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Robust adaptive asymptotic tracking control of a class of nonlinear systems with unknown input dead-zone
Author’s Accepted Manuscript Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone Wenxiang Deng, Jianyong Yao, Dawei Ma www.elsevier.com
PII: DOI: Reference:
S0016-0032(15)00360-9 http://dx.doi.org/10.1016/j.jfranklin.2015.09.013 FI2447
To appear in: Journal of the Franklin Institute Received date: 26 January 2015 Revised date: 12 August 2015 Accepted date: 24 September 2015 Cite this article as: Wenxiang Deng, Jianyong Yao and Dawei Ma, Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.09.013 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.
1
Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone Wenxiang Deng, Jianyong Yao*, Dawei Ma Abstract—This paper considers the tracking control for a class of uncertain single-input and single-output (SISO) nonlinear strict-feedback systems with unknown input dead-zone nonlinearity, parametric uncertainties and unknown bounded disturbances. By constructing a smooth dead-zone inverse and applying the backstepping recursive design technique, a robust adaptive backstepping controller is proposed, in which adaptive control law is synthesized to handle parametric uncertainties and a novel robust control law to attenuate disturbances. The robust control law is developed by integrating a sufficiently smooth positive integral function at each step of the backstepping design procedure. In addition, a smooth projection mapping is used and assumptions are made that the prior knowledge of the extents of parametric uncertainties and the variation ranges of the bounds of disturbances is known to facilitate the backstepping recursive design. However, the exact bounds of disturbances are not required. The major feature of the proposed controller is that it can theoretically guarantee asymptotic output tracking performance, in spite of the presence of unknown input dead-zone nonlinearity, various parametric uncertainties and unknown bounded disturbances via Lyapunov stability analysis. Comparative simulation results are obtained to illustrate the effectiveness of the proposed control strategy. Index Terms—dead-zone inverse; robust control; adaptive control; backstepping; smooth projection mapping; disturbance suppression
I. INTRODUCTION
C
ontrol of nonlinear systems is of great significance since the majority of practical systems exhibit nonlinear dynamic behaviors, such as the dynamic behaviors of electrical motors [1-2], robot manipulators [3], hydraulic systems [4-5], and so on. However, these systems are typically subjected to various parametric uncertainties and disturbances, which could severely deteriorate the achievable control performance, leading to undesirable control accuracy, limit cycles and even instability [6]. In order to handle parametric uncertainties and disturbances in nonlinear systems and improve the tracking performance, extensive research has been devoted to the design of high performance nonlinear controller. During the past several decades, adaptive control design for nonlinear systems to achieve globally asymptotic stability of the closed loop system has been proposed and undergone rapid developments, with plenty of publications presented, such as [8-10]. Specially, under the assumption that the nonlinear systems are subjected to parametric uncertainties only, systematic approaches of adaptive control were proposed in [9], which can achieve asymptotic output tracking performance. And the inherent overparameterization problem was overcome by introducing the idea of tuning function in [10]. In fact, no matter how accurate the practical system model and parameter identification are, modeling errors and external disturbances always exist in physical systems. Hence, some robust adaptive tracking controllers were developed for nonlinear systems with consideration of parametric uncertainties and disturbances simultaneously in [11-13]. In addition, an adaptive robust control (ARC) strategy was proposed by Yao et al. in [14] for uncertain nonlinear systems in the presence of various modeling uncertainties. As an effective control strategy, adaptive robust control has been widely employed in many applications [15-18]. However, most of the existing results in the literature can only achieve bounded-error trajectory tracking performance rather than the strongly expected asymptotic tracking with zero steady-state error in the presence of parametric uncertainties and disturbances. To cope with this issue, adaptive sliding mode control was presented for uncertain nonlinear systems with perturbations in [19], which assures that the tracking error converges to zero asymptotically. It is worth to note that the discontinuous function contained in the controller [19] may cause severe chattering, which limits the employment of the controller in engineering practice. In [21, 22], adaptive control design was integrated with a novel robust integral of the sign of the error (RISE) control proposed in [20] to obtain asymptotic tracking performance, meanwhile ensuring the continuity of the final control input. However, all these RISE based adaptive controllers can *Corrosponding author, phone: 86-25-84315125; fax: 86-25-84303248; e-mail: jerryyao.buaa@ gmail.com; Manuscript received xxx xx, 2015. This work was supported in part by the National Nature Science Foundation of China (Grant no. 51305203), in part by Project funded by China Postdoctoral Science Foundation (Grant no. 2015T80553), and in part by the Natural Science Foundation of Jiangsu Province (Grant no. BK20141402). Wenxiang Deng, Jianyong Yao and Dawei Ma are with the School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; e-mail: jerryyao.buaa@ gmail.com; email: [email protected]).
2 only tackle matched disturbances while many practical systems indeed contain the so-called mismatched disturbances. Therefore, new approaches need to be developed to handle the asymptotic tracking problem of uncertain nonlinear systems with parametric uncertainties, matched and mismatched disturbances. Besides the above discussed parametric uncertainties, matched and mismatched disturbances, input dead-zone nonlinearity also usually exists in physical applications, such as hydraulic servo valve, mechanical transmissions, DC servo motors, and other devices. Dead-zone is often a source of instability and performance deterioration [27], and hence needs more attention to handle it. To effectively attenuate the effect of dead-zone nonlinearity, a dead-zone inverse model in conjunction with adaptive control was first proposed in [23] for linear systems with unmeasured dead-zone outputs. However, the controller in [23] can only guarantee the convergence of the tracking error to a residual bounded set. Asymptotic output tracking performance was derived in [24] under the assumption that the dead-zone output is measurable, which might be unreasonable in practice. Recently, an integrated direct/indirect adaptive robust control scheme was developed in [25] for a class of nonlinear systems with unknown dead-zone. By constructing a novel dead-zone inverse and making full use of the fact that the unknown dead-zone can be linearly parameterized perfectly within certain known intervals, indirect adaptation laws are designed within every interval to obtain accurate parameter estimation of the unknown dead-zone and then achieve perfect dead-zone compensation in the design of adaptive robust controller. However, all aforementioned approaches use discontinuous dead-zone inverse, which may cause unavoidable chattering of the control input. Hence, a smooth dead-zone inverse was introduced in [7], and utilized in the development of an adaptive output feedback backstepping controller for a class of uncertain nonlinear system preceded by unknown dead-zone to guarantee uniformly bounded tracking performance. Another method to cope with the attenuation problem of input dead-zone is to model the dead-zone as a time-varying disturbance-like term, examples like in [26-28, 30]. With this formulation, many robust control laws can be utilized to handle the disturbance-like term, such as sliding mode control [31] and RISE feedback control [1, 4, 20-22]. In [30], a feasible adaptive controller was proposed to tackle the tracking control of nonlinear systems with unknown input dead-zone and time delays. And asymptotic tracking performance was obtained in [26] by applying a novel robust adaptive controller. However, the dead-zone characteristics are not explicitly considered in those controllers, which might lead to not precise enough tracking performance, such as the steady-state performance [7], especially when facing severe input dead-zone. Moreover, high gain feedback problem may also be caused by treating the dead-zone as a disturbance-like term if the nonlinear system is subjected to sever input dead-zone, i.e., large slopes or breakpoints. In viewing the above observations, the control design for uncertain nonlinear systems with unknown dead-zone nonlinearity, parametric uncertainties and disturbances deserves further work to improve the tracking performance. In this paper, a robust adaptive backstepping control strategy is proposed for a class of uncertain SISO nonlinear strict-feedback systems which contain unknown input dead-zone, parametric uncertainties as well as unknown bounded disturbances. In the controller design, a smooth dead-zone inverse in [7] is referred to compensate the effect of input dead-zone, with expecting to effectively compensate severe input dead-zone effects. The adaptation laws are synthesized via backstepping recursive design technique to obtain parameter estimation of unknown dead-zone parameters, other unknown system parameters and the bounds of terms corresponding to disturbances. And disturbances are handled by novel robust control laws which are derived by incorporating a sufficiently smooth positive integral function into the controller design at each step when backstepping. A smooth projection mapping is used to facilitate the backstepping design since differential operation on virtual control law is needed. The main contributions of this paper are given as follows: (i) The proposed control approach does not require exact bounds of the unknown disturbances, instead, they are estimated by adaptive laws; (ii) the robust control concept in [26] are integrated and extended into a model-based adaptive compensation case for a class of uncertain SISO nonlinear systems with unknown input dead-zone, parametric uncertainties, and unknown bounded disturbances via backstepping method; (iii) to the best of our knowledge, it may be the first time in the literature that the asymptotic tracking performance is achieved for the nonlinear system considered in this paper meanwhile with a continuous control input. This paper is arranged as follows. Problem statement and some preliminaries are presented in Section II. Section III gives the robust adaptive backstepping controller design procedure and its global stability analysis. Comparative simulation results are obtained in Section IV. And some conclusions can be found in Section V. II. PROBLEM STATEMENT AND PRELIMINARIES A. Problem statement In this paper, we consider a class of uncertain SISO nonlinear strict-feedback systems with unknown input dead-zone given as follows: xi xi 1 T i ( xi , t ) di ( xn , t ), 1 i n 1
xn Eu (v) T n ( xn , t ) d n ( xn , t )
(1)
y x1 where x=[x1, x2, … , xn] ∈ R is the state vector and y is the system output, xi [ x1 , x2 , ... , xi ]T ∈ Ri is the partial state vector, θ ∈ Rp is an unknown constant parameter vector, φi ∈ Rp , (i=1, 2, … , n), are known nonlinear functions, di, (i=1, 2, … , n), are unknown T
n
3 time-varying disturbances, v is the control signal to be designed and also the input of the dead-zone, u(v) is the output of the dead-zone nonlinearity and also actual control input to the plant, E≠0 represents an unknown constant gain. The dead-zone characteristic can be described as follows [23]: v br mr (v br ) (2) u (v) DZ (v) 0 bl v br m (v b ) v bl l l where DZ(∙) represents the considered dead-zone; mr >0, ml >0, br ≥0 and bl ≤0 are unknown constants, which represent the right slope, left slope, right break-point and left break-point of the dead-zone, respectively. Given the desired smooth trajectory yd(t)=x1d(t), the objective is to design a control input v(t) such that the output y=x1 tracks x1d(t) as closely as possible and all closed loop signals are bounded in the presence of unknown dead-zone, parametric uncertainties and disturbances. Since the control input v(t) is the input of the nonsmooth dead-zone, it could not be designed directly. In order to compensate the dead-zone effect, a smooth inverse proposed in [7] for the dead-zone is integrated in this paper, which could help to prevent the control input signal v(t) from chattering. Let DI(∙) denote the smooth inverse of the dead-zone DZ(∙), which is given as follows: u (t ) ml bl u (t ) mr br v(t ) DI (u (t )) r (u ) l (u) (3) mr ml where Φr(u) and Φl(u) are smooth continuous indicator functions defined as r (u)
eu / eu / , l (u) u / u / e e e eu /
(4)
u /
where ε is a positive constant to be selected. Without loss of generality, let E=1 and consider its effect in the unknown slopes mr and ml. Define the unknown parameter set θd=[θd1, θd2, θd3, θd4]T, where θd1=mr, θd2=mrbr, θd3=ml, θd4=mlbl. Thus, the dead-zone characteristic in (2) can be rewritten as (5) u(t ) dT (t ) where
(t ) [ r (t )v, r (t ), l (t )v, l (t )]T 1 if u 0 1 if u 0 , l (t ) 0 else 0 else
(6)
r (t )
Noting that the vector ω(t) contains the functions χr(t) and χl(t) related to the signal u(t) which is considered unmeasurable, thus it is unavailable in the controller design. Therefore, the smooth continuous indicator functions defined in (4) are utilized to approximate the two functions χr(t) and χl(t) as follows (7) ˆ (t ) [r (v)v, r (v), l (v)v, l (v)]T With above formulations, we first design the actual control input to the plant ud(t) as
ud (t ) ˆdT ˆ (t )
(8)
where ˆd [ˆd1 , ˆd 2 , ˆd 3 , ˆd 4 ]T is the real-time estimate of θd via the designed adaptive laws which will be given later. Then, viewing (3), the control input v(t) could be calculated by v(t ) DI (ud (t )) u (t ) ˆd 2 u (t ) ˆd 4 d r (ud ) d l (ud ) ˆ ˆ d1
(9)
d3
From (5) and (8), the error between u(t) and ud(t) can be shown as u(t ) ud (t ) dT ˆ (t ) d (t )
(10)
where d is the estimation error of θd, and d (t ) dT (ˆ (t ) (t ))
.
