www.ietdl.org Published in IET Control Theory and Applications Received on 26th March 2013 Accepted on 5th March 2014 doi: 10.1049/iet-cta.2014.0059
ISSN 1751-8644
Adaptive non-backstepping fuzzy tracking control for a class of multiple-input–multiple-output pure-feedback systems with unknown dead-zones Rui Wang1,2 , Fu ShengYu1 , JiaYin Wang1 1 School
of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China of Mathematical and Biological Engineering, Inner Mongolia University of Science andTechnology, Baotou, 014000, People’s Republic of China E-mail:
[email protected] 2 School
Abstract: In this study, the authors present a novel adaptive non-back-stepping alleviating computation tracking control algorithm for a class of MIMO pure-feedback systems with unknown dead-zone inputs. Based on the introduced new state variables and coordinate transforms, the pure-feedback form is converted into a normal form where back-stepping scheme is not necessary. Owing to the unavailability of the new states, the tracking control problem is changed from a state-feedback one to an output-feedback one. Therefore, observers need to be designed to estimate the indirect un-measurable states. In the controller and observer design procedure, the control method is based on non-back-stepping scheme and for each subsystem only three adaptive adjusted parameters are needed to be updated on-line. Thus, different from the back-stepping-based ones, this new algorithm can not only alleviate on-line computation burden but also can considerably simplify the control procedure. According to Lyapunov stable analysis methods, the bounded of all the signals in the closed-loop system can be guaranteed and the tracking error is proven to converge to a small neighbourhood of the origin. Simulation results are presented to illustrate the effectiveness of this proposed approach.
1
Introduction
It is well-known that unknown dead-zone non-linearity as one of the most important non-smooth non-linear characteristics is frequently encountered in many industrial practical processes, actuators and sensors such as mechanical connections, hydraulic actuators and electric servomotors. Their existence is a source of instability and usually severely limits the control performance of system. Hence, investigations [1–10] on how to mitigate the effect of unknown dead-zone inputs have become one of the significant and challenging research topics during the past decades. How to compensate the unknown dead-zone inputs, the usually adopted approaches were based on constructing dead-zone inverse operators [1] scheme until 2004, the approximationbased robust adaptive back-stepping design technique was extended to deal with unknown dead-zone problem by Wang et al. [2]. Subsequently, significant progress has been made in handling the unknown dead-zone non-linearity following these approaches. We know that many approximations (FLSs [3]), radial basis function neural network (NN), fuzzy NN (FNN [4]), multilayer NN or high-order NN have been usually utilised to model the unknown continuous functions and non-linearity. In addition, these approximations are always combined together with the traditional back-stepping approaches to cope with the control problem of unknown dead-zone uncertainties. Several adaptive IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
control schemes were given to cope with the unknown deadzone non-linearity (in symmetric or non-symmetric form) in single-input–single output (SISO) form and in multi-input– multi-output (MIMO) form [5, 6]); the non-linear systems with unknown dead-zone input were considered in strictfeedback form [7] or in pure-feedback form [8]; when the system states are assumed to be measurable or unavailable, many researchers study state-feedback technique [9], and output-feedback schemes [10, 11]. Afterwards, these methods have been extended to more complex non-linear systems: time-delay stochastic systems [12, 13], large-scale systems [14, 15], even discrete-time systems [16]. However, all of these aforementioned robust adaptive control approaches coping with the unknown dead-zone input were based on the traditional back-stepping design methodologies. These schemes always inherit the open problem of ‘explosion of complexity’ in the recursive design procedure, many parameters need to be adjusted by updating the fuzzy optimal approximation vector or the NN optimal weight vector. The tedious and complex procedure exhibits an exponential increase with the increase order of the controlled systems. Therefore the control performance of these approaches is affected, even resulting in increasing running cost, especially in dealing with MIMO or large-scale non-linear systems. In order to reduce the computation burden caused by the traditional back-stepping control methods, dynamic surface control design technique [17, 18] and the novel 1293 © The Institution of Engineering and Technology 2014
www.ietdl.org alleviating computation design technique [19] were introduced to develop the problem of ‘explosion of complexity’ and decrease the large number of approximate parameters. However, these methods are still based on back-stepping technique with tedious procedure. It should be mentioned that, in [20], a novel adaptive non-back-stepping technique was proposed for a class of discrete pure-feedback system and the stability analysis procedure was considerably simpler than the aforementioned back-stepping-based methodologies. Afterwards, based on NN and FLSs approximation, this non-back-stepping method was extended to dealing with continuous strict-feedback systems [21] and state-time-delay systems [22], even for input-delay systems [23] and hypersonic aircraft model [24]. These non-back-stepping-based methods can avoid repeatedly differentiation and the explosion of computation. However, all of these control strategies did not consider the unknown dead-zone input, especially for MIMO non-linear systems in pure-feedback form. Motivated by the above analysis, here we address a novel non-back-stepping alleviating computation fuzzy tracking control algorithm for a class of MIMO pure-feedback nonlinear system with unknown dead-zone inputs. Firstly, we introduce some new state variables and the corresponding coordinate transforms. Secondly, because of the new states variables are indirectly unavailable, observers need to be designed and FLSs are utilised to approximate the unknown non-linearities. Thirdly, aiming at alleviating the computation, at each design procedure, we use one parameter to be the bound of the norm of optimal approximation vectors of FLSs, this enables us need only to adjust this parameter rather than the elements of the optimal approximation vectors of FLSs. As a result, compared with the results in [3–10, 12–18, 25–27], the computational burden is significantly alleviated for the fewer adjusted parameters and the simplifier design procedure. At last, according to Lyapunov-function stable analysis, the developed controller can guarantee all the signals in the closed-loop systems to be bounded and the tracking error to convergence to a small neighbourhood of zero. The remainder of this paper is organised as follows. The problem formulation and preliminaries are given in Section 2. A novel non-back-stepping adaptive decentralised fuzzy tracking control scheme is presented in Section 3. A simulation to illustrate the effectiveness of the proposed approach is presented in Section 4, and followed by Section 5, which concludes the work of this paper.
