Robust adaptive neural network control of a class of uncertain strict-feedback nonlinear systems with unknown dead-zone and disturbances

Author's Accepted Manuscript

Robust adaptive neural network control of a class of uncertain strict-feedback nonlinear systems with unknown dead-zone and disturbances Gang Sun, Dan Wang, Mingxin Wang

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S0925-2312(14)00686-9 http://dx.doi.org/10.1016/j.neucom.2014.05.039 NEUCOM14243

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Received date: 28 October 2013 Revised date: 16 March 2014 Accepted date: 17 May 2014 Cite this article as: Gang Sun, Dan Wang, Mingxin Wang, Robust adaptive neural network control of a class of uncertain strict-feedback nonlinear systems with unknown dead-zone and disturbances, Neurocomputing, http://dx. doi.org/10.1016/j.neucom.2014.05.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust adaptive neural network control of a class of uncertain strict-feedback nonlinear systems with unknown dead-zone and disturbances✩ Gang Suna,b , Dan Wangb,∗, Mingxin Wanga a

Department of Mathematics and Physics, Hunan Institute of Technology, Hengyang 421002, China b School of Marine Engineering, Dalian Maritime University, Dalian 116026, China

Abstract In this paper, a robust adaptive neural control design approach is presented for a class of perturbed strict-feedback nonlinear systems with unknown deadzone. In the controller design, different from existing methods, all the virtual control laws need not be actually implemented at intermediate steps, and only one actual robust adaptive control law is constructed by approximating the lumped unknown function of the system with a single neural network at the last step. By this approach, the structure of the designed controller is much simpler since the causes for the problem of complexity growing in existing methods are eliminated. Stability analysis shows that the proposed scheme can guarantee the uniform ultimate boundedness of all the closed-loop system signals, and the steady-state tracking error can be made arbitrarily small by appropriately choosing control parameters. Simulation studies demonstrate the effectiveness and merits of the proposed approach. Keywords: robust adaptive control, single neural network, uncertain strict-feedback nonlinear systems, dead-zone, disturbances

✩

This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 61074017, 61273137, 61374009, and 51209026), Scientific Research Fund of Hunan Institute of Technology. ∗ Corresponding author. Email addresses: [email protected] (Gang Sun), [email protected] (Dan Wang), [email protected] (Mingxin Wang)

Preprint submitted to Neurocomputing

May 27, 2014

1. Introduction Adaptive backstepping has been a powerful method for synthesizing controllers for lower-triangular nonlinear systems with uncertainties. In [1–4], several adaptive backstepping control design methods were developed for strict-feedback and pure-feedback nonlinear systems with linearly parameterized uncertainties. For nonlinear systems with uncertainties that cannot be linearly parameterized or are completely unknown, online approximation based adaptive control techniques have been found to be particularly useful, such as neural networks (NN) control [5–20], fuzzy control [21–32], and so on. In [5–10, 28, 30–32], online approximation based adaptive backstepping control methods were presented for uncertain strict-feedback nonlinear systems. While in [11–18, 22], adaptive backstepping control schemes were developed for uncertain pure-feedback nonlinear systems by using online approximation techniques. A drawback exists in aforementioned adaptive control design methods. That is, the complexity of the designed controller grows drastically as the system order increases due to two reasons. One is the repeated differentiations of certain nonlinear functions in the controller design process, and the other is the use of multiple online approximators. It is difficult to implement a controller if the controller is complex. To deal with the complexity growing problem, a dynamic surface control (DSC) technique was proposed by introducing a first-order filter of the synthetic input at each step in the traditional backstepping approach [33]. In [34], by incorporating the DSC technique into the online approximation based adaptive control design framework, an adaptive backstepping DSC approach was presented for a class of uncertain strict-feedback nonlinear systems. In recent years, a lot of online approximation based adaptive DSC design methods were developed for uncertain nonlinear systems in strict-feedback and pure-feedback forms, such as [35–44], and some references therein. By using the DSC technique, the repeated differentiations of certain nonlinear functions in traditional design process were avoided. That is, one of the reasons behind the complexity growing problem is eliminated. However, the complexity growing problem is not solved completely because many online approximators are employed in these designs. The use of too many approximators makes the complexity of the both controller structure and computation grow significantly. To eliminate the two reasons behind the complexity growing problem simultaneously, single neural network (SNN) approximation 2

based adaptive control design methods were developed in [45, 46]. In these methods, the designed controllers can be given directly, and the structures of the controllers are simpler. Dead-zone nonlinearity, which is a common nonsmooth nonlinear characteristic, exists widely in many components of the actuators such as valves, DC servo motors, and other devices. The presence of the characteristic may cause deterioration of the control system performance. For a long time, the study about the dead-zone nonlinearity has been drawing much interest of the researchers [28, 30, 32, 35, 47–52]. For some classes of systems with unknown dead-zones, adaptive dead-zone inverse methods were developed in [47–50], etc. For a class of nonlinear systems in which the dead-zone slopes are the same in the positive and negative regions, a robust adaptive control approach was proposed without constructing the inverse of the dead-zone in [51]. For nonlinear systems with nonsymmetric dead-zone, online approximation based adaptive backstepping control methods are investigated in [28, 30, 32, 35, 52], and the references therein. In some of the recent work, such as [28, 30], etc., state observer based adaptive output feedback control approaches are developed for nonlinear systems with unknown dead-zones and immeasurable states. In this paper, the adaptive tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with nonsymmetric dead-zone and unknown bounded disturbances. In order to ensure that the designed controller possess good robustness, compensation techniques are used for the nonsymmetric dead-zone and disturbances of the systems. And with the purpose of substantially eliminate the complexity growing problem, the technique of SNN approximation is used to get rid of the two reasons behind the problem. By combining these techniques, an effective control design approach is developed for the class of the systems under study. By this approach, the problem of complexity growing can be dealt, and the robust performance of the control system can be guaranteed because the impact of the disturbances and the dead-zone is compensated effectively. The controller design contains n steps for an nth-order system. At the intermediate step i (i = 1, ..., n − 1), the disturbance is firstly compensated, and then a virtual robust control law is given to stabilize the ith subsystem. It is important to note that the virtual control law needs not be implemented actually. At the last step (step n), after compensating the disturbance and dead-zone, a desired robust control law is designed to stabilize the nth subsystem. Then, by replacing the lumped unknown part of the desired control law with an 3