Based on (10), the system model can be rewritten as
xi xi 1 T i ( xi , t ) di ( xn , t ),
1 i n 1
xn ud (t ) ˆ n ( xn , t ) d ( xn , t ) T d
T
(11)
where d ( xn , t ) d (t ) dn ( xn , t ) is the lumped disturbance which contains dω(t) and the inherent time-varying disturbance of the nth channel of the original model.
4 Throughout the paper, “•i” represents the ith component of the vector “•” and the operation < for two vectors is performed in terms of the corresponding elements of the vectors. To facilitate the robust adaptive backstepping design, the following lemmas and assumptions are presented. Lemma 1 [7]: With above definition and transformation, the approximation error dω(t) in (10) is bounded for all t≥0 and the upper bound can be given as 1 1 mr br ml bl , v br e mr ml e2br / 1 2 (12) bl v br max{mr , ml } br bl , 1 e1 m m mr br ml bl , v bl r l e2br / 1 2 Furthermore, dω(t)→0 as ˆ and 0 . Lemma 2 [26]: The following inequalities always hold
0 tanh( / a) , R, a 0
(13)
b / (b c) 1, b 0, c 0 or b 0, c 0
(14)
Assumption 1: The desired trajectory yd(t) has bounded derivatives up to nth order, i.e., there exist ξ0i such that for all t,
yd(i ) (t ) 0i , i=1, 2, … , n. Assumption 2: The time-varying disturbances di ( xn , t ) , i=1, 2, … , n are all bounded. In addition, the following inequalities are satisfied: d1 ( xn , t ) 1 , d 2 ( xn , t ) 1 d1 ( xn , t ) 2 , ... , x1 i 1
di ( xn , t )
j 1
i 1 d j ( xn , t ) i , ... , x j
n 1
n 1
d ( xn , t )
j 1
x j
(15)
d j ( xn , t ) n
where αi (i=1, 2, … , n-1) are virtual control laws which will be synthesized later in the backstepping design, and Θi (i=1, 2, ... , n) are unknown constants. Assumption 3: The unknown parameter vectors θ and Θ lie in known bounded sets Ωθ and ΩΘ, respectively, i.e., { : min max } (16) { : min max } where θmin=[θ1min, … , θpmin]T, θmax=[θ1max, … , θpmax]T and Θmin =[Θ1min, … , Θnmin]T, Θmax=[Θ1max, … , Θnmax]T are known. Assumption 4: There exists positive sufficiently smooth and integrable functions δi(t) such that i( j ) (t ) i*, j , (1 i n, 1 j n 1) t
0 where
* i, j
i ( )d i , t 0
(17)
and i are some positive constants.
Remark 1: Assumption 2 is reasonable since the conditions in (15) can be easily satisfied in many practical systems. Actually, the state-related disturbances di contain not only the external disturbances but also the uncertainties arising from the internal uncertainties of the system model. For general mechanical systems, such as hydraulic systems [4-5] and electrical motors [2], the internal uncertainties of the system model are mainly unmodeled nonlinear friction, which is naturally bounded. Remark 2: The functions δi(t), i=1, 2, … , n in Assumption 4 are similar to that of [26]. But besides that being positive integrable in [26], a stronger assumption is made in (17), which requires these functions to have bounded derivatives up to order n-1. However, it is not difficult to obtain these functions. Noting that δi(t) are required to be derivable and have bounded integrals, which implies that they are sufficiently smooth non-increasing functions. For example, the following given smooth function δ(t) indeed satisfies the requirements in (17), a (18) (t ) 2 3 a1t a2 where a1, a2, a3 are arbitrary positive constants.
5 B. Smooth projection mapping and parameter adaptation Let ˆ denote the estimate of θ, and the estimation error (i.e., ˆ ). Let ˆ (ˆ) where π is the vector of bounded smooth projection mappings defined later. Define ˆ as the projected parameter estimation errors. Viewing (16), a smooth projection for the estimated parameter vector ˆ [ˆ , ... , ˆ ]T can be defined as [14] 1
p
( ) [1 (1 ), ... , p ( p )]T T
(19)
T
where υ=[υ1, … , υp] . Let εθ=[εθ1, … , εθp] be a vector of arbitrary small positive real numbers. Then there exists a real-valued, sufficiently smooth nondecreasing function πi such that i (i ) i , i [i min , i max ] (20) i (i ) ˆ [i min i , i max i ], i R i
with bounded derivatives up to order n-1. Thus
(ˆ) ˆ, ˆ (ˆ) ˆ
{ : min max }
{ : min max }, ˆ R p (21)
( j ) (ˆ) j
{ : i c i j }, ˆ R p , 1 j n
where ˆ is a known bounded compact set and j is a bounded compact set; ( j ) (ˆ) R p is a vector and c i j is a positive constant. Define (i ) [ (ˆ), ... , (i ) (ˆ)]T , i=1, … , n-1. Based on (16) and (21), the following defined function is positive definite and proper w. r. t for each θ ∈ Ωθ [29]. p
V ( , )
i
1
i 1
0 [ i (i i ) i ]di , i
i 0
(22)
where Γ=diag{Γ1, … , Γp} is a positive definite diagonal adaptation rate matrix. In addition, the following property will be utilized in the final Lyapunov stability analysis for the closed loop system [29].
1 1 V ( , ) [1 (ˆ1 ) 1 ], ... , [ p (ˆp ) p ] p 1 (23) T 1 ˆ [ ˆ ,..., ˆ ]T has similar definition of smooth projection in (19) and some properties Then, the estimated parameter vector 1 n in (21) , (22) and (23), which are given as follows: Define smooth projection as (24) ( ) [1 (1 ), ... , n ( n )]T
where к=[к1, … , кn]T. Let εΘ=[εΘ1, … , εΘn]T, the following properties hold: ˆ) ˆ , ˆ ( { : min max } ˆ ˆ R n (25) () ˆ { : min max },
ˆ ) ( j ) ( j
ˆ Rn , 1 j n { : i s i j },
ˆ ) Rn is a vector, s is a positive constant, and where ( j ) ( i j n
V (, )
i 1
1
i
i 0
[ i ( i i ) i ]d i , i 0
(26)
is positive definite and proper w. r. t for each Θ∈ ΩΘ. In addition, similar to (23), the following property holds
1 ˆ ) ], ... , 1 [ ( ˆ ) ] V (, ) [1 ( 1 1 n n n 1 (27) T 1 ˆ , ˆ ( ˆ), ˆ . Similarly, we where =diag{γ1, … , γn} is a positive definite adaptation rate matrix, and (i ) (i ) ˆ T ˆ also define [ (), ... , ()] , i=1, … , n-1. Remark 3: The smooth projections defined in (19) and (24) can guarantee that the estimates of uncertain parameters θ and Θ always lie in known bounded sets. More important, they have high order bounded derivatives, which can be used to ensure the boundness of
6 the proposed controller since the derivatives of the virtual control laws are required in the final controller. The specific expression of the projection function π(∙) can be chosen as in [14], which contains an exponential function and has bounded derivatives up to order n-1. III. ROBUST ADAPTIVE BACKSTEPPING CONTROLLER DESIGN AND STABILITY ANALYSIS In this section, robust adaptive control is proposed for the considered uncertain SISO nonlinear system with unknown input dead-zone. Global asymptotic tracking performance and the bounded stability of all signals in the closed loop system are achieved. The design follows the recursive backstepping design procedure [8] due to the existence of various mismatched disturbances in the system model. Step 1: Define error variables z1=x1-x1d and z2=x2-α1, where z1 denotes the output tracking error and α1 is the virtual control law for x2 to be synthesized later. Viewing (11), differentiating z1 with respect to time, we have z1 x1 x1d (28) z2 1 T 1 ( x1 , t ) d1 ( xn , t ) x1d Since the time derivative of the virtual control law at the previous step is needed at the next design step during backstepping design procedure, the resulting control law at each step has to be sufficiently smooth. Hence, by referring the ideas in [26], a sufficiently smooth positive integrable function in Assumption 4 is used to synthesize the control law. The resulting virtual control law α1 is given by ˆ , t) , 1 ( x1 ,ˆ , 1a 1s 1s 1s1 1s 2 T ˆ ˆ (x , , t) x (x , t) 1a
1
1d
1
1
1s1 ( x1 , t ) k1 z1 ˆ2 ks1 z1 1 ˆ , t) 1s 2 ( x1 , z1 ˆ ks1 z1 tanh( ) (t ) 1 (t ) 1 1
(29)
where k1 and ks1 are positive feedback gains. In (29), α1a is a model-based feedforward control law utilized to improve model compensation, α1s is a robust control law in which α1s1 is a linear robust feedback term to stabilize the nominal system and α1s2 is a nonlinear robust term to tackle the disturbance d1 ( xn , t ) . Substituting (29) into (28), we have (30) z1 z2 k1 z1 T 1 ( x1 , t ) 1s 2 d1 ( xn , t ) Consider the Lyapunov function candidate
V1
1 2 c1 z1 2
(31)
where c1 is a positive constant. The time derivative of (31) along with (30) is given by
V1 k1c1 z12 c1 z1 z2 c1T 1 ( x1 , t ) z1 c1 z11s 2 c1 z1d1 ( xn , t )
(32)
Invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have V1 k1c1 z12 c1 z1 z2 c1T 1 ( x1 , t ) z1 c1 z1 1
ˆ2 ks1c1 z12 1 z1 ˆ ks1 z1 tanh( ) (t ) 1 (t ) 1 1
k1c1 z12
c1 z1 z2 c1 1 ( x1 , t ) z1 c1 z1 1
ˆ2 ks1c1 z12 1 ˆ (t ) ks1 z1 1 1
Step 2: Define the error variable z3=x3-α2. Viewing (11) and differentiating z2 with respect to time, we have z2 x2 1
z3 2 T 2 ( x2 , t ) d2 ( xn , t ) 1c 1u where
(33)
T
