Adaptive H∞ control in finite frequency domain for uncertain linear systems

Information Sciences 314 (2015) 14–27

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Adaptive H1 control in finite frequency domain for uncertain linear systems Xiao-Jian Li a, Guang-Hong Yang a,b,⇑ a b

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110004, China State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, China

a r t i c l e

i n f o

Article history: Received 20 May 2014 Received in revised form 13 March 2015 Accepted 27 March 2015 Available online 1 April 2015 Keywords: Uncertain systems Adaptive control Finite frequency domain H1 performance

a b s t r a c t This paper is concerned with the problem of adaptive H1 controller design in finite frequency domains for uncertain linear systems. The uncertainties are assumed to be time-invariant, unknown, but bounded, which appear affinely in the matrices of system models. An adaptive mechanism is introduced to construct a novel finite frequency H1 controller with time-varying gains. By using Lyapunov theory and Parseval’s Theorem, the controller design conditions are given in terms of a set of linear matrix inequalities (LMIs). It is shown that the proposed finite frequency controller with time-varying gains can achieve better H1 performance than the traditional ones with fixed gains. Finally, a numerical example of the F-18 aircraft model is given to illustrate the presented theoretical results. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction In many applications, modelling errors and system uncertainties in plant models are inevitable. To achieve satisfactory system performances, robust control problems have received increasing attention during the past decades [7]. An effective tool for robust controller designs is based on the use of LMI techniques [9], which are computationally simple and numerically reliable. Within this framework, a number of controller design approaches have been given in the literature. In [11], a robust controller was designed for the uncertain time-invariant (LTI) retarded system. In [34], the robust H1 controller design for uncertain stochastic nonlinear systems was considered. In [35], the L2  L1 controller design was investigated for uncertain two dimensional systems. In [31], a fuzzy adaptive backstepping design procedure was proposed for a class of uncertain nonlinear systems. In [16], the problem of insensitive tracking control for a class of complex networked systems with controller additive coefficient variations was addressed. In [32], the state observer-based adaptive fuzzy control techniques were developed. Note that the norm bounded uncertainties have been widely studied in the above results. For the polytopic type uncertainties, there also exist various controller design approaches. In [1], the problem of robust H1 dynamic output feedback controller design for uncertain continuous-time linear systems was addressed. The delay-dependent robust H1 filtering for uncertain discrete-time singular systems was studied in [17]. An approach was provided in [4] to design robust dynamic output feedback controller for linear systems with polytopic uncertainties. In [39], the problems of robust stability and stabilization were investigated for a class of continuous-time uncertain systems. In [6], the problem of robust static output feedback control was studied for uncertain continuous-time linear systems. In [29,2], the authors considered the problems of ⇑ Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110004, China. E-mail addresses: [email protected] (X.-J. Li), [email protected] (G.-H. Yang). http://dx.doi.org/10.1016/j.ins.2015.03.059 0020-0255/Ó 2015 Elsevier Inc. All rights reserved.

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

15

robust controller designs of linear discrete-time periodic systems and linear switched systems with polytopic-type uncertainties. In [33], a robust control storage function method was developed for solving the H1 control problem of single-input polytopic nonlinear systems. It should be pointed out that, although the stability and robust H1 performance of the closed-loop systems can be ensured by using the above methods, the designed controllers with fixed gains may lead to conservatism when large parameter uncertainties are encountered. Based on this observation, the controllers and filters with varying gains have been designed in [37,36,23,3,24–26,19] by using adaptive or switching techniques. The comparison results have shown the superiority of the adaptive controllers and filters with varying gains. While adaptive H1 control approaches are more effective in addressing the large parameter uncertainties, the methods in [37,36] are not completely compatible with practical requirements, because they overlook a vital fact that external inputs may belong to known finite frequency ranges, which include low/middle/high frequency (LF/MF/HF) ranges. For instance, the information of incipient faults is always contained within low frequency bands as the fault development is slow [20]; in addition, the frequency ranges of reference inputs generally lie in known finite intervals. For these finite frequency external inputs, the full frequency controller design approaches will be much conservative due to overdesign. Recently, a milestone in the road of investigating finite frequency specifications of LTI systems is the generalized Kalman–Yakubovich–Popov (GKYP) lemma developed in [15,14], where the equivalence between a frequency domain inequality and an LMI over a finite frequency range was established. Based on the generalized KYP lemma, a number of controller and filter design results have been derived, allowing designers to impose different performance requirements over chosen finite frequency ranges. In [38], the low frequency positive real control was considered for delta operator systems, and the generalized KYP lemma was used in [27] to develop new stability tests for differential linear repetitive processes. In [5], the model reduction problem of two-dimensional digital filters over finite frequency ranges was investigated. In [18,30], the finite frequency H1 controllers were designed for LTI systems. In [10,21], the finite frequency H1 filters were designed for linear time-delayed and two dimensional discrete-time systems, respectively. The advantages of finite frequency controllers and filters have been proven in [18,30,38,27,10,5,21]. However, up to now, the design problem of adaptive finite frequency controller has not been solved. A key challenge is that the generalized KYP lemma [15,14] cannot be used for the control systems with adaptive mechanism, which are time-varying in nature. Therefore, it is necessary to develop a new approach to resolve this problem, which motivates the present investigation. This paper is concerned with the problem of adaptive finite frequency controller design for uncertain linear systems. The uncertainties are assumed to be time-invariant, unknown, but bounded, which appear affinely in the matrices of system models. An adaptive mechanism is introduced to construct a robust H1 controller with varying gains. Instead of using generalized KYP lemma, the problem of finite frequency performance analysis is studied via the Parseval’s Theorem [40] and Lyapunov function approach. Towards this direction, the controller design conditions are then given in terms of a set of LMIs. It is shown that the proposed adaptive finite frequency controller with varying gains can achieve better H1 performance than the traditional finite frequency controllers with fixed gains. The rest of the paper is organized as follows: the problem statement and preliminaries are presented in Section 2. The problem of finite frequency performance analysis is summarized in Section 3. The finite frequency controller design conditions are provided in Section 4. In Section 5, the effectiveness and advantages of the proposed method are illustrated by a numerical example of the F-18 aircraft model, and some conclusions are given in Section 6. Notations: Rn denotes the n-dimensional Euclidean space; I represents the identity matrix. In block symmetric matrices or long matrix expressions, we use a star ðÞ to represent a term that is induced by symmetry. A block diagonal matrix with matrices X 1 ; X 2 ; . . . ; X n on its main diagonal is denoted as diagfX 1 ; X 2 ; . . . ; X n g. For a matrix A, its complex conjugate trans . For a symmetric matrix, A > 0ðP 0Þ and A < 0ð6 0Þ denote positive pose is denoted by A ; HeðAÞ ¼: A þ A , HeðAÞ ¼: AþA 2 (semi) definiteness and negative (semi) definiteness. L2 denotes the Hilbert space of square integrable functions with the R 1 1 following norm: kv ðtÞk2 ¼ 0 v  ðtÞv ðtÞdt 2 . 2. Problem statement and preliminaries 2.1. System model Consider the following uncertain linear system

