PD Control of a Second-Order System with Hysteretic Actuator
52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy
PD Control of a Second-order System with Hysteretic Actuator Ruiyue Ouyang, Bayu Jayawardhana, Jacquelien M. A. Scherpen
Abstract— In this paper, we design a proportional and derivative (PD) controller for a mass-damper-spring system interconnected with a hysteretic actuator, such as, the piezoactuated stage. The hysteretic actuator is assumed to have a counterclockwise (CCW) input-output (I/O) dynamics (e.g. piezo-actuator) and is modeled by a Duhem operator. Based on the CCW of the hysteretic actuator, we provide sufficient conditions on the controller gains that depend on known intervals where the plant and hysteresis parameters lie such that the velocity of the linear plant converges to the origin. Robustness analysis and experimental results for tracking a constant reference velocity are also presented.
I. I NTRODUCTION Nowadays, many actuators are made of “smart material”, such as piezoceramic material, magnetostrictive material and magnetic shape memory alloy. While they can offer a novel type of actuation, these smart materials exhibit hysteresis phenomena which need to be taken into account in the control design. Without a proper compensation, it has been known that hysteresis can cause problems in applications, such as, limiting the accuracy and performance degradation [?]. There are many different approaches proposed in recent literatures for compensating the hysteresis in the feedback loop, see, among many others, [7], [19], [20], [6]. In [7], an inverse Preisach operator is used to compensate a hysteretic system. In [19], an adaptive control design for compensating an unknown backlash-type hysteresis for systems with known plant is proposed. In [20], the stability of a linear system with an backlash operator is analyzed by using generalized sector conditions of the backlash-type hysteresis and the stability conditions are given based on linear matrix inequalities (LMIs). In [6], the dissipativity of the Preisach operator is derived and a controller which is strictly passive is designed for the smart actuators. In this paper we are focusing on the Duhem hysteresis operator (the definition of the Duhem hysteresis operator will be given in Section II). Our main results are based on our previous results in [9],[16] and [17], where we study the dissipativity property of the Duhem hysteresis operator with counterclockwise (CCW) and clockwise (CW) inputoutput (I/O) behavior, and the absolute stability of a feedback interconnection between a linear system and a Duhem hysteresis operator based on the CCW or CW property of each subsystem. In particular, we design a PD controller for a second-order mass-damper-spring system interconnected R. Ouyang, B. Jayawardhana and J. M. A. Scherpen are with the Dept. Discrete Technology and Production Automation, University of Groningen, Groningen 9747AG, The Netherlands e-mail: [email protected], [email protected], [email protected]
978-1-4673-5716-6/13/$31.00 ©2013 IEEE
with a CCW Duhem hysteretic actuator. We provide the sufficient conditions on the controller based on the parameters of the linear plant such that the velocity of the mass-damperspring system converges to the origin. Based on this analysis, we can provide a method to determine the control gains based on the parameters of the linear plant. The robustness of the closed-loop system is also discussed with respect to an input disturbance using the integral input-to-state stability concept. Our main results are different from our previous result in [14] in the following aspects: (i). The actuator in [14] is considered to be clockwise which is motivated by frictionbased actuator and has a limited number of applications since this class of actuators is rather small. In this paper, we consider CCW actuators which include many well-known actuators, such as, piezoactuator which we will use in the experimental results; (ii). In [14], one of the main tools in the control design is to make the cascaded linear system counterclockwise which is rather straightforward. Adopting this to our present context, designing a linear controller, such that the cascaded linear system is clockwise, is not trivial. In particular, a PID controller with a non-zero integrator gain cannot make the cascaded system clockwise; One motivating application for our control design is the raster scanning in an atomic force microscopy [11]. For such scanning, the sample is placed on the stage which is driven by a piezo-actuator and a fixed cantilever is placed on the top of the sample. The stage follows a wave signal (move back and forth at a constant speed) and the roughness of the sample will be measured by the deflection of the cantilever using laser. II. P RELIMINARIES 1
Denote C (R+ ) the space of continuously differentiable functions f : R+ → R. Denote AC(R+ ) the space of absolutely continuous functions f : R+ → R. Define dz(t) z(t+h)−z(t) . dt := limhց0+ h A function α : R+ → R+ belongs to class K if it is continuous, strictly increasing and α(0) = 0. A function α belongs to class K∞ if α ∈ K and lims→∞ α(s) = ∞. A function β : R+ × R+ → R+ belongs to class KL if for each fixed ς ∈ R+ , β(·, ς) ∈ K and for each fixed s ∈ R+ , β(s, ·) is decreasing and limς→∞ β(s, ς) = 0. A. integral input-to-state stability The concept of integral input-to-state stability is introduced for the stability analysis of nonautonomous nonlinear systems given a bounded-energy input signal in [18], [2]. It is an integral variant of the input-to-state stability (ISS) property and is closely related to the L2 stability of the
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C. Duhem hysteresis operator The Duhem operator Φ : AC(R+ )×R → AC(R+ ), uΦ 7→ Φ(uΦ , yΦ0 ) =: yΦ is described by the following differential equation y˙ Φ (t) = f1 (yΦ (t), uΦ (t))u˙ Φ+ (t)+f2 (yΦ (t), uΦ (t))u˙ Φ− (t), yΦ (0) = yΦ0 , (6) where f1 , f2 : R2 → R+ are assumed to be C 1 functions, u˙ Φ+ (t) := max{0, u˙ Φ (t)}, u˙ Φ− (t) := min{0, u˙ Φ (t)} (see also, [10], [12], [23]). Assume that there exist a unique solution to (6). −5
−1
−150
(a) 1.5
0
For a nonlinear operator G, inequality (2) holds if there exists a function H : R2 → R+ such that for every input signal u ∈ AC(R+ ), the inequality (3)
holds for almost every t where the output signal y := G(u). Definition 2.3: [13] A (nonlinear) operator G : AC(R+ ) → AC(R+ ) is clockwise (CW) if for every u ∈ AC(R+ ) with the corresponding output map y := G(u), the following inequality holds Z T lim inf y(t)u(t)dt ˙ > −∞. (4) 0
Consequently, for a nonlinear operator G, inequality (4) holds if there exists a function H : R2 → R+ such that for every input signal u ∈ AC(R+ ), the inequality dH (y(t), u(t)) ≤ y(t)u(t), ˙ dt holds for a.e. t where the output signal y := G(u).
10
−5
(5)
x 10
yΦ (t)
In the following, we provide the definitions of the CCW and CW dynamics which are based on the work by Angeli in [1] and by Padthe, Oh and Bernstein in [13]. Definition 2.2: [1] A (nonlinear) operator G : AC(R+ ) → AC(R+ ) is counterclockwise (CCW) if for every u ∈ AC(R+ ) with the corresponding output map y := G(u), the following inequality holds Z T lim inf y(t)u(t)dt ˙ > −∞. (2)
T →∞
t
0
(1)
B. Counterclockwise and Clockwise dynamics
dH (y(t), u(t)) ≤ y(t)u(t), ˙ dt
yΦ (t)
1
yΦ (t)
uΦ (t)
where γ is referred to as A-iISS gain.
