Automatica 49 (2013) 1295–1303
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Brief paper
B-spline-decomposition-based output tracking with preview for nonminimum-phase linear systems✩ Haiming Wang a , Kyongsoo Kim b,1 , Qingze Zou a,2 a
Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, United States
b
Reform Office Korean Army HQ, Daejun, Chung-nam, 321-759, South Korea
article
info
Article history: Received 19 April 2012 Received in revised form 21 November 2012 Accepted 27 November 2012 Available online 1 March 2013 Keywords: B-spline Trajectory-decomposition System inversion Iterative learning control
abstract This article proposes a B-spline-decomposition-based approach to output tracking with preview for nonminimum-phase (NMP) systems. It has been shown that when there exists a finite (in time) preview of future desired trajectory, precision output tracking of NMP systems can be achieved by using the preview-based stable-inversion technique. The performance of this approach, however, can be sensitive to system dynamics uncertainty, and the computation involved can be demanding. We propose to address these challenges by integrating the notion of trajectory decomposition with the iterative learning control (ILC) technique. Particularly, the B-splines are used to construct a library of output elements and the corresponding input elements a priori, and the ILC techniques are used to obtain the input elements for precision tracking of the output elements. During the tracking with preview, the previewed future desired trajectory is decomposed by using the output elements, and the input is synthesized by using the corresponding input elements with chosen pre- and post-actuation times. The required pre-/postactuation times are quantified based on the stable-inversion theory. The use of B-splines substantially reduces the number of output elements in the library, and the decomposition-synthesis occurs only at time instants separated by the difference between the preview time and pre-actuation time. The proposed approach is illustrated through a simulation study of nanomanipulation application using a NMP piezo actuator model. © 2013 Published by Elsevier Ltd
1. Introduction In this article, we propose a B-spline-decomposition-based approach to the problem of output tracking with preview for nonminimum-phase (NMP) linear systems. It is noted that exact tracking of NMP systems, although not achievable by using feedback control alone (Chen, Ren, Hara, & Qiu, 2001), can be attained by using the stable-inversion theory (Devasia, Chen, & Paden, 1996). The obtained solution, however, is noncausal—the unique stable (bounded) control input (called the inverse input)
✩ This work was supported by the NSF CAREER grant CMMI-1066055. The material in this paper was partially presented at the 2012 American Control Conference (ACC12), June 27–29, 2012, Montréal, Canada. This paper was recommended for publication in revised form by Associate Editor Abdelhamid Tayebi under the direction of Editor Toshiharu Sugie. E-mail addresses:
[email protected] (H. Wang),
[email protected] (K. Kim),
[email protected] (Q. Zou). 1 Kyongsoo Kim was with Qingze Zou’s research group during the early stage of
this work. 2 Tel.: +1 8484453268; fax: +1 8484453241. 0005-1098/$ – see front matter © 2013 Published by Elsevier Ltd doi:10.1016/j.automatica.2013.01.044
depends on the entire future desired trajectory, thereby cannot be implemented to applications where the desired trajectory is online specified (e.g., robotics manipulation, or autonomous vehicle guidance). The dependence of the inverse input on the future desired trajectory has been quantified through the development of the preview-based stable-inversion technique (Zou, 2009). Such a quantification enables the stable-inversion techniques to be implemented to precision output tracking of NMP systems, i.e., the tracking error can be rendered arbitrarily small with a large enough preview time. The performance of the preview-based stableinversion technique in practical implementations, however, can be sensitive to uncertainties and variations of system dynamics (Wu & Zou, 2009). Moreover, the online computation involved can be demanding (Sogo, 2010; Zou, 2009). The proposed approach aims to achieve precision tracking of output trajectory with preview while addressing these challenges. Current approaches to output tracking with preview of NMP systems need to be improved for practical implementations. The benefits of preview for NMP tracking have been well recognized and exploited in various approaches including the LQ-, the H∞ - and the ℓ2 -optimal preview control (Hoover, Longchamp, & Rosenthal, 2004; Marro & Zattoni, 2005). Particularly, the preview
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mechanism has been utilized to alleviate the fundamental performance limits of feedback control to NMP systems (Chen et al., 2001). It is shown that the limit of the tracking error exponentially decays with the increase of the preview time. The quantification, however, is limited to SISO systems (Halpern, 1994) and special functions only (either step functions, sinusoidal functions, or exponential functions) (Chen et al., 2001), and does not lead to the design of the ‘‘optimal’’ feedback controller that attains the lower limit. Although the tracking performance limit was extended to general functions in (Chen et al., 2001), the extension is constrained to SISO systems only with simple zeros, and it is difficult to directly quantify the tracking error from the limit bound due to the use of frequency-dependent weight functions. These drawbacks in feedback-based approaches have been addressed in the preview-based stable-inversion approach (Zou, 2009), which explicitly quantifies, in a systematically and conceptually straightforward fashion, the tracking error that exponentially decays as the preview time increases (Zou, 2009). The practical implementation of the preview-based stable-inversion approach, however, still faces challenges arising from the system dynamics variations and the demanding online computation, particularly, as convolution operation is involved in computing the control input at each sampling instant (Marro, Prattichizzo, & Zattoni, 2002). Thus, it is needed to further develop techniques for output tracking with preview of NMP systems. These uncertainty and computation related challenges can be avoided by using the iterative learning control (ILC) techniques (Xu & Tan, 2003). By exploiting the repetitive nature of the operation, the ILC techniques can compensate for repetitive disturbances and system dynamics uncertainties—due to either the nonlinear dynamics effect on the linear part or the part of dynamics difficult to model and address otherwise (Xu & Tan, 2003)—without compromising the tracking performance. For example, it has been demonstrated in experiments that by using a ILC technique, precision tracking can be achieved for complex trajectories with power spectrum similar to a band-limited white-noise and cut-off frequency beyond the dominant resonance of the system (Kim & Zou, 2012). Also, the iteration mechanism allows the ILC to be noncausal, achieving exact output tracking of NMP systems in repetitive applications. Moreover, online computation can be largely avoided. Two constraints, however, limit the ILC for output tracking with preview: one, the entire desired trajectory needs to be know a priori; and secondly, the operation needs to be repetitive. Therefore, challenges still exist to achieve highly-efficient and precision tracking with preview for NMP systems. The main contribution of this article is the development of a trajectory-decomposition-based approach. The proposed approach utilizes the decomposition concept (Hoelzle, Alleyne, & Johnson, 2011) to enable the use of ILC techniques for nonrepetitive output tracking. Specifically, the desired output trajectory is decomposed into a finite summation of output elements, and a library consisting of pairs of input–output elements is constructed a priori offline, where the output elements are generated by using B-splines, and the corresponding input elements are obtained by using the ILC techniques (e.g., Yan, Wang, & Zou, 2012). Then, the control input is synthesized online via the superposition principle. To account for the NMP zeros and the truncation of the extended B-spline elements (needed in decomposing the previewed output with nonzero starting and/or end values), finite pre- and post-actuation times are needed to implement each input element. The needed pre- and post-actuation times are quantified based on the stable-inversion theory. The proposed approach, illustrated via a simulation example of nanomanipulation, extends the notion of trajectory decomposition (Hoelzle et al., 2011) to more general and broader tracking of arbitrary, non-repetitive output trajectories through the utilization of B-splines, and clarifies and quantifies the pre- and post-actuation effects.
