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Performance guaranteed tracking control of nonlinear systems under anomaly actuation: A neuro-adaptive fault-tolerant approach
Accepted Manuscript
Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach Ye Cao, Yongduan Song, Kai Zhao, Liu He PII: DOI: Reference:
S0925-2312(17)31709-5 10.1016/j.neucom.2017.10.058 NEUCOM 19038
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
4 June 2017 22 September 2017 31 October 2017
Please cite this article as: Ye Cao, Yongduan Song, Kai Zhao, Liu He, Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.10.058
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Ye Cao
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, Kai Zhao
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, Liu He
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The key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, Chongqing University, Chongqing 400044, PR China 2 School of Automation, Chongqing University, Chongqing 400044, PR China
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, Yongduan Song
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Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach
Abstract
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In this work, we investigate the performance guaranteed tracking control problem of a class of multi-input multi-output (MIMO) nonlinear systems with anomaly actuation. By introducing new forms of parameter estimation error together with Lyapunov function, and using neural network approach, the obstacles caused by anomaly actuation can be handled gracefully and the assumptions on control gain matrices in existing results are significantly relaxed. Furthermore, a strictly increasing function is introduced to form a scaling speed transformation, which directly impacts both the controller law and the adaptive law, accelerating the learning rate and thus enhancing the tracking performance. It is shown that all the closed-loop signals are uniformly bounded and the tracking error converges to an adjustable residual set with a pre-assignable decay rate during the tracking process. Simulation results are presented to illustrate the effectiveness of the proposed scheme.
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Keywords: neuro-adaptive control, anomaly actuation, performance guaranteed.
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1. Introduction Actuators play a crucial role in any control system, but most existing works are based on the assumption that the actuator operates normally all Email address: [email protected] (Yongduan Song
Preprint submitted to Elsevier
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November 3, 2017
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the time. However, actuator might become defective suddenly or unnoticeably, resulting in so called anomaly actuation during a long term operation. Anomaly actuation arises from uncertain actuator nonlinearities, unknown input saturation or faulty actuation. Among them nonsmooth nonlinearities occur in many actuators, such as piezoelectric translators, mechanical connections, electric servomotors and other areas [1]. As we all known, the typical actuator nonlinearities include saturation, deadzone, backlash and hysteresis [2] [3]. In practice, actuator saturation is one of most important nonsmooth nonlinearities which is unavoidable in most actuators in many industrial control systems, and when an actuator has reached an input saturation limit, any efforts to further increase the actuator output would not make any variation of the output, so the saturation nonlinearity often limits system performance or even results in instability of the system. There is a rich collection of methods on dealing with actuator saturation. In [4], a robust adaptive control method is proposed for a class of uncertain nonlinear systems in the presence of input saturation and external disturbance. The Nussbaum function is introduced to compensate for the nonlinear term arising from the actuator saturation. In [5], a robust fault-tolerant control is developed for the strict-feedback systems with parametric uncertainties and asymmetric nonsmooth saturation. An adaptive fuzzy backstepping control approach for a class of uncertain nonlinear time-delay systems in the presence of actuator saturation is presented in [6]. The aforementioned results investigate the control problem of single-input single-output (SISO) nonlinear systems, and the nonlinear functions in [4] and [5] must satisfy linear parameterized condition. However, most practical multi-input multi-output (MIMO) systems, such as electrical machines, robotic manipulator, high speed trains, etc., are strongly coupled, making the control problem rather complicated. Certain prior knowledge or constraint on the control gain matrices (CGMs) is required in order to handle such systems. In [7] [8] under the assumption that the bounds of input constraint and the nonlinear functions in the saturation are exactly known, an adaptive tracking control is proposed for a class of multi-input multi-output (MIMO) nonlinear systems. In [9], a network-based actuator saturation compensation scheme is presented for nonlinear systems in Brunovsky canonical form, but the method in [9] can only deal with linear asymmetric saturation and the CGMs must be completely known for control design. To facilitate the stability analysis and control design, [10][11] consider a class of MIMO nonlinear systems with a lower triangular structure CGMs under actuator dead-zones. In the more general case as studied in 2
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[12], a neuro-adaptive PI control for nonlinear systems with unknown and nonsmooth actuation characteristics is developed on condition that the matrix B(x, t) + B T (x, t) is symmetric positive, where B(x, t) = G(x, t)L(·), G(x, t) represents the CGM and L(·) is a diagonal matrix due to actuator saturation. It is noted that this condition is not easy to satisfy in practical situations. At the same time, actuator failure is one of the most common phenomenons in anomaly actuation. In the past decades, there is a rich collection of methods on dealing with actuator failure. In [13][14][15], based on fault detection and diagnosis, reconfigurable control schemes are proposed such that the impacts of failure can be compensated and the stability as well as the acceptable performance of the system can be maintained. Whereas if the actuation faults are time-varying and undetectable faults, the robust control theory by using unchangeable controller throughout the failure-free case and the failure case is a good way to solve the actuator failure. The typical approaches have been developed in [5][16][17]. However, the assumption on the matrix B(x, t) + B T (x, t) is also required, where B(x, t) = G(x, t)ρ(·), G(x, t) represents the positive definite control gain matrix, ρ(·) is a diagonal matrix due to actuator failure. In addition, these results do not take into account the non-smooth actuator nonlinearities, and thus cannot be applied directly to systems with limited actuator input. Few researchers investigate actuator faults and actuator output constraints simultaneously. In [18] these two issues are considered for a particular spacecraft system model, and in [19] an adaptive fault-tolerant controller are provided for a linear system with actuator saturation, whereas the controllers require large computation power and are difficult to implement. Thus this paper will investigate more general MIMO nonlinear systems under undetectable failure and nonsmooth actuator saturation. Another important issue that should be explicitly addressed in control design for MIMO nonlinear systems under anomaly actuation is the tracking control performance in terms of the transient behavior and the steady-state response. When actuator failures occur or the actuator reach an saturation limit, tracking performance degradation is inevitable. Earlier contributions addressing the performance issue include those by [20][21][22]. In [20], by using a switching strategy for non-decreasing adaptive gain and requiring a prior bound on the initial data, controllers are designed for minimum-phase linear time invariant systems which guarantee the error to be less than an arbitrary prespecified constant after a transient with an overshoot below 3
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a prespecified upper bound. The funnel method [21] provides arbitrarily assignable transient rate through the shape of the funnel, but it only guarantees approximate performance for the system output and it has no adaption and uses an error-dependent gain which lead to infinity control gain once the error approaches the funnel boundary. Thus, how to enhance the transient and steady-state performance needs to be further investigated. Note that, on one hand, with the common sense that the enhanced performance can be obtained by speeding up the parameter updating rate and/or cranking up the feedback control gain, and on the other hand, it is unclear by how much and in what way to increase the control gain and the parameter updating rate to achieve improved tracking performance. As a matter of fact, simply choosing a high constant control gain or simply increasing the parameter updating rate by a large constant value in the adaptive law does not necessarily produce better performance. The objective of this work is to design an intelligent nonlinear adaptive controller that can enhance the system performance in the presence of actuator saturation and actuator failure. A neuro-adaptive prescribed performance control is proposed for a class of MIMO nonlinear systems with unknown asymmetric input saturation as well as actuation faults, achieving adjustable tracking precision with pre-specifiable decay rate during transient period. Based upon new forms of parameter estimation error and Lyapunov function and combining neural network, the the requirement on the symmetric positive definite condition of matrix B(x, t) + B T (x, t) is not needed, which significantly relaxes the assumption on CGMs. The proposed control also allows the mode and the rate of convergence during the transient period to be pre-specifiable explicitly by choosing the rate function and the design parameter properly. The rest of this paper is organized as follows. In section 2, the control problem and some preliminaries are introduced. Section 3 gives controller design and stability analysis. Simulation results are presented in section 4 to illustrate the effectiveness of the proposed scheme. Finally, we conclude the paper in Section 5.
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2. Problem formulation and preliminaries
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2.1. System description Consider the following class of nonlinear MIMO systems described by: y1 (n1 ) .. x)H(u) (1) = F (x) + G(¯ . ym (nm ) (n −1)
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where y = [y1 , · · · , ym ]T ∈ Rm and x = [y1 , y˙ 1 , · · · , y1 1 , · · · , ym , y˙ m , · · · , P (n −1) ym m ] ∈ Rn with n = m k=1 nk are the output vector and the state vector, respectively; F (x) ∈ Rm are continuous unknown nonlinear function vector, and G(¯ x) ∈ Rm×m are unknown nonlinear C1 function gain matrix (n −2) (n −2) of x¯ = [y1 , · · · , y1 1 , · · · , ym , · · · , ym m ] ∈ Rn−m , and H(u) = [h1 (u1 ), · · · , hm (um )]T ∈ Rm represents the control vector of the system with unknown actuation characteristics, where u is the actual control design. In this paper, two typical actuation models are shown in Fig. 1. Model A. Asymmetric nonsmooth saturation with unknown slope [5] ¯ ui > uma1 δ, l(ui ), − uma2 ≤ ui ≤ uma1 i = 1, · · · , m (2) hi (ui ) = −δ, ui < −uma2
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where uma1 > 0 and −uma2 < 0 represent the break points. Model B. Asymmetric nonsmooth saturation with dead zone [5] ¯ δ, ui > umb1 l1 (ui − b1 ), b1 < ui ≤ umb1 0, − b2 ≤ ui ≤ b1 i = 1, · · · , m hi (ui ) = l2 (ui + b2 ), − umb2 ≤ ui < −b2 −δ, ui < −umb2
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where l1 and l2 are the slope of the dead-zone characteristic, b1 > 0, −b2 < 0, umb1 > 0 and −umb2 < 0 represent the break points. Remark 1. The MIMO system (1) can describe many physical systems such as robot manipulators, satellites and electrical machines. Even without actuator nonlinearity, the system was studied in [24][25] or the special case in 5
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(−uma2, −δ)
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Model A
(−umb2, −δ)
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Model B
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Figure 1: The approximation to nonsmooth saturation functions (solid-line: asymmetric nonsmooth saturation function hi (ui ); dot-line: the smooth approximating function Γi (ui ))
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[22][23]. It means that F (x) can encompass many time-varying uncertainties and external disturbances. The control gain matrix G(¯ x) depends only on the state vector x¯.
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To deal with the nonsmooth and asymmetric actuator nonlinearities, a well-defined smooth function is used to approximate the saturation function, which takes the form as [5]
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hi (ui ) = Γi (ui ) + $i (ui ), i = 1, · · · , m ¯ (+ηui ) − δe−(+ηui ) δe Γi (ui ) = (+ηui ) e + e−(+ηui )
(4) (5)
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¯ and η > 0 is a design parameter. Fig. 1 shows the with = 0.5 ln(δ/δ) approximation to the two type nonsmooth saturation functions. As the bounded property of the function Γi (ui ) and saturation function hi (ui ), we know that function $i (ui ) is bounded, i.e. |$i (ui )| < Di where Di is a positive and unknown constant. With the help of Γ(u), the original system (1) can be expressed as y (n) = F (x) + G(¯ x)(Γ(u) + $(u)) (n )
(n )
where y (n) = [y1 1 , · · · , ym m ]T , [$1 (u1 ), · · · , $m (um )]T .
