Adaptive Dynamic Surface Output-Feedback Control for a Class of Hysteric Nonlinear Systems with Prespeeified Tracking Performance

Proceedings of the 2nd IFAC Conference on Embedded Systems, Computational and Telematics Control Systems, Proceedings of Intelligence the 2nd IFAC Conference oninEmbedded Proceedings of the 2nd Conference on Systems, Proceedings of Intelligence theMaribor, 2nd IFAC IFAC Conference oninEmbedded Embedded June 22-24, 2015. Slovenia Computational and Telematics Control Available online at Systems, www.sciencedirect.com Computational Intelligence and Telematics in Control Computational Intelligence and Telematics in Control June 22-24, 2015. Maribor, Slovenia June 22-24, 2015. Maribor, Slovenia June 22-24, 2015. Maribor, Slovenia

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IFAC-PapersOnLine 48-10 (2015) 288–293 Adaptive Dynamic Surface Output-Feedback Control for a Class of Hysteric Adaptive Dynamic Surface Output-Feedback Control a AdaptiveNonlinear Dynamic Systems Surface with Output-Feedback Control for for a Class Class of of Hysteric Hysteric Prespecified Tracking Performance Nonlinear Systems with Prespecified Tracking Performance Nonlinear Systems with Prespecified Tracking Performance Xiuyu Zhang* Chunyi Su** Peng Yan***

