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Journal ArticleDOI

Switching LPV control design with MDADT and its application to a morphing aircraft

06 Feb 2017-Kybernetika (Institute of Information Theory and Automation)-Vol. 52, Iss: 6, pp 967-987

TL;DR: A novel switching strategy, mode dependent average dwell time (MDADT), is adopted to investigate the flight control of a morphing aircraft in its morphing phase to accommodate different performance goals in different sweep wing configurations.
Abstract: In flight control of a morphing aircraft, the design objective and the dynamics may be different in its various configurations. To accommodate different performance goals in different sweep wing configurations, a novel switching strategy, mode dependent average dwell time (MDADT), is adopted to investigate the flight control of a morphing aircraft in its morphing phase. The switching signal used in this note is more general than the average dwell time (ADT), in which each mode has its own ADT. Under some simplified assumptions the control synthesis condition is formulated as a linear matrix optimization problem and a set of mode-dependent dynamic state feedback controllers are designed. Afterwards the proposed approach is applied to a morphing aircraft with a variable sweep wing to demonstrate its validity.
Topics: Morphing (55%), Wing configuration (53%), Dwell time (52%)

Summary (1 min read)

Introduction

  • Kybernetika Yong He; Chunjuan Li; Weiguo Zhang; Jingping Shi; Yongxi Lü Switching LPV control design with MDADT and its application to a morphing aircraft Kybernetika, Vol. 52 (2016), No. 6, 967–987 Persistent URL: http://dml.cz/dmlcz/146000.
  • © Institute of Information Theory and Automation AS CR, 2016 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use, also known as Terms of use.
  • Each copy of any part of this document must contain these Terms of use.
  • This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz.

SWITCHING LPV CONTROL DESIGN WITH MDADT

  • AND ITS APPLICATION TO A MORPHING AIRCRAFT Yong He, Chunjuan Li, Weiguo Zhang, Jingping Shi and Yongxi Lü.
  • In flight control of a morphing aircraft, the design objective and the dynamics may be different in its various configurations.
  • It has been recognized that the property in the ADT switching is still not anticipated, since the average time interval between any two consecutive switching is at least τa, which is independent of the system mode.
  • So far there is no result available yet on control of switching LPV systems with MDADT based on parameter-dependent Lyapunov functions, which will reduce the conservatism, enhance flexibility and improve the disturbance attenuation performance in the analysis and synthesis of a switched LPV system.
  • A1. (A(ρ),B2(ρ),C2(ρ)) triple is parameter-dependent stabilizable and detectable for all ρ; A2.

PjPi Sij

  • Could be different and even conflicting for different parameter regions.
  • 2. Parameter of the morphing aircraft configurations.
  • Whereas the dashed line has different variation rate in different region in order to maintain different dwell time in its subregion, as the dwell time is mode-dependent, the switching signal, which is determined by the sweep signal, is mode-dependent.
  • Bei Lu and Fen Wu: Switching lpv control designs using multiple parameterdependent lyapunov functions.
  • IET Control Theory Appl. 4 (2010), 5, 817–826.

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Kybernetika
Yong He; Chunjuan Li; Weiguo Zhang; Jingping Shi; Yongxi
Switching LPV control design with MDADT and its application to a morphing aircraft
Kybernetika, Vol. 52 (2016), No. 6, 967–987
Persistent URL: http://dml.cz/dmlcz/146000
Terms of use:
© Institute of Information Theory and Automation AS CR, 2016
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents
strictly for personal use. Each copy of any part of this document must contain these Terms of use.
This document has been digitized, optimized for electronic delivery and stamped
with digital signature within the project DML-CZ: The Czech Digital Mathematics
Library http://dml.cz

