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Output-Feedback Adaptive Neural Control of a Compliant Differential SMA Actuator Yongping Pan, Member, IEEE, Zhao Guo, Member, IEEE, Xiang Li, and Haoyong Yu, Member, IEEE Abstract— This brief focuses on modeling and neural-networkbased control of a novel compliant differential shape memory alloy (SMA) actuator characterized by reduced total stiffness and increased compliance. A fourth-order strict-feedback nonlinear model with an internal dynamics is derived to fully describe the SMA actuator. Due to nonlinearity, parametric uncertainty, and state-measurement difficulty of the SMA actuator, an adaptive observer-based output-feedback adaptive neural control method is developed to rigorously guarantee closed-loop stability. An experimental device is constructed to test the performance of the SMA actuation control system, where load changes and control tasks with various frequencies are considered during experiments. Experimental results have demonstrated effectiveness and superiority of the proposed approach. Index Terms— Adaptive control, adaptive observer, compliant actuator, neural network, output feedback, shape memory alloy.

I. I NTRODUCTION

T

HE application of shape memory alloy (SMA) actuators to robotic systems has attracted great attention owing to the prominent merits of SMA materials, such as high force-to-mass ratio, biocompatibility, light weight, and quiet operation compared with conventional electric, hydraulic, and pneumatic actuators [1]–[4]. The SMA achieves actuation via the transformation between martensitic (cold) to austenitic (hot) phases [1]. The SMA shape can be easily deformed by external stress in the martensitic phase, and the original shape can be recovered by simply heating SMA to the austenitic phase. In spite of the prominent advantages, SMA actuators also have some undesirable properties, such as low energy efficiency, nonlinearity, and slow responses compared with the conventional actuators. Typically, SMA actuators are classified into two types: a bias type and a differential type [3]. The

Manuscript received June 5, 2015; revised May 4, 2016; accepted December 3, 2016. Manuscript received in final form December 9, 2016. This work was supported in part by the Future System Directorate, Ministry of Defence, Singapore, under Grant MINDEF-NUS-DIRP/2012/02, and in part by the Natural Science Foundation of Jiangsu Province, China, under Grant SBK2015040859. Recommended by Associate Editor D. Vrabie. (Corresponding Author: H. Yu.) Y. Pan and H. Yu are with the Department of Biomedical Engineering, National University of Singapore, Singapore 117583 (e-mail: biepany@ nus.edu.sg; [email protected]). Z. Guo was with the Department of Biomedical Engineering, National University of Singapore, Singapore 117583, and also with the Suzhou Research Institute, National University of Singapore, Suzhou 215123, China. He is now with the School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China (e-mail: [email protected]). X. Li was with the Department of Biomedical Engineering, National University of Singapore, Singapore 117583. He is now with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2016.2638958

former, composed of an SMA element and a bias spring, has slower response speed, whereas the latter, consisted of two antagonistic SMA elements, has faster response speed at the cost of high energy consumption and restricted output angles. The main challenges of controlling SMA actuators stem from the following aspects [1], [4]: 1) hysteretic nonlinearity in the SMA phase transformation process; 2) parametric uncertainty and variation from loads, SMA physical features, and ambient conditions; and 3) measurement difficulty of martensitic fraction and temperatures. Some linear control methods, such as proportional–integral–derivative (PID) control [5], [6], variable-gain inversion control [2], and H ∞ robust control [5], have been applied to the SMA actuators. However, as only simplified or linearized models are considered, linear control cannot ensure performances for nonlinear and time-varying SMA actuation processes. Owing to the robustness against parametric uncertainty and external disturbances, sliding mode control (SMC) became a popular nonlinear control method for SMA actuators [1], [7]–[9]. The limitations of SMC result from the requirement of model bounds and the tradeoff between tracking accuracy and control chattering. To enhance the performance of PID/SMC control, hybrid feedback-feedforward (HFF) control methods were developed for various SMA actuators in [10]–[19], where feedforward models are applied to cancel out hysteresis, and PID/SMC feedback terms are employed to ensure system stability. The models for estimating SMA hysteresis include multilayer neural networks (NNs) [10], hysteretic recurrent NNs [11]–[13], neurofuzzy networks [14], classical Preisach models [15], fuzzy Preisach models [16]–[18], and Prandtl– Ishlinskii models [19]. Despite better tracking results, the above methods have some drawbacks: 1) tedious data acquisition and off-line training are needed; 2) fixed data-driven models lack adaptability and robustness guarantee; and 3) theoretical stability analysis is largely ignored. By exploiting adaptability and universal approximation capability of NNs, adaptive neural control (ANC) methods have also been applied to SMA actuators to relax the drawbacks of off-line training-based NN control. An outputfeedback ANC (OFANC) method with discontinuous compensation was developed for an SMA actuator in [4], where the actuator is modeled by a simplified two-order model, and a Kalman filter is applied to estimate plant states. In [20], an HFF-based ANC method was developed for an SMA actuator, where the actuator model is simplified to be the first order so that the estimation of plant states is not needed. It is worth noting that in general, SMA actuators are modeled by the fourth-order models [5]. Thus, the stability results in [4] and [20] are not rigorous as internal information is largely omitted

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the minimal and maximal eigenvalues of A, respectively, min(·), max(·), and sup(·) denote the minimum, maximum, and supremum operators, respectively, L ∞ denotes the space of bounded signals, diag(·) denotes a diagonal matrix, sat(·) := min{1, | · |}sgn(·) denotes a saturation function, and C k is the space of functions whose k-order derivatives all exist and are continuous, where x ∈ Rn , A ∈ Rn×m , and n, m, k ∈ N. Note that in the subsequent contents, the arguments of a function may be omitted while the context is sufficiently explicit. II. ACTUATION S YSTEM D ESCRIPTION