T →∞
x 10
150
uΦ (t)
nonlinear dynamical systems. It is shown in [2] that a system is integral input-to-state stable (iISS) if it is (a) 0-GAS and (b) dissipative with respect to supply rate σ(u), where σ is class K function. The relation between the iISS gain γ and the supply rate σ(u) is then discussed in [8] and it is shown that for a class of dissipative systems, the supply rate σ(u) is essentially the iISS gain function. In this paper, a modified iISS, so-called A-iISS, is used which will be defined below. Compare to the standard iISS definition, the solution x of the system converges to an invariant set A instead of the origin in the notion of A-iISS when a bounded-energy input signal is used. Definition 2.1: Consider a system ζ˙ = f (ζ, u), ζ(0) = ζ0 , where f : Rn × Rm → Rn is locally Lipschitz. Let A ⊂ Rn be a nonempty and closed set. It is said to be integral input-to-state stable with respect to A (A-iISS) if there exist functions α ∈ K∞ , β ∈ KL and γ ∈ K such that, for every ζ0 ∈ Rn and u ∈ L∞ (Rm ), the unique maximal solution ζ is global and Z t α(kζ(t)kA ) ≤ β(kζ0 kA , t) + γ(ku(s)k)ds ∀t ∈ R+ ,
−1.5 −150
uΦ (t)
150
(b) Fig. 1. The input-output dynamcis of the Bouc-Wen model with kx = 1.2117 × 10−7 , kw = −5.08 × 10−6 , n = 1.27, α = 8.93 × 10−3 , β = 6.6082 × 10−3 and λ = 2.321810−3 .
An example of the CCW hysteresis model is the BoucWen model [4], which is commonly used to model the hysteresis phenomena in piezo-actuators and will be used in our numerical and experimental results. The general representation of the Bouc-Wen model is given by yΦ (t) = kx uΦ (t) + kw h(t) ˙ h(t) = ρu˙ Φ (t) − ρσ|u˙ Φ (t)||h(t)|n−1 h − ρ(1 − σ)u˙ Φ (t)|h(t)|n ,
where uΦ denotes the voltage, yΦ denotes the displacement, n ≥ 1 and kx , kw , ρ, σ are the parameters determine the shape of the hysteresis curve. The Bouc-Wen model can be described by the Duhem hysteresis operator (6) with x υ n−1 γ−kx υ f1 (γ, υ) = kx + kw ρ − kw ρσ| γ−k kw | kw xυ n −kw ρ(1 − σ)| γ−k kw | γ−kx υ n−1 γ−kx υ f2 (γ, υ) = kx + kw ρ + kw ρσ| kw | kw xυ n −kw ρ(1 − σ)| γ−k kw | . (7)
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d +
u
y
P
the following inequalities hold. Q CT >0 C D
yc
C
−
yΦ
Φ
AT Q + QA + LT L ≤ 0
uΦ
T
Fig. 2. Negative feedback interconnection with a linear system P, a controller C and a Duhem hysteresis operator Φ.
In Figure 1, we illustrate the behavior of the Bouc-Wen model with kx = 1.2117 × 10−7 , kw = −5.08 × 10−6 , n = 1.27, ρ = 8.93 × 10−3 and σ = 0.74. III. M AIN
RESULTS
Consider a feedback interconnection of a mass-damperspring system with a PD controller and a hysteretic actuator as shown in Figure 2, where P represents the linear plant, C is the PD controller and Φ represents the Duhem hysteresis operator. In Figure 2, the exogeneous input disturbance signal is represented by d. The closed-loop system as shown in Figure 2, are given by P: C: Φ:
0 1 0 x + 1 u, b k − − m m m y = c1 c2 x yc = kp y + kd y, ˙ y˙ Φ = f1 (yΦ , uΦ )u˙ Φ+ + f2 (yΦ , uΦ )u˙ Φ− , u Φ = yc , u = d − yΦ ,
x˙ =
(8)
where x = [x1 x2 ]T with x1 be the position and x2 be the velocity, and c1 , c2 are the measurement gains of the position and velocity, respectively. For the PD controller C, kp > 0 and kd > 0 are the controller gains. First, we consider the case where d = 0, i.e. there is no input disturbance in the closed-loop system. Let G denote the cascaded system of P and C, then we have G:
where A
=
h
x˙ yc
0 0 k b −m −m
= Ax + Bu, = Cx + Du, i , B
=
(9) h
kd c2 m .
0
1 m
i , C
=
] and D = Theorem 3.1: Consider the closed-loop system in (8) with d = 0. Assume that for the Duhem hysteresis operator Φ there exists H : R2 → R+ such that for every uΦ ∈ AC(R+ ) and for every admissible yΦ0 , (3) holds, where yΦ = Φ(uΦ , yΦ0 ). Suppose that either (i) the function H (·, ξ) is proper (or radially unbounded) for every ξ; or (ii) the solution yΦ = Φ(uΦ , yΦ0 ) is bounded for every uΦ ∈ AC(R+ ) and every admissible yΦ0 . If there exists Q = QT > 0, L = [ l1 l2 ], w ,kp > 0 and kd > 0 such that [
k c k kp c1 − dm2
k c b kp c2 +kd c1 − dm2
T
(10) (11) T
QB + A C = L w
(12)
2
(13)
2CB = −w ,
l2 b + wm −m
< 0. Then for every initial conditions x0 ∈ R2 and and every admissible yΦ0 ∈ R, the state trajectories of the closed-loop system (8) are bounded and converge to A := {(x1 , x2 , yΦ ) ∈ R3 |x2 = 0}, i.e. the closed-loop system is globally asymptotic stable with respect to A (A-GAS).
Proof: By the assumption of the theorem, the Duhem operator Φ is CCW and there exists a function H : R2 → R+ such that H˙ (yΦ (t), uΦ (t)) ≤ y˙ Φ (t)uΦ (t). (14)
2 Using VG = 12 xT Qx+(yc −Du)u+ D 2 u as the Lyapunov function for the linear systems G, and (11)-(13), we have 1 V˙ G = xT (AT Q + QA)x + xT QBu 2 + xT AT C T u + uCBu + yc u˙ 1 ≤ yc u˙ − (Lx − wu)2 . 2 Based on Definition 2.3, it can be easily checked that G is CW. Now take Vcl (x, yΦ ) = VG + H (yΦ , yc ) as the Lyapunov function of the closed-loop system (8) and substituting the interconnection conditions uΦ = −yc and u = yΦ , we obtain
V˙ cl = V˙ G + H˙ , 1 ≤ yc u˙ − (Lx − wu)2 + y˙ Φ uΦ , 2 1 = − (Lx − wu)2 . 2 Using (10), it can be checked that VG is proper in x and yΦ . Based on the LaSalle’s invariance principle and the compactness of (x, yΦ ), the trajectories converge to the largest invariant set O contained in M := {(x, yΦ ) ∈ R3 |Lx − wyΦ = 0} as t → ∞. L L In the invariant set O in M, yΦ = − w x, i.e., u = w x. L Substituting u = w x in P, we have that in the invariant set O 0 1 x (15) x˙ = l1 l2 b k + wm −m + wm −m √ 2kd c2 b−2m(kp c2 +kd c1 ) . From (13), we have that w = ± m Using (11) and (12), it can be computed that l has a 1 √ 2kd c2 b−2m(kp c2 +kd c1 ) if w = m √ k 2kd c2 b−2m(kp c2 +kd c1 ) 2kd c2 b−2m(kp c2 +kd c1 ) or l1 = − m m √ 2kd c2 b−2m(kp c2 +kd c1 ) w =− . Substituting these w and m
unique solution l1 = √
k
if l1 into (15), we get that in the invariant set O 0 1 x x˙ = l2 b + wm 0 −m
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(16)
holds which implies that x2 = 0 in the invariant set O. By La-Salle’s principle, this implies that x2 (t) → 0 as t → ∞ l2 b if − m + wm < 0. Hence, the closed-loop system is GAS with respect to A := {(x1 , x2 , yΦ ) ∈ R3 |x2 = 0}. 2 Remark 3.2: If all the hypotheses in Theorem 3.1 are satisfied, then the LMI conditions given in (10)-(13) indicates the interval of the control parameters such that the 2k closed-loop system is A-GAS. For instance, if kp ≤ kcd1cm , 3 2 2 b c2 −b c1 m−kc1 m kd c2 b−kd c1 m and kp > kd b2 c2 m+kc2 m2 −bc1 m2 then the kp ≤ c2 m √ 2qk closed-loop system (8) satisfies (10)-(13) with L = [ m 0 ], √ 2q w = m and Q=
kd c2 k2 −kp c1 k m qk m
qk m
kd c2 k−kp c1 m− qb m
where q = (kd c2 b − kp c2 m − kd c1 m). In the rest of this section, we proceed our analysis with d 6= 0, i.e., we study the robustness of the closed-loop system (8) by adding disturbances d to the input of the plant. The concept of A-iISS is applied here to show the robustness of the closed-loop system. Let us first study the dissipativity property of (8). Lemma 3.3: Consider the closed-loop system in (8) with d ∈ C 1 (R+ ) s.t. d˙ ∈ AC(R+ ). Assume that the Duhem hysteresis operator satisfies the hypotheses given in Theorem 3.1. If there exists Q = QT > 0, L = [ l1 l2 ], w, kp > 0 and kd > 0 such that the following inequalities hold.