K Y B E R N E T I K A V O L U M E 5 2 ( 2 0 1 6 ) , N U M B E R 6 , P A G E S 9 6 7 9 8 7
SWITCHING LPV CONTROL DESIGN WITH MDADT
AND ITS APPLICATION TO A MORPHING AIRCRAFT
Yong He, Chunjuan Li, Weiguo Zhang, Jingping Shi and Yongxi L
¨
u
In flight control of a morphing aircraft, the design objective and the dynamics may be
different in its various configurations. To accommodate different performance goals in different
sweep wing configurations, a novel switching strategy, mode dependent average dwell time
(MDADT), is adopted to investigate the flight control of a morphing aircraft in its morphing
phase. The switching signal used in this note is more general than the average dwell time
(ADT), in which each mode has its own ADT. Under some simplified assumptions the control
synthesis condition is formulated as a linear matrix optimization problem and a set of mode-
dependent dynamic state feedback controllers are designed. Afterwards the proposed approach
is applied to a morphing aircraft with a variable sweep wing to demonstrate its validity.
Keywords: switching linear parameter-varying system, flight control, morphing aircraft,
mode dependent average dwell time
Classification: 93C95, 93D09
1. INTRODUCTION
With the development of new material and technology, aircraft could improve the flight
performance by morphing wings, in which “morphing” means the aircraft can change
aerodynamic shape to obtain optimal flight performance [2, 3]. One morphing con-
cept is using variable-sweep wing to optimize flight performance. During the morphing
aircraft
0
s wing shape-varying process, the dynamic responses will be governed by time-
varying aerodynamic forces and moments, which will be related to the wing
0
s shape.
Due to the significant wing reconfiguration, aerodynamic parameters, which are varied
dramatically, will make the morphing aircraft be a complicated system with strong non-
linearity and uncertainties. Therefore, analysis and control of morphing aircraft are more
challenging than those for traditional flight vehicles [10, 15]. On the other hand, com-
pared with conventional wing-fixed aircraft, the morphing aircraft has multi-objective
adaptability, wider flight envelop and higher combat effectiveness [4]. Moreover, in the
flight control of a morphing aircraft, different performance goals are often desirable for
different wing configurations. In such a circumstance, it is sometimes difficult to design
a single controller to satisfy different performance in its all configurations. Typically,
the controller is designed by compromising the performance in some wing configuration.
DOI: 10.14736/kyb-2016-6-0967

968 YONG HE, CHUNJUAN LI, WEIGUO ZHANG, JINGPING SHI AND YONGXI L
¨
U
In recent years, the issue of LPV system has been widely investigated due to its
merits of compensating for the shortages of traditional gain-scheduling techniques (see
for example [1, 18, 21, 23, 24] and references therein). LPV control theory, whose state
matrices depend on (measurable) time-varying parameters [8, 16], provides a systematic
gain-scheduling design technique [1] and has been extensively used in the fields ranging
from aerospace to process control industries [1, 22]. In Ref. [17], the conditions that
guarantee stability, robustness and performance properties of the global gain-scheduled
designs are given using a quadratic Lyapunov functions, but the quadratic Lyapunov
function, which is independent of the scheduled parameters, showed its conservatism.
In Ref. [20], a parameter-dependent Lyapunov function method, which leads to a less
conservative result, has been proposed to analyze and synthesize the LPV system. How-
ever, for an LPV system with a large parameter variation range, a single Lyapunov
function, quadratic or parameter-dependent, may not exist, even if it does exist, it is
often necessary to sacrifice the performance in some parameter subregions in order to
obtain a single LPV controller over the entire parameter region. In such a case, the
concept of switching LPV system, by generalizing the switched LTI systems to LPV
ones, is put forward [9, 12, 13, 21]. In Ref. [13], two parameter-dependent switching
logics, hysteresis switching and average dwell time (ADT) switching, are applied to an
F-16 aircraft model with different design objectives and aircraft dynamics in its low and
high angle of attack regions. In Ref. [14], a switching LPV controller is used to regu-
late the air-fuel ratio of an internal combustion engine, and all of them have improved
system performance in certain extent. Meanwhile, for the purpose of improving design
performance and transient responses as switching occurs, a smooth switching strategy
has also been developed and various successful control applications have been reported
[5, 6, 10].
On another research front, switching signal, which is used to distinguish the switched
systems from the other systems, has played a vital role to the system performance [19].
As a type of switching signals, ADT switching logic means that the number of switches
is bounded in a finite interval and the average time between consecutive switching is not
less than a constant [11], which is more general than Dwell Time (DT) switching logic
[7]. However, it has been recognized that the property in the ADT switching is still not
anticipated, since the average time interval between any two consecutive switching is at
least τ
a
, which is independent of the system mode. To release the restrictions of ADT
to the switched control system, a mode-dependent ADT switching strategy is proposed
by providing two mode-dependent parameters to ADT switching strategy [26].
So far there is no result available yet on control of switching LPV systems with
MDADT based on parameter-dependent Lyapunov functions, which will reduce the con-
servatism, enhance flexibility and improve the disturbance attenuation performance in
the analysis and synthesis of a switched LPV system. This motives us for this investiga-
tion. The main contribution of this paper is that a novel notion of parameter-dependent
MDADT switching scheme and a group of parameter-dependent Lyapunov functions
are used to investigate the problem of control of switching LPV systems, and then the
proposed result is applied to a switching LPV representation of the morphing aircraft
to accommodate multiple control objectives in different sweep wing configurations.
This paper is organized as follows. The system description and some preliminaries