2 2.1
System descriptions and preliminaries System descriptions
We focus on novel adaptive fuzzy tracking controller design for a class of uncertain MIMO non-linear pure-feedback systems with unknown dead-zone inputs which are consisted of N interconnected subsystems. Each subsystem model can be expressed as follows ⎧ ⎪ x˙ i1 = xi2 + fi1 (xi1 , xi2 ) ⎪ ⎪ ⎪ ⎪ ⎨x˙ i2 = xi3 + fi2 (¯xi2 , xi3 ) (1) ... ... ⎪ ⎪ ⎪ x˙ i,ni = ui + fi,ni (¯xi,ni , ui ) + di (t, X ) ⎪ ⎪ ⎩ yi = xi1 , i = 1, 2, . . . , N where x¯ ij = [xi1 , xi2 , . . . , xij ]T ∈ Rj (j = 1, 2, . . . , ni ), xi = [xi1 , xi2 , . . . , xi,ni ]T ∈ Rni , ui ∈ R and yi ∈ R denote the state 1294 © The Institution of Engineering and Technology 2014
vector, control input and the output for the ith subsystem respectively; X = [x1 , x2 , . . . , xN ]T and y = [y1 , y2 , . . . , yN ]T are state vector and output for the whole MIMO non-linear system. fij (·) denote unknown smooth non-linear functions and di (t, X ) ∈ R denote the external disturbances of the ith subsystems. The non-symmetric unknown dead-zones with inputs ωi (t) and outputs ui (t) can be described as follows ⎧ ⎨mir (ωi (t) − bir ) if ωi (t) bir ui (t) = Di (ωi (t)) = 0 if bil < ωi (t) < bir ⎩ mil (ωi (t) − bil ) if ωi (t) bil (2) with mir (·) and mil (·) are the general slope of the unknown dead-zone; bir and bil express unknown right and left deadzone break points parameters. Assumption 1 [8]: For each ith subsystem, assume that the unknown dead-zone outputs ui (t) are unavailable and the dead-zone slops in positive region and in negative region are not the same, that is, mir = mil , the dead-zone parameters bir , bil are unknown but with known signs bir 0 and bil 0, that is, there exist known constants bilmin , birmin , birmax , bilmax . mmin , mmax , such that bil ∈ [bilmin , bilmax ], bir ∈ [birmin , birmax ] and mir ∈ [mmin , mmax ], |mil | ∈ [mmin , mmax ]. Based on Assumption 1, from a practical point of view, the non-symmetric dead-zone model (2) can be represented as ui (t) = Di (ωi (t)) = mmin [1 + ηi (t)]ωi (t) + hi (ωi (t))
(3)
where ηi (t) is a small piecewise positive function satisfying mi (t) mir if ωi (t) > 0 1 + ηi (t) = ; mi (t) = mil if ωi (t) 0 mmin and hi (t) can be concluded from ⎧ ⎨−mir bir hi (ωi (t)) = −mi (t)ωi (t) ⎩ −mil bil
(2) and (3) as follows if ωi (t) bir if bil < ωi (t) < bir if ωi (t) bil
(4)
Remark 1: Many practical systems can be expressed or transformed into this pure-feedback form (1) with unknown dead-zone inputs. Such as: hydraulic actuators or electric servomotors and so on. From Assumption 1, we conclude that hi (ωi (t)) is bounded and there exists constant ρ ∗ satisfying ρ ∗ = supt 0 |hi (ωi (t))|, where ρ ∗ can be chosen as ρ ∗ = max{mmax birmax , −mmax bilmin }; similarly, we define η∗ as η∗ = supt 0 |ηi (t)|. Assume the external disturbance di (t, X ) is bounded and satisfying |di (t, X )| di∗ , with di∗ > 0, (i = 1, 2, . . . , N ) [8]. The control objective is to design novel adaptive fuzzy controllers ωi (t)(i = 1, 2, . . . , N ), such that the system output y = [y1 , y2 , . . . , yN ]T can follow the desired trajectory reference signal Yd = [y1d , y2d , . . . , yNd ]T to a small neighbourhood of zero and all the signals in the closed-loop system are ultimately uniformly bounded. Remark 2: It is well known that extensive researches have been studied for uncertain MIMO non-linear systems with unknown dead-zone inputs [4–6, 10, 11, 15, 27], under different assumptions and conditions, but all of these techniques were all based on the traditional back-stepping technique and the approximation properties of FLSs, NN, FNN IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
www.ietdl.org and so on. In addition, these methods always inherited the problem of ‘explosion of computation’ and the problem of larger numbers of the online updating parameters and to be designed. So, in order to conquer these problems, in this paper, we firstly try to extend the non-back-stepping technique (combined with the alleviating computation technique) rather than back-stepping scheme for MIMO non-linear systems not only with unknown dead zone but also in pure-feedback form. 2.2