Q C
CT D
>0
AT Q + QA + LT L + ǫ1 I ≤ 0 T
T
Young’s inequality, u = yΦ and ǫ3 > 0, we have η 1 1 1 V˙ cl ≤ d˙2 + yc2 − ǫ1 xT x − ǫ2 u2 2 2η 2 2 1 2 2 1 T T 1 1 η ˙2 ≤ d + D u + x C Cx − ǫ1 xT x − ǫ2 u2 2 2η 2η 2 2 η η ≤ d˙2 + d2 2 2 where the last inequality is obtained since η is an arbitrary positive constant. This implies that system
(8) is dissipative ˙ = µ d˙ 2 and µ := η/2. with respect to σ where σ(d, d) d 2 Theorem 3.4: Consider the closed-loop system in (8) with d ∈ C 1 (R+ ) s.t. d˙ ∈ AC(R+ ). Assume that the hypotheses in Lemma 3.3 hold. Then (8) is iISS with
2 respect to A (A iISS), with iISS gain γ dd˙ = µ dd˙ , where µ > 0.
Proof: The proof of the Theorem 3.4 is similar to the proof of Theorem 3.1 in [8]. First of all, consider the system (8) with d = 0 which has the following form " # Ax−ByΦ x˙ f1 (yΦ ,uΦ ) (CAx−CBy ) Φ + 1+Df1 (yΦ ,uΦ ) , (21) y˙ Φ = f (yΦ ,uΦ ) (CAx−CByΦ )− 2 (yΦ ,uΦ )
2 + 1+Df
where u = −yΦ . It can be checked that the RHS of equation (21) is locally Lipschitz. Let us write (21) by ζ˙ = f (ζ) where ζ = [xT yΦ ]T . Based on the converse Lyapunov theorem [21, Corollary 2], Theorem 3.1 implies that there exists a smooth Lyapunov function V : R3 → R+ such that • there exist K∞ functions α1 and α2 such that
(17)
α1 (kζkA ) ≤ V (ζ) ≤ α2 (kζkA )
(18) T
QB + A C = L w
(19)
2
(20)
2CB = −w − ǫ2 ,
for some positive constants ǫ1 and ǫ2 . Suppose l2 b + wm < 0, then there exists µ > 0 such that −m the closed-loop system is dissipative with respect to the
2
˙ = µ d˙ . supply function σ(d) Proof: Let Vcl = VQ + H be the Lyapunov function for the closed-loop system, where VQ and H have the same descriptions as in the proof of Theorem 3.1. Then using (17)(20) we have
V˙ cl = V˙Q + H˙ , 1 ≤ yc u˙ − (Lx − wu)T (Lx − wu) + y˙ Φ uΦ 2 1 1 − ǫ1 xT x − ǫ2 u2 , 2 2 1 1 1 T T = − (Lx − wu) (Lx − wu) + yc d˙ − ǫ1 x x − ǫ2 u2 2 2 2 where the last equality is obtained since u = d − yΦ . Using
•
∀ζ ∈ R3 .
there exists a continuous, positive definite function α3 such that dV (ζ) f (ζ) ≤ −α3 (kζkA ) ∀ζ ∈ R3 . dζ
In the case when the input disturbance d 6= 0, the system ˙ where (8) can be written as ζ˙ = f (ζ, d, d)
˙ = f (ζ, d, d)
Ax+B(d−yΦ ) f1 (yΦ ,uΦ ) ˙ 1+Df1 (yΦ ,uΦ ) (CAx+CB(d−yΦ )+D d)+ f2 (yΦ ,uΦ ) ˙− + 1+Df (y ,u ) (CAx+CB(d−yΦ )+Dd) 2 Φ Φ
,
u = d − yΦ . Since f1 (yΦ , uΦ ) ≥ 0 and f2 (yΦ , uΦ ) ≥ 0 according to the definition of the Duhem operator given in Secf1 (yΦ ,uΦ ) f2 (yΦ ,uΦ ) tion II, then 1+Df ∈ [0, 1] and 1+Df ∈ [0, 1]. 1 (yΦ ,uΦ ) 1 (y Φ ,uΦ )
x x ] Hence, we have that for every BA = [ ≤ l , yΦ l yΦ there exists c > 0 such that
˙ ≤ c 1 + µ d˙ 2 kf (ζ, d, d)k d
˙ ∈ BA × R2 . (22) ∀(ζ, d, d) l
The rest of the proof can use the same arguments as in [8] where the compact set K ⊂ Rn in [8] is replaced by BA l . It can be checked that the same lemmas as given in
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AND EXPERIMENTAL RESULTS
In the previous sections, stability for a second-order hysteretic system was shown by using a PD controller. In particular, we give the sufficient conditions on the control parameters such that the closed-loop system is A-GAS. In case the linear plant is the mass-damper-spring system and the position is measured output, then applying our approach, the velocity of the linear plant converges to the origin. The robustness property has also been discussed with respect to the input disturbance. In this section, we apply these results to develop a PD controller for a piezo-actuated stage. To evaluate the controller experimentally, the piezoactuated stage P 611.2S from Physik Instrumente (PI) is used combined with the piezo amplifier E610.S0 from Physik Instrumente (PI). The displacement of the stage is measured by a strain gauge sensor. The input voltage is controlled by a PC computer where the MATLAB Real-Time Workshop is used. The PD controller is implemented within Simulink. The piezo-actuated stage can be considered as a hysteretic actuator connected to a mass-damper-spring plant as shown in Figure 3.
1.5
1
0.5
0
−0.5
−1
−1.5 −2
−1.5
−1
−0.5
0 Voltage v
0.5
1
1.5
2
(a) Comparison between the model output and the measured output in input-output plane −5
x 10
measured output model output
1.5
1
k
0.5
0
−1
m
0
b Fig. 3.
x 10
−0.5
x Φ
−5
2
Displacement m
IV. S IMULATION
Based on these two steps, the identified parameters for the Bouc-Wen model (7) are: kx = 0.2379, kw = −0.0723, ρ = 2.57, σ = 3.53 and n = 1.46. Based on the parameters of the Bouc-Wen model, the parameters of the mass-damper-spring system can also be identified by using lscurvefit, since boundaries of the parameters can be given. The parameters we obtained are m = 0.0352, b = 80.0543 and k = 2.5541 × 104 . The identified parameters are validated by a mixed frequency signal as shown in Figure 4.
