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The recovery of sparse initial state based on compressed sensing for discrete-time linear system
Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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The recovery of sparse initial state based on compressed sensing for discrete-time linear system Zhongmei Wang, Huanshui Zhang n School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2014 Received in revised form 11 May 2015 Accepted 24 June 2015 Communicated by Rongni Yang
This paper considers the recovery of sparse initial state for deterministic discrete-time linear timeinvariant systems based on the compressed sensing theory. A class of deterministic linear systems with the global observation matrices satisfying the restricted isometry property (RIP) is characterized. Sufficient conditions on the measurement time instants that guarantee the global observation matrix to be a RIP matrix are obtained. With respect to the recovery of the sparse initial state of a highdimensional linear system, it is worth mentioning that the number of measurements can be significantly decreased in terms of compressed sensing theory. & 2015 Elsevier B.V. All rights reserved.
Keywords: Discrete-time linear system Compressed sensing Restricted isometry property Restricted isometry constant
1. Introduction The methods based on compressed sensing (CS) have been attracting attention since the pioneering works of Candès et al. [1] and Donoho [2]. This theory has been widely applied in the area of signal processing. The active studies include the stability of sparse signal recovery [3,4], the construction of RIP matrix [5–7], and the recovery algorithm [8–11]. Recently, compressed sensing theory has been applied to the field of control theory to design sparse feedback gain matrices in networked control systems [12–15]. Different from the traditional network control design [16,17], the works of [12–15] took into account the price and feasibility of communication among system components. In [12], the control vectors transmitted through rate-limited channels were compressed without much deterioration of control performance. In [13], sparse control packets for networks were designed by adopting an l1 optimization. A sparsity-promoting optimal control algorithm was introduced in [14], where a linear quadratic optimal control problem with an additional penalty on the communication links was considered and a relaxation of the problem using the l1norm was proposed. In [15], alternating direction method of multipliers was applied to identify sparsity of feedback gains.
n
Corresponding author. E-mail addresses: [email protected] (Z. Wang), [email protected] (H. Zhang).
Aforementioned works are all concerned with the sparsepromoting control problem. In this paper, we investigate the observability of linear system with sparse initial state. In traditional control theory, a linear system of dimension n is said to be observable if the global observation matrix has rank n. It means that reconstructing the initial state of a high-dimensional system requires a potentially large number of measurements in case that the measurement matrix is an m n matrix with m⪡n. Differently, the problem of reconstructing the sparse initial state of a linear system by means of l1-minimization is studied in [18]. It is proposed in [18] that the sparse initial state of a linear system can be reconstructed by applying compressed sensing theory. The deficiency lies in that the existence of such linear system is not confirmed. In this paper, we consider the problem as to characterize such a class of deterministic linear systems that the observation matrices satisfy the RIP with proper parameters. Furthermore, the sufficient conditions on the measurement time instants that guarantee that the global observation matrix satisfies RIP are investigated. The following notations will be used throughout this paper. Z þ (resp. Z Z 0 ) denotes the set of positive (resp. nonnegative) integers. Rmn denotes the family of m n dimensional real matrices. In denotes n n dimensional identity matrix. For a given vector or matrix X, XT denotes its transpose. For a given vector x; J x J i denotes its li-norm ði ¼ 0; 1; 2Þ. For a given field F and a given integer r, Pr denotes the set of polynomials with degree r r on F. The rest of this paper is organized as follows. In Section 2, we review some basic knowledge about the CS theory and formulate
http://dx.doi.org/10.1016/j.neucom.2015.06.042 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
Z. Wang, H. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
the problem to be investigated. In Section 3, we present the main results of this paper. Finally, Section 4 concludes the paper.
2. Preliminaries and problem formulation 2.1. Preliminaries on compressed sensing Compressed sensing is a signal processing technique being applied to reconstruct a sparse signal x from the following stationary linear system: y ¼ Φx
ð1Þ
with significantly reduced number of samples y, where Φ A Rmn is referred to as the measurement matrix and m⪡n. We recall some basic facts about compressed sensing theory. Definition 1. Let Ψ A Rnn be an orthogonal basis, i.e., Ψ contains n orthogonal columns. A vector x A Rn is said to be k-sparse under Ψ if x ¼ Ψ c for some c A Rn with J c J 0 r k. Mathematically, we say that a vector x is k-sparse if it has at most k nonzero elements, i.e., J x J 0 r k. Generally, it is almost impossible to reconstruct x exactly from measurement equation (1) because the equation is under-determined. But the problem can reverse if x A Rn is sufficiently sparse and Φ satisfies the property known as restricted isometry property (RIP) that first introduced in [19] as follows: Definition 2. A matrix Φ A Rmn is said to satisfy the restricted isometry property (RIP) of order k if there exists a constant δk ð0 r δk o 1Þ such that for any k-sparse signal x A Rn ð1 δk Þ‖x‖22 r ‖Φx‖22 r ð1 þ δk Þ‖x‖22 : The smallest nonnegative number isometry constant of order k.