Fig. 1. Structure of the compliant differential SMA actuator. (a) CAD model. (b) Prototype for experiments.

in those excessively simplified actuation models. Inspired by the biological architecture of human joints driven by antagonistic muscles with tendons, a novel compliant differential SMA actuator was developed in [21], where a soft torsion spring is integrated into an antagonistic SMA actuator to decrease the total stiffness and to increase the compliance (see Fig. 1). Compared existing other SMA actuator designs, the design of [21] has the following advantages: 1) the usage of differential SMA wires increases the response speed and 2) the usage of the torsion spring improves the output range. However, these new elements also complicate the actuator structure resulting in the increased control difficulty. This brief focuses on modeling and control of the developed SMA actuator. The procedure of the control design is as follows: first, the strictfeedback nonlinear SMA model is transformed into a controllable canonical form; second, a state-feedback ANC law is given to facilitate the output-feedback design; third, an NN observer-based OFANC scheme is presented; finally, practical asymptotic stability of the closed-loop system is established by using the time-separation principle and Lyapunov synthesis. The contributions of this brief are summarized as follows. 1) The SMA actuator is modeled by a fourth-order strictfeedback nonlinear system with an internal dynamics. 2) An OFANC method based on an adaptive NN observer is developed to rigorously guarantee closed-loop stability. 3) Experimental results are provided to demonstrate a favourable performance of the SMA actuator. In the rest of this brief, the SMA actuator is described and modeled in Sections II and III, respectively. The control problem is formulated in Section IV. The OFANC is designed in Section V. Experiments are provided in Section VI. The conclusions are summarized in Section VII. Throughout this brief, N denotes the set of natural numbers, R, R+ , Rn , and Rn×m denote the spaces real numbers, positive real numbers, real n-vectors, and real n × m-matrices, respectively, x denotes the Euclidean-norm of x, λmin (A) and λmax (A) denote

Human elbow joints are actuated by antagonistic skeletal muscles that connect to the bone through the tendon. Inspired by this biological structure, we proposed a novel compliant differential SMA actuator in [21] to mimic the extension/flexion motions of human joints. As shown in Fig. 1(a), the actuator is composed of two antagonistic SMA wires, a soft torsion spring, and two cylindrical couplers, where the SMA wires connected to the couplers are used in providing the active force of bidirectional motion, the torsion spring packaged inside the couplers is used for mimicking the human tendon, and the load is applied to Coupler 1. The torsion spring that has a lower stiffness than the SMA wires provides two functions, where one is to store energy and to provide recovery forces for the SMA wires, and the other is to reduce the total stiffness in the joint. A prototype of the SMA actuator is shown in Fig. 1(b). The actuator is assembled into a base, Coupler 1 is fixed on the axis, Coupler 2 is able to freely rotate around the axis, and an encoder is connected to the axis to measure the rotational angle. When the upper SMA wire is activated by Joule heating, Coupler 1 rotates due to the active contraction force of the SMA wire, the load follows the rotation of Coupler 1, thereby the torsion spring starts to twist and transmit the active force from Couplers 1 to 2, and the lower SMA wire will be stretched to restrict the rotation. On the contrary, when the lower SMA wire is heated to contract, the active force is transmitted in the opposite direction. Based on this principle, we can heat the antagonist SMA wire and cool the opposite one to speed up the actuation response. It has been shown in [21] that the developed SMA actuator provides the largest output angle with the smallest passive stiffness compared with traditional bias and differential SMA actuators. The potential applications benefitted from this SMA actuation design include different kinds of robotic systems, such as flexible surgical tools and rehabilitation fingers. III. ACTUATION S YSTEM M ODELING A. SMA Wire Model 1) Constitutive Model: The constitutive model describes a relationship among stress σ , strain ε, temperature T , and martensite fraction ξ ∈ [0, 1], which is expressed by [22] σ˙ = E ε˙ + ξ˙ + T˙

(1)

where E is Young’s modulus, is a transformation tensor, and is a thermoelastic tensor. For the purpose of simplification, and E can be given by = −ε L E and E = (E M + E A )/2, respectively [2], where ε L is the maximal recoverable

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PAN et al.: OFANC OF A COMPLIANT DIFFERENTIAL SMA ACTUATOR

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(i.e., initial) strain, and E M and E A are Young’s moduli of martensite and austenite phases, respectively. 2) Phase Transformation Model: SMA has inherent hysteresis during a transformation between martensite and austenite phases, where ξ = 1 implies SMA is totally in the martensite (cold) phase, and ξ = 0 implies SMA is totally in the austenite (hot) phase. Let Ms and M f be initial and final temperatures of the martensite phase, respectively, and As and A f be initial and final temperatures of the austenite phase, respectively. The entire transformation process with both cooling and heating transition phases is given as follows [2]: ξ˙ = ξT (T, σ )T˙ + ξσ (T, σ )σ˙