where Fi3 (¯xi3 ) = 2k=1 (∂Fi2 (¯xi2 )/∂xik ) + (∂Gi2 /∂xik )xi3 ) (xi,k+1 + fik (¯xi,k+1 ))+ Gi2 (¯xi3 )fi3 (¯xi3 )+(∂Gi2 (¯xi3 )/∂xi3 )xi3 and Gi3 (¯xi4 ) = ((∂Gi2 (¯xi3 )/∂xi3 )xi3 + Gi2 (¯xi3 ))(1 + gi3 (¯xi3 , ξi4 )). Step 3: Similarly to the above analysis based on (6)–(8) in Steps 1–2, in general, if we define Fi1 (xi1 ) = fi1 (xi1 ), Gi1 (xi1 , xi2 ) = 1 + gi1 (xi1 , xi2 ), then, we have x˙ i1 = Fi1 (xi1 ) + Gi1 (xi1 , xi2 ) and for j = 1, 2, . . . , ni , i = 1, 2, . . . , N , we conclude that zij = Fi,j−1 (¯xi,j−1 ) + Gi,j−1 (¯xij )xij
Original systems states transformation
For the conciseness of presentation, in the following analysis, we only give the detailed design procedure for the ith(i = 1, 2, . . . , N ) subsystem. To facilitate the control design procedure, in this section, new state variables and the corresponding coordinate transformations will be introduced to change the system (1) in pure-feedback form into the system in normal form. Since unknown functions fij (¯xij , xi,j+1 ) are smooth and continuously differential with respect to the whole variables xi1 , xi2 , . . . , xi,j+1 . For i = 1, 2, . . . , N ; j = 1, 2, . . . , ni , it is reasonable for us to define gij (¯xij , xi,j+1 ) = ∂fij (¯xij , xi,j+1 )/∂xi,j+1
zi,j+1 = Fij (¯xij ) + Gij (¯xi,j+1 )xi,j+1
xi,j−1 )/∂xik ) + (∂Gi,j−1 (¯xij )/ where Fij (¯xi,j−1 ) = j−1 k=1 (∂Fi−1 (¯ ∂xik )xij )(xi,k+1 + fik (¯xi,k+1 )) + Gi,j−1 (¯xij )fij (¯xij ) + (∂Gi,j−1 (¯xij )/ ∂xij )xij and Gij (¯xi,j+1 ) = ((∂Gi,j−1 (¯xij )/∂xij )xij + Gi,j−1 (¯xij )) × (1 + gij (¯xij , ξi,j+1 )). Step 4: Now, based on (6)–(9) in Steps 1–3, it is clear that the pure-feedback system (1) now can be described as the following normal form with respect to the newly state variables zi = [zi1 , zi2 , . . . , zi,ni ]T ⎧ z˙ = zi2 ⎪ ⎪ ⎪ i1 ⎪ ⎪ z˙ = zi3 ⎪ ⎪ i2 ⎨ ... ⎪ z˙i,ni −1 = zi,ni ⎪ ⎪ ⎪ θ ⎪ ⎪z˙i,ni = Fi (xi ) + Gi (xi , ui )ui + di (t, X ) ⎪ ⎩ yi = zi1 ; (i = 1, 2, . . . , N )
(5)
According to mean value theorem [18], there must exist ξi,j+1 = θxi,j+1 (0 < θ < 1), satisfying fij (¯xij , xi,j+1 ) = fij (¯xij ) + gij (¯xij , ξi,j+1 )|ξi,j+1 =θ xi,j+1 xi,j+1
(6)
The details of the states transformation procedure for each ith subsystem will be given in the following. Step 1: Define: zi1 = yi = xi1 , zi2 = z˙i1 = xi2 + fi1 (xi1 , xi2 ). Based on (5), with j = 2, we have fi2 (¯xi2 , xi3 ) = fi2 (¯xi2 , 0) + gi2 (¯xi2 , ξi3 )|ξi3 =θ xi3 (xi3 − 0) Along the derivative of zi2 , we obtain that ∂fi1 (fi2 (¯xi2 ) + (1 + gi2 (¯xi2 , ξi3 ))xi3 ) z˙i2 = 1 + ∂xi2 ∂fi1 ∂fi1 + (xi2 + fi1 (¯xi2 )) + 1 + ∂xi1 ∂xi2 = Fi2 (¯xi2 ) + Gi2 (¯xi3 )xi3
(7)
where Fi2 (¯xi2 ) = (1 + ∂fi1 /∂xi2 )fi2 (¯xi2 ) + ∂fi1 /∂xi1 (xi2 + fi1 (¯xi2 ), Gi2 (¯xi3 ) = Gi2 (¯xi2 , ξi3 ) = (1 + ∂fi1 /∂xi2 )(1 + gi2 (¯xi2 , ξi3 )). Step 2: Define zi3 = z˙i2 = Fi2 (¯xi2 ) + Gi2 (¯xi3 )xi3 , combing with the definition of fij (¯xij , xi,j+1 ) for j = 3, its time derivative is derived as 2 2
∂Fi2 (¯xi2 )
∂Gi2 (¯xi3 ) z˙i3 = + xi3 ∂xik ∂xik k=1 k=1 × (xi,k+1 + fik (¯xi,k+1 )) ∂Gi2 (¯xi3 ) + xi3 + Gi2 (¯xi3 ) ∂xi3 × (fi3 (¯xi3 ) + (1 + gi3 (¯xi3 , ξi4 )xi4 )) = Fi3 (¯xi3 ) + Gi3 (¯xi4 )xi4 IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
(8)
(9)
(10)
where Fi (xi ) = Fi,ni (¯xi,ni ), Gi (xi , uiθ ) = Gi,ni (¯xi,ni , uiθ ), zi = [zi1 , zi2 , . . . , zi,ni ]T and Z = [z1 , z2 , . . . , zN ]T . From the newly defined state vector zi = [zi1 , zi2 , . . . , zi,ni ], and the corresponding coordinate transformations form (7)–(10) in steps 1–4, we conclude that the former pure-feedback system (1) is now converted into the normal one with output yi = zi1 = xi1 . Remark 3: From Steps 1–4, the system form is converted from pure-feedback form (1) into the normal form (10), hence, it is not necessary for us to consider the traditional back-stepping technique. In this paper, we will extend the novel non-back-stepping technique to this pure-feedback non-linear systems with unknown dead-zone inputs. At the same, Although the state vector X = [x1 , x2 , . . . , xN ]T of the system (1) is available, the newly state vector of system (10) is not directly measurable and functions F(xi ) = Fi,ni (¯xi,ni ), G(xi ) = Gi,ni (¯xi,ni , uiθ ) are totally unknown. Hence, in the next section, we need to construct observers to estimate the unknown states, and we will incorporate FLSs to approximate the unknown functions.