_ xðtÞ ¼ AðhÞxðtÞ þ BðhÞuðtÞ þ Bd ðhÞdðtÞ

ð1Þ

zðtÞ ¼ CðhÞxðtÞ þ DðhÞuðtÞ

where xðtÞ 2 Rn is the state space vector, zðtÞ 2 Rp is the performance output, dðtÞ 2 L2 denotes the external input and the frequency of dðtÞ resides in a known but finite frequency set X, and uðtÞ is the control input. Here

AðhÞ ¼ A1 þ

N N N X X X hi Ai ; BðhÞ ¼ B1 þ hi Bi ; Bd ðhÞ ¼ Bd1 þ hi Bdi ; i¼2

i¼2

N N X X hi C i ; DðhÞ ¼ D1 þ hi Di CðhÞ ¼ C 1 þ i¼2

i¼2

i¼2

ð2Þ

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X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

where A1 ; Ai ; B1 ; Bi ; Bd1 ; Bdi ; C 1 ; C i ; D1 and Di are known constant matrices of appropriate dimensions. hi are unknown time-invariant uncertainties, which satisfy

hi 6 hi 6 hi

ð3Þ

hi are known lower and upper bounds of hi , respectively. and hi ;  Remark 1. The system (1) includes a general polytopic system as special case. For example, considering the following polytopic system:

_ xðtÞ ¼ Ap ðhÞxðtÞ þ Bp ðhÞuðtÞ þ Bpd ðhÞdðtÞ

ð4Þ

zðtÞ ¼ C p ðhÞxðtÞ þ Dp ðhÞuðtÞ where

Ap ðhÞ ¼

N N N X X X hi Api ; Bp ðhÞ ¼ hi Bpi ; Bpd ðhÞ ¼ hi Bpdi i¼1

i¼1

i¼1

N N N X X X C p ðhÞ ¼ hi C pi ; Dp ðhÞ ¼ hi Dpi ; hi ¼ 1 i¼1

i¼1

ð5Þ

i¼1

and Api ; Bpi ; Bpdi ; C pi and Dpi are known constant matrices of appropriate dimensions. Let

Ap1 ¼ A1 ; Bp1 ¼ B1 ; Bpd1 ¼ Bd1 ; C p1 ¼ C 1 ; Dp1 ¼ D1 Api  Ap1 ¼ Ai ; Bpi  Bp1 ¼ Bi ; Bpdi  Bpd1 ¼ Bdi ; C pi  C p1 ¼ C i ; Dpi  Dp1 ¼ Di ; i ¼ 2; 3 . . . ; N hi ¼ 1. then the system (4) is in the same of system (1) with hi ¼ 0 and  2.2. Closed-loop system In this paper, the following state feedback controller with adaptive mechanism is designed:

uðtÞ ¼

K1 þ

! N X ^hi ðtÞK i xðtÞ

ð6Þ

i¼2

hi ðtÞði ¼ 2; . . . ; NÞ are the estimate signals of hi , where K 1 ; K i ði ¼ 2; . . . ; NÞ are controller parameter matrices to be designed, ^ and determined according to some adaptive laws to be defined later. Then the closed-loop system is in the following form:

_ xðtÞ ¼ ðAðhÞ þ BðhÞKð^hðtÞÞÞxðtÞ þ Bd ðhÞdðtÞzðtÞ ¼ ðCðhÞ þ DðhÞKð^hðtÞÞÞxðtÞ

ð7Þ

P hðtÞÞ ¼ K 1 þ Ni¼2 ^ hi ðtÞK i . where Kð^ 3. Finite frequency performance analysis 3.1. Finite frequency adaptive H1 performance Let us now recall the existing notion of adaptive H1 performance defined in full frequency domain. Definition 1 ([37,36]). Let c > 0 be a given constant, then the system (7) is said to be with an adaptive H1 performance ^i ðtÞði ¼ 2; . . . NÞ such that the following inequality holds: bound no larger than c, if for any e > 0, there exist h

kzðtÞk2 6 ckdðtÞk2 þ e

ð8Þ

Remark 2. The quantitative relation between the adaptive H1 performance and standard H1 performance has been discussed in [37,36], and the scalar e is introduced due to the existence of adaptive mechanism. ^ xÞ, viewed as a function of frequency x 2 R, is square integrable in the sense that Now suppose that dð

^ xÞk2 ¼ 1 kdð 2 2p

Z

þ1

1

^ ðxÞdð ^ xÞdx < 1 d

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

17

^ xÞ is the Fourier transform of external input signal dðtÞ in that dð ^ xÞ ¼ R þ1 dðtÞejxt dt Using Parseval’s theorem [40], dð 0 and we have that

^ xÞk ¼ kdðtÞk kdð 2 2

ð9Þ

In view of (9), the full frequency adaptive H1 performance defined in (8) is equivalent to