2. B-spline-decomposition-based approach to output-tracking with preview 2.1. Problem formulation Consider a square linear time invariant (LTI) system x˙ (t ) = Ax(t ) + Bu(t ),
y(t ) = Cx(t )
(1)
where x(t ) ∈ ℜn is the state, and u(t ) ∈ ℜq , y(t ) ∈ ℜq are the input and the output, respectively. We assume that Assumption 1. System (1) is stabilizable and detectable, and hyperbolic (i.e., has no zeros on the imaginary axis) with a well defined relative degree r = r1 , r2 , . . . , rq (Devasia et al., 1996). By Assumption 1, we consider with no loss of generality that system (1) is stable. Moreover, although the proposed approach is applicable to right invertible systems, square systems are considered below for ease of presentation. Assumption 2. The desired output trajectory is sufficiently smooth, i.e., the kth desired output, for each k = 1, 2, . . . , q, is differentiable at least to the rk th order. Moreover, preview of the future desired output trajectory exists for a finite preview time of Tp < ∞, i.e., at any time instant tc , the desired output trajectory is known for t ∈ [tc , tc + Tp ]. Definition 3. Output tracking of NMP-LTI systems with preview. Let Assumptions 1 and 2 be satisfied, then for any given desired tracking precision ε > 0, the output tracking of system (1) with preview is to obtain the control input upre (·) such that at any given time instant t during the tracking course, the tracking error is less than ε for a large enough preview time Tp , i.e.,
∥eout (t )∥2 , ∥yd (t ) − ypre (t )∥2 ≤ ε,
(2)
where ∥a∥2 is the standard 2-norm of a ∈ ℜ , yd (·) denotes the desired output trajectory, and ypre (·) denotes the output tracking trajectory obtained by the preview-based input. n
2.2. Decomposition-based output tracking approach The proposed approach comprises three main steps. 2.2.1. Library of the desired input–output elements Le The library Le of pairs of output–input elements is given as Le = {[y∗k,i (·), u∗k,i (·)]|k = 1, 2, . . . , q; i = 1, 2, . . . , Ne }
(3)
where Ne is the total number of different base output elements y∗e,i (·) ∈ ℜ, and for ease of input synthesis, the output element y∗k,i (·) ∈ ℜq×1 for given k and i = 1, 2, . . . , Ne has only the kth output channel nonzero, i.e., only one base output element y∗e,i (·) appears in the kth output channel, i.e.,
T
y∗k,i (·) = 0 · · · y∗e,i (·) · · · 0
u∗k,i (·)
,
with i = 1, 2, . . . , Ne ,
(4)
q×1
and ∈ ℜ is the corresponding desired input element, i.e., for i = 1, 2, . . . , Ne , u∗k,i (·) = uk,i,1 (·) uk,i,2 (·) · · · uk,i,q (·)
T
.
(5)
Thus, the total number of output–input elements NL is given by NL = q × Ne . Furthermore, the base output elements y∗e,i (·)s are sufficiently smooth with a compact support starting at time t = 0, i.e., for i = 1, 2, . . . , Ne , y∗e,i (t ) = 0
for t ̸∈ [0, ti ], with ti < ∞.
(6)
As the entire output elements are known a priori, ILC approach is ideal to obtain the input elements in the library Le offline
H. Wang et al. / Automatica 49 (2013) 1295–1303
a priori. Note that the input elements are noncausal for NMP systems (Devasia et al., 1996), i.e., to achieve exact tracking of the output element (starting from tc = 0), the corresponding input needs to be applied from t → −∞ (Devasia, 2011; Perez & Devasia, 2003), i.e., an infinitely long pre-actuation time. Similarly, an infinitely long post-actuation time is needed for exact tracking of each output element—in order to maintain the necessary state condition when adding the input elements together (Devasia, 2011; Perez & Devasia, 2003). Thus in practice, truncation to finite pre- and post-actuation times becomes necessary (addressed later in Section 2.5). In the following, for α ∈ ℜ and x a vector (or matrix), α x denotes the vector (or matrix) obtained by scaling each entry of x with α . 2.2.2. Online desired trajectory decomposition We denote the pre- and post-actuation times at any given jth decomposition instant as Tpa,j and Tpst,j , respectively, then Definition 4. For given preview time Tp and pre-actuation time Tpa,j , the jth decomposition instant, tdec,j , is the time instant at which the previewed desired trajectory yd (t ) for t ∈ [tdec,j + Tpa,j , tdec,j + Tp ) is decomposed, and is given by (see Fig. 1) tdec,j = tdec,j−1 + Tp − Tpa,j ,
for j > 1, and tdec,1 = 0.