(34)
7
ˆ , t) 1 ( x1 , ˆ ,
1 1 1 ˆ x1 1 ˆ ˆ ˆ t x1
1c 1u
1c
1 1 1 ˆ (35) [ x2 ˆT 1 ( x1 , t )] 1 ˆ ˆ ˆ t x1
1u
1 [T 1 ( x1 , t ) d1 ( xn , t )] x1
In (35), 1c represents the calculable part of 1 and can be used in the control law design while 1u is the incalculable part due to the existence of time-varying disturbance and has to be handled by the robust feedback term. For the error dynamic in (34), the resulting virtual control law is given by
2 ( x2 , (1) , (1) , t ) 2 a 2 s , 2 s 2 s1 2 s 2 2 a ( x2 , (1) , t ) 1c ˆT 2 ( x2 , t ) 2 s1 ( x2 , t ) k2 z2 ˆ2 k s 2 z2 2 ˆ , t) 2 s 2 ( x2 , z2 ˆ ks 2 z2 tanh( ) 2 (t ) 2 (t ) 2
(36)
where k2 and ks2 are positive feedback gains. In (36), α2a is a model-based feedforward control law utilized to improve model compensation, α2s is a robust control law in which α2s1 is a linear robust feedback term to stabilize the nominal system and α2s2 is a nonlinear robust term to tackle the disturbance d2 ( xn , t ) . Substituting (36) into (34), we have z2 z3 k2 z2 T 2 ( x2 , t ) 2 s 2 d 2 ( xn , t ) 1u
z3 k2 z2 T 2 ( x2 , t ) 2 s 2 d 2 ( xn , t )
(37)
1 T [ 1 ( x1 , t ) d1 ( xn , t )] x1
Consider the Lyapunov function candidate
1 V2 V1 c2 z22 2
(38)
where c2 is a positive constant. The time derivative of (38) along with (37) is given by
V2 V1 k2 c2 z22 c2 z2 z3 c2 z2 2 s 2 c2T 2 ( x2 , t ) z2 c2 z2 d 2 ( x2 , t ) c2 z2
(39) 1 T [ 1 ( x1 , t ) d1 ( xn , t )] x1
Invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have V2 V1 k2 c2 z22 c2 z2 z3 T {c2 [2 ( x2 , t )
ˆ2 (40) ks 2 c2 z22 1 2 1 ( x1 , t )]}z2 c2 z2 2 ˆ (t ) x1 k s 2 z2 2 2 Step i (1≤ i ≤ n-1): Define the error variable zi+1=xi+1-αi. Viewing (11) and differentiating zi with respect to time, we have zi zi 1 i T i ( xi , t ) di ( xn , t ) (i 1)c (i 1)u (41) where
8
i 1 ( xi 1 , (i 2) , (i 2) , t )
i 1 i 1 i 1 xj j 1 x t j
i 1 ˆ i 1 ˆ (i 1) c (i 1)u ˆ ˆ
(i 1) c
i 1 i 1 i 1 [ x j 1 ˆT j ( x j , t )] j 1 x t j
(42)
i 1 ˆ i 1 ˆ ˆ ˆ
i 1
(i 1)u
j 1
i 1 [T j ( x j , t ) d j ( xn , t )] (2 i n) x j
In (42), (i 1)c represents the calculable part of i 1 and can be used in the control law design while (i 1)u is the incalculable part due to the existence of time-varying disturbance and has to be handled by the robust feedback term. For (41), the resulting virtual control law is given by i ( xi , (i 1) , (i 1) , t ) ia is , is is1 is 2 ( x , (i 1) , t ) ˆT ( x , t ) ia
i
( i 1) c
i
i
is1 ( xi , t ) ki zi ˆ2 ksi zi i is 2 ( xi , (i 1) , t ) zi ˆ ksi zi tanh( ) i (t ) i (t ) i
(43)
where ki and ksi are positive feedback gains. In (43), αia is a model-based feedforward control law utilized to improve model compensation, αis is a robust control law in which αis1 is a linear robust feedback term to stabilize the nominal system and αis2 is a nonlinear robust term to tackle the disturbance di ( xn , t ) . Substituting (43) into (41), we have zi zi 1 ki zi T j ( x j , t ) is 2 di ( xn , t ) (i 1)u zi 1 ki zi T j ( x j , t ) is 2 di ( xn , t ) i 1
j 1
(44)
i 1 T [ j ( x j , t ) d j ( xn , t )] x j
Consider the Lyapunov function candidate
1 Vi Vi 1 ci zi2 2
(45)
where ci is a positive constant. Differentiating (45) along with (44) and invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have Vi Vi 1 ki ci zi2 ci zi zi 1 ci zi is 2 ciT i ( xi , t ) zi
i 1
i 1
ci zi di ( xn , t ) ci zi
x j
j
[T j ( x j , t ) d j ( xn , t )]
Vi 1 ki ci zi2 ci zi zi 1 ci zi i
i 1
i 1
T {ci [i ( xi , t ) j
x j
ˆ2 (46) ksi ci zi2 i ˆ ksi zi i i (t )
j ( x j , t )]}zi
Step n: The time derivative of zn is given by
zn ud Td ˆ T n ( xn , t ) d ( xn , t ) (n 1)c (n 1)u (47) Based on (42), the actual control input ud is synthesized as follows
9
ud uda uds , uds uds1 uds 2 u ˆT ( x , t ) ( n 1) c
da
n
n
uds1 kn zn uds 2
ˆ2 ksn zn n zn ˆ ksn zn tanh( ) n (t ) n (t ) n
(48)
where kn and ksn are positive feedback gains. In (48), uda is a model-based feedforward control law utilized to improve model compensation, uds is a robust control law in which uds1 is a linear robust feedback term to stabilize the nominal system and uds2 is a nonlinear robust term to tackle the disturbance d ( xn , t ) . Substituting (48) into (47), we have zn kn zn dT ˆ T n ( xn , t ) uds 2 d ( xn , t ) ( n 1)u
kn zn dT ˆ T n ( xn , t ) uds 2 d ( xn , t )
n 1
n 1
j 1
x j
(49)
[T j ( x j , t ) d j ( xn , t )]
Theorem 1: Provided that the full state information is available, with the following adaptation laws n
i 1
i 2
j 1
ˆ {c11 ( x1 , t ) z1 ci [i ( xi , t ) ˆ c z , i
i 1 j ( x j , t )]zi } (50) x j
z [ z1 , z2 ,... zn ]T
ˆd cnˆ zn
(51) (52)
where M is a positive definite diagonal adaptation rate matrix, and selecting feedback gains k1, … , kn large enough and proper c1, … , cn and ks1, … , ksn such that the matrix Λ defined as follows is positive definite, c1 / 2 0 0 k1c1 c / 2 k c c2 / 2 2 2 1 (53) 0 c2 / 2 k3c3 0 cn / 2 0 0 c / 2 kn cn n then the proposed control law (9) and (48) guarantees that all signals in the closed loop system are bounded, and global asymptotic output tracking performance is also obtained, i.e., z1→0 as t→∞. ◊ Proof : Consider the Lyapunov function candidate 1 1 Vn Vn 1 cn zn2 V ( , ) V (, ) dT 1d (54) 2 2 Differentiating (54) along with (49) and invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have
Vn Vn 1 kn cn zn2 cn zn uds 2 cn dT ˆ zn cn T n ( xn , t ) zn cn zn d ( xn , t )
n 1
n 1
cn zn
j 1
x j
[ T j ( x j , t ) d j ( xn , t )]
ˆ T 1ˆ T 1ˆ T 1 d d 2ˆ2 k sn zn n Vn 1 kn cn zn2 c z ˆ (t ) n n n k sn zn n n
n 1
n 1
T {cn [n ( xn , t )
j 1
x j
j ( x j , t )]}zn
ˆ T 1ˆ cn dT ˆ zn T 1ˆ T 1 d d
(55)
10 Considering (46) and the adaptation law in (50), (51) and (52), we obtain n
n 1
n
i 1
i 1
i 1
ˆ2 k si ci zi2 i ˆ (t ) ksi zi i i
Vn ki ci zi2 ci zi zi 1 [
ci zi i ] cn dT ˆ zn T {c11 ( x1 , t ) z1 n
i 1
i 2
j 1
ci [i ( xi , t )
i 1 j ( x j , t )]zi } T 1ˆ x j
ˆ T 1ˆ T 1 d d
(56)
n
n 1
n
i 1
i 1
i 1
ˆ2 k si ci zi2 i
ki ci zi2 ci zi zi 1 [
ˆ (t ) k si zi i i
ˆ ] ci zi i ˆ (t ) ci zi i i ˆ (t ) i 1 k z n
Z T Z
si
i
i
i
where Z is defined as Z=[z1, … , zn]T. Invoking the inequality in (14), noting that the matrix Λ defined in (53) is positive definite, we have n c (t ) Vn Z T Z i i i 1 k si
2 min ( )( z1
...
zn2 )
n c (t ) c (t ) i i W i i i 1 k i 1 k si si n
(57)
where W min ()( z12 ... zn2 ) is a positive function and λmin(Λ) is the minimal eigenvalue of matrix Λ. Integrating (57) with respect to time, we have t
n
Vn (t ) W ( )d Vn (0) 0
ci
i 1 k
t
0 i ( )d si
n
c Vn (0) i i i 1 k si
(58)
It can be concluded from (58) that Vn(t) ∈ L∞ and W∈ L2, which implies z1, … , zn and , , d are bounded. From Assumption 1, we can infer that all system states x1, … , xn are bounded, and with Assumption 4 we know that the control input v(t) is bounded. Therefore, all the signals in the closed loop system are bounded. In addition, viewing (30), (37), (44) and (49), we know that z1 , ... , zn are bounded, which leads to the boundness of W , thus W is uniformly continuous. By applying Babalat’s lemma [8], W→0 as t→∞, which leads to the results in Theorem 1. Remark 4: Results of Theorem 1 indicate that the proposed strategy can ensure that all the signals in the closed loop system are bounded and the output tracking error converges to zero asymptotically, and the converging rate can be enhanced by increasing the control gains k1, … , kn and ks1, … , ksn. Such a theoretical result of the proposed robust adaptive backstepping controller is strongly expected in many practical systems subject to unknown input dead-zone nonlinearity, parametric uncertainties as well as disturbances. IV. COMPARATIVE SIMULATION RESULTS In this section, we will present two examples to demonstrate the effectiveness of the proposed control strategy. Example 1: we firstly consider the uncertain nonlinear system subject to unknown input dead-zone as in [7, 26] 1 e x (59) x bu (v) a d (t ) 1 e x where u is output of the dead-zone and d(t) represents disturbance. To simplify the simulation process, we consider the symmtric input dead-zone here and assume that the slopes of the dead-zone are known, i.e., mr=ml=1, and b=1 are kown. But the breakpoints are unknown, and the actual values of unknown parameters are a=1, br=-bl=1. The disturbance is set as d(t)=0.6sint. Define θd=br, θ=a, and Θ as the bound of the lumped disturbance. The desired trajectory is a sinusoidal-like curve given by yd(t)=arctan(sint)[1-exp(-0.01t3)]. The desired trajectory is shown in Fig. 1. Four controllers are compared to verify the effectiveness of the proposed controller in this paper.
