Adaptive integral robust control and application to electromechanical servo systems

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Research article

Adaptive integral robust control and application to electromechanical servo systems$ Wenxiang Deng a, Jianyong Yao a,b,n a b

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, Qinhuangdao 066004, China

art ic l e i nf o

a b s t r a c t

Article history: Received 24 April 2016 Received in revised form 1 December 2016 Accepted 23 January 2017

This paper proposes a continuous adaptive integral robust control with robust integral of the sign of the error (RISE) feedback for a class of uncertain nonlinear systems, in which the RISE feedback gain is adapted online to ensure the robustness against disturbances without the prior bound knowledge of the additive disturbances. In addition, an adaptive compensation integrated with the proposed adaptive RISE feedback term is also constructed to further reduce design conservatism when the system also exists parametric uncertainties. Lyapunov analysis reveals the proposed controllers could guarantee the tracking errors are asymptotically converging to zero with continuous control efforts. To illustrate the high performance nature of the developed controllers, numerical simulations are provided. At the end, an application case of an actual electromechanical servo system driven by motor is also studied, with some specific design consideration, and comparative experimental results are obtained to verify the effectiveness of the proposed controllers. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Uncertain nonlinear system RISE feedback Adaptive Electrical motor servo system

1. INTRODUCTION HIGH performance tracking controller design for uncertain nonlinear system is an unending pursuit in control community. How to handle various modelling uncertainties, which can be categorized into parametric uncertainties and additive disturbances [1,2], is the research hotspot that continues to challenge control theoreticians and engineers, since these uncertainties are always bring undesirable influence on the performance specification. To name a few, see hydraulic motion platforms [3,4], suspension vehicles [5,6], motor drive systems [7–10]. To deal with this troublesome problem, lots of design methods have been proposed and integrated during the last three decades for various classes of uncertain nonlinear systems. For example, adaptive control [11,12] is often considered to be the prior choice if modelling uncertainty can be linearly parameterized (i.e., parametric uncertainty); nonlinear robust control, such as sliding mode control [13,14], on the ☆ This work was supported in part by the National Natural Science Foundation of China under Grant 51675279, in part by the Natural Science Foundation of Jiangsu Province, China, under Grant BK20141402, and in part by the Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control under Grant HBSZKF2016-1. n Corresponding author at: School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China E-mail addresses: [email protected] (W. Deng), [email protected] ( J. Yao).

other hand, has been widely concerned to be the method of choice to handle additive disturbance (i.e., unstructured disturbance) with assuming that the disturbance can be upper bounded by a prior known norm-based inequality. These two fundamental methodologies can both theoretically achieve asymptotic tracking performance in their corresponding circumstances. However, in many practical systems, the mathematical model is poorly known or heavily uncertain, that is to say, the uncertain nonlinear system both exists considerable parametric uncertainties and unstructured disturbances, which makes matters even more difficult and complicated. Specifically, additive disturbances may cause adaptive system unstable and widely discussed solutions are various robust adaptive controls [10,15–17] which however, can only ensure the tracking error be driven into a residual bounded set with size of the order of the disturbance magnitude, and the excellent asymptotic tracking performance will disappear in this case; on the other hand, parametric uncertainties may lead to large design conservatism of nonlinear sliding mode control, even the design prerequisite, i.e., a prior known bound of uncertainty, no longer exists. Moreover, although sliding mode control could achieve excellent asymptotic tracking control, it typically results in a discontinuous control effort, which may cause chattering problem in physical systems. In addition, sliding mode controllers are often hard to tune, and may waste energy and cause unnecessary machine wear due to their aggressive nature, and may not be robust in the face of time-varying system parameters. To avoid these disadvantages, the authors in [18] proposed a judicious robust

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feedback control strategy called the robust integral of the sign of the error (RISE) to accommodate for sufficiently smooth bounded disturbances. As long as the matched additive disturbance is smooth enough with known bounds of its time derivatives, the RISE feedback control can achieve asymptotic tracking performance. More important, the resulting control effort is always keeping continuous, and thus the chattering problem is greatly alleviated. In recent years, the RISE feedback control methodology has been greatly developed. In [2,19], gradient adaptive and modular adaptive extension of RISE control were developed respectively. The authors in [20] proposed a RISE-based adaptive backstepping design method for uncertain nonlinear systems with mismatched parametric uncertainties, and the authors in [21] gave an integration of direct adaptive and indirect adaptive control with RISE feedback. Above adaptive designs with RISE feedback control reveal that the inclusion of an adaptive feed-forward term can reduce the need for high-gain feedback and improve the tracking performance compared with the traditional RISE feedback controller [18]. In addition, RISE-based control was also successfully applied to various physical nonlinear systems [22–26]. For a more in depth review of RISE-based control approaches of uncertain nonlinear systems, the reader is referred to the related references in the aforementioned literatures. However, all above RISE feedback designs are based on a common assumption that the matched additive disturbances are C2 with bounded time derivatives, and the bounds have to be definitely known [2,18–26]. A significant outcome of this assumption is that one can use the bound information to build a unique continuous term with a judicious RISE feedback gain, which can perfectly accommodate for sufficiently smooth disturbances to result in asymptotic tracking performance. The prior known bounds on the time derivatives of disturbances are the basis of the choice of RISE feedback gain and an important clue to guarantee the stability of the closed-loop system. However, this specific assumption of known bounds imposes a strong restriction on the considered additive disturbance, which is often not satisfied with physical systems. Assuming uncertain disturbance be C2 with bounded time derivatives is acceptable, but how does one investigate the definite bounds of its time derivatives in practice? Typically, the additive disturbance is poorly known, to seek exact bounds of its time derivatives is thus very difficult and complicated, even impossible in practice, due to the complexity of structure of the disturbance. Even if the bounds can sometimes be obtained, they are usually very conservative. Previous RISE-based controller just utilized a fixed RISE feedback gain. However, too large selection of the RISE feedback gain will lead to severe design conservatism while too small selection may cause performance deterioration and even instability. In this paper, we present a new development of RISE feedback control for a class of uncertain nonlinear systems in which the RISE feedback gain is adopted online via an appealing adaptation method. With this design, the prior needs for knowing the upper bounds of the time derivatives of the additive disturbances is eliminated. Asymptotically stable feature of the closed-loop system has been proved via Lyapunov analysis. The resulting controller possesses the advantages of continuous control effort, automated gain tuning without the loath and conservative investigation on the disturbances, and asymptotic tracking capability, all these features are the pressing demands of industrial applications. Comparing with the traditional RISE feedback design, the contributions can be summarized as: the proposed method greatly reduces the design conservatism, and is more feasible for physical systems. In addition, motivated by the desire to reduce the need for high-gain feedback, we also consider the case that the uncertain nonlinear system both exists parametric uncertainties and additive disturbances, and propose an adaptive compensation