(7)
Assumption 5. At any decomposition instant tdec,j for j = 1, 2, . . . , the available preview time Tp is greater than the required preactuation time Tpa,j , i.e., Tp > Tpa,j
∀j = 1, 2, . . . .
(8)
We present below the decomposition at the jth decomposition instant tdec,j and consequently the synthesis and update of the input between the jth and (j + 1)th decomposition instants. The added part of the previewed desired output trajectory, i.e., yd (t ) for t ∈ [tdec,j + Tpa,j , tdec,j + Tp ] (see Fig. 1), will be decomposed into a finite sum of output elements with given desired precision εa , i.e., we approximate the previewed desired output trajectory in the kth channel, yd,k (·) in
T
yd (·) = yd,1 (·) yd,2 (·) · · · yd,q (·)
yd,k (t ) ≈
Nd,k
,
as
(9)
pk,i y∗e,j (t − tdec,j − tsep,k,i )
(10)
Fig. 1. The proposed decomposition-based output tracking scheme.
where u∗k,i (t ) ∈ ℜp×1 is given by (5). Note that the tracking of the original desired trajectory yd (·) is guaranteed by the superposition property of LTI systems (see (3)–(5)). Then for a given required tracking precision ε , the corresponding pre-actuation time Tpa,j and post-actuation time Tpst,j can be determined (discussed later in Section 2.5), and the truncated control input to track the decomposed part of the previewed desired output trajectory is obtained as utrt,j (t ) = Wtj,1 ,tj,2 (t )ud,j (t ), for tracking yd,k ∈ [tdec,j + Tpa,j , tdec,j + Tp ],
=
upre,j (t ) =
(14)
Ey,apox (τ )2 (dτ )
≤ εa ,
where
2
2.2.3. Online input synthesis Based on the decomposition in Eq. (11), the desired input to track yd,k (t ) for t ∈ [tdec,j + Tpa,j , tdec,j + Tp ] is synthesized
i =1
pk,i u∗k,i (t
− tdec,j − tsep,k,i )
(15)
With knots chosen above, the B-spline basis functions are obtained recursively from
where pk,i s ∈ ℜ are constant coefficients, and the positive constants tsep,k,i ≥ 0 denote any additional shift needed in the decomposition.
N q d,k
pk,i Bi,s (t ) , papx,k (t )
i=−s+1
0 ≤ t−s+1 ≤ t−s+2 ≤ · · · ≤ tm−1 ≤ Td .
tdec,j +Tpa,j
Nd,k q ∗ Ey,apox (t ) , yd (τ ) − pk,i yk,j (t − tdec,j − tsep,k,i ) k=1 i=1
where Bi,s (t ) are the sth-degree basis functions, and m ≥ s + 2 is the total number of knots with ti s for i = −s + 1, . . . , m − 1, chosen below in the approximation
1/2
k=1
t = tdec,j t ∈ (tdec,j , tdec,j+1 ).
2.3.1. B-spline-based trajectory decomposition For completeness we present below the construction of B-splines as a LQ-optimal control problem (Xu & Basset, 2012) as in Kano, Nakata, and Martin (2005). With no loss of generality, we consider the decomposition of the kth channel desired output trajectory, yd,k (t ) for time t ∈ I , [0, Td ]. By (10),
(11)
i =1
ud,j (t ) =
upre,j (t ) + utrt,j (t ), upre,j (t ),
m−1
pk,i y∗k,j (t − tdec,j − tsep,k,i ),
tdec,j+Tp
2.3. B-spline-based desired trajectory decomposition
yd,k (t ) ≈
such that at any given decomposition instant tdec,j
(13)
where Wtj,1 , tj,2 (·) is the window function (i.e., Wt1 , t2 (t ) = 1 for t ∈ [t1 , t2 ], and Wt1 , t2 (t ) = 0 otherwise) with tj,1 = tdec,j , tj,2 = tdec,j + tj,max + Tpst,j , and tj,max is the upper bound of the supports of all output elements selected for the jth decomposition. Finally, the control input is updated,
i=1 Nd,k
1297
(12)
Bi,s (t ) =
t − ti ti+s − ti
Bi,s−1 (t ) +
ti+s+1 − t ti+s+1 − ti+1
i = −s + 1, −s + 2, . . . , m − 1,
Bi+1,s−1 (t ), (16)
with initially Bi,0 (t ) = 1 for ti ≤ t < ti+1 , and 0 otherwise. Thus, the total number of B-splines used in the approximation Nd,k is Nd,k = m + s − 1 ≥ 2s + 1. We select NB number of sampled desired output values, {yd,k (γ1 ), yd,k (γ2 ), . . . , ydk (γNB )}, in the approximation, 0 ≤ γ1 ≤ γ2 · · · ≤ γNB ≤ Td .
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Then the approximation problem in (15) can be formulated as an optimal control problem, min J (γ ) , min
N Pk ∈ℜ d,k
Pk
+
I NB
2 wj papx,k (γj ) − yd,k (γj ) ,
(17) Fig. 2. Two of the seven output elements in the library Le generated by using the 3rd-degree uniform B-spline basis functions.
