A Unified Framework for the H∞ Mixed-Sensitivity Design of Fixed Structure Controllers through Putinar Positivstellensatz

Abstract

In this paper, we present a novel technique to design fixed structure controllers, for both continuous-time and discrete-time systems, through an H∞ mixed sensitivity approach. We first define the feasible controller parameter set, which is the set of the controller parameters that guarantee robust stability of the closed-loop system and the achievement of the nominal performance requirements. Then, thanks to Putinar positivstellensatz, we compute a convex relaxation of the original feasible controller parameter set and we formulate the original H∞ controller design problem as the non-emptiness test of a set defined by sum-of-squares polynomials. Two numerical simulations and one experimental example show the effectiveness of the proposed approach.

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machines
Article
A Unified Framework for the H∞Mixed-Sensitivity Design of
Fixed Structure Controllers through Putinar Positivstellensatz †
Valentino Razza and Abdul Salam *


Citation: Razza, V.; Salam, A. A
Unified Framework for the
H∞Mixed-Sensitivity Design of Fixed
Structure Controllers through Putinar
Positivstellensatz. Machines 2021,9,
176. https://doi.org/10.3390/
machines9080176
Academic Editors: Mingcong Deng,
Hongnian Yu and Changan Jiang
Received: 15 July 2021
Accepted: 16 August 2021
Published: 20 August 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Dipartimento di Automatica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24,
10129 Torino, Italy; valentino.razza@polito.it
*Correspondence: abdul.salam@polito.it
† This paper is an extended version of our paper published in V. Cerone, V. Razza, D. Regruto. H∞
mixed-sensitivity design with fixed structure controller through Putinar positivstellensatz. In Proceedings of
the American Control Conference (ACC), Philadelphia, PA, USA, 10–12 July 2019; pp. 1806–1811.
Abstract:
In this paper, we present a novel technique to design fixed structure controllers, for
both continuous-time and discrete-time systems, through an
H∞
mixed sensitivity approach. We
first define the feasible controller parameter set, which is the set of the controller parameters that
guarantee robust stability of the closed-loop system and the achievement of the nominal performance
requirements. Then, thanks to Putinar positivstellensatz, we compute a convex relaxation of the
original feasible controller parameter set and we formulate the original
H∞
controller design problem
as the non-emptiness test of a set defined by sum-of-squares polynomials. Two numerical simulations
and one experimental example show the effectiveness of the proposed approach.
Keywords: mixed sensitivity control; discrete time H∞control; fixed structure H∞control
1. Introduction
The development of a worst-case control design for a linear plant subjected to un-
known parameter uncertainties and disturbances has attracted the interest of the control
community for many years. In [
1
], within the context of sensitivity reduction, Zames
introduces the
H∞
norm minimization to formulate the control design problem. The mixed-
sensitivity approach, introduced in [
2
,
3
], is a general control design formulation where the
H∞
norm is used to define constraints on both the sensitivity and complementary sensitiv-
ity function. These constraints are defined by suitable weighting functions to ensure good
performances and the robustness of the system to be controlled. The books [
4
,
5
] and the
paper [
6
] provide a deep discussion about the underlying theory and, starting from robust-
ness and time-domain requirements, the way to suitably formulate the mixed-sensitivity
control design problem.
Nominal
H∞
mixed-sensitivity control design problem can be solved through algo-
rithms based on linear matrix inequalities (LMI) (see, e.g., [
7
,
8
]) or on the algebraic Riccati
equation (see, e.g., [
9
,
10
]). Most of the
H∞
mixed-sensitivity control design approaches are
developed for continuous-time systems, while a few approaches deal with the discrete-
time systems. In [
11
,
12
], discrete-time controllers are designed through the solution of two
Riccati equations, while a convex optimization approach is proposed in [
13
]. The interested
reader is referred to [
14
], and references therein, for a deeper discussion on discrete-time
H∞mixed-sensitivity control design.
In general, algorithms for
H∞
control synthesis cannot take into account the order
of the controller, which instead depends on the order of the transfer functions defining
the underlying optimization problem. However, in several practical applications, like PI
and PID controllers or embedded control systems, the controller structure is a-priori fixed
and cannot be modified. In [
15
], the authors show that controller structure constraints
make the
H∞
control design problem non-convex and NP-hard to be solved. The main
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Machines 2021,9, 176 2 of 24
difficulty is that structural constraints produce bilinear matrix inequalities (BMI) [
16
],
that are non-convex. Convexification methods to transform the BMIs constraints in LMIs
through variable change (see, e.g., [
17
]) or inner convex approximations (see, e.g., [
18
])
are proposed in the literature. However, the success of these methods depends on the
specific structure of the constraints and cannot be generalized. A common approach to
BMIs problems is represented by iterative algorithms (see, e.g., [
19
–
22
] and references
therein) that finds local optimum solutions in polynomial time.
To avoid numerical difficulties related to BMIs, some techniques based on the con-
troller or the plant order reduction have been proposed in [
23
–
25
]. However, the plant
order reduction leads to higher conservatism in the uncertainty model, while the controller
order reduction leads to performances degradation. Moreover, these techniques still cannot
ensure a specific controller structure (e.g., PID).
Burke et al., in [
26
], propose a gradient sampling algorithm for the design of a fixed-
order
H∞
controller, which is implemented in the HIFOO Matlab toolbox (see [
27
]). Another
Matlab toolbox for the design of fixed-structure controllers is Hinfstruct, which implements
the algorithm proposed in [
28
] and is based on the Clarke sub-differential approach pre-
sented in [
29
]. Both these Matlab packages are based on local optimization techniques,
which have no guarantees about the convergence to the global optimal solution.
A few approaches based on global optimization have also been proposed in the
literature to design fixed structure controllers. These approaches require a parametric
representation of uncertain plants and exploit interval arithmetic tools to synthesize the
H∞
control problem. In [
30
], the authors present a remarkable result by providing a branch-
and-bound based algorithm to compute inner and outer approximations of the controller
parameter set. Other global optimization-based approaches rely on quantifier elimination
techniques (see, e.g., [31]).
Among the several structures, the
H∞
mixed-sensitivity design of PID controllers is the
most investigated in the literature. Convex optimization techniques have been proposed
in [32,33]
to tune continuous-time PID controllers, while bilinear transformation is used to
compute discrete-time regulators in [
34
]. The main difficulty related to the fixed-structure
controller design is related to the non-convexity of the stabilizing parameter set. For linear
parametrized controllers, inner convex approximations are proposed in [35–37].
In this paper, we propose a unified framework to design continuous and discrete-time
fixed structure controllers in the framework of the mixed-sensitivity approach. Starting
from the work [
38
], we extend the previous results to the most general case by considering
both continuous and discrete-time systems. In the proposed algorithm, we first define the
set of controller parameters that achieve robust stability and nominal performances of the
feedback control system. Then, we rewrite the controller design problem as the positivity
test over a bounded domain. By exploiting Putinar positivstellensatz
theorem [39]
, we
formulate the
H∞
mixed sensitivity controller design as the non-emptiness test of a convex
set defined through a number of sum of squares (SOS) polynomial constraints. The
problem to be solved is a convex semi-definite problem (SDP), whose solution can be found
in polynomial time.
The paper is organized as follows: Section 2reviews
H∞
mixed-sensitivity notations
and backgrounds fundamentals, while the problem formulation is given in Section 3. In
Section 4, we present the proposed
H∞
control design approach based on the Putinar
positivstellensatz. Numeric examples are provided in Section 5, to show the effectiveness
of the proposed methods to design both continuous and discrete-time controllers, together
with the results obtained with the Matlab function Hinfstruct. Section 6shows experimental
results of the controller design problem for a magnetic suspension system, and Section 7
concludes the paper.
2. Notations and Background
In this section, we introduce the notations that are used in the paper and review some
basics on
H∞
mixed sensitivity controller design. We define the transfer functions through