SNN approximator, an actual robust adaptive control law is constructed. Stability analysis shows that the uniform ultimate boundedness (UUB) of all the closed-loop system signals can be guaranteed, and the steady state tracking error can be made arbitrarily small by appropriately choosing control parameters. Simulation results demonstrate the effectiveness of the approach. This paper is organized as follows. Section 2 presents the problem formulation and preliminaries. Some assumptions and notation are also given in this section. Section 3 proposes the SNN based robust adaptive control design procedure. Section 4 gives the stability analysis. Some simulation examples are used to demonstrate the effectiveness and merits of the approach in Section 5. Section 6 concludes this paper. 2. Problem formulation and preliminaries 2.1. Systems description Consider a class of uncertain nonlinear dynamical systems as follows, ⎧ xi )xi+1 + fi (¯ xi ) + Δi (¯ xn , t), 1 ≤ i ≤ n − 1, ⎪ ⎪ x˙ i =gi (¯ ⎪ ⎨ x˙ =g (¯ xn ) + Δn (¯ xn , t), n n xn )u + fn (¯ (1) ⎪ u =D(v), ⎪ ⎪ ⎩ y =x1 , where x¯i = [x1 , ..., xi ]T ∈ Ri , i = 1, ..., n, are system state variables; u ∈ R and y ∈ R are system input and output, respectively; v is the output from the controller; the dead-zone characteristic of the actuator is described as D(v); fi (¯ xi ) and gi (¯ xi ), i = 1, ..., n, are unknown smooth nonlinear functions; Δi (¯ xn , t), i = 1, ..., n, are system uncertainties which could be come from measurement noise, modeling errors, external disturbances, modeling simplifications or changes due to time variations, etc [10, 31]. The control objective is, for a given reference input signal yr (t), where (n) yr (t), y˙ r (t), ..., yr (t) are bounded for t ≥ 0, to design a robust adaptive controller for the system (1) such that all the signals in the close-loop system remain uniformly ultimately bounded, and the system output y follows the reference input signal yr (t). xi ), i = 1, ..., n, are known, and there exist Assumption 1. The signs of gi (¯ positive constants g i and g¯i such that (i) |gi (¯ xi )| ≥ g i , ∀¯ xi ∈ Ri , and (ii) |gi (¯ xi )| ≤ g¯i , ∀¯ xi ∈ Ωx¯i ⊂ Ri , where Ωx¯i is a compact region. 4

Assumption 1 implies that gi (¯ xi ), i = 1, ..., n, are strictly either positive xi ) > 0. or negative. Without loss of generality, it is assumed that gi (¯ Assumption 2. There exist smooth positive functions ϕi (¯ xi ), i = 1, ..., n, xn , t)| ≤ ϕi (¯ xi ), ∀(¯ xn , t) ∈ Rn × R+ . such that |Δi (¯ Remark 1. Assumption 2 implies that the allowed class of uncertainties Δi (¯ xn , t), i = 1, ..., n, satisfy a triangularity condition in terms of x¯n . This will be exploited for the ease of the controller design. Similar assumptions to assumption 2 are required in [9, 10, 35], etc. It is noted that the exact expressions of ϕi (¯ xi ), i = 1, ..., n, are not needed in the controller design process in this paper. 2.2. Dead-zone characteristic The dead-zone characteristic of the actuator can be represented as follows [35], ⎧ ⎪ ⎨ Dl (v), v ≤ dl , u = D(v) = 0, dl < v < dr , (2) ⎪ ⎩ Dr (v), v ≥ dr , where dl ≤ 0 and dr ≥ 0 are unknown bounded constants, Dl (v) and Dr (v) are unknown smooth functions. Assumption 3. There exist positive constants Dl0 , Dl1 , Dr0 and Dr1 such that  0 < Dl0 ≤ Dl (v) ≤ Dl1 , ∀v ∈ (−∞, dl ], (3) 0 < Dr0 ≤ Dr (v) ≤ Dr1 , ∀v ∈ [dr , +∞), where Dl (v) = dDl (v)/dv and Dr (v) = dDr (v)/dv. Let Dmin = min{Dl0 , Dr0 } and Dmax = max{Dl1 , Dr1 }. Using mean value theorem, the dead-zone characteristic (2) can be rewritten as follows, u = D(v) = D(v)v + d(v), where

⎧  ⎪ ⎨ Dl (ξl ), v ≤ dl ,  Dlr dl < v < dr , , D(v) = ⎪ ⎩  Dr (ξr ), v ≥ dr , ⎧  v ≤ dl , ⎪ ⎨ − Dl (ξl )dl ,  − Dlr v, dl < v < dr , d(v) = ⎪ ⎩  − Dr (ξr )dr , v ≥ dr , 5

(4)

(5)

(6)

 here ξl ∈ (v, dl ), if v ≤ dl ; ξr ∈ (dr , v), if v ≥ dr ; constant Dlr satisfies  Dmin ≤ D lr ≤ Dmax . From (3), (5) and (6), it can be obtained that Dmin ≤ D(v) ≤ Dmax and |d(v)| ≤ d∗ := max {−Dmax dl , Dmax dr }.