Displacement m
[8, Lemma 3.2, Lemma 3.3 and Lemma 3.4] can also be obtained in this case. Applying the Lemmas and the converse Lyapunov function for A-GAS system, we can obtain the AiISS Lyapunov function similar to the proof of [8, Theorem 3.1]. 2
Φ : y˙ Φ = f1 (yΦ , uΦ )u˙ Φ+ + f2 (yΦ , uΦ )u˙ Φ− ,
where x = [ x1 x2 ]T , x1 denotes the displacement and x2 denotes the velocity. The hysteresis system Φ is represented by the Bouc-Wen model as introduced in (7) in Section II. The identification process is separated into two parts: the identification of the Bouc-Wen model and the identification of the mass-damper-spring system. First of all, we identify the hysteresis system by applying low frequent input signals, in which case the linear plant can be considered as a DC gain. To identify the Bouc-Wen model, we first determine the initial parameters of the BoucWen model based on the limit cycle approach as proposed in [5]. Then we use the initial parameters and apply the least square optimization toolbox (lsqcurvefit) in MATLAB to estimate the real parameters of the Bouc-Wen model.
4
6
8
10 Time s
12
14
16
18
(b) Comparison between the model output and the measured output in time (zoom in)
Mass-damper-spring system connected with a hysteretic actuator.
Then the piezo-actuated stage system can be given as follows 0 1 0 u, x + P : x˙ = 1 b k −m −m m y= 1 1 x (23)
2
Fig. 4. Identification results for the piezo-actuated stage, where the red line is the output of the model and the blue line is the measured output from the setup.
The control parameters for the PD controller can then be obtained based on the parameters of the linear plant. It can be checked that with kp = 2000, kd = 1, L = [ 3.1981×106 0 ] and w = 125.215, we can obtain the matrix 1.85 × 1010 7.04 × 106 Q= , 7.04 × 106 3.4 × 103 which satisfies the conditions given in (10)-(13). The simulation result is given in Figure 5 (a) with initial condition T x0 = [ 0 10−5 ] . To show the robustness of the system, we add a distur−2 bance signal d(t) = 10t2 , which satisfies d, d˙ ∈ L2 , to the input of the linear plant. The simulation result is shown in Figure 5 (b). We will now apply our results to the velocity tracking problem where the position of piezoactuator must track a ramp reference signal. Consider again the closed-loop system (8) with d = 0. Let x ¯2 = v¯ denote the desired constant velocity and the corresponding steady-state displacement is
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−5
x 10
16
1
14
x 10
linear plant and the PD controller. Furthermore, the robustness of the closed-loop system with respect to the input disturbance is also studied by applying the concept of AiISS. To evaluate our approach, we conduct an experiment on the piezo-actuated stage where a constant velocity is tracked.
0.5
12
0
Velocity m/s
Velocity m/s
10 8 6
−0.5 −1 −1.5
4 −2
2
−2.5
0 −2
0
1
2
3
4
−3
5
0
1
2
Time s
3
4
R EFERENCES
5
Time s
Fig. 5. Simulation results of the piezo-actuated stage system with initial condition x0 = [0 10−5 ]T . (a) without disturbance; (b) with disturbance −2 d(t) = 10t2
then given by x ¯1 = v¯t (up to a constant) with t ∈ R+ . It follows from the state equations of the closed-loop system in (8) that in the steady-state, y¯Φ = −k¯ v t − b¯ v (up to a T T constant). Denote ζ¯ = [ x¯1 x¯2 y¯Φ ] , e = [ e1 e2 eΦ ] , where e1 = x1 − x ¯1 , e2 = x2 − x¯2 and eΦ = yΦ − y¯Φ . Let u ˜Φ = uΦ − u ¯Φ where u ¯Φ satisfies −kw + kx u ¯Φ = y¯Φ . It can be computed that, in the neighborhood of (eΦ , u ˜Φ ) = (0, 0), ˜ uΦ ) can be approximated by a Duhem the operator eΦ = Φ(˜ operator. Hence the error dynamics can be described by e2 k b 1 − e − e − e 1 2 Φ m m m f (e) = (24) f1 (eΦ ,˜ uΦ )(u ˜˙ Φ )+ +f2 (eΦ ,˜ uΦ )(u ˜˙ Φ )− ,
Applying Theorem 3.1, we can obtain that the error system (24) is locally asymptotically stable with respect to A := {(e1 , e2 , eΦ ) ∈ R3 |e2 = 0}. We remark that in the original coordinate, the applied control input is given by uΦ = u ˜Φ + u¯Φ where u ˜Φ := kp (y − v¯t − v¯) + kd (y˙ − v¯) is the output of the PD controller. The simulation result of tracking a constant velocity v¯ = 5µm/s is shown in Figure 6(a). The experimental result is shown in Figure 6(b) where we use constant reference velocity of v¯ = 5µm/s. −6
6
−6
x 10
7
x 10
6 5 5 4 Velocity m/s
Velocity m/s
4
3
2
3 2 1 0
1
0
Simulation output Reference velocity 0
5
10 Time s
15
20
−1 −2
Measured output Reference velocity 0
5
10
15
20
25
Time s
Fig. 6. (a) Simulation results of the piezo-actuated stage system for tracking a constant velocity 5µm/s; (b) Experimental results for tracking a constant velocity 5µm/s
V. C ONCLUSION In this work we investigate the stability for a second-order mass-damper-spring system driven by a hysteretic actuator which can be represented by a CCW Duhem hysteresis operator. The hysteretic actuator is controlled by a PD controller, where we give the sufficient conditions on the control parameters such that the closed-loop is A-GAS. The sufficient conditions on the control parameters are obtained based on the CW property of the cascaded system of the
[1] D. Angeli, “Systems with counterclockwise input-output dynamics,” IEEE Trans. Aut. Contr., Vol. 51, no. 7, pp. 1130-1143, 2006. [2] D. Angeli, E. D. Sontag, Y. Wang, “A characterization of integral input to state stability,” IEEE Trans. Aut. Contr., Vol. 45, no. 6, pp. 1082-1097, 2000. [3] G. Bertotti & I. D. Mayergoyz, The Science of Hysteresis: Mathematical Modeling and Applications, Academic press, San Diego, 2006. [4] F. Ikhoune, J. Rodeller, Systems with Hysteresis: analysis, identification and control using the Bouc-Wen model, Wiley, 2007. [5] F. Ikhoune, O. G. Bellmunt, A Limit Cycle Approach for the Parametric Identification of Hysteretic Systems, System and Control Letters, vol. 57, no. 8, pp. 663-669, 2008. [6] R.B. Gorbet, K.A. Morris, D.W.L.Wang Passivity-Based Stability and Control of Hysteresis in Smart Actuators, IEEE Transaction on Control Systems Technology, vol. 9, no. 1, pp. 5-16, 2001. [7] R.V.Iyer, X. Tan Control of Hysteretic Systems through Inverse Compensation, IEEE Control Systems Magazine, vol. 29, no. 1, pp. 83-99, 2009. [8] B. Jayawardhana, E.P. Ryan, Andrew R. Teel, “Bounded-Energy-Input Convergent-State Property of Dissipative Nonlinear Systems: An iISS Approach,” IEEE Trans. Aut. Contr., vol. 55, no. 1, pp. 159-164, 2010. [9] B. Jayawardhana, Ruiyue Ouyang, V. Andrieu, “Stability of Systems with Duhem Hysteresis Operator: Dissipativity Approach,” Automatica, vol. 48, no. 10, pp. 2657-2662, 2012. [10] J.W. Macki, P. Nistri, P. Zecca, “Mathematical models for hysteresis,” SIAM Review, vol. 35, no. 1, pp. 94–123, 1993. [11] B. Mokaberi, A. G. Requicha, Compensation of Scanner Creep and Hysteresis for AFM Nanomanipulation, IEEE Transactions on Automation Science and Engineering, vol. 5, no. 2, pp. 197-206, 2008. [12] J. Oh, D.S. Bernstein, “Semilinear Duhem model for rate-independent and rate-dependent hysteresis,” IEEE Trans. Automat. Contr., vol. 50, pp. 631–645, 2005. [13] A.K. Padthe, J-H. Oh, D.S. Bernstein, “Counterclockwise Dynamics of a Rate-independent Semilinear Duhem Model,” Proc. 44th IEEE Conf. Dec. Contr. and Eur. Contr. Conf., Seville, 2005. [14] R. Ouyang, B. Jayawardhana, V. Andrieu, “On the robustness of hysteretic second-order systems with PID: iISS approach,” Proc. 51st IEEE Conference on Decision and Control, Maui, December 2012. [15] Ruiyue Ouyang, Bayu Jayawardhana, “Stability analysis and controller design for a linear system with Duhem hysteresis nonlinearity,” Proc. IEEE American Control Conference, Montreal, June 2012. [16] R. Ouyang, V. Andrieu, B. Jayawardhana, “On the Characterization of the Duhem Hysteresis Operator with Clockwise Input-Output Dynamics”, System & Control Letters, vol.62, no. 3, pp. 286-293, 2013. [17] Ruiyue Ouyang, Bayu Jayawardhana, “Stability Analysis and Controller Design for a Linear System with Duhem Hysteresis Nonlinearity,” Online pre-printed is available at: http://arxiv.org/abs/1206.3065. [18] E. D. Sontag, “Comments on integral variants of ISS,” Syst. Contr. Lett., Vol. 34, pp. 93–100, 1998. [19] G. Tao, P. V. Kokotovic ,“Adaptive Control of Systems with Unknown Output Backlash“, IEEE Transaction On Automatic Control, vol. 40, no. 2, pp. 326–330, 1995. [20] S. Tarbouriech, C. Prieur, I. Queinnec ,“Stability Analysis for linear systems with Input Backlash through Sufficient LMI Conditions“, Automatica, vol. 46, no. 11, pp. 1911–1915, 2010. [21] A. R. Teel, L. Praly ,“A Smooth Lyapunov Function from A ClassKL Estimate Involving two Positive Semidefinite functions’, ESAIM: Contr., Opt. and Cal. of Variations, vol. 5, pp. 313–367, 2000. [22] V. Andrieu, L. Praly, “A Unifying Point of View on Output Feedback Designs for Global Asymptotic Stabilization,” Automatica, Survey paper, Vol. 45, no. 8, Pages 1789-1798, 2009. [23] A. Visintin, Differential Models of Hysteresis, Springer-Verlag, New York, 1994.