Machines 2021,9, 176 3 of 24
a generic variable
ξ∈C
which is
ξ=s
when dealing with continuous time (CT) systems
and
ξ=z
for discrete time (DT) systems. Given a transfer function
C(ξ)
, we denote with
C(jω)
the frequency response computed by assigning
ξ=jω
for CT systems and
ξ=ejωTs
for DT systems, where Tsis the sampling time.
Let us consider the feedback control system depicted in Figure 1, where
Gn(ξ)
and
K(ξ)
are the nominal plant and the controller transfer functions, respectively,
w∈R
is the
reference signal,
u∈R
is the control input,
y∈R
is the measured output and
z1∈Rn1
and
z2∈Rn2are the controlled outputs associated to the assigned performance requirements.
K(ξ)
W1(ξ)
W2(ξ)
Gn(ξ)
∆(ξ)
w+u
+
+y
−
z1
z2
Figure 1. Block diagram of feedback system.
Let G(ξ)be the uncertain model of the plant described by
G(ξ) = Gn(ξ)(1+∆(ξ)) (1)
where
∆(ξ)∈C
is unstructured multiplicative uncertainty, which is bounded by a given
transfer function Wu(ξ), i.e.,
|∆(ξ)|≤|Wu(ξ)|,∀ω∈Ω(2)
such that Ω= [0, +∞)for CT systems and Ω=h0, π
Tsifor DT systems.
W1(ξ)
and
W2(ξ)
are suitable weighting functions that describe the performance con-
straints on the nominal sensitivity
Sn(ξ)
and nominal complementary sensitivity transfer
function
Tn(ξ)
, respectively. For a given nominal plant
Gn(ξ)
and a controller
K(ξ)
, the
nominal loop transfer function is defined as
Ln(ξ) = K(ξ)Gn(ξ), (3)
the nominal sensitivity function and complementary sensitivity function are defined as
Sn(ξ) = (1+Ln(ξ))−1(4)
and
Tn(ξ) = Ln(ξ)(1+Ln(ξ))−1(5)
respectively. Nominal closed loop system performances constraints are met if
kSn(ξ)W1(ξ)k∞≤1
kTn(ξ)W2(ξ)k∞≤1(6)
where
k·k∞
is the
H∞
norm of a dynamical system, which, for a generic single-input
single-output (SISO) system H(ξ), is
kH(ξ)k∞=sup
ω∈Ω
|H(jω)|. (7)