3 3.1
FLSs controller design and ability analysis Observer design
As the newly defined state vector zi (t) = [zi1 (t), zi2 (t), . . . , zi,ni (t)]T of system (10) is indirectly unavailable, in this section, we need to construct state observer vector zˆ i (t) to estimate this unmeasured state vector zi (t). Now, define zˆ i (t) = [ˆzi1 (t), zˆi2 (t), . . . , zˆi,ni (t)]T as the estimation vector of the state vector zi (t), the estimate error vector z˜ i (t) as z˜ i (t) = zˆ i (t) − zi (t), with z˜ i (t) = [˜zi1 (t), z˜i2 (t), . . . , z˜i,ni (t)]T . 1295 © The Institution of Engineering and Technology 2014
www.ietdl.org ε, ∀Z ∈ ⊂ Rn , where W ∗T = [W1∗ , W2∗ , . . . , WN∗ ]T is the optimal fuzzy approximate weight parameter vector, here, the optimal fuzzy parameter vector W ∗T is artificial quantity required only for analytical purpose. ε ∈ R is a bounded error, ξ(Z) = [ξ 1 , ξ 2 , . . . , ξ m ]T is fuzzy basis function and m is the number of rules [28]. Owing to newly functions Fi (xi ) = Fi,ni (¯xi,ni ) and Gi (xi , uiθ ) = Gi,ni (¯xi,ni , uiθ ) are unknown smooth functions, here, we combine FLSs to approximate these two unknown functions on the compact as
If we choose the following matrices ⎡ 0 1 ⎢0 0 ⎢ Ai = ⎢ · · · ⎣0 0 0 0 ⎡ ⎤ 1 ⎢0⎥ ⎢.⎥ ⎥ Ci = ⎢ ⎢ .. ⎥ ⎣0⎦ 0
0 1 ··· 0 0
··· ··· ··· ··· ···
⎤ 0 0 0 0⎥ ⎥ ⎥, 0 1⎦ 0 0
⎡ ⎤ 0 ⎢0⎥ ⎢.⎥ ⎥ Bi = ⎢ ⎢ .. ⎥ ⎣0⎦ 1
Fi (xi ) = WiF∗T ξiF (xi ) + εi(1) Gi (xi , uiθ )
Then, system (10) can be rewritten as follows
z˙ i = Ai zi + Bi [Fi (xi ) + Gi (xi , uiθ )ui + di (t, X )] yi = CiT zi = zi1
Choose the states vector observer as z˙ˆ i = (Ai − Bi KiT )ˆzi + Bi (yid(ni ) + KiT Yid ) − Li CiT z˜ i yˆ i = CiT zˆ i = zˆi1
(11)
(12)
Now, for each ith subsystem, define the tracking error as ei (t) = yi − yid (i = 1, 2, . . . , N ), with yid as the reference signal. Define the tracking error vector e¯ i (t), the estimation error vector eˆ¯ i (t), and the estimation error vector e˜¯ i (t) as follows (see equation at the bottom of the page) where Yid = [yid , y˙ id , . . . , yid(ni −1) ], along the time derivatives of e¯ i (t), eˆ¯ i (t) and e˜¯ i (t) in (11) and (12), we have (see (13)) with Ki = [ki1 , ki2 , . . . , ki,ni ]T and Li = [li1 , li2 , . . . , li,ni ]T denote observer and feedback gain vector, they are chosen such that the characteristic polynomials of (Ai − Bi KiT ) and (Ai − Li CiT ) are Hurwitz. 3.2
Controller design and stability analysis
In this section, adaptive fuzzy tracking controller and the corresponding adaptive adjusted laws will be constructed. According to universe approximation theorem [28], FLSs can approximate any non-linear smooth function Q(Z) over a compact set [3, 11] as Q(Z) = W ∗T ξ (Z) +
=
∗T WiG ξiG (xi , uif
)+
(14) εi(2)
+
εi(3)
(15)
where xi ∈ Rni , [xi , uif ]T ∈ Rni +1 are the inputs of FLSs. ∗ 1 2 m T WiF∗ = [WiF1 , WiF2 , . . . , WiFm ]T , WiG = [WiG , WiG , . . . , WiG ] denote the optimal approximation parameter vector. and εi(1) , εi(2) , εi(3) are fuzzy approximation errors, with εi(3) = ∗ WiG (ξiG (xi , uiθ ) − ξiG (xi , uif )). ξiF (xi ), ξiG (xi , uiθ ) are fuzzy basis function vector. Assumption 2: For each ith subsystem, assume that the ∗ and the optimal approximation parameters vectors WiF∗ , WiG (1) (2) (3) fuzzy approximation error εi = εi + εi + εi satisfy the following inequality ∗ θiG ; WiF∗ θiF , WiG
|εi | δi (i = 1, 2, . . . , N ) (16)
parameters θiF > 0, θiG > 0, δi > 0 are unknown constants. Define θˆiF , θˆiG , δˆi as the estimations of θiF , θiG , δi respectively. The estimation error are selected as θ˜iF = θˆiF − θiF , θ˜iG = θˆiG − θiG , δ˜i = δˆi − δi . Remark 4: In (15), uiθ is unavailable, so, we introduce a filtered signal defined as uif = (1/s + λ)ui ≈ uiθ = θui to approximate uiθ . Similarly, we use this filtered uif as approximation [17]. We know that a series of significant results [2–10, 12–18] based on adaptive fuzzy (or NN) control approaches for non-linear uncertainty systems have been obtained by updating the estimations of the optimal FLSs or NN parameter vectors. But, the higher approximation accuracy is always accompanied the larger quantity numbers of fuzzy rules or NN hidden nodes, which always
e¯ i (t) = zi (t) − Yid = [ei , e˙ i , . . . , ei(ni −1) ]T = [zi1 − yid , zi2 − y˙ id , . . . , zi,ni − yid(ni −1) ] eˆ¯ i (t) = zˆ i (t) − Yˆ id = [ˆei , e˙ˆ i , . . . , eˆ i(ni −1) ]T = [ˆzi1 − yˆ id , zˆi2 − y˙ˆ id , . . . , zˆi,ni − yˆ id(ni −1) ] e˜¯ i (t) = e¯ i (t) − eˆ¯ (t) = [˜ei , e˙˜ i , . . . , e˜ (ni −1) ]T = [ˆei − ei , e˙ˆ i − e˙ i , . . . , eˆ (ni −1) − e(ni −1) ]T i
i
i
e˙¯ i = Ai e¯ i + Bi [Fi (xi ) + Gi (xi , uiθ )ui − yid(ni ) + di (t, X ); e˙ˆ¯ i = (Ai − Bi KiT )eˆ¯ i + Pi CiT z˜ i eˆ i = CiT eˆ¯ i ; e˙˜¯ i = (Ai − Li CiT )e˜¯ i + Bi [KiT eˆ¯ i − yid(ni ) + Fi (xi ) + Gi (xi , uiθ )ui + di (t, X )] e˜ i = CiT e˜¯ i
1296 © The Institution of Engineering and Technology 2014
(13)
IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
www.ietdl.org result in the heavy on-line computation burden and will not be conducive to the practical application. According to Assumption 2, · is defined as the 2-norm, we need only to approximate the norm of the optimal parameters 1 2 m rather than each WiF1 , WiF2 , . . . , WiFm , WiG , WiG , . . . , WiG and (1) (2) (3) each εi , εi , εi (i = 1, 2, . . . , N ). That is to say, no matter how many states in the design procedure are investigated and how many rules in the FLSs are utilised, for each ith(i = 1, 2, . . . , N ) subsystem, the number of adjusted parameters is only 3, with only 3N for the whole systems. Hence, distinguish with the resent researches [2–10, 12–18, 25–31], the number of on-line adaptive adjust parameters in this paper is decreased. Now, substituting the fuzzy approximation functions (14) and (15) and the dead-zone model (2) back into the estimation error vector e˜¯ i (t) in (13), we have e˙¯˜ i = (Ai − Li CiT )e˜¯ i + Bi [KiT eˆ¯ i − yid(ni ) + WiF∗T ξiF (xi ) + εi + di (t, X )] ∗T + Bi WiG ξiG (xi , uif )(mmin (1 + ηi (t))ωi (t) + hi (t)) (17)
