nonholonomic system given in Eq. (4) can be converted to the chained form. Since most of non-holonomic robotic systems (such as wheeled mobile robots) often include two pseudo-velocities, we only consider two independent generalized velocities ðn m ¼ 2Þ case for the sake of simplicity; however, the results can be extended to more general case. That is, the nonholonomic chained system considered in this paper is the 2-input, single-chain, single-generator chained form T
x_ 1 ¼ u1 x_ i ¼ u1 xi þ 1 x_ n ¼ u2
ð2 r ir n 1Þ ð6Þ
The necessary and sufficient condition for the existence of the transformation can be founded in [41–43]. Based on the above
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transformation, the dynamic model is converted as _ þ G2 ðxÞ þ d2 ðtÞ ¼ B2 ðxÞτ M 2 ðxÞu_ þ C 2 ðx; xÞu where
ð7Þ
_ ¼ T T2 ðqÞðM 1 ðqÞ M 2 ðxÞ ¼ T T2 ðqÞM 1 ðqÞT 2 ðqÞjq ¼ T 1 ðxÞ ; C 2 ðx; xÞ 1
To facilitate the subsequent control development, the new error variables are defined as follows: y1 ¼ x1e xj1e i xjd j ¼ i þ 1 ðj iÞ! n
_ 2 ðqÞÞjq ¼ T 1 ðxÞ , G2 ðxÞ ¼ T T2 ðqÞG1 ðqÞjq ¼ T 1 ðxÞ , B2 ðxÞ ¼ T_ 2 ðqÞ þ C 1 ðq; qÞT
yi ¼ xie
T T2 ðqÞB1 ðqÞjq ¼ T 1 ðxÞ , d2 ðtÞ ¼ T T2 ðqÞd1 ðtÞjq ¼ T 1 ðxÞ . The following properties of the dynamic model (7) can be easily proved.
ð2 r i r n 1Þ yn ¼ xne
Property 3. Matrix M 2 ðxÞ is symmetric positive definite; M 2 ðxÞ; _ G2 ðxÞ and d2 ðtÞ are bounded. C 2 ðx; xÞ;
Then, the following error model for tracking system can be attained:
_ 2 2C 2 is a skew-symmetric matrix. Property 4. M
y_ 1 ¼ u1 u1d
Given a differential desired trajectory qd(t) satisfies the constraint equation (1), it can be easily proved that there exists a time-varying reference control vd(t) satisfying
z_ ¼ u1d Az þ ðu1 u1d ÞAz þ Bðu2 u2d Þ ΔBðy1 ; u2d Þy1
1
1
1
1
q_ d ¼ Sðqd Þvd
ð8Þ
then there must exist a coordinate transformation xd ¼ T 1 ðqd Þ and a state feedback vd ¼ T 2 ðqd Þud , so that x_ 1d ¼ u1d x_ id ¼ u1d xði þ 1Þd
ð2 r ir n 1Þ
x_ nd ¼ u2d
ð9Þ
Then according to the above transformations, the tracking problem considering in this paper is equivalent to design the control law τ for the system (6) and (7) such that lim ðxðtÞ xd ðtÞÞ ¼ 0:
ð10Þ
t-1
∑
ð12Þ
ð13Þ
ð14Þ
where z ¼ ½z1 ; z2 ; …; zn 1 ¼ ½y2 ; y3 ; …; yn , ΔBðy1 ; u2d Þ ¼ ðn 2Þ!; yn1 4 =ðn 3Þ!; …; 1; 0T u2d , and (A, B) is in the controllable canonical form, that is 2 3 2 3 0 1 0 … 0 0 60 0 1 … 07 607 6 7 6 7 6 7 6 7 7 6 7 A¼6 6 ⋮ ⋮ ⋮ ⋱ ⋮ 7; B ¼ 6 ⋮ 7 6 7 6 7 0 0 … … 1 4 5 405 0 0 … … 0 1 T
T
Hereafter, we consider system (13) and (14) as our starting point. For subsystem (14), we define α ¼ ½α1 ; α2 ; …; αn 1 T with α1 ¼ 0; α2 ¼ 0 and for 2 r ir n 2 ∂α i zj þ 1 ki 1 ðzi 1 αi 1 Þ j ¼ 1 ∂zj i1
αi þ 1 ¼ ∑
3. Model-based control In this section, under the assumption that the dynamic model of the nonholonomic system is exactly known without considering disturbances, a model-based control system shown in Fig. 1 is developed. The control design procedure can be divided into two steps: first, a kinematic controller for subsystem (6) is designed, and then the conventional computed-torque control method is adopted to develop a toque law using backstepping approach.
½yn1 3 =
ð15Þ
where ki ð1 r i r n 3Þ are positive constants. From the definition of α, it can be easily checked that αi ð1 r i r n 1Þ are linear functions with respect to z1 ; z2 ; …; zi 1 , that is α ¼ Ψ z, where Ψ is a constant strictly lower triangular matrix. Let z ¼ z α, we have z_ 1 ¼ u1d z 2 þ ðu1 u1d Þz 2 u2d yn1 2 =ðn 2Þ! z_ ¼ u ð k z þz Þ u yn i 1 =ðn i 1Þ! i
i1 i1
1d
iþ1
2d 1
∂α i u2d y1n j 1 =ðn j 1Þ! þ ∑ j ¼ 1 ∂zj i1
∂α i zj þ 1 j ¼ 1 ∂zj i1
!
