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Control design for multi-variable fuzzy systems with application to parallel hybrid electric vehicles
Abstract
This paper studies the fuzzy logic controller (FLC) design for multi-variable fuzzy systems based on the semi-tensor product of matrices, and presents several new results. A new expression of fuzzy rules for multi-variable FLC is introduced, which is very convenient to use in fuzzy logic inference. Based on the new expression of fuzzy rules, the complex fuzzy reasoning is converted into simple algebraic equations by constructing structural matrices of the FLC. A simulation example is given to demonstrate the effectiveness of the proposed approach. A set of least in-degree controls that remove possible fabricated variables are constructed, and an algorithm is given to design the least in-degree controls when the control rules are incomplete. Principles are proposed for dealing with the inconsistency of fuzzy control rules. Finally, the results obtained in this paper are applied to the design of fuzzy controller for energy management and control strategy of parallel hybrid electric vehicles (PHEV).
... Similar to the proof of theorem in paper [23], we can obtain Theorem 3.1. ...
This paper mainly studied the problem of solving interval type-2 fuzzy relation equations . First, to solve the interval type-2 fuzzy relation equations, we extend the semi-tensor product of matrices to interval matrices and give its specific definition. Second, the interval type-2 fuzzy relation equation was divided into two parts: primary fuzzy matrix equation and secondary fuzzy matrix equation . Since all elements of equal to one, only the principal fuzzy matrix equation needs to be considered. Furthermore, it was proved that all solutions can be obtained from the parameter set solutions if the primary fuzzy matrix equation is solvable. Finally, with semi-tensor product of interval matrices, the primary fuzzy matrix equation was transformed into an algebraic equation and the specific algorithm for solving an interval type-2 fuzzy relation equation was proposed.
... In addition, it possesses some incomparable advantages over the latter, such as interchangeability and complete compatibility. Due to these advantages, the STP is widely used in various fields, such as engineering [2], image encryption [3,4], Boolean networks [5,6], networked games [7,8], classical logic and fuzzy mathematics [9,10], finite state machines [11,12], finite systems [13] and others. ...
We investigate the Riccati matrix equation in which the conventional matrix products are generalized to the semi-tensor products . When A and B are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of A and B . We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.
... Lv et al. 16 proposed to optimize the fuzzy relation using semi-tensor product matrices, which can take the fuzzy inference language into the matrix. Ge et al. 17 proposed an approach that applies the semi-tensor product theory in order to design a multivariable controller, which is then used in the control field of hybrid cars. Duan et al. 8 and Lv et al. 18 studied the fuzzy control of air conditioning in an indoor thermal comfort environment, which builds the fuzzy relationship matrix and calculates a result based on the semitensor product theory. ...
A multivariable fuzzy controller was designed to solve the control problem of vegetable waste fermentation. The knowledge-based fuzzy reasoning was constructed according to the sample data from both the fermentation process and the rule of temperature change in the control process. It is represented using the mathematical method of semi-tensor product matrices. The structural matrix of the fuzzy logic was constructed and the complex fuzzy reasoning was converted into simple matrix operations. In addition, a new algorithm based on the least in-degree method is put forward, solving the problem of control rules being incomplete and inconsistent. It was used for the design of the fuzzy controller on vegetable waste fermentation. The result of experiment proved that this control method is effective; the control method improves the adjustment speed as well as the control precision.
... Ref. [150] solved the fuel efficiency optimization problem for commuting vehicles. In addition, Ref. [169] studied the fuzzy logic controller design for multi-variable fuzzy systems based on STP method and applied the theoretical results to the design of fuzzy controller for energy management and control strategy of parallel hybrid electric vehicles. ...
Semi-tensor product (STP) of matrices has attracted more and more attention from both control theory and engineering in the last two decades. This paper presents a comprehensive survey on the applications of STP method in engineering. Firstly, some preliminary results on STP method are recalled. Secondly, some applications of STP method in engineering, including gene regulation, power system, wireless communication, smart grid, information security, combustion engine and vehicle control, are reviewed. Finally, some potential applications of STP method are predicted. © 2017, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.