According to e˜ i = ei − eˆ i = zi1 − zˆi1 = yi − xˆ i1 , corporate the strict-positive-real (SPR) Lyapunov design approach to construct the adaptive output feedback update laws for θiF , θiG , and the error (17) can be described as e˜ i = Hi (s)Li (s)[Li−1 (s)(KiT eˆ¯ i − yid(ni ) ) + WiF∗T ξ¯iF (xi ) ∗T ¯ + εif + dif + WiG (18) ξiG (xi , uif )ωif (t)] where Hi (s) = CiT (sI − (Ai − Li CiT ))−1 Bi is the stable transfer function of (20) and s is the Laplace variable, with εif = Li−1 (s)εi , dif = Li−1 di , ξ¯iF = Li−1 (s)ξiF , ξ¯iG = Li−1 (s)ξiG , ωif (t) = Li−1 (s)mmin (1 + ηi (t))ωi (t) + hi (t). Li−1 (s) is a proper stable transfer function and Hi (s)Li−1 (s) is proper SPR transfer function. Assume that Li (s) = sni + bi1 sni −1 + . . . + bi,ni −1 s + bi,ni . Equation (13) can be redescribed as (see (19)) with Aic = Ai − Li CiT , Bi = [1, bi1 , . . . , bi,ni ]T . Choose the robust adaptive controller as follows (see (20)) where i = 1, 2, . . . , N , τi(1) > 0, τi(2) > 0 are design parameters, with m∗i = mmin (1 + ηi ) > 0, di∗ > 0, ρi∗ > 0 are given in Section 1. Consider the adaptive fuzzy laws as follows θ˙ˆiF = i(1) [−θˆiF + |˜ei |ξ¯iF (xi )] θˆ˙iG = i(2) [−θˆiG + |˜ei |/θˆiG Hi (Zi )] δˆ˙ = γ [−γˆ δˆ + |˜e |] i
i
i i
(21) (22) (23)
i
with Hi (Zi ) = (−KiT eˆ¯ i − yid(ni ) ) − Li (s)θˆiF2 e˜ i ξ¯iF (xi )2 / ¯ ˆ and (θiF |˜ei |ξiF (xi ) + τi(1) ) − δˆi2 e˜ i /(δˆi |˜ei | + τi(2) ) − di∗
i(1) > 0, i(2) > 0, γi > 0 are positive design constants to be designed later. Now, let us consider the whole system, the following theorem is used to show the stability of the whole closed-loop system and control performance. Theorem 1: Consider non-linear pure-feedback systems (1) and the transformed normal form systems (10). Assume that the racking reference signal vector Yd (t) = [y1d (t), y2d (t), . . . , yNd (t)]T (t 0) to be sufficiently smooth functions with continuously and bounded kth derivatives. Under Assumptions 1 and 2, assume the state observer vector zˆ = [ˆz1 , zˆ 2 , . . . , zˆ N ] are chosen as (12), the controller ω = [ω1 , ω2 , . . . , ωN ]T is chosen as (20), and the adaptive laws are chosen as (21)–(23). Then, according to Lypunov analysis methods, this proposed approach can guarantee that all the signals in the overall closed-loop system are ultimately uniformly bounded and the outputs y = [y1 , y2 , . . . , yN ]T can track the reference signals Yd to a small neighbourhood of zero. Proof: First, let consider the following Lyapunov function for each ith (i = 1, 2, . . . , N ) subsystem. Vi (t) =
1 ˜T ˜ 1 T 1 T 1 2 θ˜ θ˜ + θ˜ θ˜ + δ˜i (24) e¯ Pi e¯ i + (1) iF iF (2) iG iG 2 i 2γ 2 i 2 i i
Along the time derivative of Vi , we have 1 1 T 1 V˙ i (t) = e˜˙¯ Ti Pi e˜¯ i + e˜¯ i Pi e˙˜¯ i + (1) θ˜iF θ˙ˆiF 2 2
i 1 ˙ 1 T˙ + (2) θ˜iG θˆiG + δ˜i δˆi γi
i
Substituting (19), the non-back-stepping fuzzy tracking controller (20) and adaptive adjusted laws (21)–(23) into the derivative of Vi in (25), we have 1 V˙ i (t) = e˜¯ iT ((ATic Pi )T + Pi Aic )e˜¯ i + e˜¯ iT Pi Bic [Li−1 (s) 2 × (KiT eˆ¯ i − yid(ni ) ) + WiF∗T ξ¯iF + εif + dif 1 1 T˙ 1 ∗T ¯ + WiG θˆiG + δ˜i δ˙ˆi ξiG ωif ] + (1) θ˜iF θˆ˙iF + (2) θ˜iG γi
i
i 1 − e¯˜ iT Qi e¯˜ i + |e¯˜ iT Pi Bic |WiF∗T ξ¯iF (xi ) 2 θˆiF2 e˜ i2 ξ¯iF (xi )2 + |e¯˜ iT Pi Bic |εif − θˆiF |˜ei |ξ¯iF (xi ) + τi(1) −
δˆi2 e˜ i2 δˆi |˜ei | + τi(2)
+
1
1 T˙ 1 ˜ ˙ˆ + θ˜ θˆ + δ˜i δ˙ˆi (2) iG iG γ
i i (26)
θiF θiF
i(1)
Because of Hi (s)Li−1 (s) is the SPR proper transfer function, there must exist positive-definite matrices Pi , Qi such that
∗T ¯ e˙˜¯ i = Aic e˜¯ i + Bic [Li−1 (s)(KiT eˆ¯ i − yid(ni ) ) + WiF∗T ξ¯iF + εif + dif + WiG ξiG ωif (t)] T˜ e˜ i = Ci e¯ i
ωi (t) =
1 θˆiG ξiG (xi )m∗i
(−KiT eˆ¯ i
−
yid(ni ) )
IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
(25)
ρ∗ Li (s)θˆiF2 e˜ i ξ¯iF (xi )2 δˆi2 e˜ i ∗ − − − di − i∗ (1) (2) mi θˆiF |˜ei |ξ¯iF (xi ) + τi δˆi |˜ei | + τi
(19)
(20)
1297 © The Institution of Engineering and Technology 2014
www.ietdl.org (ATic Pi )T + Pi Aic = −Qi , e˜¯ iT Pi Bic = e˜¯ i Ci = e˜ i . Using Yong’ inequality, together with adaptive laws (21)–(23), we have
d(V (t)e−μt )/dt le−μt
θˆiF2 e˜ i2 ξ¯iF (xi )2 1 + (1) θ˜iF θ˙ˆiF e˜ i WiF∗T ξ¯iF (xi ) − ˆθiF |˜ei |ξ¯iF (xi ) + τi(1)
i 2 2 θˆiF e˜ i ξ¯iF (xi )2 |˜ei |θˆiF ξ¯iF (xi ) − θˆiF |˜ei |ξ¯iF (xi ) + τi(1) 1 − |˜ei |θ˜iF ξ¯iF (xi ) + (1) θ˜iF θ˙ˆiF
i τi(1) − θ˜iF θˆiF τi(1) − θ˜iF (θ˜iF + θiF ) τi(1) 1 1 − θ˜iF2 + θiF 2 2 2 δˆi2 e˜ i2 1 e˜ i |εif | − + δ˜i δ˙ˆi (2) γi δˆi |˜ei | + τi |˜ei |δˆi −
δˆi2 e˜ i2 δˆi |˜ei | +
τi(2)
− |˜ei |δ˜i +
0 < V (t) < l/μ + [V (0) − l/μ]e−μt
1 ˙ δ˜i δˆi γi (28) (29)
1 1 2 + δ˜i2 ) V˙ i − e˜¯ iT Qi e˜¯ i + τi(1) + τi(2) − (θ˜iF2 + θ˜iG 2 2 1 + (θiF 2 + θiG 2 + δi2 ) −μi Vi + li (30) 2 where μi = min{λmin (Qi ), i(1) , i(2) , γi } and l = 1/2(θiF 2 + θiG 2 + δi2 ) are positive constants to be design later. Now, let consider the overall Lyapunov function candidate for the whole systems
V (t) =
N
V˙ i (t)
i=1
where μ =
N i=1
˜z
N
(−μi Vi + li ) −μV + l
i=1
μi and l =
N
i=1 li .