Xiuyu Zhang* Chunyi Su** Peng Yan*** Xiuyu Xiuyu Zhang* Zhang* Chunyi Chunyi Su** Su** Peng Peng Yan*** Yan*** * School of Automation, Northeast Dianli University, Jilin City, Jilin Province, CO132012 * School of Automation, Northeast Dianli e-mail: University, Jilin City, Jilin Province, CO132012 China (Tel: 15804325994; [email protected]). * School School of of Automation, Automation, Northeast Dianli University, University, Jilin City, City, Jilin Jilin Province, Province, CO132012 CO132012 * Northeast Dianli Jilin China (Tel: 15804325994; e-mail: [email protected]). ** Department of China Mechanical and Engineering, Concordia University, Montreal, QC, H3G 1M8 (Tel: 15804325994; 15804325994; e-mail: [email protected]). China (Tel: e-mail: [email protected]). ** Department of Mechanical and (e-mail: Engineering, Concordia University, Montreal, QC, H3G 1M8 Canada [email protected]). ** Department Department of of Mechanical Mechanical and Engineering, Engineering, Concordia University, University, Montreal, Montreal, QC, QC, H3G H3G 1M8 1M8 ** and Concordia Canada (e-mail: [email protected]). *** School of Automation, Northeast Dianli University, Jilin City, Jilin Province, CO132012 Canada (e-mail: [email protected]). Canada (e-mail: [email protected]). *** School of Automation, Northeast [email protected]). University, Jilin City, Jilin Province, CO132012 China (e-mail: *** Dianli University, *** School School of of Automation, Automation, Northeast Northeast [email protected]). University, Jilin Jilin City, City, Jilin Jilin Province, Province, CO132012 CO132012 China (e-mail: China (e-mail: [email protected]). China (e-mail: [email protected]). Abstract: In this paper, a high-gain observer based adaptive dynamic surface output-feedback control is Abstract: In this paper, high-gainsystems observerpreceded based adaptive dynamicbacklash-like surface output-feedback control is proposed for a class of aanonlinear by unknown hysteresis. The main Abstract: In paper, high-gain observer based dynamic surface output-feedback control is Abstract: In1)this this paper, anonlinear high-gain observer based adaptive adaptive dynamicbacklash-like surface output-feedback control is proposed for athe class ofneural systems preceded byapproximate unknown hysteresis. The main features are RBF networks are employed to the unknown smooth functions; 2) proposed for a class of nonlinear systems preceded by unknown backlash-like hysteresis. The main proposed for a class of nonlinear systems preceded by unknown backlash-like hysteresis. The main features the RBF neural to approximate the unknown smooth functions; 2) by usingare the1) controlnetworks scheme are andemployed the tracking error transformation functions, the tracking features are 1) proposed the networks are employed to the smooth 2) features the RBF RBF neural neural are employed to approximate approximate the unknown unknown smooth functions; functions; 2) by usingare the1)could proposed controlnetworks scheme and the tracking error transformation functions, thefused tracking performance be prespecified; 3) the derivative-explosion problem when the hysteresis is with by using using the the proposed proposed control control scheme scheme and and the the tracking tracking error error transformation transformation functions, functions, the the tracking tracking by performance could be prespecified; 3) the derivative-explosion problem when the hysteresis is fused with backstepping could designbecan be eliminated, greatly simplifies the control 4) by combining performance prespecified; 3) the thewhich derivative-explosion problem when law; the hysteresis hysteresis is fused fused with with performance could becan prespecified; 3) derivative-explosion problem when the is backstepping design be eliminated, which greatly simplifies the control law; 4) by combining with the estimation of vector norm of the unknown parameters, the computational burden is greatly reduced. backstepping design design can can be be eliminated, eliminated, which which greatly greatly simplifies simplifies the the control control law; law; 4) 4) by by combining with backstepping combining with the estimation of vector norm of the unknown the computational burden is greatly reduced. Simulation results show the effectiveness of the parameters, proposed scheme. the estimation of norm of parameters, the the estimation of vector vector norm of the the unknown unknown the computational computational burden burden is is greatly greatly reduced. reduced. Simulation results show the effectiveness of the parameters, proposed scheme. Simulation results show the effectiveness of the proposed scheme. © 2015, IFAC (International Automatic Control) Hosting by Elsevier(PI) Ltd. hysteresis; All rights reserved. Simulation results show the Federation effectiveness of the feedback; proposed scheme. Keywords: dynamic surface control;ofoutput Prandtl-Ishilinskii high-gain Keywords: dynamic surface outputRBF feedback; Prandtl-Ishilinskii (PI) hysteresis; high-gain observer; prespecified tracking control; performance; neural network approximation. Keywords: dynamic surface surface control; output feedback; feedback; Prandtl-Ishilinskii (PI) hysteresis; hysteresis; high-gain high-gain Keywords: dynamic output Prandtl-Ishilinskii (PI) observer; prespecified tracking control; performance; RBF neural network approximation. observer; prespecified tracking performance; RBF neural network approximation. observer; prespecified tracking performance; RBF neural network approximation. Consider the following nonlinear systems with an unknown 1. INTRODUCTION Consider thehysteresis: following nonlinear systems with an unknown saturated PI 1. INTRODUCTION Consider the following Consider PI thehysteresis: following nonlinear nonlinear systems systems with with an an unknown unknown 1. 