þðu1 u1d Þ zi þ 1 ∑
3.1. Kinematic Control As a precursor to the full dynamic system study, let us first neglect the dynamics of system (6) and (7) and consider the kinematic tracking problem only. Now u is assumed to be the virtual control, the object is to design the control input u¼ η such that (10) holds. Define xe ¼ x xd , it can be directly checked that the xe dynamics satisfy the following differential equations:
ð2 r i r n 2Þ n 2 ∂α n1 z_ n 1 ¼ u2 u2d u1d ∑ zj þ 1 j ¼ 1 ∂zj n2 ∂
þ ∑
j¼1
αn 1 ∂zj
u2d yn1 j 1 =ðn j 1Þ!
n2 ∂
ðu1 u1d Þ ∑
j¼1
x_ 1e ¼ u1 u1d x_ ie ¼ u1d xði þ 1Þe þ xi þ 1 ðu1 u1d Þ
αn 1 ∂zj
ð16Þ
zj þ 1
With the aid of z, we have the following theorem.
ð2 r ir n 1Þ x_ ne ¼ u2 u2d
ð11Þ
ud
Theorem 1. Consider the kinematic tracking problem of subsystem (6). Assume that u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0;
d dt
xe
xd
Kinematic Controller (17)
u
Torque Controller (19)
τ
Dynamic Model (7)
u
x Fig. 1. Structure of model-based control system.
Kinematic Model (6)
x
x
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Consider the system
then the virtual control law u ¼ η ¼ ζ þ ud k0 y1
2
ζ¼6 4 k
ξ_ ¼ f ðt; ξÞ þ gðt; ξÞ
3
n 2 u1d z n 2 kn 1 z n 1 þ u1d
n 2∂
∑
αn 1
j¼1
makes (10) holds, where k0 ; kn 2 ; kn 1 parameters.
∂zj
7 zj þ 1 5
ð17Þ
are positive control
3.2. Dynamic control Now the control law for τ is designed to make the outputs of the dynamic subsystem u tend to the auxiliary signal η. To design the torque input, denote: u~ ¼ u η, the subsystem (7) can be rewritten as _ u~ ¼ B2 ðxÞτ M 2 ðxÞu~_ þC 2 ðx; xÞ _ η þ G2 ðxÞ þ d2 ðtÞÞ ðM 2 ðxÞη_ þ C 2 ðx; xÞ
ð18Þ
Assume that the dynamics of the system is exactly known and dðtÞ ¼ 0. If the control law is chosen as _ η þG2 ðxÞ τ ¼ B2þ ðxÞ½ K p u~ þ ðM2 ðxÞη_ þC 2 ðx; xÞ
ð19Þ
B2þ ðxÞ
is the right pseudo inverse of B2 ðxÞ, Kp is a positive where definite matrix, and η is defined in Eq. (17). Then substituting (19) into (18), which yields _ u~ K p u~ M 2 ðxÞu~_ ¼ C 2 ðx; xÞ
ð20Þ
Substituting (17) into (13) and (16) and rewriting them in a more compact form: y_1 ¼ k0 y1 þ u~ 1
ð21Þ
z_ ¼ Aðu1d Þz þ ðu~ 1 k0 y1 ÞðI Ψ ÞAðI Ψ Þ 1 z þ Bu~ 2 ðI Ψ ÞΔBðy1 ; u2d Þy1
0
6 k u 1 1d 6 6 0 Aðu1d Þ ¼ 6 6 6 ⋮ 4 0
where ξ A R , f ðt; ξÞ and gðt; ξÞ are piecewise continuous in t and locally Lipschitz in ξ. We think of this system as a perturbation of the nominal system:
ξ_ ¼ f ðt; ξÞ
ð24Þ
The perturbation term gðt; ξÞ satisfies
Remark 2. Since z ¼ ðI Ψ Þz, it follows that the new z dynamics can be obtained by a linear coordinate transformation of the zcoordinates. This idea was used and the explicit expression of Ψ was given in [20,11]. In our method, Ψ does not occur in the controller expression, however, it will be used in the following stability analysis for ease of notation.
2
ð23Þ
n
Remark 1. The proof of this theorem can be easily validated in the following procedure of proving Theorem 2. The new tracking error used in our paper is first introduced in [44]. Another different tracking error is given in [45] via the use of Lie group operation. However, it can be easily checked that the definitions of tracking error in [44,45] are equivalent. It should be mentioned that the persistency of excitation (PE) assumption on u1d ðtÞ made in our method is also needed in most of the results on tracking control of nonholonomic systems. Roughly speaking, the fulfillment of this condition implies that the desired reference trajectory is a moving trajectory, instead of fixed set point.
where
219
ð22Þ
u1d
0
…
0
0
u1d
…
0
k2 u1d
0
⋱
⋮
⋮
⋮
⋱
u1d
0
…
kn 2 u1d
kn 1
3 7 7 7 7 7 7 5
Based on the above analysis, the closed-loop system of (6), (7), (17) and (19) consists of three subsystems (20)–(22). Before giving the completely stability analysis, we present a technical lemma on the stability of nonlinear cascaded systems with vanishing disturbances that will be used in the proof of our main results.