ð2Þ
δk in (2) is called restricted
Remark 1. If a sensing matrix satisfies the RIP and its restricted isometry constant is small enough, the recovery of sparse signals can be achieved by l1 -minimization [3], i.e., min J x J 1
s:t: y ¼ Φx:
ð3Þ
Furthermore, RIP also guarantees that the recovery process is robust to noise and stable in case that the signal is not precisely sparse [3,4]. 2.2. Problem formulation Consider the following discrete-time linear system (with zero control input): xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
ð4Þ
yðtÞ ¼ ηt CxðtÞ;
ð5Þ n
where t A Z þ denotes the discrete instant, xðtÞ A R and yðtÞ A Rm are the state and measurement, respectively; the matrices A A Rnn and C A Rmn are the state transfer matrix and measurement matrix, respectively; the scalar variable ηt takes value either 0 or 1 (ηt ¼ 1 means an observation at time t is available and ηt ¼ 0 otherwise). Assume that the measurement y(t) can be obtained at the time instants t 1 ; t 2 ; …; t m . Let T m ¼ ft 1 ; t 2 ; …; t m g; yT m ¼ ½yT ðt 1 Þ; yT ðt 2 Þ; …; yT ðt m ÞT ; OT m ¼ ½ðCAt 1 ÞT ; ðCAt2 ÞT ; …; ðCAtm ÞT T : Then the global observation equation of system (4) and (5) can be
written as yT m ¼ OT m x0 :
ð6Þ
As to the recovery of the initial state x0, a sufficient and necessary condition in control theory is that rankðOT m Þ ¼ n. In this paper, we consider the observability problem of the discrete-time linear system with sparse initial state. According to the compressed sensing theory, if the initial state x0 is sufficiently sparse and the global observation matrix OT m satisfies RIP, then x0 can be reconstructed perfectly by l1-minimization, i.e., min J x J 1
s:t: y ¼ OT m x:
ð7Þ
In [18], the sparse initial state recovery of deterministic and stochastic discrete-time linear system employing l1-minimization was considered. Different from [18], the existence problem of such discrete-time linear system that the observation matrix satisfies RIP is considered in this paper, which is described as follows. Problem 1. In this paper, our aim is to characterize suitable matrices A and C for system (4) and (5), such that the global t1 t2 tm observation matrix O T m ¼ ½ðC A ÞT ; ðC A ÞT ; …; ðC A ÞT T of the following discrete-time linear system xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt C xðtÞ
ð8Þ ð9Þ
satisfies RIP. Furthermore, a sufficient condition for guaranteeing O T m to be a RIP matrix will be investigated. 3. Main results 3.1. The construction of discrete-time linear systems with the global observation matrix satisfying RIP To solve Problem 1, it is necessary for us to recall DeVore's deterministic construction for RIP matrix [7]. Given a prime integer p and any integer r ð1 o r opÞ, let n ¼ pr þ 1 p2 . Given a field F with prime order p, for simplicity, F is considered as the field of integers modulo p. There are p2 elements in the set F F of ordered pairs. The elements of F F are ordered lexicographically as ð0; 0Þ; ð0; 1Þ; …; ðp 1; p 1Þ. The set of the polynomials on F with degree r is denoted by Pr . For any Q A Pr and x A F, the graph GðQ Þ is defined as the set of ordered pairs ðx; Q ðxÞÞ; x A F. Obviously, GðQ Þ D F F. Hence, Q can decide a 2 column vector vQ ðvQ D Rp Þ as follows. vQ is indexed on F F which takes value 1 at any ordered pair of GðQ Þ and takes value 0 otherwise. Thus the entries of vQ are either 1 or 0. It is clear that there are p2 linear polynomials in the form of axþ b ða; b A FÞ and pr þ 1 p2 nonlinear polynomials in Pr . Following [7], all these nonlinear polynomials can be classified by the equivalence relation described as follows. Two polynomials P; Q of degree r over F are said to be equivalent if there exist a; b A F such that PðxÞ ¼ Q ðx þaÞ þ b;
ð10Þ
for any x A F. Then if P A Pr is a nonlinear polynomial, its corresponding equivalence class consists of p2 polynomials shaped as Pðx þ aÞ þ b due to the p2 choices of a; b. Let Λr denote the set of representatives selected from each of the equivalence classes. Then the cardinality of Λr is pr 1 1. These pr 1 1 polynomials in Λr decide pr 1 1 vectors. These vectors are written in lexicographic order as vQ 1 ; vQ 2 ; …; vQ l , where Q i A Λr ði ¼ 1; 2; …; pr 1 1Þ. Let 2 r1 Ω ¼ ½vQ 1 ; vQ 2 ; …; vQ l A Rp ðp 1Þ , by Theorem 3.4 [7], we have the following conclusion: Lemma 1. The cyclic matrix Φ ¼ p1ffiffipΦ0 has the RIP with the restricted isometry constant being δ ¼ 4ðk 1Þr=p whenever
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
Z. Wang, H. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
k 1 o p=4r, where 0
Ω1
B Ω B 2 B B Ω3 B Φ0 ¼ B ⋮ B B B Ωp2 1 @
Ωp2 Ω1 Ω2
⋯ ⋯
⋮
⋱
Ω2 Ω3 Ω4
Ωp2 2 ⋯ Ωp2 1 ⋯
Ωp2
and Ωi A R1ðp
r1
1Þ
1 ¼ pffiffiffi Ω2 ; Ω1 ; Ωp2 ; Ωp2 1 ; …; Ω4 ; Ω3 : p
1
C C C C C : ⋮ C C C Ωp2 C A
⋯
ð11Þ
i
Ω.
T
T
T
T
T
T
Φ1p2 ¼ ½Ω Ω Ω Ω are the first block, the 2nd block, …, and the p th block, respectively. From DeVore's construction, we learn that the ith vector in mth ð2 r m r p2 Þ block is obtained from vQ i by m 1 cyclic shifts, and we denote it by vQ i ½m 1. The polynomial corresponding to the ith vector in each block is equivalent to Q i ði ¼ 1; 2; …; pr 1 1Þ, and all such polynomials construct an equivalence class with Qi as a representative. Thus, Φ0 can be rewritten as T p2 ;
T T 1
Φ0 ¼ ½vQ 1 ; …; vQ pr 1 1 ; …; vQ 1 ½p2 1; …; vQ pr 1 1 ½p2 1: Let
A ¼ A0 I ðpr 1 1Þ A Rðp
C ¼ p1ffiffipðΩ1 ; Ωp2 ; …; Ω2 Þ A R 0 1 0 1 0 ⋯ 0 B0 0 1 ⋯ 0C B C B C C A Rp2 p2 ; ⋮ ⋮ ⋮ ⋱ ⋮ A0 ¼ B B C B C @0 0 0 ⋯ 1A 1 0 0 ⋯ 0
1ðpr þ 1 p2 Þ
rþ1
p2 Þðpr þ 1 p2 Þ
rþ1
xðt þ 1Þ ¼ AxðtÞ;
ð13Þ and
where
and ðΩ1 ; Ωp2 ; …; Ω2 Þ A R1ðp p Þ is the first row of obtain the following dynamic system: 2
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt C xðtÞ;
p2 1
T
ð12Þ
where Φ11 ¼ ½Ω1 ; Ω2 ; …; Ωp2 T ; Φ12 ¼ ½Ωp2 ; Ω1 ; …; Ωp2 1 T ; …, and T 3 ; …;
ð16Þ
for i ¼ 2; …; p2 1. It is obvious that
Φ0 into the following block form:
Φ0 ¼ ½Φ11 ; Φ12 ; …; Φ1p2 ; T 2; 2
The computation process from matrix C to matrix C A can be regarded as the shifting of the submatrices Ωi ði ¼ 1; 2; …; p2 Þ of C . We denote this shifting by C ½1, i.e., C ½1 ¼ C A. Similarly, we have C ½i ¼ C A 1 ¼ pffiffiffiðΩi þ 1 ; …; Ω1 ; Ωp2 ; Ωp2 1 ; …; Ωi þ 2 Þ; p
Ω1
ði ¼ 1; 2; …; p2 Þ is the ith row of
Remark 2. We partition
3
O ¼ ðC ; ðC AÞT ; …; ðC A ÞT ÞT T ¼ ðC ÞT ; ðC ½1ÞT ; …; ðC ½p2 1ÞT 0 1 Ω1 Ωp2 ⋯ Ω2 BΩ Ω1 ⋯ Ω3 C C 1 B 2 C: ¼ pffiffiffiB C ⋮ ⋮ ⋱ ⋮ pB @ A Ωp2 Ωp2 1 ⋯ Ω1
ð17Þ
By Lemma 1, we have 1 O ¼ pffiffiffiΦ0 : p
ð18Þ
Hence, O satisfies the RIP of order k and its restricted isometry constant is δ ¼ 4ðk 1Þr=p. Under Assumption 1, it follows from Theorem 1.2 [3] that x0 can be exactly reconstructed by l1 -minimization. Hence, we complete the proof.□ Next, we investigate the sufficient condition on the measurement instants that guarantee the recovery of the sparse initial state.
