Content uploaded by Dan Wang

Author content

All content in this area was uploaded by Dan Wang on May 07, 2020

Content may be subject to copyright.

1

H∞-based Selective Inversion of

Nonminimum-phase Systems for Feedback Controls

Dan Wang, Xu Chen†

Abstract—Stably inverting a dynamic system model is funda-

mental to subsequent servo designs. Current inversion techniques

have provided effective model matching for feedforward controls.

However, when the inverse models are to be implemented

in feedback systems, additional considerations are demanded

for assuring causality, robustness, and stability under closed-

loop constraints. To bridge the gap between accurate model

approximations and robust feedback performances, this paper

provides a new treatment of unstable zeros in inverse design. We

provide ﬁrst an intuitive pole-zero-map-based inverse tuning to

verify the basic principle of the unstable-zero treatment. From

there, for general nonminimum-phase and unstable systems, we

propose an optimal inversion algorithm that can attain model

accuracy at the frequency regions of interest while constraining

noise ampliﬁcation elsewhere to guarantee system robustness.

Along the way, we also provide a modern review of model

inversion techniques. The proposed algorithm is validated on

motion control systems and complex high-order systems.

Index Terms—model inversion, nonminimum-phase zeros, un-

stable systems, H∞formulation

I. INTRODUCTION

Given a linear time-invariant system model G, the inver-

sion of Ghas numerous practical implementations including

iterative learning control (ILC) [1], [2], [3], repetitive control

[4], [5], two-degree-of-freedom servo in feedforward control

[6], [7], as well as Youla parameterization and disturbance

observer in feedback control [8], [9], [10], [11], [12]. Here,

Gcan be an open-loop plant model or a closed-loop control

system. For a minimum-phase system, G−1is stable and ready

to be implemented. However, for a system with nonminimum-

phase (NMP or unstable) zeros, G−1is unstable and cannot

be implemented directly. To ﬁnd a stable, rational, and causal

replacement ˆ

G−1such that Gˆ

G−1approximates 1 is thus

a fundamental challenge in inversion-based control designs.

Such a challenge is more pronounced in discrete-time systems

since 1) integrator-type plant dynamics1, common in motion

control, generate NMP zeros in their zero-order-hold (ZOH)

equivalents when the sampling time is sufﬁciently small; 2)

fractional-order delays induce unstable zeros after discretiza-

tion [13].

Considering the importance and the challenge of model in-

version, numerous strategies have been established in modern

literature. Based on system representations and scopes of ap-

plication, we can classify these strategies into two categories:

The authors are with the Department of Mechanical Engineering, Uni-

versity of Washington, Seattle, WA, 98195, USA (emails: {daw1230,

chx}@uw.edu). †: corresponding author.

1When actuators take forces or torques as the input and linear/angular

position as the output, integrator-type plant dynamics with a relative degree

not less than two show up.

frequency- and time-domain model inversions. The frequency-

domain strategies focus on expressing the transfer functions of

the stable inverses and hence can be used in both feedback and

feedforward controls. Examples in this category include the

approximate (e.g., NPZ-ignore, ZPETC, and ZMETC) [14],

[15], [16], [17], the ILC-based [18], [19], [20], and the H∞-

based [21], [22], [23] model inversions. On the other hand, the

time-domain strategies [24], [25], [26], [27] aim at identifying

the optimal control signal that minimizes the error between a

given reference and the output. These time-domain algorithms

are mainly used as feedforward techniques since a preview of

the reference is generally not available in feedback design.

This paper studies the analysis and design of model in-

version strategies in the frequency domain. Current strategies

in this category aim at achieving effective model matching

between ˆ

Gand G. Compared with the approximate and the

ILC-based model inversions, the H∞-based model inversion

can automatically identify the inverse model without knowing

the exact NMP zeros, which particularly beneﬁts systems

with complicated pole-zero distributions. However, when the

inverses are to be implemented in feedback systems, additional

considerations are needed for assuring closed-loop stability

and robustness. In pursuit of bridging the gap between accurate

model approximations and robust feedback performances, this

study builds a new H∞-based optimal inversion algorithm

that advances the ﬁeld by 1) mitigating control efforts at cus-

tomized frequencies and thereby enhancing system robustness;

2) reaching high efﬁciency for complex high-order systems

and unstable systems.

Before presenting the main algorithm, we ﬁrst provide a

pole-zero-map-based NMP-zero modulation by replacing high-

frequency NMP zeros with stable ones in motion control

applications. We verify the feasibility and limit of this intuitive

modulation in achieving a stable inverse model and mean-

while capturing the low-frequency system dynamics for high-

performance motion control. Then we extend this intuitive

modulation to an optimal design of model inversion. There,

replacing the manual adjustment with an automatic and opti-

mal search, we develop a new H∞-based algorithm that can

attain model accuracy at the frequency regions of interest while

constraining noise ampliﬁcation elsewhere to guarantee system

robustness. The design goals are achieved by a multi-objective

formulation and an all-pass factorization that consider model

matching, gain constraints, causality of transfer functions, and

factorization of unstable system modes in a uniﬁed scheme.

The proposed algorithm is validated on motion control systems

and complex high-order systems. Moreover, along the path,

we unveil previously ignored features of existing inversion

2

strategies by developing a general frequency-domain analysis

method, which also gives new insights into comparing the

performances of different strategies.

The main contributions of this paper are:

a) conducting an up-to-date review of model inversion

strategies and proposing a new frequency-domain anal-

ysis method;

b) analyzing the effect of an intuitive NMP-zero modula-

tion and developing a new H∞-based inversion algo-

rithm;

c) validating the proposed algorithm by presenting detailed

case studies with high-ﬁdelity experimental data.

The remainder of this paper is structured as follows. Section

II conducts an in-depth review of literature and proposes the

new frequency-domain analysis method. Section III elucidates

the effect of modulating NMP zeros. The proposed optimal

inversion is presented and veriﬁed in Section IV. Section V

concludes this paper.