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Γ(u) = [Γ1 (u1 ), · · · , Γm (um )]T , $(u) = 6
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According to mean value theorem [26], there exist constants αi (0 < αi < 1) such that Γi (ui ) = Γi (ui0 ) + li (ξi )(ui − ui0 ) (7)
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i (ui ) where li (ξi ) = ∂Γ∂u |ui =ξi and ξi = αi ui + (1 − αi )ui0 . As Γi (ui ) are noni decreasing functions, there exist some positive constants lm and lM such that
0 < lm < li (ξi ) < lM < ∞, i = 1, · · · , m.
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By choosing ui0 = 0 and using the fact that Γi (0) = 0, it is easy to get that Γi (ui ) = li (ξi )ui .
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With the aid of (9), the system (6) can be expressed as
y (n) = F (x) + G(¯ x)L(ξ)u + G(¯ x)$(u)
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Remark 2. As can be seen from Figure 1, the control input hi (ui ) involves non-smooth variation at the break points, which poses technical challenge for direct control design and analysis. Here in this work, motivated by [5], we introduce a bounded and smooth function Γi (ui ) to approximate hi (ui ). As Γi (ui ) is a smooth function of ui , we can employ the mean value theorem on function Γi (ui ), so that u can be separated explicitly in system (10). This treatment facilitates the development of the control scheme to be presented shortly.
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As unanticipated actuator faults may occur, inevitable in practice for long-term operation, we additionally include such scenario in the model in which the actual control vector u and the designed control vector ua are no longer the same anymore. Instead, they are linked via u = ρ(tρ , t)ua + ur (tr , t)
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where ρ(·) = diag{ρi } ∈ Rm×m , i = 1, . . . , m, is the ‘healthy indicator’ reflecting the effectiveness of the actuator, ur is the uncontrollable portion of 7
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actuation input, tρ and tr denote the time instant at which the loss of actuation effectiveness fault and the additive actuation fault occur, respectively. Both ρ(tρ , t) and ur (tr , t) are assumed to be unknown bounded, time-varying and undetectable. In this work, we consider the case that 0 < ρi ≤ 1, i.e., although loosing its effectiveness the actuation is still functional such that u can be influenced by the control input ua all the time, as considered in several studies [5][16][17]. To continue, we need the following assumptions, which are quite standard in the literature.
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Assumption 1. The desired tracking trajectory ydi , i = 1, · · · , m, as well as their derivatives up ni are known smooth functions of time and bounded. And the state vector x is available for control design. Assumption 2. The control gain matrix G(¯ x) is known to be either positive definite or negative definite. Without losing generality, this paper assumes that G(¯ x) is positive definite.
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Assumption 3. ρ(·) and ur (·) are unknown, possibly fast time-varying and unpredictable but bounded, i.e., ||ur (·)|| ≤ u¯r < ∞, 0 < ρm ≤ ρi ≤ 1
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for i = 1, . . . , m, where ρm and u¯r are unknown positive constants.
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Remark 3. Assumption 1 is commonly imposed in most existing works in addressing the tracking control problems [28]. The Assumption 2 guarantees that the system as described in (1) is controllable, which has been commonly used in most existing works in addressing MIMO systems [22][23]. As for Assumption 3, it is noted that most fault detection and diagnosis (FDD) or fault detection and identification (FDI) based fault tolerant control implicitly assume that the faults vary with time slowly enough to allow for timely fault identification and diagnosis [27] or that one has enough information on the faults to carry out parametric decomposition [28], while Assumption 3 imposes no such restriction, thus seems more practical. 2.2. Rate function Definition 1. [29][30] A real function κ(t) is qualified as a rate function as long as the following properties hold: 8
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(1) κ(t) is a positive and monotonously increasing function of time for all t ∈ [0, ∞) such that κ(t)−1 is positive and strictly decreasing and lim κ(t)−1 = t→∞ 0; (2) κ(0) = 1; (3) κ(t) is C n for t ∈ [0, ∞).
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Clearly, there are many functions such as 1 + t2 , et and (1 + t2 )et bear these properties, and a pool of rate functions has been established in [29]. Based upon the rate function κ(t) we construct the following time-varying scaling function 1 , (14) β(t) = (1 − bf )κ(t)−1 + bf
where 0 < bf ≤ 1 is a design parameter. From the definition of β(t) as given in (14), it can be seen that β is a monotonically increasing function of time with upper bound, and exhibits the following easily verifiable properties: (1) β monotonically increases from 1 to b1f and β ∈ [1, b1f ) for all t ≥ 0; (2) β 2 monotonically increases from 1 to b12 and β 2 ∈ [1, b12 ) for all t ≥ 0; f
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(3) It holds that β(0) = 1 and
(1 − bf )κ˙ (1 − bf + bf κ)2 (1 − bf )[¨ κ(1 − bf + bf κ) − 2bf κ˙ 2 ] β¨ = (1 − bf + bf κ)3
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β˙ =
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Since β is bounded and smooth for t ∈ [0, ∞), then β˙ is bounded, note that κ˙ exists and smooth, thus β˙ is smooth and bounded, which implies that β¨ is bounded and smooth as κ˙ and κ ¨ are well-defined. In this matter, it can be (j) found that β (j = 0, 1, · · · , ∞) is bounded and smooth for t ∈ [0, ∞).
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2.3. Neural networks and function approximation Neural network has been widely used for control design under various conditions [31]. In this paper, radial basis function neural networks (RBFNNs) will be employed to approximate the unknown functions. Based on the wellknown approximation property, given a continuous nonlinear scalar function
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Π(x) on a compact set Ωx , there exists an ideal RBFNN capable of approximating the smooth nonlinear function to an arbitrary degree of accuracy as follows: Π(x) = W T S(x) + (x), |(x)| ≤ ¯, ∀x ∈ Ωx , (17)
(x − τi )T (x − τi ) ), i = 1, · · · , p, ψ2
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where W ∈ Rp is the weight vector with p the number of nodes, S(x) = [S1 (x), · · · , Sp (x)]T ∈ Rp is the basis function vector, (x) denotes the approximation error, and ¯ is a constant. We choose Si (x) in S(x) as Gaussian functions, given by
where τi ∈ Rn and ψ ∈ R are constants called the center and width of the basis function, respectively. 3. Control Design and Stability Analysis
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We now start designing a control scheme that is able to achieve adjustable tracking precision at the prescribed rate convergence. Define e = y − yd as the output tracking error with yd = [yd1 , · · · ydm ]T ∈ Rm being the desired trajectories. In order to improve tracking performance, we introduce a error transformation εi (t) = β(t)ei (t)
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and we further define a new filtered variable Ei (t) as d + λi )ni −1 εi (t), i = 1, · · · , m dt
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Ei (t) = (
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where λi (i = 1, · · · , m) are positive constants to be selected, so that if Ei (t) goes to zero or within a bound, so does εi (t). It is worth noting that the filter variable Ei (t) as defined in (20) is not directly based on ei (t), but rather, on the transformed error εi (t). Such treatment, together with other design skills, allows for the aforementioned control objectives to be achieved concurrently, as seen shortly.
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Differentiating (20) with respect to time we have n i −1 X j=1
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where Cnj i =
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(t) − ydi i (t))
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(ni − 1)! , j!(ni − j − 1)! (ni −j)
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Combining (10) and (21) yields that (n) E˙ = ν + β(F (x) + G(¯ x)L(·)u + G(¯ x)$(u) − yd )
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(n −1) where E˙ = [E˙ 1 (t), · · · , E˙ m (t)]T and ν = [v1 (e1 , · · · , e1 1 , t), (n −1) · · · , vm (em , · · · , em m , t)]T is computable. By adding and subtracting −c1 E on the right side on (22), we have
E˙ = −c1 E + βG(¯ x)L(·)u + β(F (x) + G(¯ x)$(u)) + βη
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where c1 is a positive design parameter and η = β −1 ν + β −1 c1 E − yd available in constructing the controller.
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3.1. Control under Healthy Actuation Now we consider the control design under healthy actuation, where ρi = 1, i = 1, · · · , m and ur = 0, i.e., ua = u in (12). 11
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By making use of the β function as defined in (14), we construct the following accelerated neuro-adaptive control scheme u = −(δˆ aφ2 (x)ϕ2 )βE a ˆ˙ = −µˆ a + δβ 2 φ2 (x)ϕ2 ||E||2 ,
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with the adaptive law
(24)
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where ϕ = 1 + ||η|| + ||β −1 E||, φ(x) = ||S(x)|| + 1 with S(x) being defined as in (18), δ > 0 and µ > 0 are chosen by the designer. Note that the proposed control exhibits several appealing structural features: 1) it bears the feedback control form, in which the feedback gain is composed of constant gain and adaptive gain. 2) it is built upon the feedback of βE, the scaled version of E. 3) it blends the rate function β into the adaptive gain to accelerate the tracking process. These combined features play a vital role in establishing the following result.
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Theorem 1. Consider the nonlinear MIMO system (1) with asymmetric nonsmooth actuator saturation (2) or (3). Under Assumption 1-2, if the control scheme (24) and the adaptive law (25) are applied, then for any bounded initial conditions, the following objectives are achieved. O1 ) All the internal signals are bounded; the control action is uniformly continuous everywhere; and no excessively large initial control effort is involved; O2 ) The tracking error e converges to an adjustable residual set with the assignable decay rate during the main tracking process. O3 ) Not only the decay rate of the tracking error are independent on system initial value and can be pre-assignable but also the compact set can be made small enough by properly adjusting the design parameter.