1. INTRODUCTION INTRODUCTION Hysteresis exists widely in diverse devices and physical saturated saturated PI PI hysteresis: hysteresis: saturated p Hysteresis exists widely in diverse devices and physical systems smartin materials, ferromagnetic and Hysteresissuch existsaswidely widely diverse devices devices and physical physical p a f ( y) Hysteresis exists diverse and x1 = x2 + f 0,1 ( y ) + ∑ systems such asfields, smartin materials, ferromagnetic and pp i i ,1 superconductive electronic relay circuits systems such such as as smart smart materials, materials, ferromagnetic ferromagnetic and and 1 = x x + f ( y ) + = i 1 ai f i ,1 ( y ) systems 2 + f 0,1 ( y ) + ∑ superconductive fields, electronic relay circuits and  x = x aai ffi ,1 (( yy )) geosciences. Whenfields, a control system preceded hysteresis, x11 = x22 + f 0,1 superconductive electronic relay by circuits and 0,1 ( y ) + ∑ = i superconductive fields, electronic relay circuits and # ii ==111 i i ,1 geosciences. When a control system preceded by hysteresis, it makes the controlled systems cause the problems of in geosciences. When aa control system preceded by geosciences. control system preceded by hysteresis, hysteresis, # it makes oscillation the When controlled systems cause the problems in p accurate, and instability. Therefore, modellingof and ## it makes the controlled systems cause the problems of in it makes the controlled systems cause the problems of in p a f accurate, oscillation and instability. Therefore, modelling and xρ −1 = xρ + f 0, ρ −1 ( y ) + ∑ ( y) control of hysteresis have attracted a lot of attention in recent accurate, oscillation and instability. Therefore, modelling and pp i i , ρ −1 accurate, oscillation and instability. Therefore, modelling and  x = x + f ( y ) + control of hysteresis have attracted a lot of of attention in recent i =1 ai f i , ρ −1 ( y ) ∑ − − 1 0, 1 ρ ρ ρ years owing to the increasing applications smart material= + ff 0, ( y ) + aaii ffii ,, ρρ −−11 (( yy )) control of have aa lot in xxρρ −−11 = xxρρ + control of hysteresis hysteresis have attracted attracted lot of of attention attention in recent recent −11 ( y ) +p∑ 0, ρ ρ− i =1 years owing to the increasing applications smart materialbased actuators (Lyer et al. (2005)). In this of paper, inspired by =11 years owing to the increasing applications of smart materialii = years owing to the increasing applications of smart materialp  x x f y a f ( y ) + bn − ρ g ( y ) w = + ( ) + based actuators (Lyer et al. (2005)). In this paper, inspired by Esfandiari et al.(Lyer (1992), observer based adaptive ∑ 0, ρ ρ ρ +1 pp i i , ρ based actuators actuators et al. al.high-gain (2005)). In In this paper, paper, inspired by based (Lyer et (2005)). this inspired by  x x f y ai fi , ρ ( y ) + bn − ρ g ( y ) w (1) = + ( ) + = 1 i Esfandiari et al. (1992), high-gain observer based adaptive ∑ ρ ( y) + dynamic output-feedback is employed for a xxρρρ = xxρρρ +++111 + ff 0, a f ρ (( yy )) + Esfandiarisurface et al. al. (1992), (1992), high-gaincontrol observer based adaptive adaptive y w (1) = + ( ) + + bbnn −− ρρ gg ( yy ) w 0, ρ Esfandiari et high-gain observer based ∑ 0, ρ i =1 aii f ii ,, ρ dynamic surface output-feedback control is employed forPIa (1) # class of nonlinear systems preceded by unknown saturated (1) 1 = i dynamic surface surface output-feedback output-feedback control control is is employed employed for for aa i =1 dynamic # class of nonlinear systems preceded by unknown saturated PI hysteresis with the following features: p # class of nonlinear systems preceded by unknown saturated PI # class of nonlinear preceded by unknown saturated PI hysteresis with thesystems following features: p a f xn −1 = xn + f 0, n −1 ( y ) + ∑ ( y ) + b1 g ( y ) w hysteresis with the following features: hysteresis withthe thetracking following features: z By using error transformation functions, the pp i i , n −1 xn −1 = xn + f 0, n −1 ( y ) + ∑ i =1 ai fi , n −1 ( y ) + b1 g ( y ) w w = + ff 0, ( y ) + aaii ffii ,, nn −−11 (( yy )) + + bb11 gg ( yy ) w z By usingperformance the tracking could error transformation the tracking be prespecified.functions, xxnn −−11 = xxnn + −p11 ( y ) + ∑ 0, nn − i =1 z using the error functions, the z By By usingperformance the tracking tracking could error transformation transformation 1 i= tracking be prespecified.functions, the = 1 i p a f ( y) + b g ( y ) w tracking performance be xn = f 0, n ( y ) + ∑ tracking performance could be prespecified. prespecified. 0 z The explosion of could complexity problem when the pp i i , n  x f y = ( ) + = 1 ai f i , n ( y ) + b0 g ( y ) w i ∑ 0, n n z The explosionhysteresis of complexity problem when the saturated-type is fused with backstepping  x f y a f y b g ( y) w = ( ) + ( ) + xnn = f 0, n ( y ) + ∑ z of problem when z The The explosion explosionhysteresis of complexity complexity problem when the the i =1 aii f ii ,, nn ( y ) + b00 g y w saturated-type iswhich fusedgreatly with simplifies backstepping y = x1 0, n design can be eliminated, the 1 = i i =1 saturated-type hysteresis hysteresis is fused fused with with backstepping backstepping saturated-type y = x1 design can eliminated, iswhich greatly simplifies the control law. be yy = xx11 design can can be eliminated, eliminated, which which greatly greatly simplifies simplifies the the = design be T control law. where x = x1 , x2 ," , xn T ∈ \ nn is the state vector; ρ is control law. control law. z By estimating the vector norm of the unknown where x = xx2 ,," , xn TT ∈ the state vector; ρ is z By estimating the vector of the unknown parameters, the number of thenorm estimated parameters is the where = x xx111 ,,relative ∈\ \ nn ais is the is wheresystem isi ∈ the\,state state vector; is 22 " , xnndegree. i= 1,vector; " , p ρand z estimating the vector norm of the unknown z By By estimating the vector norm of the unknown parameters, the number of thecomputational estimated parameters is the system relative degree. a ∈ \, i = reduced which implies the burden 1, " , p and parameters, the number of the estimated parameters is parameters, the number estimated parameters system aiii ∈ \ ,i = 1," , p and the system relative degree. and reduced which implies of thethecomputational burden is is the b= , i 0,1,relative " , n − ρdegree. are unknown constant parameters; greatly reduced. i ∈\ reduced which implies the computational burden is reduced which implies the computational burden is b = ∈ \ , i 0,1, " , n − ρ are unknown constant parameters; greatly reduced. i are unknown constant parameters; greatly b= 0,1, unknown parameters; fiiij∈, \i ,=i 0,1, "" , p, n,− jρ=are 1," , n are constant unknown smooth greatly reduced. reduced. 2. PROBLEM STATEMENT ffij ,, ii = 0,1, " , p , j = 1," , n are unknown smooth = 0,1, " , p ,, and j = 1,g" smooth 2. PROBLEM STATEMENT areareunknown unknown nonlinear functions ( g, ≠n 0)are known smooth ij ij , 2. 2. PROBLEM PROBLEM STATEMENT STATEMENT nonlinear functions and g ( g ≠ 0) are known smooth nonlinear functions functions and are output known of smooth y∈\ the functions; and 2.1 System nonlinear g ( gis≠ 0)theare known smooth nonlinear functions; y ∈ \ is the output of the 2.1 System is the the output output of of the the nonlinear functions; functions; y ∈ \ is 2.1 System System nonlinear 2.1