2.3. Notation (i) · denotes the Euclidean norm of a vector; λmax (·) denotes the largest eigenvalue of a square matrix. (i) (i) (ii) y¯r = [yr , y˙ r , ..., yr ]T , i = 1, ..., n. (iii) Ci,j = ci ci−1 ...ci−j+1 , j ≤ i, where cl , l = 1, ..., i, are constants. For example: C3,1 = c3 , C3,2 = c3 c2 , C3,3 = c3 c2 c1 . The following notation is frequently used in the process of subsequent controller design, Ci,j = ci Ci−1,j−1. (7) 2.4. Radial basis function neural networks In this paper, a radial basis function (RBF) NN is employed to approximate the lumped unknown function of the system at the last step. Before introducing the control design method, the approximation property of the RBF network should be firstly recall. The RBF network takes the form θT ξ(x), where θ ∈ RN (N is the number of network nodes) is called weight vector and ξ(x) ∈ RN is a vector valued function defined in Rn . Denote the components of ξ(x) by ρi (x), i = 1, ..., N, which are called basis functions. In this work, ρi (x) is chosen as the commonly used Gaussian function, which has the following form, 2 /σ 2

ρi (x) = μe−x−ςi 

, i = 1, ..., N,

(8)

where ςi ∈ Rn is a constant vector called as the center of the basis function, σ > 0 is a real number called as the width of the basis function, and μ > 0 is the amplification factor of the basis function. Let Ω ⊂ Rn be a given compact set, then there is an RBF neural network θT ξ(x) which can approximate an unknown continuous real-valued function f (x) on the compact set Ω [53, 54]. And according to the approximation property of the RBF network, we have that for any ε∗ > 0, by appropriately choosing μ, σ, ςi , i = 1, ..., N, for some sufficiently large integer N, there exists an ideal weight vector θ ∈ RN such that the approximation error bounded by ε∗ > 0, i.e., f (x) = θT ξ(x) + ε, x ∈ Ω, 6

(9)

with |ε| ≤ ε∗ , where ε represents the network reconstruction error. Since θ is unknown, we need to estimate θ on-line. The notation θˆ is used to denote the estimation of θ and an adaptive law will be developed to update it. For RBF network, the following lemma provides an upper bound on the Euclidean norm of vector ξ(x), which is essential in proving our main result. Lemma 1 [13, 55]. Consider the RBF network. Let p := (1/2)mini=j ςi −ςj , and let q be the dimension of input x, μ and σ are the amplification factor and width of basis function, respectively. Then one has ξ(x) ≤

∞ 

3μq(i + 2)q−1 e−2p

2 i2 /σ 2

:= χ.

(10)

i=0

3. SNN based robust adaptive control design In this section, an SNN based robust adaptive controller will be established for the class of uncertain strict-feedback nonlinear systems (1). At step i (i = 1, ..., n − 1), a virtual robust feedback control law is given in which an unknown function is contained. At step n, a desired robust control law is firstly given, and then an actual robust adaptive control law is implemented by replacing the lumped unknown function of the system with an SNN approximator. Step 1: Let z1 = x1 − yr . The derivative of z1 is z˙1 =g1 (x1 )x2 + f1 (x1 ) + Δ1 (¯ xn , t) − y˙ r =g1 (x1 )x2 + F1 (x1 , y˙ r ) + Δ1 (¯ xn , t),

(11)

where F1 (x1 , y˙ r ) = f1 (x1 ) − y˙ r is an unknown smooth function. Choose a Lyapunov function candidate as V1 = (1/2)z12 . Using assumption 2 and Young’s inequality, we can obtain that V˙ 1 =z1 (g1 (x1 )x2 + F1 (x1 , y˙ r ) + Δ1 (¯ xn , t)) ≤z1 (g1 (x1 )x2 + F1 (x1 , y˙ r )) + z12 ϕ21 (x1 ) +  1 =z1 g1 (x1 ) x2 + F1 (x1 , y¯r(1) ) + , 4 (1)

1 4 (12)

where F1 (x1 , y¯r ) = (1/g1(x1 )) (F1 (x1 , y˙ r ) + z1 ϕ21 (x1 )) is an unknown smooth function. 7

The virtual robust control law α2 is chosen as follows, α2 = −c1 z1 − F1 (x1 , y¯r(1) ),

(13)

where c1 is a positive real constant which will be specified later. Letting z2 = x2 − α2 , we have 1 V˙ 1 ≤ −c1 g1 (x1 )z12 + g1 (x1 )z1 z2 + . 4

(14)

Substituting α2 into z2 , we can obtain that z2 =x2 + c1 z1 + F1 (x1 , y¯r(1) ) =x2 − y˙ r + C1,1 (x1 − yr ) + F1∗ (x1 , y¯r(1) ), (1)

(15)

(1)

where F1∗ (x1 , y¯r ) = F1 (x1 , y¯r ) + y˙ r is an unknown smooth function. Step 2: The derivative of z2 is z˙2 =g2 (¯ x2 )x3 + f2 (¯ x2 ) + Δ2 (¯ xn , t) − y¨r + C1,1 (g1 (x1 )x2 + f1 (x1 ) + Δ1 (¯ xn , t) − y˙ r ) ∗ ∂F1 ∂F1∗ ∂F1∗ + (g1 (x1 )x2 + f1 (x1 ) + Δ1 (¯ xn , t)) + y˙ r + y¨r ∂x1 ∂yr ∂ y˙ r =g2 (¯ x2 )x3 + F2 (¯ x2 , y¯r(2) ) + Δ2 (¯ xn , t) + C1,1 Δ1 (¯ xn , t) ∂F1∗ + Δ1 (¯ xn , t), ∂x1 (2)