11 1) RABC: This is the robust adaptive backstepping controller proposed in this paper. The control gains are given as follows: k1=100, c1=1, ks1=1. And choosing ε=0.01, δ(t)=5000/(t2+1). The bounds of uncertain rangs are given by θmax=5, Θmax=30, θmin=0, Θmin=0. ˆ (0) 0 . Paramter adaptation rates are set The initial estimates of the unknown parameters are chosen as: ˆ(0) 0.5 , ˆ (0) 0.5 , d
as: Γ=500, γ=500, and M=100. 2) RAC: This is the robust adaptive controller proposed in [26] to tackle unknown dead-zone and achieve asymptotic tracking performance. The dead-zone characteristics and model compensation are not considered in this controller design [26]. Hence, it is used here to verify the effectiveness of the compensation of dead-zone and parametric uncertainties in the proposed RABC controller. The robust adaptive controller in [26] with adaptation of ˆ has the following form:
1 ˆ2 2 z1 f ( x, t ) 2 v(t ) 1 1 z1 tanh[ z1 / (t )]ˆ f ( x, t ) (t ) 2 2 1 ˆ z1 f ( x, t ) 2
(60)
(61)
where
f ( x, t ) (
1 e x 2 ) h1 z12 h 1 x 1 e
d 0 D sup yd , max{ a , d 0 }
(62)
t 0
D sup D(t ) , D(t ) bd1 (v) d (t ) t 0
For RAC, the control parameters are chosen as h1=2, h=2, α=6, σ(t)= 5000/(t2+1), and ˆ (0) 0 . 3) ABC: This is an adaptive backstepping controller same as the RABC controller but without robust control law. This ABC controller is similar to that in [25]. It is compared to demonstrate the effectiveness of the robust feedback term proposed in this paper. The control parameters of ABC controller are the same as the corresponding parameters in RABC. 4) AVSC: This is the adaptive variable structure controller proposed in [7]. The dead-zone inverse and model compensation are also considered in this controller design. The actual control input to the plant ud can be given as follows:
ud
1 u b
u (l1 1)( z1 1 ) sg ( z1 ) aˆ
1 e x ˆ (z ) x1d Dsg 1 1 e x
(63)
and the parameter adaptation laws:
ˆ a
1 e x ( z1 1 ) f1sg ( z1 ) 1 e x
ˆd b (l (v) r (v))( z1 1 ) f1sg ( z1 )
(64)
Dˆ d ( z1 1 ) f1sg ( z1 ) where Dˆ is the estimate of D, which denotes the upper bound of the lumped disturbance, and
z1 z1 1 z1 1 f1 , sg ( z1 ) z13 0 z1 1 ( 2 z 2 )2 z 3 1 1 1 -4 The control parameters of ASVC controller are chosen as: l1=100, Γa=200, Γb=40, Γd=30, and ρ1=5×10 .
z1 1 (65)
z1 1
The comparative tracking errors of the four controllers are shown in Fig. 2 and Fig. 3. As shown, the proposed RABC controller achieves the best tracking performance in both transient and steady state. The transient tracking error of RAC is rather large because of lacking model compemsation. And without the robust feedback term to attenuate the effect of the disturbance, the ABC controller has larger trakcing error than that of RABC. The AVSC controller can only guarantee uniformly bounded tracking performance and the steady state tracking error is dominated by the selecting of ρ1. From Fig. 3, we can see that the tracking error of ASVC becomes
12 to present chattering since the value of ρ1 has been selected as small as possible. The parameter estimation and control input of RABC are shown in Fig. 4 and 5, respectively. As seen, the designed control input v is continuous, which is easier to be implemented in pracitce. 0.8 0.6
Desired Trajectory
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
0
10
20
30
40
50
60
70
80
times(s) Fig. 1 The desired trajectory 1 0 -1
0
-0.01
0
20
40
0.01
0
10
AVSC
RAC
0.01
20
60
-0.01
80
0
20
40
60
80
0
20
40
60
80
0.01
RABC
ABC
0.01
0
-0.01
0
0
20
40
60
80
0
-0.01
times(s)
Fig. 2 Comparative tracking errors of four controllers -3
1
x 10
RABC ABC AVSC RAC
Tracking Error
0.5
0
-0.5
-1 70
72
74
76
times(s)
Fig. 3 Tracking errors of four controllers during the last ten minitues
78
80
13 2
d
1 0
0
10
20
30
40
50
60
70
80
60
70
80
60
70
80
70
80
4 2 0
0
10
20
30
40
50
40
Θ 20 0
0
10
20
30
40
50
times(s)
Fig. 4 Parameter estimation of RABC 3
2
Control Input
1
0
-1
-2
-3
0
10
20
30
40
50
60
times(s)
Fig. 5 Control input of RABC
Example 2: Consider a class of nonlinear mechanical systems having the following form: x1 x2
(66) x2 u (v) x2 d ( x, t ) where u denotes the output of the unknown dead-zone nonlinearity. It is worth to note that the system model (66) can be used to describe many mechanical systems such as motor driven system [1, 2, 17, 21, 25], active suspensions [16] etc. Similar to example 1, we also assume that the slopes of the dead-zone are known, i.e., mr=ml=1. The actual values of the unknown breakpoints of the dead-zone are chosen as br=5 and bl=-3. And the actual values of is chosen as =3. The disturbance d(x,t)=0.5sint and the desired trajectory is the same as in example 1. To verify the effectiveness of the inverse dead-zone compensation, the following three controllers are compared. 1) RABC: The control gains are given as k1=200, k2=20, ks2=1, c2=1. And choosing ε=0.1, δ(t)=5000/(t2+1). The bounds of uncertain rangs are given by max 10 , Θmax=60, min 0 , Θmin=0. The initial estimates of parameters are chosen as: ˆ (0) 0 . Paramter adaptation rates are set as: Γ=80, γ=diag{15, 7}, and M=50. mr br (0) 0 , ml bl (0) 0 , ˆ(0) 0 and
2) RAC: For (66), the robust adaptive controller in [26] with adaptation of ˆ has the following form:
14
v(t )
( z1 / 2 z2 ) ˆ 2 f ( x, t )
( z1 / 2 z2 ) tanh[( z1 / 2 z2 ) / (t )]ˆ (t ) (67)
ˆ f ( x, t ) z1 / 2 z2 where
z2 x2 x1d
f ( x, t ) x22 h2 z12 z22 h 1 The control parameters are chosen as h2=2, h=2, α=2, σ(t)= 5000/(t2+1), and the initial estimation of β is set at zero.
(68)
3) RABC1: This is the robust adaptive backstepping controller proposed in this paper but without compensating the unknown dead-zone. The corrosponding control parameters are the same as that of RABC. The output tracking performance of three controllers are given in Fig. 6 and Fig. 7, respectively. The results show that the tracking error of RAC is much lager than that of the other two controllers. Furthermore, the proposed RABC controller outperforms the RABC1 controller in terms of transient and final tracking errors. This indicates that the dead-zone inverse compensation in the proposed RABC controller can apparently improve the tracking performance, and is more effective than the robust method which deals with the dead-zone by simplifying its model as a linear term and a disturbance term, in this severe input dead-zone case. The parameter estimation and control input of RABC are presented in Fig. 8 and 9, respectively.
0.7
Tracking Error of RAC
0.6 0.5 0.4 -3
x 10
0.3
2
0.2
0
0.1
-2 50
60
70
80
0 -0.1
0
10
20
30
40
50
60
70
80
times(s)
Fig. 6 Tracking error of RAC
-3
1.5
x 10
-5
2
1
x 10
0
Tracking Error
0.5 -2 50
60
70
80
0
-0.5
RABC RABC1
-1
-1.5 0
10
20
30
40
50
60
times(s)
Fig. 7 Tracking errors of RABC (with inverse compensation) and RABC1 (without inverse compensation)
70
80
15 5 mrb r
0 -5
0
10
20
30
40
50
60
70
80
60
70
80
60
70
80
60
70
80
60
70
80
0 mlb l
-2 -4
0
10
20
30
40
50
5 0 -5
0
10
20
30
40
50
50
0
0
10
20
30
40
50
times(s)
Fig.8 Parameter estimation of RABC 10 8 6
Control Input
4 2 0 -2 -4 -6 -8
0
10
20
30
40
50
times(s) Fig. 9 Control input of RABC
V. CONCLUSION In this paper, a robust adaptive backstepping controller is proposed for a class of uncertain SISO nonlinear strict-feedback systems subject to unknown input dead-zone nonlinearity, parametric uncertainties and unknown bounded disturbances. A smooth dead-zone inverse is constructed to compensate the effect of input dead-zone, and the adaptation laws are synthesized via backstepping recursive design technique to tackle parametric uncertainties. Disturbances are handled by novel robust control laws which derived by incorporating a sufficiently smooth positive integral function into the controller design at each step when backstepping. A smooth projection mapping is used to facilitate the backstepping design since differential operation on virtual control law is needed. The proposed robust adaptive backstepping controller can theoretically guarantee that all closed loop signals are bounded and the output tracking error asymptotically converges to zero, meanwhile ensuring the continuity of the control input. Comparative simulation results are obtained to illustrate the effectiveness of the proposed scheme. REFERENCES [1] [2] [3]
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PII: DOI: Reference:
S0016-0032(15)00360-9 http://dx.doi.org/10.1016/j.jfranklin.2015.09.013 FI2447
To appear in: Journal of the Franklin Institute Received date: 26 January 2015 Revised date: 12 August 2015 Accepted date: 24 September 2015 Cite this article as: Wenxiang Deng, Jianyong Yao and Dawei Ma, Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.09.013 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.
