IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 202055

Improving Intersample Behavior in
Discrete-Time System Inversion: With
Application to LTI and LPTV Systems
Jurgen van Zundert , Wataru Ohnishi , Member, IEEE, Hiroshi Fujimoto, Senior Member, IEEE,and Tom Oomen , Senior Member, IEEE

Abstract—Discrete-time system inversion for perfect
tracking goes at the expense of intersample behavior. The
aim of this article is the development of a discrete-timeinversion approach that improves continuous-time performance by also addressing the intersample behavior. The
approach balances the on-sample and intersample behavior and provides a whole range of new solutions, with stable inversion and multirate inversion as special cases. The
approach is successfully applied to a linear periodicallytime-varying system in both simulations and experiments.
The approach improves the intersample behavior throughdiscrete-time system inversion and outperforms existing
approaches.
Index Terms—Discrete-time inversion, intersample behavior, linear periodically time varying (LPTV), linear timeinvariant (LTI), multirate inversion, stable inversion.

I. INTRODUCTIONRACKING control finds application in many areas, suchas atomic force microscopes [1], wafer stages [2], andspectrometers [3]. The physical systems evolve in continuous
time and, hence, their performance is naturally defined in continuous time. Many approaches for tracking control, including
inverse model feedforward and iterative learning control, arebased on system inversion. For continuous-time systems, system inversion approaches such as [4] can be used. However,controllers are often implemented in a digital environment since
this provides a large flexibility at a low cost [5]. Due to the digital
implementation, discrete-time control is often used.

T

Manuscript received March 30, 2019; revised September 10, 2019;
accepted November 10, 2019. Date of publication November 15, 2019;
date of current version February 13, 2020. This work was supported
in part by the Netherlands Organization for Scientific Research underresearch programs Robust Cyber-Physical Systems (12694) and VIDI
(15698) and in part by the JSPS KAKENHI under Grant 18H05902.
Recommended by Technical Editor Prof. J. Yi. (Corresponding author:
Jurgen van Zundert.)
J. van Zundert was with the Control Systems Technology Group,
Department of Mechanical Engineering, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands. He is now with DEMCON,
Best, The Netherlands. (e-mail: j.c.d.v.zundert@tue.nl).
W. Ohnishi and H. Fujimoto are with the University of Tokyo, Kashiwa
277-0882, Japan (e-mail: ohnishi@ieee.org; fujimoto@k.u-tokyo.ac.jp).
T. Oomen is with the Control Systems Technology Group, Departmentof Mechanical Engineering, Eindhoven University of Technology, 5612
AZ Eindhoven, The Netherlands (e-mail: t.a.e.oomen@tue.nl).
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2019.2953829

One of the main challenges in system inversion
is nonminimum-phase behavior. Causal inversion ofnonminimum-phase systems yields unbounded inputs. To avoid
unbounded inputs, many discrete-time inversion approaches
have been proposed, see, e.g., [6], for a recent overview.
Approximate inversion approaches, such as zero-phase error
tracking control [7], zero-magnitude error tracking control, andnonminimum-phase zero-ignore [8], are well known, but yield
limited performance since an approximation is used. Optimal
approaches, such as norm-optimal feedforward, H2 -preview
control, and H∞ -preview control [6, Sec. 4.3 and 4.4], yield highperformance in discrete time. Discrete-time stable inversion[6, Sec. 4.2] yields exact tracking at the discrete-time samples.
Typically, discrete-time inversion approaches focus on theon-sample performance, i.e., at the discrete-time samples, resulting in poor intersample behavior, i.e., in between the samples,
especially for zeros close to z = −1 [9]. This is observed for bothlinear time-invariant (LTI) and linear periodically time-varying(LPTV) systems [10]. As a consequence, the continuoustime behavior is poor. Indeed, the best on-sample performance does not necessarily leads to the best continuous-time
performance.
Multirate inversion [11], [12] provides an interesting alternative to improve intersample behavior by sacrificing on-sample
performance. However, the approach does not take into accountthe system dynamics when balancing the intersample and onsample performance. As a consequence, the continuous-time
performance will in general be suboptimal.
Although there exist many discrete-time inversion approaches, the balance between on-sample performance and intersample behavior is often not addressed. The main contributionof this article is a discrete-time inversion approach that finds the
optimal balance between on-sample performance and intersample behavior for the purpose of continuous-time performance,
for both LTI and LPTV systems. LPTV systems are of interest
since they occur frequently, including sampled-data systems [5];multirate systems [5], [13]; position-dependent systems withperiodic tasks [14]; and nonequidistant sampling [15]. For both
LTI and LPTV systems, the stable inversion and multirate inversion approaches are recovered as special cases. Related work
includes [5], [16], and [17] where synthesis-based approaches
are presented. The approach presented in this article does not
require synthesis.

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56

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 2020

Fig. 1. Tracking control diagram with continuous-time system Hc ,sampler S, and hold H. Given continuous-time reference trajectory r(t),the objective is to minimize continuous-time error e(t) through design of
discrete-time controller F while control input u[k] remains bounded.

The outline of this article is as follows. In Section II, thecontrol diagram is presented and the control objective is formulated. The main idea of the approach and preliminary results arepresented in Section III. The approach is presented in Section IV.
The advantages of the approach are demonstrated by application
to an LPTV motion system in simulations and experiments in
Section V. Section VI concludes this article.
Notation: For notational convenience, single-input, singleoutput systems are considered. The results can directly
be generalized to square multivariable systems. Let s(i) didti s denote the ith time-derivative of s, B(·) a bilinear transformation, and Rb>a = {x ∈ Rb |x[k] > a for allz
k = 0, 1, . . . , b − 1}. Let Σ = (A, B, C, D) be a discretetime state-space model and define the state transformationz
T (Σ, T ) = (T AT −1 , T B, CT −1 , D).