Pk = [pk,−s+1 pk,−s+2 · · · pk,m−1 ] , T
(18)
γ = {γ1 γ2 · · · γNB }T ,
via extension and smooth transition as
and λ > 0 and wj ∈ [0, 1] for ∀j are the weights. The solution is readily obtained as Pk∗ = (λQ + M T WM )−1 M T W ζy,k ,
where
W = diag([w1 w2 · · · wNB ]),
(20) Nd,k ×Nd,k
Q = [qkr ,kc ] ∈ ℜ
,
(21)
with kr , kc = −s + 1, −s + 2, . . . , m − 1, (2)
I
(2)
Bi,s (t )Bj,s (t )dt ,
(22)
and the matrix M ∈ ℜNB ×Nd,k given by B−s+1,s (γ1 ) B−s+1,s (γ2 )
B−s+2,s (γ1 ) B−s+2,s (γ2 )
B−s+1,s (γNB )
B−s+2,s (γNB )
B=
.. .
.. .
··· ··· .. . ···
Bm−1,s (γ1 ) Bm−1,s (γ2 )
.. .
.
tsep,k,i = ti+1 − ti =
m−1
= ∆tsep,j ,
Te,i (t ) ∗ y ∗e,i (0) ye,i (t )
∗ y∗e,i (t + 3Tpa )=
Bm−1,s (γNB )
i = 0, 1, . . . , m − 2; (24)
(2) The truncation locations of the B-spline basis are fixed and uniformly spaced. For example, for the 3rd uniform B-splines, the truncation instants are at s × ∆tsep,j (s = 1, 2, 3) ahead (or after) of the starting (or the end) instant of the basis function, respectively, and the library Le only consists of seven base output elements, i.e., y∗e,i (·) for i = 1, 2, . . . , 7 (see Fig. 2). As the sth-degree B-splines is up to s − 1 order continuously differentiable, we assume Assumption 6. The base output elements y∗e,i (·) for i = 1, 2, . . . , NL in the library Le are generated from the sth-degree of uniform B-splines with s ≥ rmax + 1, where rmax = maxk rk is the highest relative degree among all output channels.
2.4. Output elements extension and input elements approximation The following output-extension, input-approximation scheme is proposed as the truncated B-splines cannot be used directly as the output elements. Output elements extension. The output elements generated by the truncated B-splines with non-zero starting values are obtained
t t t t
< −3T∗pa ∈ [−3T∗pa , − 2T∗pa ], ∈ (−2T∗pa , 0), ∈ [0, ti ],
(25)
where T∗pa is the maximum pre-actuation time large enough to guarantee the tracking precision for all tracking in the given application, and Te,i (·) denotes a smooth function to transit the output element y∗e,i (t ) from 0 to the truncated value at y∗e,i (0) (e.g., an exponential function). Note the above extended output elements are filtered to render the extension points smooth enough while keeping the filtering-caused deviation small enough under Assumption 2. Similarly, the output elements generated from the truncated B-splines with non-zero ending values are obtained via extension and smooth transition for t > ti :
(23)
Note that in (19), all the matrices (vectors) except the sampled values of the desired output are known a priori. Uniform B-splines. The uniform B-splines, obtained when the knots selected are evenly spaced within the time interval, i.e., ti+1 − ti = Td /(m − 1) for i = 0, . . . , m − 2, possess two features toward a small size library. (1) The untruncated B-splines are nothing but time-shift copies of each other, with the shift time, tsep,k,j , given by Td
0
(19)
ζy,k = [yd,k (γ1 ) yd,k (γ2 ) · · · yd,k (γNB )]T ,
qkr ,kc =
b
2) 2 λ(p(apx ,k (t )) dt
j=1
a
y∗e,i (t )
=
∗ y (t ) ye∗,i (t ) e,i
i
Te,i (t ) 0
t t t t
∈ [0, ti ), ∈ (ti , ti + 2T∗pst ), ∈ [ti + 2T∗pst , ti + 3T∗pst ], > ti + 3T∗pst
(26)
where T∗pst is the maximum post-actuation time. Input elements approximation. At any given jth decomposition instant, the input elements u∗k,i (t ) are approximated as follows,
∗ uk,i (−Tpa,j ), ∗ for t ∈ [−2Tpa,j , − T∗ ], u∗k,i (t + 3Tpa )= u∗ (t + T∗ ), for t ≥pat + T . i pst,j k,i k,i pst
(27)
Offset of the previewed output for decomposition. At each decomposition instant, the previewed desired output to be decomposed is offset to restart at zero. In the following, we denote the subset of output elements obtained from un-truncated B-spline elements, and those of truncated-B-spline elements with extension at the beginning or at the end, as Bo , Bbe , and Bfe , respectively. 2.5. Quantification of the pre- and post-actuation times We first represent system (1) in the output tracking form. Note that the quantification of the pre- and post-actuation times via the stable-inversion theory (Devasia, 2011; Devasia et al., 1996) below is general—under Assumptions 1 and 2, the input to achieve exact tracking of a given desired trajectory is unique. Output tracking form (Devasia et al., 1996). Under Assumptions 1 and 2, system (1) can be transformed into the following output tracking form through a state transformation Γ : ℜn → ℜn
ξ (t ) ξ (t ) x(t ) = Γ ηs (t ) = Γξ Γη,s Γη,u ηs (t ) , ηu (t ) ηu (t )
(28)
H. Wang et al. / Automatica 49 (2013) 1295–1303
along with the inverse control input uinv (·) uinv (t ) = MY Yd (t ) + Ms ηs (t ) + Mu ηu (t ),
such that
(29)
ξ˙ (t ) = ξ˙d (t ), η˙ s (t ) = As ηs (t ) + Bs Yd (t ), η˙ u (t ) = Au ηs (t ) + Bu Yd (t ), where dr1 −1 y1 dr2 −1 y2 ξ (t ) = y1 , y˙ 1 , . . . , r −1 , y2 , y˙ 2 , . . . , r −1 , . . . dt 1 dt 2 T r p −1 d yp yp , y˙ p , . . . , rp −1
(30)
(r )
is the output and its derivatives, and Yd (t ) = ξd (t )yd (t )
(31)
T
.
