The University of Manchester Research
Dynamic Output Feedback Controller Synthesis using an
LMI-based Strictly Negative Imaginary Framework
DOI:
10.1109/MED.2019.8798493
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Accepted author manuscript
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Citation for published version (APA):
Kurawa, S., Bhowmick, P., & Lanzon, A. (2019). Dynamic Output Feedback Controller Synthesis using an LMI-
based Strictly Negative Imaginary Framework. In Proceedings of the 27th Mediterranean Conference on Control
and Automation https://doi.org/10.1109/MED.2019.8798493
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Proceedings of the 27th Mediterranean Conference on Control and Automation
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Download date:09. Aug. 2022
Dynamic Output Feedback Controller Synthesis using an LMI-based
α− Strictly Negative Imaginary Framework
Suleiman Kurawa, Parijat Bhowmick and Alexander Lanzon
Abstract— This paper deals with the dynamic output feed-
back controller synthesis utilizing the α− strictly negative
imaginary systems property. The proposed scheme ensures
robust stability in closed-loop against the set of all stable, strictly
proper negative imaginary uncertainties that satisfies the DC
loop gain condition pertaining to negative imaginary stability.
In addition to that, a prescribed decay rate in the closed-
loop time response is also enforced via α− pole placement.
Two illustrative examples have been studied to demonstrate
the usefulness of the proposed controller synthesis scheme.
I. INTRODUCTION
The theory of negative imaginary (NI) systems was moti-
vated by the study of inertial systems with colocated force
actuator and position sensor [1], [3]. In a single-input-single-
output (SISO) setting, a stable transfer function is said to
have the NI (resp., strictly NI or SNI) frequency response
if its Nyquist plot lies below (resp., strictly below) the real
axis of the complex plane in the open frequency interval from
zero to infinity [1]. In contrast to positive real (PR) systems,
NI systems can have relative degree of up to two and they
can also have non-minimum phase zeros [8]. Due to virtue
of the simple internal stability condition for interconnected
NI and SNI systems that depends primarily on the DC loop
gain information [1], the NI system theory finds immense
applications in controller design especially for mechanical
and mechatronic systems. For example, it has been applied
in vibration control of a flexible robotic arm [10], in motion
control of a robotic arm with unknown parameters [6], in
vehicle platooning [4], in control of a DC servo motor [7],
in nano-positioning of an atomic force microscope [11], and
in position control of a swing-arm hard disk drive [15].
So far in the literature, a handful of effort has been
dedicated towards the problem of NI controller synthesis. For
example, an LMI-based state feedback synthesis imposing
closed-loop NI property is explored in [3], [6]. While in [11],
the state feedback synthesis has been done to enforce closed-
loop NI/SNI property using an algebraic Riccati equation
(ARE) based approach. In [13], the authors introduce a static
state feedback and a dynamic output feedback controller
synthesis schemes imposing closed-loop NI property. In [15],
This work was supported by the Engineering and Physical Sciences Re-
search Council (EPSRC) [grant number EP/R008876/1] and the Petroleum
Technology Development Fund overseas scholarship sheme. All research
data supporting this publication are directly available within this publication.
Suleiman Kurawa, Parijat Bhowmick and Alexander Lanzon are with the
Control Systems Centre, School of Electrical and Electronic Engineering,
University of Manchester, Sackville Street, Manchester M13 9PL, U.K.
Suleiman.Kurawa@postgrad.manchester.ac.uk,
Parijat.Bhowmick@manchester.ac.uk,
Alexander.Lanzon@manchester.ac.uk.
a dynamic output feedback control framework is proposed
for systems with PR uncertainty which is then applied
to systems with stable NI uncertainty by transforming the
NI uncertainty into PR framework. Very recently, in [16]
and [17], strictly negative imaginary controller synthesis
problems considering both state and output feedback have
been addressed applying an ARE-based approach.