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Proof. Firstly we consider the following quadratic form 1 x)E V1 = E T G−1 (¯ 2
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whose derivative along (23) can be expressed as
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1 x)E V˙ 1 =E T G−1 (¯ x)E˙ + E T G˙ −1 (¯ 2 = − c1 E T G−1 (¯ x)E + βE T L(·)u + βE T G−1 (¯ x)η 1 + βE T G−1 (¯ x)(F (x) + G(¯ x)$(u)) + E T G˙ −1 (¯ x)E 2 = − c1 E T G−1 (¯ x)E + βE T L(·)u + βE T G−1 (¯ x)η 1 + βE T ∆(x) + E T G˙ −1 (¯ x)E, 2
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where c1 is a positive design parameter, η = β −1 ν + β −1 c1 E − yd is available in constructing the controller and ∆(x) = G−1 (¯ x)(F (x) + G(¯ x)$(u)). According to the fact that $i (u) in (4) is bounded, it can be concluded that there exist continuous functions Π(x) ∈ R such that max{|∆(x)|, ||G−1 (¯ x)||, ||G˙ −1 (¯ x)||} ≤ Π(x)
(28)
for all x ∈ Rn and t ≥ 0. It follows from (27) and (28) that
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V˙ 1 ≤ − c1 E T G−1 (¯ x)E + βE T L(·)u 1 x)E] + βE T [∆(x) + G−1 (¯ x)η + β −1 G˙ −1 (¯ 2 ≤ − c1 E T G−1 (¯ x)E + βE T L(·)u + β||E||Π(x)ϕ,
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where
1 ϕ = 1 + ||η|| + β −1 ||E||. 2
(29)
(30)
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Since Π(x) are unknown, we employ RBFNNs to approximate them as follows: Π(x) =W T S(x) + (x)
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(31) ≤||W T ||||S(x)|| + ||¯|| ≤aφ(x), where |(x)| ≤ ¯, a = max ||W T ||, ||¯|| , φ(x) = ||S(x)|| +1 for all x ∈ Ωx and Ωx is a compact set. Substituting (31) into (29), it becomes V˙ 1 ≤ −c1 E T G−1 (¯ x)E + βE T L(·)u + β||E||aφ(x)ϕ. 13
(32)
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Now we define a Lyapunov function candidate for the closed-loop system as follow: 1 2 a ˜ , (33) V = V1 + 2lm where V1 is given by (26). Note that a ˆ is the estimation of the unknown weight a, and we introduce a parameter estimation error of the form a ˜ = a − lm a ˆ, where lm is defined as (8). By blending such error into second part of Lyapunov function candidate, the unknown actuator saturation can be processed gracefully. Differentiating (33) and using (24), we have
From Young’s inequality we have
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V˙ ≤ − c1 E T G−1 (¯ x)E − δβ 2 lm a ˆφ2 (x)ϕ2 ||E||2 + β||E||aφ(x)ϕ − a ˜a ˆ˙ . βφ(x)ϕ||E|| ≤ δβ 2 φ2 (x)ϕ2 ||E||2 + Substituting (35), it becomes
1 . 4δ
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V˙ ≤ − c1 E T G−1 (¯ x)E + δβ 2 (a − lm a ˆ)φ2 (x)ϕ2 ||E||2 a + −a ˜a ˆ˙ 4δ
(34)
(35)
(36)
Inserting (25) into (36) generates
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a V˙ ≤ −c1 E T G−1 (¯ x)E + µ˜ aa ˆ+ . 4δ
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Note that
1 1 a ˜(a − a ˜) = (˜ aa − a ˜2 ) lm lm 1 1 1 2 ≤ ( a2 + a ˜ −a ˜2 ) lm 2 2 1 2 ≤ (a − a ˜2 ) 2lm
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a ˜a ˆ=
(38)
Thus, we get µ 2 µ 2 a V˙ ≤ −c1 E T G−1 (¯ x)E − a ˜ + a + 2lm 2lm 4δ ≤ −lV + Θ 14
(39)
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l = min{2c1 , µ} a µ 2 a + . Θ= 2lm 4δ
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with
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Consequently, from (39), it can be concluded that V enters the set Ω1 = {V |||V || ≤ Θl } as time goes by. Once V outside the set Ω1 , we have V˙ < 0. Therefore, there exists a finite time T0 such that V ∈ Ω1 for ∀t > T0 and the signals E and a ˜ are semiglobally uniformly ultimately bounded [32]. Then we establish the following important results. First we show that objective O1 is achieved. (n −1) 1) We first prove that E, εi , ε˙i , · · · , εi i , e, x and a ˆ for i = 1, · · · , m are lt bounded. Multiplying (39) by e yields d (V (t)elt ) ≤ Θelt . dt
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Integrating (40) over [0, t] leads to
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Therefore, V ∈ L∞ for any bounded initial conditions, which indicates that E ∈ L∞ and a ˆ ∈ L∞ . According to the definition of Ei in (20), it is seen that (ni −1) εi , ε˙i , · · · , ε ∈ L∞ (i = 1, · · · , m). Note that ei = β −1 εi , it is ensured (ni −1) that ei , e˙i , · · · , ei are bounded as β −1 is bounded, which further implies (n−1) that yi , y˙ i , · · · , yi , x¯, x are also bounded as desired signal ydi and its up to ni th order derivatives are bounded. ˙ ε(ni ) , e(ni ) , y (n) , a 2) Next we prove E, ˆ˙ and the control signal u are bounded. i i As x and E are bounded, then from (30) and (31) it follows that ϕ and φ(x) are bounded. From (24) and (25), it is ensured that u ∈ L∞ and a ˆ˙ ∈ L∞ . In (n ) (n ) addition, one can conclude from (1) that y (n) , ei i and εi i are also bounded. 3) Finally, we show that the proposed control scheme does not involve excessively large initial control effort. Note that a ˆ(0) can be set as 0, then from (24) the initial control signals u(0) is u(0) = 0, 15
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thus excessively large initial control signals are avoided. Next we show that objective O2 is obtained. We examine the tracking performance of the developed control strategy. x)E ≤ V , then we have from (41) that Note that 12 σ(G−1 )||E||2 ≤ 12 E T G−1 (¯ s 2Θ + 2V (0) l ||E|| ≤ := BE . (43) σ(G−1 ) We have from (20) that
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Ei (t) = (
Now, we transform (44) into the following form: wi,1 (t) = εi (t)
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wi,2 (t) = w˙ i,1 (t) + λi wi,1 (t) = (
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wi,ni −1 (t) = e−λi t wi,ni −1 (0) +
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|wi,1 (t)| ≤ |wi,1 (0)| +
|wi,2 (0)| |wi,ni −1 (0)| BE + ··· + + ni −1 := Bεi . (48) ni −2 λi λi λi 16
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Therefore, there exists a bound Bε =
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Since e = β −1 ε, by the definition of β as in (14), we further have ||e|| ≤ (1 − bf )κ(t)−1 Bε + bf Bε .
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There are two parts in (50): the second part bf Bε represents the adjustable residual region, the first term (1 − bf )κ(t)−1 Bε is explicitly controlled at the rate no less than κ−1 , and we call κ−1 as the “accelerated decay rate”. 0 Therefore, the tracking error converges to the compact set Ω := {e : ||e|| ≤ bf Bε } at the decay rate no less than κ−1 . Finally we show that objective O3 is achieved. 0 As bf is free parameter chosen by the designer, the compact set Ω can be made arbitrarily small by properly selecting bf > 0 sufficiently small. Furthermore, the rate function κ is chosen from the pool of rate functions in [29] which independent of initial condition and any other parameters, thus can be pre-assignable. The proof is completed.
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Remark 4. It is interesting to note that the proposed control schemes (24) and (25) involve β in the control law and β 2 in the adaptive law, respectively. Here β and β 2 act as the ”soft” accelerator to control gain and adaptive gain because they strictly but gently increase with time and are upper bounded. It is such feature that gives rise to the desired control performance: achieving adjusting tracking precision at pre-assignable decay rate.
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3.2. Control under Actuation Failure In this section, we consider the control design for MIMO system (1) with partial loss of effectiveness and additive actuation faults, both time varying and undetectable.
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Theorem 2. Consider the nonlinear MIMO system (1) with asymmetric nonsmooth actuator saturation (2) or (3) and actuation faults (12) simultaneously. Under Assumption 1-3, let the control strategy be ua = −(δˆ aφ2 (x)ϕ2 )βE
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Proof: We first rewrite the dynamics (23) as
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E˙ = − c1 E + βG(¯ x)L(·)ρ(·)ua + β(F (x) + G(¯ x)$(u) + G(¯ x)L(·)ur ) + βη
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whose derivative along (52) can be expressed as
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where c1 is a positive design parameter, ∆(x) = G−1 (¯ x)(F (x) + G(¯ x)$(u) + G(¯ x)L(·)ur ). By using the similar analysis in Theorem 1, the time derivative of V1 is V˙ 1 ≤ −c1 E T G−1 (¯ x)E + βE T L(·)ρ(·)ua + β||E||aφ(x)ϕ.
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The Lyapunov function candidate is chosen as follow: V = V1 +
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where V1 is given by (53), a ˜ = a − lm ρm a ˆ, and ρm is defined as in (13). Furthermore, by using the analysis similar to that used in the proof of Theorem 1 (We omit it to save space), it is not difficult to achieve the control objectives O1 ), O2 ), O3 ). The proof is completed. Remark 5. Both the healthy actuation and actuation failure are considered in this paper and the control structure do not need to reconfigure before and after the occurrence of actuator faults. In addition, a separate fault detection, isolation and identification (FDII) unit does not required in our control approach. 18
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4. Simulation results
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To verify the effective of the proposed method to a real world, a two-joint rigid-link robotic manipulator system is used for simulation verification. The dynamic equations of this MIMO system are given by [12]: M (q)¨ q + C(q, q) ˙ q˙ + G(q) + τd = τ
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where the state vector q ∈ R2 denotes the joint position, τd is the external disturbances, M (q) ∈ R2×2 is the inertia matrix of the manipulator, C(q, q) ˙ ∈ R2×2 is the centripetal Coriolis matrix of the manipulator, G(q) ∈ R2 is gravitational vector and τ ∈ R2 is the manipulators torque input vector, where %1 + %2 + 2%3 cos(q2 ) %2 + %3 cos(q2 ) M (q) = , (58) %2 + %3 cos(q2 ) %2 −%3 q˙2 sin(q2 ) −%3 (q˙1 + q˙2 ) sin(q2 ) C(q, q) ˙ = , (59) −%3 q˙2 sin(q2 ) 0 %4 cos(q1 ) + %5 cos(q1 + q2 ) G(q) = , (60) %5 cos(q1 + q2 ) 0.2 sin(t) h1 (u1 ) τd = , τ= , (61) 0.2 sin(t) h2 (u2 ) % = [%1 , %2 , %3 , %4 , %5 ] = [3.5, 0.76, 0.87, 3.04, 0.87].
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Note that hi (ui ) for i = 1, 2 are the considered Actuation Mode B (3) which represent nonsmooth asymmetric saturation with dead-zone. The control objective is to force the system state q1 and q2 to track the desired trajectories qd1 = 1.0 sin(t) and qd2 = 0.85 cos(t), respectively. To verify the effectiveness of the proposed method, two different cases to be considered in the simulation. Case 1: Under healthy actuation In this case, the proposed neural-adaptive control is compared with the PI control in [12]. To verify the proposed control gives rise to much better control performance as compared with the PI control method, we use the same shared parameters and initial conditions. The actuation parameters being chosen as: δ¯ = 8, δ = 9, b1 = 0.3, b2 = 0.6, and l1 = l2 = 1. The initial conditions are selected as q1 (0) = 0.6, q2 (0) = 0.5, q˙1 (0) = 0.3, q˙2 (0) = 0.5. Controller u = [u1 , u2 ]T is determined according to Theorem 1. The control 19
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parameters in this simulation are chosen as: δ = 0.1, µ = 1, κ = et , bf = 0.05, and choose the total number of neurons p = 89, the center of the neural network τ = 0, the same width for each Gaussian function φ2 = 9. The simulation results are shown in Fig. 2, where Fig.2 (a) and Fig. 2 (b) are the evolution of each component of the tracking error, |e1 | and |e2 |, under different control methods. It is seen that with the PI control in [12] the state can follow the desired trajectories, despite the external disturbance and actuator saturation. Whereas, the convergence precision is not high. For the control method proposed in this paper, the high tracking precision and fast rate of convergence are obtained. Fig.2 (c) and Fig. 2 (d) show that the actuator outputs are saturated, but the actuator outputs generated by the proposed control are much smoother than the PI control in [12]. After the comparison of the simulation results between the PI control in [12] and the proposed control method, it is easy to observe that the proposed neuroadaptive control has much better transient and steady-state performance. Case 2: Under actuation failure In this case, we consider the trajectory tracking problem for robotic manipulator system (57) with asymmetric non-smooth saturation as well as undetectable actuation faults simultaneously. Both additive and actuation effectiveness faults are determined by
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Controller u = [u1 , u2 ]T is determined according to Theorem 2. The control parameters in this simulation are chosen as: δ = 0.05, µ = 5, bf = 0.05, and choose the total number of neurons p = 89, the center of the neural network τ = 0, the same width for each Gaussian function φ2 = 9. The simulation results are shown in Fig. 3, where Fig.3 (a) and Fig. 3 (b) are the evolution of each component of the tracking error, |e1 | and |e2 |, under different rate functions (1, t2 + 1, et ). Among these, κ = 1 is the traditional case without acceleration. As can be seen, the high tracking precision and well-shaped transient response are obtained, as compared with the tradition control method. Moreover, it is able to achieve assignable accelerated decay rate by adjusting the rate function κ, which confirms the theoretical prediction. Thus, we can choose proper decay speed as required by adjusting the rate function κ. The control input h1 (u1 ) and h2 (u2 ) are presented in Fig.3 (c) and Fig. 3 (d). It is interesting to observe that no 20
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more control effort is needed and the proposed neural-adaptive control in this paper smoother and less saturation as compared with the traditional control, implying less wearing to the actuator. As predicted theoretically, although the MIMO nonlinear systems under unknown anomaly actuation, the proposed neuro-adaptive control has much better transient and steadystate performance. 5. Conclusion
Acknowledgments
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In this paper, performance guaranteed tracking control for a class of MIMO nonlinear systems with unknown actuator saturation and failure is studied in this paper. The developed method relaxes the restrictions on CGMs and allows the actuator nonlinearities to be unknown. We have shown that the tracking error converges to a small residual set with the assignable decay rate which is determined by designer. Furthermore, all the internal signals are bounded and the control action is uniformly continuous everywhere. Extending the design method to a wider class of nonlinear systems (such as non-affine and strict feedback systems) represents an interesting topic for future research.