1298 © The Institution of Engineering and Technology 2014
2/λmin (Q) l/μ + [V (0) − l/μ]e−μt
(36)
where λmin (Q) denotes the minimum eigenvalue of the matrix Q . We assume that there exists constant α satisfying eˆ¯ α, then, we conclude |e| |¯e| |eˆ¯ | + ˜z 2/λmin (Q) l/μ + [V (0) − l/μ]e−μt + h
(31)
Along its trajectory with time t, according to the above analysis from (18)–(30) in Section 3, combing with the controller ω = [ω1 , ω2 , . . . , ωN ]T in (20) and the adaptive update parameters θˆF = [θˆ1F , θˆ2F , . . . , θˆNF ]T , θˆG = [θˆ1G , θˆ2G , . . . , θˆNG ]T , δˆ = [δˆ1 , θˆ2 , . . . , θˆN ]T in (21)–(23). Based on the analysis results for each ith(i = 1, 2, . . . , N ) subsystem in (30), we conclude that
V˙ (t) =
Since matrixes Qi , Pi , Q, P are all positive definite matrixes, ∗ , εi(1) , εi(2) , εi(3) are all positive constants, including WiF∗ , WiG li , hi (i = 1, 2, . . . , N ). Then, according to the extend Lyapunov theorem [32], the Lemmas 2.1, 2.2 in [33]. For any bounded initial condition Vi (0)(i = 1, 2, . . . , N ) and V (0) over the compact , it can be shown that all the signals including the tracking error vector e˜¯ = [e˜¯ 1 , e˜¯ 2 , . . . , e˜¯ N ]T , the observer error z˜ = [˜z1 , z˜ 2 , . . . , z˜ N ]T , the adaptive laws θ˜F = [θ˜1F , θ˜2F , . . . , θ˜NF ]T , θ˜G = [θ˜1G , θ˜2G , . . . , θ˜NG ]T , and δˆ = [δˆ1 , θˆ2 , . . . , θˆN ]T in the closed-loop systems (10) are ultimately uniformly bounded; in addition, according to the definition of zˆ = [ˆz1 , zˆ 2 , . . . , zˆ N ]T , we known that z = [z1 , z2 , . . . , zN ]T is ultimately uniformly bounded, and combing with the desired reference signals Yd = [y1d , y2d , . . . , yNd ]T and their bounded kth(k = 1, 2, . . . , ni ) derivatives yid(k) we conclude that the signal X = [x1 , x2 , . . . , xN ]T and adaptive adjusted parameters θF = [θ1F , θ2F , . . . , θNF ]T and θG = [θ1G , θ2G , . . . , θNG ]T are all ultimately uniformly bounded. Hence, are the estimation eˆ¯ = zˆ − Yd and the controller ω(t) = [ω1 (t), ω2 (t), . . . , ωN (t)]T . From (34), we have
N
1 ˜T ˜ 1 T θ˜iF θ˜iF e¯ i Pi e¯ i + 2 2 i(1) i=1 i=1 1 2 1 T ˜ ˜ ˜ θiG θiG + δ + 2γi i 2 i(2) Vi (t) =
(34)
Note that 0 < e−μt < 1 and (l/μ)e−μt > 0, then, (34) satisfying 0 < V (t) < l/μ + V (0) (35)
Substituting inequalities (27)–(29) back into the derivative of Vi in (26), we have
N
(33)
Integrating (33) over [0, t], we have
(27)
1 1 τi(2) + δ˜i δˆi τi(2) − δ˜i2 + |δi |2 2 2 1 1 2 1 2 θ˜iG /θˆiG ωi + (2) θ˜iG θ˙ˆiG −θ˜iG θiG − θ˜iG + θiG 2 2
i
Multiplying both sides of equation (32) by e−μt , we rewrite (32) as
(32)
There must such that for any t > T , limt→T e−μt = exist T 0, |e| 2/λmin (Q) l/μ + h. At last, we get that the tracking error can convergence to a small area and all the signals in the closed-loop systems are ultimately uniformly bounded. This completes the proof.