Pertaining to NI synthesis that facilitates some tran-
sient performance (e.g. decay rate) via closed-loop pole-
placement, the work of [12] can be referred which provides
a state feedback NI controller synthesis technique exploiting
the notion of α− and D − pole placement introduced in
[19]. However, in many practical applications, some of
the states may not be accessible for external measurement
and hence, output feedback sometimes outperforms state
feedback. Nevertheless, regional pole placement, or pole
placement in general, provides a way of achieving time
domain performance such as reducing peak overshoot and
settling time, increasing decay rate, etc. [20].
Being motivated by the aforementioned developments, this
paper proposes an LMI-based procedure to design a dynamic
output feedback controller applying the α− SNI framework
that ensures robust stability against all stable, strictly proper,
NI uncertainties. Furthermore, it is shown via examples that
the designed controller can also satisfy a certain level of
robust H
∞
performance applying µ-analysis.
II. PRELIMINARIES
This section recalls some definitions, lemmas and techni-
cal results from the literature important for developing the
main results. The first definition for NI systems was given
in [1] for stable systems. This was then extended in [2] to
marginally-stable systems having poles in the closed left-half
plane excluding the origin. In this paper, we have used the
definition of NI systems according to [2].
Definition 1: (NI System) [2] A square, real, rational and
proper transfer function matrix M(s) is said to be NI if
1) M(s) has no poles in ℜ{s} > 0;
2) j[M( jω) − M( jω)
∗
] ≥ 0 for all ω ∈ (0, ∞) such that
jω is not a pole of M(s);
3) If jω
0
with ω
0
∈ (0, ∞) is a pole of M(s), it is at most
a simple pole and the residue matrix K
0
= lim
s→ jω
0
j(s −
jω
0
)M(s) is Hermitian and positive semidefinite.
The definition of NI systems have been further extended in
[8], [10] to account for NI systems with single and double
poles at the origin. See also [14] for some recent feedback
stability analysis results allowing poles at the origin. Let us
recall the definition of SNI systems.
Authors' Camera Ready Manuscript. To appear in the
Proceedings of the 27th Mediterranean Conference on Control and Automation, Akko, Israel, July 2019,
Please cite using bibliographic data of the associated published version.
Definition 2: (SNI System) [1] Let M(s) be a square, real,
rational and proper transfer function matrix. M(s) is said to
be SNI if M(s) has no poles in ℜ[s] ≥ 0 and j[M( jω) −
M( jω)
∗
] > 0 for all ω ∈ (0, ∞).
The following lemma characterizes the strongly strict nega-
tive imaginary (SSNI) systems introduced in [5].
Lemma 1: [5] Let
A B
C D
be a state-space realiza-
tion of a real, rational, proper and square transfer func-
tion matrix M(s). Assume, M(s) − M(−s)
T
has full normal
rank and the pair (A,C) is observable. Then, A is Hurwitz
and M(s) is SNI with lim
ω→0
j
1
ω
[M( jω) − M( jω)
∗
] > 0 and
lim
ω→∞
jω [M( jω) − M( jω)
∗
] > 0 if and only if D = D
T
and
there exists Y = Y
T
> 0 such that
AY +YA
T
< 0 and B + AYC
T
= 0. (1)
The conditions necessary and sufficient for the internal
stability of a positive feedback interconnection of stable NI
and SNI systems are next presented.
Theorem 1: [1] Let M(s) be SNI and γ > 0. Then,
the positive feedback interconnection of M(s) and ∆(s) is
internally stable for all stable NI systems ∆(s) that satisfy
∆(∞) ≥ 0, ∆(∞)M(∞) = 0 and λ
max
[∆(0)] < γ
−1
if and only
if λ
max
[M(0)] ≤ γ.
1
y
( )
M s
( )
s
2
e
1
e
2
y
2
w
1
w
(a)
( )
s
z
w
( )
G s
u
( )
K s
y
( )
M s
(b)
Fig. 1: (a) Positive feedback interconnection of two NI
systems; (b) M −∆ configuration for robust stability analysis.