(16)

x2 , y¯r ) = f2 (¯ x2 ) − y¨r + C1,1 (g1 (x1 )x2 + f1 (x1 ) − y˙ r ) + (∂F1∗ /∂x1 )· where F2 (¯ yr is an unknown smooth func(g1 (x1 )x2 + f1 (x1 ))+(∂F1∗ /∂yr )y˙ r +(∂F1∗ /∂ y˙ r )¨ tion. Choose a Lyapunov function candidate as V2 = (1/2)z22 . Using assumption 2 and Young’s inequality, we can obtain that

 ∗ ∂F 1 (2) V˙ 2 =z2 g2 (¯ x2 )x3 + F2 (¯ x2 , y¯r ) + Δ2 (¯ xn , t) + C1,1 Δ1 (¯ xn , t) + Δ1 (¯ xn , t) ∂x1  1 1 2 x2 )x3 + F2 (¯ x2 , y¯r(2) ) + z22 ϕ22 (¯ x2 ) + + z22 C1,1 ϕ21 (x1 ) + ≤z2 g2 (¯ 4 4

∗ 2 ∂F1 1 ϕ21 (x1 ) + + z22 ∂x1 4  3 (17) x2 ) x3 + F2 (¯ x2 , y¯r(2) ) + , =z2 g2 (¯ 4 8

 (2) (2) 2 where F2 (¯ x2 , y¯r ) = (1/g2 (¯ x2 )) F2 (¯ x2 , y¯r ) + z2 ϕ22 (¯ x2 ) + z2 C1,1 ϕ21 (x1 ) + z2 · (∂F1∗ /∂x1 )2 ϕ21 (x1 ) is an unknown smooth function. The virtual robust control law α3 is chosen as follows, α3 = −c2 z2 − F2 (¯ x2 , y¯r(2) ),

(18)

where c2 is a positive real constant which will be specified later. Letting z3 = x3 − α3 , we have 3 V˙ 2 ≤ −c2 g2 (¯ x2 )z22 + g2 (¯ x2 )z2 z3 + . (19) 4 Substituting α3 into z3 and using the notation (7), we can obtain that x2 , y¯r(2) ) z3 =x3 + c2 z2 + F2 (¯ =x3 − y¨r +

2 

 C2,j x3−j − yr(2−j) + F2∗ (¯ x2 , y¯r(2) ),

(20)

j=1 (2)

(1)

(2)

x2 , y¯r ) = c2 F1∗ (x1 , y¯r ) + F2 (¯ x2 , y¯r ) + y¨r is an unknown smooth where F2∗ (¯ function. Step i (3 ≤ i ≤ n − 1): Consider zi =xi −

yr(i−1)

+

i−1 

 ∗ Ci−1,j xi−j − yr(i−1−j) + Fi−1 (¯ xi−1 , y¯r(i−1) ),

(21)

j=1 (i−1)

∗ where Fi−1 (¯ xi−1 , y¯r is

) is an unknown smooth function. The derivative of zi

xi )xi+1 + fi (¯ xi ) + Δi (¯ xn , t) − yr(i) z˙i =gi (¯ +

i−1 

 Ci−1,j gi−j (¯ xi−j )xi+1−j + fi−j (¯ xi−j ) + Δi−j (¯ xn , t) − yr(i−j)

j=1

+

i−1 ∗  ∂Fi−1 j=1

∂xj

(gj (¯ xj )xj+1 + fj (¯ xj ) + Δj (¯ xn , t)) +

=gi (¯ xi )xi+1 + Fi (¯ xi , y¯r(i) ) + Δi (¯ xn , t) +

i−1 

i−1 ∗  ∂Fi−1 (j) j=0 ∂yr

y˙ r(j)

Ci−1,j Δi−j (¯ xn , t)

j=1

+

i−1  j=1

∗ ∂Fi−1 Δj (¯ xn , t), ∂xj

(22)

9

 (i) (i) where Fi (¯ xi , y¯r ) = fi (¯ xi )−yr + i−1 xi−j )xi+1−j + fi−j (¯ xi−j ) − j=1 Ci−1,j gi−j (¯ i−1 (i−j) (j) (j) ∗ ∗ + j=1 (∂Fi−1 /∂xj )(gj (¯ xj )xj+1 + fj (¯ xj )) + i−1 yr j=0 (∂Fi−1 /∂yr )y˙ r is an unknown smooth function. Choose a Lyapunov function candidate as Vi = (1/2)zi2 . Using assumption 2 and Young’s inequality, we can obtain that

i−1  (i) ˙ Vi =zi gi (¯ xi )xi+1 + Fi (¯ xi , y¯ ) + Δi (¯ xn , t) + Ci−1,j Δi−j (¯ xn , t) r

+

i−1 ∗  ∂Fi−1 j=1

∂xj

j=1

 Δj (¯ xn , t)

 1 xi )xi+1 + Fi (¯ xi , y¯r(i) ) + zi2 ϕ2i (¯ xi ) + ≤zi gi (¯ 4     i−1

i−1

∗  1 1 2 2 2 2 ∂Fi−1 2 2 + xi−j ) + ϕj (¯ xj ) + zi Ci−1,j ϕi−j (¯ zi + 4 ∂xj 4 j=1 j=1  2i − 1 xi ) xi+1 + Fi (¯ xi , y¯r(i) ) + , =zi gi (¯ 4

(23)

(i) (i) 2 2 xi , y¯r ) = (1/gi (¯ xi ))(Fi (¯ xi , y¯r )+zi ϕ2i (¯ xi )+zi i−1 xi−j )+ where Fi (¯ j=1 Ci−1,j ϕi−j (¯ i−1 2 ∗ zi j=1 (∂Fi−1 /∂xj ) ϕ2j (¯ xj )) is an unknown smooth function. The virtual robust control law αi+1 is chosen as follows, xi , y¯r(i) ), αi+1 = −ci zi − Fi (¯