Fig. 2. Block diagram of the approach. The discrete-time system H isdecomposed into H1 and H2 . System H1 is inverted such that thereis exact state tracking of the desired state x̂1 [k] every n1 samples
for the purpose of intersample behavior. System H2 is inverted suchthat there is exact output tracking every sample for the purpose of
on-sample behavior.

both on-sample (t = kδ) and intersample (t = kδ) performance.
Importantly, u[k] should remain bounded, even in the presenceof nonminimum-phase behavior. Trajectory r(t) is assumed tobe known a priori.
In the next section, the main idea of the approach and preliminary results are presented.

III. CONCEPTUAL IDEA AND PRELIMINARY RESULTS
In this section, the conceptual idea of the approach andpreliminary results are presented. The results form the basis forthe complete approach presented in Section IV.

II. PROBLEM FORMULATIONIn this section, the control problem is formulated. The considered tracking control configuration is shown in Fig. 1, withreference trajectory r(t) ∈ R, control input u(t) ∈ R, output
y(t) ∈ R, digital controller F , sampler S, and zero-order holdH. The continuous-time, LTI system Hc is given by the minimal
realizationẋ(t) = Ac x(t) + Bc u(t)(1a)y(t) = Cc x(t)(1b)

with x(t) ∈ Rn , n ∈ N and can be either an open-loop or closedloop system. It is assumed that Hc is stable.
In conventional discrete-time control, the focus is on onsample performance. The discrete-time system H = SHc H
with Hc in (1) and sampling time δ is given byx[k + 1] = Ax[k] + Bu[k]y[k] = Cx[k]withA = eAc δ ,

B=

 δ

eAc τ Bc dτ,

(2a)
(2b)

C = Cc .

(2c)0

In this setting, perfect on-sample tracking, i.e., e[k] = 0, for allk, is achieved for F = H −1 . However, this does not provide any
guarantees for the intersample performance e(t), t = kδ. Hence,the continuous-time performance in terms of e(t), for all t, may
be poor as observed in, e.g., [10].
The control objective considered in this article is to minimize the continuous-time error e(t). Note that this includes

A. Conceptual Idea
In the proposed approach, the system is decomposed into twoparts and both parts are inverted separately according to Fig. 2,
where H is decomposed as H = H1 H2 . The inversion of system
H1 aims at the intersample behavior. More specific, let n1 be thestate dimension of H1 , then H1 is inverted such that there isexact state tracking of a desired state x̂1 [k] every n1 samples.
The inversion of H2 aims at the on-sample behavior throughperfect output tracking for every sample.
Exact state tracking is experienced to yield good intersample behavior in multirate inversion [11], whereas exact output
tracking yields good on-sample behavior in stable inversion [6].
Hence, the choice of the decomposition into H1 and H2 canbe used to balance the on-sample behavior and the intersample
behavior to the benefit of the continuous-time performance.
The idea is conceptually illustrated in Fig. 3. An important
observation is that a small on-sample error does not necessarily yield a small continuous-time error. The figure shows thatthe approach provides a whole range of solutions that were
nonexisting before. The stable inversion and multirate inversion solution are recovered as the two extreme cases, see also
Section IV-C.
The approach requires the decomposition H = H1 H2 interms of state-space realizations and the desired state x̂1 [k]for H1 , see also Fig. 2. In Section III-B, the desired state forthe continuous-time system Hc is presented. In Section III-C,
the state-space decomposition H = H1 H2 is presented. The
results form the basis for the complete approach presented in
Section IV.

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VAN ZUNDERT et al.: IMPROVING INTERSAMPLE BEHAVIOR IN DISCRETE-TIME SYSTEM INVERSION

where

57

⎡

⎤
0 ···
0
0
⎢
0⎥
1 ···
0
⎢
⎥
.. ⎥
.. . .
..
⎢
⎢
⎥
.
.
.
.
=⎢
⎥
⎢ 0
⎥
0
0
0
·
·
·
1
⎢
⎥
⎣ −a0 −a1 −a2 · · · −an−1 b0 ⎦1 bb01 bb20 · · ·0

Accf Bccf
Cccf

0
0
..
.

1
0
..
.

and
Fig. 3. Qualitative plot of the continuous-time error versus the onsample error. The approach balances the intersample behavior and theon-sample behavior for the purpose of continuous-time performance. It
provides a whole range of solutions ( ) that were nonexisting before.
Importantly, the smallest on-sample error does not necessarily yield thesmallest continuous-time error. The relative performance depends on
the particular settings, e.g., the system dynamics, and may vary. Stable
inversion ( ) and multirate inversion ( ) are recovered as special cases.

⎡a

−1
Tccf
= Bc

Ac Bc

⎡

⎤
y (0) (t)⎢ (1)⎥⎢ y (t) ⎥⎢⎥.
ȳ(t) = ⎢
..
⎥
⎣
⎦
.
(n−1)
y
(t)

B(s)
A(s)

(7)

with all poles ps ∈ C of Fs (s) such that (ps ) < 0 and all polespu ∈ C of Fu (s) such that (pu ) > 0. Let
fu (t) = L−1 (Fu (−s))
fs (t) = L−1 (Fs (s)),
 tx̂ccf,s (t) =fs (t − τ )r̄(τ ) dτ

(8a)
(8b)

−∞

x̂ccf,u (t) = ∞fu (t − τ )r̄(τ ) dτ

(8c)

t

(3)

where L−1 (·) is the inverse unilateral Laplace transform[19, Sec. 9.3]. Let Hc in (4) have realization (5), theny(t) = Cc x̂(t) where−1(x̂ccf,s (t) + x̂ccf,u (t))x̂(t) = Tccf