II. REVIEW AND COMPARISON OF FREQUENCY-DOM AI N

INVERSION ALGORITHMS—AN E ND -POINT PERSPECTIVE

The frequency-domain inversion algorithms aim at express-

ing the stable inverse models F=ˆ

G−1in the s- or z-domain

(sand zare complex numbers in the Laplace transform and

z-transform, respectively). ˆ

Gis the minimum-phase system

model that approximates Gand has a stable inverse. An

optimal inverse model is desired for Gˆ

G−1to approximate 1.

In this section, we review and compare three typical types of

frequency-domain inversion algorithms. In addition, we unveil

new features of existing algorithms by developing a general

frequency-domain analysis method.

A. H∞-based model inversion

1) Algorithm

The model inversion problem for NMP systems has been

solved using the H∞formulation [22], [23], [21]. For a

continuous-time NMP system G(s) = (b−s)/(b+s)with

b > 0, under a cost function J=||W(s)(1−G(s)ˆ

G−1(s))||∞,

where the weighting W(s)=(k+ξs)/(k+s)is a low-pass

ﬁlter with k > 0and 0≤ξ < 1, the optimal inverse of G(s)

that minimizes Jis a lead ﬁlter [22]:

ˆ

G−1(s) = k(1 −ξ)(b+s)

(k+b)(k+ξs)(1)

that has high gains at high frequencies. The frequency response

of the optimal G(s)ˆ

G−1(s)is

G(jΩ) ˆ

G−1(jΩ) = k(1 −ξ)(b−jΩ)

(k+b)(k+jξΩ),(2)

where Ωis in rad/s.

2) Frequency-domain analysis

To quickly capture the essence of Gˆ

G−1, we examine the

frequency response of Gˆ

G−1at the two frequency endpoints

(0 and ∞for a continuous-time system or 0 and πin rad for a

discrete-time system) and evaluate the characteristics of model

matching.

-20

-15

-10

-5

0

5

Magnitude (dB)

10-3 10-2 10-1 100101102

-180

-135

-90

-45

0

Phase (deg)

k=5

k=0.6

k=0.1

Bode Diagram

Frequency (Hz)

Fig. 1. Frequency responses of G(jΩ) ˆ

G−1(jΩ) with b= 2,ξ= 0.3, and

different values of k

Considering b= 2,ξ= 0.3, and different k’s, we depict

in Fig. 1 the frequency responses of (2). As Ωincreases from

0 to ∞, the phase of G(jΩ) ˆ

G−1(jΩ) always goes from 0to

−180◦(the bottom plot of Fig. 1), and its magnitude goes

from b1−ξ

k+b(<0 dB) to k

ξ1−ξ

k+b, monotonically. Therefore,

depending on the values of kand ξ,G(s)ˆ

G−1(s)is a high-

pass ﬁlter when k

ξ1−ξ

k+b> b 1−ξ

k+b, i.e., k > ξb (0.6 in this

example), a low-pass ﬁlter when k < ξb, and has a constant

magnitude when k=ξb (the top plot of Fig. 1).

B. Approximate model inversions

1) Algorithms

For discrete-time NMP systems, to obtain the basic structure

of the inverse model, approximate model inversions [16], [15],

[17], [14] ﬁrst factor out the unstable zeros of the system as

G(z) = N(z)

D(z)=Ns(z)Nu(z)

D(z),(3)

where N(z)and D(z)are coprime polynomials of z, and

Ns(z)and Nu(z)contain, respectively, the stable and the

unstable zeros. Here, we deﬁne Nu(z)as

Nu(z)=(z−z1)(z−z2)· · · (z−zn),(4)

where z1, z2,· · · , znare outside the unit circle. Note that

Nu(z−1)=(z−1−z1)(z−1−z2)· · · (z−1−zn)

has stable zeros. In addition, Nu(z)Nu(z−1)is zero-phase.

In the general case, the approximate inverse model of G(z)

in (3) has a structure of

ˆ

G−1(z) = D(z)

Ns(z)˜

Nu(z),(5)

where ˜

Nu(z)is a design parameter.

Table I summarizes three approximate model inversions

with different designs of ˜

Nu(z). The NMP zeros ignore

method (NPZ-ignore) [16], [17] replaces Nu(z)with ˜

Nu(z) =

Nu(1) at the cost of magnitude and phase mismatch in

G(z)ˆ

G−1(z). The zero-phase-error-tracking control (ZPETC)

3

TABLE I

˜

Nu(z),G(z)ˆ

G−1(z),AN D Y(z)

R(z)IN APPROXIMATE MODEL INVERSIONS.Y(z)AND R(z)ARE TRANSFER FUNCTIONS OF THE OUTPUT AND REFERENCE

SI GNA LS SH OWN I N FIG. 2 .

Methods NPZ-ignore ZPETC ZMETC

˜

Nu(z)Nu(1) [Nu(1)]2

Nu(z−1)Nu(z−1)

G(z)ˆ

G−1(z)Nu(z)

Nu(1)

Nu(z)Nu(z−1)

[Nu(1)]2

Nu(z)

Nu(z−1)

Y(z)

R(z)z−mNu(z)

Nu(1) z−mNu(z)Nu(z−1)

[Nu(1)]2z−mNu(z)

Nu(z−1)

Y(ejω )

R(ejω )e−jmω Nu(ej ω )

Nu(1) e−jmω Nu(ejω )Nu(e−j ω )

[Nu(1)]2e−jmω Nu(ejω )

Nu(e−jω )

Y(ejπ )

R(ejπ )

Nu(−1)

Nu(1) hNu(−1)

Nu(1) i21

[14] assigns instead ˜

Nu(z)=[Nu(1)]2/Nu(z−1)and

achieves zero-phase error dynamics since G(z)ˆ

G−1(z) =

Nu(z)Nu(z−1)/[Nu(1)]2is zero-phase. The zero-magnitude-

error-tracking control (ZMETC) [16], on the other hand,

eliminates all magnitude errors by converting the unstable

zeros to their stable reciprocals, namely, ˜

Nu(z) = Nu(z−1).

