Optimal scheduling of multiple sensors in continuous time

Optimal scheduling of multiple sensors in continuous time

ISA Transactions 53 (2014) 793–801 Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Op...

815KB Sizes 0 Downloads 67 Views

ISA Transactions 53 (2014) 793–801

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Optimal scheduling of multiple sensors in continuous time Xiang Wu a,b,c, Kanjian Zhang a,b, Changyin Sun a,b,n a b c

School of Automation, Southeast University, Nanjing 210096, PR China Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing 210096, PR China School of Electrical and Information Engineering, Hunan Institute of Technology, Hengyang 421002, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 19 October 2013 Received in revised form 9 December 2013 Accepted 20 December 2013 Available online 12 March 2014 This paper was recommended for publication by Dr. Jeff Pieper

This paper considers an optimal sensor scheduling problem in continuous time. In order to make the model more close to the practical problems, suppose that the following conditions are satisfied: only one sensor may be active at any one time; an admissible sensor schedule is a piecewise constant function with a finite number of switches; and each sensor either doesn’t operate or operates for a minimum nonnegligible amount of time. However, the switching times are unknown, and the feasible region isn’t connected. Thus, it’s difficult to solve the problem by conventional optimization techniques. To overcome this difficulty, by combining a binary relaxation, a time-scaling transformation and an exact penalty function, an algorithm is developed for solving this problem. Numerical results show that the algorithm is effective. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Optimal control Sensor scheduling Stochastic systems Time-scaling transformation Exact penalty function

1. Introduction With the development of wireless communication, digital electronics and micro-electromechanical systems, wireless sensor networks have been attracting wide applications, such as pollution monitoring [1], smart grid [2], health care [3] and mobile robotic [4]. However, new challenges in system analysis and design arise due to the unprecedented characteristics [5–7]. Recently, optimal sensor scheduling problems for state estimation have received considerable attention in the control community. The main objective is to minimize the variance of the estimation error, and the motivation arises from resource limitations introduced by wireless sensor networks [8]. In [9], the optimal sensor scheduling problem is modeled in continuous time. Then, the optimal scheduling policy is obtained by solving a quasi-variational inequality. However, the formulation is much too complex. Lee et al. [10] consider the optimal sensor scheduling problem in continuous time, where the control variables are restricted to take values from a discrete set but the switching times are to take place over a continuous time horizon. This formulation leads to an optimal discrete-valued control problem. Ref. [11] discusses the continuous-time optimal sensor scheduling n Corresponding author at: School of Automation, Southeast University, Nanjing 210096, PR China. Tel.: þ 86 2583792720. E-mail addresses: [email protected] (X. Wu), [email protected] (K. Zhang), [email protected] (C. Sun).

problem by a combination of a branch and cut technique and a gradient-based method. A optimal sensor scheduling problem is considered in [12]. A new heuristic approach, which incorporates the discrete filled function algorithm into standard optimal control software, is proposed for finding a global solution of this problem. For the case of discrete time, the optimal sensor scheduling problem is solved by stochastic strategies [13], the tree search type of algorithms [14], and a combination of a branch and bound and a gradient-based method [15]. In this paper, the optimal sensor scheduling problem in continuous time is reconsidered. Compared with [9–12], the model discussed in the paper is more close to practical problems. However, the switching times are unknown, and the feasible region is not connected. Thus, it is difficult to solve such a problem by standard optimization algorithms, e.g., sequential quadratic programming [17–19], constrained Quasi-Newton method, multiplier penalty function [20], Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [21], etc. To overcome this difficulty, by combining a binary relaxation, a time-scaling transformation and an exact penalty function, an efficient computational approach is developed for solving this problem. Finally, two numerical examples are provided to illustrate the effectiveness of the developed algorithm. The rest of the paper is organized as follows. Section 2 presents the optimal scheduling problem of multiple sensors in continuous time. Then, in Section 3, by introducing new binary variables and the time-scaling technique, the sensor optimal scheduling problem is transformed into an equivalent problem with fixed

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.024

794

X. Wu et al. / ISA Transactions 53 (2014) 793–801

switching times. However, the feasible region is not connected, which poses a challenge for standard optimization algorithms. Thus, in Section 4, by an exact penalty function, the problem is transformed into a sequence of unconstrained problems. Each of these unconstrained problems can be effectively solved by any gradient-based optimization technique. Section 5 provides the gradient formulae of the cost function and our algorithm. A numerical example in Section 6 provides evidence that our method is effective. 2. Problem formulation Consider the optimal sensor scheduling problem in Fig. 1. Let ðΩ; F ; PÞ be a given probability space. The process is the following stochastic linear system: _ ¼ AðtÞxðtÞ þ BðtÞV_ ðtÞ; xðtÞ

t A ½0; T;

xð0Þ ¼ x0 ;

In order to make the model more close to the practical problems, suppose that the following three conditions are satisfied: Assumption 1. In some applications, e.g., robotics, operating several sensors simultaneously causes interference in the system and affects the measurement accuracy [16]. Thus, suppose that only one sensor may be active at any one time. Assumption 2. Since a sensor schedule with infinitely many switches is not suitable for application in engineering, suppose that u(t) is a piecewise constant function with at most M switches. Assumption 3. Note that running every potential sensor may not be optimal. Thus, suppose that each sensor either does not operate or operates for a minimum non-negligible amount of time.