(2)

with ξT and ξσ being given by ⎧ ξM ⎪ ⎪ − sin(a A (T A − As ))a A , ⎪ ⎪ 2 ⎪ ⎪ ⎪ if As < T A < A f &T˙ A > 0 ⎨ 1 − ξ A ξT (T, σ ) = sin(a M (TM − M f ))a M , ⎪ ⎪ ⎪ 2 ⎪ ⎪ if M f < TM < Ms &T˙M < 0 ⎪ ⎪ ⎩ 0, otherwise ⎧ 1 ⎪ ⎪ − ξ (T, σ ), if As < T A < A f &T˙ A > 0 ⎪ ⎨ CA T 1 ξσ (T, σ ) = ξT (T, σ ), if M f < TM < Ms &T˙M < 0 ⎪− ⎪ C ⎪ M ⎩ 0, otherwise with a A := π/(A f − As ), b A := −a A /C A , a M := π/(Ms − M f ), b M := −a M /C M , T A := T − σ/C A and TM := T − σ/C M , where C A and C M are the fitting parameters, and ξ M and ξ A are the maximums of ξ at the martensite and austenite phases, respectively. 3) Heat Transfer Model: The heat for phase transformation is generated by applying the voltage to SMA wires. The heat transfer, which consists of natural convection and electrical heating, is described by [22] m w C p T˙ = Ri 2 − h c Ac (T − Ta )

(3)

where m w is a mass per unit length, C p is a specific heat constant, R is an SMA’s resistance per unit length, i is a current applied to SMA wires, Ac is a circumferential area of SMA wires, Ta is an ambient temperature, and h c := h 0 + h 2 T 2 is a heat convection coefficient with h 0 and h 2 being some constants. Let α := R/(m w C p ) denote a power coefficient, β := h c Ac /m w C p denote a heat loss coefficient, and Te := T − Ta denote a temperature discrepancy. By regarding Ta as a constant, (3) is rewritten as follows: T˙e = −βTe + αi 2 .

(4)

B. SEA Actuator Model In our SMA actuator, the couplers are actuated by upper and lower SMA wires. For simplification, it is assumed that the two SMA wires share the same properties such that they have the same parameter values. A schematic of the SMA actuator is given in Fig. 2, where Fk = Aσk is an SMA contraction force, Fs is a spring force, rs is a spring radius, A is a crosssectional area of the SMA wires, Jk is a coupler inertia, kk and bk are the stiffness and damping factors of the SMA wires,

Fig. 2. Schematic of the compliant differential SMA actuator. (a) Perspective with applied forces. (b) Equivalent mechanical model.

respectively, ks and bs are the stiffness and damping factors of the spring, respectively, rw and rl are the moment arms of the SMA wires and the load, respectively, m k and m l are the masses of the SMA wires and the load, respectively, and the subscript k = 1, 2 denotes a series number of the couplers. Note that m k is negligible as it is much smaller than m l . By considering the damping and stiffness in the actuation system, the coupler dynamics is modeled as follows: θ¨1 = f 21 (θ ) + (Arw /J1 )σ1 (5) θ¨2 = f 22 (θ ) − (Arw /J2 )σ2 with θ := [θ1 , θ˙1 , θ2 , θ˙2 ]T , where f 21 (θ ) = (−b1 θ˙1 − ks (θ1 − θ2 ) − bs (θ˙1 − θ˙2 ) −m l gv rl cos θ1 )/J1 , f22 (θ ) = (ks (θ1 − θ2 ) + bs (θ˙1 − θ˙2 ) − b2 θ˙2 )/J2 . Applying (2) to (1), one obtains σ˙ k =

E ξT (Tk , σk ) + ˙ ε˙ k + Tk 1 − ξσ (Tk , σk ) 1 − ξσ (Tk , σk )

where the subscript k = 1, 2 is added to differ the two couplers. The kinematic model is given by ε˙ k = −rw θ˙k /l0 , where l0 is an initial length of the SMA wires [21]. Substituting ε˙ k = −rw θ˙k /l0 and (4) into the above equality yields σ˙ k = f 3k (θ˙k , Tk , σk ) + g3k (Tk , σk )i k2

(6)

where f 3k and g3k are given by −rw θ˙k E/l0 − βTek (ξT (Tk , σk ) + ) 1 − ξσ (Tk , σk ) ξT (Tk , σk ) + k α. g3 (Tk , σk ) = 1 − ξσ (Tk , σk )

f 3k (θ˙k , Tk , σk ) =

By combining (4) and (5) with (6), the SMA actuator is expressed in a strict-feedback form as follows: ⎧ k k ⎨ θ¨k = f2 (θ ) + g2 σk (7) σ˙ = f 3k (θ˙k , Tk , σk ) + g3k (Tk , σk )i k2 ⎩ ˙k Tek = q(Tek , i k ) with g2k = Arw /J1 for k = 1 and g2k = −Arw /J2 for k = 2, where q(Tek , i k ) = −βTek +αi k2 , θ1 is the actuator output, and the last row of (7) denotes the internal dynamics.



IV. C ONTROL P ROBLEM FORMULATION As the models of the actuation processes for the two couples have the same form as in (7), let χ := [χ1 , χ2 , χ3 ]T = [θk , θ˙k , σk ]T ∈ R3 , η := Tek ∈ R+ , and u := i k2 ∈ R+ , such that the two models are unified into one formula as follows: ⎧ ⎪ ⎨ χ˙ 1 = χ2 χ˙ 2 = f 2 (χ1 , χ2 ) + g2 χ3 (8) ⎪ ⎩ χ˙ 3 = f3 (χ2 , χ3 , η) + g3 (χ3 , η)u η˙ = q(η, u). The control challenges of the system (8) come from the high order and the immeasurable χ2 , χ3 , and η. From Section III, it is easy to verify the following properties of (8) [2]. Property 1: f 2 , f 3 , g3 , and q are of C 1 . Property 2: One has g1 = 0, g2 = 0, and g3 (χ3 , χ4 ) > 0. Property 3: η˙ = q(η, u) is input (u)-to-state (η) stable. It follows from the result in [23, Sec. II] that there exists a diffeomorphism x = T (χ ) with x = [x 1 , x 2 , x 3 ]T and x 1 = χ1 that transforms (8) into a Brunovsky canonical form: x˙ = Ax + b( f (xe ) + g(xe )u) (9) η˙ = q(η, u) with xe := [xT , η]T , where f and g are some functions, and 0 0 1 0 A = 0 0 1 ,b = 0 . 1 0 0 0 Select kc = [kc1 , kc2 , kc3 ]T ∈ R3 such that Ac := A − bkcT ∈ R3×3 is strictly Hurwitz. Therefore for any given Q c ∈ R3×3 satisfying Q c = Q cT > 0, there exists a unique solution Pc ∈ R3×3 satisfying Pc = PcT > 0 for the Lyapunov equation: AcT