Note that Nu(1) in NPZ-ignore and [Nu(1)]2in ZPETC are

added to create a unity DC gain of G(z)ˆ

G−1(z).

Furthermore, to make the approximate inverse model

ˆ

G−1(z)in (5) realizable and ready to be implemented as a

block during feedback/feedforward implementation, a causal

inverse model is obtained by multiplying ˆ

G−1(z)with z−m:

F(z) = z−mˆ

G−1(z) = z−mD(z)

Ns(z)˜

Nu(z),(6)

where

m=Order[Denominator of ˆ

G(z)] −Order[N umerator of ˆ

G(z)]

(7)

is the relative degree of ˆ

G(z)and the Order function cal-

culates the highest exponent in a transfer function. Next we

will prove that mis always larger than 0 in the NPZ-ignore,

ZPETC, and ZMETC.

Proof: We can tell from Table I that the relative degree

of ˜

Nu(z)is 0 in each of the three designs. Thus, from (5), the

expression of min (7) can be reduced to

m=Order[D(z)] −Order[Ns(z)].(8)

Also, we have Order[D(z)] ≥Order[Ns(z)]+Order[Nu(z)]

from (3) and Order[Nu(z)] >0for NMP systems, yielding

Order[D(z)] > Order[Ns(z)], that is, m > 0in (8).

Here, the result m > 0means the delay z−mshould always

be accounted for to make the inverse model realizable in

the feedback/feedforward applications of approximate model

inversions. In feedforward applications where a preview of the

desire output yd(k)is available, the delay z−mcan be canceled

out by letting r(k) = yd(k+m).

2) Frequency-domain analysis

Fig. 2 shows a block diagram to illustrate the goal of

the model inversion design, where r,u, and yrepresent the

reference, the input, and the output signals, respectively. Note

that subsequently Fcan be implemented as a block in the

feedback/feedforward controller designs, such as the examples

in Section IV-C. In Fig. 2, the overall transfer function

F: approximate inverse of GG

r u y

Fig. 2. Block diagram to illustrate the goal of the model inversion design.

Note that Fcan be implemented as a feedback/feedforward controller.

from the reference signal r(k)to the output signal y(k)is

Y(z)

R(z)=F(z)G(z) = z−mG(z)ˆ

G−1(z), which reﬂects the

accuracy of the causal inverse F(z). Table I lists the transfer

functions of Y(z)

R(z)in the three approximate model inversions.

We take the hard disk drive (HDD) system in Section III as an

illustrative example. The transfer function of the system with

a sampling frequency (1/Ts) of 26.4 kHz is

G(z) = z−31.447663(z+ 0.050852)(z+ 2.494311)

z2−1.978354z+ 0.978808 .(9)

Here, G(z)has one NMP zero at around −2.5,Nu(z) =

z+ 2.494311, and min (8) is 4. Y(z)

R(z)are z−4(z+2.494311)

3.494311

for NPZ-ignore, z−4(z+2.494311)(z−1+2.494311)

3.4943112for ZPETC, and

z−4(z+2.494311)

z−1+2.494311 for ZMETC. Fig. 3 plots the frequency re-

sponses of Y(z)

R(z)of the three approximate designs. At low

frequencies close to 0, i.e., z=ejω →1, we get the

desired result Y(z)

R(z)→1for all three methods, and thereby

the magnitude and phase responses of Y(z)

R(z)largely overlap

with each other (Fig. 3). At the Nyquist frequency πrad (i.e.,

13.2 kHz), where z=ejπ ,

Y(ejπ )

R(ejπ )

equals

Nu(−1)

Nu(1)

for NPZ-

ignore and equals hNu(−1)

Nu(1) i2

for ZPETC; that is to say, in log

scale, Y(ejπ )

R(ejπ )in ZPETC (−14.72 dB) has twice the magnitude

of Y(ejπ )

R(ejπ )in NPZ-ignore (−7.36 dB) (the top plot of Fig. 3).

Moreover, in this HDD example, since the NMP zero is a real

one at around −2.5and m= 4, all three Y(ejπ)

R(ejπ )have zero

phase at the Nyquist frequency (the bottom plot of Fig. 3).

C. ILC-based model inversion

1) Algorithm

ILC, originally developed for output tracking in repetitive

tasks, can be extended to the ﬁeld of model inversion [19],

[20], [18]. Here, the inverse model F(z)is constructed by

4

10110210 3104

-15

-10

-5

0

Magnitude (dB)

NPZ-ignore

ZPETC

ZMETC

10110210 3104

Frequency (Hz)

-200

-100

0

100

Phase (deg)

Fig. 3. Frequency responses of Y(z)/R(z)(= z−mG(z)ˆ

G−1(z)) (indicat-

ing tracking performances) for different approximate model inversions used

in the example of the HDD system in (9)

designing its impulse response f(k)as the feedforward signal

in the following ILC:

F(z) =

N/2

X

k=−N/2

f(k)z−k,

f(k),lim

i→∞ ui(k),(10)

where ui(k)is the learned input at the i-th iteration:

ui(k) = ui−1(k) + L(z) [r(k)−G(z)ui−1(k)]

=I−(I−L(z)G(z))iG−1(z)r(k).(11)

Here, the training reference r(k)is designed as the delta

impulse δ(k). The ILC learning ﬁlter L(z)is built from the

approximate model inversions (Section II-B) such that the

stability condition k1−L(z)G(z)k∞<1is satisﬁed. With

i→ ∞, from (10) and (11), f(k)→u∞(k)→G−1(z)δ(k),

that is, f(k)approximates the impulse response of the unstable

G−1(z). Recall that f(k)is the impulse response of F(z).