ð1Þ

These assumptions indicate that adjacent switching times satisfy the following constraint:

ð2Þ

τi  τi  1 A f0g [ ½εi ; 1Þ;

where T 4 0 is a given terminal time; for each t Z 0, AðtÞ A Rnn and BðtÞ A Rnp are uniformly bounded measurable matrix-valued functions; x0 is a Rn-valued Gaussian random vector on ðΩ; F ; PÞ with mean Eðx0 Þ ¼ x 0 and covariance Eðx0  x 0 Þðx0  x 0 ÞT ¼ P 0 ; the process fVðtÞ; t Z 0g is an Rp-valued Brownian motion on ðΩ; F ; PÞ with mean EðV ðtÞÞ ¼ 0 and covariance EðVðtÞ  VðsÞÞðVðtÞ  VðsÞÞT ¼ Q ðt  sÞ, where Q A Rpp is a symmetric, positive semi-definite matrix; and fxðtÞ; t Z 0g is an Rn-valued square integrable process. Suppose that xðtÞ can be estimated on the basis of measurement data obtained by N sensors. Then, a sensor schedule can be denoted by a function uðtÞ : ½0; T-f1; …; Ng, and uðtÞ ¼ i indicates that the sensor i is used at time t. Thus, u(t) is completely determined by specifying:

 Switching sequence: the order in which it assumes the different values in f1; …; Ng.

ð3Þ

where τi ; i ¼ 1; …; M, are the ith switching time; and τ0 ¼ 0, τM þ 1 ¼ T. Clearly, constraint (3) implies that τi ; i ¼ 0; 1; …; M þ 1, also satisfy the following conventional ordering constraints: 0 ¼ τ0 r τ1 r ⋯ r τM þ 1 ¼ T:

ð4Þ

Clearly, constraint (3) is more complex than conventional constraint (4), which is a simple ordering constraint on the switching times. It must be pointed out that constraint (4) is convex, but constraint (3) is non-convex, and imposing (3) results in a nonconnected feasible region for the sensor durations. Let U be the set of all such sensor schedules which are measurable, and any uðtÞ A U is referred to as an admissible sensor schedule. Suppose that the state information can be measured using these sensors. Then, for any uðtÞ A U, the output equation is given by N

 Switching time: the time at which it switches from one value in f1; …; Ng to another.

_ i ðtÞχ _ ¼ ∑ ½C i ðtÞxðtÞ þ Di ðtÞW yðtÞ fuðtÞ ¼ ig ðtÞ; i¼1

t A ½0; T;

yð0Þ ¼ 0;

For example, consider an optimal sensor scheduling problem with three sensors and two switches. Suppose that a sensor schedule is defined by 8 > < 2; t A ½0; 0:3Þ; uðtÞ ¼ 1; t A ½0:3; 0:7Þ; > : 3; t A ½0:7; 1: Then, (2,1,3) is the switching sequence, 0.3 and 0.7 are the switching time.

ð5Þ ð6Þ

mn

mk

and Di ðtÞ A R are uniformly where for each t Z0, C i ðtÞ A R bounded measurable functions; the process fW i ðtÞ; t Z0g is an Rk-valued Brownian motion on ðΩ; F ; PÞ with mean EðW i ðtÞÞ ¼ 0 and covariance EðW i ðtÞ  W i ðsÞÞðW i ðtÞ W i ðsÞÞT ¼ Ri ðt  sÞ, where Ri A Rkk is a symmetric, positive definite matrix; and the characteristic function χ fuðtÞ ¼ ig ðtÞ is defined by ( 1 if uðtÞ ¼ i; χ fuðtÞ ¼ ig ðtÞ ¼ ð7Þ 0 if uðtÞ a i:

Fig. 1. Block diagram of optimal scheduling of multiple sensors in continuous time.

X. Wu et al. / ISA Transactions 53 (2014) 793–801

Suppose that Di ðtÞRi DTi ðtÞ 40. Furthermore, suppose that x0, V(t), and W 1 ðtÞ; …; W N ðtÞ are mutually independent. Then, there exits a unique solution y(t) of (5) and (6), which is referred to as the observation process associated with the scheduling policy uðtÞ A U. Let uðtÞ A U be given and let F yt ¼ sfyðsÞ; 0 r s r tg denote the s-algebra generated by the observation process y(t) associated with u(t). For a given F yt , the optimal mean-square estimate of x(t) y ^ ¼ EfxðtÞjF ^ and the associated error covariance are given by xðtÞ tg T ^ ^ and PðtÞ ¼ EðxðtÞ  xðtÞÞðxðtÞ  xðtÞÞ , respectively. Then, by similar arguments given in [10,22,23], we obtain the following lemma on ^ the optimal xðtÞ. ^ Lemma 1. For a given uðtÞ A U, the optimal xðtÞ is the solution of the following stochastic nonlinear system: " # N 1 ^ x^_ ðtÞ ¼ AðtÞ  PðtÞ ∑ C Ti ðtÞR~ i ðtÞC i ðtÞχ fuðtÞ ¼ ig ðtÞ xðtÞ i¼1

"

N

# 1

þ PðtÞ ∑ C Ti ðtÞR~ i i¼1

_ ðtÞC i ðtÞχ fuðtÞ ¼ ig ðtÞ yðtÞ;

t A ½0; T;

^ xð0Þ ¼ x0;

ð8Þ ð9Þ

where R~ i ðtÞ ¼ Di ðtÞRi DTi ðtÞ; and P(t) satisfies the following matrix Riccati differential equation: P_ ðtÞ ¼ AðtÞPðtÞ þ PðtÞAT ðtÞ þ BðtÞQBT ðtÞ N

1  ∑ PðtÞC Ti ðtÞR~ i ðtÞC i ðtÞPðtÞχ fuðtÞ ¼ ig ðtÞ; i¼1

t A ½0; T;

Pð0Þ ¼ P 0 :

ð10Þ ð11Þ

Clearly, the solution of (10) and (11) depends on uðtÞ A U. Let PðtjuðtÞÞ be the solution corresponding to uðtÞ A U. By choosing uðtÞ A U, the estimation error can be reduced. Thus, the optimal sensor scheduling problem can be formulated as follows. Problem 1. Choose an admissible sensor schedule uðtÞ A U such that Z T JðuðtÞÞ ¼ η TrðPðTjuðtÞÞÞ þ TrfωðtÞPðtjuðtÞÞg dt ð12Þ