Φ0. Then we
Theorem 1. The s-sparse initial state x0 of system (14) and (15) can be reconstructed by l1-minimization in case that the measurement data at instants t 1 ; t 2 ; …; t p2 can be received, where t 1 ; t 2 ; …; t p2 satisfy the following conditions: ð1Þ t 1 ot 2 o ⋯ o t p2 ;
ð19Þ
ð14Þ
ð2Þ t 1 ¼ up2 ; t i þ 1 t i ¼ vp2 þ 1;
ð20Þ
ð15Þ
where u; v A Z Z 0 and 1 r i r p2 1:
, yðtÞ A R, and the scalar variable ηt is the same where xðtÞ A R as aforementioned. pr þ 1 p2
Proof. If the measurement data at instants t 1 ; t 2 ; …; t p2 can be obtained, then the global observation matrix can be written as t1
t2
O T p2 ¼ ððC A ÞT ; ðC A ÞT ; …; ðC A 3.2. The RIP of the global observation matrix Having constructed the dynamic system (14) and (15), in this section, we will investigate the RIP of the global observation matrix for system (14) and (15). The following assumption is needed for the next discussions of this paper.
If successive measurements can be obtained, then we have the following conclusion with respect to the global observation matrix. Lemma 2. For system 2(14) and (15), the global observation matrix p T O ¼ ½C ; ðC AÞT ; …; ðC A ÞT T satisfies the RIP of order k and its restricted isometry constant is δ ¼ 4ðk 1Þr=p. Furthermore, the initial state vector x0 can be exactly reconstructed by l1-minimization.
Þ Þ :
ð21Þ
In order to investigate the RIP of O T p2 , we need to give the expression of O T p2 . To this end, we now give the expression of ti C A ; ði ¼ 1; 2; …; p2 Þ by means of the principle of mathematical induction. 0 It naturally holds that C A ¼ C when u ¼0. When u ¼1 p2 1
p2
t1
Assumption 1. The initial state vector x0 is an s-sparse vector with pffiffiffi 1Þr s ¼ 2k, where k A Z þ and satisfies that 4ðk o 2 1. p
t p2 T T
CA ¼ CA ¼ CA A ¼ ðΩp2 ; Ωp2 1 ; …; Ω1 Þ
¼ ðΩ1 ; Ωp2 ; Ωp2 1 ; …; Ω3 ; Ω2 Þ ¼ C:
ð22Þ
For any j Z 2 ðjA Z þ Þ, suppose C A CA
jp2
¼ CA
ðj 1Þp2
A
p2
¼ CA
p2
ðj 1Þp2
¼ C , then we have
¼ C:
ð23Þ
It follows that t1
CA ¼ CA
up2
¼C
ð24Þ
Proof. Note that
for any u A Z Z 0 . Similarly, the expression of C A
1 C A ¼ pffiffiffi Ω1 ; Ωp2 ; Ωp2 1 ; …; Ω4 ; Ω3 ; Ω2 A p
can ¼ CA
be
given
ðvp2 Þ þ t 1
¼ CA
as
follows.
ðv þ uÞp2
Notice
that
when
tm
ðm Z 2Þ
m ¼ 2; C A
A ¼ C A.
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
t2
Z. Wang, H. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Suppose C A tj
CA ¼ CA ¼ CA ¼ CA
tj 1
¼ CA
t j 1 þ vp2 þ 1 j2 vp2
A
A
j2
¼ CA
4. Conclusions
, then we have
tj 1
A
vp2 þ 1
vp2 þ 1
j1
¼ CA
j1
:
ð25Þ
Combining (24) and (25), we obtain the expression of O T p2 as follows: T t1 t2 t 2 O T p2 ¼ ðC A ÞT ; ðC A ÞT ; …; ðC A p ÞT T
¼ ðC ; ðC AÞT ; …; ðC A
p2 1 T T
Þ Þ :
ð26Þ
In this paper, compressed sensing theory is applied to deterministic dynamic systems. The recovery of sparse initial state for a class of deterministic linear systems is discussed. As we have presented, it is not necessary to obtain successive measurements to reconstruct the sparse initial state and the sparse initial state can be reconstructed provided that the measurements in suitable time instants are obtained. It is an attempt on reconstructing the sparse initial state for a dynamic linear system by means of compressed sensing theory. Our future work will focus on the recovery of the sparse initial state for more general deterministic linear systems.