1
Robust Adaptive Asymptotic Tracking Control of A Class of Nonlinear Systems with Unknown Input Dead-Zone Wenxiang Deng, Jianyong Yao*, Dawei Ma Abstract—This paper considers the tracking control for a class of uncertain single-input and single-output (SISO) nonlinear strict-feedback systems with unknown input dead-zone nonlinearity, parametric uncertainties and unknown bounded disturbances. By constructing a smooth dead-zone inverse and applying the backstepping recursive design technique, a robust adaptive backstepping controller is proposed, in which adaptive control law is synthesized to handle parametric uncertainties and a novel robust control law to attenuate disturbances. The robust control law is developed by integrating a sufficiently smooth positive integral function at each step of the backstepping design procedure. In addition, a smooth projection mapping is used and assumptions are made that the prior knowledge of the extents of parametric uncertainties and the variation ranges of the bounds of disturbances is known to facilitate the backstepping recursive design. However, the exact bounds of disturbances are not required. The major feature of the proposed controller is that it can theoretically guarantee asymptotic output tracking performance, in spite of the presence of unknown input dead-zone nonlinearity, various parametric uncertainties and unknown bounded disturbances via Lyapunov stability analysis. Comparative simulation results are obtained to illustrate the effectiveness of the proposed control strategy. Index Terms—dead-zone inverse; robust control; adaptive control; backstepping; smooth projection mapping; disturbance suppression
I. INTRODUCTION
C
ontrol of nonlinear systems is of great significance since the majority of practical systems exhibit nonlinear dynamic behaviors, such as the dynamic behaviors of electrical motors [1-2], robot manipulators [3], hydraulic systems [4-5], and so on. However, these systems are typically subjected to various parametric uncertainties and disturbances, which could severely deteriorate the achievable control performance, leading to undesirable control accuracy, limit cycles and even instability [6]. In order to handle parametric uncertainties and disturbances in nonlinear systems and improve the tracking performance, extensive research has been devoted to the design of high performance nonlinear controller. During the past several decades, adaptive control design for nonlinear systems to achieve globally asymptotic stability of the closed loop system has been proposed and undergone rapid developments, with plenty of publications presented, such as [8-10]. Specially, under the assumption that the nonlinear systems are subjected to parametric uncertainties only, systematic approaches of adaptive control were proposed in [9], which can achieve asymptotic output tracking performance. And the inherent overparameterization problem was overcome by introducing the idea of tuning function in [10]. In fact, no matter how accurate the practical system model and parameter identification are, modeling errors and external disturbances always exist in physical systems. Hence, some robust adaptive tracking controllers were developed for nonlinear systems with consideration of parametric uncertainties and disturbances simultaneously in [11-13]. In addition, an adaptive robust control (ARC) strategy was proposed by Yao et al. in [14] for uncertain nonlinear systems in the presence of various modeling uncertainties. As an effective control strategy, adaptive robust control has been widely employed in many applications [15-18]. However, most of the existing results in the literature can only achieve bounded-error trajectory tracking performance rather than the strongly expected asymptotic tracking with zero steady-state error in the presence of parametric uncertainties and disturbances. To cope with this issue, adaptive sliding mode control was presented for uncertain nonlinear systems with perturbations in [19], which assures that the tracking error converges to zero asymptotically. It is worth to note that the discontinuous function contained in the controller [19] may cause severe chattering, which limits the employment of the controller in engineering practice. In [21, 22], adaptive control design was integrated with a novel robust integral of the sign of the error (RISE) control proposed in [20] to obtain asymptotic tracking performance, meanwhile ensuring the continuity of the final control input. However, all these RISE based adaptive controllers can *Corrosponding author, phone: 86-25-84315125; fax: 86-25-84303248; e-mail: jerryyao.buaa@ gmail.com; Manuscript received xxx xx, 2015. This work was supported in part by the National Nature Science Foundation of China (Grant no. 51305203), in part by Project funded by China Postdoctoral Science Foundation (Grant no. 2015T80553), and in part by the Natural Science Foundation of Jiangsu Province (Grant no. BK20141402). Wenxiang Deng, Jianyong Yao and Dawei Ma are with the School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]; e-mail: jerryyao.buaa@ gmail.com; email: [email protected]).
2 only tackle matched disturbances while many practical systems indeed contain the so-called mismatched disturbances. Therefore, new approaches need to be developed to handle the asymptotic tracking problem of uncertain nonlinear systems with parametric uncertainties, matched and mismatched disturbances. Besides the above discussed parametric uncertainties, matched and mismatched disturbances, input dead-zone nonlinearity also usually exists in physical applications, such as hydraulic servo valve, mechanical transmissions, DC servo motors, and other devices. Dead-zone is often a source of instability and performance deterioration [27], and hence needs more attention to handle it. To effectively attenuate the effect of dead-zone nonlinearity, a dead-zone inverse model in conjunction with adaptive control was first proposed in [23] for linear systems with unmeasured dead-zone outputs. However, the controller in [23] can only guarantee the convergence of the tracking error to a residual bounded set. Asymptotic output tracking performance was derived in [24] under the assumption that the dead-zone output is measurable, which might be unreasonable in practice. Recently, an integrated direct/indirect adaptive robust control scheme was developed in [25] for a class of nonlinear systems with unknown dead-zone. By constructing a novel dead-zone inverse and making full use of the fact that the unknown dead-zone can be linearly parameterized perfectly within certain known intervals, indirect adaptation laws are designed within every interval to obtain accurate parameter estimation of the unknown dead-zone and then achieve perfect dead-zone compensation in the design of adaptive robust controller. However, all aforementioned approaches use discontinuous dead-zone inverse, which may cause unavoidable chattering of the control input. Hence, a smooth dead-zone inverse was introduced in [7], and utilized in the development of an adaptive output feedback backstepping controller for a class of uncertain nonlinear system preceded by unknown dead-zone to guarantee uniformly bounded tracking performance. Another method to cope with the attenuation problem of input dead-zone is to model the dead-zone as a time-varying disturbance-like term, examples like in [26-28, 30]. With this formulation, many robust control laws can be utilized to handle the disturbance-like term, such as sliding mode control [31] and RISE feedback control [1, 4, 20-22]. In [30], a feasible adaptive controller was proposed to tackle the tracking control of nonlinear systems with unknown input dead-zone and time delays. And asymptotic tracking performance was obtained in [26] by applying a novel robust adaptive controller. However, the dead-zone characteristics are not explicitly considered in those controllers, which might lead to not precise enough tracking performance, such as the steady-state performance [7], especially when facing severe input dead-zone. Moreover, high gain feedback problem may also be caused by treating the dead-zone as a disturbance-like term if the nonlinear system is subjected to sever input dead-zone, i.e., large slopes or breakpoints. In viewing the above observations, the control design for uncertain nonlinear systems with unknown dead-zone nonlinearity, parametric uncertainties and disturbances deserves further work to improve the tracking performance. In this paper, a robust adaptive backstepping control strategy is proposed for a class of uncertain SISO nonlinear strict-feedback systems which contain unknown input dead-zone, parametric uncertainties as well as unknown bounded disturbances. In the controller design, a smooth dead-zone inverse in [7] is referred to compensate the effect of input dead-zone, with expecting to effectively compensate severe input dead-zone effects. The adaptation laws are synthesized via backstepping recursive design technique to obtain parameter estimation of unknown dead-zone parameters, other unknown system parameters and the bounds of terms corresponding to disturbances. And disturbances are handled by novel robust control laws which are derived by incorporating a sufficiently smooth positive integral function into the controller design at each step when backstepping. A smooth projection mapping is used to facilitate the backstepping design since differential operation on virtual control law is needed. The main contributions of this paper are given as follows: (i) The proposed control approach does not require exact bounds of the unknown disturbances, instead, they are estimated by adaptive laws; (ii) the robust control concept in [26] are integrated and extended into a model-based adaptive compensation case for a class of uncertain SISO nonlinear systems with unknown input dead-zone, parametric uncertainties, and unknown bounded disturbances via backstepping method; (iii) to the best of our knowledge, it may be the first time in the literature that the asymptotic tracking performance is achieved for the nonlinear system considered in this paper meanwhile with a continuous control input. This paper is arranged as follows. Problem statement and some preliminaries are presented in Section II. Section III gives the robust adaptive backstepping controller design procedure and its global stability analysis. Comparative simulation results are obtained in Section IV. And some conclusions can be found in Section V. II. PROBLEM STATEMENT AND PRELIMINARIES A. Problem statement In this paper, we consider a class of uncertain SISO nonlinear strict-feedback systems with unknown input dead-zone given as follows: xi xi 1 T i ( xi , t ) di ( xn , t ), 1 i n 1
xn Eu (v) T n ( xn , t ) d n ( xn , t )
(1)
y x1 where x=[x1, x2, … , xn] ∈ R is the state vector and y is the system output, xi [ x1 , x2 , ... , xi ]T ∈ Ri is the partial state vector, θ ∈ Rp is an unknown constant parameter vector, φi ∈ Rp , (i=1, 2, … , n), are known nonlinear functions, di, (i=1, 2, … , n), are unknown T
n
3 time-varying disturbances, v is the control signal to be designed and also the input of the dead-zone, u(v) is the output of the dead-zone nonlinearity and also actual control input to the plant, E≠0 represents an unknown constant gain. The dead-zone characteristic can be described as follows [23]: v br mr (v br ) (2) u (v) DZ (v) 0 bl v br m (v b ) v bl l l where DZ(∙) represents the considered dead-zone; mr >0, ml >0, br ≥0 and bl ≤0 are unknown constants, which represent the right slope, left slope, right break-point and left break-point of the dead-zone, respectively. Given the desired smooth trajectory yd(t)=x1d(t), the objective is to design a control input v(t) such that the output y=x1 tracks x1d(t) as closely as possible and all closed loop signals are bounded in the presence of unknown dead-zone, parametric uncertainties and disturbances. Since the control input v(t) is the input of the nonsmooth dead-zone, it could not be designed directly. In order to compensate the dead-zone effect, a smooth inverse proposed in [7] for the dead-zone is integrated in this paper, which could help to prevent the control input signal v(t) from chattering. Let DI(∙) denote the smooth inverse of the dead-zone DZ(∙), which is given as follows: u (t ) ml bl u (t ) mr br v(t ) DI (u (t )) r (u ) l (u) (3) mr ml where Φr(u) and Φl(u) are smooth continuous indicator functions defined as r (u)
eu / eu / , l (u) u / u / e e e eu /
(4)
u /
where ε is a positive constant to be selected. Without loss of generality, let E=1 and consider its effect in the unknown slopes mr and ml. Define the unknown parameter set θd=[θd1, θd2, θd3, θd4]T, where θd1=mr, θd2=mrbr, θd3=ml, θd4=mlbl. Thus, the dead-zone characteristic in (2) can be rewritten as (5) u(t ) dT (t ) where
(t ) [ r (t )v, r (t ), l (t )v, l (t )]T 1 if u 0 1 if u 0 , l (t ) 0 else 0 else
(6)
r (t )
Noting that the vector ω(t) contains the functions χr(t) and χl(t) related to the signal u(t) which is considered unmeasurable, thus it is unavailable in the controller design. Therefore, the smooth continuous indicator functions defined in (4) are utilized to approximate the two functions χr(t) and χl(t) as follows (7) ˆ (t ) [r (v)v, r (v), l (v)v, l (v)]T With above formulations, we first design the actual control input to the plant ud(t) as
ud (t ) ˆdT ˆ (t )
(8)
where ˆd [ˆd1 , ˆd 2 , ˆd 3 , ˆd 4 ]T is the real-time estimate of θd via the designed adaptive laws which will be given later. Then, viewing (3), the control input v(t) could be calculated by v(t ) DI (ud (t )) u (t ) ˆd 2 u (t ) ˆd 4 d r (ud ) d l (ud ) ˆ ˆ d1
(9)
d3
From (5) and (8), the error between u(t) and ud(t) can be shown as u(t ) ud (t ) dT ˆ (t ) d (t )
(10)
where d is the estimation error of θd, and d (t ) dT (ˆ (t ) (t ))
.