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This work was supported in part by the Technology Transformation Program of Chongqing Higher Education University under Grant KJZH17102, Natural Science Foundation of China (NSFC) under Grant 61773081, and in part by the Graduate Scientific Research and Innovation Foundation of Chongqing under Grant CYB17048.
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References [1] G. Tao, P.V. Kokotovoc, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York, John Willey & Sons, 1996.
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[2] Z. Zhang, S. Xu, B. Zhang, Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity. IEEE Trans. Autom. Control, 59 (5) (2014) 1336–1341.
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[3] L. Ma, Z. Wang, Y. Liu Y, et al. A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time-delays. Int. J. Robust and Nonlinear Control, (2017) DOI: 10.1002/rnc.3809. [4] C.Y. Wen, J. Zhou, Z.T. Liu, Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Trans. Autom. Control, 56 (7) (2011) 1672–1678.
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[5] K. Zhao, Y.D. Song, C.Y. Wen, Computationally inexpensive fault tolerant control of uncertain non-linear systems with non-smooth asymmetric input saturation and undetectable actuation failures. IET Control Theory & Appl., 10 (15) (2016) 1866–1873.
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[6] C. Ting, A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs. Int. J. Fuzzy Syst., 10 (1) (2008) 50–60.
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[7] M. Chen, S.S. Ge, B.B. Ren, Adaptive tracking control of uncertain MIMO nonlinear systems with input saturation. Automatica, 47 (3) (2011) 452–465.
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[8] M. Chen, S.S. Ge, B. How, Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans. Neur. Networks, 21 (5) (2010) 796–812.
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[9] W. Gao, R. Selmic, Neural network control for a class of nonlinear systems with actuator saturation. IEEE Trans. Neur. Networks, 17 (1) (2006) 147–156.
[10] T.P. Zhang, S.S. Ge, Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs. Automatica, 43 (6) (2007) 1021–1033. 22
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[11] T.P. Zhang, Y. Yi, Adaptive fuzzy control for a class of MIMO nonlinear systems with unknown dead-zones. Acta Automatica Sinica, 33 (1) (2007) 1021–1033.
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[12] Y. D. Song, J. X. Guo, X. C. Huang, Smooth neuro-adaptive PI tracking control of nonlinear systems with unknown and non-smooth actuation characteristics. IEEE Trans. Neur. Netw. Lear. Syst., (2016) DOI: 10.1109/TNNLS.2016.2575078.
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[13] J. D. Boskovic, J. A. Jackson, R. K. Mehra, et al, Multiple-model adaptive fault-tolerant control of a planetary lander. J. Guid. Control Dyn, 32 (6) (2009) 1812-1826.
[14] P. Panagi, M. M. Polycarpou, Decertralized fault tolerant control of a class of interconnected nonlinear systems. IEEE Trans. Autom. Control, 56 (1) (2011) 178-184.
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[15] M. M. Polycarpou, Fault accommodation of a class of multivariable nonlinear dynamical systems using a learning approach. IEEE Trans. Autom. Control, 46 (5) (2001) 736-742.
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[16] Y. D. Song, J. X. Guo, Neuro-adaptive fault-tolerant tracking control of Lagrange systems pursing targets with unknown trajectory. IEEE Trans. Industrial Electronics, 64 (5) (2017) 3913-3920.
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[17] Y. D. Song, X. C. Huang, C. Y. Wen, Robust adaptive fault-tolerant PID control of MIMO nonlinear systems with unknown control direction. IEEE Trans. Industrial Electronics, 64 (6) (2017) 4876-4884.
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[18] Q. Hu, B. Xiao, M. I. Friswell, Robust fault-tolerant control for spacecraft attitude stabilisation subject to input saturation. IET Control Theory and Applications, 5 (2) (2011) 271-282.
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[19] W. Guan, G.H. Yang, Adaptive fault-tolerant control of linear systems in the presence of actuator saturation and L-2-disturbances, in: Proceedings of the 20th Chinese Control and Decision Conference, Yantai, Peoples Republic of China, 2008, pp. 2977C2982. [20] D.E. Miller, Davison, E.J, An adaptive controller which provides arbitrarily good transient and steady-state response, IEEE Trans. Autom. Control, 36 (1) (1993) 68–81. 23
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[21] A. Ilchmann, E.P. Ryan, P. Townsend, Tracking with prescribed transient behavior for nonlinear systems of known ralative degree, SIAM J. Control Optim., 46 (1) (2007) 210–230.
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[22] C. P. Bechlioulis, G.A. Rovithakis, Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance, IEEE Trans. Autom. Control, 53 (9) (2008) 2090–2099. [23] G. Besan¸con, Global output feedback tracking control for a class of Lagrangian systems, Automatica, 36 (12) (2000) 1915–1921.
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[24] M. M. Arefi, M.R. Jahed-Motlagh, Observer-based adaptive neural control of uncertain MIMO nonlinear systems with unknown control direction, Int. J. Adapt. Control Signal Process, 27 (9) (2013) 741–754.
[25] A. Boulkroune, M. Tadjine, M. M’Saad, et al, Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction, Fuzzy Set. Syst., 161 (6) (2010) 797–820.
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[26] T. M. Apostol, Mathematical Analysis. MA, USA: Addison-Wesley, 1963.
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[27] S. N. Huang, K. K. Tan, and T. H. Lee, Automated fault detection and diagnosis in mechanical systems, IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., 37 (6) (2007) 1360-1364.
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[28] X. Tang, G. Tao, and S. M. Joshi, Adaptive actuator failure compensation for nonlinear MIMO systems with an aircraft control application, Automatica, 43 (11) (2007) 1869-1883.
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[29] Y. D. Song, K. Zhao, Accelerated adaptive control of nonlinear uncertain systems[C]//American Control Conference (ACC), 2017: 2471-2476.
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[30] Y. D. Song, K. Zhao, M. Krstic, Adaptive Control with Exponential Regulation in the Absence of Persistent Excitation, IEEE Trans. Autom. Control, (2016) DOI: 10.1109/TAC.2016.2599645. [31] R. M. Sanner, J.J.E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neur. Networks, 3 (6) (1992) 837–863. [32] S. S. Ge, C. Wang. Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans. Neur. Networks, 15 (3) (2004) 674-692. 24
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Kai Zhao received the M.S. degree from the School of Automation, Chongqing University, Chongqing, China, in 2015, where he is currently pursuing the Ph.D. degree. His current research interests include intelligent control, adaptive control, robust control, and fault
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Liu He received the M.S. degree from tolerant control. University of Electronic Science and Technology of China, Chengdu, China, in 2014. He is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. His current research interests include intelligent control, machine learning, and computer vision.
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Ye Cao received the M.S. degree from Shaanxi normal university, Xian, China, in 2016. She is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. Her current research interests include robust adaptive control, neural net-
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Yongduan Song work, and event-triggered control. received his Ph.D. degree in electrical and computer engineering from Tennessee Technological University, Cookeville, USA, in 1992. He held a tenured Full Professor position with North Carolina A&T State University, Greensboro, from 1993 to 2008 and a Langley Distinguished Professor position with the National Institute of Aerospace, Hampton, VA, from 2005 to 2008. He is now the Dean of School of Automation, Chongqing University, and the Founding Director of the Institute of Smart Systems and Renewable Energy, Chongqing University. He was one of the six Langley Distinguished Professors with the National Institute of Aerospace (NIA), Founding Director of Cooperative Systems at NIA. He has served as an Associate Editor/Guest Editor for several prestigious scientific journals. Prof. Song has received several competitive research awards from the National Science Foundation, the National Aeronautics and Space Administration, the U.S. Air Force Office, the U.S. Army Research Office, and the U.S. Naval Research Office. His research interests include intelligent systems, guidance navigation and control, bio-inspired adaptive and cooperative systems, rail traffic control and safety, and smart grid.
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Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach Ye Cao, Yongduan Song, Kai Zhao, Liu He PII: DOI: Reference:
S0925-2312(17)31709-5 10.1016/j.neucom.2017.10.058 NEUCOM 19038
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
4 June 2017 22 September 2017 31 October 2017
Please cite this article as: Ye Cao, Yongduan Song, Kai Zhao, Liu He, Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.10.058
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Ye Cao
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, Kai Zhao
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, Liu He
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The key Laboratory of Dependable Service Computing in Cyber Physical Society, Ministry of Education, Chongqing University, Chongqing 400044, PR China 2 School of Automation, Chongqing University, Chongqing 400044, PR China
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, Yongduan Song
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Performance Guaranteed Tracking Control of Nonlinear Systems under Anomaly Actuation: A Neuro-adaptive Fault-tolerant Approach
Abstract
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In this work, we investigate the performance guaranteed tracking control problem of a class of multi-input multi-output (MIMO) nonlinear systems with anomaly actuation. By introducing new forms of parameter estimation error together with Lyapunov function, and using neural network approach, the obstacles caused by anomaly actuation can be handled gracefully and the assumptions on control gain matrices in existing results are significantly relaxed. Furthermore, a strictly increasing function is introduced to form a scaling speed transformation, which directly impacts both the controller law and the adaptive law, accelerating the learning rate and thus enhancing the tracking performance. It is shown that all the closed-loop signals are uniformly bounded and the tracking error converges to an adjustable residual set with a pre-assignable decay rate during the tracking process. Simulation results are presented to illustrate the effectiveness of the proposed scheme.
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Keywords: neuro-adaptive control, anomaly actuation, performance guaranteed.