^ xÞk þ e; x 2 ð1; þ1Þ k^zðxÞk2 6 ckdð 2

ð10Þ

^ xÞ with kdð ^ xÞk < 1. This observation will be exploited to define the following finite frequency adaptive H1 for all inputs dð 2 performance. Definition 2. Let c > 0 be a given constant, then the system (7) is said to be with a finite frequency adaptive H1 performance ^i ðtÞði ¼ 2; . . . NÞ such that the following inequality holds: bound no larger than c, if for any e > 0, there exist h

^ xÞk þ e; x 2 X k^zðxÞk2 6 ckdð 2

ð11Þ

where X is defined in Table 1, and -l ; -1 ; -2 ; -h are known scalars. Based on the Definition 2, the problem considered in this paper is formulated as follows: Problem. Given the system (1), design a controller in the form of (6) such that for the closed-loop system (7), the following specifications are satisfied: (1) the closed-loop system (7) is asymptotically stable with dðtÞ ¼ 0 and (2) the closed-loop system (7) has a finite frequency adaptive H1 performance bound no larger than c. 3.2. Finite frequency performance analysis Before presenting the main result of this section, denote

"

K¼

AðhÞ Bd ðhÞ I

0

#

"

;  ¼

CðhÞ 0 0

#

I



; P¼

I

0 2

0 c I

 ; ð21Þ

AðhÞ ¼ AðhÞ þ BðhÞKð^hðtÞÞ; CðhÞ ¼ CðhÞ þ DðhÞKð^hðtÞÞ;   ~hi ¼ ^hi  hi ; n ðtÞ ¼ x ðtÞ d ðtÞ  A new result of the finite frequency performance analysis is derived in the following lemma.

Lemma 1. Let c > 0 be a given constant. Then for the closed-loop system (7), the following finite frequency adaptive H1 performance

^ xÞk þ k^zðxÞk2 6 ckdð 2

N ~2 X h ð0Þ i

i¼2

li

; x2X

ð22Þ

is satisfied if there exist positive definite matrices P and Q such that

n ðtÞðK NK þ   P ÞnðtÞ þ

_ N X 2~hi ðtÞ~hi ðtÞ i¼2

li

<0

ð23Þ

holds, where li ði ¼ 2; 3 . . . ; NÞ are given positive constants, and  Q P P -2l Q ,   Q P , (ii) for high frequency range x P -h : N ¼ 2 P -h Q (iii) for middle frequency range -1 6 x 6 -2 : (i) for low frequency range x 6 -l : N ¼



Table 1 X for different frequency ranges. LF

MF

HF

X

x 6 -l

-1 6 x 6 -2

x P -h

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X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27



N¼

Q

P þ j-0 Q

P  j-0 Q

-1 -2 Q

 ;

-1 þ -2

-0 ¼

2

:

Proof. Let us first consider the middle frequency case. In fact, n ðtÞðK NK þ   P ÞnðtÞ is equivalent to

_ þ j-0 x_  ðtÞQxðtÞ  j-0 x ðtÞQ xðtÞ _  -1 -2 x ðtÞQxðtÞ þ z ðtÞzðtÞ  c2 d ðtÞdðtÞ 6 0 2x_  ðtÞPxðtÞ  x_  ðtÞQ xðtÞ

ð24Þ

Note that for any vectors / and u, the following equality holds:

/ Q u ¼ trðu/ Q Þ

ð25Þ

Based on (24) and (25), (23) can be rewritten as 

 _ x_  ðtÞQ  j-0 xðtÞx_  ðtÞQ þ j-0 xðtÞx _ 2x_  ðtÞPxðtÞ þ z ðtÞzðtÞ  c2 d ðtÞdðtÞ  trðxðtÞ ðtÞQ þ -1 -2 xðtÞx ðtÞQ Þ

þ

_ N X 2~hi ðtÞ~hi ðtÞ li

i¼2





_ _ ¼ 2x_  ðtÞPxðtÞ þ z ðtÞzðtÞ  c2 d ðtÞdðtÞ  tr½Heðð-1 xðtÞ þ jxðtÞÞð -2 xðtÞ þ jxðtÞÞ ÞQ  þ

_ N X 2~hi ðtÞ~hi ðtÞ i¼2

li

<0

ð26Þ

Under zero initial condition, integrating (26) from 0 to 1, we have

Z

_ N X 2~hi ðtÞ~hi ðtÞ

1

2x_  ðtÞPxðtÞ þ

0

li

i¼2

! dt þ

Z

1

z ðtÞzðtÞdt < c2

0

Z

1



d ðtÞdðtÞdt þ tr½HeðSÞQ ;

ð27Þ

0

where

S¼

Z

1



_ _ ð-1 xðtÞ þ jxðtÞÞð -2 xðtÞ þ jxðtÞÞ dt:

ð28Þ

0

By using Parseval’s theorem [40], we have

S¼

1 2p

Z

1

ð-1  xÞð-2  xÞ^xðxÞ^x ðxÞdx;

1

where ^ xðxÞ is the Fourier transform of xðtÞ. Note that S is Hermitian, hence (27) is equivalent to

Z

_ N X 2~hi ðtÞ~hi ðtÞ

1

2x_  ðtÞPxðtÞ þ

0

li

i¼2

For middle frequency range

! dt þ

Z

1

z ðtÞzðtÞdt < c2

0

Z

1



d ðtÞdðtÞdt þ trðSQÞ;

ð29Þ

0

-1 6 x 6 -2 , we have

ð-1  xÞð-2  xÞ 6 0 which follows that S 6 0. Consequently, the following inequality can be derived:

Z

_ N X 2~hi ðtÞ~hi ðtÞ

1

_

2x ðtÞPxðtÞ þ

0

li

i¼2

! dt þ

Z

1

z ðtÞzðtÞdt < c2

0

Z

1



d ðtÞdðtÞdt; for

-1 6 x 6 -2 :