In (30), ηs (·) and ηu (·) are the stable and the unstable internal dynamics (Devasia et al., 1996), respectively, i.e., all the eigenvalues of As and Au are on the open left hand side and the open right hand side of the complex plane, respectively. Particularly, the stable and unstable internal dynamics are solved by flowing the state forward and backward in time, respectively, i.e.,
ηs (t ) = Ls (0, t , Yk,i (·)),
ηu (t ) = Lu (t , ∞, Yk,i (·)),
(32)
where Ls (tℓ , tµ , U (·)) ,
tµ
eAs (tµ −τ ) Bs U (τ )dτ
tℓ
Lu (tℓ , tµ , U (·)) , −
tµ
(33) e−Au (τ −tℓ ) Bu U (τ )dτ .
The above (33) implies that truncation to a finite pre- and postactuation time is needed when using the input elements to synthesize the input (Devasia, 2011; Perez & Devasia, 2003) (see Section 2.2.3 and (13)). The lemma below quantifies the pre- and post-actuation times for one output element in the library Le . Lemma 7. Let Assumptions 1, 2, 5 and 6 be satisfied, and let y∗k,i (·) be any given output element in the library Le specified by (4). Then the error due to a finite pre- and post-actuation time, Tpa,i and Tpst,i respectively, in the tracking of y∗k,i (·), ey,i (t )2 , is bounded as 2
∞ Ye,i Ey,i (t , Tpa,i , Tpst,i ) + Ka,s e−(α+λ)Tpa,i −λt , y∗e,i (·) ∈ Bbe y∗e,i (·) ∈ Bo ≤ Y∞ e,i Ey,i (t , Tpa,i , Tpst,i ), ∞ Ye,i Ey,i (t , Tpa,i , Tpst,i ) + Λ(Kt ,u , Kˆt ,u , t )e−β Tpst,i , y∗e,i (·) ∈ Bfe
Ka,u e−(β+λ)Tpa,i −λt
+ Λ(Kt ,s , Kˆt ,s , t )e−αTpst,i with, Λ0 (K1 , K2 , t ) , K1 − K2 e−λt µ(t − tk,i − Tpst,i ),
(35) (36)
where µ(t ) is the unit step function, and
Ka,u , K t ,s , K t ,u ,
1
β
Theorem 8. Let conditions in Lemma 7 be satisfied, and at any given jth decomposition time instant tdec,j , let the previewed desired output yd,k (t ) for t ∈ [tdec,j + Tpa,k,j , tdec,j + Tp ] and k = 1, 2, . . . , q be given, and be decomposed into Nd,k number of desired output elements generated by the sth-degree B-splines. Then the finite pre- and postactuation caused output tracking error can be bounded as
1/2 q ey,k (t ) 2 ey (t ) = , 2 2 where for k = 1, . . . , q,
j j −β Tpa,k,1 −λ(Tpa,k,1 +t ) ∥ey,k (t )∥2 ≤ Yˆ ∞ e j,k K1 e j −β T + Λ1 (K2 , K3 , t )e pst,k,Nd,k + × Λ1 (K4 , K5 , t )e
∞ ˆ∞ Y j,k = sup |pk,i |Ye,i ,
αλ βλ
e−β Γi + Kt ,s
K4 = Kt ,s
i=Nd,k −s+1
(37)
MA MAs ∥Ms ∥2 ∥Bs ∥2 ∥B∥2 ∥C ∥2 ,
λ(tk,i +Tpst,i )
K2 = sKt ,u ,
j Nd,k
,
(40)
(41)
−β Γi +λ(T
K5 = Kt ,s
Nd,k −s
K3 = sKˆt ,u , e−α Γi + sKt ,s ,
i=s+1
Nd,k
e
pst,k,N
j +Tp −Ωi ) d,k
i=Nd,k −s+1
MA MAu ∥Mu ∥2 ∥Bu ∥2 ∥B∥2 ∥C ∥2 ,
Kˆt ,s , Kt ,s e
d,k
with Y∞ e,i given by (35), and for j = 1, . . . , 5, Kj s are
K1 = (s + 1)Ka,u ,
MA MAu ∥Γη,u ∥2 ∥Bu ∥2 ∥C ∥2 ,
2
j
−α Tpst,k,N
i = 1, 2, . . . , Nd,k ,
i
MA MAs ∥Γη,s ∥2 ∥Bs ∥2 ∥C ∥2 ,
1 + MAs
(39)
k=1
j
Ey,i (t , Tpa,i , Tpst,i ) ,
α
Next, we consider at any given jth decomposition instant, the tracking error due to the finite pre- and post-actuation of the synthesized input utrt (t ) for yd (t ) with t ∈ [tdec,j + Tpa,j , tdec,j + Tp ). For clarification, for any given kth (k = 1, 2, . . . , q) output channel yd,k (·) we order the output elements by the time they first appear in the decomposition of the previewed desired output in that channel, and denote the pre- and post-actuation times of the j j ith output element as Tpa,k,i and Tpst,k,i , respectively.
and Tpst,k,Nd,k are the pre-actuation time of the first input element and the post-actuation time of the last input element involved in the decomposition, respectively,
τ
K a ,s ,
Proof. The proof follows by applying the analysis of preview-time and pre/post-actuation time effects in the preview-based stableinversion (Zou, 2009) and the optimal output transition (Perez & Devasia, 2003) to the three different types of output elements in the library (i.e., y∗e,i (·) ∈ Bo , y∗e,i (·) ∈ Bbe , y∗e,i (·) ∈ Bfe ), respectively.
where Λ1 (·, ·, ·) has the same structure as Λ0 (·, ·, ·) in (36) with the j j j step function µ(·) replaced by µ(t − Tp + Tpa,k,1 − Tpst,k,Nd,k ), Tpa,k,1
Y∞ e,i , sup Ye,i (τ ) ∞ ,
2
(38)
u
(34)
for t ≥ 0, where
At A t e ≤ MA e−λt , e s ≤ MA e−αt , s 2 2 −A t e u ≤ MA e−β t .
j
tℓ
ey,i (t )
and the constants MA , MAu , α and β satisfy the following Hurwitz inequality:
2
dt
1299
,
Kˆt ,u , Kt ,u e
λ(tk,i +Tpst,i )
Nd,k −s
,
+ K t ,s
i=s+1
e
−α Γi +λ(Tpst,k,Nd,k +Tp −Ωi )
+ sKˆt ,s
(42)
1300
H. Wang et al. / Automatica 49 (2013) 1295–1303
where Ka,u , Kt ,s , Kˆt ,s , Kt ,u and Kˆt ,u are given by (37), and j
Γi , (Nd,k − i − 1)∆tsep,j ,
Ωi , (i − 1)∆tsep,j .