By (17) and (18), we have Acknowledgment
1 O T p2 ¼ O ¼ pffiffiffiΦ0 : p
ð27Þ
It follows from Lemma 2 that O T p2 satisfies the RIP of order k and the restricted isometry constant is δ ¼ 4ðk 1Þr=p. Furthermore, ssparse initial state x0 can be reconstructed by l1 -minimization. We complete the proof.□ Remark 3. With respect to the linear systems we have characterized, the sparse initial vector x0 can be exactly reconstructed by no less than p2 measurements. But in classical control theory, the measurement frequency is at least nðn ¼ pr þ 1 p2 Þ. Under the assumption that 1 o r o p, it is obvious that n 4 p2 . The significant message is that for a high-dimensional linear system with sufficiently sparse initial state, the number of the measurements can be significantly decreased according to the compressed sensing method. 3.3. The robust recovery for the initial state In practice, the measurement process of the system (14) and (15) is often corrupted with measurement noise. In this case, the dynamic system can be described as xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt ½C xðtÞ þ vðtÞ;
ð28Þ ð29Þ
where vðtÞ is an unknown noise term with J vðtÞ J 2 r ε. In this section, we will show that the sparse initial state x0 can be reconstructed robustly under the same hypotheses as in Theorem 1 and the recovery error can be characterized. For the time instants t 1 ; t 2 ; …; t p2 , the overall observation equation of system (28) and (29) can be described as yT
p2
¼ O T p2 x0 þvT p2 ;
ð30Þ
where vT p2 ¼ ½vT ðt 1 Þ; vT ðt 2 Þ; …; vT ðt p2 ÞT . Note that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p2 qffiffiffiffiffiffiffiffiffiffi uX ‖vðt i Þ‖22 r p2 ε2 ¼ pε: J vT p2 J 2 ¼ t
ð31Þ
i¼1
Hence, x0 can be reconstructed as the solution to the following convex optimal problem: min J x~ 0 J 1
x~ 0 A Rn
s:t: J yT 2 O T p2 x~ 0 J 2 rpε: p
ð32Þ
pffiffiffi 1Þr o 2 1, By Theorem 1.2 [3], we obtain the Because 4ðk p following conclusion:
Proposition 1. The optimal solution of (32) is denoted by xn0 , then the recovery error of (32) can be described as jxn0 x0 j2 rc1 pε; pffiffiffiffiffiffiffiffi 4 1þδ 1Þr p whenever k 1 o 4r . where c1 ¼ 1 ð1 þ pffiffi2Þδ with δ ¼ 4ðk p
The authors would like to acknowledge funding support from the Taishan Scholar Construction Engineering by Shandong Government, the National Natural Science Foundation of China under Grants 61120106011, 61203029. References [1] E. Candès, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math. 59 (8) (2006) 1207–1223. [2] D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory 52 (4) (2006) 1289–1306. [3] E. Candès, The restrict isometry property and its implications for compressed sensing, C. R. Math. 346 (3) (2008) 589–592. [4] Z. Yang, C. Zhang, L. Xie, Robustly stable signal recovery in compressed sensing with structured matrix perturbation, IEEE Trans. Singal Process. 60 (9) (2012) 4658–4670. [5] A. Amini, F. Marvasti, Deterministic construction of binary, bipolar and ternary compressed sensing matrices, IEEE Trans. Inf. Theory 57 (4) (2011) 2360–2370. [6] L. Applebaum, S. Howard, S. Searle, R. Calderbank, Chirp sensing codes: deterministic compressed sensing measurements for fast recovery, Appl. Comput. Harmon. Anal. 26 (2) (2009) 283–290. [7] R. DeVore, Deterministic constructions of compressed sensing matrices, J. Complex. 23 (4–6) (2007) 918–925. [8] Z. Yang, L. Xie, C. Zhang, Off-grid direction of arrival estimation using sparse Bayesian inference, IEEE Trans. Singal Process. 61 (1) (2013) 38–43. [9] Y. Song, W. Cao, Y. Shen, G. Yang, Compressed sensing image reconstruction using intra prediction, Neurocomputing 151 (2015) 1171–1179. [10] M. Jing, X. Zhou, C. Qi, Quasi-Newton iterative projection algorithm for sparse recovery, Neurocomputing 144 (2014) 169–173. [11] E. Candes, M. Wakin, S. Boyd, Enhancing sparsity by reweighted l1 minimization, J. Fourier Anal. Appl. 14 (2008) 877–905. [12] M. Nagahara, D.E. Quevedo, T. Masuda, K. Hayashi, Compressive sampling for networked feedback control, in: Proceedings of IEEE International Conference on Acoustics, Speech Signal Processing, 2012, pp. 2733–2736. [13] N. Masaaki, Q. Daniel E, Sparse representions for packetized predictive networked control, in: Proceedings of 18th International Federation of Automatic Control World Congress, vol. 18, 2011, pp. 84–89. [14] M. Fardad, F. Lin, M.R. Jovanovic, Sparsity-promoting optimal control for a class of distributed system, in: Proceedings of American Control Conference, 2011, pp. 2050–2055. [15] F. Lin, M. Fardad, M.R. Jovanovic, Design of optimal sparse feedback gains via the alternating direction method of multipliers, IEEE Trans. Autom. Control 58 (9) (2013) 2426–2431. [16] C. Wang, Adaptive tracking control of uncertain MIMO switched nonlinear systems, Int. J. Innov. Comput. Inf. Control 10 (3) (2014) 1149–1159. [17] R. Yang, G. Liu, P. Shi, C. Thomas, M.V. Basin, Predictive output feedback control for networked control systems, IEEE Trans. Ind. Electron. 61 (1) (2014) 512–520. [18] W. Dai, S. Yüksel, Observation of a linear system under sparsity constraints, IEEE Trans. Autom. Control 58 (9) (2013) 2372–2376. [19] E. Candès, T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory 51 (2005) 4203–4215. Zhongmei Wang received the B.S. degree in Mathematics from Shandong Normal University, Jinan, China, in 2002, and the M.S. degree in Pure Mathematics from Xiamen University, Xiamen, China, in 2005. She is currently pursuing the Ph.D. degree in Control Theory and Engineering at Shandong University. Her research interests include compressed sensing, multi-agent systems.
ð33Þ
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
Z. Wang, H. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Huanshui Zhang graduated in Mathematics from the Qufu Normal University in 1986 and received his M.Sc. and Ph.D. degrees in Control Theory from Heilongjiang University, China, and Northeastern University, China, in 1991 and 1997, respectively. He worked as a Postdoctoral Fellow at Nanyang Technological University from 1998 to 2001 and Research Fellow at Hong Kong Polytechnic University from 2001 to 2003. He is currently a Changjiang Professorship at Shandong University, China. He was a Professor in Harbin Institute of Technology from 2003 to 2006. He held visiting appointments as Research Scientist and Fellow with Nanyang Technological University, Curtin University of Technology and Hong Kong City University from 2003 to 2006. His interests include optimal estimation and control, time-delay systems, stochastic systems, signal processing and wireless sensor networked systems.