integrated with the proposed adaptive RISE feedback control to further reduce the design conservatism. Here, the adaptation is used to compensate for parametric uncertainties and the proposed adaptive RISE feedback is taken as a robustifying mechanism to compensate for additive disturbances, hence recovering the asymptotic tracking property of the traditional adaptive controller when disturbance-free. With the help of adaptive compensation, the lumped disturbance is reduced and thus the need for highgain RISE feedback is alleviated. A Lyapunov-type stability analysis is also present to prove the proposed robust adaptive controller yields semi-global asymptotic tracking. To illustrate the effectiveness of the proposed controllers, an application to electromechanical servo system driven by electrical motor is investigated. Electromechanical servo is widely employed in industrials, examples can be found in [8,10]. Various modelling uncertainties are the main obstacle of developing advanced controllers for electromechanical servo systems. The methods in [9,10] are a combination of adaptive compensation and nonlinear robust control, which can only guarantee the tracking error be bounded. Although the controllers in [8] achieved asymptotic tracking performance with the help of RISE feedback, the bound information of the additive disturbance should be known. How to handle various uncertainties and disturbances for electromechanical servo systems with as weak as possible assumptions is still an open issue in control community. In this application, the comparative experimental results show that the proposed controllers are effective and suitable choices to complete this control mission. This paper is organized as follows. Section II gives the problem statement. Section III constructs the adaptive RISE feedback design procedure and its theoretical results. Section IV presents an adaptive extension when system exists parametric uncertainties. Section V carries out the comparative simulation and experimental certification. And some conclusions can be found in section VI.

2. Problem statement In this paper, we first consider a class of nth-order, single-input–single-output (SISO) nonlinear systems having the general form:

xi̇ = xi + 1, i = 1, ... , n − 1, xṅ = u − F (x, t ) + d y = x1

(1) T

n

where x(t): ¼[x1(t), x2(t),…, xn(t)] ∈R denotes the system state, y (t)∈R is the output, u∈R is the control input, F(x, t) is a smooth nonlinear function, and d(t) represents the lumped additive disturbances. We make the following assumption regarding the considered uncertain nonlinear system. Assumption 1. d(t)∈C2, |d(̇ t )| ≤ δ1 and |d¨ (t )| ≤ δ2, where are unknown positive constants.

δ1 and δ2

Remark 1. This assumption is very common in traditional RISE based design. However, in previous RISE feedback control, δ1 and δ2 have to be known for controller design and stability analysis. This prerequisite may lay a strong restriction on the considered nonlinear systems and be not suitable for physical applications. In this paper, we just need the time derivatives of d(t) to be bounded, and this greatly release the strength of assumption. Given the desired smooth motion trajectory yd ¼ x1d(t), the objective is to synthesize a continuous control input u such that the output x1-x1d(t) as t-1. Assumption 2. The desired position trajectory yd∈R is smooth

Please cite this article as: Deng W, Yao J. Adaptive integral robust control and application to electromechanical servo systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.024i

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where kr 40 is a feedback gain, sign(∙) denotes the standard signum function, and β^ is an estimated robust feedback gain up-

enough such that.

dated by

yd(i) ∈ L ∞, i = 0, .... , n + 1

(2)

(i)

and (∙) (t) denotes the ith derivative with respect to time.