Machines 2021,9, 176 4 of 24
In the remainder of this section, we review some definitions and results about feedback
systems properties.
Definition 1.
A feedback system is said to be well-posed if all closed-loop transfer functions, defined
from any exogenous input to all internal signals, are well-defined and proper.
Result 1.
A necessary and sufficient condition for well-posedness is that
Sn(ξ)
exists and is proper,
i.e., 1
+K(ξ)Gn(ξ)
is not strictly proper. A stronger condition for well-posedness is that either
K(ξ)or Gn(ξ)be strictly proper transfer functions (see, e.g., [4]).
Definition 2.
A well-posed feedback system is internally stable if, and only if, all the transfer
functions from any input to any output are BIBO stable (see, e.g., [40]).
Result 2.
Necessary and sufficient conditions for the internal stability of feedback systems are that
(i) the nominal sensitivity function
Sn(ξ)
is BIBO stable and (ii) there are no unstable zero/pole
cancellations while forming the nominal loop function Ln(ξ). [40] provides a detailed proof.
Definition 3.
A feedback system is robustly stable if the controller
K(ξ)
makes the system inter-
nally stable for all possible uncertain plants.
Result 3.
By applying the small gain theorem (see, e.g., [
40
]), the system depicted in Figure 1is
robustly stable if the nominal sensitivity function Sn(ξ)is stable and
kTn(ξ)Wu(ξ)k∞≤1 (8)
Further details can be found in [4].
3. Problem Formulation
In this section, we formulate the
H∞
controller design problem for both CT and
DT systems. In this work, we propose a methodology to design a
H∞
controller
K(ξ
,
p)
which guarantees robust stability to unstructured multiplicative uncertainty bounded by
the function
Wu(ξ)
, and fulfils the nominal performance defined through the weighting
functions
W1(ξ)
and
W2(ξ)
. The controller is assumed to have a fixed structure, i.e.,
to belong to a certain class
K
, which guarantees the well-posedness condition, and is
characterized by an nk-th order transfer function
K(ξ,p) = ∑nk
i=0βi(p)ξi
ξnk+∑nk−1
j=0αj(p)ξj=Nk(ξ,p)
Dk(ξ,p)(9)
where the denominator and numerator coefficients,
αj(p)∈R
and
βi(p)∈R
, are polyno-
mial functions in a suitable parameter vector p∈Rnpto be designed.
We assume to know the transfer functions
W1(ζ)
,
W2(ζ)
and
Wu(ζ)
, which take into
account the design constraints, as well as the nominal plant transfer function defined as
Gn=Ng(ζ)
Dg(ζ)(10)
where
Ng(ζ)
and
Dg(ζ)
are polynomial functions and
Ng(ζ)
has no roots at
s=
0 or
z=
1.
Remark 1.
It is worth noting that, since
K(ξ
,
p)
depends on
p
, functions
(3)
–
(5)
involving
K
depend on the parameter vector
p
as well. However, for the sake of simplicity, we omit
p
as a
parameter in all functions except for K(ξ,p).

Machines 2021,9, 176 5 of 24
Definition 4. We define the stabilizing controller parameter set
S={p∈Rnp|K(ζ,p)internally stabilizes Gn(ζ)}(11)
as the set of all the controller parameters which guarantees the internal stability of the feedback
control system depicted in Figure 1.
Definition 5. By applying Result 3, we define the robust stabilizing controller parameter set
DS={p∈ S|kTn(ξ)Wu(ξ)k∞≤1}(12)
as the set of all the controller parameters which guarantees the internal robust stability of the
uncertain plant G(ξ).
We can derive some properties of the selected controller class
K
through the analysis
of the sets Sand DS.
Result 4.
If the set
S
is empty then the chosen controller structure
K
is not suitable to provide
stability of the nominal plant Gn(ξ).
Result 5.
If the set
Ds
is empty then the chosen controller structure
K
is not suitable to provide
robust stability of the uncertain plant G(ξ).
Definition 6. We define the feasible controller parameter set D
D={p∈ DS| kSn(ξ)W1(ξ)k∞≤1,
kTn(ξ)W2(ξ)k∞≤1}(13)
as the set of parameter
p
which guarantee robust stability for the plant
G(ξ)
and the achievement of
the nominal performances described by the given weighting functions W1(ξ)and W2(ξ).
It is worth noting that, by considering Equations
(6)
and
(13)
, the set
D
can be written
equivalently as
D={p∈ S | kSn(ξ)W1(ξ)k∞≤1,
kTn(ξ)ˆ
W2(ξ)k∞≤1(14)
where ˆ
W2(ξ)is such that
|ˆ
W2(jω)|=max {|W2(jω)|,|Wu(jω)|},∀ω∈Ω. (15)
The emptiness of the set
D
highlights that the chosen controller class structure
K
is not suitable to achieve the closed-loop stability and desired closed-loop performance
specifications. Instead, a large or unbounded set
D
may suggest that the controller structure
may fulfil more demanding specifications.
Remark 2.
Through the procedure described in the next Section, it is possible to test several
controller structures, e.g., a commercial solution, and to select the cheapest solution that guarantees
the non-emptiness of the feasible controller parameters set.
4. An SOS Approach to Mixed Sensitivity Design with Fixed Structure Controller
In this section, we consider the problem of looking for a parameter vector
p
belonging
to the feasible controller parameters set
D
. We rewrite this problem as the positivity
check of a number of multivariate polynomials over a bounded semi-algebraic set. This
problem, which is known to be NP-hard, can be efficiently solved by applying the Putinar
positivstellensatz (see, e.g., [39] for details), through which the polynomial positivity check
is reformulated in terms of SDP.

Machines 2021,9, 176 6 of 24
We rewrite
D
as the intersection of two sets
D=S ∩ P
, where the performance
controller parameters set Pis defined as
P={p∈Rnp| kSn(ξ)W1(ξ)k∞≤1,
kTn(ξ)ˆ
W2(ξ)k∞≤1(16)
The properties of the set Dcan be obtained through the analysis of the sets Sand P.
4.1. Mathematical Description of the Set S
At first, we look for an explicit mathematical formulation of the set
S
. For internal
stability, both conditions of Result 2must be satisfied. The first condition requires that the
nominal sensitivity function
Sn(ξ)
is stable, which is achieved if the roots of 1
+Ln(ξ)
have
negative real part when dealing with CT systems, or have the module less than one when
DT systems are considered.
4.1.1. Routh’s Stability Criterion
For CT systems, we can evaluate the sign of the real part of the roots of a polynomial
function
A(s) = ansn+an−1sn−1+. . . +a1s+a0(17)
by applying the Routh’s stability criterion, which is based on the Routh’s Table reported in
Table 1(further details on Routh’s stability criterion can be found in book [41]).
Table 1. Routh’s coefficients table.
anan−2an−4· · ·
an−1an−3an−5· · ·
b1b2b3· · ·
c1c2c3· · ·
d1d2d3· · ·
.
.
..
.
..
.
....
Coefficients biin the Routh’s Table are given by
b1=an−1an−2−anan−3
an−1
b2=an−1an−4−anan−5
an−1
b3=an−1an−6−anan−7
an−1
.
.
.
(18)