A similar approach as in [11] is used based on the controllable
canonical form given by Lemma 1, see also [18, Sec. 17.6]. The
desired state is given by Theorem 2. The results are recapitulated
for completeness and reading convenience.
Lemma 1. (Controllable canonical form): Let the transfer
function of Hc in (1) be given by
Hc = Cc (sI − Ac )−1 Bc =An−1
Bc ⎢ . · · . ⎥ . (6)c⎣ .. ·· ·· .. ⎦1
b0 0 · · · 0

B −1 (s) = Fs (s) + Fu (s)In this section, the desired state for the continuous-time system is presented. Given a continuous-time reference trajectoryr(t) together with its n − 1 time derivatives and system Hc
in (1), the objective is to determine a bounded state x̂(t) suchthat y(t) = Cc x̂(t) yields y (i) (t) = r(i) (t), i = 0, 1, . . . , n − 1,
where (·)(i) denotes the ith time derivative of (·), i.e., such thatr̄(t) = ȳ(t) where⎤r(0) (t)⎢ (1)⎥⎢ r (t) ⎥⎢⎥,r̄(t) = ⎢
..
⎥
⎣
⎦
.
(n−1)
r
(t)

⎤

1
1 a2
b b ··· b⎢ a02 a03 ·· 0 ⎥⎢ b0 b0 · 0 ⎥⎥⎢Theorem 2. (Desired continuous-time state): Let B −1 (s) in(4) be decomposed as

B. Desired State for Continuous-Time System

⎡

···

(5c)

(4a)

with

(9)

is bounded and such that ȳ(t) = r̄(t), with ȳ(t) and r̄(t) in (3).
Proof: See [11, Sec. IV-B].

Theorem 2 provides the desired bounded state for optimal
state tracking. Together with the state-space decomposition presented in the next section, Theorem 2 forms the basis of theapproach presented in Section IV.
Remark 1: If poles of B −1 (s) in Theorem 2 have (p) = 0,i.e., B −1 (s) is nonhyperbolic, similar techniques as in [20] canbe used.
C. State-Space Decomposition

A(s) =
B(s) =

s + an−1 snn−1+ · · · + a0b0
bm sm + bm−1 sm−1 + · · · + b0
b0

then
the
controllable
b0 = 0,
Hccf = T (Hc , Tccf ) is given bycanonicalẋccf (t) = Accf xccf (t) + Bccf u(t)y(t) = Cccf xccf (t)

(4b)
(4c)
form

(5a)
(5b)

In this section, the multiplicative state-space decomposition
is presented. A multiplicative decomposition, in contrast to an
additive decomposition, enables exact on-sample tracking every
few samples. Together with Theorem 2, the decomposition forms
the basis of the approach in Section IV.
Given the state-space system H in (2), the interest is inminimal realizations H1 , H2 such that H = H1 H2 in terms ofstate-space realization, where the zeros and poles of H canbe arbitrarily assigned to H1 or H2 . The starting point is themultiplicative decomposition H = H1 H2 in terms of transfer
functions as given by Lemma 3.

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58

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 2020z

Lemma 3. (Transfer function decomposition): Let H = (A,B, C, D) be a state-space realization with n states and invertible
D. Let V ∈ Rn×n1 be a column space of an invariant subspaceof A and let V× ∈ Rn×n2 be a column space of an invariant
subspace of A× = A − BD−1 C, such that S = V V× hasfull rank n = n1 + n2 . LetIn1
0n1 ×n2
S −1 .
Π=S(10)
0n2 ×n1 0n2 ×n2
Then, the realizations
z

H1f =

A ΠBD−1ABz, H2f =C
I
C(I − Π) D(11)are such that H = H1f H2f in terms of transfer functions, i.e., C(zI − A)−1 B + D = (C(zI − A)−1 ΠBD−1 +I)(C(I − Π)(zI − A)−1 B + D).
Proof: Follows directly from extending [21, Corrollary 11]
to D = I.

If the D matrix in Lemma 3 is singular, a bilinear transformation [5, Sec. 3.4], [19, Sec. 10.8.3] can possibly be employed
to obtain an equivalent system with nonsingular D matrix.
A multiplicative decomposition for the transformed system isobtained through Lemma 3. Applying the inverse transformationon the decomposed system yields the decomposition for theoriginal system since B(H1 H2 ) = B(H1 )B(H2 ).
Importantly, Lemma 3 guarantees equivalence in terms oftransfer functions, but not in terms of state-space realizations.
Indeed, the decomposition of Lemma 3 yields nonminimal
realizations of H1f , H2f as both have state dimension n. By
exploiting the modal form and using a suitable state transformation, the desired state-space decomposition for the approach is
obtained as given by Theorem 4.
Theorem 4. (State-space decomposition): Let Tmod ∈ C n×n
be such that Hmod = T (H, Tmod ) = (A, B, C, D) is in modal
form [22, Sec. 7.4] with nonsingular D. Let H1f H2f = Hmodbe the decomposition given by Lemma 3. Let Tper ∈ Rn×n be
such that⎤⎡
A1 0 B1z(12)
T (H1f , Tper ) = ⎣ 0 A2 0 ⎦C1 C1r I⎤⎡
A1 0 B2rz(13)
T (H2f , Tper ) = ⎣ 0 A2 B2 ⎦0 C2 Dwith A1 ∈ Rn1 ×n1 , A2 ∈ Rn2 ×n2 , n1 + n2 = n, and define
z

H1 =

A1 B1
,
C1 I

Furthermore, let X ∈ R

n1 ×n2

z

H2 =

A2 B2
.
C2 D

(14)