ð30Þ

0

Now define the following Lyapunov function,

VðtÞ ¼ x ðtÞPxðtÞ þ

N ~2 X h ðtÞ i

i¼2

ð31Þ

li

Then (30) can be rewritten as

Vð1Þ  Vð0Þ þ

Z 0

1

z ðtÞzðtÞdt < c2

Z

1



d ðtÞdðtÞdt; for

-1 6 x 6 -2

ð32Þ

0

Combining (9) and (31), it is known that (32) yields (22) with X ¼ fx j -1 6 x 6 -2 g. Similar to the middle frequency case, the proof of Lemma 1 can be completed for low frequency and high frequency ranges, which is omitted here for brevity. h

19

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

Remark 3. Note that if xð1Þ ¼ 0, then the matrix P is not required to be positive definite in Lemma 1. In this case, when the adaptive laws ^ hi ðtÞ are not considered in (6), the condition (23) will reduced to

n ðtÞðK NK þ   P ÞnðtÞ < 0

ð33Þ

According to Theorem 4 of [14], (33) is a necessary and sufficient condition for guaranteeing the finite frequency H1 performance

^ xÞk ; x 2 X k^zðxÞk2 6 ckdð 2

ð34Þ

4. Control design conditions In this section, the controller design conditions will be given based on the result of Lemma 1. In fact, the design of adaptive state feedback controller is to determine the feedback gains K 1 ; K i and adaptive laws such that the closed-loop system (7) is asymptotically stable and with a finite frequency adaptive H1 performance bound no larger than c. In this paper, the adaptive laws are defined by

_ h^i ðtÞ ¼ Proj½hi ;hi  fLi g; i ¼ 2; 3; . . . N ¼

(

if ^hi ¼ hi and Li 6 0 or ^hi ¼ hi and Li P 0

0;

ð35Þ

Li ; otherwise

where Li ¼ li ðx ðtÞPBi K 1 xðtÞÞ, and li ði ¼ 2; 3 . . . NÞ are constants which can be chosen according to practical applications. hi ðtÞ to the intervals ½hi ;  hi . Some properties Projfg denotes the projection operator, whose role is to project the estimations ^ of the projection type operator are given in [12]. 4.1. Controller design in middle frequency domain The following theorem presents a sufficient condition for the solvability of the considered adaptive controller design problem in middle frequency domain in terms of LMI and adaptive laws. Before giving the main result, we introduce the notations:

b i ¼ f^hi ðtÞj^hi ðtÞ ¼ hi or ^hi ðtÞ ¼ hi g D

Di ¼ fhi jhi ¼ hi or hi ¼ hi g;

Theorem 1. For a given constant c > 0, consider the closed-loop system (7), if there exist positive definite matrices Q > 0; Y > 0 b i , the following linear matrix inequality holds and general matrices L1 ; Li ði ¼ 2; 3 . . . ; NÞ such that for all hi 2 Di ; ^ hi ðtÞ 2 D

2

Q 6  6 6 4 

jxQ  Y

0

a22

a23



c2 I







0

3

a24 7 7 7<0 0 5

ð36Þ

I

with

a22 ¼ He 2 A1 þ

! ! ! ! ! N N N N X X X X h^i Li þ B1 þ hi Ai Y þ 2 B1 þ hi Bi ð^hi þ hi ÞBi L1  -1 -2 Q i¼2

a23 ¼ 2 Bd1 þ

N X

i¼2

!

a24 ¼

i¼2

hi Bdi

i¼2

YC 1

i¼2

N N X X ^hi L þ hi YC i þ L1 þ i i¼2

! D1

N X þ hi Di

i¼2

!

i¼2

hi ðtÞ determined according to the adaptive laws (35) render the closedthen the controller parameters K 1 ¼ L1 Y ; K i ¼ Li Y and ^ loop system (7) satisfying adaptive H1 performance (22) in middle frequency domain X ¼ fx j -1 6 x 6 -2 g. 1

1

Proof. Based on Lemma 1, we just need to show that the LMI (36) and the adaptive laws in (35) can ensure the inequality (23). Note that

BðhÞKð^hÞ ¼

B1 þ

N X hi Bi i¼2

! K1 þ

N X ^hi K i i¼2

! ¼

B1 þ

! ! N N N N X X X X ^hi K i þ B1 þ ^hi Bi K 1 þ hi Bi ðhi  ^hi ÞBi K 1 i¼2

i¼2

i¼2

i¼2

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X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

Then we have

2x ðtÞPBðhÞKð^hÞxðtÞ ¼ 2x ðtÞP

B1 þ

! ! ! N N N N X X X X ^hi K i þ B1 þ ^hi Bi K 1 xðtÞ þ hi Bi ðhi  ^hi Þ2x ðtÞPBi K 1 xðtÞ i¼2

i¼2

i¼2

ð37Þ

i¼2

From (35), we know that

_ N N X X 2~hi ðtÞ~hi ðtÞ ðhi  ^hi Þ2x ðtÞPBi K 1 xðtÞ þ 60 li i¼2 i¼2

ð38Þ

_ _ _ hi ðtÞ we have hi ¼ hi and Li 6 0 or ^ hi ¼  hi and Li P 0, it follows ^ hi Þ2x ðtÞPBi K 1 xðtÞ 6 0. Then by ^ If ^ hi ðtÞ ¼ 0 and ðhi  ^ hi ðtÞ ¼ ~

_ N X 2~hi ðtÞ~hi ðtÞ li

i¼2

N X ¼ 0 6  ðhi  ^hi Þ2x ðtÞPBi K 1 xðtÞ

ð39Þ

i¼2

_ hi ðtÞ ¼ Li . Therefore, it is easy to see hi ðtÞ is in other cases, from (35) it follows ~ If ^ N X 2Li ðhi  ^hi Þ

li

i¼2

þ

_ N X 2~hi ðtÞ~hi ðtÞ li

i¼2

¼0

ð40Þ

Combining (39) and (40), it follows that (38) holds. hi ðtÞ are determined by adaptive laws (35), then the inequality (23) can be rewritten as Based on (37) and (38), if ^

n ðtÞðK NQ K þ WÞnðtÞ < 0

ð41Þ

where



NQ ¼ "

W¼

Q

j- 0 Q

j-0 Q

-1 -2 Q

HeðPðAðhÞ þ BÞÞ þ CðhÞ CðhÞ PBd ðhÞ

B1 þ

B¼



!