(43)
Proof. The proof follows by (a) applying the superposition principle to Lemma 7, and (b) noting that (I) there are s number of output elements (generated by the truncated-at-the-beginning B-splines) involved in decomposing yd,0,k (t ) for t ∈ [0, ∆tsep,j ), and s number of output elements (generated by the truncated-at-the-end B-splines) involved in decomposing yd,0,k (t ) for t ∈ [tk −∆tsep,j , tk ], and (II) the weighted sum of the output elements in des involved ∗ composing yd,k (t ) equals zero at t = 0, i.e., i=1 pk,i ye,i (t ) t =0 = 0. As shown by (40), the finite pre- and post-actuation times caused tracking error exponentially depends on the length of the first pre- and the last post-actuation times, Tpa,k,1 and Tpst,k,Nd,k , respectively, and the above Theorem 8 shows that the required pre- and post-actuation times (for given tracking precision) are ultimately determined by the largest ones (among all output channels), respectively. Thus we have j
j
Tpa,j = max Tpa,k,1 ,
Tpst,j = max Tpst,k,Nd,k ,
k
k
k = 1, 2, . . . , q.
(45)
k=1
where for k = 1, . . . , q, and t > m(Tp − Tpa,m ), ∗
∗ Ey,k (t ) ≤ Yˆ ∞ K ˆ 1 e−(β Tpa,m +λt ) + Λ2 (mK2 , K ˆ 3, t ) m,k 2 ∗ ∗ ˆ 5 , t )e−αTpst,m × e−β Tpst,m + Λ2 (mK4 , K
= max Yˆ ∞ j ,k ,
(46)
T∗pst,m = min Tpst,j ,
j
ˆ 3 = ETp ,Tpa K3 , K
ˆ 1 = ETp ,Tpa K1 , K
ˆ 5 = ETp ,Tpa K5 , K
1 − em(Tp −Tpa,m )λ 1−e
,
,
with
∥
eay,k
(t )∥2 ≤
s+1
(51)
|pk,j |εe ,
j =1
where pk,j are the coefficients of the output elements used in the decomposition of the previewed desired output in the kth output channel at time instant t. 2.6. ILC approach to construct the library Le Various ILC techniques can be used for constructing the library (Xu & Tan, 2003). Below we present the recently-developed multiaxis inversion-based control (MAIIC) technique (Yan et al., 2012), 1 ˆ ˆ Uˆ k (jω) = Uˆ k−1 (jω) + ρ(jω)G− I ,md (jω)(Yd (jω) − Yk−1 (jω))
(52)
GI ,md (jω) = diag G11,md (jω), G22,md (jω), . . . , Gnn,md (jω) ,
ρ(jω) = diag [ρ1 (jω), ρ2 (jω), . . . , ρn (jω)] Uˆ k (jω) = [ˆu1,k (jω), uˆ 2,k (jω), . . . , uˆ n,k (jω)]T ,
with
and
(53)
(54)
(47)
Yˆk (jω) = [ˆy1,k (jω), yˆ 2,k (jω), . . . , yˆ n,k (jω)]T .
(48)
It has been demonstrated via experiments that the MAIIC technique can achieve precision output tracking in repetitive operations (Yan et al., 2012).
(49)
and Λ2 (·, ·, ·) has the same structure as Λ0 (·, ·, ·) in (36) with the step function µ(·) replaced by µ(t − Tp + Tpa,tot,m − Tpst,k ), with m
(t )∥
with ρp (jω) ∈ ℜ+ for each p ∈ Z, and,
∗
(Tp −T∗pa,m )λ
∥
2 2
j
j
1/2 eay,k
Tpa,m = min Tpa,j , ∗
q
for k ≥ 1, and Uˆ 0 (jω) = 0 initially, where GI ,md is a diagonal matrix with diagonal elements being the model of the diagonal subsystems of system G(jω),
where for j = 1, 2, . . . , m,
Tpa,tot,m =
Lemma 11. Let conditions in Theorem 8 be satisfied, and the practically achievable desired input element be specified by (50). Also, let the pre- and post-actuation times Tpa,1 and Tpst,Nk be as in Theorem 8. Then at any given time instant tc , the tracking error of using the synthesized input is bounded as
k=1
1/2 q Ey,k (t ) 2 Ey (t ) = , 2 2
(50)
By noting that there are at most s + 1 number of input elements summed together at any given time instant when uniform Bsplines are used, the following lemma follows immediately.
∥ (t )∥2 ≤ ∥Ey (t )∥2 +
Theorem 10. Let conditions in Theorem 8 and Assumption 9 be satisfied, and at the current time instant tc , there are m number of decomposition instants, tdec,j with j = 1, . . . , m, and at each tdec,j , there are Nj number of the sth-degree uniform B-splines based output elements used in the decomposition. Then the finite pre- and postactuation caused output tracking error can be bounded as
ETp ,Tpa =
∥yak,i (t ) − y∗k,i (t )∥2 ≤ εe .