Thus, we obtain F(z)≈G−1(z).

2) Frequency-domain analysis

In the ILC-based model inversion, the transfer function

1−L(z)G(z)determines not only the stability condition

but also the convergence rate. Fig. 4 shows the frequency

responses of (1 −L(z)G(z))i, taking again the HDD system

in (9) for example. Here, L(z)is built from ZPETC. With in-

creasing iteration number i, the magnitudes of (1−L(z)G(z))i

at low frequencies start to converge to zero. Moreover, a larger

iyields a wider low-frequency region with zero magnitude.

Therefore, under ﬁnite implementation of i,F(z)represents a

low-pass approximation of G−1(z)with a tunable bandwidth.

One drawback, however, is that system hardware (or a very

accurate model G) is needed for iterative experiments to run.

D. Summary of literature review and motivations of this paper

Table II summarizes the three model inversion strategies.

It is noteworthy that these frequency-domain strategies can

10110 2103104

-400

-300

-200

-100

0

Magnitude (dB)

i=1

i=3

i=5

i=9

i=13

i=17

i=21

10110 2103104

Frequency (Hz)

-200

-100

0

100

200

Phase (deg)

Fig. 4. Frequency responses of (1 −L(z)G(z))ifor the example of the

HDD system in (9), where L(z)is the learning ﬁlter built from ZPETC

be implemented in both feedback and feedforward controls.

Application of each method certainly depends on the speciﬁc

problem at hand. Compared with the other two methods,

the H∞-based model inversion can automatically identify the

inverse model without knowing the exact NMP zeros, which

particularly beneﬁts unstable systems and high-order systems

with complicated pole-zero distributions.

For inverse-based feedback control, all the surveyed al-

gorithms have considered accurate model inversion but not

robustness against model mismatch that is also crucial for

closed-loop performance. In contrast, the algorithm to be

proposed in Section IV enhances the system robustness by

limiting the magnitude of the inverse model at frequency

regions where large model mismatches exist. Before discussing

the main algorithm, we provide in Section III some preparatory

work on the effect of the NMP zeros.

III. FRE QUENCY-DOMAIN IMPLICATIONS OF MODULATING

NMP ZE ROS

This section studies the inﬂuence of modulating the NMP

zeros (i.e., shifting the locations of the NMP zeros) on the

frequency response of a system. For concreteness, we take the

HDD system in [10] as an example, where model inversion

underpins servo designs that control precisely the position of

the read/write head to provide reliable storage.

The solid line in Fig. 5 shows the frequency response of an

experimentally measured HDD system. The nominal model of

the motors and actuators in the system is [10]:

Gc(s) = e−10−5s3.74488 ×109

s2+ 565.487s+ 3.19775 ×105.(12)

The ZOH equivalent of Gc(s)sampled at 26.4 kHz, namely

G(z), is expressed in (9) and has one unstable zero at around

−2.5. As plotted in Fig. 5, the frequency response of the NMP

G(z)matches well with the actual system dynamics (solid

line).

We investigate next the frequency-domain implications of

the NMP-zero locations by analyzing Nu(ejω ) = ejω +

5

TABLE II

OVE RVIE W OF FR EQ UEN CY-DOM AI N INV ER SIO N ST RATEG IE S:AP PRO XIM ATE [14], [15], [16], [17], ILC-BAS ED [18], [19], [20], A ND H∞-BA SED

METHODS [21], [22], [23]. DT AN D CT ARE SHORT FOR DISCRETE TIME AND CONTINUOUS TIME,RES PEC TI VELY.

Method DT or CT Basic structure or design goal

Approximate DT F(z) = z−mD(z)

Ns(z)˜

Nu(z)

ILC-based DT F(z) = PN/2

k=−N/2f(k)z−k, f(k),limi→∞ ui(k)

H∞-based CT/DT min ||W(s)(1 −G(s)ˆ

G−1(s))||∞

Proposed H∞-based DT min

F(z)∈S

"W1(z)(F(z)G(z)−z−m)

W2(z)F(z)G(z)#

∞

with F(z) = z−mˆ

G−1(z)

101102103104

-20

0

20

40

60

80

100

Magnitude (dB)

101102103104

Frequency (Hz)

-180

-90

0

90

180

Phase (deg)

Fig. 5. Frequency responses of actual system dynamics from experiments

and nominal system models in the HDD system

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Pole-Zero Map

Real Axis

Imaginary Axis

° zeros

poles

unstable zero shift to

the inside of the unit circle

Unstable zero shift to the

inside of the unit circle

0°~17.72°

()* + 2.494311 ()*

Fig. 6. Illustration of modulating the experimentally identiﬁed NMP zero in

the HDD system

2.494311 in (9). Consider the rule of thumb that the closed-

loop bandwidth Bpis around 10% of the Nyquist frequency

(1

2TsHz) or ωp= 2πBpTs≈2π0.1

2TsTs= 18◦; in this example,

Bp= 1300 Hz, and ωp= 2π×1300/26400 = 17.72◦. In other

words, ωsweeps only a small arc on the unit circle from 0 to

17.72◦in the main performance region, yielding mild changes

to the vector ejω + 2.494311, as shown in Fig. 6. Therefore,

when shifting the NMP zero to a stable one, e.g., at −0.8(Fig.

6), we can get a minimum-phase nominal model ˆ

G0(z)that

has a stable inverse and largely maintains frequency response

of the system in desired low-frequency regions:

ˆ

G0(z) = z−31.447663(z+ 0.050852)(z+ 0.8)

z2−1.978354z+ 0.978808 .

Normalizing ˆ

G0(z)to retain the DC gain of G(z)in (9),

we get

ˆ

G(z) = z−3(z+ 0.050852)(z+ 0.8)

0.355831z2−0.703959z+ 0.348290.(13)

As shown in Fig. 5, ˆ

G(ejω )(dashed line) matches well with

the NMP G(ejω )(dotted line) and the actual system dynamics

(solid line) below 3000 Hz. This frequency is large enough for

most servo-enhancement schemes in single-stage HDDs.