Machines 2021,9, 176 7 of 24
and we stop if we achieve a zero coefficient. The remaining coefficients are computed in a
similar way, by multiplying the terms of the two previous rows
c1=b1an−3−an−1b2
b1
c2=b1an−5−an−1b3
b1
c3=b1an−7−an−1b4
b1
.
.
.
(19)
d1=c1b2−b1c2
c1
d2=c1b3−b1c3
c1
d3=c1b4−b1c4
c1
.
.
.
(20)
Result 6.
All the roots of a polynomial function have negative real part if, and only if, all the
coefficients in the first column of the Routh’s table show the same sign, i.e.,
g1(p) = an>0
g2(p) = an−1>0
g3(p) = b1>0
g4(p) = c1>0
.
.
.
(21)
4.1.2. Jury’s Stability Criterion
The Jury’s stability criterion [
42
] is used to check that the roots of a DT polynomial
function
A(z) = anzn+an−1zn−1+. . . +a1z+a0(22)
are located inside the unitary circle, and it is based on the Jury’s Table (see Table 2), which
is characterized by 2
n−
3 rows. The even numbered rows are the elements of the preceding
row in reverse order, while the odd numbered rows coefficients are computed as
bk=



a0an−k
anak



ck=



b0bn−k−1
bn−1bk



dk=



c0cn−k−2
cn−2ck



.
.
.
(23)

Machines 2021,9, 176 8 of 24
Table 2. Jury’s coefficients table.
Row Number z0z1z2. . . zn−k. . . zn−1zn
1a0a1a2. . . an−k. . . an−1an
2anan−1an−2. . . ak. . . a1a0
3b0b1b2. . . bn−k. . . bn−1
4bn−1bn−2bn−3. . . bk−1. . . b0
5c0c1c2. . . cn−k. . .
6cn−2cn−3cn−4. . . ck−2. . .
.
.
..
.
..
.
..
.
..
.
..
.
. . . .
2n−2p4p3p2p1
2n−3q0q1q2
Result 7.
All the roots of the polynomial function
(22)
are inside the unitary circle if, and only if,
all the following conditions occur
g1(p) = A(1)>0
g2(p) = (−1)nA(−1)>0
g3(p) = |an|−|a0|>0
g4(p) = |b0|−|bn−1|>0
g5(p) = |c0|−|cn−2|>0
g6(p) = |d0|−|dn−3|>0
.
.
.
(24)
The stability constraints for the nominal sensitivity transfer function
Sn(ξ)
are obtained
by applying the Result 6or 7, if the system is DT or CT, respectively, to the numerator of
1+Ln(ξ), which is
A(ξ) = Nk(ξ,p)Ng(ξ) + Dk(ξ,p)Dg(ξ)(25)
The second condition in Result 2requires to avoid unstable zero/pole cancellations
while multiplying K(ξ,p)and Gn(ξ). If Gn(ξ)does not show unstable zeros or poles, this
requirement is automatically achieved. If
Gn(ξ)
has unstable poles or unstable zeros, we
impose effective constraints to force controller numerator and denominator functions to
have only stable roots. This is obtained by applying the Routh’s, or the Jury’s, criterion to
Nk(ξ,p)or to Dk(ξ,p).
Remark 3.
It may seems that, by imposing
Dk(ξ
,
p)
to have only stable roots, the controller cannot
have poles at
s=
0or
z=
1. However, these poles are needed to guarantee zero steady-state tracking
error either to polynomial reference signals or to polynomial disturbance signals. We rewrite the
controller denominator as
Dk(ξ,p) = Zk(ξ)D0
k(ξ,p)(26)
where
Z(ξ) = sµ
for CT systems or
Z(ξ) = (z−
1
)µ
for DT systems and
µ
is the multiplicity
of the roots at
s=
0or
z=
1of
Dk(ξ
,
p)
. Instead of imposing the controller denominator
Dk
to
have only stable roots, we impose stability constraints only to the polynomial function
D0
k
. In fact,
by assumption, the plant has no roots at
s=
0or
z=
1and no unstable cancellations can occur
between Zkand Ng.
Remark 4.
The set
S
is defined by the set of conditions coming from the application of the
Routh/Jury stability criterion on
(25)
. From the definition of
Nk(ξ
,
p)
and
Dk(ξ
,
p)
, it follows that