Eya
Assumption 9. At any given decomposition instant, the pre- and j j post-actuation times, Tpa,k,1 and Tpst,k,Nd,k are the same across all output channels.
Quantification for practical implementation. To account for noise and other disturbances existing in practical implementations, we consider practically achievable desired input element and output element, uai,k (·) and yak,i (·), respectively, such that for ∀k = 1, . . . , q, and i = 1, . . . , NL ,
(44)
With no loss of generality, we assume
ˆ∞ Y m,k
Proof. The proof follows by applying the superposition principle to Theorem 8, and noting that at the current time instant tc , the actual effective pre-actuation time related to the pth decomposition instant (for p ≤ m) is given by tact,p = t − pTp − p j=1 Tpa,j−1 (with Tpa,0 = 0), i.e., the proof follows by replacing t in the expression on the right hand side of (40) with tact,p .
Tpa,j ,
j =1
and Ka,s , Ka,u , Kt ,s and Kt ,u given by (37), and Ki s for i = 1, 2, 3, 4 defined in (42), respectively.
3. Nanomanipulation simulation example 3.1. Output tracking with preview in nanomanipulation Preview-based output tracking is needed in nanomanipulation using scanning probe microscopy (SPM), where a micro-machined cantilever probe driven by piezoelectric actuators is utilized to manipulate nanoscale subjects, both horizontally and perpendicularly, to, for example, build integrated circuits using nanotubes
H. Wang et al. / Automatica 49 (2013) 1295–1303
a
1301
b
c
Fig. 3. Desired manipulation contour and previewed X –Y desired trajectories in simulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
or conduct surgical operations on a single live cell. Similar output tracking with preview also exists in areas such as nanofabrication and robotic operation. We focus, in the following, on x–y axes positioning of nanomanipulation. Consider the dynamics of the piezoelectric actuator for x-axis (the y-axis dynamics is similar) positioning of a SPM 4
(s − zq )
G(s) =
x(s) ux (s)
= Kx
q =1 6
(55)
(s − pr )
r =1
where the input and output are the voltage applied to the piezoelectric actuator (in volts) and the displacement of the piezo actuator (in µm), respectively, the Laplace transform variable s is in rad/ms, and Kx = 29.28, zq = 0.9274 ± 41.659i, − 0.2484 ± 30.434i, pr = −0.188 ± 31.326i, − 0.857 ± 24.570i, −20.26, − 15.20. 3.2. Implementation of the B-spline-decomposition-based output tracking The proposed approach was implemented to track the contour plotted in Fig. 3, by following the three steps described in Section 2.2. We mimicked the experimental conditions by augmenting a band-limited white noise similar to that observed in experiments (Yan et al., 2012) (amplitude: ±0.02 µm, cut-off frequency: 5 kHz) to both the input and the output. A preview time of 4 ms was quantified by Theorem 10 for precision tracking (i.e., with the size of the tracking error close to the added noise level) and used in the simulation. First, the output elements y∗k,i (·) were constructed by using the 7th-degree uniform B-splines with knot number chosen at m = 8, resulting in a total of NL = 15 output elements (see Section 2.4). The corresponding input elements u∗k,i (·) for k = 1, 2 and i = 1, . . . , 15 were obtained by using the MAIIC technique. The relative tracking error of each output element in 2-norm, E2 (%), was very close to the noise level at ∼0.5%. Secondly, with the previewed desired output trajectory, the required pre- and postactuation times can be quantified by Theorem 10 and Lemma 11, which, in turn, can be used to determine the decomposition instants by (7). For knot number m = 8, a total of 15 output
elements were used in the approximation at each decomposition instant (the 2-norm approximation error <0.001%). Finally, at each decomposition instant tdec,j , the control input was synthesized and updated according to Eqs. (12) and (14) accordingly. 3.3. Simulation results and discussion The output tracking with two different pre-actuation times Tpa,j and two different post-actuation times Tpst,j , Tpa,j = 0.05, 3 ms, and Tpst,j = 0.05, 1 ms, respectively, were investigated (the same pre- and post-actuation times were applied at all decomposition instants for simplicity). Specifically, the y-axis output tracking results are compared in Fig. 4 for the pre-actuation time of Tpa,j = 3 ms, and 0.05 ms, where the post-actuation time was kept the same at Tpst ,j = 1 ms for all j = 1, 2, . . . , 6. Moreover, the manipulation contour tracking results are compared in Fig. 5 for the pre- and post-actuation times Tpa = 0.05 ms, Tpst = 0.05 ms and the pre- and post-actuation times Tpa = 3 ms, Tpst = 1 ms. The simulation results showed that by using the proposed trajectory-decomposition-based output tracking technique, precision output tracking can be maintained throughout the entire output tracking. For example, by using the proposed technique with the pre-actuation time Tpa at 3 ms, the relative RMS tracking error E2 (%) was only 0.54% (comparable to the noise level), which dramatically increased to 22.67% when Tpa was shortened to 0.05 ms (see Fig. 4). Similar observations can be drawn for the effect of post-actuation on the tracking precision (see Fig. 5). Simulation results also illustrated the effects of the size of the output and the increase of the number of decomposition instants on the tracking performance. As shown in both Figs. 4 and 5, the finite pre- or post-actuation time caused tracking error became much larger in the later half of the tracking than that in the earlier one. Such an increase of the tracking error reflected the combined effect of these two factors as described in Theorem 10. The accumulation of decomposition inˆ 1 and a larger value of Λ2 (·, ·, ·), stants led to a larger constant K resulting in a larger finite pre- and post-actuation time caused tracking error (see (45)). However, simulation results also showed that the increase of pre- and post-actuation times exponentially dominated the effects of these two factors (omitted due to space limitations). Thus, the simulation results demonstrated that precision output tracking can be achieved (down to the noise level) by choosing a sufficiently large pre- and post-actuation time in the proposed approach.