N X

hi Bi

c2 I



! ^hi Bi K 1

N X

N X

i¼2

i¼2

^hi K i þ B1 þ

i¼2

#

Next, we proof that the inequality (36) is a sufficient condition for

K N Q K þ W < 0

ð42Þ

Let

  ¼

NQ

0

0

W

 ;

then (42) can be reformulated as

ð43Þ

which has the same form of the inequality (55) with

ð44Þ

and b22 ¼ -1 -2 Q þ HeðPðAðhÞ þ BÞÞ þ CðhÞ CðhÞ. On the other hand,

2

I

0

3 2

I

0

3

 Q 6 7 6 7 4 0 0 5 H4 0 0 5 ¼ 0 0 I 0 I

0 c2 I



<0

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

21

which has the same form of inequality (56). The following null space bases calculations yields

2

3?

I

6 7 4 AðhÞ 5 ¼  Bd ðhÞ

"

AðhÞ

I

Bd ðhÞ

0

2 3 #  0  I 0 0 ? 6 7 ; ¼4I5 0 0 I I 0

0

From Lemma 2 in Appendix A, the following inequality is a sufficient condition for (43)

2 6

I

3

2

7

6

3

I

7

H þ 4 AðhÞ 5P½ 0 I 0  þ ½ 0 I 0  P 4 AðhÞ 5 < 0 Bd ðhÞ



Bd ðhÞ

ð45Þ



which is equivalent to

2

Q 6  6 6 4 

jxQ  P

0

2PBd ðhÞ CðhÞ 7 7 7<0 c2 I 0 5 

c22 



3

0





ð46Þ

I

with c22 ¼ -1 -2 Q þ HeðPðAðhÞ þ B þ AðhÞÞÞ. Define J ¼ diagfP1 ; P1 ; I; Ig. We perform a congruence transformation to (46) by the full rank matrices J and J T on the left and right, and further let Q ¼ P 1 QP1 ; Y ¼ P1 ; L1 ¼ K 1 Y; Li ¼ K i Y, then the inequality (46) is equivalent to (36). h hi ðtÞ. Moreover, ^ hi ðtÞ and hi are independent. Thus from Remark 4. Noted that the condition in (36) is linearly dependent on ^ b i is equivalent to that (36) holds for all hi 2 ½hi ; hi  and ^ b i . Then ^i ðtÞ 2 D hi ðtÞ 2 D the Lemma 3.2 in [8], (36) holds for all hi 2 Di ; h ^i ðtÞ by using LMI control toolbox [9]. the inequality (36) can be solved in the vertices of the uncertain parameters hi and h Remark 5. From Remark 3, if both uncertainties and adaptive laws are not considered in this paper, a necessary and sufficient state-feedback controller synthesis condition can be derived for the system (1) based on (33) and (42)–(46). Also, the same conclusion has been drawn in the Corollary 2 of [13]. 4.2. Controller design in low frequency and high frequency domains The previous subsection presents the result of controller design in middle frequency domain. Similar to Theorem 1, the results of controller design in low and high frequency domains can be derived. See the following two theorems. Theorem 2. For a given constant c > 0, consider the closed-loop system (7), if there exist positive definite matrices Q > 0; Y > 0 b i , the following linear matrix inequality holds ^i ðtÞ 2 D and general matrices L1 ; Li ði ¼ 2; 3 . . . ; NÞ such that for all hi 2 Di ; h

2

Q

6  6 6 4  

Y

0

a22

a23 2



c I





3

0

a24 7 7 7<0 0 5

ð47Þ

I

with

a22

! ! ! ! ! N N N N X X X X ^ ^ hi Li þ B1 þ ¼ He 2 A1 þ hi Ai Y þ 2 B1 þ hi Bi ðhi þ hi ÞBi L1 þ -2l Q i¼2

a23 ¼ 2 Bd1 þ

N X

i¼2

!

a24 ¼

N N X X ^hi L þ hi YC i þ L1 þ i i¼2

i¼2

hi Bdi

i¼2

YC 1

i¼2

! D1

N X þ hi Di

i¼2

!

i¼2

hi ðtÞ determined according to the adaptive laws (35) render the closedthen the controller parameters K 1 ¼ L1 Y ; K i ¼ Li Y and ^ loop system (7) satisfying adaptive H1 performance (22) in low frequency domain X ¼ fx jj x j6 -l g. 1

1

Proof. The main idea for proving Theorem 2 is the same as that of Theorem 1, and it is omitted for brevity.

22

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

Theorem 3. For a given constant c > 0, consider the closed-loop system (7), if there exist positive definite matrices Q > 0; Y > 0 b i , the following linear matrix inequality holds ^i ðtÞ 2 D and general matrices L1 ; Li ði ¼ 2; 3 . . . ; NÞ such that for all hi 2 Di ; h

2 6 6 6 6 4

PN

Q þ HeðYÞ a12

Bd1 þ



a22

Bd1 þ





c2 I







with

a12 ¼

A1 þ

i¼2 hi Bdi PN i¼2 hi Bdi

7 a24 7 7<0 7 0 5 I

ð48Þ

i¼2

i¼2

! ! ! ! ! N N N N X X X X ^hi Li þ B1 þ ^hi Bi L1  -2 Q A1 þ hi Ai Y þ B1 þ hi B i h i¼2

a24

3

! ! ! N N N X X X ^hi Li hi Ai Y þ B1 þ hi B i L1 þ i¼2

a22 ¼ He

0

i¼2

N N X X ^hi L ¼ YC 1 þ hi YC i þ L1 þ i i¼2

i¼2

i¼2

!