4
Simulation example
To verify the effectiveness of the proposed non-backstepping control approach in Section 3, the following practical and numerical simulation examples will be illustrated in details. 1. Realistic example: The reference model is taken as the famous practical two inverted pendulums [4] that are connected by a spring with unknown dead-zone inputs and the pendulums can be described by the following differential equation (see (37) on the bottom of next page) IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
www.ietdl.org with the unknown dead-zone model is ⎧ ⎨kir (ωi (t) − bir ) ui (t) = Di (ωi (t)) = 0 ⎩ kil (ωi (t) − bil )
described as
1.5
ωi (t) bir bil < ωi (t) < bir ωi (t) bil ; i = 1, 2
IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
reference signal y1d
1
where x = [x11 , x12 , x21 , x22 ]T = [θ1 , θ˙1 , θ2 , θ˙2 ]T , with θ1 , θ2 denote angular position, θ˙1 , θ˙2 denote angular rate. Unknown functions are selected as fi1 (x) = 0, fi2 (x) = (Mi gr/Ji − Kr 2 /4Ji ) sin(xi1 xi2 ), di1 (t, x) = 0, di2 (t, x) = Kr/2Ji (l − b) + 1/J1 u1 + Kr 2 /4Ji sin(xi1 x21 x22 ), i = 1, 2. The desired reference tracking signals are chosen as y1d = sin(t), y2d = cos (t). The parameters of the unknown deadzone are chosen as: k1r = 1.3 − 0.03 sin(ω1 (t)), k1l = 1.8 − 0.92 cos(ω1 (t)), k2r = 3 − 0.09 cos(ω2 (t)), k2l = 1.2 − 0.007 sin(ω2 (t)), b1l = −0.15, b1r = 2.5, b2r = 0.5, b2l = −0.003. With system parameters are selected as: M1 = 4, M2 = 2 kg are the pendulum end masses, J1 = 1, J2 = 0.5 kg denote the moments of inertia, K = 10 μ/m is the spring constant of the connection spring, r = 0.2 m is the pendulum height, l = 0.5 m is the natural length of the spring. g = 9.8 m/s2 is the gravitational acceleration. The distance between the pendulum hinges is defined as b = 0.3 m . Based on the proposed scheme in this paper, the fuzzy tracking controllers and the adaptive laws are constructed as (20)–(23). Define z11 = x11 , z12 = z˙11 , z21 = x21 , z22 = z˙21 , e˜ i = ei − eˆ i , ei = yi − yid . The observers are defined as in Section 3, with initial values x11 (0) = x21 (0) = 1, x12 (0) = x22 (0) = 5; zˆ11 (0) = zˆ21 (0) = 0, zˆ21 (0) = zˆ22 (0) = 2, θˆ1F (0) = 1.6, θˆ2F (0) = 0.8, θˆ1G (0) = 30, θˆ2G (0) = 2, δˆ1 (0) = 10.9, δˆ2 (0) = 1.67. The design parameters are chosen as k11 = k12 = 0.1, k21 = k22 = 5, l11 = l12 = 10, l21 = l22 = 66. The robust adaptive adjusted parameters are chosen as τ1(1) = τ2(1) = 0.01, τ1(2) = τ2(2) = 0.13, d1∗ = 803, d2∗ = 981, m∗1 = 5667, m∗2 = 74, ρ1∗ = 802, ρ2∗ = 900, and the design adjusted parameters are chosen as 1(1) = 2, 2(1) = 0.05, 1(2) = 9, 2(2) = 0.25 and γ1 = γ2 = 2.0. The simulation results are shown in Figs. 1–6. It can be seen from Figs. 1 and 2, that the system output y = [y1 , y2 ]T can track the desired reference trajectory signal Yd = [y1d , y2d ]T very well and the tracking error can convergence to a small neighbourhood of zero. Based on Figs. 3 and 4, we conclude that the controller inputs and the unknown dead-zone inputs are all ultimately uniformly bounded. From Figs. 5 and 6, the ultimately uniformly bounded-ness of adaptive update parameters and be required. By comparing the simulation results in [4], we obtained the similar results that all the signals in the closed-loop systems are ultimately uniformly bounded, however, the design procedure in this paper is rather simple because of the novel non-back-stepping technique, in addition, based on Assumption 2, we only need to update the norms of the optimal fuzzy weight parameters rather which ensure that only six on-line update parameters need to be tuned. 2. Numerical example: Consider the following second-order non-linear system and it is just referred to the model in [29] ⎧ x˙ 11 = x12 ⎪ ⎪ ⎪ ⎪ M1 gr Kr 2 ⎪ ⎪ = − sin(x11 x12 ) + x ˙ ⎪ 12 ⎪ ⎨ J1 4J1 x˙ 21 = x22 ⎪ ⎪ M2 gr Kr 2 ⎪ ⎪ ⎪ x ˙ = − sin(x21 x22 ) + 22 ⎪ ⎪ J2 4J2 ⎪ ⎩ y1 = x11 , y2 = x21
output y1
0.5
0
−0.5
−1
−1.5
0
Fig. 1
5
10
20
15
30
25
System output y1 , the tracking reference signal y1d
1.5 output y
reference signal y
2
2d
1
0.5
0
−0.5
−1
−1.5
0
Fig. 2
5
10
15
20
25
30
System output y2 , the tracking reference signal y2d
and we discuss the situation with the unknown dead-zone inputs. ⎧ 2 ⎪ x˙ 11 = x12 + (x12 + x11 e−0.5x11 )x12 + 11 (t, x) ⎪ ⎪ ⎪ 2 ⎪ ⎨x˙ 12 = u1 + (2 + x11 x12 + cos(x11 x12 ))u1 + 12 (t, x) x˙ 12 = x22 + (1 + x11 + sin2 (x21 x12 ))x22 + 21 (t, x) ⎪ ⎪ ⎪ x˙ 22 = u2 + (1 + cos(x21 x12 ))u2 + (x21 x22 + x11 x12 ) 22 (t, x) ⎪ ⎪ ⎩ y1 = x11 , y2 = x21 (38) with the unknown dead-zone model described as ⎧ ⎨kir (ωi (t) − bir ) ωi (t) bir ui (t) = Di (ωi (t)) = 0 bil < ωi (t) < bir ⎩ kil (ωi (t) − bil ) ωi (t) bil ; i = 1, 2
Kr 1 Kr 2 (l − b) + u1 + sin(x11 x21 x22 ) 2J1 J1 4J1 (37) Kr 1 Kr 2 (l − b) + u2 + sin(x12 x21 x22 ) 2J2 J1 4J2 1299 © The Institution of Engineering and Technology 2014
www.ietdl.org 5
dead−zone input ω1
4
input u
0.2
the estimation of θ2F the estimation of θ
0.15
1
2G
the estimation of δ2
3 0.1
2 0.05
1 0
0
−1
−0.05
−2
−0.1
−3 −0.15
−4 −0.2
−5
0
Fig. 3
5
10
15
20
25
0
5
10
15
20
25
30
30
Fig. 6
System input u1 and the dead-zone input ω1
Adaptive update parameters θˆ2F , θˆ2G , δˆ2
1
10
output y1
dead−zone input ω
2
8
input u2
reference signal y1d
0.5
6 0
4 2
−0.5
0
2
4
6
8
10
Time(sec)
0
a 0.5
−2 −4
0
−6 −0.5
−8
output y
2
−10
−1
0
Fig. 4
5
10
15
30
25
20
0
the estimation of θ
1F
2d
8
10
Outputs tracking the reference signals