795

Clearly, (15) and (16) indicate υi A f0; 1g. In addition, to ensure that υi is consistent with the definition given in (13), we impose the following additional constraints: g i ðυi ; τi Þ ¼ ðεi  ðτi  τi  1 ÞÞυi r0;

i ¼ 1; …; M þ 1;

ð17Þ

Gi ðυi ; τi Þ ¼ ðτi  τi  1 Þð1  υi Þ r 0;

i ¼ 1; …; M þ 1:

ð18Þ

Next, we can show that the definition (13) is equivalent to (15)–(18). Suppose that (17) and (18) are satisfied. Note that (15) and (16) imply υi A f0; 1g. If υi ¼ 0, from (18), we have

τi  τi  1 ¼ ðτi  τi  1 Þð1  υi Þ r 0:

ð19Þ

Constraint (4) indicates

τi  τi  1 Z 0:

ð20Þ

Then, by (19) and (20), we derive τi  τ i  1 ¼ 0. On the other hand, If υi ¼ 1, from (17), we have

εi ðτi  τi  1 Þ ¼ ðεi  ðτi  τi  1 ÞÞυi r 0:

ð21Þ

That is, τi  τi  1 Z εi . Thus, (13) holds. Conversely, suppose that (13) is satisfied. Then, if υi ¼ 1, by (13), we have τi  τi  1 Z εi , g i ðυi ; τi Þ ¼ εi  ðτi  τi  1 Þ r 0, and Gi ðυi ; τi Þ ¼ 0. Similarly, if υi ¼ 0, by (13), we have τi  τi  1 ¼ 0, g i ðυi ; τi Þ ¼ 0, and Gi ðυi ; τi Þ ¼ 0. Thus, (17) and (18) hold. Above all, the definition (13) is equivalent to (15)–(18). Note that constraint (3) can be equivalently written as the following inequalities: ðτi  τi  1 Þðεi  τi þ τi  1 Þ r 0;

τi  τi  1 Z 0;

i ¼ 1; …; M þ 1;

i ¼ 1; …; M þ 1:

ð22Þ ð23Þ

Let τ ¼ ½τ1 ; …; τM T . Then, Problem 1 can be transformed into the following problem: Problem 2. Choose ðυ; τÞ A RM þ 1  RM such that Z T TrfωðtÞPðtjυ; τÞg dt J~ ðυ; τÞ ¼ η TrðPðTjυ; τÞÞ þ

ð24Þ

0

is minimized subject to (11), (14)–(18), (22) and (23), where Pðtjυ; τÞ is the solution of (11) and (14). Clearly, Problems 1 and 2 are equivalent.

0

is minimized subject to (10) and (11), where TrðÞ denotes the matrix trace; η is a non-negative constant; and ωðtÞ A Rnn is a positive definite matrix-valued measurable function. 3. Problem transformation To solve Problem 1, we introduce binary decision variables

υi ; i ¼ 1; …; M þ 1, defined as follows: (

υi ¼

1 0

if τi  τi  1 Z εi ;

ð13Þ

if τi  τi  1 ¼ 0:

P_ ðtÞ ¼ AðtÞPðtÞ þ PðtÞAT ðtÞ þ BðtÞQBT ðtÞ M 1 i¼1

t A ½0; T:

ð14Þ

Since it is difficult to handle binary variables (13) by standard numerical optimization algorithms [24,25], the binary requirements are dropped and each υi is treated as a continuous variable subject to the following constraints: 0 r υi r1;

i ¼ 1; …; M þ 1;

hi ðυi Þ ¼ υi ð1  υi Þ r0;

i ¼ 1; …; M þ 1:

Mþ1

t_ ðsÞ ¼ ∑ θi χ ½i  1;iÞ ðsÞ;

ð25Þ

tð0Þ ¼ 0;

ð26Þ

i¼1

Let υ ¼ ½υ1 ; …; υM þ 1 T . Then, the matrix Riccati differential equation (10) can be written as

1  ∑ PðtÞC Tυi ðtÞR~ υi ðtÞC υi ðtÞPðtÞχ ½τi  1 ;τi Þ ðtÞ;

To solve Problem 2 by using standard numerical optimization algorithms, the gradient of the cost function is required. However, since the switching times are unknown, it is difficult to integrate (11) and (14) numerically [17,26,27]. Thus, the time-scaling transformation described in [28–30] is applied to derive a more tractable equivalent problem, in which the variable switching times are replaced by conventional decision parameters. Define a function tðsÞ : ½0; M þ 1-R by the following differential equation:

ð15Þ ð16Þ

where θi ¼ τi  τi  1 is the duration of the ith sensor; and for a given interval I  ½0; M þ 1, the characteristic function χ I ðsÞ is defined by  1 if s A I; χ I ðsÞ ¼ ð27Þ 0 otherwise: Let θ ¼ ½θ1 ; …; θM þ 1 T and let Θ be the set containing all such θ. Integrating (25) with initial condition (26) yields that, for any s A ½k  1; kÞ; k ¼ 1; …; M þ 1, k1

tðsÞ ¼ ∑ θi þ θk ðs  k þ 1Þ: i¼1

ð28Þ

796

X. Wu et al. / ISA Transactions 53 (2014) 793–801

Thus, for each i ¼ 1; …; M þ 1, we have i

i

k¼1

k¼1

tðiÞ ¼ ∑ θk ¼ ∑ ðτk  τk  1 Þ ¼ τi :