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PD Control of a Second-order System with Hysteretic Actuator Ruiyue Ouyang, Bayu Jayawardhana, Jacquelien M. A. Scherpen
Abstract— In this paper, we design a proportional and derivative (PD) controller for a mass-damper-spring system interconnected with a hysteretic actuator, such as, the piezoactuated stage. The hysteretic actuator is assumed to have a counterclockwise (CCW) input-output (I/O) dynamics (e.g. piezo-actuator) and is modeled by a Duhem operator. Based on the CCW of the hysteretic actuator, we provide sufficient conditions on the controller gains that depend on known intervals where the plant and hysteresis parameters lie such that the velocity of the linear plant converges to the origin. Robustness analysis and experimental results for tracking a constant reference velocity are also presented.
I. I NTRODUCTION Nowadays, many actuators are made of “smart material”, such as piezoceramic material, magnetostrictive material and magnetic shape memory alloy. While they can offer a novel type of actuation, these smart materials exhibit hysteresis phenomena which need to be taken into account in the control design. Without a proper compensation, it has been known that hysteresis can cause problems in applications, such as, limiting the accuracy and performance degradation [?]. There are many different approaches proposed in recent literatures for compensating the hysteresis in the feedback loop, see, among many others, [7], [19], [20], [6]. In [7], an inverse Preisach operator is used to compensate a hysteretic system. In [19], an adaptive control design for compensating an unknown backlash-type hysteresis for systems with known plant is proposed. In [20], the stability of a linear system with an backlash operator is analyzed by using generalized sector conditions of the backlash-type hysteresis and the stability conditions are given based on linear matrix inequalities (LMIs). In [6], the dissipativity of the Preisach operator is derived and a controller which is strictly passive is designed for the smart actuators. In this paper we are focusing on the Duhem hysteresis operator (the definition of the Duhem hysteresis operator will be given in Section II). Our main results are based on our previous results in [9],[16] and [17], where we study the dissipativity property of the Duhem hysteresis operator with counterclockwise (CCW) and clockwise (CW) inputoutput (I/O) behavior, and the absolute stability of a feedback interconnection between a linear system and a Duhem hysteresis operator based on the CCW or CW property of each subsystem. In particular, we design a PD controller for a second-order mass-damper-spring system interconnected R. Ouyang, B. Jayawardhana and J. M. A. Scherpen are with the Dept. Discrete Technology and Production Automation, University of Groningen, Groningen 9747AG, The Netherlands e-mail: [email protected], [email protected], [email protected]
978-1-4673-5716-6/13/$31.00 ©2013 IEEE
with a CCW Duhem hysteretic actuator. We provide the sufficient conditions on the controller based on the parameters of the linear plant such that the velocity of the mass-damperspring system converges to the origin. Based on this analysis, we can provide a method to determine the control gains based on the parameters of the linear plant. The robustness of the closed-loop system is also discussed with respect to an input disturbance using the integral input-to-state stability concept. Our main results are different from our previous result in [14] in the following aspects: (i). The actuator in [14] is considered to be clockwise which is motivated by frictionbased actuator and has a limited number of applications since this class of actuators is rather small. In this paper, we consider CCW actuators which include many well-known actuators, such as, piezoactuator which we will use in the experimental results; (ii). In [14], one of the main tools in the control design is to make the cascaded linear system counterclockwise which is rather straightforward. Adopting this to our present context, designing a linear controller, such that the cascaded linear system is clockwise, is not trivial. In particular, a PID controller with a non-zero integrator gain cannot make the cascaded system clockwise; One motivating application for our control design is the raster scanning in an atomic force microscopy [11]. For such scanning, the sample is placed on the stage which is driven by a piezo-actuator and a fixed cantilever is placed on the top of the sample. The stage follows a wave signal (move back and forth at a constant speed) and the roughness of the sample will be measured by the deflection of the cantilever using laser. II. P RELIMINARIES 1
Denote C (R+ ) the space of continuously differentiable functions f : R+ → R. Denote AC(R+ ) the space of absolutely continuous functions f : R+ → R. Define dz(t) z(t+h)−z(t) . dt := limhց0+ h A function α : R+ → R+ belongs to class K if it is continuous, strictly increasing and α(0) = 0. A function α belongs to class K∞ if α ∈ K and lims→∞ α(s) = ∞. A function β : R+ × R+ → R+ belongs to class KL if for each fixed ς ∈ R+ , β(·, ς) ∈ K and for each fixed s ∈ R+ , β(s, ·) is decreasing and limς→∞ β(s, ς) = 0. A. integral input-to-state stability The concept of integral input-to-state stability is introduced for the stability analysis of nonautonomous nonlinear systems given a bounded-energy input signal in [18], [2]. It is an integral variant of the input-to-state stability (ISS) property and is closely related to the L2 stability of the
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0
C. Duhem hysteresis operator The Duhem operator Φ : AC(R+ )×R → AC(R+ ), uΦ 7→ Φ(uΦ , yΦ0 ) =: yΦ is described by the following differential equation y˙ Φ (t) = f1 (yΦ (t), uΦ (t))u˙ Φ+ (t)+f2 (yΦ (t), uΦ (t))u˙ Φ− (t), yΦ (0) = yΦ0 , (6) where f1 , f2 : R2 → R+ are assumed to be C 1 functions, u˙ Φ+ (t) := max{0, u˙ Φ (t)}, u˙ Φ− (t) := min{0, u˙ Φ (t)} (see also, [10], [12], [23]). Assume that there exist a unique solution to (6). −5
−1
−150
(a) 1.5
0
For a nonlinear operator G, inequality (2) holds if there exists a function H : R2 → R+ such that for every input signal u ∈ AC(R+ ), the inequality (3)
holds for almost every t where the output signal y := G(u). Definition 2.3: [13] A (nonlinear) operator G : AC(R+ ) → AC(R+ ) is clockwise (CW) if for every u ∈ AC(R+ ) with the corresponding output map y := G(u), the following inequality holds Z T lim inf y(t)u(t)dt ˙ > −∞. (4) 0
Consequently, for a nonlinear operator G, inequality (4) holds if there exists a function H : R2 → R+ such that for every input signal u ∈ AC(R+ ), the inequality dH (y(t), u(t)) ≤ y(t)u(t), ˙ dt holds for a.e. t where the output signal y := G(u).