Pc + Pc Ac = −Q c .

(10)

Let x d (t) ∈ R be a desired output, xˆ (t) ∈ R3 be an estimate of x(t), xd (t) := [x d (t), x˙d (t), x¨d (t)]T ∈ R3 , and xde (t) := (4) [xdT (t), x d (t)]T ∈ R4 . Define a position tracking error e(t) := x 1 (t) − x d (t). Let e(t) := x(t) − xd (t) and eˆ (t) := xˆ (t) − xd (t). The following assumptions are exploited to facilitate the subsequent control synthesis. Assumption 1: There exist constants g0 , g1 ∈ R+ such that 0 < g0 ≤ |g(xe )| ≤ g1 , ∀xe ∈ Dxe ⊂ R4 . Assumption 2: x d(i) (t) ∈ L ∞ for i = 0, 1, · · · , 4. As the internal dynamics η˙ = q(η, u) in (9) is input-to-state stable, we only consider the first row of (9) as follows: x˙ = Ax + b( f (x) + g(x)u)

(11)

for simplifying control synthesis. As only x 1 is measurable, the objective is to develop an OFANC law for the system (11) such that the position tracking error e is as small as possible. Remark 1: From the definition of g in [23, Sec. II], it has g > 0 and g < 0 for the models of Couples 1 and 2, respectively. In the subsequence, only the control design under g > 0 is presented, as the control law under g < 0 can be easily obtained based on the control law under g > 0. Remark 2: In practice, the temperature of SMA wires is disturbed so that it is difficult to be measured in open environments [9]. In addition, a compact design of SMA actuators usually does not allow the measurement of T [15].

Therefore, like most existing SMA actuators such as those in [1]–[12] and [14]–[20], our SMA actuator (see Fig. 1) does not include the measurement of T. The original intentions of only considering (11) for the control synthesis include: 1) the difficulty of the measurement of T can be avoided and 2) the inputs of NNs applied to approximate f and g can be decreased resulting in a considerable reduction of computational cost. The rationality of only considering (11) is supported by the fact that a system in the form of (9) with an input-to-state stable internal dynamics is stabilizable by a control law with only the feedback of x [24]. V. A DAPTIVE N EURAL C ONTROL D ESIGN A. Neural-Network Approximation (3) Let ν := x d − kcT e. A state-feedback certainty-equivalent control law of (8) is presented as follows: u = (− fˆ(x|Wˆ f ) + ν)/g(x| ˆ Wˆ g ) −κc eT Pc b

ua

(12)

uc

where u a is an indirect ANC, u c is a proportional-derivative (PD) controller, κc ∈ R+ is a PD gain parameter, and fˆ and gˆ are C 1 linearly parameterized NNs given by [25]

fˆ(x|Wˆ f ) = Wˆ Tf (x) = Nj=1 wˆ f j φ j (x) (13) g(x| ˆ Wˆ g ) = Wˆ gT (x) = Nj=1 wˆ g j φ j (x) in which Wˆ f = [wˆ f 1 , wˆ f 2 , · · · , wˆ f N ]T ∈ R N and Wˆ g = [wˆ g1, wˆ g2 , · · · , wˆ g N ]T ∈ R N are the vectors of NN weights, : R3 → R N ( ≤ ψ) is a vector of basis functions, N ∈ N is the number of NN nodes, and ψ ∈ R+ is a constant. For facilitating control design, define compact sets e0 d Dx f g

:= := := := :=

{e|eT Pc e/2 ≤ ce0 }, e := {e|eT Pc e/2 ≤ ce } {xd |xd ≤ cd }, de := {xde |xde ≤ cde } {x|x = e + xd , e ∈ e , xd ∈ d } {Wˆ f | − m f ≤ wˆ f j ≤ m f , j = 1, · · · , N} {Wˆ g | 0 < m g ≤ wˆ g j ≤ m g , j = 1, · · · , N}

where ce > ce0 , and ce0 , cd , cde , m f , m f , m g , and m g ∈ R+ are some constants. Then, define optimal approximation errors ε f and εg as follows: ε f (x) := f (x) − fˆ(x|W ∗f ) (14) ∗ εg (x) := g(x) − g(x|W ˆ g) with W ∗f where Wg∗ are optimal weight vectors given by W ∗f := arg min ( sup | f (x) − fˆ(x|Wˆ f )|) Wg∗

Wˆ f ∈ f x∈Dx

:= arg min ( sup |g(x) − g(x| ˆ Wˆ g )|). Wˆ g ∈g x∈Dx

B. State-Feedback Control Results Using (9) and (12)–(14), one gets the tracking error dynamics under state feedback as follows: (15) e˙ = Ac e + b W˜ Tf (x) + W˜ gT (x)u a + g(x)u c + ε with W˜ f := W ∗f − Wˆ f and W˜ g := Wg∗ − Wˆ g , where ε is a lumped optimal approximation error given by ε(x, u a ) := ε f (x) + εg (x)u a .