In this section, the nonlinearity F(x, t) in (1) is assumed to be known, and represents precisely modeled dynamics, the term d(t) denotes the lumped modelling errors. The mission of this section is to focus on the robust control design for compensation of the additive disturbance d(t). 3.1. Error dynamics Before presenting the control law, we introduce the following filtered error signals ei(t) ∈R, i ¼2, 3,…,n

(3)

where e1 ¼ x1-x1d is the output tracking error; ki 40, i¼ 1,…,n  1 are feedback gains. It is worth noting that en in (3) is measurable since it is a function of the system states and the reference trajectory. Besides the defined error signals in (3), to facilitate the subsequent analysis, a filtered tracking error denoted by r(t) ∈ R, is defined as

r: = eṅ + k nen

(4)

where kn 40 a feedback gain. In (4), we defined an auxiliary error signal r(t) to get an extra design freedom, and this ingeniously introduced auxiliary error signal is the core difference with traditional robust designs, and plays an important role in the following controller design procedure. It is worth noting that the filtered tracking error r(t) is not measurable since it depends on the time derivative of xn, and is just introduced to assist the following controller design. By borrowing the virtual control idea in backstepping design, for each channel of (1), we introduce the following virtual control law αi, i¼2, 3,…,n, which can be calculated online, as

α1: = yd(1) − k1e1 (5)

Applying above virtual control law into (3), we have

ei = xi − αi − 1, i = 2, ... , n en = xn − αn − 1 r = xṅ − αn = u − αn − F (x, t ) + d(t )

u = u a + us , ua = αn + F (x, t ), us = us1 + us2 , k rk nen + β^sign(en)dv

⎞ k n en dv⎟, β^(0) ≥ 0 ⎠

(9)

Remark 2. Traditional RISE feedback designs employed a fixed RISE feedback gain β to hold a deterministic inequality relationship with δ1 and δ2, i.e., β 4 δ1 þ δ2/kn, to ensure the stability of the closed-loop system. However, this gain determining method is not suitable here since this paper consider the unknown case of δ1 and δ2. Hence, we propose an adaptive tuning method (8) to automatically determine a suitable RISE feedback gain. In (7), ua functions as a feed-forward control law used to achieve an improved model compensation, and us as a robust control law in which us1 is a linear robust feedback law to stabilize the nominal model of the considered nonlinear system and us2 is a nonlinear robust term used to attenuate the effect of additive disturbances whose gain is adapted via (8). Substituting (7) into the expansion formula of r in (6), we have

r = − k ren −

∫0

t

k rk nen + β^sign(en)dv + d(t )

(10)

Differentiating (10) and noting the definition of r(t) in (4), we can obtain the following dynamics of r(t):

r ̇ = − k rr − β^sign(en) + d(̇ t )

(11)

3.3. Stability analysis Before presenting the main results of this section, we state the following lemma which will be invoked later. Lemma 1. From the Assumption 1, it can be seen that there indeed exists a constant β (although it is unknown) satisfying the following condition:

1 δ2 kn

(12)

then the following defined function P(t) is always positive,

∫0

t

L(ν )dν

(13)

L(t ) = r ⎡⎣ d ̇ − βsign(en)⎤⎦

(14)

⬨. Proof: The proof of this lemma is not presented here due to space limitation, the readers please refer to [18] for the technical details. Theorem 1. If choosing feedback gains k1, k2,…, kn and kr large enough such that the matrix Λ defined in (15) is positive definite, then the proposed control law (7) guarantees that all system states are bounded under closed-loop operation and that asymptotic output tracking is also achieved, i.e., e1-0 as t-1.

us1 = − k ren , t

t

where the auxiliary function L(t) is defined by:

(6)

Based on the subsequent stability analysis, we propose the following control input to achieve the stated control objective:

∫0

∫0

P (t ) = β en(0) − en(0)d(̇ 0) −

3.2. Controller design

us2 = −

⎛ β^(t ) = β^(0) + γ ⎜ en + ⎝

β ≥ δ1 +

αi: = αi̇ − 1 − kiei αn: = αṅ − 1 − k nen

(8)

where γ 40 is an adaptation gain; to avoid using r(t), the estimation law (8) can be calculated by

3. Construction of adaptive RISE feedback

ei : = ei̇ − 1 + ki − 1ei − 1 en : = eṅ − 1 + k n − 1en − 1

̇ β^ = γrsign(en)

(7)

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⎤ ⎡ 1 0 ⋯ ⋯ 0 ⎥ ⎢ k1 − 2 ⎥ ⎢ 1 ⎥ ⎢ 1 − − ⋯ k 0 0 2 ⎥ ⎢ 2 2 ⎥ ⎢ 1 ⎢ 0 − ⋱ ⋱ 0 ⋮ ⎥ ⎥ ⎢ 2 Λ: = ⎢ ⎥ 1 ⎢ ⋮ 0 ⋱ ⋱ − 0 ⎥ 2 ⎥ ⎢ ⎢ 1 1⎥ ⋮ kn − ⎥ 0 − ⎢ ⋮ 2 2⎥ ⎢ ⎥ ⎢ 1 kr ⎥ 0 ⋯ 0 − ⎢⎣ 0 ⎦ 2

(15)

⬨. Proof: Consider the following Lyapunov function candidate

V (x, t ) =

1 2

n

∑ ei2 + i=1

ẋi = xi + 1, i = 1, ... , n − 1,

1 2 1 r + P (t ) + γ −1β˜2 2 2

(16)

where β˜ : = β^ − β represents the deviation of gain estimation. Based on the error dynamics (3) and (11), the time derivative of V can be summarized as n

V̇ =

˜^ ∑ eiei̇ + rr ̇ + P (̇ t ) + γ −1ββ

̇

i=1 n

=−

n− 1

∑ kiei2+ ∑ eiei + 1 + enr i=1

i=1

˜ ^̇ −r[k rr + β^sign(en) − d]̇ − L(t ) + γ −1ββ

(17)