Switching LPV control design with MDADT and its application to a morphing aircraft 969
are given in section 2. In section 3 we investigate switching LPV control design problem
under a novel notion of MDADT switching approach and the switching control synthesis
condition will be formulated as matrix optimization problem, whereas in section 4,
the LPV model of a sweepback morphing aircraft is deduced at first, and then the
flight control is designed by the presented method, at last the corresponding simulation
illustrates the effectiveness. Finally the conclusion remarks are drawn in section 5.
The notation in this paper is standard. R stands for the set of real numbers and
R
+
for the nonnegative real numbers. R
m×n
is the set of m × n real matrices. The
transpose of a real matrix M is denoted by M
T
. ker(M) is used to denote the orthogonal
complement of M. S
n×n
is used to denote the real symmetric matrices and if M S
n×n
,
then M > 0(M 0) indicates that M is positive definite (positive semidefinite) and
M < 0(M 0) denotes a negative definite (negative semidefinite) matrix. For x R
n
,
its norm is defined as kxk
2
=
2
x
T
x. The space of square integrable function is denoted
by, that is, for any u(t) l
2
, ku(t)k
2
=
2
p
u
T
(t)u(t) is finite.
2. PRELIMINARIES
An open-loop LPV system to be investigated is described as:
˙x
z
y
=
A
i(ρ)
B
1,i(ρ)
B
2,i(ρ)
C
1,i(ρ)
D
11,i(ρ)
D
12,i(ρ)
C
2,i(ρ)
D
21,i(ρ)
D
22,i(ρ)
x
ω
u
ρ P
i
(1)
where x, ˙x R
n
, z R
n
z
is the controller output, and ω R
n
ω
is the disturbance
input, y R
n
y
is the measurement for control, u R
n
u
is the control input. All of
the statespace data are continuous functions of the parameter ρ. It is assumed that ρ is
in a compact set P R
s
with its parameter variation rate bounded by v
k
˙ρ
k
v
k
,
for k = 1, 2, . . . , s, and the parameter value is measurable in real-time. The following
assumptions are also needed.
A1. (A
(ρ)
,B
2(ρ)
,C
2
(ρ)) triple is parameter-dependent stabilizable and detectable for all
ρ;
A2. The matrix functions [B
T
2
(ρ) D
T
12
(ρ)] and [C
2
(ρ) D
21
(ρ)] have full row ranks for
all ρ;
A3. D
22(ρ)
= 0.
Supposing the parameter set P is covered by a number of closed subsets {P
i
}
iZ
N
by means of a family of switching surfaces S
ij
, where the index set Z
N
= {1, 2, . . . , N },
and
Z
N
i=1
P
i
= P , P
i
P
j
= S
ij
, (i, j) Z
N
×Z
N
,i 6= j. In this paper, we are interested
in the problem of designing a group of LPV controllers in the form of
˙x
k
u
=
A
k,i(ρ, ˙ρ
B
k,i(ρ)
C
k,i(ρ)
D
k,i(ρ)
x
k
y
, i Z
N
(2)
and each of them is suitable for a specific parameter subset P
i
. The state dimension of
each controller is x
k
R
n
k
. The control design requirement at each parameter subregion

970 YONG HE, CHUNJUAN LI, WEIGUO ZHANG, JINGPING SHI AND YONGXI L
¨
U
Pj
Pi
Sij
Fig. 1. Switching regions with dwell time.
could be different and even conflicting for different parameter regions. Each controller,
also a function of the parameter ρ, stabilizes the open-loop system with best achievable
performance in a specific parameter region, and meanwhile maintains the closed-loop
system stability under the given switching strategy.
The switching event occurs when the parameter trajectory hits the switching surfaces,
so it is obvious that the switching event is parameter-dependent. A switching signal σ
is defined as a piecewise constant function. It is assumed that σ is continuous from the
right everywhere, and only limited number of switches occur in any finite time interval.
Then the switching closed-loop LPV system can be described by:
˙x
cl
u
=
A
cl,σ(ρ, ˙ρ
B
cl,σ(ρ)
C
cl,σ(ρ)
D
cl,σ(ρ)
x
cl
ω
, ρ P
i
, i Z
N
(3)
where x
T
cl
= [x
T
x
T
k
] R
n+n
k
. It is straightforward to show that the resulting closed-
loop system is a switched LPV system, which could have discontinuity and multiple
state space gains at switching surfaces due to the use of multiple LPV controllers.
In this paper, the aim is to design a set of switching signal ρ with mode-dependent
average dwell time (MDADT) property based on parameter-dependent Lyapunov func-
tions, such that
C1. When ω = 0,the switched LPV system (3) is parameter-dependent quadratically
stable;
C2. When x
0
= 0 and ω 6= 0.ω l
2
,kzk
2
< γkωk
2
.
For this purpose, the definition of the MDADT switching is given as follow:
Definition 2.1. (Zhao et al. [26]) For a switching signal σ and any 0 t T , let
N
σp
(T, t) be the switching numbers that the p
th
subsystem is activated over the time
interval [t, T ] and T
p
(T, t) denote the total running time of the p
th
subsystem over the
time interval [t, T ], p Z
N
, we say thatσ has a mode-dependent average dwell time τ
ap
if there exist positive numbers N
0p
and τ
ap
such that
N
σp
(T, t) N
0p
+
T
p
(T, t)
τ
ap
, T t 0.
(4)