In the previous section, the global idea and preliminary results on the desired state and the state-space decomposition arepresented. Based on these results, the approach is presented.
First, the approach for LTI systems is presented. Second, theapproach for LPTV systems is presented. Finally, special cases
are recovered.
A. Approach for LTI Systems
The approach consists of two steps. First, stable inversion isapplied to H2 in (14) to obtain u[k] such that y2 [k] = u1 [k], forall k, see also Fig. 2. The solution is given by Theorem 5 andprovides exact output tracking every sample. See [6, Sec. 4.2]
for a proof.
Theorem 5. (Inversion of H2 ): Consider Fig. 2 and let H2−1
be given byxs [k + 1]
xs [k]As 0
Bs
u1 [k] (17a)=+Bu
xu [k + 1]
0 Au xu [k]
xs [k]
u[k] = Cs Cu(17b)+ Du1 [k]xu [k]
with |λ(As )| < 1 and |λ(Au )| > 1. Then, y2 [k] = u1 [k], for allk, if
u[k] = Cs xs [k] + Cu xu [k] + Du1 [k](18)

xs [k + 1] = As xs [k] + Bs u1 [k], xs [−∞] = 0(19)

forward in time and xu follows from solving(15)−1
Then, the state-space realization of T (H1 H2 , Tper
T12 ) with

In1
X
(16)
T12 =
0n1 ×n2 In2

is identical to that of Hmod .

IV. INVERSION FOR INTERSAMPLE PERFORMANCE

which is bounded for bounded u1 and where xs follows fromsolvingsatisfy

A1 X − XA2 = B1 C2 .

Proof: See Appendix A.
Theorem 4 yields a state-space decomposition H = H1 H2with identical state-space realizations. Note that such a decomposition always exists. Together with Theorem 2, Theorem 4
forms the basis for the approach presented in the next section.
Remark 2: Note that V in Lemma 3 is related to the poles of
H, whereas V× is related to the zeros of H. Hence, V, V× canbe used to assign the poles and zeros to either H1 or H2 .
Remark 3: The column spaces of the invariant subspaces inLemma 3 can be constructed from eigenvectors. Note that forcomplex eigenvectors, the real and imaginary parts should be
used. For eigenvalues with multiplicity larger than one, generalized eigenvectors obtained from the Jordan form can be used
to ensure S has full rank.
Remark 4: Sylvester (15) has a unique solution X if theeigenvalues of A1 and −A2 are distinct [23].

xu [k + 1] = Au xu [k] + Bu u1 [k], xu [∞] = 0(20)backward in time.
If u1 [k] is bounded, u[k] in Theorem 5 is bounded by construction of xs [k], xu [k], even if H2 is nonminimum phase. The
stable inversion solution in Theorem 5 achieves exact output
tracking every sample and has infinite preactuation. Regular

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VAN ZUNDERT et al.: IMPROVING INTERSAMPLE BEHAVIOR IN DISCRETE-TIME SYSTEM INVERSION

causal inversion is recovered as special case if the system isminimum phase (xu is nonexisting), see also [6].
Second, multirate inversion is applied to H1 in (14) to obtainu1 [k]. Note that by Theorem 5, y2 [k] = u1 [k], for all k. The
solution is based on lifting the state equation over n1 samples.
The solution is adopted from [11] and given by Theorem 6. The
solution provides exact state tracking every n1 samples.
Theorem 6. (Inversion of H1 ): Consider Fig. 2 withy2 [k] = u1 [k], for all k, and let x̂1 [k] be the desired state
for system H1 in (14). Consider the state equation lifted over τ
samples given by
x1 [q + 1] = A1 x1 [q] + B 1 u1 [q](21)

with A1 = An1 1 , B 1 = [An1 1 −1 B1 An1 1 −2 B1 . . . B1 ], u1 [q] =[u1 [kn1 ] u1 [kn1 + 1] . . . u1 [(k + 1)n1 − 1]] , and x1 [q] =x1 [kn1 ]. Then, x1 [q] = x̂1 [q], for all q, where x̂1 [q] is x̂1 [k] lifted
over τ samples, if
u1 [q] = B −1
1 (x̂1 [q + 1] − A1 x̂1 [q])(22)which is bounded for bounded x̂1 .
Proof: See [11, Sec. IV-C].

Importantly, the inversion approach in Theorem 6 is basedon the continuous-time system Hc , rather than the discrete-timesystem H. The approach yields exact state tracking, and, hence,exact output tracking, every n1 samples and has n1 samples
preactuation. Note that u1 is bounded if x̂1 is bounded, evenif H1 is nonminimum phase. More details can be found in, forexample, [11] and [12]. The desired state x̂1 in Theorem 6 isobtained by Procedure 7, which follows from Sections III-B and
III-C. An example of the application of Procedure 7 to a motion
system is provided in Appendix B.
Procedure 7. (Desired state of H1 ): Given Hc in (5), H in(2), and the decomposition H = H1 H2 in Theorem 4, the following steps yield the desired state x̂1 [k] in Theorem 6.
1) Obtain the controllable canonical form Hccf = T (Hc ,Tccf ) according to Lemma 1.
2) Obtain the desired state x̂(t) of Hc using Theorem 2.
3) Set the desired state of H to x̂[k] = x̂(kδ).
4) Obtain the desired state of Hmod : x̂mod [k] = Tmod x̂[k],with Hmod , Tmod in Theorem 4.
5) Given H1 , H2 in (14), let⎤⎡
A1 B1 C2 B1 D2z
A2
B2 ⎦ .
(23)
H12 = H1 H2 = ⎣ 0
C1 D1 C2 D1 D2−16) Obtain the desired state of H12 : x̂12 [k] = T12
Tperx̂mod [k], with T12 in (16) and Tper satisfying (12)and (13).
7) Obtain
thedesired
state
for
H1 :
x̂1 [k] = [In1 0n1 ×n2 ]x̂12 [k].
The combination of the inversion of H2 in Theorem 5 and theinversion of H1 in Theorem 6 constitutes the control input u[k] inFig. 2, which is bounded by design, also for nonminimum-phasesystems. The design freedom is in the decomposition of H into
H1 and H2 in Theorem 4. Equation (23) shows that the output isgiven by y[k] = C1 x1 [k] + D1 C2 x2 [k] since either D1 = 0 or
D2 = 0 as D = 0 in (2). If D1 = 0, y[k] = C1 x1 [k] and since59