Machines 2021,9, 176 9 of 24
the coefficients of
A(ξ)
in
(25)
, as well as the coefficients in Tables 1and 2, are polynomial functions
of the parameter vector
p
. Then, for both CT and DT systems,
S
is a semi-algebraic set defined by a
number of polynomial inequalities gi(p)>0.
4.2. Polynomial Description of the Set P
A polynomial description of the performance controller parameter set
P
is obtained
by the following result.
Result 8.
Through a suitable choice of a variable
φ
and a set
Φ
, the inequalities in
(16)
can be
equivalently written as
hi(φ,p)>0, i=1, 2, φ∈Φ(27)
where hi(φ,p)are polynomial functions of both pand φ
Proof. Let us consider the two rational transfer functions
Hi(ξ,p) = Ni(ξ,p)
Di(ξ,p),i=1, 2 (28)
where
H1(ξ
,
p) = Sn(ξ)W1(ξ)
and
H2(ξ
,
p) = Tn(ξ)ˆ
W2(ξ)
. Then, by applying the
H∞
norm definition (7), we can rewrite conditions (16) as
hi(ω,p) = |Di(jω,p)|2− |Ni(jω,p)|2≥0, i=1, 2, ∀ω∈Ω. (29)
For CT systems,
H1(jω
,
p)
and
H2(jω
,
p)
are complex rational functions and their
magnitudes are polynomial functions in
p
and
ω
. Therefore, by setting
φ=ω
and
Φ=Ω
we have (27).
Instead, the magnitude of a DT transfer function depends on
ejωTs=cos(ωTs) +
jsin(ωTs)
. Since
cos(ωTs):Ω→[−
1, 1
]
is a bijective function for
ω∈Ω
, we can rewrite
ξ=ejωTs,∀ω∈Ωas
ξ=ejωTs=a+jb,a∈[−1, 1]⊂R,a2+b2=1, b≥0 (30)
where
a
and
b
are scalar variables. Therefore, for DT systems, through
(29)
and
(30)
, we ob-
tain
(27)
by choosing
φ= [a b]T
and
Φ=φ∈R2:−1≤a≤1, a2+b2+1=0, b≥0
.
4.3. SOS Relaxation of the Set D
From Result 8, the closed loop system achieves the performance specifications defined
by
W1(ξ)
and
W2(ξ)
if the polynomial functions
h1(φ
,
p)
and
h2(φ
,
p)
are positive over
the semi-algebraic set
Φ
. It is well known from the literature that testing the global
non-negativity of a polynomial function is an NP-hard problem. In this subsection, by
exploiting the Putinar’s Positivstellensatz, we compute a SOS decomposition of polynomial
functions
h1(φ
,
p)
, and
h2(φ
,
p)
, and we also show that if a non-negative polynomial
has a SOS representation, then one can compute polynomial positivity by using SDP
optimization methods.
A polynomial f(x)is SOS if it can be written as
f(x) = ∑
i
f2
i(x),x∈R[x](31)
where,
R[x]
denotes the ring of polynomials in
x= (x1
,
x2
,
. . .
,
xn)
. Suppose that
vδ(x)
is
the vector of all the monomials of degree less than or equal to δ, given by
vδ(x) = 1, x1, . . . , xn,x2
1,x1x2, . . . , xn−1xn,xn
n. . . , xδ
1, . . . , xδ
nT∈R`δ(32)
where
`δ=(n+δ
δ)
. The polynomial
f(x)
can be expressed as a quadratic form in the
monomial vector vδ(x)thanks to the following result.

Machines 2021,9, 176 10 of 24
Result 9.
A polynomial
f∈R[x]2δ
has a SOS decomposition if, and only if, there exists a real
symmetric and positive semi-definite matrix
Q∈R`δ×`δ
, such that
f(x) = vδ(x)TQvδ(x)
, for all
x∈Rn(see [43] for a detailed proof).
Thus, the problem of checking whether a polynomial
f(x)
is SOS is equivalent to the
problem of finding a symmetric positive definite matrix Q∈R`δ×`δ.
The Putinar’s Positivstellensatz, which is reviewed below, can be applied to
(27)
to derive
sufficient conditions to verify that the inequalities are satisfied.
Result 10. (Putinar’s Positivstellensatz [39])
Consider a compact semi-algebraic set
Φ={φ∈Rn:q1(φ)≥0, q2(φ)≥0, . . . , qm(φ)≥0}(33)
where
q1(φ)
,
q2(φ)
,
. . .
,
qm(φ)
are
m
polynomial functions. If a polynomial
f
is positive in
Φ
then there are polynomials σv, such that
f(φ) = σ0(φ) +
m
∑
ν=1
σν(φ)qν(φ),
for some σν(φ)∈Σδ[φ]
(34)
where
Σδ[φ]
is the set of SOS polynomials in
φ
up to the degree 2
δ
. The integer
δ
is called
relaxation order.
Based on result 10, we state the following result.
Result 11.
For
some σν(φ)∈Σδ[φ]
, where
Σδ[φ]
is the set of SOS polynomials in
φ
up to the
degree 2δ, if
f(φ)−
m
∑
ν=1
σν(φ)qν(φ)is SOS,(35)
then f (φ)is positive on semi-algebraic set Φ.
Proof. The proof is rather trivial and based on the fact that σ0(φ)is a SOS polynomial.
The feasible controller parameters set
D
can be relaxed to a convex set
Dδ
for a
suitable value of the relaxation order
δ
. In fact, the Result 11 can be applied to polynomial
inequalities which define the set
S
and
P
, to replace the polynomial constraints defined
by (21), (24) and (27) with a set of SDP constraints in the form (35).
Remark 5.
If the set
(14)
is not empty, then the relaxed problem obtained by applying Result 11
admits a feasible solution for any relaxation order
δ≥δmin
, where
δmin
is an integer value large
enough (see [
44
] and reference therein for further details). Therefore, the problem of extracting a
controller parameter vector
p
from the the feasible controller parameters set
D
in
(14)
is replaced by
a convex SDP problem.
5. Numeric Examples
In this section, we show the efficiency of the proposed controller design approach
through three simulation examples.

Machines 2021,9, 176 11 of 24
5.1. Design of CT Controller
Consider a CT SISO system characterized by the following nominal transfer function
Gn(s) = 700
s(s+100)(36)
which is subjected to the multiplicative uncertainty with the following weighting filter
Wu(s) = 0.3(s+49)
s+101 . (37)
The goal is to design a controller, such that
kSn(s)W1(s)k∞≤1 and kTn(s)ˆ
W2(s)k∞≤1
,
where
W1(s) = s2+13.68s+64
s(1.995s+15.96)(38)
and
W2(s) = s2+37.74s+625
1247 . (39)
Structure of the desired controller is known a priori and is given by:
K(s,p) = c1s2+c2s+c3
s2+c4s(40)
where,
p= [c1
,
c2
,
c3
,
c4]T∈R4
is the vector of unknown controller parameters. By
following the design procedure described in Section 4, we derive a description of the
set
S
that guarantees the stability of the nominal sensitivity transfer function, whose
denominator is described by
A(s) = 700(c1s2+c2s+c3) + (s2+c4s)(s2+100s). (41)
Through the Routh’s stability criterion, the conditions, such that the roots of
A(s)
have negative real part, are
g1(p) = 100 +c4>0
g2(p) = 100(100 +c4)(c4+7c1)−c2>0
g3(p) = 100(100 +c4)(c4+7c1)c2−c2
2−c3(100 +c4)2>0
g4(p) = c3>0.
(42)
Since
Gn(s)
has neither zeros nor poles in the right half plane (RHP), no further
conditions are needed to guarantee the stability of the nominal closed loop system. Nominal
performance and robust stability conditions are taken into account by the performance
controller set defined in Equation
(16)
, which requires the selection of a suitable weighting
function
ˆ
W2(jω)
. According to
(15)
, from the comparison between
|W2(jω)|
and
|Wu(jω)|
shown in Figure 2, we select
ˆ
W2(s) = W2(s). (43)
The set Pis defined by the following constraints
kSn(s)W1(s)k∞=