J gðt; ξÞ J r γ ðtÞ J ξ J þ δðtÞ
ð25Þ
where γ ðtÞ A R; δðtÞ A R are nonnegative, differentiable, and bounded for all t Z 0. For the perturbed system (23), we have the following. Lemma 1. Let ξ ¼0 be a globally exponentially stable equilibrium point of the nominal system (24), and suppose gðt; ξÞ is vanishing, that is, it satisfies (25) with lim γ ðtÞ ¼ 0;
t-1
lim δðtÞ ¼ 0
ð26Þ
t-1
Then the origin of perturbed system (23) is globally asymptotically stable. Proof. Since ξ_ ¼ f ðt; ξÞ is globally exponentially stable, using converse Lyapunov theory [46], there exists a Lyapunov function Vðt; ξÞ that satisfies c1 ξ2 r Vðt; ξÞ r c2 ξ2
∂V ∂V þ f ðt; ξÞ r c3 ξ2 ∂t ∂ξ ∂V J J r c4 J ξ J ∂ξ
ð27Þ
for all ðt; ξÞ A ½0; 1Þ Rn and some positive constants c1 ; c2 ; c3 and c4. The derivative of Vðt; ξÞ along the trajectories of (23) is given by ∂V ∂V ∂V þ f ðt; ξÞ þ gðt; ξÞ V_ ðt; ξÞ ¼ ∂t ∂ξ ∂ξ ∂V 2 r c3 ‖ξ‖ þ J J J gðt; ξÞ J ∂ξ r ð c3 c4 γ ðtÞÞ‖ξ‖2 þ c4 δðtÞ J ξ J Using (27), we can find an upper bound on V_ as sffiffiffiffiffi c3 c4 V V þc4 δðtÞ V_ r γ ðtÞ c1 c2 c1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi By defining WðtÞ ¼ Vðt; ξðtÞ, we have c4 _ ðtÞ r 1 c3 γ ðtÞc4 WðtÞ þ p ffiffiffiffiffiδðtÞ W 2 c2 c1 2 c1
ð28Þ
ð29Þ
ð30Þ
Since γ ðtÞ tends to zero as t-1, there exists a time moment T 0 4t 0 and a positive constant c5 such that γ ðtÞ oðc1 c3 c5 Þ=c2 c4 for all t Z T 0 . Which implies c4 _ ðtÞ r c5 WðtÞ þ p W ffiffiffiffiffiδðtÞ; 2c1 c2 2 c1
t ZT 0
ð31Þ
By the comparison theorem, WðtÞ r UðtÞðt Z T 0 Þ, where c5 c4 U_ ðtÞ ¼ UðtÞ þ pffiffiffiffiffiδðtÞ; 2c1 c2 2 c1
Uðt 0 Þ ¼ Wðt 0 Þ
ð32Þ
On the other hand, by assumption, for an arbitrary constant ε1 4 0, pffiffiffiffiffi there exists T 1 ðε1 Þ 4 0 such that δðtÞ r ðc5 =2 c1 c2 c4 Þε1 for all t Z T 0 þ T 1 . Solving the scalar differential equation (32) yields c5 UðtÞ ¼ UðT 0 þT 1 Þ exp ðt T 0 T 1 Þ 2c1 c2 Z t c4 c5 pffiffiffiffiffiδðsÞ exp ðt sÞ ds þ 2c1 c2 T þ T 2 c1 0 1 c5 ε1 ð33Þ r exp ðt T 0 T 1 Þ UðT 0 þ T 1 Þ þ 2c1 c2 2
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δðtÞ ¼ J B J ju~ 2 ðtÞj þ J I Ψ J J ΔBðy1 ; u2d Þ J jy1 ðtÞj
It follows that UðtÞ r ε1 for all t Z ð2c1 c2 =c5 Þ ln½2UðT 0 þ T 1 Þ=ε1 þ T 0 þ T 1 . Observing that ε1 can beffi arbitrarily small, then U(t) tends pffiffiffiffiffiffiffiffi to zero as t-1, hence VðtÞ and J ξðtÞ J are asymptotically convergent to zero. □
Since Ψ , A and B are bounded, ΔBðy1 ; u2d Þ is bounded as y1 ðtÞ and ~ u2d are bounded, moreover, uðtÞ and y1 ðtÞ tend to zero as t-1, it follows that γ ðtÞ and δðtÞ satisfy the vanishing condition (26). Thus, if the origin of system
We now view our closed-loop system as a form of (23) in ~ Lemma 1, with uðtÞ and y1 ðtÞ acting as vanishing perturbations of the dynamics of zðtÞ. Then we have the following theorem.