(16)


5

The adaptive laws of Wˆ f and Wˆ g are given as follows: W˙ˆ f = P(γ f (x)eT Pc b) (17) W˙ˆ = P(γ (x)eT P bu ) g

g

c

a

in which γ f , γg ∈ R+ are learning rates, and P(•) = [P(•1), · · · , P(• N )]T is a projection operator given by [26]–[29] ⎧ if wˆ j = m and • j < 0 ⎨0 if wˆ j = m and • j > 0 P(• j ) := 0 ⎩ • j otherwise where wˆ j = wˆ f j or wˆ g j , m = −m f or m g , and m = m f or m g . Choose a Lyapunov function candidate 1 1 ˜T ˜ 1 ˜T ˜ Vz (z) = eT Pc e + Wf Wf + W Wg 2 2γ f 2γg g

(18)

with z := [eT , W˜ Tf , W˜ gT ]T ∈ R3+2N for the closed-loop system composed by (12), (15), and (17), and define constants c f := maxWˆ f ,W ∗ ∈ f W˜ Tf W˜ f /(2γ f ) f T (19) W˜ g W˜ g /(2γg ) . cg := max ˆ ∗ Wg ,Wg ∈g

Now we give the stability result of the state-feedback case. Lemma 1 [30, Th. 1]: Consider the system (11) with Assumptions 1 and 2 driven by the state-feedback control law composed of (12), (13), and (17). If one has e(0) ∈ e0 , Wˆ f (0) ∈ f , Wˆ g (0) ∈ g , and xde (t) ∈ de , and the choice of Dx is subjected to ce ≥ ce0 + c f + cg

(20)

then, there exist suitably large control parameters κc , γ f , and γg such that the closed-loop system achieves practical asymptotic stability in the sense that all involved signals are uniformly bounded and e(t) converges to a small neighborhood of 0 dominated by κc , γ f , and γg . C. Observer-Based Control Scheme To avoid the use of the immeasurable x in (12), an adaptive NN observer is introduced as follows: x˙ˆ = Axˆ + b( fˆ(ˆx|Wˆ f ) + g(ˆ ˆ x|Wˆ g )uˆ s ) + H (x 1 − cT xˆ )

(21)

with H := ko , := diag(1/, 1/ 2 , 1/ 3 ), and c = [1, 0, 0]T , where ∈ (0, 1) is an observer parameter, ko = [ko1, ko2 , ko3 ]T ∈ R3 is selected so that s 3 + ko1s 2 + ko2s + ko3 is Hurwitz, and uˆ s is a saturated control input defined later. Based on the observer (21), an output-feedback certaintyequivalent control law is introduced as follows: ˆ g(ˆ ˆ x|Wˆ g ) −κc eˆ T Pc b uˆ = (− fˆ(ˆx|Wˆ f ) + v)/

uˆ a

(22)

uˆ c

with νˆ := x d(3) − kcT eˆ . Define compact sets eˆ := {e|eT Pc e/2 ≤ ceˆ } and Dxˆ := {ˆx|ê ∈ e , xd ∈ d } with ceˆ > ce . To avoid peaking responses resulted from the observer (21), the control input uˆ is saturated outside of Dxˆ as follows: ˆ = Ssat((uˆ a + uˆ c )/S) uˆ = Ssat(u/S) s

(23)

with S ≥ max |u(e, xde , Wˆ f , Wˆ g )|, where the maximization is taken over e ∈ eˆ , xde ∈ de , Wˆ f ∈ f , and Wˆ g ∈ g .

To apply the result of high-gain observers in [29], define a parameter estimation error x˜ = [x˜1 , x˜2 , x˜3 ]T as follows: x˜i = (x i − xî )/ 3−i , i = 1, 2, 3.

(24)

It follows from (24) that x − xˆ = ˜x with := and = 1. Applying u = uˆ s to (11), subtracting (21) from (11) and noting x − xˆ = ˜x, one obtains the estimation error dynamics as follows: diag( 2 , , 1)

˜ xˆ )uˆ s ) x˙˜ = Ao x˜ + b( f˜(x, xˆ ) + g(x, with Ao := −1 (A − H cT ), where f˜(x, xˆ ) := f (x) − fˆ(ˆx|Wˆ f ) g(x, ˜ xˆ ) := g(x) − g(ˆ ˆ x|Wˆ g ).

(25)

(26)

Lemma 2 is introduced to facilitate control analysis. Lemma 2 [29]: For the system (25), there exists a constant 1∗ ∈ (0, 1) such that if ∈ (0, 1∗ ), then ˜x(t) ≤ c , ∀t ∈ [T1 (), T2 ), where c , T1 () ∈ R+ are constants, T2 > T1 () is the moment when x(t) leaves the domain x for the first time, and lim→0 T1 () = 0. D. Output-Feedback Control Results It is obtained from the expression of uˆ a in (22) that ˆ x|Wˆ g )uˆ a . yd(n) = kcT eˆ + fˆ(ˆx|Wˆ f ) + g(ˆ Using (14) and (26), one obtains f˜(x, xˆ ) = W˜ Tf (ˆx) + υ f (˜x) + ε f (x) g(x, ˜ xˆ ) = W˜ gT (ˆx) + υg (˜x) + εg (x)

(27)

(28)

where υ f and υg are augmented estimation errors given by υ f (˜x) := W ∗f ((x) − (ˆx)) (29) υg (˜x) := Wg∗ ((x) − (ˆx)). Letting u = uˆ in (9), subtracting (27) from (11), and applying (28) to the resulting expression, one obtains e˙ = Ac eˆ + b W˜ Tf (ˆx) + W˜ gT (ˆx)uˆ a + g(x)uˆ c + δ (30) where δ is a lumped perturbation given by δ(x, xˆ , uˆ a ) := ε(x, uˆ a ) + υ(˜x, uˆ a )

(31)

and υ is a lumped estimation error given by υ(˜x, uˆ a ) := υ f (˜x) + υg (˜x)uˆ a .