s[s2+ (100 +c4)s+100c4] (s2+13.68s+64)
[s4+ (100 +c4)s3+ (100c4+700c1)s2+700c2s+700c3] (1.995s+15.96)


∞
=



N1(s,p)
D1(s,p)


∞
≤1
kTn(s)ˆ
W2(s)k∞=



700 (c1s2+c2s+c3) (s2+37.74s+625)
1247 [s4+ (100 +c4)s3+ (100c4+700c1)s2+700c2s+700c3]


∞
=



N2(s,p)
D2(s,p)


∞
≤1.
(44)

Machines 2021,9, 176 12 of 24
that are written in the polynomial form
(29)
. The polynomial constraints defining the
feasible controller parameters set are relaxed thanks to the Result 11, with
δ=
1 and
Ω= [
0, 10
5]
. The resulting SDP problem has been formulated with Yalmip (see [
45
]) and
solved with Mosek (see [46]).
100101102103
-20
-10
0
10
20
30
40
50
Frequency (rad/s)
Figure 2. Comparison between |Wu(jω)|(dotted) and |W2(jω)|(solid).
The controller parameter extracted from the feasible controller parameters set leads to
K(s) = 0.398s2+10.281s+21.347
s2+10s(45)
which guarantees the stability of the nominal closed loop system, in fact
A(s) = (s+97.64)(s+3.915)(s2+8.441s+39.09)(46)
has all negative real-part roots.
Since the sensitivity function and the complementary sensitivity function are below
their weighting functions (see Figures 3and 4), the controller achieves robust stability
and nominal performance requirements. Numerically,
kSn(jω)W1(jω)k∞=
0.7396 and
kTn(jω)ˆ
W2(jω)k∞=0.6839.

Machines 2021,9, 176 13 of 24
10-1 100101102103
-60
-50
-40
-30
-20
-10
0
Frequency (rad/s)
Figure 3. Comparison between |W1−1(jω)|(solid) and |Sn(jω)|(dashed).
10-1 100101102103
-70
-60
-50
-40
-30
-20
-10
0
Frequency (rad/s)
Figure 4. Comparison between |W2−1(jω)|(solid) and |Tn(jω)|(dashed).
5.2. DT Controller Design
Consider a DT SISO system
Gn(z) = 3z+2.25
4z2−2.8z+1(47)
which is subjected to multiplicative uncertainty with a weighting filter
Wu(z) = 0.3944z2−0.143z−0.05305
z2+0.5162z−0.3177 . (48)

Machines 2021,9, 176 14 of 24
The objective is to design a robust PI controller, such that
kSn(z)W1(z)k∞≤
1 and
kTn(z)W2(z)k∞≤1, where
W1(z) = 0.606z2−0.96z+0.3875
(z−0.7787)(z−1), (49)
W2(z) = z2−1.254z+0.4595
0.1636z+0.1261 (50)
and
K(z,p) = kp+ki
z−1. (51)
The unknown controller parameter vector is p= [kp,ki]T∈R2.
Similar to the previous example, we derive the description of the nominal stability
parameter set
S
. The Jury’s stability criterion is applied to the denominator of the nominal
sensitivity function
A(z) = 4z3+ (3kp−6.8)z2+ (3ki−0.75kp+3.8)z+2.25ki−2.25kp−1, (52)
leading to the following polynomial constraints
g1(p) = ki>0
g2(p) = 15.6 −1.5ki+0.75kp>0
g3(p) = 16 −(2.25ki−2.25kp−1)2>0
g4(p) = h5.0625k2
i+k2
p+4.5kp−ki−10.125kikpi2
−6.75kp(ki−kp)−27.3ki+15.3kp−8.42>0.
(53)
From Figure 5, we see that
|W2(z)|
is greater than
|Wu(z)|
for all the frequencies, thus
we select
ˆ
W2(z) = W2(z). (54)
10-4 10-2 100
-20
-10
0
10
20
30
40
50
Frequency (rad/s)
Figure 5. Comparison between |Wu(ejω)|(dotted) and |W2(ejω)|(solid).
The performance set Pis defined by the rational functions

Machines 2021,9, 176 15 of 24
kSn(z)W1(z)k∞=



(4z2−2.8z+1) (0.606z2−0.96z+0.3875)
[(3z+2.25) (kpz+ki−kp) + (4z2−2.8z+1) (z−1)] (z2−0.7787)


∞
=



N1(z,p)
D1(z,p)


∞
≤1
kTn(z)ˆ
W2(z)k∞=




(3z+2.25) (kpz+ki−kp) (z2−1.254 +0.4595)
[(3z+2.25) (kpz+ki−kp) + (4z2−2.8z+1) (z−1)] (0.1636 +0.1261)



∞
=



N2(z,p)
D2(z,p)