... Based on the algebraic form, many fundamental results on the analysis and control of Boolean networks have been presented, which include the controllability and observability [7][8][9][10][11][12][13] , the stability and stabilization [14][15][16][17][18][19][20] , the disturbance decoupling [21][22][23] , the optimal control [24][25] and the synchronization [26][27] . Besides, the semi-tensor product method has also been used in many other fields such as the general logical system [28][29] , the fuzzy control [30][31][32] , the calculation of Boolean derivative [33][34] , the finite automata [35][36] , the graph coloring [37] and the game theory [38][39] . ...
The stability analysis of gene regulatory networks is a hot topic in systems biology. We investigate the stability of switched singular Boolean networks (SSBNs) by using the semi-tensor product of matrices. First, the dynamics of SSBNs is converted to an algebraic form, based on which a necessary and sufficient condition is established for the uniqueness of solution of SSBNs. Second, several necessary and sufficient conditions are presented for the stability of SSBNs under arbitrary switching signal and the switching stabilizability of SSBNs, respectively, by converting an SSBN into an equivalent switched Boolean network. Two illustrative examples are presented to show that the main results obtained in this paper are effective in analyzing the stability of SSBNs.
The process of atmospheric pollutants traceability based on unmanned aerial vehicles (UAVs) is affected by many factors that can impact and increase the complexity of the traceability of atmospheric pollutants. In this study, we proposed a new algorithm called the fuzzy control traceability (FCT) to track odor plumes. Our proposed algorithm combined the characteristics and fuzzy control of the UAV and designed a controller based on the actual environment of the UAV. The fuzzy controller fuzzed the input gas concentration information, established fuzzy control rules by imitating human brain thinking, and outputted the turning angle and the move length according to rules, thus realizing intelligent tracking of the odor plume by the UAV. We compared the FCT algorithm with the bio-inspired “ZigZag” algorithm to validate its performance. Various concentration fields were constructed, and ten sets of experiments are performed using the two algorithms in different concentration fields. The average success rate of the FCT algorithm under different concentration fields was 95.4% higher than that of the ZigZag algorithm.
Implications: Fuzzy control logic is applied to the field of air pollutant traceability of drones, and a single drone traceability algorithm based on fuzzy control is proposed; and in view of the shortcomings of a single traceability subject in the traceability, multiple traceability subjects are introduced to optimize fuzzy control.
This paper provides a comprehensive survey on semi-tensor product (STP) of matrices and its applications to different disciplines. First of all, the STP and its basic properties are introduced. Meanwhile, its inside physical meaning is explained. Second, its application to conventional dynamic systems is presented. As an example, the region of attraction of stable equilibriums is discussed. Third, its application to logical systems is presented. Particularly, the algebraic state space representation of logical systems and the important role it plays in analysis and control of logical systems are emphasized. Fourth, its application to finite games is discussed. The most interesting problems include potential game, evolutionary game, and game theoretic control. Finally, the mathematical essence of STP is briefly introduced.
The logical dynamic system in this paper stands for the systems where the state variables can take only finite values. Particularly, when the number is 2 it is a classical logic (or Boolean logic); k-valued logic, and general finitely valued (general) logic. In recent years, using semi-tensor product of matrices the algebraic state space approach to logical dynamic systems has been developed and widely appreciated. It has been used to many engineering problems and to theoretical researches. Parallel to the Kalman state space approach to continuous state space dynamics where the differential equations or difference equations are used to describe the dynamic systems, the algebraic state space approach may provide a convenient platform for analyzing and control design of logical systems. The purpose of this paper is two fold: First, we give a brief survey on this new approach; then we introduce its current research topics and main results. Finally many applications and predict the potential of its further applications are presented. ©, 2014, South China University of Technology. All right reserved.
A new kind of scheme is proposed to obtain hierarchical fuzzy control of multivariable systems. It is based on algebraic fuzzy reasoning equations of fuzzy system using semi-tensor product framework recently described. First, the bi-decomposition of multivariable fuzzy systems is considered, and some new results are obtained. Using these results, the algorithm of hierarchical fuzzy systems realization is developed such that one can easily design the middle layers of the hierarchical structure. The resulting hierarchical system is totally equivalent to the conventional single layer fuzzy logic system. At last, an example is given to demonstrate that the algorithm is effective and feasible.