z_ ¼ Aðu1d Þz
V 3 ¼ 12 z T Kz
V_ 3 ¼ kn 1 z 2n 1
Along the trajectories of (20), the time derivative of V1 is given as ð35Þ
Remark 3. Unlike the standard backstepping method, in order to reduce complexity of the controllers, the control law is chosen without canceling the coupling terms. Based on cascaded systems theory, and exploiting the inherent cascade interconnect structure of the nonholonomic system, we show that it is possible to design controllers for subsystems independently. Comparing with the existing results [20,21,23], our controller is simpler, since the virtual velocity controller is linear static state feedback law, and the actual torque controller is a conventional computed-torque law.
where λmin ðK p Þ is the minimum eigenvalue of matrix Kp. The above ~ inequality implies that uðtÞ globally asymptotically converges to zero. For y_ 1 ¼ k0 y1 þ u~ 1 , define Lyapunov function ð36Þ
The time derivative of V2 along the trajectory of y1 is V_ 2 ¼ k0 y21 þ u~ 1 y1 r 2k0 V 2 þju~ 1 j
pffiffiffiffiffiffiffiffiffi 2V 2
ð37Þ
Then the following inequality holds: pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi dð V 2 ðtÞÞ=dt r k0 V 2 ðtÞ þ ju~ 1 ðtÞj= 2
ð38Þ
By the comparison theorem, we have pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ðtÞ r V 2 ðt 0 Þ expð k0 ðt t 0 ÞÞ Z t pffiffiffi ju~ 1 ðsÞj expð k0 ðt sÞÞ ds= 2 þ
ð39Þ
4. RNN-based control In practice, however, uncertainties and external disturbances, such as mechanical parameter variation, external load disturbance, and unstructured uncertainty do exist in the dynamics of the nonholonomic systems. Thus, the model-based controller cannot be precisely obtained. To solve this problem, a RNN-based control system (shown in Fig. 2) is developed for the nonholonomic systems without requiring explicit knowledge of the system dynamics, and can deal with bounded disturbances and unstructured unmodeled dynamics. The description of the recurrent neural network and the development of the control system are described as follows.
t0
pffiffiffiffiffiffiffiffiffiffiffi Since u~ 1 ðtÞ tends to zero as t-1, it follows that V 2 ðtÞ as well as y1 ðtÞ are asymptotically convergent. Step 2: For (22), we view it as a form of (23), that is z_ ¼ Aðu1d Þz þ gðt; zÞ
ð44Þ
It is well known [47] that the origin of the system (42) is globally exponentially stable if the pair ðAðu1d Þ; CÞ with C ¼ ½0; 0; …; kn 1 is uniformly completely observable (UCO). It follows immediately from [16, Corollary 2.3.4] that the pair ðAðu1d Þ; CÞ is UCO, which completes the proof. □
ð34Þ
V 2 ¼ 12 y21
ð43Þ
where K ¼ diag½ðk1 k2 ⋯kn 2 Þ; ðk2 k3 ⋯kn 2 Þ; …; kn 2 ; 1. From the skew symmetric property of KAðu1d Þ, we have
Proof. The whole proof process is divided into two steps: first we ~ prove that uðtÞ and y1 ðtÞ tend to zero as t-1, then by means of Lemma 1, we prove zðtÞ-0 as t-1. Step 1: For (20), consider the following Lyapunov function candidate:
~ 2 V_ 1 ¼ u~ T K p u~ r λmin ðK p Þ‖u‖
ð42Þ
is globally exponentially stable, then by means of Lemma 1, Theorem 2 can be proved. Now, for system (42), consider the following Lyapunov function candidate:
Theorem 2. Consider the dynamic tracking problem of system (6) and (7). Assume that the dynamics of the system is exactly known and dðtÞ ¼ 0, u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0; then the closed-loop system (20)–(22) is globally asymptotically stable, hence, the control law defined by (19) makes (10) hold.
V 1 ¼ 12 u~ T M 2 u~
ð41Þ
ð40Þ
where
4.1. Recurrent neural network
y1 ðtÞ. It can be easily checked that gðt; zÞ satisfies (25) with
A three-layer RNN, as shown in Fig. 3, which has n1 neurons in the input layer, n2 neurons in the hidden layer and n3 neurons in the output layer, is adopted to implement the proposed control
gðt; zÞ ¼ ðu~ 1 ðtÞ k0 y1 ðtÞÞðI Ψ ÞAðI Ψ Þ 1 z þBu~ 2 ðtÞ ðI Ψ ÞΔBðy1 ; u2d Þ
γ ðtÞ ¼ J I Ψ J J A J J ðI Ψ Þ 1 J ðju~ 1 ðtÞj þ jk0 y1 ðtÞjÞ
ud xd
RNN
_
x
xe
Kinematic Controller (17)
η_
u~
fˆ
Kp
_+
τ
Dynamic Model (7)
u
K s sgn( )
Fig. 2. Structure of RNN-based control system.