∞
≤1
(55)
that, according to Result 8, are rewritten as
hi(φ,p) = |Di(a+jb,p)|2− |Ni(a+jb,p)|2≥0, i=1, 2 (56)
where
φ= [a b]T
and
Φ=φ∈R2:−1≤a≤1, a2+b2+1=0, b≥0
. The polyno-
mial constraints that define the feasible controller parameters set are relaxed by applying
Result 11
with
δ=
1. The relaxed SDP problem is solved with Yalmip (see [
45
]) and Mosek
(see [46]) leading to the controller
K(z) = 0.1408 +0.1266
z−1. (57)
This controller achieves nominal stability since
A(z) = (z−0.6253)(z2−0.9691z+0.4126)(58)
has all the roots inside the unitary circle. Moreover, the graphical comparisons between
|Sn(z)|
and
|Tn(z)|
with the weighting functions
|W1(z)|
and
|ˆ
W2(z)|
, respectively, reported
in Figures 6and 7, show that the controller achieves desired performance specifications.
Numerically, kSn(z)W1(z)k∞=0.8725 and kTn(z)ˆ
W2(z)k∞=0.99.
10-4 10-2 100
-70
-60
-50
-40
-30
-20
-10
0
Frequency (rad/s)
Figure 6. Comparison between |W−1
1(ejω)|(solid) and |Sn(ejω)|(dotted).

Machines 2021,9, 176 16 of 24
Figure 7. Comparison between |ˆ
W−1
2(ejω)|(solid) and |Tn(ejω)|(dotted).
5.3. Comparison with Hinfstruct
In this subsection, we compare the algorithm proposed in this paper with the common
library function Hinfstruct (see [
29
]), which is included in Matlab. Through Hinfstruct, the
controller parameter vector
p
is computed as the solution to the optimization problem
defined as
p=arg min
p∈Rnpγ
s.t.
kS(s)W1(s)k∞≤γ,
kT(s)W2(s)k∞≤γ,
K(s,p)stabilizes the closed loop system.
(59)
It is worth noting that, if the solution to
(59)
is such that
γ≤
1, Hinfstruct provides a
solution that is also feasible for our approach. However, since Hinfstruct is based on local
optimization techniques, the solver may find local minimum solution to
(59)
which do not
guarantee the feasibility of the solution.
Let us consider the CT SISO system described by the following nominal transfer
function
Gn(s) = 2s+100
s2+3s+2. (60)
The goal is to design a PI controller, such that
kSn(s)W1(s)k∞≤1 and kTn(s)W2(s)k∞≤1
,
where
W1(s) = 2.25s2+5.4s+5.063
4.489s2+6.734s(61)
and
W2(s) = 50s2+13750s+125000
0.4988s+2.494e05 . (62)
The controller computed by means of Hinfstruct toolbox is
K(s,) = 0.00523s+0.00891
s(63)

Machines 2021,9, 176 17 of 24
having γ=1.0488, where
γ=max {kS(s)W1(s)k∞,kT(s)W2(s)k∞}. (64)
Since
γ≥
1, the requirements specified by the
W1(s)
and
W2(s)
are not achieved and
Hinfstruct provided an unfeasible controller.
Through the procedure described in Section 4, we solve this control design problem.
As we have shown in Section 5.1, we explicitly define the stability constraints thanks to
the Routh theorem and, by exploiting the results 11, we define an SDP problem with a
relaxation order
δ=
1 that is formulated with Yalmip and solved by Mosek. The controller
extracted from the feasible controller parameters set is
K(s) = 0.0127s+0.0158
s. (65)
The controller achieves the nominal performances since
Sn(jω)
and
Tn(jω)
are smaller
than
W1(jω)
and
W2(jω)
,respectively(seeFigures 8and 9). Numerically,
kSn(jω)W1(jω)k∞=0.56
and kTn(jω)W2(jω)k∞=0.74.
Figure 8. Comparison between |W−1
1(jω)|(solid) and |Sn(jω)|(dotted).
In this example, even if a feasible solution exists to the control design problem, the
iterative algorithm implemented in Hinfstruct stops to an unfeasible solution. On the
other hand, our approach, based on convex optimization techniques, finds the controller
parameter vector that satisfies all of the requirements.

Machines 2021,9, 176 18 of 24
Figure 9. Comparison between |W−1
2(jω)|(solid) and |Tn(jω)|(dotted).
6. Experimental Example
In this section, we apply the proposed control design technique to design a controller
for the magnetic levitation system shown in Figure 10.
Figure 10. Magnetic levitation system.
In this system, a transconductance amplifier regulates the current through an elec-
tromagnet coil proportional to the input voltage
u
. The magnetic field, generated by the
current, exerts a force on a light ball in the opposite direction to the gravity force. An
optical transducer measures the ball position and produced the output voltage signal
y
.
The
book [47]
provides a detailed description of the considered system. Magnetic levitation
systems are highly non-linear unstable systems. In order to design a low order fixed struc-
ture controller, the control schema depicted in Figure 1is considered, where a
Gn(s)
is the