ð29Þ

In particular, tðM þ 1Þ ¼ T. Now, by the time-scaling transform (25), (26), (14), (17), (18), (22), (23) and (24) are transformed into M 1 T _ ^ P^ ðsÞ þ P^ ðsÞA^ T ðsÞ þ BðsÞQ ^ B^ ðsÞ P^ ðsÞ ¼ ∑ θi ½AðsÞ i¼1

T 1  P^ ðsÞC^ υi ðsÞR^ υi ðsÞC^ υi ðsÞP^ ðsÞχ ½i  1;iÞ ðsÞ;

i ¼ 1; …; M þ 1;

ð30Þ

g^ i ðυi ; θi Þ ¼ ðεi  θi Þυi r 0;

i ¼ 1; …; M þ 1;

ð31Þ

G^ i ðυi ; θi Þ ¼ θi ð1  υi Þ r 0;

i ¼ 1; …; M þ 1;

ð32Þ

θi ðεi  θi Þ r0; θi Z 0;

i ¼ 1; …; M þ 1;

i ¼ 1; …; M þ 1;

ð34Þ

J^ ðυ; θÞ ¼ η TrðP^ ðM þ1jυ; θÞÞ Z Mþ1 Mþ1 ^ ðsÞP^ ðsjυ; θÞgχ ½i  1;iÞ ðsÞ ds; þ ∑ θi Trfω 0

ð33Þ

Problem 4. Choose ðυ; θ; ρÞ A RM þ 1  RM þ 1  R such that (36) is minimized subject to (11), (25), (26), and (30) and the bounds on the variables given by (15), (34) and (38). Clearly, during the process of minimizing J^ δ ðυ; θÞ, if the penalty parameter γ is increased, ρβ should be reduced, which implies that ρ should be reduced as β is fixed. Thus, ρ  α will be increased, and the constraint violation will be reduced, which indicate that for each i ¼ 1; …; M þ 1, ½maxf0; hi ðυi Þg2 , ½maxf0; g i ðυi ; θi Þg2 , ½maxf0; Gi ðυi ; θi Þg2 and tðM þ 1Þ  T must go down. When the penalty parameter γ is sufficiently large, any solution of Problem 4 is also an optimal solution of Problem 3 [17–19]. Problem 4 is an optimal parameter selection problem with simple bounds on the variables given by (15), (34) and (38). Such problems can be solved efficiently using any gradient-based optimization technique.

5. Gradient formulae Before presenting the gradient formulae of the cost function of Problem 4, the functional (36) will be rewritten into the following form: Z M þ1 J^ γ ðυ; θ; ρÞ ¼ ΦðP^ ðM þ1Þ; υ; θ; ρÞ þ Lðs; P^ ðsÞ; υ; θÞ ds; ð39Þ 0

ð35Þ

i¼1

where, for simplicity, we have written PðtðsÞÞ; AðtðsÞÞ; BðtðsÞÞ; ^ ^ C υi ðtðsÞÞ; R~ υi ðtðsÞÞ and ωðtðsÞÞ as P^ ðsÞ; AðsÞ, BðsÞ; C^ υi ðsÞ; R^ υi ðsÞ and ω^ ðsÞ. Suppose that (15), (16), (31), (32) and (34) are satisfied and 0 o θi o εi . If υi ¼ 0, by (32) and (34), we have θi ¼ 0, which is a contradiction. If υi ¼ 1, (31) indicates θi Z εi , which is a contradiction. Thus, if (15), (16), (31), (32) and (34) hold, then 0 o θi o εi is impossible. That is, constraint (33) is redundant. Then, a new optimal control problem is defined as follows. Problem 3. Choose ðυ; θÞ A RM þ 1  RM þ 1 such that (35) is minimized subject to (11), (15), (16), (25), (26), (30)–(32) and (34). Clearly, Problems 2 and 3 are equivalent.

where

ΦðP^ ðM þ 1Þ; υ; θ; ρÞ ¼ η TrðP^ ðM þ 1ÞÞ þ ρ  α Δðυ; θÞ þ γρβ ; Mþ1

^ ðsÞP^ ðsjυ; θÞgχ ½i  1;iÞ ðsÞ: Lðs; P^ ðsÞ; υ; θÞ ¼ ∑ θi Trfω i¼1

Define the Hamiltonian function Hðs; P^ ðsÞ; υ; θ; ρ; λÞ by Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ¼ Lðs; P^ ðsÞ; υ; θÞ þ λðsÞT f~ ðs; P^ ðsÞ; υ; θÞ; where " f~ ðs; P^ ðsÞ; υ; θÞ ¼

Since the constraints (15) and (16) define a disjoint feasible region for υi ; i ¼ 1; …; M þ 1, it is difficult to find an optimal solution of Problem 3 by standard numerical optimization algorithms. To overcome this difficulty, by the idea proposed in [17–19], an exact penalty function is defined as follows: 8 ^ > if ρ ¼ 0 and Δðυ; θÞ ¼ 0; > < J ðυ; θÞ ^J ðυ; θ; ρÞ ¼ J^ ðυ; θÞ þ ρ  α Δðυ; θÞ þ γρβ if ρ 4 0; γ > > : þ1 otherwise; ð36Þ where ρ is a new decision variable; γ 4 0 is the penalty parameter; α and β are positive constants satisfying 1 r β r α; and Δðυ; θÞ is the constraint violation given by Mþ1

Δðυ; θÞ ¼ ∑ f½maxf0; hi ðυi Þg2 þ ½maxf0; gi ðυi ; θi Þg2 i¼1

The new decision variable 0 r ρ r ρ~ ;

Mþ1

T

T

^ P^ ðsÞ þ P^ ðsÞA^ ðsÞ þ BðsÞQ ^ ∑ θi ½AðsÞ B^ ðsÞ

i¼1

i T 1  P^ ðsÞC^ υi ðsÞR^ υi ðsÞC^ υi ðsÞP^ ðsÞχ ½i  1;iÞ ðsÞ ;