(24)

where ci is a positive real constant which will be specified later. Letting zi+1 = xi+1 − αi+1 , we have 2i − 1 . xi )zi2 + gi (¯ xi )zi zi+1 + V˙ i ≤ −ci gi (¯ 4

10

(25)

Substituting αi+1 into zi+1 and using the notation (7), we can obtain that xi , y¯r(i) ) zi+1 =xi+1 + ci zi + Fi (¯  i−1   =xi+1 + ci xi − yr(i−1) + Ci−1,j xi−j − yr(i−1−j) j=1



∗ (¯ xi−1 , y¯r(i−1) ) + Fi (¯ xi , y¯r(i) ) +Fi−1

=xi+1 −

yr(i)

+

i 

 Ci,j xi+1−j − yr(i−j) + Fi∗ (¯ xi , y¯r(i) ),

(26)

j=1 (i)

(i−1)

∗ where Fi∗ (¯ xi , y¯r ) = ci Fi−1 (¯ xi−1 , y¯r smooth function. Step n: The derivative of zn is

(i)

(i)

) + Fi (¯ xi , y¯r ) + yr is an unknown

z˙n =gn (¯ xn )u + fn (¯ xn ) + Δn (¯ xn , t) − yr(n) +

n−1 

 Cn−1,j gn−j (¯ xn−j )xn+1−j + fn−j (¯ xn−j ) + Δn−j (¯ xn , t) − yr(n−j)

j=1

+

n−1 ∗  ∂Fn−1 j=1

∂xj

(gj (¯ xj )xj+1 + fj (¯ xj ) + Δj (¯ xn , t)) +

n−1 ∗  ∂Fn−1 j=0

=gn (¯ xn )u + Fn (¯ xn , y¯r(n) ) + Δn (¯ xn , t) +

n−1 

(j) ∂yr

y˙ r(j)

Cn−1,j Δn−j (¯ xn , t)

j=1

+

n−1 ∗  ∂Fn−1 j=1

∂xj

Δj (¯ xn , t),

(27)

 (n) (n) where Fn (¯ xn , y¯r ) = fn (¯ xn )−yr + n−1 xn−j )xn+1−j + fn−j (¯ xn−j ) j=1 Cn−1,j gn−j (¯ n−1 (n−j) n−1 (j) (j) ∗ ∗ + j=1 (∂Fn−1 /∂xj ) (gj (¯ xj )xj+1 + fj (¯ xj ))+ j=0 (∂Fn−1 /∂yr )y˙ r −yr is an unknown smooth function. Choose a Lyapunov function candidate as Vn = (1/2)zn2 . Using assump-

11

tion 2 and Young’s inequality, we can obtain that

xn )D(v)v + gn (¯ xn )d(v) + Fn (¯ xn , y¯r(n) ) + Δn (¯ xn , t) V˙ n =zn gn (¯ +

n−1  j=1

Cn−1,j Δn−j (¯ xn , t) +

n−1 ∗  ∂Fn−1 j=1

∂xj

 Δj (¯ xn , t)

 1 1 xn )D(v)v + Fn (¯ xn , y¯r(n) ) + zn2 g¯n2 d∗2 + + zn2 ϕ2n (¯ xn ) + ≤zn gn (¯ 4 4     n−1

n−1

∗  1 1 2 2 2 2 ∂Fn−1 2 2 zn Cn−1,j ϕn−j (¯ zn + xn−j ) + ϕj (¯ xj ) + + 4 ∂x 4 j j=1 j=1

 2n (28) xn ) D(v)v + Fn (¯ xn , y¯r(n)) + , =zn gn (¯ 4  (n) (n) where Fn (¯ xn , y¯r ) = (1/gn (¯ xn )) Fn (¯ xn , y¯r ) + zn g¯n2 d∗2 + zn ϕ2n (¯ xn )+  n−1 2 2 2 ∗ xn−j ) + zn n−1 xj ) is an unknown smooth zn j=1 Cn−1,j ϕ2n−j (¯ j=1 (∂Fn−1 /∂xj ) ϕj (¯ function. From Young’s inequality, we have 1 zn gn (¯ xn )Fn (¯ xn , y¯r(n)) ≤ zn2 gn2 (¯ xn )Fn2 (¯ xn , y¯r(n) ) + . 4

(29)

Substituting inequality (29) into inequality (28), it can be obtained that xn ) 2n gn (¯ V˙ n ≤zn gn (¯ + xn )D(v)v + zn2 gn (¯ xn )D(v)Fn2 (¯ xn , y¯r(n) ) + 4D(v) 4  g¯n 2n xn )D(v) v + zn Fn2 (¯ xn , y¯r(n) ) + + . ≤zn gn (¯ 4Dmin 4

(30)

The desired robust control law v ∗ is chosen as follows, xn , y¯r(n)), v ∗ = −cn zn − zn Fn2 (¯

(31)

where cn is a positive real constant which will be specified later. Substituting

12

zn into v ∗ and using the notation (7), we can obtain that   n−1   ∗ v ∗ = − cn xn − yr(n−1) + Cn−1,j xn−j − yr(n−1−j) + Fn−1 (¯ xn−1 , y¯r(n−1) ) j=1

− =−

zn Fn2 (¯ xn , y¯r(n)) n 

 Cn,j xn+1−j − yr(n−j) − Fn∗ (¯ xn , y¯r(n) ),

(32)

j=1 (n)

(n−1)

(n)

∗ where Fn∗ (¯ xn , y¯r ) = cn Fn−1 (¯ xn−1 , y¯r ) + zn Fn2 (¯ xn , y¯r ) is an unknown smooth function. Given a compact set Ω ⊂ R2n+1 , and let θ and ε be such that for any (n) (¯ xn , y¯r ) ∈ Ω, Fn∗ (¯ xn , y¯r(n) ) = θT ξ(¯ xn , y¯r(n) ) + ε, (33)

with |ε| ≤ ε∗ . The actual robust adaptive control law is chosen as follows, v =−

n 

 Cn,j xn+1−j − yr(n−j) − θˆT ξ(¯ xn , y¯r(n)),

(34)

j=1

where θˆ is the estimation of θ and is updated as follows,   ˙ xn , y¯r(n) ) − η θˆ , θˆ = Γ z1 ξ(¯