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
5
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Brief Papers
The recovery of sparse initial state based on compressed sensing for discrete-time linear system Zhongmei Wang, Huanshui Zhang n School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 December 2014 Received in revised form 11 May 2015 Accepted 24 June 2015 Communicated by Rongni Yang
This paper considers the recovery of sparse initial state for deterministic discrete-time linear timeinvariant systems based on the compressed sensing theory. A class of deterministic linear systems with the global observation matrices satisfying the restricted isometry property (RIP) is characterized. Sufficient conditions on the measurement time instants that guarantee the global observation matrix to be a RIP matrix are obtained. With respect to the recovery of the sparse initial state of a highdimensional linear system, it is worth mentioning that the number of measurements can be significantly decreased in terms of compressed sensing theory. & 2015 Elsevier B.V. All rights reserved.
Keywords: Discrete-time linear system Compressed sensing Restricted isometry property Restricted isometry constant
1. Introduction The methods based on compressed sensing (CS) have been attracting attention since the pioneering works of Candès et al. [1] and Donoho [2]. This theory has been widely applied in the area of signal processing. The active studies include the stability of sparse signal recovery [3,4], the construction of RIP matrix [5–7], and the recovery algorithm [8–11]. Recently, compressed sensing theory has been applied to the field of control theory to design sparse feedback gain matrices in networked control systems [12–15]. Different from the traditional network control design [16,17], the works of [12–15] took into account the price and feasibility of communication among system components. In [12], the control vectors transmitted through rate-limited channels were compressed without much deterioration of control performance. In [13], sparse control packets for networks were designed by adopting an l1 optimization. A sparsity-promoting optimal control algorithm was introduced in [14], where a linear quadratic optimal control problem with an additional penalty on the communication links was considered and a relaxation of the problem using the l1norm was proposed. In [15], alternating direction method of multipliers was applied to identify sparsity of feedback gains.
n
Corresponding author. E-mail addresses: [email protected] (Z. Wang), [email protected] (H. Zhang).
Aforementioned works are all concerned with the sparsepromoting control problem. In this paper, we investigate the observability of linear system with sparse initial state. In traditional control theory, a linear system of dimension n is said to be observable if the global observation matrix has rank n. It means that reconstructing the initial state of a high-dimensional system requires a potentially large number of measurements in case that the measurement matrix is an m n matrix with m⪡n. Differently, the problem of reconstructing the sparse initial state of a linear system by means of l1-minimization is studied in [18]. It is proposed in [18] that the sparse initial state of a linear system can be reconstructed by applying compressed sensing theory. The deficiency lies in that the existence of such linear system is not confirmed. In this paper, we consider the problem as to characterize such a class of deterministic linear systems that the observation matrices satisfy the RIP with proper parameters. Furthermore, the sufficient conditions on the measurement time instants that guarantee that the global observation matrix satisfies RIP are investigated. The following notations will be used throughout this paper. Z þ (resp. Z Z 0 ) denotes the set of positive (resp. nonnegative) integers. Rmn denotes the family of m n dimensional real matrices. In denotes n n dimensional identity matrix. For a given vector or matrix X, XT denotes its transpose. For a given vector x; J x J i denotes its li-norm ði ¼ 0; 1; 2Þ. For a given field F and a given integer r, Pr denotes the set of polynomials with degree r r on F. The rest of this paper is organized as follows. In Section 2, we review some basic knowledge about the CS theory and formulate
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Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
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2
the problem to be investigated. In Section 3, we present the main results of this paper. Finally, Section 4 concludes the paper.
2. Preliminaries and problem formulation 2.1. Preliminaries on compressed sensing Compressed sensing is a signal processing technique being applied to reconstruct a sparse signal x from the following stationary linear system: y ¼ Φx
ð1Þ
with significantly reduced number of samples y, where Φ A Rmn is referred to as the measurement matrix and m⪡n. We recall some basic facts about compressed sensing theory. Definition 1. Let Ψ A Rnn be an orthogonal basis, i.e., Ψ contains n orthogonal columns. A vector x A Rn is said to be k-sparse under Ψ if x ¼ Ψ c for some c A Rn with J c J 0 r k. Mathematically, we say that a vector x is k-sparse if it has at most k nonzero elements, i.e., J x J 0 r k. Generally, it is almost impossible to reconstruct x exactly from measurement equation (1) because the equation is under-determined. But the problem can reverse if x A Rn is sufficiently sparse and Φ satisfies the property known as restricted isometry property (RIP) that first introduced in [19] as follows: Definition 2. A matrix Φ A Rmn is said to satisfy the restricted isometry property (RIP) of order k if there exists a constant δk ð0 r δk o 1Þ such that for any k-sparse signal x A Rn ð1 δk Þ‖x‖22 r ‖Φx‖22 r ð1 þ δk Þ‖x‖22 : The smallest nonnegative number isometry constant of order k.