In summary, a stable inverse is readily achievable through

modulating the NMP zeros as long as the NMP zeros do not

occur in the desired low-frequency regions. This result justiﬁes

the basic idea of the H∞-based optimal inversion, where the

manual modulation is upgraded to an automatic and optimal

search, as shall be proposed next.

IV. PROPOSED H∞-BASED OPTIMAL INVERSION

Based on the frequency-domain analysis in Section III,

this section develops an H∞-based optimal inversion. The

design principle is to automatically search for the optimal

inverse model to selectively ﬁt different frequency regions.

At frequencies where no NMP zeros exist and no large model

uncertainties occur, we impose an accurate model matching

between the minimum-phase model ˆ

G(z)and the original

NMP model; at other frequencies, we limit the magnitude re-

sponse of the inverse model to increase the system robustness.

We explore the design procedures, case studies, and frequency-

domain analyses of the proposed algorithm, ﬁrst for NMP

systems and then for unstable systems.

A. H∞-based optimal inversion for NMP systems

1) Algorithm

Let Sdenote the set of stable, proper, and rational discrete-

time transfer functions. We search among Sto ﬁnd the optimal

inverse model F(z) = z−mˆ

G−1(z)that satisﬁes:

a) F(z)is realizable/proper. This relates to the z−mterm

in F(z). To minimize the delays, mcan be tuned and

usually equals the relative degree of G(z).

b) model matching:min ||W1(z)(F(z)G(z)−z−m)||∞.

Namely, we minimize the maximfum magnitude of the

6

-80

-60

-40

-20

0

20

Magnitude (dB)

50 100 150 200 250 300 350 400

-90

-45

0

45

90

Phase (deg)

Bode Diagram

Frequency (Hz)

Fig. 7. Frequency responses of the weightings W1and W2in the active

suspension system

F G W1

z−m

W2

r u y

+

e1

e2

−

Fig. 8. Block diagram for the H∞-based optimal inverse design

model mismatch F(z)G(z)−z−mweighted by W1(z).

The weighting W1(z)determines the frequency regions

for accurate model matching. If G−1(z)is stable, the

direct solution is F(z) = z−mG−1(z).

c) gain constraint:min ||W2(z)F(z)G(z)||∞. Here, the

magnitude of F(z)G(z)is scaled by the weighting

W2(z). For instance, W2(z)can be a high-pass ﬁlter to

constrain noise ampliﬁcation at high frequencies. The

solution for this condition alone is F(z) = 0, that is,

F(z)does not amplify any input signals.

Integrating the above three goals yields the multi-objective

optimization principle:

min

F(z)∈S

W1(z)(F(z)G(z)−z−m)

W2(z)F(z)G(z)

∞

.(14)

The optimal inverse model F(z)given by (14) preserves

accurate model information in the frequency regions speciﬁed

by W1(z)and, on the other hand, penalizes excessive high

gains of F(z)at frequencies determined by W2(z). Typically,

W1(z)is a low-pass ﬁlter, and W2(z)is a high-pass one, as

shown in the example of Fig. 7. For one system model, the

weightings can be ﬂexibly designed, yielding different inverse

models F(z).

The optimization principle in (14) can be solved within the

framework of H∞controls. F(z)can be solved by the hinfsyn

function in the robust control toolbox of MATLAB and tuned

for the target performance by changing the input arguments

gamTry and gamRange of the function. Fig. 8 shows the

block diagram realization of (14). Here, the hinfsyn function

minimizes the two error signals e1and e2. The solution of

F(z)exists as long as G(z),W1(z), and W2(z)are stable.

After (14) is solved, a lower-order F(z)can be reached by

applying standard model-reduction techniques, if needed.

F G W1

WI∆I

z−m

W2

ry

+

e1

e2

−

u

+

−

Fig. 9. Block diagram for the H∞-based optimal inverse design considering

uncertainty

-1 -0.5 0 0.5 1 1.5

-4

-3

-2

-1

0

1

2

3

4

Pole-Zero Map

Real Axis

Imaginary Axis

Fig. 10. Pole-zero plot of the experimentally identiﬁed system model and its

minimum-phase approximation of the active suspension system

Remark: When the system model is subjected to perturba-

tions, we can use a multiplicative uncertainty model to lump

the various dynamic uncertainties:

Gp(z) = G(z)(1 + WI(z)∆I(z)),(15)

where k∆Ik∞≤1[28]. Fig. 9 shows the block diagram of the

proposed H∞-based optimal inverse with uncertainties taken

into consideration. The problem now is to ﬁnd a stabilizing

inverse model F(z)such that the H∞norm of the transfer

function between rand [e1, e2]Tis less than 1 for all ∆I, that

is,

min

F(z)∈S

W1(z)(F(z)Gp(z)−z−m)

W2(z)F(z)Gp(z)

∞

,(16)

which is no longer a standard H∞optimization but a robust

performance problem. The µ-synthesis and DK-iteration pro-

cedures can be utilized to solve the problem [28], [23].

2) Case study with frequency-domain analysis

This case study shows efﬁciency of the proposed algorithm

for high-order NMP systems with complicated pole-zero dis-

tributions. We take for example the active suspension system

in [29] that serves as a benchmark on adaptive regulation. The

control goal there is to attenuate the vibrations transmitted to

the base frame, and model inversion is critical for the best

results achieved in the benchmark [30]. Although the system

is open-loop stable, the existence of the NMP zeros challenges

model inversion in general feedback and feedforward control.