4. An exact penalty function

þ ½maxf0; Gi ðυi ; θi Þg2 g þ ½tðM þ 1Þ  T2 :

ð40Þ

ð37Þ

ρ satisfies ð38Þ

where ρ~ is a small positive number. Then, an unconstrained penalty problem is defined as follows.

and λðsÞ is the costate that satisfies the following system:

λ_ ðsÞT ¼ 

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ; ∂P^ ðsÞ

λðM þ 1Þ ¼ ηI:

ð41Þ

ð42Þ

Then, the gradient formulas of the cost function (36) are given in the following theorem. Theorem 1. The gradient formulae of the cost function (39) with respect to υi, θi and ρ are given by ∂J^ γ ðυ; θ; ρÞ ¼ ∂υi

Z

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ds ∂υi  Mþ1  ∂hi ðυi Þ þ 2ρ  α ∑ max 0; hi ðυi Þ ∂υi i¼1 i

i1

 ∂g i ðυi ; θi Þ þ max 0; g i ðυi ; θi Þ ∂υi   ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ ; ∂υi i ¼ 1; …; M þ 1;

ð43Þ

X. Wu et al. / ISA Transactions 53 (2014) 793–801

∂J^ γ ðυ; θ; ρÞ

Z

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ¼ ds ∂θi ∂θ i i1  M þ1  ∂g i ðυi ; θi Þ þ 2ρ  α ∑ max 0; g i ðυi ; θi Þ ∂θ i i¼1   ∂Gi ðυi ; θi Þ ; i ¼ 1; …; M þ 1; þ max 0; Gi ðυi ; θi Þ ∂θ i

  ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ δθi ∂θ i Z M þ1 ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ þ δθi ds ∂θ i 0 Z M þ1 ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ þ δρ ds ∂ρ 0

i

∂J^ γ ðυ; θ; ρÞ M þ 1 ¼ ∑ ∂ρ i¼1

Z

ð44Þ

þ ðγβρβ  1  αρ  α  1 Δðυ; θÞÞδρ ! Z M þ1 ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ _ T þ λ ðsÞ δP^ ðsÞ ds: þ ∂P^ ðsÞ 0

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ds ∂ρ i1 i

þ ðγβρβ  1  αρ  α  1 Δðυ; θÞÞ:

ð45Þ

Proof. By (30) and (40), we have J^ γ ðυ; θ; ρÞ ¼ ΦðP^ ðM þ 1Þ; υ; θ; ρÞ Z Mþ1 _ ðHðs; P^ ðsÞ; υ; θ; ρ; λÞ  λðsÞT P^ ðsÞÞ ds: þ

ð46Þ

 ∂g i ðυi ; θi Þ þ max 0; g i ðυi ; θi Þ ∂ υi  ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ : ∂ υi ! Z i ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ds δυi þ ∂υi i1  Mþ1  ∂g i ðυi ; θi Þ þ 2ρ  α ∑ max 0; g i ðυi ; θi Þ ∂θ i i¼1   ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ ∂θi ! Z i ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ds δθi þ ∂θ i i1 Z i M þ1 ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ þ ∑ ds ∂ρ i ¼ 1 i1  þ ðγβρβ  1  αρ  α  1 Δðυ; θÞÞ δρ:

_ Integrating by parts the term λðsÞT P^ ðsÞ yields J^ γ ðυ; θ; ρÞ ¼ η TrðP^ ðM þ 1ÞÞ þ ρ  α Δðυ; θÞ þ γρβ

ð47Þ

0

Then, the first order variation of (47) is given by d TrðP^ ðM þ 1ÞÞ ^ δP ðM þ 1Þ dP^ ðM þ 1Þ  Mþ1  ∂hi ðυi Þ δυi þ 2ρ  α ∑ max 0; hi ðυi Þ ∂υi i¼1  ∂g i ðυi ; θi Þ þ max 0; g i ðυi ; θi Þ δυi ∂υi  ∂g i ðυi ; θi Þ þ max 0; g i ðυi ; θi Þ δθi ∂θ i  ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ δυi ∂υi   ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ δθi ∂θ i

δJ^γ ðυ; θ; ρÞ ¼ η

By (50), (43)–(45) is derived. The proof is complete.

Z

Mþ1

0

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ^ δP ðsÞ ∂P^ ðsÞ

∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ δυi þ δθi ∂υi ∂θ i ! T ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ δρ þ λ_ ðsÞδP^ ðsÞ ds: þ ∂ρ

þ

Collecting terms in (48) gives

δJ^γ ðυ; θ; ρÞ ¼ ðηI  λT ðM þ 1ÞÞδP^ ðM þ1Þ 

 ∂hi ðυi Þ ∑ max 0; hi ðυi Þ þ 2ρ ∂υi i¼1  ∂g i ðυi ; θi Þ þ max 0; g i ðυi ; θi Þ ∂υi   ∂Gi ðυi ; θi Þ þ max 0; Gi ðυi ; θi Þ δυi ∂υi Z Mþ1 ∂Hðs; P^ ðsÞ; υ; θ; ρ; λÞ þ δυi ds ∂υi 0  Mþ1  ∂g i ðυi ; θi Þ þ 2ρ  α ∑ max 0; g i ðυi ; θi Þ ∂θ i i¼1 α

Mþ1

ð50Þ



Based on the above discussions, the following algorithm for solving Problem 1 is proposed:

T þ ðγβρβ  1  αρ  α  1 Δðυ; θÞÞδρ  λ ðM þ 1ÞδP^ ðM þ 1Þ T þ λ ð0ÞδP^ ð0Þ þ

ð49Þ

By (41) and (42), from (49), we have  M þ1  ∂h ðυ Þ δJ^ γ ðυ; θ; ρÞ ¼ 2ρ  α ∑ max 0; hi ðυi Þ i i ∂ υi i¼1

0

T T  λ ðM þ 1ÞP^ ðM þ 1Þ þ λ ð0ÞP^ ð0Þ Z Mþ1 T ðHðs; P^ ðsÞ; υ; θ; ρ; λÞ þ λ_ ðsÞP^ ðsÞÞ ds: þ

797

ð48Þ

Algorithm

1. Step

1.