inversion of H1 in Theorem 5 ensures perfect state tracking of
x1 [k] every n1 samples, there is perfect output tracking every
n1 samples. If D1 = 0, y[k] also depends on x2 [k] of H2 andsince inversion of H2 in Theorem 6 does not provide perfectstate tracking, there is no perfect output tracking for y[k] every
n1 samples. Hence, to guarantee exact on-sample tracking every
n1 samples, V, V× in Theorem 4 are preferably chosen such that
D1 = 0.
In this section, the approach for LTI systems is presented. In
the next section, the approach for LPTV systems is presented.
Remark 5: For strictly proper systems H2 , Theorem 5 can beapplied to the biproper system H̄2 obtained through time shifts
H̄2 = z d2 H2 , where d2 is the relative degree of H2 , see also
[6, Remark 1]. If there are eigenvalues on the unit circle, i.e.,there exist λi such that |λi (A)| = 1, then similar techniques asin [20] can be followed.
Remark 6: The decomposition of H −1 given by (17) can beobtained through an eigenvalue decomposition.
Remark 7: Note that B 1 in Theorem 6 is the controllability
matrix of H1 , and hence, B −1
1 exists if H1 is controllable.
B. Approach for LPTV Systems
In this section, the approach for LPTV systems is presented.
Let the LPTV system H with period τ ∈ N be given byx[k + 1] = A[k]x[k] + B[k]u[k]y[k] = C[k]x[k]

(24a)
(24b)

with A[k + τ ] = A[k], B[k + τ ] = B[k], and C[k + τ ] = C[k],for all k. LPTV systems may result from nonequidistant sampling as in Example 1.
Example 1. (Nonequidistant sampling): Let the sampling in
Fig. 1 be nonequidistant in time and given by the sampling
sequence Δne ∈ R∞
>0 with periodicity τ ∈ N defined as
Δne = (δ1 , δ2 , . . . , δτ , δ1 , δ2 . . .)(25)

with δi = γi δb , δb ∈ R>0 , γi ∈ N, i = 1, 2, . . . , τ . Then, thediscretized system H = SHc H is given by (24) with
A[i] = eAc δi, B[i] = δieAc τ Bc dτ, C = Cc

(26a)

0

i = 1, 2, . . . , τ , where A[k + τ ] = A[k] and B[k + τ ] = B[k],for all k. By linearity of Hc and periodicity of Δne , H is LPTV
with period τ .
The approach for LPTV systems is similar to that for LTI
systems, with the key difference that an additional lifting step isused. The lifting step turns the LPTV system into a (multivariable) LTI system as given by Lemma 8.
Lemma 8: Lifting the input of H in (24) over τ samples
yields the LTI system H given by
x[q + 1] = A x[q] + B u[q](27a)y[q] = C x[q] + D u[q](27b)Authorized licensed use limited to: University of Tokyo. Downloaded on February 16,2020 at 14:07:26 UTC from IEEE Xplore. Restrictions apply.

60

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 2020

where

⎡

x[q] = x[kτ ],u[kτ ]u[kτ + 1]..
.

⎢⎢
u[q] = ⎢
⎣

⎤
⎥
⎥
⎥
⎦

(27c)

u[(k + 1)τ − 1]
A B
C D
⎡
Φτ +1,1 Φτ +1,2 B[1]
⎢ C[1]0⎢⎢ C[2]Φ2,1 C[2]B[1]
=⎢
⎢
..
..
⎣
.
.

⎤
. . . B[τ ]··· 0 ⎥⎥··· 0 ⎥
⎥
.. ⎥
..
. . ⎦
C[τ ]Φτ,1 C[τ ]Φτ,2 B[1] C[τ ]Φτ,3 B[2] · · · 0(27d)Φτ +1,3 B[2]
0
0
..
.

with transition matrixI,Φk2 ,k1 =A[k2 − 1]A[k2 − 2] . . . A[k1 ],k 2 = k1(28)
k2 > k1 .

For the lifted system H in (27), the same approach as for
the LTI system illustrated in Fig. 2 is used. The state-spacedecomposition H = H 1 H 2 is obtained using Theorem 4. System H 2 is inverted using Theorem 5 and H 1 is inverted using
Theorem 6, where the desired state x̂1 [q] follows along the samelines as in Procedure 7. The result is the lifted input signalu[q], which, after inverse lifting, yields input u[k], for the LPTV
system H in (24).
In the previous and present section, the approaches for LTI
and LPTV are presented, respectively. Next, special cases arerecovered.
C. Special Cases
The approach provides a whole range of solutions as illustrated in Fig. 3. The stable inversion and multirate inversion
solution are recovered as the two extreme cases and given by
Corollaries 9 and 10. The results hold for both LTI and LPTV
systems.
Corollary 9. (Special case: Stable inversion [6], [24]): The
stable inversion solution for H is recovered from the approachin Section IV as special case if H = H2 , i.e., H1 = I and
n1 = 0.
Corollary 10. (Special case: Multirate inversion [11], [12]):The multirate inversion solution for H is recovered from theapproach in Section IV as special case if H = H1 , i.e., H2 = I
and n1 = n.
Importantly, although Theorem 5 yields exact output trackingof H2 for every sample, the inversion of H1 does not reduce toconventional multirate inversion of H1 since the desired statex̂1 [k] depends on the full system Hc and not only on H1 .
Finally, the approach for LTI systems is recovered from thatfor LPTV systems as given by Corollary 11. Indeed, for τ = 1,
(2) is recovered from (24).
Corollary 11: The approach is applicable to both LTI and
LPTV systems. Indeed, the approach for LTI systems in

