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

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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

}

i∈Z

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)