Based on (10), the system model can be rewritten as
xi xi 1 T i ( xi , t ) di ( xn , t ),
1 i n 1
xn ud (t ) ˆ n ( xn , t ) d ( xn , t ) T d
T
(11)
where d ( xn , t ) d (t ) dn ( xn , t ) is the lumped disturbance which contains dω(t) and the inherent time-varying disturbance of the nth channel of the original model.
4 Throughout the paper, “•i” represents the ith component of the vector “•” and the operation < for two vectors is performed in terms of the corresponding elements of the vectors. To facilitate the robust adaptive backstepping design, the following lemmas and assumptions are presented. Lemma 1 [7]: With above definition and transformation, the approximation error dω(t) in (10) is bounded for all t≥0 and the upper bound can be given as 1 1 mr br ml bl , v br e mr ml e2br / 1 2 (12) bl v br max{mr , ml } br bl , 1 e1 m m mr br ml bl , v bl r l e2br / 1 2 Furthermore, dω(t)→0 as ˆ and 0 . Lemma 2 [26]: The following inequalities always hold
0 tanh( / a) , R, a 0
(13)
b / (b c) 1, b 0, c 0 or b 0, c 0
(14)
Assumption 1: The desired trajectory yd(t) has bounded derivatives up to nth order, i.e., there exist ξ0i such that for all t,
yd(i ) (t ) 0i , i=1, 2, … , n. Assumption 2: The time-varying disturbances di ( xn , t ) , i=1, 2, … , n are all bounded. In addition, the following inequalities are satisfied: d1 ( xn , t ) 1 , d 2 ( xn , t ) 1 d1 ( xn , t ) 2 , ... , x1 i 1
di ( xn , t )
j 1
i 1 d j ( xn , t ) i , ... , x j
n 1
n 1
d ( xn , t )
j 1
x j
(15)
d j ( xn , t ) n
where αi (i=1, 2, … , n-1) are virtual control laws which will be synthesized later in the backstepping design, and Θi (i=1, 2, ... , n) are unknown constants. Assumption 3: The unknown parameter vectors θ and Θ lie in known bounded sets Ωθ and ΩΘ, respectively, i.e., { : min max } (16) { : min max } where θmin=[θ1min, … , θpmin]T, θmax=[θ1max, … , θpmax]T and Θmin =[Θ1min, … , Θnmin]T, Θmax=[Θ1max, … , Θnmax]T are known. Assumption 4: There exists positive sufficiently smooth and integrable functions δi(t) such that i( j ) (t ) i*, j , (1 i n, 1 j n 1) t
0 where
* i, j
i ( )d i , t 0
(17)
and i are some positive constants.
Remark 1: Assumption 2 is reasonable since the conditions in (15) can be easily satisfied in many practical systems. Actually, the state-related disturbances di contain not only the external disturbances but also the uncertainties arising from the internal uncertainties of the system model. For general mechanical systems, such as hydraulic systems [4-5] and electrical motors [2], the internal uncertainties of the system model are mainly unmodeled nonlinear friction, which is naturally bounded. Remark 2: The functions δi(t), i=1, 2, … , n in Assumption 4 are similar to that of [26]. But besides that being positive integrable in [26], a stronger assumption is made in (17), which requires these functions to have bounded derivatives up to order n-1. However, it is not difficult to obtain these functions. Noting that δi(t) are required to be derivable and have bounded integrals, which implies that they are sufficiently smooth non-increasing functions. For example, the following given smooth function δ(t) indeed satisfies the requirements in (17), a (18) (t ) 2 3 a1t a2 where a1, a2, a3 are arbitrary positive constants.
5 B. Smooth projection mapping and parameter adaptation Let ˆ denote the estimate of θ, and the estimation error (i.e., ˆ ). Let ˆ (ˆ) where π is the vector of bounded smooth projection mappings defined later. Define ˆ as the projected parameter estimation errors. Viewing (16), a smooth projection for the estimated parameter vector ˆ [ˆ , ... , ˆ ]T can be defined as [14] 1
p
( ) [1 (1 ), ... , p ( p )]T T
(19)
T
where υ=[υ1, … , υp] . Let εθ=[εθ1, … , εθp] be a vector of arbitrary small positive real numbers. Then there exists a real-valued, sufficiently smooth nondecreasing function πi such that i (i ) i , i [i min , i max ] (20) i (i ) ˆ [i min i , i max i ], i R i
with bounded derivatives up to order n-1. Thus
(ˆ) ˆ, ˆ (ˆ) ˆ
{ : min max }
{ : min max }, ˆ R p (21)
( j ) (ˆ) j
{ : i c i j }, ˆ R p , 1 j n
where ˆ is a known bounded compact set and j is a bounded compact set; ( j ) (ˆ) R p is a vector and c i j is a positive constant. Define (i ) [ (ˆ), ... , (i ) (ˆ)]T , i=1, … , n-1. Based on (16) and (21), the following defined function is positive definite and proper w. r. t for each θ ∈ Ωθ [29]. p
V ( , )
i
1
i 1
0 [ i (i i ) i ]di , i
i 0
(22)
where Γ=diag{Γ1, … , Γp} is a positive definite diagonal adaptation rate matrix. In addition, the following property will be utilized in the final Lyapunov stability analysis for the closed loop system [29].
1 1 V ( , ) [1 (ˆ1 ) 1 ], ... , [ p (ˆp ) p ] p 1 (23) T 1 ˆ [ ˆ ,..., ˆ ]T has similar definition of smooth projection in (19) and some properties Then, the estimated parameter vector 1 n in (21) , (22) and (23), which are given as follows: Define smooth projection as (24) ( ) [1 (1 ), ... , n ( n )]T
where к=[к1, … , кn]T. Let εΘ=[εΘ1, … , εΘn]T, the following properties hold: ˆ) ˆ , ˆ ( { : min max } ˆ ˆ R n (25) () ˆ { : min max },
ˆ ) ( j ) ( j
ˆ Rn , 1 j n { : i s i j },
ˆ ) Rn is a vector, s is a positive constant, and where ( j ) ( i j n
V (, )
i 1
1
i
i 0
[ i ( i i ) i ]d i , i 0
(26)
is positive definite and proper w. r. t for each Θ∈ ΩΘ. In addition, similar to (23), the following property holds
1 ˆ ) ], ... , 1 [ ( ˆ ) ] V (, ) [1 ( 1 1 n n n 1 (27) T 1 ˆ , ˆ ( ˆ), ˆ . Similarly, we where =diag{γ1, … , γn} is a positive definite adaptation rate matrix, and (i ) (i ) ˆ T ˆ also define [ (), ... , ()] , i=1, … , n-1. Remark 3: The smooth projections defined in (19) and (24) can guarantee that the estimates of uncertain parameters θ and Θ always lie in known bounded sets. More important, they have high order bounded derivatives, which can be used to ensure the boundness of
6 the proposed controller since the derivatives of the virtual control laws are required in the final controller. The specific expression of the projection function π(∙) can be chosen as in [14], which contains an exponential function and has bounded derivatives up to order n-1. III. ROBUST ADAPTIVE BACKSTEPPING CONTROLLER DESIGN AND STABILITY ANALYSIS In this section, robust adaptive control is proposed for the considered uncertain SISO nonlinear system with unknown input dead-zone. Global asymptotic tracking performance and the bounded stability of all signals in the closed loop system are achieved. The design follows the recursive backstepping design procedure [8] due to the existence of various mismatched disturbances in the system model. Step 1: Define error variables z1=x1-x1d and z2=x2-α1, where z1 denotes the output tracking error and α1 is the virtual control law for x2 to be synthesized later. Viewing (11), differentiating z1 with respect to time, we have z1 x1 x1d (28) z2 1 T 1 ( x1 , t ) d1 ( xn , t ) x1d Since the time derivative of the virtual control law at the previous step is needed at the next design step during backstepping design procedure, the resulting control law at each step has to be sufficiently smooth. Hence, by referring the ideas in [26], a sufficiently smooth positive integrable function in Assumption 4 is used to synthesize the control law. The resulting virtual control law α1 is given by ˆ , t) , 1 ( x1 ,ˆ , 1a 1s 1s 1s1 1s 2 T ˆ ˆ (x , , t) x (x , t) 1a
1
1d
1
1
1s1 ( x1 , t ) k1 z1 ˆ2 ks1 z1 1 ˆ , t) 1s 2 ( x1 , z1 ˆ ks1 z1 tanh( ) (t ) 1 (t ) 1 1
(29)
where k1 and ks1 are positive feedback gains. In (29), α1a is a model-based feedforward control law utilized to improve model compensation, α1s is a robust control law in which α1s1 is a linear robust feedback term to stabilize the nominal system and α1s2 is a nonlinear robust term to tackle the disturbance d1 ( xn , t ) . Substituting (29) into (28), we have (30) z1 z2 k1 z1 T 1 ( x1 , t ) 1s 2 d1 ( xn , t ) Consider the Lyapunov function candidate
V1
1 2 c1 z1 2
(31)
where c1 is a positive constant. The time derivative of (31) along with (30) is given by
V1 k1c1 z12 c1 z1 z2 c1T 1 ( x1 , t ) z1 c1 z11s 2 c1 z1d1 ( xn , t )
(32)
Invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have V1 k1c1 z12 c1 z1 z2 c1T 1 ( x1 , t ) z1 c1 z1 1
ˆ2 ks1c1 z12 1 z1 ˆ ks1 z1 tanh( ) (t ) 1 (t ) 1 1
k1c1 z12
c1 z1 z2 c1 1 ( x1 , t ) z1 c1 z1 1
ˆ2 ks1c1 z12 1 ˆ (t ) ks1 z1 1 1
Step 2: Define the error variable z3=x3-α2. Viewing (11) and differentiating z2 with respect to time, we have z2 x2 1
z3 2 T 2 ( x2 , t ) d2 ( xn , t ) 1c 1u where
(33)
T
(34)
7