a System output y1 (dot line), reference signal y1d (solid line) b System output y2 (dot line), reference signal y2 d (solid line)
the estimation of θ1G the estimation of δ
4
1
3.5 3 2.5 2 1.5 1 0.5 0
6
b
Fig. 7
4.5
4 Time(sec)
System input u2 and the dead-zone input ω2
5
2
reference signal y
0
Fig. 5
5
10
15
20
25
30
Adaptive update parameters θˆ1F , θˆ1G , δˆ1
We note that the aforementioned example can be viewed as special cases of system (38), in fact, the practical mechanical model are mainly in the form as the above realistic model, such as the two inverted pendulums on spring mounted cars 1300 © The Institution of Engineering and Technology 2014
[9] and the mechanical system of mass–spring–damper [9] 2 and so on. with 11 (t, x) = 0.5(x11 x21 x22 ) sin(t), 12 (t, x) = 2 2 0.2(x21 x12 ) cos(t), 21 (t, x) = 0.6 sin(x21 x11 x12 ) sin(t), 22 2 2 (t, x) = 0.5 sin(x21 x22 ) sin2 (t). Dead-zone parameters are chosen as b1l = 1.125, k1l = 1 − 0.05 sin(ω1 (t)), k1r = 2.5 cos(ω1 ), b1r = 0.05, k2r = 1.5 sin(ω2 (t)), k2l = 3.5 − 0.01 sin(ω2 (t)), b2l = −0.125 and b2r = 0.125. The desired reference tracking signals are chosen as y1d = 1/3 sin(3t) and y2d = 1/2 sin(2t). The actual controllers and adaptive laws are chosen as (20)–(23). With the initial values zˆ11 (0) = −0.25, zˆ12 (0) = 0; zˆ21 (0) = −0.25, zˆ22 (0) = 0.1, θˆ1F (0) = 0.1, θˆ2F (0) = 10, θˆ1G (0) = 1.0, θˆ2G (0) = 5 and δˆ1 (0) = δˆ2 (0) = 1. The design parameters are chosen as k11 = k12 = 1, k21 = k22 = 0.5, l11 = l12 = 0.1 and l21 = l22 = 89. The adaptive parameters are chosen as τ1(1) = τ2(1) = 0.1, τ1(2) = τ2(2) = 1, d1∗ = 82.3, d2∗ = 99; m∗1 = 31.9, m∗2 = 97.1; h∗1 = 80, h∗2 = 8 and the design adjusted parameters are chosen as 1(1) = 2(1) = 3, 1(2) =
2(2) = 9; γ1 = γ2 = 0.9. The simulation results are shown in Figs. 7–9, similar to the results in [29, 30]. From these results, we conclude IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
www.ietdl.org 5
10 5 0 −5 controll input u1 −10
0
2
6
4
dead−zone input ω1 10
8
Time(sec)
a 10 5 0 −5 controll input u2 −10
0
2
4
6
dead−zone input ω2 8
10
Time(sec)
b
Fig. 8
Control inputs and dead-zone inputs
a System input u1 (solid line), dead-zone input ω1 (dot line) b System input u2 (solid line), dead-zone input ω2 (dot line)
40
the estimation of θ1F
the estimation of θ2F
20 0
0
2
4
6
8
10
Time(sec) a
40
the estimation of θ
1G
the estimation of θ
2G
20 0
0
2
4
6
8
10
Time(sec) b
0.4
the estimation of δ
1
the estimation of δ
2
Conclusions
By means of FLSs and non-back-stepping technique, we present a novel adaptive alleviating computation observer design and tracking control algorithm for a class of purefeedback MIMO non-linear systems with unknown deadzone inputs. This proposed control algorithm introduces novel state coordinate transformation, converts the statefeedback control to an output-feedback control. Moreover, assumes that only one unknown parameter is used as the bound of the norm of the optimal approximation vectors of the FLSs and we need to adjust this parameter. As a result, the developed controller has a simpler form with only three adaptive parameters to be updated on-line. According to Lyapunov stable analysis methods, it can be guaranteed that all the signals in the closed-loop systems are ultimately uniformly bounded and the outputs can track the reference control signals very well. In addition, the proposed approach can mitigate the effect of dead-zone, alleviate the on-line computation burden and develop the robust performance of the whole system. Hence, this proposed algorithm is rather simple. From the viewpoint of practical application, the fuzzy observer, the controller and control schemes are implemented online without recursive calculations and numerical adjusted adaptive parameters. Hence, the computational costs are alleviated compared with the backstepping-based methods coping with unknown dead-zones non-linearity problems. The effectiveness of this proposed approach is demonstrated via the simulation results.
6
Acknowledgment
This work is partially supported by National Natural Science Foundation of China (nos. 60775032 and 10971243) and partially supported by Beijing Natural Science Foundation (no. 4112031). It is also sponsored by the priority discipline of Beijing Normal University.
7
References
0.2
1 0
0
2
4
6
8
10
Time(sec) c
2
Trajectories of adaptive update laws
3
a Adaptive law θˆ1F (solid line), adaptive law θˆ2F (dot line) b Adaptive law θˆ1G (solid line), adaptive law θˆ2G (dot line) c Adaptive law δˆ1 (solid line), adaptive law δˆ2 (dot line)
4
Fig. 9
5
that all the signals in the closed-loop system are ultimately uniformly bounded. Including the outputs y = [y1 , y2 ]T can track the reference signals very well. Different from the previous results in [2–18, 29, 30, 34], in this paper, we did not consider the traditional back-stepping design technique, we introduce the non-back-stepping approach combing with the novel alleviating computation technique, the adaptive adjusted parameters here (only six parameters need to be adjusted online). The design procedure is rather simpler, the number of adaptive parameters is rather fewer, and more convenient for online computation than the traditional back-stepping-based methodologies. IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
6 7 8 9 10
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IET Control Theory Appl., 2014, Vol. 8, Iss. 14, pp. 1293–1302 doi: 10.1049/iet-cta.2014.0059