Towards this end, we set the notations for α− strictly
negative imaginary (abbreviated as α− SNI) systems. The
α− SNI subclass is closely related to the SSNI class of
systems [5] and is defined by a state-space characterization.
Definition 3: Let D = D
T
and α > 0. Then, R(s) =
A B
C D
is said to be α− SNI if there exists a real matrix
Y = Y
T
> 0 such that
AY +YA
T
+ 2αY ≤ 0 and B + AYC
T
= 0. (2)
We now provide some remarks which explore the properties
of α− SNI systems and find the connections between α−
SNI and SSNI systems properties.
Remark 1: The α− SNI systems are inherently stable.
This can be readily established from (2), which implies
AY +YA
T
< 0 for α > 0 and Y > 0, which in turn ensures
Hurwitzness of A applying [18, Lemma 3.19].
Note that the definition of α− SNI systems does not require
a minimal state-space realisation of the underlying system.
In this context, the literature [6], [16] may be referred where
it is shown that most of the analysis and synthesis results
associated with NI, SNI and SSNI systems theory remain
applicable in case of non-minimal system realization.
Remark 2: From Definition 3, one may think that the set
of the α− SNI systems is a subset of the SSNI class having
poles in ℜ[s] ≤ −α. But, unlike SSNI systems [5], α− SNI
system property does not impose any restrictions on the state-
space realisation. It can be concluded that the set of α−
SNI systems, say R(s), with a completely observable state-
space realisation and R(s)− R(−s)
T
having full normal rank
belongs to the SSNI class.
Note that in case of α− SNI systems the full normal rank
constraint on R(s) − R(−s)
T
is implied by (2) when the B
matrix has full column rank. It is proved in the following
lemma. The same conclusion applies to Lemma 1 as well.
Lemma 2: Let R(s) =
A B
C D
be an (m × m) α− SNI
system with rank[B] = m. Then, R(s) − R(−s)
T
has full
normal rank.
Proof. For a given α > 0 and Y > 0, (2) implies AY +
YA
T
< 0. Then there exists a square and non-singular matrix
L such that AY + YA
T
= −L
T
L. For these L and Y , the
transfer function matrix N(s) =
A B
LY
−1
A
−1
0
acquires
full column rank at s = jω for all ω ∈ R since A is Hurwitz
and rank[B] = m via assumption and rank[LY
−1
A
−1
] = n. It
implies from [2, Corollary 1]
jω[R( jω) − R( jω)
∗
] = ω
2
N( jω)
∗
N( jω) > 0 (3)
for all ω ∈ R\{0} and R(0) − R(0)
T
= 0 since R(0) =
CYC
T
+D = R(0)
T
. This implies that there does not exist any
continuum interval of ω ∈ R for which det[R( jω) −R( jω)
∗
]
remains zero. This in turn ensures that R(s) − R(−s)
T
must
have full normal rank. Note minimality is not required.
In the next section, we will introduce a dynamic output
feedback controller synthesis framework exploiting the α−
SNI property.