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Copyright © 2015 IFAC 288 2405-8963 Copyright©©2015, 2015IFAC IFAC (International Federation of Automatic Control) 288 Hosting by Elsevier Ltd. All rights reserved. Peer review© of International Federation of Automatic Copyright 2015 IFAC 288 Copyright ©under 2015responsibility IFAC 288Control. 10.1016/j.ifacol.2015.08.146

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x = Ax + ΦT W + Bg ( y ) u + δ

system; w∈ \ is the unknown backlash-like hysteresis nonlinearity which can be expressed as

= w ϖ u + d (u )

289

y = CT x

(2)

(8)

with

⎡ψ 1T ( y ) ⎤ ⎢ ⎥ n×( N1 + N 2 +"+ N n ) = ΦT ( y ) ⎢ % ⎥∈\ d (u ) ≤ D (3) ⎢ ψ nT ( y ) ⎥⎦ ⎣ ⎡0( ρ −1)×1 ⎤ 2.2 Radial basis function neural network approximation ⎡ω1 ⎤ ⎢ ⎥ bn − ρ ⎥ N1 + N 2 +"+ N n )×1 ⎢ ⎥ ( ⎢ In this article, radial basis function neural network (RBFNN) ,B = = W ⎢ # ⎥∈\ ⎢ # ⎥ is employed to approximate a continuous function on a given ⎢⎣ωn ⎥⎦ ⎢ ⎥ compact set. ⎢⎣ b0 ⎥⎦ Lemma 1 (Sanner and Mears 1992): RBFNN is an universal ⎡ q1 ⎤ ⎡ − q1 1 ⎤ approximator in the sense that given any real continuous δ ⎡ ⎤ 1 ⎢ ⎥ ⎢ n q2 ⎥ − q2 % ⎥⎥ function F (ξ ) : Ωξ → \ with Ωξ ⊂ \ a compact set, ⎢ ⎥ ⎢ ⎢ , A0 = δ =⎢# ⎥ ,q= ⎢ # ⎢# ⎥ 1⎥ and any δ m > 0 , by appropriately choosing σ k ⎢⎣δ n ⎥⎦ ⎢ ⎥ ⎢ ⎥ n 0⎦ ⎣ − qn 0 ⎣ qn ⎦ and ζ k ∈ \ , k = 1," , N ,for some sufficiently large integer N, there exists an RBF network such that ⎡1 ⎤ ⎡0 I ⎤ (9) A = ⎢ n −1 ⎥ , C = ⎢ ⎥ ⎣0 0 ⎦ = F (ξ ) ϑ *Tψ (ξ ) + δ (ξ ) , ∀ξ ∈ Ω ⊂ \ n ⎣⎢0( n −1)×1 ⎦⎥ Let D denote the upper bound of d ( u ) satisfying

ξ

δ (ξ ) ≤ δ m where ϑ is an optimal weight vector of *

ϑ and is defined as

⎧⎪ ⎫⎪ = ϑ * arg minn ⎨sup Y (ξ ) − F (ξ ) ⎬ ϑ∈\ ξ ∈Ωξ ⎩⎪ ⎭⎪ and

δ (ξ )

2.4 High-gain K-filters observer

(4)

(5)

Inspired by Atassi et al. (2000), the following high-gain observer, say, high-gain K-filters, is introduced to mitigate the effect of the hysteresis nonlinearity Xu Zhang et al. (2013):

= vi kA0 vi + Ψ −1en −i g= ( y )u , i 0,1," , n − ρ

is approximation error,

δ= (ξ ) F (ξ ) − ϑ *Tψ (ξ )

(6)

Using neural network to estimate (1) p

f 0, j ( y ) + ∑ ai fi , j ( y ) = ψ Tj ( y ) ω j + δ i ( y ) , j = 1," , N

(10)

= υ0 kA0υ0 + kqy

(11)

Ξ= kA0 Ξi + Ψ −1ΦT , i = 1," , p i

(12)

where k ≥ 1 is a positive design parameter, en −i denotes the (7)

i =1

( n − i ) th

coordinate vector in R n , and

Ψ =diag {1, k ," , k n −1}

With ω j ∈ \ Ni , N i is on behalf of the number of nodes of RBF NN,ψ j ( y ) ⎡ m1 ( y ) ," , mN ( y ) ⎤ ∈ \ Ni is the weight = i ⎣ ⎦ vector, ω j is the Radial basis function, δ i ( y ) is

(13)

Then the state estimate is formed as n−ρ

xˆi = Ψυ0 + ΨΞ iW + ∑ βi Ψvi

approximation error. The approximation error of the RBF NN which was defined in (12) is bounded. There is a constant

(14)

i =0

δ iM > 0 , satisfy: δ i ≤ δ iM .