Kinematic Model (6)
x
∫
x
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
221
network approximation theory, a general function Hðχ Þ : Rn1 -Rn3 can be written as
z−1
^ χ jθn Þ þ ε Hðχ Þ ¼ Hð z χ1
ð53Þ
^ χ jθ Þ ¼ Υ ðχ jθ Þ, ε is the functional reconstruction error; where Hð n the ideal weights θ can be defined as ( ) n ^ θ ¼ arg min sup J Hðχ Þ Hðχ jθÞ J ð54Þ n
−1
1
θ
n
χ
However, the ideal weights are difficult to determine. If we denote
χn
1
z−1
^ χ jθn Þ is θ^ as the estimation of θn , the Taylor expansion of Hð
n3
^ ^ ^ χ jθn Þ ¼ Hð ^ χ jθ^ Þ þ ∂Hðχ jθ Þ θ~ þ Δ Hð ^ ∂θ n where θ~ ¼ θ θ^ , Δ is a vector of higher order terms;
ð55Þ
^ χ jθ^ Þ ∂Hð ∂H^ 1 ðχ jθ^ Þ ∂H^ 2 ðχ jθ^ Þ ∂H^ n3 ðχ jθ^ Þ T ¼ð ; ; …; Þ ^ ^ ^ ∂θ ∂θ ∂θ ∂θ^
z−1 Fig. 3. Three-layer RNN structure.
and scheme. The signal propagation and the activation function in each layer are introduced as follows: Layer 1: Input layer
∂H^ k ðχ jθ^ Þ ð1 rk rn3 Þ ∂θ^
net 1i ðNÞ ¼ χ 1i ðNÞ;
are defined as " # ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ∂H^ k ðχ jθ^ Þ ¼ ; ; ^ ∂w ∂θ^ ∂ϑ^ ∂β^
1 O1i ðNÞ ¼ f i ðnet 1i ðNÞÞ
¼
1 ; 1 þexpð net 1i ðNÞÞ
i ¼ 1; 2; …; n1 :
ð45Þ
where χ1i represents the ith input to the node of input layer; N denotes the number of iterations; f1i is the activation function, which is a sigmoid function. Layer 2: Hidden layer net 2j ðNÞ ¼
β
ϑχ
2 2 j Oj ðN 1Þ þ ∑ ij i ðNÞ; i
1 ; 1 þexpð net 2j ðNÞÞ
j ¼ 1; 2; …; n2 :
ð46Þ
where βj are the weight of the self-feedback loop in the hidden layer; ϑij are the connective weights between the input layer and the hidden layer; and f2j is the activation function, which is also a sigmoid function. Layer 3: Output layer net 3k ðNÞ ¼ ∑wjk χ 3j ðNÞ; j
3
O3k ðNÞ ¼ f k ðnet 3k ðNÞÞ ¼ net 3k ðNÞ;
ðk 1Þn2
ð57Þ
ðn3 kÞn2
2 3 ∂H^ k ðχ jθ^ Þ 6 ∂H^ k ∂H^ k 7 ¼ 4 0…0 … 0…0 5 |ffl{zffl} ∂ϑ^ ∂ϑ^ n1 k |ffl{zffl} ∂ϑ^ 1k ðk 1Þn1
2 O2j ðNÞ ¼ f j ðnet 2j ðNÞÞ
¼
2 3 ∂H^ k ðχ jθ^ Þ 6 ∂H^ k ∂H^ k 7 … 0…0 5 ¼ 4 0…0 |ffl{zffl} ∂w ^ ^ 1k ∂w ^ n2 k |ffl{zffl} ∂w
ð56Þ
ð58Þ
ðn2 kÞn1
" # ∂H^ k ðχ jθ^ Þ ∂H^ k ∂H^ k ¼ … ∂β^ 1 ∂β^ n2 ∂β^
ð59Þ
Based on the above analysis, for a arbitrarily function Hðχ Þ, we can transform it into the following form: ^ ^ ^ χ jθ^ Þ þ ∂Hðχ jθ Þ θ~ þ ðΔ þ εÞ ð60Þ Hðχ Þ ¼ Hð ^ ∂θ ^ χ jθ^ Þ can be represented by a RNN output; ½∂Hð ^ χ jθ^ Þ=∂θ^ θ~ where Hð is a parametric linear term respect to θ~ , and ðΔ þ εÞ is treated as an bounded disturbance.