(35)

with a constant matrix Γ = ΓT > 0, and a real scalar η > 0. Remark 2. For the class of uncertain strict-feedback nonlinear system (1), the designed controller only contains an actual control law and an adaptive law, and can be given directly. Though the above design procedure contains n steps, the virtual control laws at intermediate steps are not necessary to be implemented. Thus, the structure of the designed controller is much simpler than that of exiting design approaches, and the controller realization is easier. 4. Stability analysis In this section, we show that the actual control law and the adaptive law introduced in Section 3 can guarantee the UUB of all the closed-loop system signals. 13

From (31), (32), (33) and (34), we can obtain that xn , y¯r(n) ) − θ˜T ξ(¯ xn , y¯r(n)) + ε, v = −cn zn − zn Fn2 (¯

(36)

where θ˜ = θˆ − θ. Substituting (36) into (30), we have   2 T (n) ˜ ˙ Vn ≤ gn (¯ xn )D(v) −cn zn − zn θ ξ(¯ xn , y¯r ) + zn ε +

g¯n 2n + . 4Dmin 4

(37)

Theorem 1. Given ε∗ , let θ ∈ RN be such that (33) hold in the compact set Ω ⊂ R2n+1 with |ε| ≤ ε∗ . Consider the closed-loop system consisting of the system (1), the actual control law (34) and the adaptive law (35). Then, for any bounded initial conditions, all the signals in the closed-loop system remain uniformly ultimately bounded, and the steady state tracking error can be made arbitrarily small by appropriately choosing control parameters. Proof. Consider the Lyapunov function candidate of the closed-loop system ˜ The derivative of V is as V = ni=1 Vi + (1/2)θ˜T Γ−1 θ. V˙ = ≤

n 

˙ V˙ i + θ˜T Γ−1 θˆ

i=1 n−1 

xi )zi2 −ci gi (¯

i=1

2i − 1 + gi (¯ xi )zi zi+1 + 4



  xn )D(v) −cn zn2 − zn θ˜T ξ(¯ xn , y¯r(n) ) + zn ε + gn (¯  g¯n 2n ˜T  + + θ z1 ξ(¯ + xn , y¯r(n)) − η θˆ 4Dmin 4 n−1 n−1   2 2 =− ci gi (¯ xi )zi − cn gn (¯ xn )D(v)zn + gi (¯ xi )zi zi+1 i=1

i=1

˜T

− gn (¯ xn )D(v)zn θ

ξ(¯ xn , y¯r(n) )

+ gn (¯ xn )D(v)zn ε n2 + 1 g¯n xn , y¯r(n) ) − η θ˜T θ˜ − η θ˜T θ + . + z1 θ˜T ξ(¯ + 4 4Dmin

14

(38)

Using the facts that ⎧ g¯i g¯i 2 ⎪ gi (¯ xi )zi zi+1 ≤ zi2 + zi+1 , i = 1, ..., n − 1, ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ 2 ⎪ 2Dmax χ2 g¯n2 2 η ˜T ˜ ⎪ T (n) ⎪ ˜ zn + θ θ, (¯ x )D(v)z ξ(¯ x , y ¯ ) ≤ − g θ ⎪ n n n n r ⎪ ⎪ η 8 ⎪ ⎪ ⎨ 2 ε 2 xn )D(v)zn ε ≤ Dmax g¯n2 zn2 + , gn (¯ ⎪ 4 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ˜T xn , y¯(n)) ≤ 2χ z 2 + η θ˜T θ, ˜ ⎪ ⎪ z1 θ ξ(¯ r 1 ⎪ η 8 ⎪ ⎪ ⎪ T ⎪ ⎪ ⎩ − η θ˜T θ ≤ η θ˜T θ˜ + ηθ θ , 2 2

(39)

(n)

where ||ξ(¯ xn , y¯r )|| ≤ χ (see Lemma 1), we can obtain that

 n−1  2  g ¯ g¯i−1 g¯i  2 2χ 1 2 ˙ V ≤ − c1 g 1 − ci g i − − z1 − − z 2 η 2 2 i i=2

 2 g¯n−1 2Dmax χ2 g¯n2 2 2 − cn g n Dmin − − − Dmax g¯n zn2 2 η 2 T 2 ε η ηθ θ n + 1 g¯n − θ˜T θ˜ + + , + + 4 4 2 4 4Dmin

(40)

Choosing ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

g¯1 2χ2 α + + , 2 η 2 g¯i−1 g¯i α + + , i = 2, ..., n − 1, ci g i ≥ ⎪ 2 2 2 ⎪ ⎪ 2 ⎪ 2Dmax χ2 g¯n2 g¯ α ⎪ 2 ⎪ ⎩ cn g Dmin ≥ n−1 + + Dmax g¯n2 + , n 2 η 2 c1 g 1 ≥

where α > 0, and letting we have

(41)

α η ≥ , −1 4λmax (Γ ) 2

(42)

V˙ ≤ −αV + γ,

(43)

where γ = ε∗2 /4 + ηθT θ/2 + (n2 + 1)/4 + g¯n /(4Dmin). 15

Solving the inequality (43) gives γ  −αt γ  e , ∀t ≥ 0. 0 ≤ V (t) ≤ + V (0) − α α

(44)