III. MAIN RESULTS
A. Problem Formulation
Consider an LTI generalized plant G described by the
following state-space equations
G :
˙x = Ax + B
1
w + B
2
u,
z = C
1
x + D
11
w + D
12
u,
y = C
2
x + D
21
w,
(4)
where x(t) ∈ R
n
is the state vector of the generalized plant,
u(t) ∈ R
n
u
represents the control input , y(t) ∈ R
n
y
is the
measured output, w(t) ∈ R
m
is the exogenous input and z(t) ∈
R
m
is the objective signal. The matrices A ∈ R
n×n
, B
1
∈
R
n×m
, B
2
∈ R
n×n
u
, C
1
∈ R
m×n
, C
2
∈ R
n
y
×n
, D
11
and D
21
are all constant and known. Assume that D
12
= 0, (A, B
2
) is
stabilizable and (A,C
2
) is detectable. The aim is to synthesize
a full-order dynamic output feedback controller
K :
(
˙x
c
= A
c
x
c
+ B
c
y,
u = C
c
x
c
+ D
c
y,
(5)
such that the nominal closed-loop system M(s) is α− SNI
and the generalized plant G is robustly stabilized against all
stable, strictly proper, NI uncertainties ∆(s) with ∆(0) ≤ γ
−1
for a given γ > 0. The state-space realization of the controller
synthesized nominal closed-loop system M(s), as shown in
Fig. 1b, from w to z is given by
M(s) =
A
cl
B
cl
C
cl
D
cl
=
A + B
2
D
c
C
2
B
2
C
c
B
1
+ B
2
D
c
D
21
B
c
C
2
A
c
B
c
D
21
C
1
0 D
11
.
(6)
B. Controller synthesis using α− SNI framework
This subsection provides the main contribution of this
paper. Theorem 2 gives a set of sufficient conditions required
for the existence of a dynamic output feedback controller
K(s) which makes the nominal closed-loop system given
in (6) α− SNI with a given α > 0 and maintains closed-
loop stability in presence of any stable, strictly proper, NI
uncertainty satisfying the DC gain condition.
Theorem 2: Let a generalized plant G be given by (4)
with D
11
= D
T
11
, D
12
= 0, (A, B
2
) stabilizable and (A,C
2
)
detectable. Let γ > 0, α > 0 and m ≤ 2n. Suppose there exist
matrices
ˆ
A ∈ R
n×n
,
ˆ
B ∈ R
n×n
y
,
ˆ
C ∈ R
n
u
×n
,
ˆ
D ∈ R
n
u
×n
y
and
symmetric matrices P ∈ R
n×n
and X ∈ R
n×n
such that
Φ
11
+ 2αP Φ
12
+ 2αI PB
1
+
ˆ
BD
21
+
ˆ
AC
T
1
? Φ
22
+ 2αX Φ
23
? ? 0
≤ 0,
(7)
P I
I X
> 0, (8)
and C
1
XC
T
1
+ D
11
< γI, (9)
with the following shorthand
Φ
11
= PA + PA
T
+
ˆ
BC
2
+C
T
2
ˆ
B
T
, (10a)
Φ
12
=
ˆ
A + (A + B
2
ˆ
DC
2
)
T
, (10b)
Φ
22
= X A
T
+ AX + B
2
ˆ
C +
ˆ
C
T
B
T
2
, (10c)
Φ
23
= B
1
+ B
2
ˆ
DD
21
+ AXC
T
1
+ B
2
ˆ
CC
T
1
, (10d)
and the symbol ? denotes the elements due to symmetry.
Then, an internally stabilizing controller K(s) is given by
(5) where
D
c
=
ˆ
D, (11a)
C
c
= (
ˆ
C − D
c
C
2
X)M
−T
, (11b)
B
c
= N
−1
(
ˆ
B − PB
2
D
c
), (11c)
A
c
= N
−1
(
ˆ
A − PAX − NB
c
C
2
X
− PB
2
C
c
M
T
− PB
2
D
c
C
2
X)M
−T
, (11d)
and M and N are square and non-singular solutions of the
algebraic equation MN
T
= I −X P. This controller K(s) forms
a closed-loop system M(s), expressed as in (6), which is α−
SNI and is robust to all stable, strictly proper, NI uncertainty
∆(s) satisfying λ
max
[∆(0)] ≤ γ
−1
.
Proof. First note that D
cl
= D
T
cl
in (6) and m ≤ 2n. The
proof will proceed via the following steps which establishes
that for M(s) =
A
cl
B
cl
C
cl
D
cl
to be α− SNI with a given
α > 0, conditions (7)-(9) need to be satisfied.