Fig. 4. Motion system [25] used in simulations and for experimental
validation. (a) Experimental high-precision positioning stage with force
input u and output displacement y. (b) Bode diagram of a frequency
response function measurement ( ) of the motion system in Fig. 4(a)).
and the identified continuous-time model Hc (Section IV-A is recovered as a special case from the approachfor LPTV systems in Section IV-B for τ = 1.
The approach provides a whole range of solutions that were
nonexisting before, see also Fig. 3. The advantages of theapproach are demonstrated by application to an LPTV motion
system in Section V.
V. EXPERIMENTAL RESULTS
In this section, the approach outlined in Section IV is validated
experimentally on a motion system. The experiments show theapplicability of the approach in a practical setting. First, themotion system is presented. Second, the approach is validated
in simulations revealing the desired properties and providing the
optimal decomposition for use in the experiments. Finally, theapproach is validated in experiments. Both in simulations andin experiments, the approach is superior to the special cases ofstable inversion and multirate inversion.
A. Motion System
The approach is validated on the experimental high-precision
positioning stage shown in Fig. 4(a). The Bode diagram ofa frequency response function measurement of the system isshown in Fig. 4(b). The measurement is obtained through a
dedicated identification experiment with a multisine input in the

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VAN ZUNDERT et al.: IMPROVING INTERSAMPLE BEHAVIOR IN DISCRETE-TIME SYSTEM INVERSION

Fig. 5. Reference
polynomials in time.

trajectoryr(t)consistingof

61

eighth-order

range between 1–1000 Hz and a sampling frequency of 12.5 kHz.
The identified eighth-order continuous-time system Hc (n = 8
and m = 4) is given by
Hc =×

3.7232 · 106 (s2 + 7.181s + 2.507 · 104 )s(s + 2.33)(s2 + 9.132s + 3.672 · 104 )(s2 + 102.6s + 8.531 · 105 )(s2 + 37.91s + 3.12 · 105 )(s2 + 254.5s + 3.478 · 106 )(29)

and is stable and minimum phase. The Bode diagram of themodel Hc is also shown in Fig. 4(b). The continuous-timereference trajectory r(t) is shown in Fig. 5.
A nonequidistant sampling sequence with γ1 = 1, γ2 = 2,
and δb = 400 μs (fb = 2.5 kHz) is used, see also Example 1,
resulting in an LPTV system H with period τ = 2. The corresponding lifted LTI system H in (27) has one nonminimumphase (transmission) zero due to discretization.

Fig. 6. Generated input signals u[k]. The input signal for stable inversion is nonsmooth, whereas the input signals of multirate inversion
and the proposed approach are smooth. (a) Stable inversion approach.
(b) Multirate inversion approach. (c) Proposed approach.

B. Simulation: Finding the Optimal Decomposition
In this section, the approach in Section IV is evaluated insimulation to confirm its properties and to find the optimal
decomposition for use in the experiments. In Section V-C, theapproach is validated in experiments.
First, three different solutions are considered: the approach in
Section IV with n1 = 4, the special case n1 = 0, i.e., multirate
inversion in Corollary 10, and the special case n1 = 8, i.e., stable
inversion in Corollary 9. The input signals are shown in Fig. 6.
Note that the input signal of stable inversion is nonsmooth. The
error signals are shown in Fig. 7, which, for the purpose ofintersample performance evaluation, are evaluated at a sampling
frequency of 250 kHz, i.e., a factor 100 higher than fb .
The special case of stable inversion in Fig. 7(a) achieves
perfect output tracking, however, the intersample performanceis poor as a result of the nonsmooth input signal, see Fig. 6. The
special case of multirate inversion in Fig. 7(b) yields perfect state
tracking every n = 8 samples, with reasonable intersample performance. The approach in Fig. 7(c) achieves perfect state tracking every n1 = 4 samples and good intersample performance.
The proposed approach outperforms the special cases of stable
inversion and multirate inversion in terms of the continuous-timeerror e(t).

Second, the performance is evaluated for a variety of solutions. The results are shown in Fig. 8 and quantify Fig. 3. The
results show that many of the solutions provided by the approach
outperform the special cases of stable inversion and multirate
inversion. Note that Fig. 8 only shows results for even numbers
n1 due to the additional lifting step with τ = 2 that is presently
used. The solution shown in Figs. 6(c) and 7(c) is the solutionthat yields the best performance as indicated by in Fig. 8 andis also used in the experiments in Section V-C. For comparison,
the discrete-time norm-optimal solution [6, 4.3] is also shownin Fig. 8. Note that since the boundary effects are negligible, theperformance is almost identical to that of the stable inversion
approach [6, Sec. 4.2 and 4.3] and, hence, also inferior to thatof the proposed approach.
In summary, the simulations show that the intersample performance of the special case of stable inversion is poor andthat the performance of the special case of multirate inversion
is moderate. Most importantly, the proposed approach offersa variety of options that outperform the two special cases andachieves superior performance. In the next section, the solutionthat yields the best performance in simulations is validated inexperiments.

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62

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 2020

Fig. 9. Experimental results showing the error signal e(t) in the time), multirate inversion
and frequency domain for stable inversion ((), the discrete-time norm-optimal solution (), and the proposed
approach (). The proposed approach outperforms the special casesof stable inversion and multirate inversion as well as the norm-optimal
solution. (a) Time-domain response. (b) Cumulative power spectrum.

Fig. 7. Error signals over time. The intersample error signal e(t) ()
and on-sample error signal e[k] ( ) near t = 0.30 s show that the proposed approach outperforms the other approaches. (a) Stable inversion
approach achieves exact output tracking every sample ( ). (b) Multirate
inversion approach achieves exact state and output tracking every n = 8
samples ( ). (c) Proposed approach achieves output tracking every
n1 = 4 samples ( ).