D1 þ

N X

!

i¼2

hi Di

i¼2

hi ðtÞ determined according to the adaptive laws (35) render the closedthen the controller parameters K 1 ¼ L1 Y 1 ; K i ¼ Li Y 1 and ^ loop system (7) satisfying adaptive H1 performance (22) in high frequency domain X ¼ fx jj x jP -h g. Proof. The result of Theorem 3 is easily verified, and it is omitted here. h 4.3. Stability condition Note that the controller design conditions given in previous subsections cannot ensure a stable closed-loop system, so we wish to add an additional constraint to guarantee the stability of the closed-loop system (7). Theorem 4. The closed-loop system (7) is asymptotically stable with dðtÞ ¼ 0 if there exist positive definite matrices Y > 0; L1 ; Li such that the following LMI holds:

He

! ! !! N N N X X X ^ L1 þ <0 hi Li A1 þ hi Ai Y þ B1 þ hi B i i¼2

i¼2

ð49Þ

i¼2

Proof. Note that (49) is equivalent to PAðhÞ þ AðhÞ P < 0, where P ¼ Y 1 ; L1 ¼ K 1 Y; Li ¼ K i Y and AðhÞ is the system matrix defined in (21). Therefore, the condition (49) guarantees the asymptotical stability of the closed-loop system (7) with dðtÞ ¼ 0. h 4.4. Algorithm Based on Theorems 1–4, the following algorithm is provided to optimize the H1 performance bound c in different frequency domains. Algorithm 1. Let c denote the H1 performance bound, a feasible solution to the adaptive state feedback controller (6) is derived by solving the following optimization problems:

In middle frequency domain : min c s:t: ð36Þ and ð49Þ; for b i ; i ¼ 2; 3; . . . ; N: hi 2 Di ; ^hi ðtÞ 2 D

ð50Þ

In low frequency doamin : min c s:t: ð47Þ and ð49Þ; for b i ; i ¼ 2; 3; . . . ; N: hi 2 Di ; ^hi ðtÞ 2 D

ð51Þ

In high frequency doamin : min c s:t: ð48Þ and ð49Þ; for b i ; i ¼ 2; 3; . . . ; N: hi 2 Di ; ^hi ðtÞ 2 D

ð52Þ

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

23

After solving these optimization problems, the controller parameters can be constructed as K 1 ¼ L1 Y 1 ; K i ¼ Li Y

1

; i ¼ 2; 3; . . . N.

5. Example In this section, an example of H1 control for an F-18 aircraft is given to illustrate the proposed design method. The longitudinal dynamical equation of motion of the F-18 aircraft is described in [22]. The longitudinal-axis motion is in the following form:

_ xðtÞ ¼ Along xðtÞ þ Blong uðtÞ þ Bd dðtÞ

ð53Þ

where

2

1:4 þ 0:2h1

1 þ 0:1h1

1:6 0

0:78 0

0 0

0 0

6 6 Along ¼ 6 4 2

6 0 6 Blong ¼ 6 4 8:2 þ 0:5h1

3

10 8:2

4:7 0

7 7 7 5

0

7:3

3

ð54Þ

7 7 7 5

0 0

0

0:26 0:53

7:3 þ 0:3h1

xðtÞ ¼ ½ aðtÞ qv ðtÞ ds ðtÞ dF ðtÞ T ; uðtÞ ¼ ½ us ðtÞ uF ðtÞ T . The states are the angle of attack a, the pitch rate qv , the horizontal stabilizers ds , and the flaperon deflections dF , and control surfaces are actuated by hydraulic actuators. The flight parameters are given for Mach number 0.4 and 20,000-ft altitude. The uncertain parameter satisfies

0 6 h1 6 10 In addition, it is assumed that the distribution matrix of external input is

Bd ¼ ½ 0 0 1 0 T and performance output zðtÞ ¼ CxðtÞ þ DuðtÞ with

C ¼ ½ 1 1 0 0 ;

D ¼ ½1 2

According to the above system matrices, one can rewrite the system (53) as an uncertain one that has the same form of system (1). In this example, two different finite frequency external inputs are considered. Case 1. 2 6 x 6 3, that is, the frequency range of dðtÞ belongs to middle frequency domain. By solving (50) in Algorithm 1, the controller parameters are derived as follows:



139:9273



70:4636 21:2585 21:3707 2:0179  13:3847 4:0550 4:2290 0:5295

K1 ¼ K2 ¼

43:5171

6:6924 2:0275

42:7415

2:1145



4:0358

0:2647

and the optimal adaptive H1 performance bound is c ¼ 1:3631  105 . On the other hand, the finite frequency controller design with fixed gains has been considered in [30] via the generalized KYP lemma. By using the design conditions (14) and (15) in [30] and solving four LMIs in the vertices of the uncertain parameter h1 , the controller parameter can also be derived as follows:

 K¼

34:4898

20:5475

17:7449 9:7738

87:2228 28:5181 43:6114



14:2590

and the optimal adaptive H1 performance bound is c ¼ 2:7860  105 . The comparison result indicates the superiority of our adaptive method. h1 ð0Þ ¼ 0:2. The external input dðtÞ is In the following simulation results, the initial conditions are set to be xð0Þ ¼ 0; ^ assumed to be dðtÞ ¼ 10 sinð4ptÞ for 0 6 t 6 15. Just as mentioned in Definition 1 and Remark 1 of [37], the adaptive H1 perP ~h2 ð0Þ formance index is closed to the traditional H1 performance index when we choose li relatively large to make Ni¼2 i li sufficiently small, and thus in Case 1, we set l1 ¼ 100. The performance outputs derived by using the proposed method and the one in [30] are plotted in Fig. 1 with h ¼ 5, from which it can be seen that the adaptive finite frequency control approach can

24

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

5

x 10

−5

performance output in this paper performance output in [30]

4 3 2

z (t)

1 0 −1 −2 −3 −4

0

5

10

15

20

25

30

t (sec) Fig. 1. Performance output zðtÞ derived by using different methods.