Step 1: From Definition 3, M(s) =
A
cl
B
cl
C
cl
D
cl
is α−
SNI with a given α > 0 if there exists Y = Y
T
> 0 such that
A
cl
Y +YA
T
cl
+ 2αY ?
B
T
cl
+C
cl
YA
T
cl
0
≤ 0. (12)
Step 2: Since inequality (12) is not in LMI form due
to presence of the terms containing products of unknown
controller variables, a linearising change in controller vari-
ables [19], [20] is required to transform (12) into LMI form.
Partition the closed-loop Lyapunov matrix Y and Y
−1
as
follows:
Y =
X M
M
T
•
and Y
−1
=
P N
N
T
•
, (13)
where X and P are symmetric n × n matrices and the symbol
• represents matrices that are not explicitly used in the
linearization process. Note Y
−1
exists since Y > 0 via (8)
which has been explained subsequently in step 4. Note also
that X, P, M, N are not independent variables but must satisfy
XP + MN
T
= I (see [19], [20] for details). Since, M and N
are square and non-singular, the following block matrices
Π
1
=
P I
N
T
0
and Π
2
=
I X
0 M
T
(14)
are non-singular. Π
1
and Π
2
are related through the expres-
sion
Y Π
1
= Π
2
(15)
which is obtained from YY
−1
= I. The change of controller
variables are defined as
ˆ
A = PAX + NB
c
C
2
X +PB
2
C
c
M
T
+ PB
2
D
c
C
2
X +NA
c
M
T
,
ˆ
B = NB
c
+ PB
2
D
c
,
ˆ
C = D
c
C
2
X +C
c
M
T
, and
ˆ
D = D
c
.
(16)
Step 3: Applying a congruence transformation on (12)
with the block diagonal matrix diag{Π
1
, I} and using (15),
we obtain
Π
T
1
A
cl
Π
2
+ Π
T
2
A
T
cl
Π
1
+ 2αΠ
T
1
Π
2
?
B
T
cl
+C
cl
YA
T
cl
Π
1
0
≤ 0. (17)
Simplifying all the product terms and substituting into (17)
the linearizing change of controller variables given in (16),
we get back condition (7), that is,
Φ
11
+ 2αP Φ
12
+ 2αI PB
1
+
ˆ
BD
21
+
ˆ
AC
T
1
? Φ
22
+ 2αX Φ
23
? ? 0
≤ 0.
Step 4: Positive definiteness of the closed-loop Lyapunov
matrix Y =
X M
M
T
•
is guaranteed by (8) via the
congruence transformation shown below
Π
T
1
Y Π
1
=
P I
I X
> 0. (18)
Step 5: The inequality condition (9) is equivalent to
M(0) < γI since M(0) = C
cl
YC
T
cl
+ D
11
. This in turn implies
λ
max
[M(0)∆(0)] < 1 via [1]. Thus, the interconnection of
M(s) being α− SNI and ∆(s) being stable NI satisfies all
the assumptions of Theorem 1 as well as the DC loop gain
condition. Therefore, the interconnection is robustly stable.
This completes the proof.
Remark 3: In order to find square and non-singular so-
lutions of M and N from the expression MN
T
= I − XP,
methods such as Q − R factorisation, Cholesky factorisation
or Eigen-decomposition can be used. But the easiest solution
is to choose M = I and accordingly, N = I −PX . This choice
of M and N rules out the possibility of getting ill-conditioned
solutions in Matlab due to computational issues.
IV. ILLUSTRATIVE EXAMPLES
In this section, we will present two illustrative examples to
elucidate the usefulness of the proposed synthesis technique.
A. Example 1
We reconsider the model of an uncertain flexible structure
system with colocated position sensor and force actuator,
as shown in Fig. 2, originally studied in [3], [6] followed
later by [13]. The generalized plant model G of the physical
d
2
x
y
u
z
2
1
s
1
1
s
1
1
s
( )
s
y
w
3
x
1
x
Fig. 2: Block diagram of the simplified model of an uncertain
flexible structure system taken in [6].
system is governed by the following state-space equations.