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1. Introduction Actuators play a crucial role in any control system, but most existing works are based on the assumption that the actuator operates normally all Email address: [email protected] (Yongduan Song
Preprint submitted to Elsevier
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November 3, 2017
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the time. However, actuator might become defective suddenly or unnoticeably, resulting in so called anomaly actuation during a long term operation. Anomaly actuation arises from uncertain actuator nonlinearities, unknown input saturation or faulty actuation. Among them nonsmooth nonlinearities occur in many actuators, such as piezoelectric translators, mechanical connections, electric servomotors and other areas [1]. As we all known, the typical actuator nonlinearities include saturation, deadzone, backlash and hysteresis [2] [3]. In practice, actuator saturation is one of most important nonsmooth nonlinearities which is unavoidable in most actuators in many industrial control systems, and when an actuator has reached an input saturation limit, any efforts to further increase the actuator output would not make any variation of the output, so the saturation nonlinearity often limits system performance or even results in instability of the system. There is a rich collection of methods on dealing with actuator saturation. In [4], a robust adaptive control method is proposed for a class of uncertain nonlinear systems in the presence of input saturation and external disturbance. The Nussbaum function is introduced to compensate for the nonlinear term arising from the actuator saturation. In [5], a robust fault-tolerant control is developed for the strict-feedback systems with parametric uncertainties and asymmetric nonsmooth saturation. An adaptive fuzzy backstepping control approach for a class of uncertain nonlinear time-delay systems in the presence of actuator saturation is presented in [6]. The aforementioned results investigate the control problem of single-input single-output (SISO) nonlinear systems, and the nonlinear functions in [4] and [5] must satisfy linear parameterized condition. However, most practical multi-input multi-output (MIMO) systems, such as electrical machines, robotic manipulator, high speed trains, etc., are strongly coupled, making the control problem rather complicated. Certain prior knowledge or constraint on the control gain matrices (CGMs) is required in order to handle such systems. In [7] [8] under the assumption that the bounds of input constraint and the nonlinear functions in the saturation are exactly known, an adaptive tracking control is proposed for a class of multi-input multi-output (MIMO) nonlinear systems. In [9], a network-based actuator saturation compensation scheme is presented for nonlinear systems in Brunovsky canonical form, but the method in [9] can only deal with linear asymmetric saturation and the CGMs must be completely known for control design. To facilitate the stability analysis and control design, [10][11] consider a class of MIMO nonlinear systems with a lower triangular structure CGMs under actuator dead-zones. In the more general case as studied in 2
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[12], a neuro-adaptive PI control for nonlinear systems with unknown and nonsmooth actuation characteristics is developed on condition that the matrix B(x, t) + B T (x, t) is symmetric positive, where B(x, t) = G(x, t)L(·), G(x, t) represents the CGM and L(·) is a diagonal matrix due to actuator saturation. It is noted that this condition is not easy to satisfy in practical situations. At the same time, actuator failure is one of the most common phenomenons in anomaly actuation. In the past decades, there is a rich collection of methods on dealing with actuator failure. In [13][14][15], based on fault detection and diagnosis, reconfigurable control schemes are proposed such that the impacts of failure can be compensated and the stability as well as the acceptable performance of the system can be maintained. Whereas if the actuation faults are time-varying and undetectable faults, the robust control theory by using unchangeable controller throughout the failure-free case and the failure case is a good way to solve the actuator failure. The typical approaches have been developed in [5][16][17]. However, the assumption on the matrix B(x, t) + B T (x, t) is also required, where B(x, t) = G(x, t)ρ(·), G(x, t) represents the positive definite control gain matrix, ρ(·) is a diagonal matrix due to actuator failure. In addition, these results do not take into account the non-smooth actuator nonlinearities, and thus cannot be applied directly to systems with limited actuator input. Few researchers investigate actuator faults and actuator output constraints simultaneously. In [18] these two issues are considered for a particular spacecraft system model, and in [19] an adaptive fault-tolerant controller are provided for a linear system with actuator saturation, whereas the controllers require large computation power and are difficult to implement. Thus this paper will investigate more general MIMO nonlinear systems under undetectable failure and nonsmooth actuator saturation. Another important issue that should be explicitly addressed in control design for MIMO nonlinear systems under anomaly actuation is the tracking control performance in terms of the transient behavior and the steady-state response. When actuator failures occur or the actuator reach an saturation limit, tracking performance degradation is inevitable. Earlier contributions addressing the performance issue include those by [20][21][22]. In [20], by using a switching strategy for non-decreasing adaptive gain and requiring a prior bound on the initial data, controllers are designed for minimum-phase linear time invariant systems which guarantee the error to be less than an arbitrary prespecified constant after a transient with an overshoot below 3
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a prespecified upper bound. The funnel method [21] provides arbitrarily assignable transient rate through the shape of the funnel, but it only guarantees approximate performance for the system output and it has no adaption and uses an error-dependent gain which lead to infinity control gain once the error approaches the funnel boundary. Thus, how to enhance the transient and steady-state performance needs to be further investigated. Note that, on one hand, with the common sense that the enhanced performance can be obtained by speeding up the parameter updating rate and/or cranking up the feedback control gain, and on the other hand, it is unclear by how much and in what way to increase the control gain and the parameter updating rate to achieve improved tracking performance. As a matter of fact, simply choosing a high constant control gain or simply increasing the parameter updating rate by a large constant value in the adaptive law does not necessarily produce better performance. The objective of this work is to design an intelligent nonlinear adaptive controller that can enhance the system performance in the presence of actuator saturation and actuator failure. A neuro-adaptive prescribed performance control is proposed for a class of MIMO nonlinear systems with unknown asymmetric input saturation as well as actuation faults, achieving adjustable tracking precision with pre-specifiable decay rate during transient period. Based upon new forms of parameter estimation error and Lyapunov function and combining neural network, the the requirement on the symmetric positive definite condition of matrix B(x, t) + B T (x, t) is not needed, which significantly relaxes the assumption on CGMs. The proposed control also allows the mode and the rate of convergence during the transient period to be pre-specifiable explicitly by choosing the rate function and the design parameter properly. The rest of this paper is organized as follows. In section 2, the control problem and some preliminaries are introduced. Section 3 gives controller design and stability analysis. Simulation results are presented in section 4 to illustrate the effectiveness of the proposed scheme. Finally, we conclude the paper in Section 5.
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2. Problem formulation and preliminaries
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2.1. System description Consider the following class of nonlinear MIMO systems described by: y1 (n1 ) .. x)H(u) (1) = F (x) + G(¯ . ym (nm ) (n −1)
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where y = [y1 , · · · , ym ]T ∈ Rm and x = [y1 , y˙ 1 , · · · , y1 1 , · · · , ym , y˙ m , · · · , P (n −1) ym m ] ∈ Rn with n = m k=1 nk are the output vector and the state vector, respectively; F (x) ∈ Rm are continuous unknown nonlinear function vector, and G(¯ x) ∈ Rm×m are unknown nonlinear C1 function gain matrix (n −2) (n −2) of x¯ = [y1 , · · · , y1 1 , · · · , ym , · · · , ym m ] ∈ Rn−m , and H(u) = [h1 (u1 ), · · · , hm (um )]T ∈ Rm represents the control vector of the system with unknown actuation characteristics, where u is the actual control design. In this paper, two typical actuation models are shown in Fig. 1. Model A. Asymmetric nonsmooth saturation with unknown slope [5] ¯ ui > uma1 δ, l(ui ), − uma2 ≤ ui ≤ uma1 i = 1, · · · , m (2) hi (ui ) = −δ, ui < −uma2
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where uma1 > 0 and −uma2 < 0 represent the break points. Model B. Asymmetric nonsmooth saturation with dead zone [5] ¯ δ, ui > umb1 l1 (ui − b1 ), b1 < ui ≤ umb1 0, − b2 ≤ ui ≤ b1 i = 1, · · · , m hi (ui ) = l2 (ui + b2 ), − umb2 ≤ ui < −b2 −δ, ui < −umb2
(3)
where l1 and l2 are the slope of the dead-zone characteristic, b1 > 0, −b2 < 0, umb1 > 0 and −umb2 < 0 represent the break points. Remark 1. The MIMO system (1) can describe many physical systems such as robot manipulators, satellites and electrical machines. Even without actuator nonlinearity, the system was studied in [24][25] or the special case in 5
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(−uma2, −δ)
u
Model A
(−umb2, −δ)
−b2 b1
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Model B
¯ (umb1, δ)
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¯ (uma1, δ)
Figure 1: The approximation to nonsmooth saturation functions (solid-line: asymmetric nonsmooth saturation function hi (ui ); dot-line: the smooth approximating function Γi (ui ))
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[22][23]. It means that F (x) can encompass many time-varying uncertainties and external disturbances. The control gain matrix G(¯ x) depends only on the state vector x¯.
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To deal with the nonsmooth and asymmetric actuator nonlinearities, a well-defined smooth function is used to approximate the saturation function, which takes the form as [5]
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hi (ui ) = Γi (ui ) + $i (ui ), i = 1, · · · , m ¯ (+ηui ) − δe−(+ηui ) δe Γi (ui ) = (+ηui ) e + e−(+ηui )
(4) (5)
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¯ and η > 0 is a design parameter. Fig. 1 shows the with = 0.5 ln(δ/δ) approximation to the two type nonsmooth saturation functions. As the bounded property of the function Γi (ui ) and saturation function hi (ui ), we know that function $i (ui ) is bounded, i.e. |$i (ui )| < Di where Di is a positive and unknown constant. With the help of Γ(u), the original system (1) can be expressed as y (n) = F (x) + G(¯ x)(Γ(u) + $(u)) (n )
(n )
where y (n) = [y1 1 , · · · , ym m ]T , [$1 (u1 ), · · · , $m (um )]T .
(6)
Γ(u) = [Γ1 (u1 ), · · · , Γm (um )]T , $(u) = 6
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According to mean value theorem [26], there exist constants αi (0 < αi < 1) such that Γi (ui ) = Γi (ui0 ) + li (ξi )(ui − ui0 ) (7)
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i (ui ) where li (ξi ) = ∂Γ∂u |ui =ξi and ξi = αi ui + (1 − αi )ui0 . As Γi (ui ) are noni decreasing functions, there exist some positive constants lm and lM such that
0 < lm < li (ξi ) < lM < ∞, i = 1, · · · , m.
(8)
By choosing ui0 = 0 and using the fact that Γi (0) = 0, it is easy to get that Γi (ui ) = li (ξi )ui .
(9)
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With the aid of (9), the system (6) can be expressed as
y (n) = F (x) + G(¯ x)L(ξ)u + G(¯ x)$(u)
L(ξ) =
l1 (ξ1 )
l2 (ξ2 )
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where
..
.
lm (ξm )
.
(10)
(11)
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Remark 2. As can be seen from Figure 1, the control input hi (ui ) involves non-smooth variation at the break points, which poses technical challenge for direct control design and analysis. Here in this work, motivated by [5], we introduce a bounded and smooth function Γi (ui ) to approximate hi (ui ). As Γi (ui ) is a smooth function of ui , we can employ the mean value theorem on function Γi (ui ), so that u can be separated explicitly in system (10). This treatment facilitates the development of the control scheme to be presented shortly.