The problem of solving type-2 fuzzy relation equations is investigated. In order to apply semi-tensor product of matrices, a new matrix analysis method and tool, to solve type-2 fuzzy relation equations, a type-2 fuzzy relation is decomposed into two parts as principal sub-matrices and secondary sub-matrices; an r-ary symmetrical-valued type-2 fuzzy relation model and its corresponding symmetrical-valued type-2 fuzzy relation equation model are established. Then, two algorithms are developed for solving type-2 fuzzy relation equations, one of which gives a theoretical description for general type-2 fuzzy relation equations; the other one can find all the solutions to the symmetrical-valued ones. The results can improve designing type-2 fuzzy controllers, because it provides knowledge to search the optimal solutions or to find the reason if there is no solution. Finally some numerical examples verify the correctness of the results/algorithms.
This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.
This paper investigates the stochastic fuzzy logic and controller design of stochastic fuzzy systems based on the semi-tensor product of matrices. First, some concepts and properties on stochastic fuzzy logic are given. Then, the design of stochastic fuzzy controller is studied based on the semi-tensor product, and the algebraic form of stochastic fuzzy reasoning is obtained by constructing the structure matrix and the transition probability matrix. Finally, a numerical example is provided to demonstrate the new results in this paper.
The problem of solving max–min fuzzy relational equations is investigated. First, we show that if there is a solution, then there is a corresponding solution within the set of parameters [briefly, the parameter set solution (PSS)]. Then, the semitensor product of matrices is used to convert the logical equations into algebraic equations via the vector expression of logical variables. Under this form, every PSS can be obtained. It is proved that all the solutions can be revealed from their corresponding PSS. Some examples are presented to demonstrate the algorithm to solve fuzzy relational equations.
Boolean functions are of special importance, though they are the simplest class of finite-valued functions. It is widely applied to many fields, including information, control and so on. The theoretical foundation of Boolean functions is introduced in this paper, involving some main results from Boolean algebra to Boolean Calculus, applications in information and control and some recent frontiers and developments. The semi-tensor product approach to these fields is introduced emphatically.
For nonlinear control systems with multiple outputs which cannot be decoupled, we make use of the sampling data of the real system to obtain a fuzzy relation matrix model via the semi-tensor product (STP) operation of matrices, and design a novel fuzzy controller based on this model. In this method, the traditional fuzzy logic-reasoning and fuzzy rulebuilding in the fuzzy control system are converted into matrix algebraic computations, effectively reducing the complexity in modeling and control. This method has been applied to design the controller for indoor thermal comfort control. On the basis of the indoor and outdoor temperature, humidity etc., we calculate the fuzzy relation matrix and build the fuzzy controller. Experiments and simulation results show that this fuzzy controller can realize the optimal control of the thermal comfort system, without the need of decoupling multiple outputs.
This paper gives a matrix expression of logic. Under the matrix expression, a general description of the logical operators
is proposed. Using the semi-tensor product of matrices, the proofs of logical equivalences, implications, etc., can be simplified
a lot. Certain general properties are revealed. Then, based on matrix expression, the logical operators are extended to multi-valued
logic, which provides a foundation for fuzzy logical inference. Finally, we propose a new type of logic, called mix-valued
logic, and a new design technique, called logic-based fuzzy control. They provide a numerically computable framework for the
application of fuzzy logic for the control of fuzzy systems.
In the analysis and design of expert control systems, when linguistic types of control algorithms are used, it is important to model a physical system using fuzzy theoretic approach. The modeling, analysis, and synthesis of multivariable structures of fuzzy control systems is discussed. First, a multivariable open-loop fuzzy system is described by a set of fuzzy equations. Then a block diagram of the system consisting of functional blocks and intersectional blocks is presented. Series, parallel, and series-parallel connections of multivariable open-loop systems are also analyzed. Multivariable feedback structures are depicted by a set of fuzzy equations and block diagrams. Then the theory of multivariable-multilevel structures is presented. An example of two interconnected tank liquid-level system is given.