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H. Wang et al. / Automatica 49 (2013) 1295–1303
a
b
Fig. 4. Comparison of the y-axis output tracking and the tracking error obtained by using the proposed technique with pre-actuation time of (left) Tpa = 3 ms, and (right) Tpa = 0.05 ms. The post-actuation time was the same at Tpst = 1 ms for j = 1 to j = 6.
a
b
Fig. 5. Comparison of the contour tracking using the proposed technique pre-actuation time of Tpa = 3 ms vs. Tpa = 0.05 ms, and (right) comparison of pre- and postactuation times Tpa = 3 ms and Tpst = 1 ms vs. pre- and post-actuation times Tpa = 0.05 ms and Tpst = 0.05 ms. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4. Conclusions In this article, a B-spline-decomposition-based approach to output tracking with preview was proposed. A library consisting of pairs of input–output elements was constructed offline a priori and utilized online to decompose the previewed desired output and synthesized the corresponding input. Uniform B-splines were used to generate the output elements, and the corresponding input elements were obtained by using iterative learning control techniques. The effect of finite pre- and post-actuation times of the synthesized control input on the tracking precision were quantified by using the stable-inversion theory. The proposed method was illustrated by a planer nanomanipulation simulation study with two nonminimum-phase piezo actuator models.
Marro, G., & Zattoni, E. (2005). H2 -optimal rejection with preview in the continuoustime domain. Automatica, 41(5), 815–821. Perez, H., & Devasia, S. (2003). Optimal output-transitions for linear system. Automatica, 39, 181–192. Sogo, T. (2010). On the equivalence between stable inversion for nonminimum phase systems and reciprocal transfer functions defined by the two-sided Laplace transform. Automatica, 46(1), 122–126. Wu, Y., & Zou, Q. (2009). Robust-inversion-based 2DOF-control design for output tracking: piezoelectric actuator example. IEEE Transactions on Control Systems Technology, 17, 1069–1082. Xu, Y., & Basset, G. (2012). Sequential virtual motion camouflage method for nonlinear constrained optimal trajectory control. Automatica, 48, 1273–1285. Xu, J., & Tan, Y. (2003). Linear and nonlinear iterative learning control, vol. 291. Springer Verlag. Yan, Y., Wang, H., & Zou, Q. (2012). A decoupled inversion-based iterative control approach to multi-axis precision positioning: 3-d nanopositioning example. Automatica, 48, 167–176. Zou, Q. (2009). Optimal preview-based stable-inversion for output tracking of nonminimum-phase linear systems. Automatica, 45, 230–237.
References Chen, J., Ren, Z., Hara, S., & Qiu, L. (2001). Optimal tracking performance: preview control and exponential signals. IEEE Transactions on Automatic Control, 46(10), 1647–1651. Devasia, S. (2011). Nonlinear minimum-time control with pre- and post-actuation. Automatica, 47, 1379–1387. Devasia, S., Chen, D., & Paden, B. (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control, 41(7), 930–942. Halpern, M. E. (1994). Preview tracking for discrete-time siso systems. IEEE Transactions on Automatic Control, 39(3), 589–592. Hoelzle, D. J., Alleyne, A. G., & Johnson, A. J. W. (2011). Basis task approach to iterative learning control with applications to micro-robotic deposition. IEEE Transactions on Control Systems Technology, 19, 1138–1148. Hoover, D. N., Longchamp, R., & Rosenthal, J. (2004). Two-degree-of-freedom l2 optimal tracking with preview. Automatica, 40(1), 155–162. Kano, H., Nakata, H., & Martin, C. F. (2005). Optimal curve fitting and smoothing using normalized uniform B-splines: a tool for study complex systems. Applied Mathematics and Computation, 169, 96–128. Kim, K.-S., & Zou, Q. (2012). A Modeling-free inversion-based iterative feedforward control for precision output tracking of linear time-invariant systems. IEEE/ASME Transactions on Mechatronics. Marro, G., Prattichizzo, D., & Zattoni, E. (2002). Convolution profiles for rightinversion of multivariable non-minimum phase discrete-time systems. Automatica, 38(10), 1695–1703.
Haiming Wang received his B.S. degree in mechanical engineering from Hefei University of Technology, Hefei, China, in 2005, and the M.S. degree in mechanical engineering from the University of Science and Technology of China, Hefei, China, in 2008, respectively. He is currently working toward his Ph.D. degree in the Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey. His research interests include iterative learning control, inversion-based output tracking theory, and high-speed nanomanipulation.
Kyongsoo Kim received his Ph.D. in mechanical engineering from Iowa State University in 2009. He obtained a B.S. degree in mechanical engineering from Pusan National University, Pusan, Republic of Korea (ROK) in 1995, and a M.S. degree in aeronautical engineering from the Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, in 2002. He has been an officer for the ROK Army since 2010, and currently a battalion commander (Lieutenant Colonel).
H. Wang et al. / Automatica 49 (2013) 1295–1303 Qingze Zou received his Ph.D. in mechanical engineering from the University of Washington, Seattle, WA in Fall 2003. He obtained a M.S. degree in mechanical engineering from Tsinghua University, Beijing, China in 1997 and a bachelor degree in automatic control from the University of Electronic Science and Technology of China in 1994. Currently he is an Associate Professor in the Mechanical & Aerospace Engineering Department, Rutgers, The State University of New Jersey. Previously he had taught in the
1303
Mechanical Engineering Department of Iowa State University. His research interests are in inversion-based output tracking and path-following, iterative learning control, control tools for high-speed scanning probe microscope imaging, probebased nanomanufacturing, micro-machining, and rapid broadband nanomechanical measurement and mapping of soft materials. He received the NSF CAREER award in 2009, and the O. Hugo Schuck Best Paper Award from the American Automatic Control Council (AACC) in 2010. Dr. Zou is an Associate Editor for ASME Journal of Dynamic Systems, Measurement and Control (2011–2014).