Choose

γ 0 4 0, ρ~ A R, and γ max 4 γ 0 . Step 2. Set γ ≔γ 0 .

ðυ0 ; θ0 ; ρ0 Þ A RM þ 1  RM þ 1  R,

Step 3. Solve Problem 4 by steepest-descent algorithm with n Armijo0 s step-sizes. Let ðυn ; θ ; ρn Þ denote the local minimizer obtained. n Step 4. If ρn o ρ~ , then stop. Take ðυn ; θ Þ as a solution of Problem 4, and go to Step 6. Otherwise, set γ ≔10γ and go to Step 5. n Step 5. If γ r γ max , then set ðυ0 ; θ0 ; ρ0 Þ≔ðυn ; θ ; ρn Þ and go to Step 3. Otherwise stop, the algorithm can0 t find a solution of Problem 1. Step 6. Construct the solution un ðtÞ A U of Problem 1 from n ðυn ; θ Þ. Numerical results in Section 6 will show that Algorithm 1 is a very effective method for solving Problem 1, however, it should be pointed out that generally speaking, the solution obtained by Algorithm 1 is a local minimizer.

6. Numerical results In this section, the effectiveness of the proposed algorithm is shown using two numerical examples. Example 1. Consider an optimal sensor scheduling problem with six sensors and at most seven switches as discussed in [11,12].

798

X. Wu et al. / ISA Transactions 53 (2014) 793–801

The system dynamics are given by #  " #  " x1 ðtÞ x_ 1 ðtÞ 0:5 1 2 _ þ ¼ V ðtÞ: x_ 2 ðtÞ 1 0:5 x2 ðtÞ 2

diagonal elements of the covariance matrix

12

Suppose that there are six sensors described by _ i ðtÞ; y_ i ðtÞ ¼ C i ðtÞxðtÞ þ Di ðtÞW 

P 0 ðtÞ ¼ " C 1 ðtÞ ¼  C 2 ðtÞ ¼ " C 3 ðtÞ ¼

6

0

0

6



" C 6 ðtÞ ¼

1

0

0

1

;

#

0

1 þ 1:2 sin ð2tÞ

0

1 þ 0:5 sin ð2tÞ

1 þ 0:5 sin ð2tÞ

0

0

1 þ 0:5 sin ð2tÞ

0

;

1 þ 0:5 sin ð2tÞ

0

1 þ 0:5 sin ð2tÞ

1 þ 0:5 sin ð2tÞ

0

0

0

0

1 þ 0:5 sin ð2tÞ # 1 þ 1:8 sin ð2tÞ ; 1 þ 1:8 sin ð2tÞ 

Di ðtÞ ¼ Ri ðtÞ ¼

1

0

0

1

;

8

6

4

2

0

0

1

2

3

;

4

5

6

7

8

time(s)

#

Fig. 3. Diagonal elements of the covariance matrix P(t) obtained by Algorithm 1.

; #

1 þ 0:5 sin ð2tÞ 0



#

0

" C 5 ðtÞ ¼



1 þ 1:2 sin ð2tÞ

" C 4 ðtÞ ¼

 ;

7

;

6

5

i ¼ 1; …; 6:

the i−th sensor

where

i ¼ 1; …; 6;

P11(t) P22(t)

10

The cost function is defined by Z 8 TrðPðtÞÞ dt: JðuðtÞÞ ¼

4

3

0

2

Let α ¼ 3, β ¼ 2, ρ~ ¼ 10  2 , γ max ¼ 105 and εi ¼ 0:2; i ¼ 1; …; 6, where ρ~ , γ max , and εi are the tolerance, the upper bound for the penalty parameter and the minimum duration between successive sensors, respectively. Then, the optimal sensor scheduling problem is solved by Algorithm 1 and Matlab 2010a on an Intel Pentium Dual-core PC with 2.60 GB of RAM, and we have υn ¼ ½6; 1; 6T ,

1

0

0

1

2

3

4

5

6

7

8

time(s)

Fig. 4. Optimal sensor schedule obtained by the method given in [11].

7 n

θ ¼ ½0:3571; 0:2143; 7:4826T and ρn ¼ 0:0027. Thus, the optimal switching sequence vector, the optimal switching times vector and the corresponding optimal costs are υ ¼ ½6; 1; 6T , τ ¼ ½0:3571; 0:5714T and J n ¼ 16:0872, respectively. The numerical results are presented in Figs. 2 and 3.

6

the i−th sensor

5

To compare the performance of our algorithm to existing methods, the methods given in [11,12] are also applied to solve the problem without Assumption 3. The corresponding optimal costs are J n ¼ 19:4850 and J n ¼ 15:8803. The numerical results are presented in Figs. 4–7. Fig. 3 implies that our approach is effective. Figs. 2, 4 and 6 suggest that unlike the approaches described in [11,12], our method can avoid the additional switches. More importantly, if Assumption 3 which results in a non-connected feasible region is considered, the methods given in [11,12] are invalid, however, Algorithm 1 is still effective.

4

3

2

1

0 0

1

2

3

4

5

6

time(s)

Fig. 2. Optimal sensor schedule obtained by Algorithm 1.