Via standard system identiﬁcation methods, the system

model G(z)is experimentally identiﬁed with a sampling rate

of 800 Hz and has an order of 22. As shown in the pole-zero

plot in Fig. 10, four NMP zeros show up in G(z). Furthermore,

7

50 100 150 200 250 300 350 400

-20

0

20

40

Magnitude (dB)

System model

NPZ

ZPETC

ZMETC

w/o W2

w/ W2

50 100 150 200 250 300 350 400

Frequency (Hz)

-200

-100

0

100

200

Phase (deg)

Fig. 11. Frequency responses of the experimentally identiﬁed system model

G(z)and its minimum-phase approximations of the active suspension system.

Models obtained from ZMETC and ZPETC, respectively, have the same

magnitude and phase responses as the system model. Proposed H∞-based

optimal inversion: red dashed line. Previous H∞-based method without gain

constraint: magenta solid line.

with the two weighting functions designed as in Fig. 7, we

solve the optimization principle in (14) and obtain the optimal

inverse F(z). After that, we reduce the order of F(z)to 23 by

applying the model-reduction function reduce in MATLAB.

The pole-zero plot of the 23rd-order F(z)is also shown in

Fig. 10. Then the minimum-phase system model is secured

by ˆ

G(z) = z−mF−1(z)(m= 2).

As shown in Fig. 11, ˆ

G(z)obtained from the proposed

H∞-based optimal inversion (red dashed line) matches well

with the identiﬁed NMP G(z)(blue solid line). Moreover, at

high frequencies near the Nyquist frequency, ˆ

G(z)from the

proposed method (red dashed line) has higher magnitudes than

that from the existing H∞-based method without the gain-

constraint condition (magenta solid line). That is to say, the

second weighting W2has served to limit the magnitudes of the

inverse model F(z), as it was designed to. Fig. 11 also brings

the approximate methods (Section II-B) into comparison. The

minimum-phase model ˆ

G(z)from ZMETC has the same

magnitude response as the system model G(z)but has large

phase errors, whereas ZPETC yields a ˆ

G(z)with no phase

error but large magnitude mismatch. ˆ

G(z)obtained from NPZ-

ignore has large errors in both magnitude and phase. The

proposed H∞-based optimal inversion outperforms the other

methods by not only striking a balance between magnitude

and phase matches but also mitigating control efforts (i.e.,

magnitudes of F(z)) at high frequencies for system robustness.

B. H∞-based optimal inversion for unstable systems

1) Algorithm

For unstable G(z), Fig. 8 and (14) are ill conditioned, and

the MATLAB function hinfsyn returns an empty solution of

F(z). The ﬁrst intuition for applying the H∞-based optimal

inversion is perhaps to ignore the unstable poles of G(z)and

take the remaining part as a ﬁctitious system model. However,

ignoring the unstable poles alters the relative degree of the

system and may generate a non-causal system. Furthermore,

numerical issues may arise after changing the magnitudes

of the system. To overcome these difﬁculties, this section

introduces an approach by using an all-pass factorization.

We ﬁrst factor out the unstable poles of G(z):

G(z) = z−mG0(z)Y

i

1

z+pi

,(17)

where |pi|>1and G0(z)contains all the zeros and stable

poles of G(z).

Performing the all-pass factorization gives

G(z) = Gs(z)Y

i

¯piz+ 1

z+pi

,(18)

Gs(z) = z−mG0(z)Y

i

1

¯piz+ 1,(19)

where ¯piis the complex conjugate of pi. Here, the unstable

poles in G(z)are replaced by their reciprocals in Gs(z). The

product term Qi(¯piz+1)/(z+pi)in (18) has unity magnitude,

that is, the stable Gs(z)has the same magnitude response as

the unstable G(z). Then we can substitute G(z)with Gs(z)

when implementing the procedure proposed in Section IV-A.

For unstable systems, the design steps of the H∞-based

optimal model inversion are modiﬁed as:

a) Write the pole-zero representation of G(z), determine

the relative degree mof G(z), and then factor out the

unstable poles as in (17);

b) Perform the all-pass factorization by transforming G(z)

in (17) to Gs(z)in (19);

c) Substitute Gs(z)into (14), and solve (14) to ﬁnd

Fs(z) = z−mˆ

Gs

−1(z);

d) Take into account the effect of the unstable poles in (18)

by F(z) = Fs(z)Qi(z+pi)/(¯piz+ 1). The minimum-

phase system model is then ˆ

G(z) = z−mF−1(z).

2) Case study with frequency-domain analysis

In this case study, we show how to implement the H∞-

based optimal inversion in unstable systems.

Consider a discrete-time transfer function

G(z) = z−1(z+ 1.5)

z−1.2(20)

with a relative degree of m= 1 and a sampling rate of 26.4

kHz. G(z)contains an unstable pole 1.2 at low frequency

and an unstable zero -1.5 at high frequency. Following the

aforementioned design steps for unstable systems, we ﬁrst get

Gs(z) = z−1(z+ 1.5)

(1 −1.2z).(21)

Substituting the stable Gs(z)into the hinfsyn function yields

a nonempty solution of Fs(z)that satisﬁes the optimization

principle in (14): Fs(z) = z−mˆ

Gs

−1(z). Here, we design the

weighting functions as

W1(z) = 0.5138z+ 0.5137

z+ 0.0264 , W2(z) = z−0.6423

z−0.2846,

8

102103104

-50

0

50

100

150

Magnitude (dB)

102103104

Frequency (Hz)

-180

-90

0

90

180

Phase (deg)

Fig. 12. Frequency responses of the system model G(z) = z−1(z+

1.5)/(z−1.2) and its minimum-phase approximations. Proposed H∞-

based approach: red dashed line. Previous H∞-based method without gain

constraint: magenta dotted line.

using the MATLAB function makeweight. The obtained Fs(z)

is further normalized to have the same magnitude as the

unstable G−1

s(z)at 800 Hz. The inverse ﬁlter is thus given

by F(z) = Fs(z)(z−1.2)/(1 −1.2z).Using minreal in

MATLAB, we reduce the order of the inverse ﬁlter F(z)from

6 to 3 and obtain

F(z) = 0.7439z3−1.086z2+ 0.227z+ 0.006236

z3+ 0.5056z2−0.1335z−0.003618 .