ð2Þ
δk in (2) is called restricted
Remark 1. If a sensing matrix satisfies the RIP and its restricted isometry constant is small enough, the recovery of sparse signals can be achieved by l1 -minimization [3], i.e., min J x J 1
s:t: y ¼ Φx:
ð3Þ
Furthermore, RIP also guarantees that the recovery process is robust to noise and stable in case that the signal is not precisely sparse [3,4]. 2.2. Problem formulation Consider the following discrete-time linear system (with zero control input): xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
ð4Þ
yðtÞ ¼ ηt CxðtÞ;
ð5Þ n
where t A Z þ denotes the discrete instant, xðtÞ A R and yðtÞ A Rm are the state and measurement, respectively; the matrices A A Rnn and C A Rmn are the state transfer matrix and measurement matrix, respectively; the scalar variable ηt takes value either 0 or 1 (ηt ¼ 1 means an observation at time t is available and ηt ¼ 0 otherwise). Assume that the measurement y(t) can be obtained at the time instants t 1 ; t 2 ; …; t m . Let T m ¼ ft 1 ; t 2 ; …; t m g; yT m ¼ ½yT ðt 1 Þ; yT ðt 2 Þ; …; yT ðt m ÞT ; OT m ¼ ½ðCAt 1 ÞT ; ðCAt2 ÞT ; …; ðCAtm ÞT T : Then the global observation equation of system (4) and (5) can be
written as yT m ¼ OT m x0 :
ð6Þ
As to the recovery of the initial state x0, a sufficient and necessary condition in control theory is that rankðOT m Þ ¼ n. In this paper, we consider the observability problem of the discrete-time linear system with sparse initial state. According to the compressed sensing theory, if the initial state x0 is sufficiently sparse and the global observation matrix OT m satisfies RIP, then x0 can be reconstructed perfectly by l1-minimization, i.e., min J x J 1
s:t: y ¼ OT m x:
ð7Þ
In [18], the sparse initial state recovery of deterministic and stochastic discrete-time linear system employing l1-minimization was considered. Different from [18], the existence problem of such discrete-time linear system that the observation matrix satisfies RIP is considered in this paper, which is described as follows. Problem 1. In this paper, our aim is to characterize suitable matrices A and C for system (4) and (5), such that the global t1 t2 tm observation matrix O T m ¼ ½ðC A ÞT ; ðC A ÞT ; …; ðC A ÞT T of the following discrete-time linear system xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt C xðtÞ
ð8Þ ð9Þ
satisfies RIP. Furthermore, a sufficient condition for guaranteeing O T m to be a RIP matrix will be investigated. 3. Main results 3.1. The construction of discrete-time linear systems with the global observation matrix satisfying RIP To solve Problem 1, it is necessary for us to recall DeVore's deterministic construction for RIP matrix [7]. Given a prime integer p and any integer r ð1 o r opÞ, let n ¼ pr þ 1 p2 . Given a field F with prime order p, for simplicity, F is considered as the field of integers modulo p. There are p2 elements in the set F F of ordered pairs. The elements of F F are ordered lexicographically as ð0; 0Þ; ð0; 1Þ; …; ðp 1; p 1Þ. The set of the polynomials on F with degree r is denoted by Pr . For any Q A Pr and x A F, the graph GðQ Þ is defined as the set of ordered pairs ðx; Q ðxÞÞ; x A F. Obviously, GðQ Þ D F F. Hence, Q can decide a 2 column vector vQ ðvQ D Rp Þ as follows. vQ is indexed on F F which takes value 1 at any ordered pair of GðQ Þ and takes value 0 otherwise. Thus the entries of vQ are either 1 or 0. It is clear that there are p2 linear polynomials in the form of axþ b ða; b A FÞ and pr þ 1 p2 nonlinear polynomials in Pr . Following [7], all these nonlinear polynomials can be classified by the equivalence relation described as follows. Two polynomials P; Q of degree r over F are said to be equivalent if there exist a; b A F such that PðxÞ ¼ Q ðx þaÞ þ b;
ð10Þ
for any x A F. Then if P A Pr is a nonlinear polynomial, its corresponding equivalence class consists of p2 polynomials shaped as Pðx þ aÞ þ b due to the p2 choices of a; b. Let Λr denote the set of representatives selected from each of the equivalence classes. Then the cardinality of Λr is pr 1 1. These pr 1 1 polynomials in Λr decide pr 1 1 vectors. These vectors are written in lexicographic order as vQ 1 ; vQ 2 ; …; vQ l , where Q i A Λr ði ¼ 1; 2; …; pr 1 1Þ. Let 2 r1 Ω ¼ ½vQ 1 ; vQ 2 ; …; vQ l A Rp ðp 1Þ , by Theorem 3.4 [7], we have the following conclusion: Lemma 1. The cyclic matrix Φ ¼ p1ffiffipΦ0 has the RIP with the restricted isometry constant being δ ¼ 4ðk 1Þr=p whenever
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k 1 o p=4r, where 0
Ω1
B Ω B 2 B B Ω3 B Φ0 ¼ B ⋮ B B B Ωp2 1 @
Ωp2 Ω1 Ω2
⋯ ⋯
⋮
⋱
Ω2 Ω3 Ω4
Ωp2 2 ⋯ Ωp2 1 ⋯
Ωp2
and Ωi A R1ðp
r1
1Þ
1 ¼ pffiffiffi Ω2 ; Ω1 ; Ωp2 ; Ωp2 1 ; …; Ω4 ; Ω3 : p
1
C C C C C : ⋮ C C C Ωp2 C A
⋯
ð11Þ
i
Ω.
T
T
T
T
T
T
Φ1p2 ¼ ½Ω Ω Ω Ω are the first block, the 2nd block, …, and the p th block, respectively. From DeVore's construction, we learn that the ith vector in mth ð2 r m r p2 Þ block is obtained from vQ i by m 1 cyclic shifts, and we denote it by vQ i ½m 1. The polynomial corresponding to the ith vector in each block is equivalent to Q i ði ¼ 1; 2; …; pr 1 1Þ, and all such polynomials construct an equivalence class with Qi as a representative. Thus, Φ0 can be rewritten as T p2 ;
T T 1
Φ0 ¼ ½vQ 1 ; …; vQ pr 1 1 ; …; vQ 1 ½p2 1; …; vQ pr 1 1 ½p2 1: Let
A ¼ A0 I ðpr 1 1Þ A Rðp
C ¼ p1ffiffipðΩ1 ; Ωp2 ; …; Ω2 Þ A R 0 1 0 1 0 ⋯ 0 B0 0 1 ⋯ 0C B C B C C A Rp2 p2 ; ⋮ ⋮ ⋮ ⋱ ⋮ A0 ¼ B B C B C @0 0 0 ⋯ 1A 1 0 0 ⋯ 0
1ðpr þ 1 p2 Þ
rþ1
p2 Þðpr þ 1 p2 Þ
rþ1
xðt þ 1Þ ¼ AxðtÞ;
ð13Þ and
where
and ðΩ1 ; Ωp2 ; …; Ω2 Þ A R1ðp p Þ is the first row of obtain the following dynamic system: 2
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt C xðtÞ;
p2 1
T
ð12Þ
where Φ11 ¼ ½Ω1 ; Ω2 ; …; Ωp2 T ; Φ12 ¼ ½Ωp2 ; Ω1 ; …; Ωp2 1 T ; …, and T 3 ; …;
ð16Þ
for i ¼ 2; …; p2 1. It is obvious that
Φ0 into the following block form:
Φ0 ¼ ½Φ11 ; Φ12 ; …; Φ1p2 ; T 2; 2
The computation process from matrix C to matrix C A can be regarded as the shifting of the submatrices Ωi ði ¼ 1; 2; …; p2 Þ of C . We denote this shifting by C ½1, i.e., C ½1 ¼ C A. Similarly, we have C ½i ¼ C A 1 ¼ pffiffiffiðΩi þ 1 ; …; Ω1 ; Ωp2 ; Ωp2 1 ; …; Ωi þ 2 Þ; p
Ω1
ði ¼ 1; 2; …; p2 Þ is the ith row of
Remark 2. We partition
3
O ¼ ðC ; ðC AÞT ; …; ðC A ÞT ÞT T ¼ ðC ÞT ; ðC ½1ÞT ; …; ðC ½p2 1ÞT 0 1 Ω1 Ωp2 ⋯ Ω2 BΩ Ω1 ⋯ Ω3 C C 1 B 2 C: ¼ pffiffiffiB C ⋮ ⋮ ⋱ ⋮ pB @ A Ωp2 Ωp2 1 ⋯ Ω1
ð17Þ
By Lemma 1, we have 1 O ¼ pffiffiffiΦ0 : p
ð18Þ
Hence, O satisfies the RIP of order k and its restricted isometry constant is δ ¼ 4ðk 1Þr=p. Under Assumption 1, it follows from Theorem 1.2 [3] that x0 can be exactly reconstructed by l1 -minimization. Hence, we complete the proof.□ Next, we investigate the sufficient condition on the measurement instants that guarantee the recovery of the sparse initial state.