k ¼ 1; 2; …; n3 :
ð47Þ
where wjk are the connective weights between the hidden layer and the output layer; f3k is the activation function, which is set to be unit; and O3k is the output of the RNN. Moreover, we denote
4.2. Control design
w ¼ ðw11 w21 …wn2 1 ; …; w1n3 w2n3 …wn2 n3 ÞT
ð48Þ
_ η þ G2 ðxÞ þ d2 ðtÞ f ¼ M 2 ðxÞη_ þ C 2 ðx; xÞ
ϑ ¼ ðϑ11 ϑ21 …ϑn1 1 ; …; ϑ1n2 ϑ2n2 …ϑn1 n2 ÞT
ð49Þ
β ¼ ðβ1 β2 …βn2 ÞT
ð50Þ
θ ¼ ðwT ; ϑT ; βT ÞT
ð51Þ
The output of the RNN can be rewritten as follows: O ¼ Υ ðχ ; w; ϑ; βÞ ¼ Υ ðχ jθÞ 3
ð52Þ
where χ ¼ ðχ 1 ; χ 2 ; …; χ n1 Þ are the inputs of RNN, Υ ¼ ðγ 1 ; γ 2 ; …; γ n3 ÞT are the outputs of RNN. According to the neural T
From Eqs. (18) and (19), we know that the important nonlinear function ð61Þ
is required to be known to derive the control law τ. However, function f contains all the dynamic quantities of the nonholonomic system that often imperfectly known and difficult to determine, therefore, a RNN is adopted to adaptively learn this nonlinear function, that is ∂f^ ðχ jθ^ Þ ~ f ðχ Þ ¼ f^ ðχ jθ^ Þ þ θ þ ðΔ þ εÞ ∂θ^
ð62Þ
where the vector required to compute f is defined as χ ¼ ½xT ; ηT T , and we assume that ðΔ þ εÞ is bounded, i.e., J Δ þ ε J r K s , Ks is a positive constant. We have the following result:
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Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Theorem 3. Consider the robust dynamic tracking problem of system (6) and (7). Assume that u1d ; u_ 1d are bounded, and lim inf t-1 ju1d ðtÞj 4 0; then the control law ~ þ f^ ðχ jθ^ Þ τ ¼ B2þ ðqÞ½ K p u~ K s sgnðuÞ
ð63Þ
" #T ∂f^ ðχ jθ^ Þ _ ^ u~ θ ¼ Γ ∂θ^
ð64Þ
makes (10) holds, where Γ is a positive definite matrix. Proof. The closed-loop system of (6), (7), (17), (63) and (64) can be written as _ u~ K p u~ M 2 ðqÞu~_ ¼ C 2 ðq; qÞ
∂f^ ðχ jθ^ Þ ~ θ ∂θ^
~ ðΔ þ εÞ K s sgnðuÞ _
"
θ^ ¼ Γ
∂f^ ðχ jθ^ Þ ∂θ^
u~
ð66Þ ð67Þ
þ ðu~ 1 k0 y1 ÞðI Ψ ÞAðI Ψ Þ 1 z
ð68Þ
~ We will prove that for subsystem (65) and (66), uðtÞ still tends to zero as t-1, and the rest is the same as in Theorem 2. For (65) and (66), consider Lyapunov function candidate V 1 1
θ~ ¼ 12 u~ T M2 u~ þ 12 θ~ Γ 1 θ~ T
ð69Þ
the time derivative of V 1 is
! ∂f^ ðχ jθ^ Þ ~ _ T 1 V_ 1 ¼ u~ T K p u~ þ θ~ Γ u~ T θ ∂θ^ ~ þ ðΔ þ εÞÞ u~ T ðK s sgnðuÞ r u~ T K p u~
From the analysis in Sections 3 and 4, a design procedure for the proposed RNN-based control system is outlined as follows: Step 1: Compute the auxiliary tracking error (12). Step 2: Compute the virtual control law η by (17). Step 3: Compute the hybrid torque control law as (63), where the RNN weight parameters are updated as (64). Step 4: Return to Step 1.
ð65Þ
z_ ¼ Aðu1d Þz þ Bu~ 2 ðI Ψ ÞΔBðy1 ; u2d Þy1
T
Remark 7. The controller parameter Ks has a significant effect on the control performance. If Ks is selected too large, the sign function of the robust controller will result in serious chattering phenomena. On the other hand, if Ks is selected too small, the stability may not be guaranteed. Therefore, when we implement the controller, a suitable Ks is needed.
#T
y_1 ¼ k0 y1 þ u~ 1
V 1 ¼ V 1 þ 12 θ~ Γ
weights between input and hidden layer. This training scheme will increase the learning capability of the RNN.