The inequality (44) means that V (t) eventually is bounded by γ/α. That is, zi , i = 1, ..., n, and θ˜ are uniformly ultimately bounded. Since x1 = z1 + yr (1) is uniformly ultimately bounded, the smooth function F1 (x1 , y¯r ) is also (1) uniformly ultimately bounded. Then x2 = z2 + α2 = z2 − c1 z1 − F1 (x1 , y¯r ) is uniformly ultimately bounded accordingly. By analogy, x3 , ..., xn are all uniformly ultimately bounded. That is, all the closed-loop system signals are uniformly ultimately bounded. Moreover, by increasing the values of ci , i = 1, ..., n, and reducing the value of λmax (Γ−1 ), i.e., increasing the value of α, the quantity γ/α can be made arbitrarily small. Thus, the tracking error z1 can be made arbitrarily small. This concludes the proof. Remark 3. The control constants c1 , c2 , ..., cn determined by (41) are fairly conservative. They may be bigger than really need. The equation (41) just provides a guideline for choosing the control constants. In practice, these control constants can be adjusted according to the chosen neural network and other arguments. 5. Simulation examples In this section, we will present two practical examples to demonstrate the effectiveness and merits of the proposed scheme. (i−1) Reference signal processing. The big differences xi − yr , i = 1, ..., n, may result in a very big control effort at the initial time. This is harmful to equipments in real systems. To avoid output saturation of the actuator at initial phase, some filters are used to make the reference signal smooth. The filters are given as follows, Ti (t)Y˙ r(i−1) + Yr(i−1) = yr(i−1) , i = 1, ..., n, (i−1)

(i−1)

(45)

and Yr are inputs and outputs of the filters, respectively; the where yr (i−1) (0) = xi (0); the filtering parameters initial conditions of the filters are Yr −ωi t are chosen as Ti (t) = ψi e + τi , where ψi > 0, ωi > 0, τi > 0. Remark 4. The purpose of the aforementioned signal processing is to obtain a virtual reference signal which is tracked by the system output actually. Because the initial condition of the virtual reference signal is equal to the 16

initial condition of the system states, all the error terms in the actual control law are small at the beginning of control. Furthermore, if the parameters τi , i = 1, ..., n, are sufficiently small, the ultimate difference between the actual reference signal and the virtual reference signal is also sufficiently small. Thus, as long as the system output is capable to track the virtual reference signal very well, good actual tracking performance can be obtained accordingly. 5.1. Brusselator model We firstly consider a very popular nonlinear oscillatory model of chemical kinetics as follows [17, 56], ⎧ 2 x2 , t), ⎪ ⎨ x˙ 1 =x1 x2 + A − (B + 1) x1 + Δ1 (¯ 2 x˙ 2 = (2 + cos x1 ) u + Bx1 − x1 x2 + Δ2 (¯ x2 , t), (46) ⎪ ⎩ y =x , 1 where x1 and x2 denote the concentrations of the reaction intermediates; positive constants A and B are parameters which describe the supply of “reservoir” chemicals; Δ1 (¯ x2 , t) and Δ2 (¯ x2 , t) are system uncertainties, which could be come from external disturbances, etc. This model was named Brusselator model which is introduced in [57] and studied in detail in [58]. According to the proposed approach in section 3, the SNN based robust adaptive controller can be chosen as follows, ⎧ ⎨ u = − C2,1 (x2 − y˙ r ) − C2,2 (x1 − yr ) − θˆT ξ(¯ x2 , y¯r(2) ),   (47) ⎩ θˆ˙ =Γ z1 ξ(¯ x2 , y¯r(2) ) − η θˆ . x2 , t) = 0.7x21 cos(1.5t) In the simulation, it is assumed that A = 1, B = 3, Δ1 (¯ and Δ2 (¯ x2 , t) = 0 [17]. The system initial condition is [x1 (0), x2 (0)]T = T [2.7, 1] . The reference signal is yr = 3 + sin t + 0.5 sin(1.5t). The controller parameters chosen for simulation are c1 = 15, c2 = 15, Γ = diag{4}, η = 0.2, N = 243, σ = 10, μ = 2. The centers of the Gaussian functions are {1, 3, 5}×{−2, 0, 2}×{1, 3, 5}×{−2, 0, 2}×{−3, 0, 3}. The NN ˆ = 0. The parameters used for reference signal processing initial weight is θ(0) are ψ1 = ψ2 = 1, ω1 = ω2 = 1, τ1 = τ2 = 0.01. Figs. 1, 2, 3, and 4 are the obtained simulation results in this case. From Fig. 1, it can be seen that fairly good tracking performance is obtained. The 17

Tracking performance 5

y = x1

yr

4 3 2 1

0

5

10

15

20

0.2

25

30

z1 = x1 − yr

0 −0.2 −0.4 −0.6 −0.8

0

5

10

15 t(s)

20

25

30

Figure 1: The performance of the system output tracking the reference signal

Control input 15

u

10 5 0 −5 −10 −15

0

5

10

15 t(s)

20

25

30

Figure 2: Control input signal

Norm of the network weight 3

ˆ ||θ|| 2 1 0 −1

0

5

10

15 t(s)

20

25

Figure 3: Norm of the network weight

18

30

Virtual tracking performance 5

y = x1

Yr

4 3 2 1

0

5

10

15

20

25

0.1

30

x1 − Yr

0.05

0

−0.05

0

5

10

15 t(s)

20

25

30

Figure 4: The performance of the system output tracking the virtual reference signal

system output tracks the reference signal at a high precision. Fig. 2 gives the control signal of the closed-loop system. The update history of the neural network weight is given in Fig. 3. The performance of the system output tracking the virtual reference signal is shown in Fig. 4. Remark 5. The above mentioned Brusselator model is often used to test the performance of controller. Compared with controllers given in [17, 56], the controller (47) contains only one actual control law, therefore only one online approximator is actually needed in the controller implementation. 5.2. One-link manipulator with a BDC motor Now, we give a popular benchmark of application example, i.e., the trajectory tracking control of a one-link manipulator actuated by a brush DC (BDC) motor, which has been considered in [28, 39]. The dynamics of a one-link manipulator actuated by a BDC motor can be expressed as follows,  E q¨ + B q˙ + N sin q = I + ΔI , (48) ˙ M I˙ + HI = U − Km q, where q, q˙ and q¨ denote the link angular position, velocity and acceleration, respectively; I is the motor current; ΔI represents the torque disturbance; U is the input control voltage. 19