(32)

Noting eˆ = xˆ − xd , e = x − xd , and ˜x = x − xˆ , one obtains eˆ = e − ˜x. Substituting eˆ = e − ˜x into (30), one gets the tracking error dynamics under output feedback as follows: e˙ = Ac e − Ac ˜x + b W˜ Tf (ˆx) + W˜ gT (ˆx)uˆ a + g(x)uˆ c + δ . (33) New adaptive laws of Wˆ f and Wˆ g are designed as follows: W˙ˆ f = P(γ f eˆ T Pc b(ˆx)) (34) ˙ Wˆ g = P(γg eˆ T Pc buˆ a (ˆx)). Define ε¯ , an upper bound of ε, as follows: ε¯ := max |ε(x, uˆ a )| = ε¯ f + ε¯ g u¯ˆ a

(35)

with ε¯ f := max |ε f (x)|, ε¯ g := max |εg (x)|, and u¯ˆ a := max |uˆ a (ê, xde , Wˆ f , Wˆ g )|, where the maximization is done over



e ∈ e , eˆ ∈ eˆ , Wˆ f ∈ f , Wˆ g ∈ g , and xde ∈ de . Choose Vz (z) in (18) as a Lyapunov function candidate for the closed-loop system composed of (23), (33), and (34). Now, we establish the main result of this brief. Theorem 1: Consider the system (11) with Assumptions 1– 2 driven by the OFANC law composed of (13), (21)–(23), and (34). If one has e(0) ∈ e0 , Wˆ f (0) ∈ f , Wˆ g (0) ∈ g , xde (t) ∈ de , and λmin (Q c ) > 1, and the choice of Dxˆ is subject to ceˆ > ce with ce ≥ ce1 + c f + cg

(36)

where ce1 := (1/2)eT (T1 )Pc e(T1 ), then there exist suitably large control parameters κc , 1/, γ f , and γg that satisfy (41) such that the closed-loop system achieves practical asymptotic stability in the sense that all closed-loop signals are uniformly bounded and e(t) is of the order O(ε2 /κc + + 1/γ f + 1/γg ). Proof: According to Lemma 2, there exists a constant 1∗ ∈ (0, 1) such that if ∈ (0, 1∗ ), then x − xˆ ≤ ˜x ≤ c and x(t) ∈ x , ∀t ∈ [T1 (), T2 ), which implies e(t) ∈ e , ∀t ∈ [T1 (), T2 ). In addition, there exists a constant 2 such that if x − xˆ ≤ 2 , then eˆ (t) ∈ eˆ . Let 2∗ := min{1∗ , 2 }. Thus, if ∈ (0, 2∗ ), then the saturation of uˆ in (23) is not ˆ ∀t ∈ [T1 (), T2 ). Consequently, (33) effective, i.e., uˆ s = u, obtained without control saturation can be utilized during t ∈ [T1 (), T2 ). Consider Vz (z) in (18) for ∀t ∈ [T1 (), T2 ) and ∈ (0, 2∗ ). Differentiating Vz (z) along (33) on time t and noting (10), (32), and uˆ s = u, ˆ one obtains V˙z = −eT Q c e/2 + eT Q c ˜x/2 + eˆ T Pc b(ε + υ + g uˆ c ) + (eT Pc b − eˆ T Pc b) W˜ Tf + W˜ gT uˆ a (ˆx)

Applying the above results to (37) leads to V˙z ≤ −(λmin (Q c ) − 1)e2 /2 + eˆ T Pc b(ε + g uˆ c ) + ρ() with ρ() := c1 2 + c2 + c3 ∈ R+ . Substituting the of uˆ c in (22) to the above inequality yields V˙z ≤ −(λmin (Q c ) − 1)e2 /2 + ρ() −κc g0 (êT Pc b)2 + eˆ T Pc bε = −(λmin (Q c ) − 1)e2 /2 + ρ() + ε2 /4κc g0 −κc g0 (êT Pc b − ε/2κc g0 )2 . Thus, one immediately obtains V˙z ≤ −(λmin (Q c ) − 1)e2 /2 + (κc , ε, ) with (κc , ε, ) := ρ() + ε2 /(4κc g0 ) ∈ R+ . Applying (18) and (19) to the above inequality, one gets V˙z ≤ −λe Vz + λe (c f + cg ) + (κc , ε, ) with λe := (λmin (Q c ) − 1)/λmax (Pc ) ∈ R+ , which implies V˙z (t) ≤ −λe (Vz (t) − co )

(38)

with co := c f + cg + (κc , ε¯ , )/λe ∈ R+ . Thus, one gets V˙z (t) ≤ 0, ∀Vz (t) ≥ co , t ∈ [T1 (), T2 ).

(39)

According to (18), (19), and (36), one also gets Vz (T1 ) ≤ ce1 + c f + cg ≤ ce .