2

control input u (t) in this paper 1

1 0 −1 −2

0

5

10

15

20

25

30

t (sec) 2

control input u (t) in [30] 1

1 0 −1 −2

0

5

10

15

20

25

30

t (sec) Fig. 2. Control input u1 ðtÞ derived by using different methods.

achieve a better H1 performance. That is, the finite frequency controller with varying gains has more restraint external input ability than the existing finite frequency controllers with fixed gains. Also, the control inputs are plotted in Figs. 2 and 3. We note that the actuator forces in this paper are no larger than the ones in [30], which further increases the feasibility of the proposed method. Case 2. 1 6 x 6 1, that is, the external input dðtÞ belongs to low frequency range. As faults always lie in low frequency ranges, therefore, dðtÞ in Case 2 can be viewed as a fault input signal. By solving (51) in Algorithm 1, the controller parameters are derived as follows:

K1 ¼ K2 ¼



118:9043



59:9521 17:2429 17:6511 0:8261  10:5873 3:0499 3:2824 0:2850

35:4858

5:2937 1:5250

35:3021

1:6412



1:6522

0:1425

and the optimal adaptive H1 performance bound is c ¼ 1:3583  105 . By using the controller design method in [30], the controller parameter is given in the following

K¼



25:8727

15:6822

13:4363 7:3411

67:7193 22:2373 33:8596

11:1186



25

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27 1

control input u2(t) in this paper

0.5 0 −0.5

0

5

10

15

20

25

30

t (sec) 1

control input u2(t) in [30]

0.5 0 −0.5 −1

0

5

10

15

20

25

30

t (sec) Fig. 3. Control input u2 ðtÞ derived by using different methods.

1.5

x 10

−4

performance output in this paper performance output in [30]

1 0.5

z (t)

0 −0.5 −1 −1.5 −2

0

10

20

30

40

50

60

70

80

t (sec) Fig. 4. Performance output zðtÞ derived by using different methods.

2

control input u (t) in this paper 1

1 0 −1 −2

0

10

20

30

40

50

60

70

80

t (sec) 2

control input u (t) in [30] 1

1 0 −1 −2

0

10

20

30

40

50

60

70

t (sec) Fig. 5. Control input u1 ðtÞ derived by using different methods.

80

26

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27 1

control input u (t) in this paper 2

0.5 0 −0.5 −1

0

10

20

30

40

50

60

70

80

t (sec) 2

control input u (t) in [30] 2

1 0 −1

0

10

20

30

40

50

60

70

80

t (sec) Fig. 6. Control input u2 ðtÞ derived by using different methods.

and the optimal adaptive H1 performance bound is c ¼ 2:7949  105 . Assume that the initial conditions are the same as those in Case 1. The external input dðtÞ is assumed to be dðtÞ ¼ 10sinð0:2ptÞ for 10 6 t 6 50. The performance output derived by using the methods in this paper and in [30] are plotted in Fig. 4 with h ¼ 5 and l1 ¼ 100. The control inputs are plotted in Figs. 5 and 6. Similar to the Case 1, the comparison results also illustrate the merits of the adaptive finite frequency controller.

6. Conclusion In this paper, the adaptive finite frequency controller with varying gains has been designed for uncertain linear systems. The controller design conditions have been derived in terms of a set of LMIs, which guarantee the asymptotic stability and finite frequency H1 performance of the closed-loop systems. Compared with the existing finite frequency controller designs with fixed gains, the effectiveness and advantages of the proposed method have been illustrated by a numerical example of an F-18 aircraft model with different finite frequency inputs. Especially, we believe that the presented finite frequency controller design approach may be applied to other types of systems such as T–S fuzzy systems and linear time-delay systems, and produce better results than the existing methodologies in the literature.

Acknowledgement This work was supported in part by the Funds of National Science of China (Grant Nos. 61273148, 61420106016, 61403070), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201157), the Fundamental Research Funds for the Central Universities (Nos. N130405012, N140402002), China Postdoctoral Science Foundation (No. 2013M541241), and Postdoctoral Science Foundation of Northeastern University. The Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2013ZCX01). Appendix A Lemma 2 (Projection Lemma). [28] Let U; V; H be given. There exists a matrix F satisfying

U  FV þ V  FU þ H < 0 if and only if the following two conditions hold

NU HNU < 0

ð55Þ

NV HNV < 0

ð56Þ

where N U and N V are arbitrary matrices whose columns form a basis of the nullspaces of U and V, respectively.