7

8

Example 2. Consider an optimal sensor scheduling problem with eight sensors and at most eleven switches as discussed in [11]. The

X. Wu et al. / ISA Transactions 53 (2014) 793–801

9 8

P11(t) P22(t)

10

7 8 6

the i−th sensor

diagonal elements of the covariance matrix

12

799

6

4

5 4 3

2 2 0

−2

1

0

1

2

3

4

5

6

7

0

8

0

2

4

10

12

12

diagonal elements of the covariance matrix

7

6

5

the i−th sensor

8

Fig. 8. Optimal sensor schedule obtained by Algorithm 1.

Fig. 5. Diagonal elements of the covariance matrix P(t) by the method given in [11].

4

3

2

1

0

1

2

3

4

5

6

7

P11(t) P22(t) P33(t)

10

8

6

4

2

0

0 8

0

2

4

6

8

10

12

time(s)

time(s)

Fig. 6. Optimal sensor schedule obtained by the method given in [12].

Fig. 9. Diagonal elements of the covariance matrix P(t) obtained by Algorithm 1.

12

9 8

P11(t) P22(t)

10

7 8

6

the i−th sensor

diagonal elements of the covariance matrix

6 time(s)

time(s)

6

4

5 4 3 2

2 1 0

0

1

2

3

4

5

6

7

8

time(s) Fig. 7. Diagonal elements of the covariance matrix P(t) by the method given in [12].

0

0

2

4

6

8

10

time(s) Fig. 10. Optimal sensor schedule obtained by the method given in [11].

12

800

X. Wu et al. / ISA Transactions 53 (2014) 793–801

system dynamics are given by 2 3 2 cos ð3tÞ 0 x_ 1 ðtÞ 6 x_ ðtÞ 7 6 0:3 0:8 sin ð3tÞ 4 2 5¼4 0:2 0:5 x_ 3 ðtÞ 2 3 1:5 6 7 þ 4 1:5 5V_ ðtÞ:

9

32

0:4

x1 ðtÞ

3 8

76 x ðtÞ 7 54 2 5 0:5ð sin t þ cos tÞ x3 ðtÞ 0:2

7

the i−th sensor

6

2 Suppose that there are eight sensors given by

where

2

10 6 P 0 ðtÞ ¼ 4 0 0 

C 4 ðtÞ ¼  C 7 ðtÞ ¼

0

10 0

3

7 0 5; 10

0 0

0 ; 0

1

0

0

0

1

0

1 C 1 ðtÞ ¼ 1 

0

;

1 6 Q ¼40 0

 C 5 ðtÞ ¼

0:5

0

0 1 0

0 ; 1

0

0

3

0

0 ; 0

0

0

0

0

1

1

 C 8 ðtÞ ¼

2

7 0 5; 1

1

0 0:7

4 3

1 1

0 C 2 ðtÞ ¼ 0

0:7



2



0:6 ; 0

Di ðtÞ ¼ Ri ðtÞ ¼

i ¼ 1; …; 8;

1





 ;

0 0

1 ; 1

0

0

1

1

0

0

0 C 3 ðtÞ ¼ 0 C 6 ðtÞ ¼

0

2

4

6

8

10

12

Fig. 12. Optimal sensor schedule obtained by the method given in [12].

;

12

0 ; 0:6

0 0:5

0

time(s)

i ¼ 1; …; 8:

The cost function is given by Z 12 TrðPðtÞÞ dt: JðuðtÞÞ ¼ 0

Let α ¼ 3, β ¼ 2, ρ~ ¼ 10  2 , γ max ¼ 105 and εi ¼ 0:3; i ¼ 1; …; 8, where ρ~ , γ max , and εi are the tolerance, the upper bound for the penalty parameter and the minimum duration between successive sensors, respectively. Then, the optimal sensor scheduling problem is solved by Algorithm 1 and Matlab 2010a on an Intel Pentium Dual-core PC with 2.60 GB of RAM, and we have υn ¼ ½5; 7; 5; 7T , θn ¼ ½3:8042; 1:5149; 5:2818; 1:3991T and ρn ¼ 0:0053. The optimal switching sequence vector, the optimal switching times vector and the corresponding optimal costs are υ ¼ ½5; 7; 5; 7T ,

diagonal elements of the covariance matrix

_ i ðtÞ; y_ i ðtÞ ¼ C i ðtÞxðtÞ þ Di ðtÞW

5

P11(t) P22(t) P33(t)

10

8

6

4

2

0

0

2

4

6

8

10

12

time(s) Fig. 13. Diagonal elements of the covariance matrix P(t) by the method given in [12].

τ ¼ ½3:8042; 5:3191; 10:6009T and J n ¼ 102:5704, respectively. The

12

diagonal elements of the covariance matrix

numerical results are presented in Figs. 8 and 9. P11(t) P22(t) P33(t)

10

8

6

4

2

0

0

2

4

6

8

10

12

time(s) Fig. 11. Diagonal elements of the covariance matrix P(t) by the method given in [11].

To compare the performance of our algorithm to existing methods, the methods given in [11,12] are also applied to solve the problem without Assumption 3. The corresponding optimal costs are J n ¼ 116:7702 and J n ¼ 100:4693. The numerical results are presented in Figs. 10–13. Fig. 9 implies that our approach is effective. Figs. 8, 10 and 12 suggest that unlike the approaches described in [11,12], our method can avoid the additional switches. More importantly, if Assumption 3 which results in a non-connected feasible region is considered, the methods given in [11,12] are invalid, however, Algorithm 1 is still effective. Above numerical results show that Algorithm 1 is an effective alternative approach for the optimal sensor scheduling problem in continuous time, and the algorithm can avoid the additional switches. In addition, it is important to point out that the optimal sensor scheduling problem described in this paper can be effectively solved by Algorithm 1, but such problem cannot be solved by the approaches given in [11,12], due to Assumption 3 leading to a non-connected feasible region.