10
−5
(5)
x 10
yΦ (t)
In the following, we provide the definitions of the CCW and CW dynamics which are based on the work by Angeli in [1] and by Padthe, Oh and Bernstein in [13]. Definition 2.2: [1] A (nonlinear) operator G : AC(R+ ) → AC(R+ ) is counterclockwise (CCW) if for every u ∈ AC(R+ ) with the corresponding output map y := G(u), the following inequality holds Z T lim inf y(t)u(t)dt ˙ > −∞. (2)
T →∞
t
0
(1)
B. Counterclockwise and Clockwise dynamics
dH (y(t), u(t)) ≤ y(t)u(t), ˙ dt
yΦ (t)
1
yΦ (t)
uΦ (t)
where γ is referred to as A-iISS gain.
T →∞
x 10
150
uΦ (t)
nonlinear dynamical systems. It is shown in [2] that a system is integral input-to-state stable (iISS) if it is (a) 0-GAS and (b) dissipative with respect to supply rate σ(u), where σ is class K function. The relation between the iISS gain γ and the supply rate σ(u) is then discussed in [8] and it is shown that for a class of dissipative systems, the supply rate σ(u) is essentially the iISS gain function. In this paper, a modified iISS, so-called A-iISS, is used which will be defined below. Compare to the standard iISS definition, the solution x of the system converges to an invariant set A instead of the origin in the notion of A-iISS when a bounded-energy input signal is used. Definition 2.1: Consider a system ζ˙ = f (ζ, u), ζ(0) = ζ0 , where f : Rn × Rm → Rn is locally Lipschitz. Let A ⊂ Rn be a nonempty and closed set. It is said to be integral input-to-state stable with respect to A (A-iISS) if there exist functions α ∈ K∞ , β ∈ KL and γ ∈ K such that, for every ζ0 ∈ Rn and u ∈ L∞ (Rm ), the unique maximal solution ζ is global and Z t α(kζ(t)kA ) ≤ β(kζ0 kA , t) + γ(ku(s)k)ds ∀t ∈ R+ ,
−1.5 −150
uΦ (t)
150
(b) Fig. 1. The input-output dynamcis of the Bouc-Wen model with kx = 1.2117 × 10−7 , kw = −5.08 × 10−6 , n = 1.27, α = 8.93 × 10−3 , β = 6.6082 × 10−3 and λ = 2.321810−3 .
An example of the CCW hysteresis model is the BoucWen model [4], which is commonly used to model the hysteresis phenomena in piezo-actuators and will be used in our numerical and experimental results. The general representation of the Bouc-Wen model is given by yΦ (t) = kx uΦ (t) + kw h(t) ˙ h(t) = ρu˙ Φ (t) − ρσ|u˙ Φ (t)||h(t)|n−1 h − ρ(1 − σ)u˙ Φ (t)|h(t)|n ,
where uΦ denotes the voltage, yΦ denotes the displacement, n ≥ 1 and kx , kw , ρ, σ are the parameters determine the shape of the hysteresis curve. The Bouc-Wen model can be described by the Duhem hysteresis operator (6) with x υ n−1 γ−kx υ f1 (γ, υ) = kx + kw ρ − kw ρσ| γ−k kw | kw xυ n −kw ρ(1 − σ)| γ−k kw | γ−kx υ n−1 γ−kx υ f2 (γ, υ) = kx + kw ρ + kw ρσ| kw | kw xυ n −kw ρ(1 − σ)| γ−k kw | . (7)
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d +
u
y
P
the following inequalities hold. Q CT >0 C D
yc
C
−
yΦ
Φ
AT Q + QA + LT L ≤ 0
uΦ
T
Fig. 2. Negative feedback interconnection with a linear system P, a controller C and a Duhem hysteresis operator Φ.
In Figure 1, we illustrate the behavior of the Bouc-Wen model with kx = 1.2117 × 10−7 , kw = −5.08 × 10−6 , n = 1.27, ρ = 8.93 × 10−3 and σ = 0.74. III. M AIN
RESULTS
Consider a feedback interconnection of a mass-damperspring system with a PD controller and a hysteretic actuator as shown in Figure 2, where P represents the linear plant, C is the PD controller and Φ represents the Duhem hysteresis operator. In Figure 2, the exogeneous input disturbance signal is represented by d. The closed-loop system as shown in Figure 2, are given by P: C: Φ:
0 1 0 x + 1 u, b k − − m m m y = c1 c2 x yc = kp y + kd y, ˙ y˙ Φ = f1 (yΦ , uΦ )u˙ Φ+ + f2 (yΦ , uΦ )u˙ Φ− , u Φ = yc , u = d − yΦ ,
x˙ =
(8)
where x = [x1 x2 ]T with x1 be the position and x2 be the velocity, and c1 , c2 are the measurement gains of the position and velocity, respectively. For the PD controller C, kp > 0 and kd > 0 are the controller gains. First, we consider the case where d = 0, i.e. there is no input disturbance in the closed-loop system. Let G denote the cascaded system of P and C, then we have G:
where A
=
h
x˙ yc
0 0 k b −m −m
= Ax + Bu, = Cx + Du, i , B
=
(9) h
kd c2 m .
0
1 m
i , C
=
] and D = Theorem 3.1: Consider the closed-loop system in (8) with d = 0. Assume that for the Duhem hysteresis operator Φ there exists H : R2 → R+ such that for every uΦ ∈ AC(R+ ) and for every admissible yΦ0 , (3) holds, where yΦ = Φ(uΦ , yΦ0 ). Suppose that either (i) the function H (·, ξ) is proper (or radially unbounded) for every ξ; or (ii) the solution yΦ = Φ(uΦ , yΦ0 ) is bounded for every uΦ ∈ AC(R+ ) and every admissible yΦ0 . If there exists Q = QT > 0, L = [ l1 l2 ], w ,kp > 0 and kd > 0 such that [
k c k kp c1 − dm2
k c b kp c2 +kd c1 − dm2
T
(10) (11) T
QB + A C = L w
(12)
2
(13)
2CB = −w ,
l2 b + wm −m
< 0. Then for every initial conditions x0 ∈ R2 and and every admissible yΦ0 ∈ R, the state trajectories of the closed-loop system (8) are bounded and converge to A := {(x1 , x2 , yΦ ) ∈ R3 |x2 = 0}, i.e. the closed-loop system is globally asymptotic stable with respect to A (A-GAS).