ˆ , t) 1 ( x1 , ˆ ,
1 1 1 ˆ x1 1 ˆ ˆ ˆ t x1
1c 1u
1c
1 1 1 ˆ (35) [ x2 ˆT 1 ( x1 , t )] 1 ˆ ˆ ˆ t x1
1u
1 [T 1 ( x1 , t ) d1 ( xn , t )] x1
In (35), 1c represents the calculable part of 1 and can be used in the control law design while 1u is the incalculable part due to the existence of time-varying disturbance and has to be handled by the robust feedback term. For the error dynamic in (34), the resulting virtual control law is given by
2 ( x2 , (1) , (1) , t ) 2 a 2 s , 2 s 2 s1 2 s 2 2 a ( x2 , (1) , t ) 1c ˆT 2 ( x2 , t ) 2 s1 ( x2 , t ) k2 z2 ˆ2 k s 2 z2 2 ˆ , t) 2 s 2 ( x2 , z2 ˆ ks 2 z2 tanh( ) 2 (t ) 2 (t ) 2
(36)
where k2 and ks2 are positive feedback gains. In (36), α2a is a model-based feedforward control law utilized to improve model compensation, α2s is a robust control law in which α2s1 is a linear robust feedback term to stabilize the nominal system and α2s2 is a nonlinear robust term to tackle the disturbance d2 ( xn , t ) . Substituting (36) into (34), we have z2 z3 k2 z2 T 2 ( x2 , t ) 2 s 2 d 2 ( xn , t ) 1u
z3 k2 z2 T 2 ( x2 , t ) 2 s 2 d 2 ( xn , t )
(37)
1 T [ 1 ( x1 , t ) d1 ( xn , t )] x1
Consider the Lyapunov function candidate
1 V2 V1 c2 z22 2
(38)
where c2 is a positive constant. The time derivative of (38) along with (37) is given by
V2 V1 k2 c2 z22 c2 z2 z3 c2 z2 2 s 2 c2T 2 ( x2 , t ) z2 c2 z2 d 2 ( x2 , t ) c2 z2
(39) 1 T [ 1 ( x1 , t ) d1 ( xn , t )] x1
Invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have V2 V1 k2 c2 z22 c2 z2 z3 T {c2 [2 ( x2 , t )
ˆ2 (40) ks 2 c2 z22 1 2 1 ( x1 , t )]}z2 c2 z2 2 ˆ (t ) x1 k s 2 z2 2 2 Step i (1≤ i ≤ n-1): Define the error variable zi+1=xi+1-αi. Viewing (11) and differentiating zi with respect to time, we have zi zi 1 i T i ( xi , t ) di ( xn , t ) (i 1)c (i 1)u (41) where
8
i 1 ( xi 1 , (i 2) , (i 2) , t )
i 1 i 1 i 1 xj j 1 x t j
i 1 ˆ i 1 ˆ (i 1) c (i 1)u ˆ ˆ
(i 1) c
i 1 i 1 i 1 [ x j 1 ˆT j ( x j , t )] j 1 x t j
(42)
i 1 ˆ i 1 ˆ ˆ ˆ
i 1
(i 1)u
j 1
i 1 [T j ( x j , t ) d j ( xn , t )] (2 i n) x j
In (42), (i 1)c represents the calculable part of i 1 and can be used in the control law design while (i 1)u is the incalculable part due to the existence of time-varying disturbance and has to be handled by the robust feedback term. For (41), the resulting virtual control law is given by i ( xi , (i 1) , (i 1) , t ) ia is , is is1 is 2 ( x , (i 1) , t ) ˆT ( x , t ) ia
i
( i 1) c
i
i
is1 ( xi , t ) ki zi ˆ2 ksi zi i is 2 ( xi , (i 1) , t ) zi ˆ ksi zi tanh( ) i (t ) i (t ) i
(43)
where ki and ksi are positive feedback gains. In (43), αia is a model-based feedforward control law utilized to improve model compensation, αis is a robust control law in which αis1 is a linear robust feedback term to stabilize the nominal system and αis2 is a nonlinear robust term to tackle the disturbance di ( xn , t ) . Substituting (43) into (41), we have zi zi 1 ki zi T j ( x j , t ) is 2 di ( xn , t ) (i 1)u zi 1 ki zi T j ( x j , t ) is 2 di ( xn , t ) i 1
j 1
(44)
i 1 T [ j ( x j , t ) d j ( xn , t )] x j
Consider the Lyapunov function candidate
1 Vi Vi 1 ci zi2 2
(45)
where ci is a positive constant. Differentiating (45) along with (44) and invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have Vi Vi 1 ki ci zi2 ci zi zi 1 ci zi is 2 ciT i ( xi , t ) zi
i 1
i 1
ci zi di ( xn , t ) ci zi
x j
j
[T j ( x j , t ) d j ( xn , t )]
Vi 1 ki ci zi2 ci zi zi 1 ci zi i
i 1
i 1
T {ci [i ( xi , t ) j
x j
ˆ2 (46) ksi ci zi2 i ˆ ksi zi i i (t )
j ( x j , t )]}zi
Step n: The time derivative of zn is given by
zn ud Td ˆ T n ( xn , t ) d ( xn , t ) (n 1)c (n 1)u (47) Based on (42), the actual control input ud is synthesized as follows
9
ud uda uds , uds uds1 uds 2 u ˆT ( x , t ) ( n 1) c
da
n
n
uds1 kn zn uds 2
ˆ2 ksn zn n zn ˆ ksn zn tanh( ) n (t ) n (t ) n
(48)
where kn and ksn are positive feedback gains. In (48), uda is a model-based feedforward control law utilized to improve model compensation, uds is a robust control law in which uds1 is a linear robust feedback term to stabilize the nominal system and uds2 is a nonlinear robust term to tackle the disturbance d ( xn , t ) . Substituting (48) into (47), we have zn kn zn dT ˆ T n ( xn , t ) uds 2 d ( xn , t ) ( n 1)u
kn zn dT ˆ T n ( xn , t ) uds 2 d ( xn , t )
n 1
n 1
j 1
x j
(49)
[T j ( x j , t ) d j ( xn , t )]
Theorem 1: Provided that the full state information is available, with the following adaptation laws n
i 1
i 2
j 1
ˆ {c11 ( x1 , t ) z1 ci [i ( xi , t ) ˆ c z , i
i 1 j ( x j , t )]zi } (50) x j
z [ z1 , z2 ,... zn ]T
ˆd cnˆ zn
(51) (52)
where M is a positive definite diagonal adaptation rate matrix, and selecting feedback gains k1, … , kn large enough and proper c1, … , cn and ks1, … , ksn such that the matrix Λ defined as follows is positive definite, c1 / 2 0 0 k1c1 c / 2 k c c2 / 2 2 2 1 (53) 0 c2 / 2 k3c3 0 cn / 2 0 0 c / 2 kn cn n then the proposed control law (9) and (48) guarantees that all signals in the closed loop system are bounded, and global asymptotic output tracking performance is also obtained, i.e., z1→0 as t→∞. ◊ Proof : Consider the Lyapunov function candidate 1 1 Vn Vn 1 cn zn2 V ( , ) V (, ) dT 1d (54) 2 2 Differentiating (54) along with (49) and invoking the inequality (13) in Lemma 2 and (15) in Assumption 2, we have
Vn Vn 1 kn cn zn2 cn zn uds 2 cn dT ˆ zn cn T n ( xn , t ) zn cn zn d ( xn , t )
n 1
n 1
cn zn
j 1
x j
[ T j ( x j , t ) d j ( xn , t )]
ˆ T 1ˆ T 1ˆ T 1 d d 2ˆ2 k sn zn n Vn 1 kn cn zn2 c z ˆ (t ) n n n k sn zn n n
n 1
n 1
T {cn [n ( xn , t )
j 1
x j
j ( x j , t )]}zn
ˆ T 1ˆ cn dT ˆ zn T 1ˆ T 1 d d
(55)
10 Considering (46) and the adaptation law in (50), (51) and (52), we obtain n
n 1
n
i 1
i 1
i 1
ˆ2 k si ci zi2 i ˆ (t ) ksi zi i i
Vn ki ci zi2 ci zi zi 1 [
ci zi i ] cn dT ˆ zn T {c11 ( x1 , t ) z1 n
i 1
i 2
j 1
ci [i ( xi , t )
i 1 j ( x j , t )]zi } T 1ˆ x j
ˆ T 1ˆ T 1 d d
(56)
n
n 1
n
i 1
i 1
i 1
ˆ2 k si ci zi2 i
ki ci zi2 ci zi zi 1 [
ˆ (t ) k si zi i i
ˆ ] ci zi i ˆ (t ) ci zi i i ˆ (t ) i 1 k z n
Z T Z
si
i
i
i
where Z is defined as Z=[z1, … , zn]T. Invoking the inequality in (14), noting that the matrix Λ defined in (53) is positive definite, we have n c (t ) Vn Z T Z i i i 1 k si
2 min ( )( z1
...
zn2 )
n c (t ) c (t ) i i W i i i 1 k i 1 k si si n
(57)
where W min ()( z12 ... zn2 ) is a positive function and λmin(Λ) is the minimal eigenvalue of matrix Λ. Integrating (57) with respect to time, we have t
n
Vn (t ) W ( )d Vn (0) 0
ci
i 1 k
t
0 i ( )d si
n
c Vn (0) i i i 1 k si
(58)
It can be concluded from (58) that Vn(t) ∈ L∞ and W∈ L2, which implies z1, … , zn and , , d are bounded. From Assumption 1, we can infer that all system states x1, … , xn are bounded, and with Assumption 4 we know that the control input v(t) is bounded. Therefore, all the signals in the closed loop system are bounded. In addition, viewing (30), (37), (44) and (49), we know that z1 , ... , zn are bounded, which leads to the boundness of W , thus W is uniformly continuous. By applying Babalat’s lemma [8], W→0 as t→∞, which leads to the results in Theorem 1. Remark 4: Results of Theorem 1 indicate that the proposed strategy can ensure that all the signals in the closed loop system are bounded and the output tracking error converges to zero asymptotically, and the converging rate can be enhanced by increasing the control gains k1, … , kn and ks1, … , ksn. Such a theoretical result of the proposed robust adaptive backstepping controller is strongly expected in many practical systems subject to unknown input dead-zone nonlinearity, parametric uncertainties as well as disturbances. IV. COMPARATIVE SIMULATION RESULTS In this section, we will present two examples to demonstrate the effectiveness of the proposed control strategy. Example 1: we firstly consider the uncertain nonlinear system subject to unknown input dead-zone as in [7, 26] 1 e x (59) x bu (v) a d (t ) 1 e x where u is output of the dead-zone and d(t) represents disturbance. To simplify the simulation process, we consider the symmtric input dead-zone here and assume that the slopes of the dead-zone are known, i.e., mr=ml=1, and b=1 are kown. But the breakpoints are unknown, and the actual values of unknown parameters are a=1, br=-bl=1. The disturbance is set as d(t)=0.6sint. Define θd=br, θ=a, and Θ as the bound of the lumped disturbance. The desired trajectory is a sinusoidal-like curve given by yd(t)=arctan(sint)[1-exp(-0.01t3)]. The desired trajectory is shown in Fig. 1. Four controllers are compared to verify the effectiveness of the proposed controller in this paper.