Fig. 8. Quantification of Fig. 3 through simulation. The number ofstates n1 in H 1 corresponds to the number of samples between exact
on-sample tracking. The approach offers a variety of solutions ( ). The
best choice for the approach (n1 = 4, ), which is also shown in Figs. 6
and 7, outperforms the stable inversion approach (n1 = 0, ) and themultirate inversion approach (n1 = 8, ). For comparison, the discretetime norm-optimal solution () is also shown and is almost identicalto that of the stable inversion solution ( ).

C. Experimental Validation
In this section, the three solutions shown in Fig. 6 are validated
in experiments by application to the motion system shown in
Fig. 4.
For the purpose of intersample performance evaluation anddue to hardware limitations, the error signal is measured with asampling frequency of 12.5 kHz, i.e., five times larger than fb .
Besides the feedforward controllers running at the nonequidistant rate, a feedback controller running at an equidistant ratewith frequency fb is used for stabilization. The PID controller
with notch filter yields a bandwidth of 10 Hz and a modulus
margin of 6 dB. Due to the low bandwidth of the controller, thefeedback controller has limited effect on the performance.
The experimental results are shown in Fig. 9. The results showthat the proposed approach outperforms the special cases of stable inversion and multirate inversion in terms of the continuoustime error e(t). Note that due to experimental conditions, e.g.,model mismatches, there is no exact on-sample tracking every
few samples as is the case for the simulations in Section V-B.
In summary, the experiments show the practical applicabilityof the proposed approach. In particular, the proposed approach
outperforms the special cases of stable inversion, multirate inversion as well as norm-optimal control.
VI. CONCLUSION
A discrete-time inversion approach is developed that allows tobalance the on-sample and intersample behavior for the purposeof continuous-time performance. The approach is applicable toboth LTI and LPTV systems. The multirate inversion and stable
inversion approaches are recovered as special cases. Application
to an LPTV motion system in both simulations and experiments
demonstrates the advantages of the approach.
For LPTV systems, the approach currently involves an additional lifting step, which limits applicability due to constraints
on the input and state dimensions, i.e., the state dimension

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VAN ZUNDERT et al.: IMPROVING INTERSAMPLE BEHAVIOR IN DISCRETE-TIME SYSTEM INVERSION

should be an integer multiple of the input dimension. In contrast,
stable inversion and multirate inversion can be directly applied
to LPTV systems. Future work focuses on an explicit state-spacedecomposition for LPTV systems to avoid the additional lifting
step and thereby potentially increase the performance of theapproach.
Obviously, the interest is also in determining the optimal decomposition into H1 and H2 a priori in order to avoid exhaustive
simulations. Preliminary results show that the best performanceis obtained when capturing sampling zeros, introduced by zeroorder-hold discretization, in H1 , and damped (anti) resonancesin H2 . Ongoing research focuses on more detailed guidelines infinding the optimal decomposition.

63
, i.e., n = 2, the state-spaceδ = 1 ms. For state x(t) = q̇(t)q(t)
realization in (1) is given by


d
k
−m
1
−m
Ac =
, Cc = 0 1 . (33)
, Bc =
0
1
0
Next, the steps in Procedure 7 are followed.
1) By Lemma 1 follows
Hc = Cc (sI − Ac )−1 Bc =

B(s)
A(s)

(34a)

with
1

B(s) = m=1
1

APPENDIX A
PROOF OF THEOREM 4
Due to the modal form of Hmod , the A matrix of Hmod is block
diagonal and the states are decoupled per mode. The matrix Tper
is a permutation matrix and follows directly from V, V× andthe state ordering of Hmod . The last n2 states in (12) areuncontrollable and are redundant since the states are decoupled.
Similarly, the first n1 states in (13) are unobservable and are
redundant since the states are decoupled. Hence, H1 and H2
are minimal realizations such that Hmod = H1 H2 in terms oftransfer functions. The product H1 H2 with H1 andH2 in (14) is
given by⎤⎡
A1 B1 C2 B1 D
B2 ⎦ .
A2(30)
H1 H2 = ⎣ 0
C1 C2
DUsing (15)T (H1 H2 , T12 )⎤⎡A1 −A1 X + B1 C2 + XA2 B1 D + XB2
⎦
A2
B2
=⎣ 0
C1
−C1 X + C2
D⎡⎤
A1 0 B2r= ⎣ 0 A2 B2 ⎦ .
C1 C1r D

(31a)

(31b)

−1
By definition of Tper in (12) and (13), T (H1 H2 , Tper
T12 ) =−1
T (T (H1 H2 , T12 ), Tper ) = (A, B, C, D) = Hmod , which concludes the proof.

APPENDIX B
EXAMPLE OF PROCEDURE 7
This appendix demonstrates the steps in Procedure 7 for a
simple motion system.
Consider a mass-damper-spring system of which the dynamics are given bymq̈(t) + dq̇(t) + kq(t) = u(t)

(34b)

m

(32)

with q(t) the displacement, q̇(t) the velocity, q̈(t) the acceleration, and the parameters mass m = 2 kg, damping constantd = 6 N·s/m, and spring constant k = 4 N/m. The reference
trajectory is given by r(t) = sin(t) and the sample time isA(s) =dks+ ms2 + m
1
m

(34c)

1
k
d
i.e., b0 = m, a0 = m, and a1 = m. Hence, the controllable canonical form in (5) is given by
⎤
⎡
0
1 0
s
d
k
1 ⎦
(34d)
Hccf = T (Hc , Tccf ) = ⎣ − m
−m
m
1
0


−1
with Tccf= 01 01 .
2) Since B(s) = 1, it follows from Theorem 2 that sin(t)  and x̂ccf,u (t) = 00 , hence the dex̂ccf,s (t) = cos(t)sired state in (9) is given bycos(t)
x̂(t) =.
(35)
sin(t)