Machines 2021,9, 176 19 of 24
linearized model of the magnetic levitation system obtained around a suitable equilibrium
point and is given by
Gn(s) = Y(s)
U(s)=−7044
(s−29.68)(s+29.68)(66)
where
U(s)
and
Y(s)
are the Laplace transform of the input and output voltage signals,
respectively. It is worth noting that we consider the voltage transducer signal as system
output to have a comparable reference signal
w
that can be produced by a common
laboratory equipment, i.e., a signal generator. Moreover, we can directly measure the output
voltage
y
with an oscilloscope, while the ball position in meters can be only computed
through the knowledge of the mathematical model of the position transducer.
The nominal plant in Equation
(66)
is subjected to multiplicative uncertainty charac-
terized by the following weighting function
Wu(s) = 0.1993s2+6.852s+55.96
s2+46.15s+429.5 . (67)
The aim is to design a controller in the form
K(s,p) = c1s2+c2s+c3
c4s2+s, (68)
where
p= [c1
,
c2
,
c3
,
c4]T∈R4
is the unknown parameter vector, such that the closed loop
system is internally stable. Moreover, for a square wave reference signal
w(t)
with period
2 s, duty-cycle 50% and amplitude 0.1 V, the closed loop system must satisfy the following
nominal specifications: (i) zero steady-state tracking error for a step reference, (ii) rise
time
tr≤
0.015 s, and (iii) overshoot
ˆ
s≤
25%. The presence of a pole at
s=
0 in
K(s
,
p)
guarantees that the first requirement is implicitly achieved. According to the methodology
described in [
4
], the time domain requirements are mapped into the frequency domain
weighting filters
W1(s) = s2+145s+9877
s(1.646s+82.3)(69)
and
W2(s) = 0.003333s2+1.633s+414
560.7 . (70)
The constraints that define the stabilizing controller parameters set
S
are obtained by
the Routh’s stability criterion, leading to
g1(p) = c4>0
g2(p) = −140.88c3>0
g3(p) = c2c4−c1>0
g4(p) = c1−c2c4−7.9963c2
2c4+7.9963c1c2+0.0011c3>0.
(71)
Since the magnetic levitation system is unstable, to avoid unstable pole-zero cancella-
tion between the plant and the controller we consider stable Nk(s,p)and D0
k(s,p), where
Nk(s,p) = c1s2+c2s+c3(72)
and
D0
k(s,p) = c4s+1 (73)

Machines 2021,9, 176 20 of 24
Stability of
Nk(s
,
p)
and
D0
k(s
,
p)
is ensured by exploiting Routh Hurwitz criterion
which provide following additional constraints in set S.
g7(p) = −c1>0
g8(p) = −c2>0(74)
The graphical comparison between
W2(s)
and
Wu(s)
is reported in Figure 11. Since
|W2(s)|>|Wu(s)|, we choose
ˆ
W2(s) = W2(s). (75)
100101102103104
-20
-10
0
10
20
30
40
50
Frequency (rad/s)
Figure 11. Comparison between |Wu(jω)|(dotted) and |Wd2(jω)|(solid).
The performance set
P
is derived in the same way as in previous examples. Through
Result 11, we formulate the controller design as a SDP optimization problem by setting
the relaxation order
δ=
1 and
Ω= [
0, 10
5]
. The relaxed SDP problem is solved with
Yalmip (see [
45
]) and Mosek (see [
46
]). The controller extracted from the feasible controller
parameters set is
K(s) = −0.0265s2−1.226s−15.01
0.0015s2+s. (76)
The controller achieves the nominal performances as
Sn(jω)
and
Tn(jω)
are smaller
than
W1(jω)
and
W2(jω)
, respectively (see Figures 12 and 13). Numerically,
kSn(jω)W1(jω)k∞
= 0.99 and
kTn(jω)W2(jω)k∞=
0.9473. We provide the comparison between the linearized
system
Gn(s)
and the real plant in Figure 14, which shows the time-domain responses of
the closed-loop systems when the reference is a square wave with amplitude 0.1V and
frequency 0.5 Hz. The linear system
Gn(s)
achieves the time domain requirements: both the
rise time
tr≈
0.00928 s and the overshoot
ˆ
s≈
23.02%. However, the designed controller
is not able to achieve the maximum overshoot requirement on the real plant, which is
ˆ
s≈
35%. The larger overshoot is due to the model mismatch between the non-linear plant
and the approximated linear model and, thus, does not depend on the specific approach
proposed in this work. Despite this modeling error, the designed controller stabilizes the
magnetic levitation system and guarantees the rise time tr≈0.011 s.

Machines 2021,9, 176 21 of 24
10-2 100102104
-80
-70
-60
-50
-40
-30
-20
-10
0
Frequency (rad/s)
Figure 12. Comparison between |W−1
1(jω)|(solid) and |Sn(jω)|(dotted).
100101102103104
-60
-50
-40
-30
-20
-10
0
Frequency (rad/s)
Figure 13. Comparison between |W−1
2(jω)|(solid) and |Tn(jω)|(dotted).
3 3.5 4 4.5 5
Time (s)
-0.2
-0.1
0
0.1
0.2
Figure 14.
Magnetic levitation system response to square wave reference signal: reference
w(t)
(solid square-wave),
magnetic levitation system output y(t)(solid) and linearized Gn(s)system output (dashed) responses.

Machines 2021,9, 176 22 of 24
7. Conclusions
In this paper, we present a unified approach to design for the
H∞
mixed-sensitivity
design for fixed structure robust controllers for both CT and DT systems. We define the
feasible controller parameter set as a semi-algebraic set of all the controller parameters that
achieve nominal performance and robust stability for the closed-loop system. We formulate
the control design problem as the non-emptiness test of the feasible controller parameters
set, which is an NP-hard problem. Thanks to the results on the Putinar positivstellensatz,
we propose a novel SOS based approach to formulate a convex relaxation of the original
problem in terms of SDP constraints. Therefore, the achieved solution is not affected by
local minima that may be found while solving non-convex problem through iterative
methods. The proposed approach is a global optimization approach and is a powerful
tool for fixed structure
H∞
mixed sensitivity control design, whose solution can be found
efficiently in polynomial time.
We provide three simulation examples and one experimental application to show
the efficiency of the proposed algorithm on both CT and DT systems. In particular, one
example shows the comparison of the proposed approach with the state of the art algorithm
implemented in the Hinfstruct Matlab function. In this example, we show that the solution
provided by Hinfstruct does not achieve the desired requirements, while our approach
successfully solves the control design problem.
Author Contributions:
Conceptualization, V.R.; methodology, V.R.; software, A.S.; investigation,
A.S.; writing—original draft preparation, A.S.; writing—review and editing, V.R. Both authors have
read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments:
Computational resources are provided by HPC@polito, which is a project of
Academic Computing within the Department of Control and Computer Engineering at the Politecnico
di Torino.
Conflicts of Interest: The authors declare no conflicts of interest.
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