The minimum-phase system model is thereby ˆ

G(z) =

z−1F−1(z). As shown in Fig. 12, ˆ

G(z)(dashed line) matches

well with G(z)(solid line) particularly at frequencies below

5000 Hz, which is large enough for general feedback designs.

Besides, compared with the existing H∞-based method (dotted

line), near the Nyquist frequency, the high gain of ˆ

G(z)

from the proposed method (dashed line) indicates a small

magnitude of F(z), which matches with the gain-constraint

design criterion in Section IV-A.

C. Feedback applications of the proposed algorithms

Model inversion is fundamental to subsequent servo de-

signs, such as Youla-Kucera parameterization and adaptive

disturbance observers [8], [9], [10], [11], [12]. This section

provides application examples that experimentally verify the

preliminary NMP-zero modulation (Section III) and the H∞-

based optimal inversion (Section IV).

In laser-based additive manufacturing, a galvo scanner sys-

tem applies mirrors to reﬂect input laser beams to follow

predeﬁned trajectories. In [12], the authors ﬁrst identify exper-

imentally the NMP system model. After that, the minimum-

phase model is obtained by moving the unstable zero from -

4.419 to -0.6. Based on the minimum-phase model, [12] builds

an outer-loop inverse-based Youla-Kucera parameterization

scheme to reject single-frequency narrow-band disturbances.

[10] studies the track-following problem in a single-stage

HDD system. The system model in (9) has one NMP zero,

which is shifted inside the unit circle to make the inverse

model strictly stable, as shown in Fig. 6. Then with the stable

inverse model, [10] designs an adaptive disturbance observer

based on the internal model principle to reject multiple narrow-

band disturbances.

In the active suspension benchmark discussed in [30], the

minimum-phase model (red dashed line in Fig. 11) is obtained

by applying the proposed H∞-based optimal inversion. The

model is then used to build an adaptive disturbance observer

with an inﬁnite impulse response structure to reject unknown

or time-varying narrow-band vibrations.

V. CONCLUSION

In this paper, we discussed new frequency-domain analysis

and design approaches to invert a nonminimum-phase (NMP)

linear time-invariant system, with a focus on robustness and

needed design constraints in feedback implementations. We

reveal that among existing model inversion techniques, the

H∞-based method stands out by automatically identifying the

inverse model without knowing the exact NMP zeros. Further-

more, we illustrated that modulating the location of the NMP

zero only changes the system response at selective frequency

regions. Leveraging this fact, for general NMP systems, we

propose a discrete-time H∞-based optimal inversion to au-

tomatically design the inverse model for selective frequency

regions deﬁned by two weighting functions. Veriﬁcations in

complex high-order systems and unstable systems show the

strengths of the proposed algorithm.

ACKNOWLEDGMENTS

This material is based upon work supported in part by the

National Science Foundation under Grant No. 1953155. Any

opinions expressed herein are those of the authors and do not

represent those of the sponsor.

REFERENCES

[1] D. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative

learning control,” IEEE Control Systems Magazine, vol. 26, no. 3, pp.

96–114, 2006.

[2] D. Shen, “Iterative learning control with incomplete information: A

survey,” IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 5, pp.

885–901, 2018.

[3] R. de Rozario and T. Oomen, “Data-driven iterative inversion-based

control: Achieving robustness through nonlinear learning,” Automatica,

vol. 107, pp. 342–352, 2019.

[4] D. Wang and X. Chen, “A multirate fractional-order repetitive control

for laser-based additive manufacturing,” Control Engineering Practice,

vol. 77, pp. 41–51, 2018.

[5] S. Zhu, X. Wang, and H. Liu, “Observer-based iterative and repetitive

learning control for a class of nonlinear systems,” IEEE/CAA Journal

of Automatica Sinica, vol. 5, no. 5, pp. 990–998, 2017.

[6] Y. Li and M. Tomizuka, “Two-degree-of-freedom control with robust

feedback control for hard disk servo systems,” IEEE/ASME Transactions

on Mechatronics, vol. 4, no. 1, pp. 17–24, 1999.

[7] C. Wang, M. Zheng, Z. Wang, and M. Tomizuka, “Robust two-degree-

of-freedom iterative learning control for ﬂexibility compensation of

industrial robot manipulators,” in 2016 IEEE International Conference

on Robotics and Automation (ICRA). IEEE, 2016, pp. 2381–2386.

[8] T. Jiang, H. Xiao, J. Tang, L. Sun, and X. Chen, “Local loop shaping for

rejecting band-limited disturbances in nonminimum-phase systems with

application to laser beam steering for additive manufacturing,” IEEE

Transactions on Control Systems Technology, 2019.

[9] K. Ohnishi, “Robust Motion Control by Disturbance Observer,” Journal

of the Robotics Society of Japan, vol. 11, no. 4, pp. 486–493, 1993.

9

[10] X. Chen and M. Tomizuka, “A minimum parameter adaptive approach

for rejecting multiple narrow-band disturbances with application to hard

disk drives,” IEEE Transactions on Control Systems Technology, vol. 20,

no. 2, pp. 408–415, 2011.

[11] A. Apte, U. Thakar, and V. Joshi, “Disturbance observer based speed

control of pmsm using fractional order pi controller,” IEEE/CAA Journal

of Automatica Sinica, vol. 6, no. 1, pp. 316–326, 2019.

[12] D. Wang and X. Chen, “A tutorial on loop-shaping control method-

ologies for precision positioning systems,” Advances in Mechanical

Engineering, vol. 9, no. 12, pp. 1–12, 2017.

[13] K. J. Astrom, P. Hagander, and J. Sternby, “Zeros of sampled systems,”

Automatica, vol. 20, no. 1, pp. 31–38, 1984.