Proof: By the assumption of the theorem, the Duhem operator Φ is CCW and there exists a function H : R2 → R+ such that H˙ (yΦ (t), uΦ (t)) ≤ y˙ Φ (t)uΦ (t). (14)
2 Using VG = 12 xT Qx+(yc −Du)u+ D 2 u as the Lyapunov function for the linear systems G, and (11)-(13), we have 1 V˙ G = xT (AT Q + QA)x + xT QBu 2 + xT AT C T u + uCBu + yc u˙ 1 ≤ yc u˙ − (Lx − wu)2 . 2 Based on Definition 2.3, it can be easily checked that G is CW. Now take Vcl (x, yΦ ) = VG + H (yΦ , yc ) as the Lyapunov function of the closed-loop system (8) and substituting the interconnection conditions uΦ = −yc and u = yΦ , we obtain
V˙ cl = V˙ G + H˙ , 1 ≤ yc u˙ − (Lx − wu)2 + y˙ Φ uΦ , 2 1 = − (Lx − wu)2 . 2 Using (10), it can be checked that VG is proper in x and yΦ . Based on the LaSalle’s invariance principle and the compactness of (x, yΦ ), the trajectories converge to the largest invariant set O contained in M := {(x, yΦ ) ∈ R3 |Lx − wyΦ = 0} as t → ∞. L L In the invariant set O in M, yΦ = − w x, i.e., u = w x. L Substituting u = w x in P, we have that in the invariant set O 0 1 x (15) x˙ = l1 l2 b k + wm −m + wm −m √ 2kd c2 b−2m(kp c2 +kd c1 ) . From (13), we have that w = ± m Using (11) and (12), it can be computed that l has a 1 √ 2kd c2 b−2m(kp c2 +kd c1 ) if w = m √ k 2kd c2 b−2m(kp c2 +kd c1 ) 2kd c2 b−2m(kp c2 +kd c1 ) or l1 = − m m √ 2kd c2 b−2m(kp c2 +kd c1 ) w =− . Substituting these w and m
unique solution l1 = √
k
if l1 into (15), we get that in the invariant set O 0 1 x x˙ = l2 b + wm 0 −m
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(16)
holds which implies that x2 = 0 in the invariant set O. By La-Salle’s principle, this implies that x2 (t) → 0 as t → ∞ l2 b if − m + wm < 0. Hence, the closed-loop system is GAS with respect to A := {(x1 , x2 , yΦ ) ∈ R3 |x2 = 0}. 2 Remark 3.2: If all the hypotheses in Theorem 3.1 are satisfied, then the LMI conditions given in (10)-(13) indicates the interval of the control parameters such that the 2k closed-loop system is A-GAS. For instance, if kp ≤ kcd1cm , 3 2 2 b c2 −b c1 m−kc1 m kd c2 b−kd c1 m and kp > kd b2 c2 m+kc2 m2 −bc1 m2 then the kp ≤ c2 m √ 2qk closed-loop system (8) satisfies (10)-(13) with L = [ m 0 ], √ 2q w = m and Q=
kd c2 k2 −kp c1 k m qk m
qk m
kd c2 k−kp c1 m− qb m
where q = (kd c2 b − kp c2 m − kd c1 m). In the rest of this section, we proceed our analysis with d 6= 0, i.e., we study the robustness of the closed-loop system (8) by adding disturbances d to the input of the plant. The concept of A-iISS is applied here to show the robustness of the closed-loop system. Let us first study the dissipativity property of (8). Lemma 3.3: Consider the closed-loop system in (8) with d ∈ C 1 (R+ ) s.t. d˙ ∈ AC(R+ ). Assume that the Duhem hysteresis operator satisfies the hypotheses given in Theorem 3.1. If there exists Q = QT > 0, L = [ l1 l2 ], w, kp > 0 and kd > 0 such that the following inequalities hold.
Q C
CT D
>0
AT Q + QA + LT L + ǫ1 I ≤ 0 T
T
Young’s inequality, u = yΦ and ǫ3 > 0, we have η 1 1 1 V˙ cl ≤ d˙2 + yc2 − ǫ1 xT x − ǫ2 u2 2 2η 2 2 1 2 2 1 T T 1 1 η ˙2 ≤ d + D u + x C Cx − ǫ1 xT x − ǫ2 u2 2 2η 2η 2 2 η η ≤ d˙2 + d2 2 2 where the last inequality is obtained since η is an arbitrary positive constant. This implies that system
(8) is dissipative ˙ = µ d˙ 2 and µ := η/2. with respect to σ where σ(d, d) d 2 Theorem 3.4: Consider the closed-loop system in (8) with d ∈ C 1 (R+ ) s.t. d˙ ∈ AC(R+ ). Assume that the hypotheses in Lemma 3.3 hold. Then (8) is iISS with
2 respect to A (A iISS), with iISS gain γ dd˙ = µ dd˙ , where µ > 0.
Proof: The proof of the Theorem 3.4 is similar to the proof of Theorem 3.1 in [8]. First of all, consider the system (8) with d = 0 which has the following form " # Ax−ByΦ x˙ f1 (yΦ ,uΦ ) (CAx−CBy ) Φ + 1+Df1 (yΦ ,uΦ ) , (21) y˙ Φ = f (yΦ ,uΦ ) (CAx−CByΦ )− 2 (yΦ ,uΦ )
2 + 1+Df
where u = −yΦ . It can be checked that the RHS of equation (21) is locally Lipschitz. Let us write (21) by ζ˙ = f (ζ) where ζ = [xT yΦ ]T . Based on the converse Lyapunov theorem [21, Corollary 2], Theorem 3.1 implies that there exists a smooth Lyapunov function V : R3 → R+ such that • there exist K∞ functions α1 and α2 such that
(17)
α1 (kζkA ) ≤ V (ζ) ≤ α2 (kζkA )
(18) T
QB + A C = L w
(19)
2
(20)
2CB = −w − ǫ2 ,
for some positive constants ǫ1 and ǫ2 . Suppose l2 b + wm < 0, then there exists µ > 0 such that −m the closed-loop system is dissipative with respect to the
2
˙ = µ d˙ . supply function σ(d) Proof: Let Vcl = VQ + H be the Lyapunov function for the closed-loop system, where VQ and H have the same descriptions as in the proof of Theorem 3.1. Then using (17)(20) we have
V˙ cl = V˙Q + H˙ , 1 ≤ yc u˙ − (Lx − wu)T (Lx − wu) + y˙ Φ uΦ 2 1 1 − ǫ1 xT x − ǫ2 u2 , 2 2 1 1 1 T T = − (Lx − wu) (Lx − wu) + yc d˙ − ǫ1 x x − ǫ2 u2 2 2 2 where the last equality is obtained since u = d − yΦ . Using
•
∀ζ ∈ R3 .
there exists a continuous, positive definite function α3 such that dV (ζ) f (ζ) ≤ −α3 (kζkA ) ∀ζ ∈ R3 . dζ
In the case when the input disturbance d 6= 0, the system ˙ where (8) can be written as ζ˙ = f (ζ, d, d)
˙ = f (ζ, d, d)
Ax+B(d−yΦ ) f1 (yΦ ,uΦ ) ˙ 1+Df1 (yΦ ,uΦ ) (CAx+CB(d−yΦ )+D d)+ f2 (yΦ ,uΦ ) ˙− + 1+Df (y ,u ) (CAx+CB(d−yΦ )+Dd) 2 Φ Φ
,
u = d − yΦ . Since f1 (yΦ , uΦ ) ≥ 0 and f2 (yΦ , uΦ ) ≥ 0 according to the definition of the Duhem operator given in Secf1 (yΦ ,uΦ ) f2 (yΦ ,uΦ ) tion II, then 1+Df ∈ [0, 1] and 1+Df ∈ [0, 1]. 1 (yΦ ,uΦ ) 1 (y Φ ,uΦ )
x x ] Hence, we have that for every BA = [ ≤ l , yΦ l yΦ there exists c > 0 such that
˙ ≤ c 1 + µ d˙ 2 kf (ζ, d, d)k d
˙ ∈ BA × R2 . (22) ∀(ζ, d, d) l
The rest of the proof can use the same arguments as in [8] where the compact set K ⊂ Rn in [8] is replaced by BA l . It can be checked that the same lemmas as given in
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AND EXPERIMENTAL RESULTS
In the previous sections, stability for a second-order hysteretic system was shown by using a PD controller. In particular, we give the sufficient conditions on the control parameters such that the closed-loop system is A-GAS. In case the linear plant is the mass-damper-spring system and the position is measured output, then applying our approach, the velocity of the linear plant converges to the origin. The robustness property has also been discussed with respect to the input disturbance. In this section, we apply these results to develop a PD controller for a piezo-actuated stage. To evaluate the controller experimentally, the piezoactuated stage P 611.2S from Physik Instrumente (PI) is used combined with the piezo amplifier E610.S0 from Physik Instrumente (PI). The displacement of the stage is measured by a strain gauge sensor. The input voltage is controlled by a PC computer where the MATLAB Real-Time Workshop is used. The PD controller is implemented within Simulink. The piezo-actuated stage can be considered as a hysteretic actuator connected to a mass-damper-spring plant as shown in Figure 3.