Substituting (14) into (17), we can obtain

V̇ = −

n

n− 1

i=1

i=1

continuously differentiable nonlinear function with respect to the system state x, f(x, t) represents the approximation error which cannot be captured by known functions. The subsequent design and analysis illustrate how to develop an adaptive compensation term integrated with the previously proposed adaptive RISE feedback control to achieve asymptotic output tracking, in which adaptive compensation term is used to compensate for parametric uncertainties while adaptive RISE feedback is used as a robustifying mechanism to compensate for additive disturbances. The result of this adaptive extension is to reduce the need for highgain feedback, and hence the design conservatism can be greatly reduced for heavy parametric uncertain nonlinear systems. With above consideration, and re-expressing F(x, t) as F(x,t) ¼ θTφ(x)-f(x,t), the considered uncertain nonlinear system can be rewritten as

̇

˜ ^ − β˜ rsign(en) ∑ kiei2 + ∑ eiei + 1 + enr − krr 2 + γ −1ββ

(18)

Applying the gain updating law (8), and noting that the matrix

Λ defined in (15) is positive definite, thus V̇ = − Z T ΛZ ≤ − Z T λmin(Λ)Z

(19)

xṅ = u − θ T φ(x) + d y = x1

(20)

where d(x, t) is re-defined as a combination of the original disturbance in (1) and the approximation error f(x, t) to denote all modelling uncertainties which cannot be explicitly expressed by known nonlinear functions. This new re-defined additive disturbance also satisfies the Assumption 1, i.e., it is at least secondorder continuous, and its first and second time derivatives are bounded by some unknown constants δ1, δ2. With the system model (20), and applying the virtual control input αi, we can rewrite the error expression (6) as follows

ei = xi − αi − 1 en = xn − αn − 1 r = xṅ − αn = u − αn − θ T φ(x) + d

(21)

To formulate the parameter adaptation, we replace the continuously differentiable function φ(x) by φ(xd), where T

xd : = [yd , yd(1) , ... , yd(i) , ... , yd(n − 1) ] is the desired trajectory vector for x, then we can obtain

r = u − αn − θ T φ(x d ) + w + d

(22)

T

where Z is the tracking error vector defined as Z¼ [e1, …, en, r] , and λmin(Λ) is the minimal eigenvalue of Λ. Therefore, V∈L1 and Z∈L2, which means Z is bounded. From Assumption 2, we can infer that all states are bounded. Based on Assumption 1, it is easy to check that the time derivative of Z is bounded and thus Z is uniformly continuous. By applying Barbalat's lemma [12], Z-0 as t1, which leads to the results in Theorem 1.

4. Adaptive control extension 4.1. Problem formulation and controller design The adaptive RISE feedback control technique, i.e., the RISE control with adaptive gain, presented in the previous sections can be extended to an adaptive case for further design conservatism reduction. Although the proposed adaptive RISE feedback control can be utilized to compensate for both parametric uncertainty and additive disturbance since it can automatically search a suitable robust gain to achieve asymptotic tracking; since RISE control methods are based on worst-case uncertainties and disturbances, high gain feedback is often required to achieve the stated stability. For the considered uncertain nonlinear system (1), the exact knowledge of F(x,t) is critical for physical systems, it is more common that the nonlinear function F(x, t) can be expressed as F(x, t) ¼ θTφ(x)-f(x, t), where θ is unknown but constant parameter vector, φ is the known structure which is assumed to be a

where

w : = θ T φ(x d ) − θ T φ(x)

(23)

and the time derivative of w satisfies the following inequality by applying Mean Value Theorem

||ẇ || ≤ ρ(||Z ||)||Z ||

(24)

where ||∙|| denotes the Euclidean norm, and ρ: R Z 0-R Z 0 is some globally invertible, non-decreasing function. For simplicity, we use φd to represent φ(xd) in the following controller design and stability analysis. Based on (22), we design the resulting control input and parameter update law as follows: T u = ua + us , ua = αn + θ^ φd ,

us = us1 + us2 , us1 = − k ren , us2 = −

∫0

t

k rk nen + β^sign(en)dv

(25)

where the robust control action us containing us1 and us2 are same as in (7); θ^ is an online estimation of parameter vector θ, which is updated by

̇ θ^ = − Γφḋ r

(26)

where Γ is a known, constant, diagonal, positive-definite adaptation gain matrix. Since φ̇d is only a function of the known desired

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time varying trajectory, similar to (9), the update law (26) can be calculated by

θ^(t ) = θ^(0) − Γφḋ en + Γ

∫0

t

(φ¨d − k nφḋ )endv

(27)

5

Vθ̇ = − Z T ΛZ + ||φḋT Γφd||max r 2 + ||ẇ || |r| ≤ − λmin(Λ)||Z||2 + ||φḋT Γφd||max ||Z||2 + ρ(||Z||)||Z||2

(34)

then we can state that

Vθ̇ ≤ − c||Z ||2 forλmin(Λ) − ||φḋT Γφd||max > ρ(||Z ||) Remark 3. The design in (7) could only compensate for system uncertainty via the automated-tuning high-gain RISE feedback term. Through the new development in (25), an adaptive feedforward compensation term can also be used to compensate for system uncertainty. This flexibility gives a significant advantage since it allows more system dynamics to be integrated in the controller design, that is to say, if some of the system uncertainty can be represented as parametric uncertainty, then the modelbased adaptive feed-forward compensation term can be injected to compensate for it instead of just depending on the non-model based high gain RISE feedback term. Heuristically, this improvement could improve the tracking performance and reduce the design conservatism (i.e., smaller feedback gain). After applying (25) into (22), we can infer that

r = θ˜ T φd − k ren −

∫0

t

k rk nen + β^sign(en)dv + w + d

r ̇ = θ^ φd + θ˜ T φḋ − k rr − β^sign(en) + ẇ + d ̇

(28)