11 1) RABC: This is the robust adaptive backstepping controller proposed in this paper. The control gains are given as follows: k1=100, c1=1, ks1=1. And choosing ε=0.01, δ(t)=5000/(t2+1). The bounds of uncertain rangs are given by θmax=5, Θmax=30, θmin=0, Θmin=0. ˆ (0) 0 . Paramter adaptation rates are set The initial estimates of the unknown parameters are chosen as: ˆ(0) 0.5 , ˆ (0) 0.5 , d
as: Γ=500, γ=500, and M=100. 2) RAC: This is the robust adaptive controller proposed in [26] to tackle unknown dead-zone and achieve asymptotic tracking performance. The dead-zone characteristics and model compensation are not considered in this controller design [26]. Hence, it is used here to verify the effectiveness of the compensation of dead-zone and parametric uncertainties in the proposed RABC controller. The robust adaptive controller in [26] with adaptation of ˆ has the following form:
1 ˆ2 2 z1 f ( x, t ) 2 v(t ) 1 1 z1 tanh[ z1 / (t )]ˆ f ( x, t ) (t ) 2 2 1 ˆ z1 f ( x, t ) 2
(60)
(61)
where
f ( x, t ) (
1 e x 2 ) h1 z12 h 1 x 1 e
d 0 D sup yd , max{ a , d 0 }
(62)
t 0
D sup D(t ) , D(t ) bd1 (v) d (t ) t 0
For RAC, the control parameters are chosen as h1=2, h=2, α=6, σ(t)= 5000/(t2+1), and ˆ (0) 0 . 3) ABC: This is an adaptive backstepping controller same as the RABC controller but without robust control law. This ABC controller is similar to that in [25]. It is compared to demonstrate the effectiveness of the robust feedback term proposed in this paper. The control parameters of ABC controller are the same as the corresponding parameters in RABC. 4) AVSC: This is the adaptive variable structure controller proposed in [7]. The dead-zone inverse and model compensation are also considered in this controller design. The actual control input to the plant ud can be given as follows:
ud
1 u b
u (l1 1)( z1 1 ) sg ( z1 ) aˆ
1 e x ˆ (z ) x1d Dsg 1 1 e x
(63)
and the parameter adaptation laws:
ˆ a
1 e x ( z1 1 ) f1sg ( z1 ) 1 e x
ˆd b (l (v) r (v))( z1 1 ) f1sg ( z1 )
(64)
Dˆ d ( z1 1 ) f1sg ( z1 ) where Dˆ is the estimate of D, which denotes the upper bound of the lumped disturbance, and
z1 z1 1 z1 1 f1 , sg ( z1 ) z13 0 z1 1 ( 2 z 2 )2 z 3 1 1 1 -4 The control parameters of ASVC controller are chosen as: l1=100, Γa=200, Γb=40, Γd=30, and ρ1=5×10 .
z1 1 (65)
z1 1
The comparative tracking errors of the four controllers are shown in Fig. 2 and Fig. 3. As shown, the proposed RABC controller achieves the best tracking performance in both transient and steady state. The transient tracking error of RAC is rather large because of lacking model compemsation. And without the robust feedback term to attenuate the effect of the disturbance, the ABC controller has larger trakcing error than that of RABC. The AVSC controller can only guarantee uniformly bounded tracking performance and the steady state tracking error is dominated by the selecting of ρ1. From Fig. 3, we can see that the tracking error of ASVC becomes
12 to present chattering since the value of ρ1 has been selected as small as possible. The parameter estimation and control input of RABC are shown in Fig. 4 and 5, respectively. As seen, the designed control input v is continuous, which is easier to be implemented in pracitce. 0.8 0.6
Desired Trajectory
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
0
10
20
30
40
50
60
70
80
times(s) Fig. 1 The desired trajectory 1 0 -1
0
-0.01
0
20
40
0.01
0
10
AVSC
RAC
0.01
20
60
-0.01
80
0
20
40
60
80
0
20
40
60
80
0.01
RABC
ABC
0.01
0
-0.01
0
0
20
40
60
80
0
-0.01
times(s)
Fig. 2 Comparative tracking errors of four controllers -3
1
x 10
RABC ABC AVSC RAC
Tracking Error
0.5
0
-0.5
-1 70
72
74
76
times(s)
Fig. 3 Tracking errors of four controllers during the last ten minitues
78
80
13 2
d
1 0
0
10
20
30
40
50
60
70
80
60
70
80
60
70
80
70
80
4 2 0
0
10
20
30
40
50
40
Θ 20 0
0
10
20
30
40
50
times(s)
Fig. 4 Parameter estimation of RABC 3
2
Control Input
1
0
-1
-2
-3
0
10
20
30
40
50
60
times(s)
Fig. 5 Control input of RABC
Example 2: Consider a class of nonlinear mechanical systems having the following form: x1 x2
(66) x2 u (v) x2 d ( x, t ) where u denotes the output of the unknown dead-zone nonlinearity. It is worth to note that the system model (66) can be used to describe many mechanical systems such as motor driven system [1, 2, 17, 21, 25], active suspensions [16] etc. Similar to example 1, we also assume that the slopes of the dead-zone are known, i.e., mr=ml=1. The actual values of the unknown breakpoints of the dead-zone are chosen as br=5 and bl=-3. And the actual values of is chosen as =3. The disturbance d(x,t)=0.5sint and the desired trajectory is the same as in example 1. To verify the effectiveness of the inverse dead-zone compensation, the following three controllers are compared. 1) RABC: The control gains are given as k1=200, k2=20, ks2=1, c2=1. And choosing ε=0.1, δ(t)=5000/(t2+1). The bounds of uncertain rangs are given by max 10 , Θmax=60, min 0 , Θmin=0. The initial estimates of parameters are chosen as: ˆ (0) 0 . Paramter adaptation rates are set as: Γ=80, γ=diag{15, 7}, and M=50. mr br (0) 0 , ml bl (0) 0 , ˆ(0) 0 and
2) RAC: For (66), the robust adaptive controller in [26] with adaptation of ˆ has the following form:
14
v(t )
( z1 / 2 z2 ) ˆ 2 f ( x, t )
( z1 / 2 z2 ) tanh[( z1 / 2 z2 ) / (t )]ˆ (t ) (67)
ˆ f ( x, t ) z1 / 2 z2 where
z2 x2 x1d
f ( x, t ) x22 h2 z12 z22 h 1 The control parameters are chosen as h2=2, h=2, α=2, σ(t)= 5000/(t2+1), and the initial estimation of β is set at zero.
(68)
3) RABC1: This is the robust adaptive backstepping controller proposed in this paper but without compensating the unknown dead-zone. The corrosponding control parameters are the same as that of RABC. The output tracking performance of three controllers are given in Fig. 6 and Fig. 7, respectively. The results show that the tracking error of RAC is much lager than that of the other two controllers. Furthermore, the proposed RABC controller outperforms the RABC1 controller in terms of transient and final tracking errors. This indicates that the dead-zone inverse compensation in the proposed RABC controller can apparently improve the tracking performance, and is more effective than the robust method which deals with the dead-zone by simplifying its model as a linear term and a disturbance term, in this severe input dead-zone case. The parameter estimation and control input of RABC are presented in Fig. 8 and 9, respectively.
0.7
Tracking Error of RAC
0.6 0.5 0.4 -3
x 10
0.3
2
0.2
0
0.1
-2 50
60
70
80
0 -0.1
0
10
20
30
40
50
60
70
80
times(s)
Fig. 6 Tracking error of RAC
-3
1.5
x 10
-5
2
1
x 10
0
Tracking Error
0.5 -2 50
60
70
80
0
-0.5
RABC RABC1
-1
-1.5 0
10
20
30
40
50
60
times(s)
Fig. 7 Tracking errors of RABC (with inverse compensation) and RABC1 (without inverse compensation)
70
80
15 5 mrb r
0 -5
0
10
20
30
40
50
60
70
80
60
70
80
60
70
80
60
70
80
60
70
80
0 mlb l
-2 -4
0
10
20
30
40
50
5 0 -5
0
10
20
30
40
50
50
0
0
10
20
30
40
50
times(s)
Fig.8 Parameter estimation of RABC 10 8 6
Control Input
4 2 0 -2 -4 -6 -8
0
10
20
30
40
50
times(s) Fig. 9 Control input of RABC
V. CONCLUSION In this paper, a robust adaptive backstepping controller is proposed for a class of uncertain SISO nonlinear strict-feedback systems subject to unknown input dead-zone nonlinearity, parametric uncertainties and unknown bounded disturbances. A smooth dead-zone inverse is constructed to compensate the effect of input dead-zone, and the adaptation laws are synthesized via backstepping recursive design technique to tackle parametric uncertainties. Disturbances are handled by novel robust control laws which derived by incorporating a sufficiently smooth positive integral function into the controller design at each step when backstepping. A smooth projection mapping is used to facilitate the backstepping design since differential operation on virtual control law is needed. The proposed robust adaptive backstepping controller can theoretically guarantee that all closed loop signals are bounded and the output tracking error asymptotically converges to zero, meanwhile ensuring the continuity of the control input. Comparative simulation results are obtained to illustrate the effectiveness of the proposed scheme. REFERENCES [1] [2] [3]
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