The output of the system y(t) is subjected to some bounded
disturbance d(t) ∈ R.
G :
˙x
1
˙x
2
˙x
3
=
−1 0 0
1 −1 1
0 1 −1
x
1
x
2
x
3
+
0 0
0 0
1 1
w
d
+
−2
1
0
u,
z
y
=
0 1 0
0 1 0
x
1
x
2
x
3
+
0 0
1 1
w
d
,
and W (s) = ∆(s)Z(s), where W (s) and Z(s) are the Laplace
transform of w(t) and z(t) respectively. In line with [6], ∆(s)
is an SNI uncertainty with ∆(∞) = 0 and ∆(0) ≤ 1.
Part I. The control objective is to synthesize a dynamic
output feedback controller K(s) such that the nominal closed-
loop system M(s) becomes α− SNI and the generalized plant
G remains stable closed-loop in presence of any ∆(s) defined
above. Choosing α = 0.8 and applying Theorem 2, we obtain
a feasible solution set of matrices
P =
126.3650 15.0662 83.8253
15.0662 7.8543 15.0240
83.8253 15.0240 126.6103
> 0,
X =
33.3391 −2.0000 −31.8531
−2.0000 0.5764 1.5764
−31.8531 1.5764 30.8991
> 0,
and
ˆ
A,
ˆ
B,
ˆ
C,
ˆ
D using the CVX toolbox [21]. We fix M = I and
hence, N = I − PX. The controller matrices A
c
, B
c
, C
c
and
D
c
are then uniquely reconstructed for this M and N using
the relations (11a)-(11d). The controller K(s) is computed as
K(s) =
3.7491(s + 1)(s + 1.345)(s + 6.358)
(s + 36.45)(s + 1.386)(s + 1.001)
which constitutes the nominal closed-loop system
M(s) =
3.749s
5
+ 33.63s
4
+
97.04s
3
+ 126.6s
2
+ 77.97s + 18.48
s
6
+ 38.09s
5
+ 174.3s
4
+
336.1s
3
+ 328.9s
2
+ 162s + 32.06
from w to z. The closed-loop poles are given by λ
i
(A
cl
) =
{−33.1278, −1.1847, −0.9259 ± j0.1564, −0.9264, −1.00}.
It can be readily verified that M(s) is an α− SNI transfer
function with ℜ[λ
i
[A
cl
]] < −0.8 for all i. The Nyquist plot
of M(s) given in Fig. 3a also reflects that M(s) is an SNI
transfer function. Now, we find M(0) = 0.5764 and hence,
M(0)∆(0) = 0.5764 < 1, which ensures robust stability in
closed-loop against the given set of ∆(s) having ∆(0) ≤
1 via Theorem 2. In order to show the applicability of
0 0.1 0.2 0.3
Real Axis
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
Imaginary Axis
(a)
( )
s
z
w
( )
N s
d
y
( )
f
s
(b)
Fig. 3: (a) Nyquist plot of the α− SNI transfer function M(s)
obtained in Example 1; (b) LFT configuration for robust
performance problem by augmenting a fictitious uncertainty
∆
f
(s) with the stable NI uncertainty ∆(s).
the proposed synthesis scheme, we study the disturbance-
attenuation problem in presence of two arbitrarily chosen
strictly proper SNI uncertainties given by ∆
1
(s) =
1
s+2
and
∆
2
=
1
s+20
. A pulse disturbance having amplitude 0.1 and
T
on
= 1s is applied to the system under all zero initial con-
dition. Figures 4a-4d compare the closed-loop time response
corresponding to the nominal case and in presence of the
![Fig. 2: Block diagram of the simplified model of an uncertain flexible structure system taken in [6].](/figures/fig-2-block-diagram-of-the-simplified-model-of-an-uncertain-2dt69pq2.webp)