Define the estimation error

ε = x − xˆ 2.3 Problem statement after the RBF

(15)

Then it is easy to verify that

ε = Aε − k Ψqε1 + bg ( y ) d + δ

With convenience, the system matrix is expressed as the following form： 289

(16)

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3. ADAPTIVE DSC DESIGN PROCEDURE

where ε1 is the first entry of ε .

Step 1: Let the first surface error (the tracking error) be defined as (17), then

2.5 Performance and the error transformation functions

p S1 = Z (− e + x1 − y r ) p

Let the tracking error

= e x1 − yr

(17)

where

where yr is the desired trajectory. By Charalampos and George (2007), performance and error transformation functions are defined as follows. A performance function p (t ) : R+ → R− − {0} is defined as a smooth and decreasing

Z=

(18)

n−ρ

x2 = kυ0,2 + k Ξ 2W + ∑ βi kvi.2 + ε 2

−σ , lim Φ ( S1 ) = 1, if e(0) > 0 ⎧ lim Φ ( S1 ) = S1 →∞ ⎪ S1 →−∞ ⎨ Φ ( S1 ) = −1, lim Φ ( S1 ) = σ , if e(0) < 0 ⎪ S lim S1 →∞ ⎩ 1 →−∞

(21)

Therefore, viewing Equation (28), Equation (27) can be rewritten as n− ρ ⎡ p ⎤ S1 = Z ⎢− e + kυ0,2 + kΞ2W + ∑ βi kvi.2 + ε 2 +ψ1T ω1 + δ1 − yr ⎥ i =0 ⎣ p ⎦ (29)

and δ1

≤ δ1M , δ1M is the upper bound of δ1 .

{

}

Let ϑ * = max ω1Tω1，W T W , the following inequality is established

From (21), if S1 is bounded, (20) holds, which, together with p (t ) > 0 and (19), implies that

−σ p (t ) < e(t ) = p(t )Φ ( S1 )< p(t ) < 1, if e(0) > 0

(22)

− p (t ) < e(t ) = p(t )Φ ( S1 )< σ p(t ), if e(0) < 0

(23)

S1Zψ 1T ω1 ≤

α1S12 Z 2ψ 1Tψ 1ϑ *

S1Zk Ξ 2W ≤

2

+

α1S12 Z 2 ΞT2 Ξ 2ϑ * 2

1 2α1

+

k2 2α1

(30)

α1 is a positive design parameter.

that is, (18) holds. Hence, to achieve the prespecified tracking performance, one only needs to show is S1 ∈ L∞ , where the

Defined the following vector

strictly increasing property of Φ ( S1 ) guarantees that we can obtain

⎛ e(t ) ⎞ S1 = Φ −1 ⎜ ⎟ ⎝ p (t ) ⎠

(28)

i =0

where S1 is the transformed error and Φ ( S1 ) is a smooth, strictly increasing and thus invertible function has the following properties: (20)

(27)

Since x2 is not available for measurement, from Equation (14), we have

(19)

⎧−σ < Φ ( S1 ) < 1, if e(0) > 0 ⎨ ⎩−1 < Φ ( S1 ) < σ , if e(0) < 0

(26)

⎡ p ⎤ S1 = Z ⎢ − e + x2 + ψ 1T ω1 + δ1 − y r ⎥ ⎣ p ⎦

where 0 < σ < 1 and limt →∞ = p∞ ， with p∞ the maximum allowable value of steady-state tracking error. To transform (18) into an equivalent unconstrained one, define the error transformation function as

e= (t ) p (t )Φ ( S1 )

1 ∂Φ −1 p ∂ (e / p )

The derivative of (24) by considering (1) is

positive function such that for all t ≥ 0 .

⎧−σ p (t ) < e(t ) < p (t ), if e(0) > 0 ⎨ ⎩− p (t ) < e(t ) < σ p (t ), if e(0) < 0

(25)

⎡ 1 ⎤ β ϑ * β0 β θa = ⎢ , , , 1 ," , n − ρ −1 , ⎥ β n − ρ ⎦⎥ ⎣⎢ β n − ρ β n − ρ β n − ρ β n − ρ

(24)

⎡ ⎛

p

⎞

3

1

ϕ = ⎢ Z ⎜ − e + kυ0,2 − y r ⎟ + l1S1 + Z 2 S1 + k 2 Z 2 S1 , 2 2 ⎠ ⎣ ⎝ p

Note that in the case of e(0) = 0 , we can incorporate it into e(0) > 0 or e(0) < 0 without any effect on the system analysis; however, cannot be chosen to be zero due to S1 (0) being infinite. 290

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where li ,(2 ≤ i ≤ ρ − 1) are positive design parameters. Let