(29)

(30)

Theorem 2. For the considered uncertain nonlinear system presented in (20), and according to the system initial conditions, if choosing feedback gains k1, k2,…, kn and kr large enough such that the matrix Λ defined in (15) is positive definite, then the proposed robust adaptive law (25) guarantees that all system states are bounded under closed-loop operation, meanwhile, asymptotic output tracking is also achieved, i.e., e1-0 as t-1.

∑ i=1

ei2

1 1 1 + r 2 + P (t ) + γ −1β˜2 + θ˜ T Γ −1θ˜ 2 2 2

(31)

whose time derivative can be summarized as

˜ ^ ̇ + θ˜ T Γ −1θ^ ̇ ∑ eiei̇ + rr ̇ + P (̇ t ) + γ −1ββ i=1 n

=−

∑ i=1

n− 1

kiei2+

∑ eiei + 1 + enr −

φḋT Γφdr 2

− kr

r2

̇ + wr

i=1

˜ ^ ̇ − β˜ sign(en)r + θ˜ T Γ −1θ^ ̇ + θ˜ T φḋ r +γ −1ββ

(32)

Noting the estimation laws (8) and (26), we can infer n

Vθ̇ = −

n− 1

∑ kiei2+ ∑ eiei + 1 + enr − krr 2 − φḋT Γφdr 2 + wṙ i=1

i=1

Noting that the matrix

where E is defined as E:¼[Z , P (t ) , β˜ , θ ] . For the defined Lyapunov function Vθ, there exist continuous positive definite functions W1(E), W2(E) such that

W1(E ) ≤ Vθ ≤ W2(E ) W1(E ): = κ1||E||2 , W2(E ): = κ 2||E||2

(37)

where κ1:¼(1/2)min{1, γ , λmin(Γ )}, and κ2: ¼(1/2)max{1, γ  1, λmax(Γ  1)}; while the upper bound for the time derivative of Vθ is defined as 1

W (E ): = c||Z ||2

(38)

i.e.,

Vθ̇ ≤ − W (E ) > ρ(||Z ||)

max

(

or ||Z || < ρ−1 λmin(Λ) − φḋT Γφd

S: =

{ E ∈ D|W (E) < κ (ρ

max

)

(39)

Λ in (15) is positive definite, thus

2

1

−1(λ

min(Λ)

− ||φḋT Γφd||max )2

}

(40) 2

then we can now invoke the Theorem 8.4 [27] to state that ||Z|| -0 as t-1 ∀ E (0) ∈ S , which leads to the results in Theorem 2.

5. Simulation and experiment verification

In this part, we present a simulation study in this section to illustrate the robust performance nature of the proposed controller. For simplification and clear presentation of the major nature of the proposed controller, consider a simple system:

ẋ = u + d(t ) y=x

n

Vθ̇ =

(36) ˜T T

T

5.1. Numerical results

⬨. Proof: Consider the following Lyapunov function candidate n

D: = {E|||E|| < ρ−1(λmin(Λ) − ||φḋT Γφd||max }

Based on (39), and using the aforementioned boundedness statements in the proof of Theorem 1, it can be inferred that all states are bounded and Ẇ (E ) is bounded, thus W(E) is uniformly continuous. If we define the region S as follows:

4.2. Main results of adaptive extension

1 Vθ = 2

where c is some positive constant. To derive the attraction area of the initial condition, we will invoke the procedure of Theorem 8.4 in [27], i.e., we first utilize the right-most inequality in (35) to define the region D as

for λmin(Λ) − φḋT Γφd

Applying the parametric adaptation law (26), we have

r ̇ = − φḋT Γφdr + θ˜ T φḋ − k rr − β^sign(en) + ẇ + d ̇

(35)

1

where θ˜: = θ^ − θ represents the parametric estimation error; then the time derivative of r can be given by Ṫ

or ||Z || < ρ−1(λmin(Λ) − ||φḋT Γφd||max )

(33)

(41)

where x, u and d(t) are the state, control input and disturbance. The objective is to make y(t) track a desired trajectory yd(t)¼ (1e  0.2t)sint subject to a specific additive disturbance d(t) ¼0.1t. The controller gains are chosen as k1 ¼1, kr ¼1, γ ¼0.3. The initial value of β^ is set at β^(0) ¼ 0. The sampling interval is 0.2 ms in the simulation. The simulation results with the controller (7) are given in Figs. 1 and 2, which reveal that the output asymptotic tracking can be achieved via the proposed controller when the additive disturbance sufficiently satisfies its prerequisites, i.e., Assumption 1. The tracking performance with a more critical disturbance d(t) ¼ (0.1t)2 is then utilized in the simulation, which does not satisfy

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6

1

1 0.5

yd y

Tracking

Tracking

0.5 0 -0.5

0 -0.5

-1

0

5

10

15 Time(s)

20

25

-1

30

0.1

Error

Error

0

5

10

15 Time(s)

20

25

-0.05

30

20

25

30

0

5

10

15 Time(s)

20

25

30

Fig. 4. Tracking performance of controller (25) with d(t)¼ 0.1t.

exhibiting powerful learning capability of the proposed gain-updating law (8). The error offset presented in Fig. 3 can be proportionately reduced via increasing the estimation gain γ. For adaptive extension, the proposed controller (25) with adaptive feed-forward compensation is applied to the following uncertain nonlinear system

0.2 Estimation of

15 Time(s)

0.05

Fig. 1. Tracking performance of controller (7) with d(t) ¼ 0.1t.