Indeed,y(t) = Cc x̂(t) = sin(t) = r(t)andẏ(t) = cos(t) = ṙ(t).
3) The desired state of H is x̂[k] = x̂(kδ).
4) The modal form in Theorem 4 is obtained by Hmod =T (H, Tmod ), with

−2.8284 −2.8284
Tmod =.
(36)−2.2361 −4.4721
The desired state of Hmod is obtained by x̂mod [k] =Tmod x̂[k].
5) The state-space decomposition in Theorem 4 requires anonsingular D, hence a bilinear transformation is used,in particularH̃mod = B(Hmod )
(37a)
⎤
⎡
−0.0010.001
0s−0.00079 ⎦ .
−0.0005=⎣ 00.2502 −0.3164 −6.25 × 10−11(37b)
The transfer function decomposition of H̃mod is obtained
through Lemma 3 by selecting the column spaces of the

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64

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 25, NO. 1, FEBRUARY 2020 
 −0.7845 , suchinvariant subspaces as Ṽ = 01 and Ṽ× = −0.6202
 1 −1.2649 that Π̃ = 0, see also (10), which yields
0
⎤
−4
0.001
0
s
−0.0005 0 ⎦H̃1f = ⎣ 00.2502 −0.3164 1⎤⎡−0.0010.001
0s−0.00079 ⎦ .
−0.0005H̃2f = ⎣ 000.00016 −6.25 × 10−11
⎡

(38)

(39)



By selecting Tper = 10 01 , (12) and (13) are satisfied
and the minimal state-space realizations H̃1 and H̃2 are
obtained. Undoing the bilinear transformation yields
H1 = B −1 (H̃1 ) =0.998 −5.651
0.35360

H2 = B −1 (H̃2 ) =0.999−0.001117. (41)
0.0002235 −1.25 × 10−7

z

z

(40)

System H12 follows from (23).
6) The desired state of H12 is given by x̂12 [k] =T12 Tper x̂mod [k], with T12 in (16) given by
1
T12 =
0

1.26491
(42)with X = 1.2649 the solution to the Sylvester equation(15).
7) The desired state for H1 is given by x̂1 [k] = [1 0]x̂12 [k].
This concludes the application of Procedure 7 to the systemin (32).

[11] W. Ohnishi, T. Beauduin, and H. Fujimoto, “Preactuated multirate feedforward control for independent stable inversion of unstable intrinsic anddiscretization zeros,” IEEE/ASME Trans. Mechatronics, vol. 24, no. 2,
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Jurgen van Zundert received the M.Sc. and
Ph.D. degrees in mechanical engineering fromthe Eindhoven University of Technology, Eindhoven, The Netherlands, in 2014 and 2018,
respectively.
He is currently a Mechatronic System Engineer with DEMCON, Best, The Netherlands. His
research interests include feedforward motion
control, multirate control, and iterative learning
control.

Wataru Ohnishi (S’13–M’18) received the B.E.,
M.S., and Ph.D. degrees from The Universityof Tokyo, Kashiwa, Japan, in 2013, 2015, and2018, respectively, all in electrical engineeringand information systems.
He is currently a Research Associate with the
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo, Kashiwa, Japan.
His research interest includes high-precision
motion control.
Dr. Ohnishi is a member of the Institute of Electrical Engineers ofJapan.

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VAN ZUNDERT et al.: IMPROVING INTERSAMPLE BEHAVIOR IN DISCRETE-TIME SYSTEM INVERSION

Hiroshi Fujimoto (S’99–M’01–SM’12) received
the Ph.D. degree in electrical engineering fromthe Department of Electrical Engineering, The
University of Tokyo, Kashiwa, Japan, in 2001.
In 2001, he joined the Department ofElectrical Engineering, Nagaoka University ofTechnology, Niigata, Japan, as a Research
Associate. From 2002 to 2003, he was a Visiting Scholar with the School of Mechanical Engineering, Purdue University, West Lafayette,
IN, USA. In 2004, he joined the Department ofElectrical and Computer Engineering, Yokohama National University,
Yokohama, Japan, as a Lecturer and went on to become an Associate
Professor of Nanoscale servo, Electric vehicle control, Motion control
in 2005. He is currently an Associate Professor with The Universityof Tokyo since 2010. His interests include control engineering, motion
control, nanoscale servo systems, electric vehicle control, and motor
drives.
Prof. Fujimoto was the recipient of the Best Paper Awards from theIEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2001 and 2013,
Isao Takahashi Power Electronics Award in 2010, Best Author Prize of
SICE in 2010, the Nagamori Grand Award in 2016, and First Prize Paper
Award from the IEEE TRANSACTIONS ON POWER ELECTONICS in 2016. He
is a member of the Society of Instrument and Control Engineers, theRobotics Society of Japan, and the Society of Automotive Engineers ofJapan, as well as a Senior Member of the Institute of Energy Economicsof Japan.

65

Tom Oomen (SM’06) received the M.Sc. (cumlaude) and Ph.D. degrees from the Eindhoven University of Technology, Eindhoven, The
Netherlands, in 2005 and 2010, respectively,
both in mechanical engineering.
He held visiting positions with the KTH
Royal Institute of Technology, Stockholm,
Sweden, and with The University of Newcastle, Callaghan, NSW, Australia. He is currently
an Associate Professor with the Department of
Mechanical Engineering, Eindhoven Universityof Technology. His research interests include the field of data-driven
modeling, learning, and control, with applications in precision mechatronics.
Dr. Oomen was the recipient of the Corus Young Talent Graduation
Award, the 2015 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY Outstanding Paper Award, the 2017 IFAC Mechatronics Best Paper
Award, and recipient of a Veni and Vidi personal grant. He is an Associate Editor for the IEEE CONTROL SYSTEMS LETTERS, IFAC Mechatronics, and IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. He isa member of the Eindhoven Young Academy of Engineering.

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