Citations
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Journal ArticleDOI
01 Dec 2018-Applied Sciences
Abstract: This paper deals with the robust stability of a class of uncertain switched systems with possibly unstable linear subsystems. In particular, conditions for global uniform exponential stability are presented. In addition, a procedure to design a mode dependent average dwell time switching signal that stabilizes a switched linear system composed of diagonalizable subsystems is established, even if all of them are stable/unstable and time-varying (within design bounds). An illustrative example of the stabilizing switching law design and numerical results are presented.

1 citations


References
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01 Sep 1995-Automatica
TL;DR: The methodology presented in this paper is applied to the gain scheduling of a missile autopilot and is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques.
Abstract: This paper is concerned with the design of gain-scheduled controllers with guaranteed H∞ performance for a class of linear parameter-varying (LPV) plants. Here the plant state-space matrices are assumed to depend affinely on a vector θ of time-varying real parameters. Assuming real-time measurement of these parameters, they can be fed to the controller to optimize the performance and robustness of the closed-loop system. The resulting controller is time-varying and automatically ‘gain-scheduled’ along parameter trajectories. Based on the notion of quadratic H∞ performance, solvability conditions are obtained for continuous- and discrete-time systems. In both cases the synthesis problem reduces to solving a system of linear matrix inequalities (LMIs). The main benefit of this approach is to bypass most difficulties associated with more classical schemes such as gain-interpolation or gain-scheduling techniques. The methodology presented in this paper is applied to the gain scheduling of a missile autopilot. The missile has a large operating range and high angles of attack. The difficulty of the problem is reinforced by tight performance requirements as well as the presence of flexible modes that limit the control bandwidth.

1,359 citations


Journal ArticleDOI
Jeff S. Shamma1, Michael Athans2Institutions (2)
Abstract: Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the robustness, performance, or even nominal stability of the overall gain-scheduled design. An analysis is presented for two types of nonlinear gain-scheduled control systems: (1) scheduling on a reference trajectory, and (2) scheduling on the plant output. Conditions which guarantee stability, robustness, and performance properties of the global gain schedule designs are given. These conditions confirm and formalize popular notions regarding gain scheduled designs, such as that the scheduling variable should vary slowly, and capture the plant's nonlinearities. >

754 citations


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Jeff S. Shamma1, Michael Athans2Institutions (2)
01 Apr 1991-Automatica
TL;DR: Conditions are given which guarantee that the stability, robustness, and performance properties of the fixed operating point designs carry over to the global gain scheduled design, such as the scheduling variable should “vary slowly.”
Abstract: Gain scheduling has proven to be a successful design methodology in many engineering applications. However, in the absence of a sound theoretical analysis, these designs come with no guarantees on the robustness, performance, or even nominal stability of the overall gain scheduled design. This paper presents such an analysis for one type of gain scheduled system, namely, a linear parameter-varying plant scheduling on its exogenous parameters. Conditions are given which guarantee that the stability, robustness, and performance properties of the fixed operating point designs carry over to the global gain scheduled design. These conditions confirm and formalize popular notions regarding gain scheduled design, such as the scheduling variable should 'vary slowly'.

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Bei Lu1, Fen Wu1Institutions (1)
01 Nov 2004-Automatica
TL;DR: This paper studies the switching control of linear parameter-varying (LPV) systems using multiple parameter-dependent Lyapunov functions to improve performance and enhance control design flexibility.
Abstract: In this paper we study the switching control of linear parameter-varying (LPV) systems using multiple parameter-dependent Lyapunov functions to improve performance and enhance control design flexibility. A family of LPV controllers is designed, each suitable for a specific parameter subregion. They are switched so that the closed-loop system remains stable and its performance is optimized. Two switching logics, hysteresis switching and switching with average dwell time, are examined. The control synthesis conditions for both switching logics are formulated as matrix optimization problems, which are generally non-convex but can be convexified under some simplifying assumptions. The hysteresis switching LPV control scheme is then applied to an active magnetic bearing problem.

218 citations


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Fen Wu1, Karolos M. Grigoriadis2Institutions (2)
01 Feb 2001-Automatica
TL;DR: The stability and the induced L"2 norm performance of these systems using parameter-dependent Lyapunov functionals are explored and the design of parameter- dependent state-feedback controllers that guarantee desired L" 2 gain performance is examined.
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209 citations