Φ0. Then we
Theorem 1. The s-sparse initial state x0 of system (14) and (15) can be reconstructed by l1-minimization in case that the measurement data at instants t 1 ; t 2 ; …; t p2 can be received, where t 1 ; t 2 ; …; t p2 satisfy the following conditions: ð1Þ t 1 ot 2 o ⋯ o t p2 ;
ð19Þ
ð14Þ
ð2Þ t 1 ¼ up2 ; t i þ 1 t i ¼ vp2 þ 1;
ð20Þ
ð15Þ
where u; v A Z Z 0 and 1 r i r p2 1:
, yðtÞ A R, and the scalar variable ηt is the same where xðtÞ A R as aforementioned. pr þ 1 p2
Proof. If the measurement data at instants t 1 ; t 2 ; …; t p2 can be obtained, then the global observation matrix can be written as t1
t2
O T p2 ¼ ððC A ÞT ; ðC A ÞT ; …; ðC A 3.2. The RIP of the global observation matrix Having constructed the dynamic system (14) and (15), in this section, we will investigate the RIP of the global observation matrix for system (14) and (15). The following assumption is needed for the next discussions of this paper.
If successive measurements can be obtained, then we have the following conclusion with respect to the global observation matrix. Lemma 2. For system 2(14) and (15), the global observation matrix p T O ¼ ½C ; ðC AÞT ; …; ðC A ÞT T satisfies the RIP of order k and its restricted isometry constant is δ ¼ 4ðk 1Þr=p. Furthermore, the initial state vector x0 can be exactly reconstructed by l1-minimization.
Þ Þ :
ð21Þ
In order to investigate the RIP of O T p2 , we need to give the expression of O T p2 . To this end, we now give the expression of ti C A ; ði ¼ 1; 2; …; p2 Þ by means of the principle of mathematical induction. 0 It naturally holds that C A ¼ C when u ¼0. When u ¼1 p2 1
p2
t1
Assumption 1. The initial state vector x0 is an s-sparse vector with pffiffiffi 1Þr s ¼ 2k, where k A Z þ and satisfies that 4ðk o 2 1. p
t p2 T T
CA ¼ CA ¼ CA A ¼ ðΩp2 ; Ωp2 1 ; …; Ω1 Þ
¼ ðΩ1 ; Ωp2 ; Ωp2 1 ; …; Ω3 ; Ω2 Þ ¼ C:
ð22Þ
For any j Z 2 ðjA Z þ Þ, suppose C A CA
jp2
¼ CA
ðj 1Þp2
A
p2
¼ CA
p2
ðj 1Þp2
¼ C , then we have
¼ C:
ð23Þ
It follows that t1
CA ¼ CA
up2
¼C
ð24Þ
Proof. Note that
for any u A Z Z 0 . Similarly, the expression of C A
1 C A ¼ pffiffiffi Ω1 ; Ωp2 ; Ωp2 1 ; …; Ω4 ; Ω3 ; Ω2 A p
can ¼ CA
be
given
ðvp2 Þ þ t 1
¼ CA
as
follows.
ðv þ uÞp2
Notice
that
when
tm
ðm Z 2Þ
m ¼ 2; C A
A ¼ C A.
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
t2
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4
Suppose C A tj
CA ¼ CA ¼ CA ¼ CA
tj 1
¼ CA
t j 1 þ vp2 þ 1 j2 vp2
A
A
j2
¼ CA
4. Conclusions
, then we have
tj 1
A
vp2 þ 1
vp2 þ 1
j1
¼ CA
j1
:
ð25Þ
Combining (24) and (25), we obtain the expression of O T p2 as follows: T t1 t2 t 2 O T p2 ¼ ðC A ÞT ; ðC A ÞT ; …; ðC A p ÞT T
¼ ðC ; ðC AÞT ; …; ðC A
p2 1 T T
Þ Þ :
ð26Þ
In this paper, compressed sensing theory is applied to deterministic dynamic systems. The recovery of sparse initial state for a class of deterministic linear systems is discussed. As we have presented, it is not necessary to obtain successive measurements to reconstruct the sparse initial state and the sparse initial state can be reconstructed provided that the measurements in suitable time instants are obtained. It is an attempt on reconstructing the sparse initial state for a dynamic linear system by means of compressed sensing theory. Our future work will focus on the recovery of the sparse initial state for more general deterministic linear systems.