1.5
1
0.5
0
−0.5
−1
−1.5 −2
−1.5
−1
−0.5
0 Voltage v
0.5
1
1.5
2
(a) Comparison between the model output and the measured output in input-output plane −5
x 10
measured output model output
1.5
1
k
0.5
0
−1
m
0
b Fig. 3.
x 10
−0.5
x Φ
−5
2
Displacement m
IV. S IMULATION
Based on these two steps, the identified parameters for the Bouc-Wen model (7) are: kx = 0.2379, kw = −0.0723, ρ = 2.57, σ = 3.53 and n = 1.46. Based on the parameters of the Bouc-Wen model, the parameters of the mass-damper-spring system can also be identified by using lscurvefit, since boundaries of the parameters can be given. The parameters we obtained are m = 0.0352, b = 80.0543 and k = 2.5541 × 104 . The identified parameters are validated by a mixed frequency signal as shown in Figure 4.
Displacement m
[8, Lemma 3.2, Lemma 3.3 and Lemma 3.4] can also be obtained in this case. Applying the Lemmas and the converse Lyapunov function for A-GAS system, we can obtain the AiISS Lyapunov function similar to the proof of [8, Theorem 3.1]. 2
Φ : y˙ Φ = f1 (yΦ , uΦ )u˙ Φ+ + f2 (yΦ , uΦ )u˙ Φ− ,
where x = [ x1 x2 ]T , x1 denotes the displacement and x2 denotes the velocity. The hysteresis system Φ is represented by the Bouc-Wen model as introduced in (7) in Section II. The identification process is separated into two parts: the identification of the Bouc-Wen model and the identification of the mass-damper-spring system. First of all, we identify the hysteresis system by applying low frequent input signals, in which case the linear plant can be considered as a DC gain. To identify the Bouc-Wen model, we first determine the initial parameters of the BoucWen model based on the limit cycle approach as proposed in [5]. Then we use the initial parameters and apply the least square optimization toolbox (lsqcurvefit) in MATLAB to estimate the real parameters of the Bouc-Wen model.
4
6
8
10 Time s
12
14
16
18
(b) Comparison between the model output and the measured output in time (zoom in)
Mass-damper-spring system connected with a hysteretic actuator.
Then the piezo-actuated stage system can be given as follows 0 1 0 u, x + P : x˙ = 1 b k −m −m m y= 1 1 x (23)
2
Fig. 4. Identification results for the piezo-actuated stage, where the red line is the output of the model and the blue line is the measured output from the setup.
The control parameters for the PD controller can then be obtained based on the parameters of the linear plant. It can be checked that with kp = 2000, kd = 1, L = [ 3.1981×106 0 ] and w = 125.215, we can obtain the matrix 1.85 × 1010 7.04 × 106 Q= , 7.04 × 106 3.4 × 103 which satisfies the conditions given in (10)-(13). The simulation result is given in Figure 5 (a) with initial condition T x0 = [ 0 10−5 ] . To show the robustness of the system, we add a distur−2 bance signal d(t) = 10t2 , which satisfies d, d˙ ∈ L2 , to the input of the linear plant. The simulation result is shown in Figure 5 (b). We will now apply our results to the velocity tracking problem where the position of piezoactuator must track a ramp reference signal. Consider again the closed-loop system (8) with d = 0. Let x ¯2 = v¯ denote the desired constant velocity and the corresponding steady-state displacement is
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−6
−5
x 10
16
1
14
x 10
linear plant and the PD controller. Furthermore, the robustness of the closed-loop system with respect to the input disturbance is also studied by applying the concept of AiISS. To evaluate our approach, we conduct an experiment on the piezo-actuated stage where a constant velocity is tracked.
0.5
12
0
Velocity m/s
Velocity m/s
10 8 6
−0.5 −1 −1.5
4 −2
2
−2.5
0 −2
0
1
2
3
4
−3
5
0
1
2
Time s
3
4
R EFERENCES
5
Time s
Fig. 5. Simulation results of the piezo-actuated stage system with initial condition x0 = [0 10−5 ]T . (a) without disturbance; (b) with disturbance −2 d(t) = 10t2
then given by x ¯1 = v¯t (up to a constant) with t ∈ R+ . It follows from the state equations of the closed-loop system in (8) that in the steady-state, y¯Φ = −k¯ v t − b¯ v (up to a T T constant). Denote ζ¯ = [ x¯1 x¯2 y¯Φ ] , e = [ e1 e2 eΦ ] , where e1 = x1 − x ¯1 , e2 = x2 − x¯2 and eΦ = yΦ − y¯Φ . Let u ˜Φ = uΦ − u ¯Φ where u ¯Φ satisfies −kw + kx u ¯Φ = y¯Φ . It can be computed that, in the neighborhood of (eΦ , u ˜Φ ) = (0, 0), ˜ uΦ ) can be approximated by a Duhem the operator eΦ = Φ(˜ operator. Hence the error dynamics can be described by e2 k b 1 − e − e − e 1 2 Φ m m m f (e) = (24) f1 (eΦ ,˜ uΦ )(u ˜˙ Φ )+ +f2 (eΦ ,˜ uΦ )(u ˜˙ Φ )− ,
Applying Theorem 3.1, we can obtain that the error system (24) is locally asymptotically stable with respect to A := {(e1 , e2 , eΦ ) ∈ R3 |e2 = 0}. We remark that in the original coordinate, the applied control input is given by uΦ = u ˜Φ + u¯Φ where u ˜Φ := kp (y − v¯t − v¯) + kd (y˙ − v¯) is the output of the PD controller. The simulation result of tracking a constant velocity v¯ = 5µm/s is shown in Figure 6(a). The experimental result is shown in Figure 6(b) where we use constant reference velocity of v¯ = 5µm/s. −6
6
−6
x 10
7
x 10
6 5 5 4 Velocity m/s
Velocity m/s
4
3
2
3 2 1 0
1
0
Simulation output Reference velocity 0
5
10 Time s
15
20
−1 −2
Measured output Reference velocity 0
5
10
15
20
25
Time s
Fig. 6. (a) Simulation results of the piezo-actuated stage system for tracking a constant velocity 5µm/s; (b) Experimental results for tracking a constant velocity 5µm/s
V. C ONCLUSION In this work we investigate the stability for a second-order mass-damper-spring system driven by a hysteretic actuator which can be represented by a CCW Duhem hysteresis operator. The hysteretic actuator is controlled by a PD controller, where we give the sufficient conditions on the control parameters such that the closed-loop is A-GAS. The sufficient conditions on the control parameters are obtained based on the CW property of the cascaded system of the
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