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As unanticipated actuator faults may occur, inevitable in practice for long-term operation, we additionally include such scenario in the model in which the actual control vector u and the designed control vector ua are no longer the same anymore. Instead, they are linked via u = ρ(tρ , t)ua + ur (tr , t)
(12)
where ρ(·) = diag{ρi } ∈ Rm×m , i = 1, . . . , m, is the ‘healthy indicator’ reflecting the effectiveness of the actuator, ur is the uncontrollable portion of 7
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actuation input, tρ and tr denote the time instant at which the loss of actuation effectiveness fault and the additive actuation fault occur, respectively. Both ρ(tρ , t) and ur (tr , t) are assumed to be unknown bounded, time-varying and undetectable. In this work, we consider the case that 0 < ρi ≤ 1, i.e., although loosing its effectiveness the actuation is still functional such that u can be influenced by the control input ua all the time, as considered in several studies [5][16][17]. To continue, we need the following assumptions, which are quite standard in the literature.
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Assumption 1. The desired tracking trajectory ydi , i = 1, · · · , m, as well as their derivatives up ni are known smooth functions of time and bounded. And the state vector x is available for control design. Assumption 2. The control gain matrix G(¯ x) is known to be either positive definite or negative definite. Without losing generality, this paper assumes that G(¯ x) is positive definite.
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Assumption 3. ρ(·) and ur (·) are unknown, possibly fast time-varying and unpredictable but bounded, i.e., ||ur (·)|| ≤ u¯r < ∞, 0 < ρm ≤ ρi ≤ 1
(13)
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for i = 1, . . . , m, where ρm and u¯r are unknown positive constants.
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Remark 3. Assumption 1 is commonly imposed in most existing works in addressing the tracking control problems [28]. The Assumption 2 guarantees that the system as described in (1) is controllable, which has been commonly used in most existing works in addressing MIMO systems [22][23]. As for Assumption 3, it is noted that most fault detection and diagnosis (FDD) or fault detection and identification (FDI) based fault tolerant control implicitly assume that the faults vary with time slowly enough to allow for timely fault identification and diagnosis [27] or that one has enough information on the faults to carry out parametric decomposition [28], while Assumption 3 imposes no such restriction, thus seems more practical. 2.2. Rate function Definition 1. [29][30] A real function κ(t) is qualified as a rate function as long as the following properties hold: 8
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(1) κ(t) is a positive and monotonously increasing function of time for all t ∈ [0, ∞) such that κ(t)−1 is positive and strictly decreasing and lim κ(t)−1 = t→∞ 0; (2) κ(0) = 1; (3) κ(t) is C n for t ∈ [0, ∞).
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Clearly, there are many functions such as 1 + t2 , et and (1 + t2 )et bear these properties, and a pool of rate functions has been established in [29]. Based upon the rate function κ(t) we construct the following time-varying scaling function 1 , (14) β(t) = (1 − bf )κ(t)−1 + bf
where 0 < bf ≤ 1 is a design parameter. From the definition of β(t) as given in (14), it can be seen that β is a monotonically increasing function of time with upper bound, and exhibits the following easily verifiable properties: (1) β monotonically increases from 1 to b1f and β ∈ [1, b1f ) for all t ≥ 0; (2) β 2 monotonically increases from 1 to b12 and β 2 ∈ [1, b12 ) for all t ≥ 0; f
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(3) It holds that β(0) = 1 and
(1 − bf )κ˙ (1 − bf + bf κ)2 (1 − bf )[¨ κ(1 − bf + bf κ) − 2bf κ˙ 2 ] β¨ = (1 − bf + bf κ)3
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β˙ =
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(15) (16)
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Since β is bounded and smooth for t ∈ [0, ∞), then β˙ is bounded, note that κ˙ exists and smooth, thus β˙ is smooth and bounded, which implies that β¨ is bounded and smooth as κ˙ and κ ¨ are well-defined. In this matter, it can be (j) found that β (j = 0, 1, · · · , ∞) is bounded and smooth for t ∈ [0, ∞).
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2.3. Neural networks and function approximation Neural network has been widely used for control design under various conditions [31]. In this paper, radial basis function neural networks (RBFNNs) will be employed to approximate the unknown functions. Based on the wellknown approximation property, given a continuous nonlinear scalar function
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Π(x) on a compact set Ωx , there exists an ideal RBFNN capable of approximating the smooth nonlinear function to an arbitrary degree of accuracy as follows: Π(x) = W T S(x) + (x), |(x)| ≤ ¯, ∀x ∈ Ωx , (17)
(x − τi )T (x − τi ) ), i = 1, · · · , p, ψ2
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Si (x) = exp(−
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where W ∈ Rp is the weight vector with p the number of nodes, S(x) = [S1 (x), · · · , Sp (x)]T ∈ Rp is the basis function vector, (x) denotes the approximation error, and ¯ is a constant. We choose Si (x) in S(x) as Gaussian functions, given by
where τi ∈ Rn and ψ ∈ R are constants called the center and width of the basis function, respectively. 3. Control Design and Stability Analysis
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We now start designing a control scheme that is able to achieve adjustable tracking precision at the prescribed rate convergence. Define e = y − yd as the output tracking error with yd = [yd1 , · · · ydm ]T ∈ Rm being the desired trajectories. In order to improve tracking performance, we introduce a error transformation εi (t) = β(t)ei (t)
(19)
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and we further define a new filtered variable Ei (t) as d + λi )ni −1 εi (t), i = 1, · · · , m dt
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Ei (t) = (
(20)
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where λi (i = 1, · · · , m) are positive constants to be selected, so that if Ei (t) goes to zero or within a bound, so does εi (t). It is worth noting that the filter variable Ei (t) as defined in (20) is not directly based on ei (t), but rather, on the transformed error εi (t). Such treatment, together with other design skills, allows for the aforementioned control objectives to be achieved concurrently, as seen shortly.
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Differentiating (20) with respect to time we have n i −1 X j=1
= +
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(ni )
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(t)
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where Cnj i =
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=
n i −1 X
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(t) − ydi i (t))
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(ni − 1)! , j!(ni − j − 1)! (ni −j)
Cnj i −1 λji εi
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j=1
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Combining (10) and (21) yields that (n) E˙ = ν + β(F (x) + G(¯ x)L(·)u + G(¯ x)$(u) − yd )
(22)
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(n −1) where E˙ = [E˙ 1 (t), · · · , E˙ m (t)]T and ν = [v1 (e1 , · · · , e1 1 , t), (n −1) · · · , vm (em , · · · , em m , t)]T is computable. By adding and subtracting −c1 E on the right side on (22), we have
E˙ = −c1 E + βG(¯ x)L(·)u + β(F (x) + G(¯ x)$(u)) + βη
(23) (n)
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where c1 is a positive design parameter and η = β −1 ν + β −1 c1 E − yd available in constructing the controller.
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3.1. Control under Healthy Actuation Now we consider the control design under healthy actuation, where ρi = 1, i = 1, · · · , m and ur = 0, i.e., ua = u in (12). 11
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By making use of the β function as defined in (14), we construct the following accelerated neuro-adaptive control scheme u = −(δˆ aφ2 (x)ϕ2 )βE a ˆ˙ = −µˆ a + δβ 2 φ2 (x)ϕ2 ||E||2 ,
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with the adaptive law
(24)
(25)
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where ϕ = 1 + ||η|| + ||β −1 E||, φ(x) = ||S(x)|| + 1 with S(x) being defined as in (18), δ > 0 and µ > 0 are chosen by the designer. Note that the proposed control exhibits several appealing structural features: 1) it bears the feedback control form, in which the feedback gain is composed of constant gain and adaptive gain. 2) it is built upon the feedback of βE, the scaled version of E. 3) it blends the rate function β into the adaptive gain to accelerate the tracking process. These combined features play a vital role in establishing the following result.
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Theorem 1. Consider the nonlinear MIMO system (1) with asymmetric nonsmooth actuator saturation (2) or (3). Under Assumption 1-2, if the control scheme (24) and the adaptive law (25) are applied, then for any bounded initial conditions, the following objectives are achieved. O1 ) All the internal signals are bounded; the control action is uniformly continuous everywhere; and no excessively large initial control effort is involved; O2 ) The tracking error e converges to an adjustable residual set with the assignable decay rate during the main tracking process. O3 ) Not only the decay rate of the tracking error are independent on system initial value and can be pre-assignable but also the compact set can be made small enough by properly adjusting the design parameter.