[14] M. Tomizuka, “Zero phase error tracking algorithm for digital control,”

ASME Journal of Dynamic Systems, Measurement, and Control, vol.

109, no. 1, pp. 65–68, 1987.

[15] L. Dai, X. Li, Y. Zhu, and M. Zhang, “Quantitative analysis on tracking

error under different control architectures and feedforward methods,” in

2019 American Control Conference (ACC). IEEE, 2019, pp. 5680–

5686.

[16] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “Analysis and

comparison of three discrete-time feedforward model-inverse control

techniques for nonminimum-phase systems,” Mechatronics, vol. 22,

no. 5, pp. 577–587, 2012.

[17] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “The effect of

nonminimum-phase zero locations on the performance of feedforward

model-inverse control techniques in discrete-time systems,” in 2008

American control conference. IEEE, 2008, pp. 2696–2702.

[18] S. Devasia, “Iterative machine learning for output tracking,” IEEE

Transactions on Control Systems Technology, vol. 27, no. 2, pp. 516–

526, 2017.

[19] K.-S. Kim and Q. Zou, “A Modeling-Free Inversion-Based Iterative

Feedforward Control for Precision Output Tracking of Linear Time-

Invariant Systems,” IEEE/ASME Transactions on Mechatronics, vol. 18,

no. 6, pp. 1767–1777, 2013.

[20] C.-W. Chen and T.-C. Tsao, “Data-based feedforward controller re-

construction from iterative learning control algorithm,” in 2016 IEEE

International Conference on Advanced Intelligent Mechatronics (AIM).

IEEE, 2016, pp. 683–688.

[21] M. Zheng, F. Zhang, and X. Liang, “A systematic design framework

for iterative learning control with current feedback,” IFAC Journal of

Systems and Control, vol. 5, pp. 1–10, 2018.

[22] B. Francis and G. Zames, “On h∞-optimal sensitivity theory for SISO

feedback systems,” IEEE Transactions on Automatic Control, vol. 29,

no. 1, pp. 9–16, 1984.

[23] M. Zheng, C. Wang, L. Sun, and M. Tomizuka, “Design of arbitrary-

order robust iterative learning control based on robust control theory,”

Mechatronics, vol. 47, pp. 67–76, 2017.

[24] R. de Rozario, A. J. Fleming, and T. Oomen, “Finite-time learning con-

trol using frequency response data with application to a nanopositioning

stage,” IEEE/ASME Transactions on Mechatronics, 2019.

[25] J. Dewey, K. Leang, and S. Devasia, “Experimental and theoretical

results in output-trajectory redesign for ﬂexible structures,” Journal of

dynamic systems, measurement, and control, vol. 120, no. 4, pp. 456–

461, 1998.

[26] K. S. Ramani, M. Duan, C. E. Okwudire, and A. Galip Ulsoy, “Tracking

Control of Linear Time-Invariant Nonminimum Phase Systems Using

Filtered Basis Functions,” Journal of Dynamic Systems, Measurement,

and Control, vol. 139, no. 1, pp. 011001:1–11, 2017.

[27] J. van Zundert and T. Oomen, “On inversion-based approaches for

feedforward and ILC,” Mechatronics, vol. 50, pp. 282–291, 2018.

[28] S. Skogestad and I. Postlethwaite, Multivariable feedback control:

analysis and design. Wiley New York, 2007, vol. 2.

[29] I. D. Landau, A. C. Silva, T.-B. Airimitoaie, G. Buche, and M. Noe, “An

active vibration control system as a benchmark on adaptive regulation,”

in 2013 European Control Conference (ECC). IEEE, 2013, pp. 2873–

2878.

[30] X. Chen and M. Tomizuka, “Selective model inversion and adaptive

disturbance observer for time-varying vibration rejection on an active-

suspension benchmark,” European Journal of Control, vol. 19, no. 4,

pp. 300–312, 2013.

Dan Wang received her B.S. degree in material

science and engineering from Shandong University,

Jinan, China, in 2012 and her M.S. degree in

mechanical engineering from Tsinghua University,

Beijing, China, in 2015. She is currently working

toward the Ph.D. degree at the University of Wash-

ington, Seattle, WA, USA. She received the Best

Paper Award in the 2018 International Symposium

on Flexible Automation. Her main research interests

include control theory, modeling, signal processing,

robotics, artiﬁcial intelligence, FEM, and AM.

Xu Chen is an Assistant Professor in the De-

partment of Mechanical Engineering at the Univer-

sity of Washington, Seattle. He received his M.S.

and Ph.D. degrees in Mechanical Engineering from

the University of California, Berkeley in 2010 and

2013, respectively, and his Bachelor’s degree with

honors from Tsinghua University, China in 2008.

His research interests include dynamic systems and

controls, advanced manufacturing, robotics, and in-

telligent mechatronics. His work – funded by NSF,

DOE, DOD, state, and industries – has led to four

Best Paper Awards, patented and massively deployed servo algorithms in

the information storage industry, top-ranked adaptive control methods in

international benchmark evaluations, and the graduation of two University

Scholars. Dr. Chen is a recipient of the National Science Foundation CAREER

Award, the Young Investigator Award from ISCIE / ASME International

Symposium on Flexible Automation, and the inaugural UTC Institute for

Advanced Systems Engineering Breakthrough Award in 2016. He is Publicity

and Local Arrangements Chairs of the 2020 and the 2023 IEEE/ASME

International Conferences on Advanced Intelligent Mechatronics, and Exhibits

Chair of the 2021 IEEE American Control Conference. He served the ASME

Dynamic Systems and Control (DSC) Division in roles including Chair of the

Vibration Technical Committee, News Editor of the DSC Magazine, Editor

of the DSC Newsletter, and Student and Young Members Chair of the 2016

and the 2020 ASME DSC Conferences. He is a member of IEEE, ASME,

SME, and SIAM.