By (17) and (18), we have Acknowledgment
1 O T p2 ¼ O ¼ pffiffiffiΦ0 : p
ð27Þ
It follows from Lemma 2 that O T p2 satisfies the RIP of order k and the restricted isometry constant is δ ¼ 4ðk 1Þr=p. Furthermore, ssparse initial state x0 can be reconstructed by l1 -minimization. We complete the proof.□ Remark 3. With respect to the linear systems we have characterized, the sparse initial vector x0 can be exactly reconstructed by no less than p2 measurements. But in classical control theory, the measurement frequency is at least nðn ¼ pr þ 1 p2 Þ. Under the assumption that 1 o r o p, it is obvious that n 4 p2 . The significant message is that for a high-dimensional linear system with sufficiently sparse initial state, the number of the measurements can be significantly decreased according to the compressed sensing method. 3.3. The robust recovery for the initial state In practice, the measurement process of the system (14) and (15) is often corrupted with measurement noise. In this case, the dynamic system can be described as xðt þ 1Þ ¼ AxðtÞ;
xð0Þ ¼ x0 ;
yðtÞ ¼ ηt ½C xðtÞ þ vðtÞ;
ð28Þ ð29Þ
where vðtÞ is an unknown noise term with J vðtÞ J 2 r ε. In this section, we will show that the sparse initial state x0 can be reconstructed robustly under the same hypotheses as in Theorem 1 and the recovery error can be characterized. For the time instants t 1 ; t 2 ; …; t p2 , the overall observation equation of system (28) and (29) can be described as yT
p2
¼ O T p2 x0 þvT p2 ;
ð30Þ
where vT p2 ¼ ½vT ðt 1 Þ; vT ðt 2 Þ; …; vT ðt p2 ÞT . Note that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p2 qffiffiffiffiffiffiffiffiffiffi uX ‖vðt i Þ‖22 r p2 ε2 ¼ pε: J vT p2 J 2 ¼ t
ð31Þ
i¼1
Hence, x0 can be reconstructed as the solution to the following convex optimal problem: min J x~ 0 J 1
x~ 0 A Rn
s:t: J yT 2 O T p2 x~ 0 J 2 rpε: p
ð32Þ
pffiffiffi 1Þr o 2 1, By Theorem 1.2 [3], we obtain the Because 4ðk p following conclusion:
Proposition 1. The optimal solution of (32) is denoted by xn0 , then the recovery error of (32) can be described as jxn0 x0 j2 rc1 pε; pffiffiffiffiffiffiffiffi 4 1þδ 1Þr p whenever k 1 o 4r . where c1 ¼ 1 ð1 þ pffiffi2Þδ with δ ¼ 4ðk p
The authors would like to acknowledge funding support from the Taishan Scholar Construction Engineering by Shandong Government, the National Natural Science Foundation of China under Grants 61120106011, 61203029. References [1] E. Candès, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math. 59 (8) (2006) 1207–1223. [2] D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory 52 (4) (2006) 1289–1306. [3] E. Candès, The restrict isometry property and its implications for compressed sensing, C. R. Math. 346 (3) (2008) 589–592. [4] Z. Yang, C. Zhang, L. Xie, Robustly stable signal recovery in compressed sensing with structured matrix perturbation, IEEE Trans. Singal Process. 60 (9) (2012) 4658–4670. [5] A. Amini, F. Marvasti, Deterministic construction of binary, bipolar and ternary compressed sensing matrices, IEEE Trans. Inf. Theory 57 (4) (2011) 2360–2370. [6] L. Applebaum, S. Howard, S. Searle, R. Calderbank, Chirp sensing codes: deterministic compressed sensing measurements for fast recovery, Appl. Comput. Harmon. Anal. 26 (2) (2009) 283–290. [7] R. DeVore, Deterministic constructions of compressed sensing matrices, J. Complex. 23 (4–6) (2007) 918–925. [8] Z. Yang, L. Xie, C. Zhang, Off-grid direction of arrival estimation using sparse Bayesian inference, IEEE Trans. Singal Process. 61 (1) (2013) 38–43. [9] Y. Song, W. Cao, Y. Shen, G. Yang, Compressed sensing image reconstruction using intra prediction, Neurocomputing 151 (2015) 1171–1179. [10] M. Jing, X. Zhou, C. Qi, Quasi-Newton iterative projection algorithm for sparse recovery, Neurocomputing 144 (2014) 169–173. [11] E. Candes, M. Wakin, S. Boyd, Enhancing sparsity by reweighted l1 minimization, J. Fourier Anal. Appl. 14 (2008) 877–905. [12] M. Nagahara, D.E. Quevedo, T. Masuda, K. Hayashi, Compressive sampling for networked feedback control, in: Proceedings of IEEE International Conference on Acoustics, Speech Signal Processing, 2012, pp. 2733–2736. [13] N. Masaaki, Q. Daniel E, Sparse representions for packetized predictive networked control, in: Proceedings of 18th International Federation of Automatic Control World Congress, vol. 18, 2011, pp. 84–89. [14] M. Fardad, F. Lin, M.R. Jovanovic, Sparsity-promoting optimal control for a class of distributed system, in: Proceedings of American Control Conference, 2011, pp. 2050–2055. [15] F. Lin, M. Fardad, M.R. Jovanovic, Design of optimal sparse feedback gains via the alternating direction method of multipliers, IEEE Trans. Autom. Control 58 (9) (2013) 2426–2431. [16] C. Wang, Adaptive tracking control of uncertain MIMO switched nonlinear systems, Int. J. Innov. Comput. Inf. Control 10 (3) (2014) 1149–1159. [17] R. Yang, G. Liu, P. Shi, C. Thomas, M.V. Basin, Predictive output feedback control for networked control systems, IEEE Trans. Ind. Electron. 61 (1) (2014) 512–520. [18] W. Dai, S. Yüksel, Observation of a linear system under sparsity constraints, IEEE Trans. Autom. Control 58 (9) (2013) 2372–2376. [19] E. Candès, T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory 51 (2005) 4203–4215. Zhongmei Wang received the B.S. degree in Mathematics from Shandong Normal University, Jinan, China, in 2002, and the M.S. degree in Pure Mathematics from Xiamen University, Xiamen, China, in 2005. She is currently pursuing the Ph.D. degree in Control Theory and Engineering at Shandong University. Her research interests include compressed sensing, multi-agent systems.
ð33Þ
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
Z. Wang, H. Zhang / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Huanshui Zhang graduated in Mathematics from the Qufu Normal University in 1986 and received his M.Sc. and Ph.D. degrees in Control Theory from Heilongjiang University, China, and Northeastern University, China, in 1991 and 1997, respectively. He worked as a Postdoctoral Fellow at Nanyang Technological University from 1998 to 2001 and Research Fellow at Hong Kong Polytechnic University from 2001 to 2003. He is currently a Changjiang Professorship at Shandong University, China. He was a Professor in Harbin Institute of Technology from 2003 to 2006. He held visiting appointments as Research Scientist and Fellow with Nanyang Technological University, Curtin University of Technology and Hong Kong City University from 2003 to 2006. His interests include optimal estimation and control, time-delay systems, stochastic systems, signal processing and wireless sensor networked systems.
Please cite this article as: Z. Wang, H. Zhang, The recovery of sparse initial state based on compressed sensing for discrete-time linear system, Neurocomputing (2015), http://dx.doi.org/10.1016/j.neucom.2015.06.042i
5