+ (eT Pc b − eˆ T Pc b)(ε + υ + g uˆ c ) + W˜ Tf (êT Pc b(ˆx) − W˙˜ f /γ f )

(40)

From the results of (39) and (40), if one has

+ W˜ gT (êT Pc buˆ a (ˆx) − W˙˜ g /γg ).

ce ≥ c f + cg + (κc , ε¯ , )/λe

According to the result of projection modification [29], if ∀ Wˆ f (0) ∈ f and ∀Wˆ g (0) ∈ g , then P(•) in (34) guarantees Wˆ f (t) ∈ f and Wˆ g (t) ∈ g , ∀t ≥ 0, and W˜ Tf (êT Pc b(ˆx) − W˙˜ f /γ f ) ≤ 0 W˜ gT (êT Pc buˆ a (ˆx) − W˙˜ g /γg ) ≤ 0. Applying the above results to the previous expression yields V˙z ≤ −eT Q c e/2 + eT Q c ˜x/2 + eˆ T Pc b(ε + g uˆ c ) + (eT Pc b − eˆ T Pc b)(ε + g uˆ c ) + eT Pc bυ + (eT Pc b − eˆ T Pc b) W˜ Tf + W˜ gT uˆ a (ˆx).

exist some constants c1 , c2 , c3 ∈ R+ to satisfy ⎧ T T 2 2 ⎨ e Q c ˜x/2 + e Pc bυ ≤ e /2+ c1 T T T T (e Pc b − eˆ Pc b) W˜ f + W˜ g uˆ a ≤ c2 ⎩ T (e Pc b − eˆ T Pc b)(ε + g uˆ c ) ≤ c3 .

(37)

For ∈ (0, 2∗ ), as e(t) ∈ e , eˆ (t) ∈ eˆ , and xde (t) ∈ de , ∀t ∈ [T1 (), T2 ), one gets ε, υ, uˆ a , uˆ c ∈ L ∞ , ∀t ∈ [T1 (), T2 ). Applying the C 1 property of fˆ and gˆ in (13)–(29) and using x − xˆ = ˜x, one obtains that there exist some constants L f , L g ∈ R+ such that υ f (˜x) ≤ L f ˜x and υg (˜x) ≤ L g ˜x. Then, it follows from the above results that eT Q c ˜x/2 + eT Pc bυ ≤ (λmax (Q c )/2 + Pc bL a )e˜x where L a := L f + L g uˆ¯ a . By combining the above result with Lemma 2 and e − eˆ = x − xˆ , it can be shown that there

(41)

then Vz (t) ≤ ce , which implies that v := {Vz ≤ ce }∩ f × g is positively invariant under the condition in (41) [27]. Inside v , e(t) ∈ e . From the definition of , for any given Q c , there exist suitably large κc , γ f , and γg and a suitably small 3 , such that (41) holds, ∀ ∈ (0, 3 ). Let ∗ := min{2∗ , 3 }. Then, if ∈ (0, ∗ ), there exist suitably large κc , γ f , and γg , such that e(t) ∈ e , ∀t ≥ T1 (), which implies T2 → ∞. Since e(t) ∈ e , ∀t ∈ [0, T1 ()], one obtains e(t) ∈ e and x(t) ∈ x , ∀t ≥ 0. From (21) and (22), it is also easy to show xˆ , uˆ ∈ L ∞ , ∀t ≥ 0. Thus, the closed-loop system is stable in the sense that all involved signals are uniformly bounded. Now, under the stability conditions in (36) and (41), [25, Lemma A.3.2] is applied to (38) to obtain e(t)T Pc e(t)/2 ≤ (Vz (T1 ) − co )e−λe t + co which implies that e(t) is of the order O(ε2 /κc + + 1/γ f + 1/γg ) after a finite time T3 ≥ T2 according to the definitions co , c f , cg , , and λe . As co can be arbitrarily diminished by the increase of κc , 1/, γ f , and γg , the stability result obtained is practically asymptotic. Remark 3: Compared with existing studies on SMA actuator control, such as [1]–[21], this brief has the following novelties:


7

1) Initialize the NN weights Wˆ f and Wˆ g in (34) under the constraints Wˆ f (0) ∈ f and Wˆ g (0) ∈ g . 2) Initialize the observer (21) as xˆ (0) = [x 1 (0), 0, 0]T . 3) Calculate the NN outputs fˆ and gˆ in (13) using xˆ , Wˆ f , and Wˆ g in the previous steps. 4) Calculate the control input u = uˆ s in (23) using xˆ , fˆ, and gˆ in the previous steps. 5) Update xˆ in (21) using the new measured plant output x 1 and go to Step 3. Remark 6: Compared with the conventional PID control, the extra computational cost of the proposed OFANC mainly comes from the update of xˆ in the observer (21) and the calculation of the NN outputs fˆ and gˆ in the control law (12). While the computational cost on xˆ is fixed for the same plant order, the computational cost of fˆ and gˆ increases linearly as the increase in the number of NN nodes N. On the other hand, increasing N decreases the optimal approximation errors ε f and εg in (14) resulting in better tracking accuracy according to Theorem 1. Therefore, there exists a tradeoff between the computational efficiency and the NN approximation ability, as well as the tracking accuracy, for the proposed OFANC. VI. E XPERIMENTAL S TUDIES Fig. 3. An illustration of the SMA actuator control system. (a) Schematic of the control principle. (b) Experimental device for testing.