⎤ ⎛ α1 S1 Z 2 Ξ T2 Ξ 2 α1 S1 Z 2ψ 1Tψ 1 ⎞ + ⎜ ⎟ , Zkv0,2 , Zkv1,2 ," , Zkvn − ρ −1,2 ⎥ 2 2 ⎝ ⎠ ⎦ (31)

vn − ρ ,i +1 pass through the following first-order filter with time constant τ i +1 to obtain a new state variable zi +1 :

Where l1 is a positive design parameter, vn − ρ ,2 is the virtual

τ i +1 zi +1 + zi +1 = vn − ρ ,i +1 , zi +1 (0) = vn − ρ ,i +1 ( 0 )

control signal to be designed to stabilize Equation (29)

k2 1 3 2 2 + − Z S1 2α1 2α1 2

Step ρ : Define the ρ th surface error

(32)

S ρ = vn − ρ , ρ − z ρ

1 − k 2 Z 2 S12 2

(41)

whose derivative by considering vn − ρ , ρ in (10) is

Now, these is the following inequality

S1θ aϕ ≤

α 22 S12ϑaϕ T ϕ 2

+

Sρ = − kqρ vn − ρ ,1 + kvn − ρ , ρ +1 + k 1− ρ u − zρ

1

(33)

2α 22

= u k ρ −1 (kqρ vn − ρ ,1 − kvn − ρ , ρ +1 + zρ − lρ S ρ ) let ϑa = θ a and α 2 is a positive design parameter，which where lρ is positive design parameters. suggests us to choose the virtual control vn − ρ ,2 as

vn − ρ ,2 = −

α 22 S1ϑˆaϕ T ϕ

Define

yi = zi − vn − ρ ,i , i = 2," , ρ

ϑa and is updated by

⎛ α 2 S 2ϑ ϕ T ϕ ⎞  = − ηϑˆa ⎟ ϑˆa γ ϑa ⎜⎜ 2 1 a ⎟ 2 ⎝ ⎠ with

(44)

The derivatives of yi , i = 2," , ρ are

(35)

(

y α2  y 2 = − 2 − 2 S1ϑˆaϕ T ϕ + S1ϑˆaϕ T ϕ + S1ϑˆaϕ T ϕ + S1ϑˆaϕ T ϕ τ 2 2Zk

with η being a positive design parameter. Let vn − ρ ,2 pass through the following first-order filter constant τ 2 to obtain a new state variable z2 :

time

τ2

yi = −

1

τi

(

)

yi + kqvn − ρ ,1 +  zi − li Si / k ,

i= 3," , ρ

1 = − yi + Bi +1

where τ 2 is a positive design parameter.

τi

(45)

Step i (2 ≤ i ≤ ρ − 1) : Define the ith surface error where B2 and Bi +1 are continuous functions.

(37)

Now, consider the closed-loop systems, let the Lyapunov function be defined as follows:

whose time derivative by considering vn − ρ ,1 in (10) is

Si = −kqi vn − ρ ,1 + kvn − ρ ,i +1 − zi = −kqi vn − ρ ,1 + k (vn − ρ ,i +1 − vn − ρ ,i +1 ) + kvn − ρ ,i +1 − zi

ρ

V = ∑ Vi

(38)

(46)

i =1

Then the virtual control vn − ρ ,i +1 is chosen as

vn= (kqi vn − ρ ,1 + zi − li Si ) / k − ρ ,i +1

)

y = − 2 + B2

τ 2 z2 + z2 = vn − ρ ,2 , z2 (0) = vn − ρ ,2 ( 0 ) (36)

= Si vn − ρ ,i − zi

(43)

4. STABILITY AND PERFORMANCE ANALYSIS

(34)

2 Zk

(42)

The actual control u appears in this step and is chosen as

2

and ϑˆa is the estimate of

(40)

where li and τ i +1 are positive design parameters.

S1S1 ≤ −l1S12 + S1β n − ρθ aT ϕ + S1Z δ1M + S1Z ε 2 + S1Z β n − ρ kvn − ρ ,2 +

291

V= 1 (39)

βn− ρ  2 1⎛ 2 ϑ + Vε ⎜ S1 + y22 + ⎜ γ ϑa a 2⎝

⎞ ⎟ ⎟ ⎠

(47)

Where Vε is a quadratic function concerning the estimation error ε i and will be given in the following lemma, which 291

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reveals the key feature of the high-gain observer in compensation of the effect of the hysteresis nonlinearity.