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Proof. Firstly we consider the following quadratic form 1 x)E V1 = E T G−1 (¯ 2
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(26)
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whose derivative along (23) can be expressed as
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1 x)E V˙ 1 =E T G−1 (¯ x)E˙ + E T G˙ −1 (¯ 2 = − c1 E T G−1 (¯ x)E + βE T L(·)u + βE T G−1 (¯ x)η 1 + βE T G−1 (¯ x)(F (x) + G(¯ x)$(u)) + E T G˙ −1 (¯ x)E 2 = − c1 E T G−1 (¯ x)E + βE T L(·)u + βE T G−1 (¯ x)η 1 + βE T ∆(x) + E T G˙ −1 (¯ x)E, 2
(27)
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where c1 is a positive design parameter, η = β −1 ν + β −1 c1 E − yd is available in constructing the controller and ∆(x) = G−1 (¯ x)(F (x) + G(¯ x)$(u)). According to the fact that $i (u) in (4) is bounded, it can be concluded that there exist continuous functions Π(x) ∈ R such that max{|∆(x)|, ||G−1 (¯ x)||, ||G˙ −1 (¯ x)||} ≤ Π(x)
(28)
for all x ∈ Rn and t ≥ 0. It follows from (27) and (28) that
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V˙ 1 ≤ − c1 E T G−1 (¯ x)E + βE T L(·)u 1 x)E] + βE T [∆(x) + G−1 (¯ x)η + β −1 G˙ −1 (¯ 2 ≤ − c1 E T G−1 (¯ x)E + βE T L(·)u + β||E||Π(x)ϕ,
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where
1 ϕ = 1 + ||η|| + β −1 ||E||. 2
(29)
(30)
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Since Π(x) are unknown, we employ RBFNNs to approximate them as follows: Π(x) =W T S(x) + (x)
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(31) ≤||W T ||||S(x)|| + ||¯|| ≤aφ(x), where |(x)| ≤ ¯, a = max ||W T ||, ||¯|| , φ(x) = ||S(x)|| +1 for all x ∈ Ωx and Ωx is a compact set. Substituting (31) into (29), it becomes V˙ 1 ≤ −c1 E T G−1 (¯ x)E + βE T L(·)u + β||E||aφ(x)ϕ. 13
(32)
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Now we define a Lyapunov function candidate for the closed-loop system as follow: 1 2 a ˜ , (33) V = V1 + 2lm where V1 is given by (26). Note that a ˆ is the estimation of the unknown weight a, and we introduce a parameter estimation error of the form a ˜ = a − lm a ˆ, where lm is defined as (8). By blending such error into second part of Lyapunov function candidate, the unknown actuator saturation can be processed gracefully. Differentiating (33) and using (24), we have
From Young’s inequality we have
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V˙ ≤ − c1 E T G−1 (¯ x)E − δβ 2 lm a ˆφ2 (x)ϕ2 ||E||2 + β||E||aφ(x)ϕ − a ˜a ˆ˙ . βφ(x)ϕ||E|| ≤ δβ 2 φ2 (x)ϕ2 ||E||2 + Substituting (35), it becomes
1 . 4δ
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V˙ ≤ − c1 E T G−1 (¯ x)E + δβ 2 (a − lm a ˆ)φ2 (x)ϕ2 ||E||2 a + −a ˜a ˆ˙ 4δ
(34)
(35)
(36)
Inserting (25) into (36) generates
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a V˙ ≤ −c1 E T G−1 (¯ x)E + µ˜ aa ˆ+ . 4δ
(37)
Note that
1 1 a ˜(a − a ˜) = (˜ aa − a ˜2 ) lm lm 1 1 1 2 ≤ ( a2 + a ˜ −a ˜2 ) lm 2 2 1 2 ≤ (a − a ˜2 ) 2lm
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a ˜a ˆ=
(38)
Thus, we get µ 2 µ 2 a V˙ ≤ −c1 E T G−1 (¯ x)E − a ˜ + a + 2lm 2lm 4δ ≤ −lV + Θ 14
(39)
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l = min{2c1 , µ} a µ 2 a + . Θ= 2lm 4δ
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with
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Consequently, from (39), it can be concluded that V enters the set Ω1 = {V |||V || ≤ Θl } as time goes by. Once V outside the set Ω1 , we have V˙ < 0. Therefore, there exists a finite time T0 such that V ∈ Ω1 for ∀t > T0 and the signals E and a ˜ are semiglobally uniformly ultimately bounded [32]. Then we establish the following important results. First we show that objective O1 is achieved. (n −1) 1) We first prove that E, εi , ε˙i , · · · , εi i , e, x and a ˆ for i = 1, · · · , m are lt bounded. Multiplying (39) by e yields d (V (t)elt ) ≤ Θelt . dt
(40)
Integrating (40) over [0, t] leads to
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Θ . l
Θ (1 − e−lt ) l
(41)
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Therefore, V ∈ L∞ for any bounded initial conditions, which indicates that E ∈ L∞ and a ˆ ∈ L∞ . According to the definition of Ei in (20), it is seen that (ni −1) εi , ε˙i , · · · , ε ∈ L∞ (i = 1, · · · , m). Note that ei = β −1 εi , it is ensured (ni −1) that ei , e˙i , · · · , ei are bounded as β −1 is bounded, which further implies (n−1) that yi , y˙ i , · · · , yi , x¯, x are also bounded as desired signal ydi and its up to ni th order derivatives are bounded. ˙ ε(ni ) , e(ni ) , y (n) , a 2) Next we prove E, ˆ˙ and the control signal u are bounded. i i As x and E are bounded, then from (30) and (31) it follows that ϕ and φ(x) are bounded. From (24) and (25), it is ensured that u ∈ L∞ and a ˆ˙ ∈ L∞ . In (n ) (n ) addition, one can conclude from (1) that y (n) , ei i and εi i are also bounded. 3) Finally, we show that the proposed control scheme does not involve excessively large initial control effort. Note that a ˆ(0) can be set as 0, then from (24) the initial control signals u(0) is u(0) = 0, 15
(42)
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thus excessively large initial control signals are avoided. Next we show that objective O2 is obtained. We examine the tracking performance of the developed control strategy. x)E ≤ V , then we have from (41) that Note that 12 σ(G−1 )||E||2 ≤ 12 E T G−1 (¯ s 2Θ + 2V (0) l ||E|| ≤ := BE . (43) σ(G−1 ) We have from (20) that
d + λi )ni −1 εi (t), i = 1, · · · , m. dt
(44)
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Ei (t) = (
Now, we transform (44) into the following form: wi,1 (t) = εi (t)
d + λi )εi (t) dt d wi,3 (t) = w˙ i,2 (t) + λi wi,2 (t) = ( + λi )2 εi (t) dt ···
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wi,2 (t) = w˙ i,1 (t) + λi wi,1 (t) = (
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wi,ni (t) = w˙ i,ni −1 (t) + λi wi,ni −1 (t) = Ei (t) = (
d + λi )ni −1 εi (t). dt
(45)
Solving the differentiate equation (45) yields that
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wi,ni −1 (t) = e−λi t wi,ni −1 (0) +
Ei (1 − e−λi t ). λi
(46)
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Note that |Ei | ≤ ||E|| ≤ BE , it is obvious that |wi,ni −1 (t)| ≤ |wi,ni −1 (0)| +
BE := Bwi,ni −1 . λi
(47)
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By using the same analysis procedure, we have |wi,ni −2 (t)| ≤ |wi,ni −2 (0)| +
Bwi,ni −1 |wi,ni −1 (0)| BE ≤ |wi,ni −2 (0)| + + 2 λi λi λi
...
|wi,1 (t)| ≤ |wi,1 (0)| +
|wi,2 (0)| |wi,ni −1 (0)| BE + ··· + + ni −1 := Bεi . (48) ni −2 λi λi λi 16
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Therefore, there exists a bound Bε =
p Bε1 + · · · + Bεm such that
||ε|| ≤ Bε .
(49)
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Since e = β −1 ε, by the definition of β as in (14), we further have ||e|| ≤ (1 − bf )κ(t)−1 Bε + bf Bε .
(50)
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There are two parts in (50): the second part bf Bε represents the adjustable residual region, the first term (1 − bf )κ(t)−1 Bε is explicitly controlled at the rate no less than κ−1 , and we call κ−1 as the “accelerated decay rate”. 0 Therefore, the tracking error converges to the compact set Ω := {e : ||e|| ≤ bf Bε } at the decay rate no less than κ−1 . Finally we show that objective O3 is achieved. 0 As bf is free parameter chosen by the designer, the compact set Ω can be made arbitrarily small by properly selecting bf > 0 sufficiently small. Furthermore, the rate function κ is chosen from the pool of rate functions in [29] which independent of initial condition and any other parameters, thus can be pre-assignable. The proof is completed.
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Remark 4. It is interesting to note that the proposed control schemes (24) and (25) involve β in the control law and β 2 in the adaptive law, respectively. Here β and β 2 act as the ”soft” accelerator to control gain and adaptive gain because they strictly but gently increase with time and are upper bounded. It is such feature that gives rise to the desired control performance: achieving adjusting tracking precision at pre-assignable decay rate.
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3.2. Control under Actuation Failure In this section, we consider the control design for MIMO system (1) with partial loss of effectiveness and additive actuation faults, both time varying and undetectable.
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Theorem 2. Consider the nonlinear MIMO system (1) with asymmetric nonsmooth actuator saturation (2) or (3) and actuation faults (12) simultaneously. Under Assumption 1-3, let the control strategy be ua = −(δˆ aφ2 (x)ϕ2 )βE
(51)
with the update law (25), where ϕ = 1 + ||η|| + ||β −1 E||, φ(x) = ||S(x)|| + 1 with S(x) being defined as in (18), δ > 0 and µ > 0 are chosen by the designer. Then for any bounded initial conditions, the similar results as Theorem 1 can be achieved. 17
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Proof: We first rewrite the dynamics (23) as
(52)
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E˙ = − c1 E + βG(¯ x)L(·)ρ(·)ua + β(F (x) + G(¯ x)$(u) + G(¯ x)L(·)ur ) + βη
(n)
where c1 is a positive design parameter and η = β −1 ν + β −1 c1 E − yd is available in constructing the controller. By employing the same quadratic form 1 x)E V1 = E T G−1 (¯ 2
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whose derivative along (52) can be expressed as
1 V˙ 1 =E T G−1 (¯ x)E˙ + E T G˙ −1 (¯ x)E 2 = − c1 E T G−1 (¯ x)E + βE T L(·)ρ(·)ua + βE T G−1 (¯ x)η 1 + βE T ∆(x) + E T G˙ −1 (¯ x)E, 2
(53)
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where c1 is a positive design parameter, ∆(x) = G−1 (¯ x)(F (x) + G(¯ x)$(u) + G(¯ x)L(·)ur ). By using the similar analysis in Theorem 1, the time derivative of V1 is V˙ 1 ≤ −c1 E T G−1 (¯ x)E + βE T L(·)ρ(·)ua + β||E||aφ(x)ϕ.
(55)
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The Lyapunov function candidate is chosen as follow: V = V1 +
1 a ˜2 2lm ρm
(56)
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where V1 is given by (53), a ˜ = a − lm ρm a ˆ, and ρm is defined as in (13). Furthermore, by using the analysis similar to that used in the proof of Theorem 1 (We omit it to save space), it is not difficult to achieve the control objectives O1 ), O2 ), O3 ). The proof is completed. Remark 5. Both the healthy actuation and actuation failure are considered in this paper and the control structure do not need to reconfigure before and after the occurrence of actuator faults. In addition, a separate fault detection, isolation and identification (FDII) unit does not required in our control approach. 18
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4. Simulation results
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To verify the effective of the proposed method to a real world, a two-joint rigid-link robotic manipulator system is used for simulation verification. The dynamic equations of this MIMO system are given by [12]: M (q)¨ q + C(q, q) ˙ q˙ + G(q) + τd = τ
(57)
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where the state vector q ∈ R2 denotes the joint position, τd is the external disturbances, M (q) ∈ R2×2 is the inertia matrix of the manipulator, C(q, q) ˙ ∈ R2×2 is the centripetal Coriolis matrix of the manipulator, G(q) ∈ R2 is gravitational vector and τ ∈ R2 is the manipulators torque input vector, where %1 + %2 + 2%3 cos(q2 ) %2 + %3 cos(q2 ) M (q) = , (58) %2 + %3 cos(q2 ) %2 −%3 q˙2 sin(q2 ) −%3 (q˙1 + q˙2 ) sin(q2 ) C(q, q) ˙ = , (59) −%3 q˙2 sin(q2 ) 0 %4 cos(q1 ) + %5 cos(q1 + q2 ) G(q) = , (60) %5 cos(q1 + q2 ) 0.2 sin(t) h1 (u1 ) τd = , τ= , (61) 0.2 sin(t) h2 (u2 ) % = [%1 , %2 , %3 , %4 , %5 ] = [3.5, 0.76, 0.87, 3.04, 0.87].
(62)
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Note that hi (ui ) for i = 1, 2 are the considered Actuation Mode B (3) which represent nonsmooth asymmetric saturation with dead-zone. The control objective is to force the system state q1 and q2 to track the desired trajectories qd1 = 1.0 sin(t) and qd2 = 0.85 cos(t), respectively. To verify the effectiveness of the proposed method, two different cases to be considered in the simulation. Case 1: Under healthy actuation In this case, the proposed neural-adaptive control is compared with the PI control in [12]. To verify the proposed control gives rise to much better control performance as compared with the PI control method, we use the same shared parameters and initial conditions. The actuation parameters being chosen as: δ¯ = 8, δ = 9, b1 = 0.3, b2 = 0.6, and l1 = l2 = 1. The initial conditions are selected as q1 (0) = 0.6, q2 (0) = 0.5, q˙1 (0) = 0.3, q˙2 (0) = 0.5. Controller u = [u1 , u2 ]T is determined according to Theorem 1. The control 19
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parameters in this simulation are chosen as: δ = 0.1, µ = 1, κ = et , bf = 0.05, and choose the total number of neurons p = 89, the center of the neural network τ = 0, the same width for each Gaussian function φ2 = 9. The simulation results are shown in Fig. 2, where Fig.2 (a) and Fig. 2 (b) are the evolution of each component of the tracking error, |e1 | and |e2 |, under different control methods. It is seen that with the PI control in [12] the state can follow the desired trajectories, despite the external disturbance and actuator saturation. Whereas, the convergence precision is not high. For the control method proposed in this paper, the high tracking precision and fast rate of convergence are obtained. Fig.2 (c) and Fig. 2 (d) show that the actuator outputs are saturated, but the actuator outputs generated by the proposed control are much smoother than the PI control in [12]. After the comparison of the simulation results between the PI control in [12] and the proposed control method, it is easy to observe that the proposed neuroadaptive control has much better transient and steady-state performance. Case 2: Under actuation failure In this case, we consider the trajectory tracking problem for robotic manipulator system (57) with asymmetric non-smooth saturation as well as undetectable actuation faults simultaneously. Both additive and actuation effectiveness faults are determined by
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u1 = ρ1 ua1 + ur1 , ρ1 = 1 − 0.2 tanh(t), ur1 = 0.02 cos(2t), u2 = ρ2 ua2 + ur2 , ρ2 = 0.9 + 0.2 sin(πt), ur2 = 0.05 sin(3t).