5. Simulation results A model of wheeled mobile robot moving on a horizontal plane, as shown in Fig. 4, is used to verify the validity of the proposed control scheme. Let the midpoint between two wheels Q be the reference point, then the complete dynamics of the system _ ¼ 0, is given by q ¼ ½x; y; θT ; MðqÞ ¼ diag½m0 ; m0 ; I 0 , GðqÞ ¼ 0, Cðq; qÞ BðqÞ ¼ ½colf cos θ; sin θ; Lg, colf cos θ; sin θ; Lg=R, and JðqÞ ¼ ½ sin θ; cos θ; 0, where m0 is the mass of the mobile robot, I0 being its inertia moment around the vertical axis at point Q, R is the radius of the wheels, 2L is the length of axis of the front wheels. Let SðqÞ ¼ ½colf cos θ; sin θ; 0g; colf0; 0; 1g, and the coordinate transformation x ¼ T 1 ðqÞ and a state feedback u ¼ T 2 1 ðqÞv is 2 3 2 32 3 x1 0 0 1 x 6 x 7 6 sin θ cos θ 0 76 y 7 4 25¼4 54 5 "
ð70Þ
The inequality implies that V 1 is bounded. Recalling that _ ½∂f^ ðχ jθ^ Þ=∂θ^ and ðΔ þ εÞ are bounded, then it can M 2 ðqÞ; C 2 ðq; qÞ; be easily checked that V€ 1 ðtÞ is also uniformly bounded, thus by ~ Barbalat's lemma, V_ 1 ðtÞ as well as uðtÞ are asymptotically convergent. Along the same lines in the proof of Theorem 2, Theorem 3 is proved. □ Remark 4. The RNN-based control law (63) consists of two ~ is the sliding components. The first component ½ K p u~ K s sgnðuÞ mode controller which is used to guarantee the stability of the closed-loop system, and the second component ½f^ ðχ jθ^ Þ is the outputs of a RNN to compensate the uncertainties. Hence, the RNN-based law is a hybrid robust adaptive controller. Remark 5. Most of the conventional adaptive control schemes consider the uncertain dynamics of nonholonomic systems as a product of a regressor matrix and the inertia parameters. It can only deal with parameter uncertain systems in which the unknown parameters must be assumed to be constants or varying slowly. However, comparing with the existing results [20,21,23], our RNN-based controller can deal with parametric uncertainties as well as non-parametric uncertainties such as disturbances or unstructured unmodeled dynamics. Remark 6. For parameter tuning, an online parameter training methodology is derived in the sense of the Lyapunov function, so that the stability of the system can be guaranteed. Not only the connecting weights between hidden and output layer are adjusted online but also the self-feedback weights and the connecting
x3 u1 u2
#
" ¼
cos θ
v2 v1 v2 x2
sin θ #
0
θ
From the analysis in Sections 3 and 4, the virtual control law can be easily obtained as " # k0 y1 þ u1d η¼ k1 u1d z 1 k2 z 2 þu2d
η
and the adaptive law (64) is defined by Eqs. (56)–(59) after ^ Specifically, ∂f k ; ∂f k ; ∂f k are defined by the substituting f^ for H. ∂wjk ∂ϑ ∂β ij
j
Y
2R Q y
2L
x Fig. 4. Wheeled mobile robot configuration.
X
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2 x/m
1 0 −1 −2 −3 0
5
10 t/s
15
2
y/m
0 −0.5 −1 −1.5 −2 5
10 t/s
15
30
15 10 5 0
10 t/s
15
20
Model−based RNN−based
0
Error of θ /rad
θ / rad
20
5
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based Desired
25
Model−based RNN−based
0
Error of y /m
1 0.5
0
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based Desired
1.5
223
5
10 t/s
15
20
Model−based RNN−based
0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
−5 0
5
10 t/s
15
1.5
20
Norm of weight vectors
y/m
0.5 0 −0.5 −1 −1.5 −3
Start point −2
−1
0 x/m
1
2
10 t/s
15
Norm of w Norm of ϑ Norm of β
5 4 3 2 1 0
5
10 t/s
15
1
Model−based RNN−based
0.5
0
20
0
3
Control τ2 /(N.m)
Control τ1 /(N.m)
1
5
6
Model−based RNN−based Desired
1
0
20
Model−based RNN−based
0.5
0
−0.5
−0.5 0
5
10 t/s
15
20
0
5
10 t/s
15
20
Fig. 5. Simulation results of Case 1. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
224
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2
x/m
1 0 −1 −2 −3 0
5
10
15
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based
0
5
t/s
2
Model−based RNN−based Desired
Error of y /m
1
y/m
0.5 0 −0.5 −1 −1.5 5
10
15
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0
20
5
10
15
20
t/s
30
0.3
Model−based RNN−based Desired
25
Model−based RNN−based
0.2
Error of θ /rad
20
θ / rad
20
Model−based RNN−based
t/s
15 10 5
0.1 0 −0.1 −0.2
0 −5
−0.3 0
5
10
15
20
0
5
t/s
10
15
20
t/s
1.5
6
Norm of weight vectors
Model−based RNN−based Desired
1 0.5
y/m
15
t/s
1.5
−2 0
10
0 −0.5 −1
Norm of w Norm of ϑ Norm of β
5 4 3 2 1
Start point
−1.5 −3
0 −2
−1
0
1
2
3
0
5
10
15
20
t/s
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3
1
Model−based RNN−based
Control τ2 / (N.m)
Control τ1 / (N.m)
x/m
Model−based RNN−based
0.5
0
−0.5 0
5
10
t/s
15
20
0
5
10
15
20
t/s
Fig. 6. Simulation results of Case 2. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
Z. Miao et al. / Neurocomputing 142 (2014) 216–227
Model−based RNN−based Desired
3
Error of x /m
2
x/m
1 0 −1 −2 −3 0
5
10
15
225
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
20
Model−based RNN−based
0
5
t/s
2
Model−based RNN−based Desired
1.5
Error of y /m
y/m
1 0.5 0 −0.5 −1 −1.5 0
5
10
15
20
Model−based RNN−based
0
Model−based RNN−based Desired
25
Error of θ / rad
θ / rad
20 15 10 5 0 −5
5
10
15
20
20
Model−based RNN−based
0
5
10
15
20
6
Norm of weight vectors
Model−based RNN−based Desired
1 0.5
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Fig. 7. Simulation results of Case 3. (a) Response of x. (b) Tracking error of x. (c) Response of y. (d) Tracking error of y. (e) Response of θ. (f) Tracking error of θ. (g) Trajectory in X Y plane. (h) L2 norm of weight vectors. (i) Control signal τ1. (j) Control signal τ2.