In the proof of the Lemma2 implies that ε 2 = k εˆ2 , we have

Vε = ε T P1ε

(

Where P1 = Ψ1−1

)

T

ZS1ε 2 ≤

P1Ψ1−1 with = P1 P1T > 0 satisfying

β n − ρ kZS1S2 ≤

where the Hurwitz matrix A0 is defined by Equations (9). Then for any k ≥ 1 , we have (50)

(

where

ζk =

k

(51)

2λmax ( P1 )

⎡ 2 ⎛ b μ D + δ ⎞2 ⎤ M δ k = k ⎢ P1 ⎜ ⎟ ⎥ ρ ⎢ k ⎝ ⎠ ⎥⎦ ⎣

given positive constants. For V ( 0 ) ≤ ι and

yr2

+

+

 yr2

≤K,

by properly choosing the design parameters k , l1 ," , lρ ,τ 2 ," ,τ ρ ,η , γ ϑa , all signals of the closed-loop

r

r

2 r

+ y r2 +  y r2 ≤ K

+

}

ρ ρ ⎪⎧ ⎪⎫ = Ω 2 ⎨∑ Si2 + ∑ yi2 + β n − ρ β 2 + 2ε T P1ε ≤ 2ι ⎬ ⎪⎩ i 1 =i 2 ⎪⎭ =

τ i +1

Bi +1 yi +1 ≤ yi +1 M i +1 ≤

σ 2

−

2

β n2− ρ k 2 2

⎞ 2 ⎛1 1 ⎞ ⎟ S2 − ⎜ − − k ⎟ y32 ⎜ ⎟ ⎟ ⎝ τ 3 2σ 2 ⎠ ⎠

1 k⎞ k ⎛ S22 + kS32 − ⎜ lρ − ⎟ S ρ2 + S ρ2 2 2⎠ 2 ⎝

3 ⎞ 2 ⎠

⎛ 1

⎝τi+1

2σ

+

σ 2

, i= 1,", ρ − 1

−

Mi2+1 k ⎞ 2 k 2 1 2 σ ⎞ − ⎟ yi+1 − Si + kSi+1 + ⎟ 2σ 2 ⎠⎟ 2 2 2 ⎠⎟

Now, choose the following design parameters to satisfy:

k ≥ 2λmax ( P1 ) ζ +

λmax ( P1 ) λmin ( P1 )

l1 ≥ ζ

(54)

β n2− ρ k 2

3 k +ζ 2 k lρ ≥ + ζ 2

(55)

li ≥

Where σ is any positive constant. We then consider the time derivative of the Lyapunov function V . 292

(58)

M 32

l2 ≥ k + yi2+1M i2+1

2

2 2 2 k 2 + 1 ηβ n − ρϑa β n − ρ k 2 + + S2 2α1 2 2

β n2− ρ k 2

⎛ i=3 ⎝ ⎝

on Ω1 × Ω 2 say, M i +1 . From Equations (45),

+ Bi +1 y= i +1 , i 1," , ρ − 1

+

+ ∑⎜ −⎜li − k ⎟ Si2 − ⎜ ⎜ ⎜

continuous functions Bi +1 in Equations (45) have maximum

yi2+1

2

2α 22

ρ−1⎛

(53)

Note that Ω1 × Ω 2 is also compact sets. Therefore, the

yi +1 yi +1 ≤ −

βn− ρ

⎛ − ⎜ l2 − k − ⎜ ⎝

Proof: Define the following compact sets r

⎛ ⎞ k 1 σ 1 ⎟ V + + δ k + δ1M −⎜ − ⎜ 2λmax ( P1 ) 2λmin ( P1 ) ⎟ ε 2 2 ⎝ ⎠ +

system (46) are uniformly bounded. Also, prespecified tracking performance can be guaranteed as that have been shown in (22) and (23).

{( y , y , y ) : y

(57)

⎛ 1 M 2 β n2− ρ k 2 ⎞ 2 ηβ n − ρ ϑa2 ⎟ y2 − V ≤ −l1S12 − ⎜ − 2 − ⎜ τ 2 2σ ⎟ 2 2 ⎝ ⎠

(52)

y r2

)

Substituting Equations (51)-(57) into Equation (46), the following inequality can be obtained:

Theorem 1: Consider the closed-loop system and the corresponding Lyapunov function (46). Let ι and K be any

= Ω1

k 2 Z 2 S12 1 + Vε 2 2λmin ( P1 )

2 2 Z 2 S12 β n − ρ k 2 + S2 2 2 2 2 Z 2 S12 β n − ρ k 2 β n − ρ kZS1 y2 ≤ + y2 2 2 Z 2 S12 1 M 2 + δ1 ZS1δ1 ≤ 2 2 2 1 2 −ϑaϑˆa ≤ ϑa − ϑa 2

(49)

Vε ≤ −ζ kVε + δ k

(56)

Using Equation (56) and Young’s inequalities

(48)

A0 P1 + P1 A0 = −2 I

k2 k2 Vεˆ = V λmin ( P1 ) λmin ( P1 ) ε

2 ε 22 =k 2εˆ22 ≤ k 2 εˆ ≤

Lemma 2: Let the high-gain K-filters be defined by Equations (10)–(12). Let the quadratic function

2

+ζ

= ,i 3," , ρ − 1

CESCIT 2015 June 22-24, 2015. Maribor, Slovenia

1

τ2

≥

1

τ i +1 η≥

β n2− ρ k 2 2

≥

+

Xiuyu Zhang et al. / IFAC-PapersOnLine 48-10 (2015) 288–293

the unknown states in the control system. By using the proposed control scheme, the tracking performance could be prespecified, the derivative-explosion problem in backstepping method can be overcome, and the computational burden is greatly reduced due to the estimation of the vector norm of the unknown parameters. Simulation results show the validity of the proposed scheme.