0.15 0.1 0.05

0

5

10

15 Time(s)

20

25

30

ẋ = u − θ1x12 + d(t ) y=x

30

where the actual value of θ1 is set at 0.5, and its initial estimation is given as θ^1(0) = 0, the adaptation gain for θ1 estimation is chosen as 5. Other controller parameters are same as those in previous simulations. The simulation results with the controller (25) are given in Figs. 4 and 5, with the disturbance d(t)¼0.1t. Comparing the results in these two cases, the proposed controllers both achieved asymptotic tracking performance in their cases, respectively. By comparing the β estimation results in Figs. 2 and 5, it can be seen the steady value of β estimation in this case increased a little, this is just the design goal by introducing the model-based adaptive compensation to reduce the parametric uncertainty, hence the high-gain feedback is avoided and the design conservatism is reduced. The results depicted in Figs. 1–5 indicate that the proposed controllers can be used to sufficiently compensate for various

2 Control Input

10

0

-0.05

0

-2

0

5

10

15 Time(s)

20

25

Fig. 2. Estimation of β and control input of controller (7) with d(t) ¼0.1t.

2

Tracking

5

0.1

0

-4

0

0.15

0.05

0

yd y

yd y

1

(42)

0

-1

0

5

10

15 Time(s)

20

25

30

0.1

Estimation of

0.2

0.05

0.15 0.1 0.05

Error

0

0

5

10

15 Time(s)

20

25

30

0

5

10

15 Time(s)

20

25

30

0

0

5

10

15 Time(s)

20

25

30

Fig. 3. Tracking performance of controller (7) with d(t)¼ (0.1t)2.

Assumption 1, and hence traditional RISE feedback design [18] cannot handle this aggressive disturbance, although theoretical results with the proposed controller (7) are not clear either in this case, the tracking performance holds rather good from Fig. 3,

Estimation of

-0.05

1

0.6 0.4 0.2 0 -0.2

Fig. 5. Estimation of β and θ1 of controller (25) with d(t) ¼ 0.1t.

Please cite this article as: Deng W, Yao J. Adaptive integral robust control and application to electromechanical servo systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.024i

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Table 1 Specifications of the verification platform. Type

Specification

Electrical motor

Continuous Torque Max. Operating Speed Resistance Inductance Peak Current Power Supply Output Continuous Current

19.7 Nm 800 rpm 4.83 Ω 18.6 mH 18.0 amps 380 V 20.0 amps

Heidenhain-ERN180 Advantech PCI-1723 Heidenhain IK-220

Accuracy 7 13″ 16-bit 16-bit

Electrical driver Kollmorgen ServoStar 620 Rotary encoder D/A board Counter card

Unmodeled Fric tion Effec t (V)

0.02

Component

Kollmorgen DH063A

7

0.01

0

-0.01

-0.02

-0.03 -3

modelling uncertainties, and hence, output asymptotic tracking performance can be obtained.

-2

-1

0 Velocity (rad/s)

1

2

3

Fig. 6. The identified unmodeled nonlinear friction [28].

5.2. Application to electromechanical servo system The investigated electromechanical servo system is composed of a bench case, an electrical actuation system (including a Kollmorgen motor with a Kollmorgen electrical driver, a rotary encoder, an inertial load, and a shaft joint, etc.), a power supply, a measurement and control system [10]. Specifications of hardware components are listed in Table 1. The measurement and control system consists of a monitoring software and a real time control software with a D/A transition board and a counter card. More details can be found in [24]. The sampling interval is 0.5 ms. The current dynamic is neglected in comparison to our interest frequency range due to the much faster electric response, that is to say, the output torque of the motor is proportional to the control input u, hence a suitable system model is given as follows:

My¨ = Kiu − Bẏ − f (y , ẏ , t )

(43)

where y and M represent the angular displacement and the moment of inertia load, respectively; Ki is the torque constant; u is the control input; B represents the combined coefficient of the modeled damping and viscous friction on the load and the actuator rotor; and f represents other disturbances, such as unmodeled nonlinear frictions of Stribeck effects. In (43), we just considered a linear friction term Bẏ in the explicit model, and other friction effects are lumped to a generalized disturbance term f. The purpose of this arrangement is to help us verify the robustness of proposed control strategy. Choose x¼ [x1, x2]T represents the state vector of the position and velocity, and let m ¼ M/Ki, b¼B/Ki represent the system parameter with respect to the control input, i.e., the normalized inertia and damping parameters, then we can rewrite the system model (43) in a state-space form as follows:

x1̇ = x2 mx2̇ = u − bx2 + d(x, t )