The control objective is to let the link angular position track a desired 2 reference signal yr (t) = (π/2)(1 − e−0.1t ) sin t. Letting x1 = q, x2 = q, ˙ x3 = I and u = U, then the equations (48) can be expressed as ⎧ ⎪ ⎨ x˙ 1 =x2 , x˙ 2 =g2 x3 + f2 (¯ x2 ) + Δ2 (t), (49) ⎪ ⎩ x˙ =g u + f (¯ 3 3 3 x3 ), x2 ) = −(N/E) sin x1 − (B/E)x2 , g3 = 1/M, f3 (¯ x3 ) = where g2 = 1/E, f2 (¯ −(Km /M)x2 − (H/M)x3 and Δ2 (t) = ΔI /E. Same with [28], the dead-zone characteristic of the actuator is assumed as follows, ⎧ v ≤ −0.5, ⎪ ⎨ 1.3(v + 0.5), u = D(v) = 0, −0.5 < v < 0.4, (50) ⎪ ⎩ 1.5(v − 0.4), v ≥ 0.4. The robust adaptive controller can be chosen directly according to the approach proposed in section 3 as follows, ⎧ ⎨ v = − C3,1 (x3 − y¨r ) − C3,2 (x2 − y˙ r ) − C3,3 (x1 − yr ) − θˆT ξ(¯ x3 , y¯r(3) ),   ⎩ θˆ˙ =Γ z1 ξ(¯ x3 , y¯r(3) ) − η θˆ . (51) In the simulation, the parameter values of the appropriate units are the same as in [39] by E = 1, B = 1, M = 0.05, H = 0.5, N = 10 and Km = 10. The torque disturbance is ΔI = 4 sin t. The system initial condition is [x1 (0), x2 (0), x3 (0)]T = [π/4, π/2, 0]T The parameters of the controller are chosen as c1 = 15, c2 = 15, c3 = 15, Γ = diag{4}, η = 0.2, N = 128, σ = 10, μ = 5. The centers of the Gaussian functions are {−2, 2} × {−2, 2} × {−2, 2} × {−2, 2} × {−2, 2} × {−2, 2} × ˆ = 0. The parameters {−2, 2}. The initial weight of the RBF network is θ(0) used for reference signal processing are ψ1 = ψ2 = ψ3 = 1, ω1 = ω2 = ω3 = 1, τ1 = τ2 = τ3 = 0.01. Figs. 5, 6, 7, and 8 are the obtained simulation results in this case. From Fig. 5, it can be seen that fairly good tracking performance is obtained. The link angular position tracks the desired reference signal at a high precision that the maximum steady state tracking error is about 3%. Fig. 6 gives the control voltage and motor current of the actuate motor. The update history 20

Tracking performance 2

q

yr

(R)

1 0 −1 −2

0

5

10

15

20

0.8

25

30

z1 = q − yr

(R)

0.6 0.4 0.2 0 −0.2

0

5

10

15 t(s)

20

25

30

Figure 5: The performance of the link angular position tracking the desired reference signal

Control voltage and motor current 40

V 20

(V)

0 −20 −40 0

5

10

15

20

25

30

20

I

(A)

10 0

−10 −20

0

5

10

15 t(s)

20

25

30

Figure 6: Control voltage and motor current

21

Norm of the network weight

ˆ ||θ||

8 6 4 2 0 0

5

10

15 t(s)

20

25

30

Figure 7: Norm of the network weight

Virtual tracking performance 2

q

Yr

(R)

1 0 −1 −2

0

5

10

15

20

25

0.4

30

q − Yr

0.3

(R)

0.2 0.1 0 −0.1

0

5

10

15 t(s)

20

25

30

Figure 8: The performance of the link angular position tracking the virtual reference signal

22

of the neural network weight is given in Fig. 7. The performance of the link angular position tracking the virtual reference signal is shown in Fig. 8. 6. Conclusion A robust adaptive neural control design is developed for a class of uncertain strict-feedback nonlinear systems with dead-zone and disturbances. Though some virtual robust control laws are given at the intermediate steps of the design, only one actual robust adaptive control law need to be implemented by replacing the lumped unknown function of the system with an SNN approximator at the last step. Thus, the designed controller only contains an actual control law and an adaptive law, and can be given directly. Compared with existing methods, the structure of the designed controller is much simpler, and the implementation of the controller is much easier in practice. Stability analysis shows that all the closed-loop system signals are uniformly ultimately bounded, and the steady state tracking error can be made arbitrarily small by adjusting the control parameters properly. The effectiveness and merits of the proposed approach are demonstrated by simulation results. The problem of adaptive control is investigated for a class of single-input single-output (SISO) uncertain nonlinear systems in this paper. The study for a more general class of multi-input multi-output (MIMO) nonlinear systems is now under the authors’ considerations. In addition, the research of state observer based output feedback control has obtained much attention in recent years, and a lot of significative results, such as [24, 25, 29–31, 44] etc., are reported. The study of incorporating single approximator technique into a state observer based output feedback control design framework should be an interesting work. References [1] I. Kanellakopoulos, P. V. Kokotovic, A. S. Morse, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Transactions on Automatic Control, 36(11): 1241-1253, 1991. [2] I. Kanellakopoulos, P. V. Kokotovic, A. S. Morse, A toolkit for nonlinear feedback design, Systems & Control Letters, 18(2): 83-92, 1992.

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