M 22 +ζ 2σ

k M i2+1 , i 2," , ρ − 1 + +ζ = 2 2σ

293

(59)

2ζ γ ϑa

0.6

0.4

Then, substituting Equations (58) and (59) into Equation (46) yields

Trackingperform ance

0.2

V ≤ −2ζ V + C k 2 +1 1 M δk C =+ + δ1 2α1 2

2

0

-0.2

-0.4

-0.6

-0.8

βn−ρ

βn−ρ ρ −1 ηϑa2 + σ+ + 2 2 2α22

2

-1

(60)

0

10

20

30

40

50 T(sec)

0.5

C ontrol signal u

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

10

20

30

40

50 T(sec)

60

70

80

90

100

Fig.2. Tracking error of the control systems REFERENCES Lyer, R.V., Tan, X.B. and Krishnaprasad, P.S. (2005). Approximate inversion of the Preisach hysteresis operator with application to control of smart actuators’, IEEE Trans Autom Control, 50(6):798-809 Esfandiari, F. and Khalil, H.K. (1992). Output feedback stabilization of fully linearizable systems. International Journal of Control, 56, 1107-1037. Atassi, A. N., and Khalil, H. K. (2000). Separation results for the observer designs. Systems & Control Letters, 39(3), 183-191 Charalampos, P.B. and George, A.R.(2009).Adaptive Control with Guaranteed Transient and Steady State Tracking Error Bounds for Strict Feedback Systems. Automatica, 45(2), 532-538. Sanner, R.M., and Mears, M.J. (1992).Stable Adaptive Tracking of Uncertainty Systems Using NonlinearlyPara meterized On-line Approximators. IEEE Transactions on Neural Networks, 3, 837–863. Xu. Zhang and Y. Lin, (2013). An adaptive output feedback dynamic surface control for a class of nonlinear systems with unknown backlash-like hysteresis", Asian J. Contr., 15( 2), 489-500. Xiuyu. Zhang, Y. Lin, J. Wang, (2013). High-gain observer based decentralised output feedback control for interconnected nonlinear systems with unknown hysteresis input. Int. J. Control, 86(6), 1046-1059.

⎡ y2 ⎤ ⎡0 1 0 ⎤ ⎡0⎤ ⎡0.05sin(t ) ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ x ⎢0 0 1 ⎥ x + ⎢sin( y ) ⎥ θ + ⎢0⎥ w + ⎢⎢ 0.1cos(t ) ⎥⎥ = ⎢ ⎥ ⎢⎣0 0 0 ⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣0.08sin(t ) ⎥⎦ ⎣ y ⎦ y = [1 0 0] x

(62) where θ = [ 0.6 1.5 0.3] , w is the unknown hysteresis 2

described by (6) with λ = 1 , p (r ) = 0.5e −0.001( r −0.6) and

ρ ( u ) = 3tanh ( u ) , respectively. The control objective is to reference

100

0.2

In this section, we consider the following third-order plant:

the

90

0.3

5. SIMULATION RESULTS

output y = x1 track

80

Tracking error e performance funct ion p(t ) performance funct ion -p(t )

0.4

uniformly bounded. Then, according our previous work Xiuyu Zhang et al. (2013), it is easy to proof that all signals of the closed-loop adaptive system are bounded. This completes the proof.

the

70

Fig.1. Tracking performance

C Where C is defined by Equation (60). Let ζ ≥ . 2ι Then V ≤ 0 if V = ι , which implies that V ≤ ι is an invariant set. Thus, given V ( 0 ) ≤ ι , we have V ( t ) ≤ ι , ∀t > 0 .It is clear from the above analysis that for any initial conditions, by properly choosing the design parameters, V is bounded. For i = 1," , n the signals S1 ," , S ρ , y2 ," , yρ , β and ε are

make

60

signal

= yr 0.5sin(0.2t ) + 0.5cos(0.4t ) as closely as possible. Fig.1 shows the tracking performance of the control system and Fig.2 shows the tracking error of the control systems. 6. CONCLUSIONS In this paper, the neural networks based adaptive dynamic surface output-feedback control is proposed for a class of nonlinear systems preceded by unknown backlash-like hysteresis, in which the high gain observer is used to estimate 293