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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
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-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
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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
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-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.
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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.