X. Wu et al. / ISA Transactions 53 (2014) 793–801

7. Conclusion In this paper, an optimal sensor scheduling problem in continuous time is studied. With three assumptions satisfied, the model is more close to the practical problems. However, the switching times are unknown, and the feasible region is not connected. Thus, it is difficult to solve such a problem by conventional optimization techniques and the methods given in [11,12]. To overcome this difficulty, based on a combination of the binary relaxation, time-scaling transformation and exact penalty function, an efficient computational method for solving such a problem is proposed. Finally, two numerical examples are presented to show the effectiveness of the proposed algorithm. Acknowledgments The authors express their sincere gratitude to Professor A.B. Rad, the editor and the anonymous reviewers for their constructive comments in improving the presentation and quality of this manuscript. This work was supposed by the Chinese National Outstanding Youth Foundation under Grant no. 61125306, the Major Program of Chinese National Natural Science Foundation under Grant no. 91016004 and 11190015, and the Chinese National Natural Science Foundation under Grant no. 61374006. References [1] Corke P, Wark T, Jurdak R, Hu W, Valencia P, Moore D. Environmental wireless sensor networks. Proc IEEE 2010;98:1903–17. [2] Gungor VC, Lu B, Hancke GP. Opportunities and challenges of wireless sensor networks in smart grid. IEEE Trans Ind Electron 2010;57:3557–64. [3] Clark J, Fierro R. Mobile robotic sensors for perimeter detection and tracking. ISA Trans 2007;46:3–13. [4] Nordman MM. A task scheduler framework for self-powered wireless sensors. ISA Trans 2003;42:535–45. [5] Howitt I, Manges WW, Kuruganti PT, Allgood G, Gutierrez JA, Conrad JM. Wireless industrial sensor networks: framework for QoS assessment and QoS management. ISA Trans 2006;45:347–59. [6] Cao XH, Chen JM, Zhang Y, Sun YX. Development of an integrated wireless sensor network micro-environmental monitoring system. ISA Trans 2008;47:247–55. [7] Ko J, Lu C, Srivastava MB, Stankovic JA, Terzis A, Welsh M. Wireless sensor networks for healthcare. Proc IEEE 2010;98:1947–60. [8] Shi D, Chen T. Approximate optimal periodic scheduling of multiple sensors with constraints. Automatica 2013;49:993–1000.

801

[9] Baras LS, Bensoussan A. Optimal sensor scheduling in nonlinear filtering of diffusion process. SIAM J Control Optim 1999;27:786–813. [10] Lee HWJ, Teo KL, Lim AEB. Sensor scheduling in continuous time. Automatica 2001;37:2017–23. [11] Feng ZG, Teo KL, Rehbock V. Optimal sensor scheduling in continuous time. Dyn Syst Appl 2008;17:331–50. [12] Woon SF, Rehbock V, Loxton RC. Global optimization method for continuoustime sensor scheduling. Nonlinear Dyn Syst Theory 2010;10:175–88. [13] Gupta V, Chung TH, Hassibi B, Murray RM. On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage. Automatica 2006;42:251–60. [14] Oshman Y. Optimal sensor selection strategy for discrete-time state estimators. IEEE Trans Aerosp Electron Syst 1994;30:307–14. [15] Feng ZG, Teo KL, Rehbock V. Hybrid method for a general optimal sensor scheduling problem in discrete time. Automatica 2008;44:1295–303. [16] Chung TH, Gupta V, Hassibi B, Burdick J, Murray RM. Scheduling for distributed sensor networks with single sensor measurement per time step. In: IEEE International Conference on Robotics and Automation, New Orleans; 2004. [17] Wu X, Zhang K, Sun C. Parameter tuning of multi-proportional-integralderivative controllers based on optimal switching algorithms. J Optim Theory Appl 2013;159:454–72. [18] Gerdts M, Kunkel M. A nonsmooth Newton0 s method for discretized optimal control problems with state and control constraints. J Ind Manag Optim 2008;4:247–70. [19] Yu C, Teo KL, Zhang L, Bai Y. A new exact penalty function method for continuous inequality constrained optimization problems. J Ind Manag Optim 2010;6:895–910. [20] Teo KL, Goh CJ, Wong KH. A unified computational approach to optimal control problems. Essex: Longman Scientific and Technical; 1991. [21] Wu X, Zhang K, Sun C. Constrained optimal control of switched systems based on modified BFGS algorithm and filled function method. Int J Comput Math; 2013. doi:10.1080/00207160.2 013.859678. [22] Ahmed NU. Linear and nonlinear filtering for scientists and engineers. Singapore: World Scientific; 1998. [23] Anderson BDO, Moore JB. Optimal control: linear quadratic methods. Englewood Cliffs, NJ: Prentice-Hall; 1990. [24] Nocedal J, Wright SJ. Numerical optimization. New York: Springer; 2006. [25] Luenberger DG, Ye Y. Linear and nonlinear programming. New York: Springer; 2008. [26] Wu X, Zhang K, Sun C. Constrained optimal control of switched systems and its application. Optimization; 2013. doi:10.1080/02331934. 2013.784764. [27] Wu X, Zhang K, Sun C. Optimal control of switched systems with an inequality constraint based on smooth penalty function method. Sensor Lett 2013;11: 945–52. [28] Liu Y, Teo KL, Jennings LS, Wang S. On a class of optimal control problems with state jumps. J Optim Theory Appl 1998;98:65–82. [29] Teo KL, Jennings LS, Lee HWJ, Rehbock VL. The control parameterization enhancing transform for constrained optimal control problems. J Austr Math Soc Ser B: Appl Math 1999;40:314–35. [30] Lee HWJ, Teo KL. Control parametrization enhancing technique for solving a special ODE class with state dependent switch. J Optim Theory Appl 2003;118: 55–66.