1) the higher-order SMA actuation model (7) is derived, such that the dynamics of SMA actuation is described more completely and 2) the OFANC method, which is particularly fit for the developed SMA actuator, is developed to overcome the model complexity and state-measurement difficulty. On the other hand, a time-separation principle was applied to establish stability of high-gain observer-based OFANC in [29]. In this brief, the time-separation principle is extended to prove stability of adaptive NN observer-based OFANC. This feature distinguishes the proposed OFANC from existing OFANC methods, such as [23] and [27]–[29]. Remark 4: It is worth noting that the practical asymptotic stability presented in Theorem 1 is only valid from a theoretical point of view. The bandwidth and accuracy of the SMA actuator can be increased by increasing κc , 1/, γ f , and γg . However, increasing κc , 1/, γ f , and γg increases the control amplitude, i.e., the voltage on SMA wires, and the control amplitude is limited by heating-transfer characteristics of SMA wires. Due to this limitation, κc , 1/, γ f , and γg cannot be arbitrarily increased, and thus, it is impossible to arbitrarily increase the bandwidth and accuracy of the SMA actuator in practice. In addition, the bandwidth and the accuracy of the SMA actuator are also subjected to measurement noises in practice. More specifically, increasing κc increases the control amplitude resulting in possible overheat of SMA wires; increasing κc and 1/ magnifies measurement noises resulting in degraded tracking accuracy and possible closed-loop instability; and increasing γ f and γg may lead to high-frequency oscillations in system responses or even destroy stability. Remark 5: The proposed OFANC law composed of (13), (21)–(23), and (34) is computed by the following steps.

An experimental device, which is composed of a personal computer with LabVIEW software, an NI-PCI-6221 Data Acquisition (DAQ) card, a self-designed voltage amplifier (with a gain of 3.125), and a power supply (Topward Dualtracking DC 6306D), was set up to test the performance of the SMA actuator driven by the proposed OFANC [see Fig. 3]. Two NiTi SMA wires with 0.25 mm diameter and 0.37-m initial length (twinned martensite state) were applied to construct the SMA actuator, where related parameters are provided in [21, Table I]. Before experiments, the strain of each actuated SMA wire is passively extended to 0.0300. A 50-g load is connected to Coupler 1. Including the load force, the initial strain of the upper wire is 0.0333. The DAQ card provides two 16-b resolution analog voltage outputs with a full-scale range of Â±10 V to the amplifier. Two channel analog voltages are augmented by the amplifier before applied to the upper and lower SMA wires. The angular position of the actuator is measured by a shaft encoder (Omron E6B2-CWZ1X) with a resolution of 2500 pulses per turn. The high-precision performance of the SMA actuator was verified by both set-point and sinusoidal tracking tasks with a frequency fd ≤ 0.1 Hz in [21]. In this section, more challenging sinusoidal tracking tasks with x d (t) = 0.1309 sin(ωt) + 0.1309 and ω ∈ [0.6283, 0.9425] are used to test the SMA actuator, where the corresponding amplitude is 15°, and the corresponding frequency fd = ω/(2π) = [0.10, 0.15] Hz. The NN input x is normalized based on x 1 (t) = 0.15 sin t, a signal that is stronger and faster than all possible x d (t), so that Dxˆ = [−1, 1] × [−1, 1] × [−1, 1] is sufficiently large to ensure system stability. The construction of the proposed control law is as follows: 1) to construct the basis functions φ j (x) in (13), select five Gaussian functions to evenly cover each universe of Dxˆ such that j = 1 to 125 (N = 125); 2) set S = 3, κ = 1, kc = [1, 3, 3]T , and


Fig. 4.


Experimental results by the two controllers under f d = 0.10 Hz. (a) By the PI control. (b) By the proposed OFANC.

Fig. 5. Experimental results by the proposed OFANC during online training under f d = 0.15 Hz.

Fig. 6. Experimental results by the proposed OFANC with fixed parameters under f d = 0.12 Hz.

Q c = 10I for the control law (23); 3) set ko = [3, 3, 1]T and = 0.1 for the observer (21); and 4) set γ f = 10, γg = 3, m f = m f = 10, m g = 0.1, m g = 5, Wˆ f (0) = 0, and Wˆ g (0) = 0.3I for the adaptive laws (34). To make comparisons, the saturated PI controller in [21] is selected as a baseline controller, where the PI gains were optimized to minimize e. Both the controllers are programmed in the LabVIEW software. Experimental results by the two controllers under f d = 0.10 Hz are shown in Fig. 4, where the 50-g load was removed during experiments to verify the adaptability of the controllers. The observations are as follows: 1) the tracking accuracy of the PI control is satisfactory under the initial load (e ∈ [−1.711, 0.8348]), and deteriorates after the load change (e ∈ [−2.046, 1.817); 2) the proposed OFANC achieves much higher tracking accuracy than the PI control for the initial load (e ∈ [−0.5905,

0.5780]), and the tracking accuracy has a slight degradation under the load change (e ∈ [−1.181, 0.6487]); and 3) the amplitudes of u for both the controllers are comparable. Note that the load changes are seldom considered in the previous SMA actuation control experiments. To convince the learning ability of the proposed OFANC, the NNs are online trained under f d = 0.15 Hz. Control trajectories after 600 s’ training are shown in Fig. 5. Existing experiments of SMA actuation control manifest that the tracking accuracy would be sharply decreased as the increase of the frequency of x d . Yet, a high tracking accuracy is still maintained under fd = 0.15 Hz by the proposed OFANC (e ∈ [−0.6150, 0.7701]). Then, Wˆ f and Wˆ g are fixed after training, and the OFANC with fixed Wˆ f and Wˆ g is applied to explore the whole operative space, where a typical result is given in Fig. 6. It is shown that the tracking accuracies by the fixed parameter OFANC under the higher-frequency x d are


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