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following equations: ∂f k 2 1 ¼ f j ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞ ∂wjk i ∂f k 2 1 1 ¼ wjk ðf j Þ0 ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞnf i ðχ 1i ðNÞÞ ∂ϑij i ∂f k 2 1 ¼ wjk ðf j Þ0 ðβ j O2j ðN 1Þ þ ∑ϑij f i ðχ 1i ðNÞÞÞnO2j ðN 1Þ ∂β j i where ðf j Þ0 is the derivative of ðf j Þ; χ1i is the ith element of vector ½x1 ; x2 ; x3 ; η1 ; η2 T ; and i ¼ 1; 2; ‥; 5; j ¼ 1; 2; …; 10; k ¼ 1; 2. In the simulation, the parameters of the systems are assumed to be L ¼ 1; R ¼ 0:1; m 0 ¼ 1; I 0 ¼ 1, where the overbar symbol represents the system parameter in the nominal condition. To investigate the effectiveness of the proposed controllers, three simulation cases including parametric variation (parametric uncertainties) and external disturbance (non-parametric uncertainties) are considered in the following: 2
2
Case 1: m0 ¼ m 0 ; dðtÞ ¼ ½0; 0; 0T . Case 2: m0 ¼ 1:5m 0 ; dðtÞ ¼ ½0; 0; 0T . Case 3: m0 ¼ m 0 ; dðtÞ ¼ ½0:5 sin 3t; 0:5 cos 2t; cos 3tT . Given a desired trajectory qd ðtÞ ¼ ½xd ðtÞ; yd ðtÞ; θd ðtÞT , with xd ðtÞ ¼ 2 sin t; yd ðtÞ ¼ cos t, and θd ðtÞ is determined by the nonholonomic constraint x_ d sin θd y_ d cos θd ¼ 0. The initial _ condition is qð0Þ ¼ ½0; 1:2; 0T ; qð0Þ ¼ ½0; 0; 0T , and the control parameters used in the controller are set as k0 ¼ 3; k1 ¼ 5; k2 ¼ 5; K p ¼ diag½5; 5; K s ¼ 2; Γ ¼ I. In addition, the structure of RNN is 5-input, 10-hidden, and 2-output, all the weights except the self-feedback loop are initialize randomly in the range of [ 1, 1], the self-feedback units are chosen as 0, that is, there are no feedback units initially. A comparison study of the model-based controller in Theorem 2 and RNN-based controller in Theorem 3 is presented to show the effectiveness of the proposed algorithms. The simulation results are shown in Figs. 5–7. It can be seen from the results in Case 1 that both model-based controller and RNN-based controller can achieve good tracking performance when no uncertainties and disturbances exist in the dynamics of the mobile robot. However, as shown in Fig. 6 for parametric variation and Fig. 7 for external disturbance, the modelbased controller fails to cope with these cases while the RNN-based controller still works well. In addition, from the respective tracking errors shown in Fig. 7(b), (d) and (f), we can see that the tracking errors of the RBF-based controller are small enough and eventually tend to zero, thus the asymptotical stability of the tracking error dynamics is guaranteed. Thus, the effectiveness and robustness of the RNN-based control system is verified.
6. Conclusion In this paper, a robust RNN-based control system for tracking control of uncertain dynamic nonholonomic systems is developed based on the universal approximation property of recurrent neural networks. The control scheme can learn the full dynamics of nonholonomic system on-line, and can deal with parametric as well as non-parametric uncertainties, yet guarantees tracking errors asymptotically converge to zero. The application to a simplified mobile robot is described, and the simulation results verify the effectiveness of the proposed control system. References [1] I. Kolmanovsky, N.H. McClamroch, Developments in nonholonomic control problems, IEEE Control Syst. Mag. 15 (6) (1995) 20–36.
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Zhiqiang Miao received the B.S. degree in Electrical and Information Engineering from Hunan University, Changsha, China, in 2010, where he is currently working toward the Ph.D. degree with the college of Electrical and Information Engineering. His research interests include neural networks, robot control, robust control and nonlinear systems.
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Yaonan Wang received the B.S. degree in Computer Engineering from East China Science and Technology University (ECSTU), Fuzhou, China, in 1981 and the M.S. and Ph.D. degrees in Electrical Engineering from Hunan University, Changsha, China, in 1990 and 1994, respectively. From 1994 to 1995, he was a postdoctoral Research Fellow with the National University of Defence Technology. From 1981 to 1994, he worked with ECSTU. From 1998 to 2000, he was a senior Humboldt Fellow in Germany, and from 2001 to 2004, he was a visiting professor with the University of Bremen, Bremen, Germany. He has been a professor at Hunan University since 1995. His research interests include intelligent control and information processing, robot control, industrial process control, and image processing.
Yimin Yang received the B.S. and M.S. degrees in 2005 and 2009, respectively, from Xiangtan University and Hunan University, Hunan, China. Now, He is currently working toward the Ph.D. degree in Hunan University. His research interests include artificial neural networks, robot control and navigation, optimization algorithm.