X.-J. Li, G.-H. Yang / Information Sciences 314 (2015) 14–27

27

References [1] C.M. Agulhari, R. Oliveira, P. Peres, LMI relaxations for reduced-order robust H1 control of continuous-time uncertain linear systems, IEEE Trans. Autom. Control 57 (6) (2012) 1532–1537. [2] L.I. Allerhand, U. Shaked, Robust stability and stabilization of linear switched systems with dwell time, IEEE Trans. Autom. Control 56 (2) (2011) 381– 386. [3] B. Chen, X.P. Liu, K.F. Liu, P. Shi, C. Lin, Direct adaptive fuzzy control for nonlinear systems with time-varying delays, Inf. Sci. 180 (5) (2010) 776–792. [4] J. Daafouz, J. Bernussou, J.C. Geromel, On inexact LPV control design of continuous-time polytopic systems, IEEE Trans. Autom. Control 53 (7) (2008) 1674–1678. [5] D.W. Ding, X. Du, X.L. Li, Finite-frequency model reduction of two-dimensional digital filters, in: IEEE Transactions on Automatic Control, DOI:0.1109/ TAC.2014.2359305. [6] J.X. Dong, G.H. Yang, Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties, Automatica 49 (6) (2013) 1821–1829. [7] J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard H2 and H1 control problems, IEEE Trans. Autom. Control 34 (8) (1989) 831–847. [8] P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Trans. Autom. Control 41 (3) (1997) 436–442. [9] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI Control Toolbox Users Guide, Mathworks, Inc. P. Gahinet, Natick, 1995. [10] H.J. Gao, X.W. Li, H1 filtering for discrete-time state-delayed systems with finite frequency specifications, IEEE Trans. Autom. Control 56 (12) (2011) 2935–2941. [11] E. Gershon, Robust reduced-order H1 output-feedback control of retarded stochastic linear systems, IEEE Trans. Autom. Control 58 (11) (2013) 2898– 2904. [12] P.A. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall, Englewood Cliffs, NJ, 1996. [13] T. Iwasaki, S. Hara, Robust control synthesis with general frequency domain specifications: static gain feedback case, in: Proceedings of the 2004 American Control Conference, 2004, pp. 4613–4618. [14] T. Iwasaki, S. Hara, Generalized KYP lemma: unified frequency domain inequalities with design applications, IEEE Trans. Autom. Control 50 (1) (2005) 41–59. [15] T. Iwasaki, S. Hara, H. Yamauchi, Dynamical system design from a control perspective: finite frequency positive-realness approach, IEEE Trans. Autom. Control 48 (8) (2003) 1337–1354. [16] X.Z. Jin, J.H. Park, Adaptive sliding-mode insensitive control of a class of non-ideal complex networked systems, Inf. Sci. 274 (2014) 273–285. [17] J.H. Kim, Delay-dependent robust H1 filtering for uncertain discrete-time singular systems with interval time-varying delay, Automatica 46 (3) (2010) 591–597. [18] X.J. Li, G.H. Yang, Dynamic observer-based robust control and fault detection for linear systems, IET Control Theor. Appl. 6 (17) (2012) 2057–2666. [19] X.J. Li, G.H. Yang, Switching-type H1 filter design for T–S fuzzy systems with unknown or partially unknown membership functions, IEEE Trans. Fuzzy Syst. 21 (2) (2013) 385–392. [20] X.J. Li, G.H. Yang, Fault detection in finite frequency domain for Takagi-Sugeno fuzzy systems with sensor faults, IEEE Trans. Cybern. 44 (8) (2014) 1446–1458. [21] X.W. Li, H.J. Gao, Robust finite frequency H1 filtering for uncertain 2-D systems: the FM model case, Automatica 49 (8) (2013) 2446–2452. [22] S.E. Lyshevski, Aircraft flight control system design under state and control bounds, IEEE Trans. Aerosp. Electron. Syst. 34 (1) (1998) 257–263. [23] M.S. Mahmoud, H.Y. Hajeer, A globally convergent adaptive controller for robot manipulators, IEEE Trans. Autom. Control 39 (1) (1994) 148–151. [24] M.S. Mahmoud, A.Y. Al-Rayyah, Adaptive control of systems with mismatched nonlinearities and time-varying delays using state-measurements, IET Control Theor. Appl. 4 (1) (2010) 27–36. [25] M.S. Mahmoud, P. Shi, A.W.A. Saif, Stabilization of linear switched delay systems: H2 and H1 methods, J. Optim. Theory Appl. 142 (3) (2009) 583–601. [26] M.S. Mahmoud, P. Shi, Asynchronous H1 filtering of discrete-time switched systems, J. Signal Proces. 92 (10) (2012) 2356–2364. [27] W. Paszke, E. Rogers, K. Gafkowski, KYP lemma based stability and control law design for differential linear repetitive processes with applications, Syst. Control Lett. 62 (7) (2013) 560–566. [28] R.E. Skelton, T. Iwasaki, K.M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, 1997. [29] E.C. Souza, D. Coutinho, Robust stability and control of uncertain linear discrete-time periodic systems with time-delay, Automatica 50 (2) (2014) 431– 441. [30] W.C. Sun, H.J. Gao, O. Kaynak, Finite frequency H1 control for vehicle active suspension systems, IEEE Trans. Control Syst. Technol. 19 (2) (2011) 416– 422. [31] S.C. Tong, Y.M. Li, P. Shi, Fuzzy adaptive backstepping robust control for SISO nonlinear system with dynamic uncertainties, Inf. Sci. 179 (9) (2009) 1319–1332. [32] Y.F. Wang, T.Y. Chai, Y.M. Zhang, State observer-based adaptive fuzzy output-feedback control for a class of uncertain nonlinear systems, Inf. Sci. 180 (24) (2010) 5029–5040. [33] J.L. Wu, Robust H1 control for polytopic nonlinear control systems, IEEE Trans. Autom. Control 58 (11) (2013) 2957–2962. [34] J.L. Wu, K.R. Karimi, P. Shi, Network-based H1 output feedback control for uncertain stochastic systems, Inf. Sci. 232 (2013) 397–410. [35] L.G. Wu, Z.D. Wang, Robust L2  L1 control of uncertain differential linear repetitive processes, Syst. Control Lett. 57 (5) (2008) 425–435. [36] G.H. Yang, D. Ye, Adaptive reliable H1 filtering against sensor failures, IEEE Trans. Singal Proces. 55 (7) (2007) 3161–3171. [37] G.H. Yang, D. Ye, Reliable H1 control for linear systems with adaptive mechanism, IEEE Trans. Autom. Control 55 (1) (2010) 242–247. [38] H.J. Yang, Y.Q. Xia, Low frequency positive real control for delta operator systems, Automatica 48 (8) (2012) 1791–1795. [39] X.D. Zhao, L.X. Zhang, P. Shi, H.R. Karimi, Robust control of continuous-time systems with state-dependent uncertainties and its application to electronic circuits, IEEE Trans. Industr. Electron. 61 (8) (2014) 4161–4170. [40] K. Zhou, J. Doyle, Essentials of Robust Control, Prertice-Hall, New Jersey, 1998.