Context-awareness is an important characteristic for smart home. Context-aware system reacts and adapts according to the changes in the domain environment. Researchers in different fields make a lot of contribution for context-aware system. The goal of the context-aware system in smart home is to provide services that maximize the user’s comfort and safety while minimizing the user’s explicit interaction with the environment as well as the cost of the service [1] . Context-aware systems can be implemented using production rules (if-then relationships), neural networks, support vector machines, fuzzy logic, Bayesian networks, etc. [2] - [5] . Boolean network (BN) was introduced by Kauffman to formulate the cellular networks [6] . A BN network consists of a set of Boolean variables whose state is determined by other variables in the network. BN is also used for modeling some other complex systems such as neural networks, social and economic networks. The state of each variable at time t + 1 can be determined by the state of its spatial neighbors at time t. A BN network with n variables has 2ⁿ possible states and the dynamics of BN is converted into an equivalent algebraic form as a standard discrete-time linear system using the matrix expression of logic. One possible way to enrich the dynamics is to consider BN with inputs and outputs called Boolean control network (BCN). Our aim is to create a mathematical model that explains the system and represents the effects of different components and to make predictions about behavior. A context-aware system can be considered as a Boolean control networks, where context elements are treated as control inputs and different services as state variables. We used left semi-tensor matrix product to describe the dynamics of the system.

In real environment, different types of sensor is used to collect situational (user ID, location, activity, etc.) and environmental (temperature, humidity, illumination, CO₂, etc.) data. Context is formed with this data. Context- aware system offer different types of services to the user based on the context information. The set of rules (logic expression) are used for this purpose. Using left semi tensor product, we can derive linear algebraic equation of these logic expression. These linear algebraic equations express state-spaces. For smart home architecture, an important term is inference. Inference system can deduce knowledge from knowledge base which helps to provide context-aware service for home user. Matrix expression is a new and convenient approach for logic inference.

The rest of the paper is organized as follows: Section 2 presents a brief review of related topics for Boolean control network. Section 3 presents mathematical modeling of context-aware system followed by conclusion and future work are drawn in Section 4.

A Boolean control network (BCN) is a discrete-time logical dynamic control system. Its dynamics can be expressed as

where, are n + m-ary logical functions, are states, are control inputs.

Boolean control network can be represented as directed graph with n nodes and m inputs. An edge from node i to node j represents that node j is effected by node i. Figure 1 shows the graphical representation of a BCN.

D. Cheng, et al. proposed the concept of left semi-tensor product and used it to represent BCNs in a linear algebraic state-space form [7] [8] . A briefly review on some topics are represented, which is useful for studying BCNs in a control-theoretic framework.

In smart home context-aware system provides different types of services. We can define each service as a logical function of context elements and states. These context elements and states are Boolean logical variables with value either true or false. Logical equation of smart home context-aware system are expressed as

where, are n-ary logical functions, are logical functions, are states, are outputs, are control inputs.

This model realizes five output services, Morning Call Service (y₁), normal service (y₂), Entertainment Service (y₃), Sleeping Service (y₄) and Guarding Service (y₅). For each service we consider separate service state. So there are five service states, Morning Call state (x₁), normal state (x₂), Entertainment state (x₃), Sleeping state (x₄), and Guarding state (x₅). For input it uses eight context elements with Boolean logic values true and false. These context elements are categorized as person u₁ (father (p), mother (q), son (r), daughter (s)), time (morning (u₂), evening (u₇), night (u₅)), location (bedroom (u₃), sofa (u₈), outside (u₆)) and activity (lying (u₄)). Each service state can be defined by logical relation between the context elements. Figure 2 shows the state diagram of this system.

Using logic variables we can express logic Equation (9) as follows:

This logical function (11) can be converted into an algebraic function using left semi-tensor matrix product (⋉), definition and theorems described in introduction section as:

where, , M_d structure matrix, transition matrix. Table 1 listed some of the structure matrix used in Boolean control network.

Using Equation (12) logic equation can be simplified as

The logical functions in (13) can be represented by structure matrix of logical operator and the applying the properties of left semi-tensor product can be represented as:

where, M_c = structure matrix of logical conjunction operation, M_d = structure matrix of logical disjunction operation, I₂ = identity matrix and W_[2] = swap matrix.

Applying left semi-tensor product on both sides the system of logical Equation (14) can be converted into a linear algebraic equation as:

where,. The first and last few columns of L are d₃₂ [14 14 10 10 14 14 10 10 14 14 10 10 14 14 16 16 14 ××× 32 32 32 32 32 32 32 32 32].

Morning call service (y₁), sleeping service (y₄), entertainment service (y₃) is a subset of normal service. When any of the service (morning call, sleeping and entertainment) is true that time normal service will be false. We can define each services as

The logical functions in (16) can be represented by structure matrix of logical operator and the applying the properties of left semi-tensor product can be represented as:

where, M_c = structure matrix of logical conjunction operation, M_d = structure matrix of logical disjunction operation, I₂ = identity matrix and W_[2] = swap matrix.

Applying left semi-tensor product on both sides the system of logical equations can be converted into a linear algebraic equation as:

where, , H = δ₃₂ [10 10 16 16 16 16 16 16 ××× 31 32 31 32 31 32].

Every service is composed of some controlling facilities to turn on and off of home appliances and device. Morning call service (c₁) turns on the alarm device (b), room light (c), blind (d), coffee maker (e), water heater (f). Sleeping service (c₂) turns off the room light and blind. Whenever nobody is inroom that time Guarding service (c₃) turns off all the appliances and device. Entertainment service (c₄) turns on the TV (h), room light. When a user is in home then normal service (c₅) turns on light, air conditioner (g), blind. We can describe the service by the output function of home appliances and device.

Applying left semi-tensor matrix product together on both sides of algebraic equation of (18).

where, , ,

The first and last few columns L_s ared₃₂ [32 16 32 32 32 32 32 32 32 32 32 32 32 32 ××× 32 32 32 32 24 32 28]. This liner model considers vector values of logic, in real world we use scalar values for controlling. According to algorithm presented in Section 1.6 the values of different service outputs can be calculated and tabulated in Table 2.

Boolean control network is a logic-based control system. Boolean control network is realizable in Smart home context-aware control system. All the logical variables represented by values 1 or 0 in this system. In this paper, we have used Boolean control network for context-aware system. To control any system modeling is an important issue. We have defined the logic relation between context information and state variables. Then these logic relations are converted to matrix expression using semi tensor matrix product. With this matrix expression, we can control and do inference easily. At present, we have applied the network with 5 nodes, number of nodes can be extended based on the state of the service. In future, we will use k-value logic to express the logic relation more efficiently.

M. Humayun Kabir,M. Robiul Hoque, (2015) Context-Aware System Modeling Based on Boolean Control Network. Open Journal of Applied Sciences,05,661-668. doi: 10.4236/ojapps.2015.511065