(44)

where d(x, t) ¼-f(x, t)/Ki represents the normalized disturbance. Before the experimental test, we first identify the system model in (44), and the results are: m is about 0.0025, b is about 0.205, and the identified unmodeled nonlinear friction f(x, t)/Ki is curved in Fig. 6, which in this paper is taken as the disturbance to verify the robustness of the proposed methods. Remark 4. For the system parameter m, its value can be obtained from the mechanical structure design. For the parameter b and the results in Fig. 6, they are all contained in the state friction, which has to be experimental identified. The following experimental friction identification procedure is taken: first, a series of control constant velocity trajectory is applied to the servo system. Steady

signals are finally recorded to obtain the static mapping relationship between the control input u and the output velocity signal. From (44), when the steady output velocity is constant (i.e., the inertial force is zero), the control input u is equal to the normalized friction level. By polynomial curve-fitting method, the viscous coefficient b ¼0.205, and the other friction effects are presented in Fig. 6, which can help to ensure the differentiability of the unconsidered disturbances. More details of the friction identification and results can be found in [28]. Case 1.. Adaptive RISE Feedback Control Verification For the verification purpose, we first utilize the identified values of m and b, and the unmodeled nonlinear friction given in Fig. 6 and other unmodeled terms are taken as the lumped disturbance d(x, t). In this case, we verify the effectiveness of the RISE feedback design with adaptive gain. The design procedure is mainly given in part B of Section III. To make the system output x1 track the desired trajectory x1d(t) as closely as possible, like (3), we define the following tracking errors:

e1 = x1 − x1d e2 : = e1̇ + k1e1

(45)

and like (5), the virtual control input

α1: = x1̇ d − k1e1 α2: = α1̇ − k2e2

(46)

Following the design procedure of the proposed adaptive RISE feedback controller (7), the resulting control input u for (44) is given as follows:

u = u a + us , ua = mα2 + bx2 , us = us1 + us2 , us1 = − k re2 , us2 = −

∫0

t

k rk2e2 + β^sign(e2)dv

(47)

with

̇ β^ = γrsign(e2)

(48)

which can be calculated like (9). With above controller design in (47) and (48) for the system (44), similar theoretical results as in Theorem 1 can be easily derived by choosing the Lyapunov function as

V (x, t ) =

1 2 1 1 1 e1 + e22 + mr 2 + P (t ) + γ −1β˜2 2 2 2 2

Please cite this article as: Deng W, Yao J. Adaptive integral robust control and application to electromechanical servo systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.024i

Remark 5. It should be explained that how to select the specific robust control gains k1, k2, kr and γ for ARISE. Knowing the above theoretical results, there are some requirements for k1, k2, kr, see (15). We may determine the needed robust control gains by the following two ways. The first method is to pick up a set of values for k1, k2 and kr to calculate all order sequential principal minor determinants of matrix Λ in (15) to ensure it positive definite; in addition, the rigorous existence of the bounds δ1 and δ2 has to be verified. Thus all prerequisites of Theorem 1 and Theorem 2 are satisfied for a guaranteed stability and control accuracy. This approach is rigorous and can be thought as the formal approach to choose. However, it may increase the complexity of the resulting control law a little since it may need some amount of investigating work, especially when the order of the dynamic system is high. As an alternative, a pragmatic approach is to simply choose k1, k2, and kr large enough without worrying about the specific prerequisites. By doing so, prerequisites (15) will be satisfied for certain sets of values of k1, k2, and kr, at least locally around the desired trajectory to be tracked. In this paper, the second approach is used since it not only reduces the offline work significantly, but also facilitates the gain online tuning process in implementation. Consider the major contribution in this paper is the proposed control design method, the experiments are just utilized to verify the effectiveness of the controller application, all controller parameter values of k1, k2, kr and γ are settled down via online tuning process. To verify the performance of the proposed controllers, the compared controllers are first tested for a sinusoidal-like motion trajectory x1d ¼ 10[1-cos(3.14t)][1-exp(  0.5t))]° which ensures the desired trajectory smooth enough. The compared tracking errors are shown in Fig. 7 respectively. And the β estimation process and the actual control input of the ARISE controller are illustrated in Fig. 8.

RISE Error(°) RSF Error(°) ARISE Error(°)

1) ARISE: This is the proposed adaptive RISE feedback controller (47) and (48) for the considered electromechanical system which has been modeled in (44). The control gains are chosen as: k1 ¼240, k2 ¼4, kr ¼0.5, γ ¼ 0.1. The initial estimation of β is chosen as 0. 2) RISE: This is the traditional RISE feedback controller for the considered electromechanical system (44), which can be developed from the design procedure in [8,18]. The control gains are chosen the same as the corresponding gains of ARISE but without gain adaptation (i.e., γ ¼0), and the initial estimation of β is chosen as 7. 3) RSF: This is the robust state feedback controller for the considered electromechanical system (44), which can be developed from the feedback linearization control method without modelbased compensation [10]. The control gains are chosen the same as the corresponding gains of ARISE. 4) PID: The proportional-integral-derivative controller, which is commonly used in industrials and can be treated as a reference controller for comparison. The controller parameters are kP ¼120, kI ¼60, kD ¼0, which represent the P-gain, I-gain and D-gain respectively. The PID controller gains in this paper are tuned via error-and-try method. And indeed, there are many optimization methods to obtain a fine tuned PID controller, and improved performance may be achieved, some optimization methods can be found in [29–31]. In the following comparison, to verify the contribution of the proposed adaptive integral robust control method and suitable application to electromechanical servo systems, the fixed-gain PID controller mentioned above is employed.