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Controller u = [u1 , u2 ]T is determined according to Theorem 2. The control parameters in this simulation are chosen as: δ = 0.05, µ = 5, bf = 0.05, and choose the total number of neurons p = 89, the center of the neural network τ = 0, the same width for each Gaussian function φ2 = 9. The simulation results are shown in Fig. 3, where Fig.3 (a) and Fig. 3 (b) are the evolution of each component of the tracking error, |e1 | and |e2 |, under different rate functions (1, t2 + 1, et ). Among these, κ = 1 is the traditional case without acceleration. As can be seen, the high tracking precision and well-shaped transient response are obtained, as compared with the tradition control method. Moreover, it is able to achieve assignable accelerated decay rate by adjusting the rate function κ, which confirms the theoretical prediction. Thus, we can choose proper decay speed as required by adjusting the rate function κ. The control input h1 (u1 ) and h2 (u2 ) are presented in Fig.3 (c) and Fig. 3 (d). It is interesting to observe that no 20
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more control effort is needed and the proposed neural-adaptive control in this paper smoother and less saturation as compared with the traditional control, implying less wearing to the actuator. As predicted theoretically, although the MIMO nonlinear systems under unknown anomaly actuation, the proposed neuro-adaptive control has much better transient and steadystate performance. 5. Conclusion
Acknowledgments
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In this paper, performance guaranteed tracking control for a class of MIMO nonlinear systems with unknown actuator saturation and failure is studied in this paper. The developed method relaxes the restrictions on CGMs and allows the actuator nonlinearities to be unknown. We have shown that the tracking error converges to a small residual set with the assignable decay rate which is determined by designer. Furthermore, all the internal signals are bounded and the control action is uniformly continuous everywhere. Extending the design method to a wider class of nonlinear systems (such as non-affine and strict feedback systems) represents an interesting topic for future research.
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This work was supported in part by the Technology Transformation Program of Chongqing Higher Education University under Grant KJZH17102, Natural Science Foundation of China (NSFC) under Grant 61773081, and in part by the Graduate Scientific Research and Innovation Foundation of Chongqing under Grant CYB17048.
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References [1] G. Tao, P.V. Kokotovoc, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York, John Willey & Sons, 1996.
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[2] Z. Zhang, S. Xu, B. Zhang, Asymptotic tracking control of uncertain nonlinear systems with unknown actuator nonlinearity. IEEE Trans. Autom. Control, 59 (5) (2014) 1336–1341.
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[3] L. Ma, Z. Wang, Y. Liu Y, et al. A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time-delays. Int. J. Robust and Nonlinear Control, (2017) DOI: 10.1002/rnc.3809. [4] C.Y. Wen, J. Zhou, Z.T. Liu, Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Trans. Autom. Control, 56 (7) (2011) 1672–1678.
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[5] K. Zhao, Y.D. Song, C.Y. Wen, Computationally inexpensive fault tolerant control of uncertain non-linear systems with non-smooth asymmetric input saturation and undetectable actuation failures. IET Control Theory & Appl., 10 (15) (2016) 1866–1873.
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[6] C. Ting, A robust fuzzy control approach to stabilization of nonlinear time-delay systems with saturating inputs. Int. J. Fuzzy Syst., 10 (1) (2008) 50–60.
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[7] M. Chen, S.S. Ge, B.B. Ren, Adaptive tracking control of uncertain MIMO nonlinear systems with input saturation. Automatica, 47 (3) (2011) 452–465.
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[8] M. Chen, S.S. Ge, B. How, Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans. Neur. Networks, 21 (5) (2010) 796–812.
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[9] W. Gao, R. Selmic, Neural network control for a class of nonlinear systems with actuator saturation. IEEE Trans. Neur. Networks, 17 (1) (2006) 147–156.
[10] T.P. Zhang, S.S. Ge, Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs. Automatica, 43 (6) (2007) 1021–1033. 22
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[11] T.P. Zhang, Y. Yi, Adaptive fuzzy control for a class of MIMO nonlinear systems with unknown dead-zones. Acta Automatica Sinica, 33 (1) (2007) 1021–1033.
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[12] Y. D. Song, J. X. Guo, X. C. Huang, Smooth neuro-adaptive PI tracking control of nonlinear systems with unknown and non-smooth actuation characteristics. IEEE Trans. Neur. Netw. Lear. Syst., (2016) DOI: 10.1109/TNNLS.2016.2575078.
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[13] J. D. Boskovic, J. A. Jackson, R. K. Mehra, et al, Multiple-model adaptive fault-tolerant control of a planetary lander. J. Guid. Control Dyn, 32 (6) (2009) 1812-1826.
[14] P. Panagi, M. M. Polycarpou, Decertralized fault tolerant control of a class of interconnected nonlinear systems. IEEE Trans. Autom. Control, 56 (1) (2011) 178-184.
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[15] M. M. Polycarpou, Fault accommodation of a class of multivariable nonlinear dynamical systems using a learning approach. IEEE Trans. Autom. Control, 46 (5) (2001) 736-742.
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[16] Y. D. Song, J. X. Guo, Neuro-adaptive fault-tolerant tracking control of Lagrange systems pursing targets with unknown trajectory. IEEE Trans. Industrial Electronics, 64 (5) (2017) 3913-3920.
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[17] Y. D. Song, X. C. Huang, C. Y. Wen, Robust adaptive fault-tolerant PID control of MIMO nonlinear systems with unknown control direction. IEEE Trans. Industrial Electronics, 64 (6) (2017) 4876-4884.
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[18] Q. Hu, B. Xiao, M. I. Friswell, Robust fault-tolerant control for spacecraft attitude stabilisation subject to input saturation. IET Control Theory and Applications, 5 (2) (2011) 271-282.
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[19] W. Guan, G.H. Yang, Adaptive fault-tolerant control of linear systems in the presence of actuator saturation and L-2-disturbances, in: Proceedings of the 20th Chinese Control and Decision Conference, Yantai, Peoples Republic of China, 2008, pp. 2977C2982. [20] D.E. Miller, Davison, E.J, An adaptive controller which provides arbitrarily good transient and steady-state response, IEEE Trans. Autom. Control, 36 (1) (1993) 68–81. 23
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[21] A. Ilchmann, E.P. Ryan, P. Townsend, Tracking with prescribed transient behavior for nonlinear systems of known ralative degree, SIAM J. Control Optim., 46 (1) (2007) 210–230.
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[22] C. P. Bechlioulis, G.A. Rovithakis, Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance, IEEE Trans. Autom. Control, 53 (9) (2008) 2090–2099. [23] G. Besan¸con, Global output feedback tracking control for a class of Lagrangian systems, Automatica, 36 (12) (2000) 1915–1921.
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[24] M. M. Arefi, M.R. Jahed-Motlagh, Observer-based adaptive neural control of uncertain MIMO nonlinear systems with unknown control direction, Int. J. Adapt. Control Signal Process, 27 (9) (2013) 741–754.
[25] A. Boulkroune, M. Tadjine, M. M’Saad, et al, Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction, Fuzzy Set. Syst., 161 (6) (2010) 797–820.
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[26] T. M. Apostol, Mathematical Analysis. MA, USA: Addison-Wesley, 1963.
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[27] S. N. Huang, K. K. Tan, and T. H. Lee, Automated fault detection and diagnosis in mechanical systems, IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., 37 (6) (2007) 1360-1364.
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[28] X. Tang, G. Tao, and S. M. Joshi, Adaptive actuator failure compensation for nonlinear MIMO systems with an aircraft control application, Automatica, 43 (11) (2007) 1869-1883.
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[29] Y. D. Song, K. Zhao, Accelerated adaptive control of nonlinear uncertain systems[C]//American Control Conference (ACC), 2017: 2471-2476.
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[30] Y. D. Song, K. Zhao, M. Krstic, Adaptive Control with Exponential Regulation in the Absence of Persistent Excitation, IEEE Trans. Autom. Control, (2016) DOI: 10.1109/TAC.2016.2599645. [31] R. M. Sanner, J.J.E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neur. Networks, 3 (6) (1992) 837–863. [32] S. S. Ge, C. Wang. Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans. Neur. Networks, 15 (3) (2004) 674-692. 24
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Kai Zhao received the M.S. degree from the School of Automation, Chongqing University, Chongqing, China, in 2015, where he is currently pursuing the Ph.D. degree. His current research interests include intelligent control, adaptive control, robust control, and fault
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Liu He received the M.S. degree from tolerant control. University of Electronic Science and Technology of China, Chengdu, China, in 2014. He is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. His current research interests include intelligent control, machine learning, and computer vision.
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Ye Cao received the M.S. degree from Shaanxi normal university, Xian, China, in 2016. She is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. Her current research interests include robust adaptive control, neural net-
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Yongduan Song work, and event-triggered control. received his Ph.D. degree in electrical and computer engineering from Tennessee Technological University, Cookeville, USA, in 1992. He held a tenured Full Professor position with North Carolina A&T State University, Greensboro, from 1993 to 2008 and a Langley Distinguished Professor position with the National Institute of Aerospace, Hampton, VA, from 2005 to 2008. He is now the Dean of School of Automation, Chongqing University, and the Founding Director of the Institute of Smart Systems and Renewable Energy, Chongqing University. He was one of the six Langley Distinguished Professors with the National Institute of Aerospace (NIA), Founding Director of Cooperative Systems at NIA. He has served as an Associate Editor/Guest Editor for several prestigious scientific journals. Prof. Song has received several competitive research awards from the National Science Foundation, the National Aeronautics and Space Administration, the U.S. Air Force Office, the U.S. Army Research Office, and the U.S. Naval Research Office. His research interests include intelligent systems, guidance navigation and control, bio-inspired adaptive and cooperative systems, rail traffic control and safety, and smart grid.
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Figure 2: Tracking performance comparison between PI control in [12] and the proposed accelerated control (κ = et ): (a) tracking error |e1 |; (b) tracking error |e2 |; (c) control 27 input h1 (u1 ); (d) control input h2 (u2 ).
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Figure 3: Tracking performance comparison between traditional control (κ = 1) and the proposed accelerated control (κ = 1 + t2 , et ): (a) tracking error |e1 |; (b) tracking error |e2 |; (c) control input h1 (u1 ); (d) control input h2 (u2 ).