0.1 0 -0.1

0

5

10

15

20

25 time(s)

30

35

40

45

50

0

5

10

15

20

25 time(s)

30

35

40

45

50

0

5

10

15

20

25 time(s)

30

35

40

45

50

0

5

10

15

20

25 time(s)

30

35

40

45

50

0.1 0 -0.1

0.05 0 -0.05

0.02 0 -0.02

Fig. 7. Compared tracking performance.

15

Estimation of

To verify the effectiveness of the proposed controller, the following controllers are compared in the next experiment:

PID Error(°)

W. Deng, J. Yao / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

5

0

0

5

10

15

20

25 time(s)

30

35

40

45

50

0

5

10

15

20

25 time(s)

30

35

40

45

50

0.2

Control Input(V)

8

0.1 0 -0.1 -0.2

Fig. 8. Estimation of β and the actual control input in ARISE.

As seen, the proposed ARISE controller has better performance than the other three controllers in terms of both transient and final tracking errors since the ARISE controller both employed known system model to achieve accurate model compensation and an adapted RISE feedback term to attenuate modelling uncertainties, then the resulting excellent asymptotic tracking performance can be achieved without the prior knowledge of the bound information of the unmodeled disturbance, while the traditional RISE controller also employed the explicit model compensation and thus obtained better performance than RFC and PID, but worse than ARISE, which can verify the effectiveness of the proposed gain adaptation method. The RFC and PID controllers just have some robustness with respect to uncertainties and their tracking errors are relatively large. The maximum steady tracking error of the proposed ARISE scheme is about 0.012° while PID about 0.055°, RSF about 0.07°, and RISE about 0.02°. This illustrates the tracking performance is greatly improved by the proposed scheme. Above comparison and analysis demonstrate that the proposed adaptive RISE feedback control is easy to be applied to physical servo systems without strict prerequisites and difficult prior investigation work; and further ensure the excellent tracking performance nature and the effectiveness of the adaptive gain method.

Please cite this article as: Deng W, Yao J. Adaptive integral robust control and application to electromechanical servo systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.024i

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AC Error(°)

0.2 0.1 0 -0.1 -0.2

0

5

10

15

20

25

15

20

25

time(s)

AEARISE Error(°)

0.1 0.05 0 -0.05 -0.1

0

5

10

9

marked as AC in below, which is a suitable control method for parametric uncertainties, is employed in this comparison. Other control types, such as PID, RSF and RISE, are ignored in this case since the previous comparison has verified the advanced nature of the proposed method. The achieved comparative experimental results for tracking x1d(t) are shown in Fig. 9, and ensure that the proposed AEARISE controller outperforms the traditional AC controller in terms of both transient and final tracking errors, since AEARISE can effectively attenuate various disturbances while AC is sensitive to unmodeled disturbances. The estimations of β and b are given in Fig. 10 of the proposed AEARISE controller. By comparing the β estimation between ARISE and AEAIRSE, i.e., the first figures of Figs. 8 and 10, it can be seen that along with the convergence of the parameter adaptation, the β estimation goes up at the beginning and down to a suitable constant at the end, that is to say, high-β-gain feedback is successfully avoided due to parameter adaptation.

time(s) Fig. 9. The compared tracking performance between AEARISE and AC.

6. Conclusion

Estimation of

30

20

10

0

0

5

10

15

20

25

15

20

25

time(s)

Estimation of b

1.5

1

0.5

0

0

5

10 time(s)

Fig. 10. Estimation of β and b in AEARISE.

Case 2. Adaptive Extension Verification To verify the adaptive extension case of the proposed adaptive RISE feedback controller, in the next experiment, the system parameter b is assumed to be unknown, and the initial estimation of b is chosen as 1, a value far away from its actual value 0.205. Based on (25), we can design the resulting control input for (44) in this case as:

u = u a + us , ^ ua = mα2 + bx1̇ d , us = us1 + us2 , us1 = − k re2 , us2 = −

∫0

t

k rk2e2 + β^sign(e2)dv

(49)

where

^̇ b = − Γx¨1d r

(50)

which can be calculated online by the method in (27). In the next experiment, the proposed controller (49) is marked as AEARISE, and the controller parameters are set as the same as those in ARISE, with Γ ¼2.5. To verify the effectiveness of the proposed AEARISE controller, traditional adaptive controller [9,10]

This paper considered the tracking control problem for a class of uncertain nonlinear systems which contain smooth additive disturbances. Based on RISE design in [18], a novel continuous adaptive RISE feedback control was proposed to ensure asymptotic convergence of the tracking error to zero under very limited restrictions on the choice of RISE feedback gain since it can be automatically tuned via the proposed adaptation method. To further reduce the design conservatism, a continuous robust adaptive controller, whose construction is founded on the fusion of an adaptive feed-forward compensation and the proposed adaptive RISE feedback mechanism, was also developed to compensate for parametric uncertainties and additive disturbances. The resulting controller guarantees the semi-global asymptotic convergence of the tracking error to zero without employing highgain feedback effort. An electromechanical servo system was studied with some specific considerations, to verify the effectiveness of the proposed controllers. Experimental results ensure the claimed theoretical results and original design intensions are achieved, which means that the proposed controllers are suitable for practical applications.

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Please cite this article as: Deng W, Yao J. Adaptive integral robust control and application to electromechanical servo systems. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.024i