From semi-tensor product of matrices to logical control systems

In this paper, we address the problem of calculating basis siphons in Petri nets (PNs) by resorting to the semi-tensor product (STP) of matrices. The proposed appraoch is based on our previous results on the calculation of siphons and minimal siphons in PNs, whose key notion is that of the siphon equation (SE) of PNs. First, we discuss the properties of the SE of PNs in the framework of STP. Second, an efficient recursion algorithm is proposed, which can be applied to computing all basis siphons for any PNs. Some results on the computational complexity of the proposed algorithm, in this paper, are also provided, as well as comparison with the existing approach. Last, an example is presented to illustrate the theoretical results. The proposed approach proved highly effective in calculating all basis siphons of PNs and only requires matrix manipulations.

The wolf‐sheep‐cabbage puzzle and the missionaries‐cannibals problem are two classical intellectual problems, and various methods have been developed to solve them. This paper proposes a new logical control system (LCS), called the incomplete LCS, which allows controls to apply to certain particular states. Using semi‐tensor product (STP) of matrices, the incomplete LCS can be expressed into its algebraic form, and then the controls are designed to reach our goal. Using algebraic state space technique, the two intellectual problems are converted to controllability of incomplete LCSs with restricted state space. Then the solvability and related algorithms are developed.

This paper proposes an improved algorithm for enumerating siphons and minimal siphons of Petri nets (PNs) in the framework of [18]. First, the logical equation of each transition is converted into a matrix equation by using the semi-tensor product (STP) of matrices, and then the matrix equation group of the PNs is obtained. Second, an improved algorithm is proposed to calculate the siphons and minimal siphons, respectively. Finally, two illustrative examples are presented to show the efficiency and application of the improved algorithm.

In this study, we investigate the control problem of asynchronous sequential machine with adversarial input by using the semi-tensor product (STP) of matrices. First, by applying the STP of matrices, a new algebraic expression of asynchronous sequential machines with an adversarial input is deduced. Next, by investigating the control process, the controller with the minimal state set is designed, and two algorithms for assigning the values to the state transition structure matrix and the output structure matrix of the controller are proposed. Then, in the framework of the algebraic expression of the asynchronous sequential machines with adversarial input and two proposed algorithms, an algebra expression of the closed-loop system dynamic is established; the theoretical results are verified by using this algebra expression. Finally, an example is presented to illustrate the validity and application of the proposed approach.

ABSTRACTIn this paper, the static output feedback stabilisation of discrete event dynamic systems (DEDSs) is investigated via the semi-tensor product (STP) of matrices. Firstly, the dynamics of DEDSs modelled by deterministic Moore-type finite automata are converted into an matrix expression in the STP frame. Secondly, necessary and sufficient conditions for the existence of a static output feedback control pattern, stabilising the controlled discrete event dynamic systems to some equilibrium point, are given, and constructive algorithms to seek the static output feedback control pattern including an effective specific solution algorithm and an analytic solution algorithm are proposed. Thirdly, the equilibrium-based static output feedback stabilisation of DEDSs is extended to the set-based static output feedback stabilisation and necessary and sufficient conditions and constructive algorithms to seek the corresponding control pattern are provided. Finally, some examples are presented to illustrate the effectiveness of the proposed approach.

Logical dynamical systems (LDSs) have wide applications in gene regulation, game theory, digital circuits, and so on. In LDSs, phenomena such as impulsive effect, time delays, and asynchronous behavior are not negligible, which generate complex LDSs. This paper presents a detailed survey on models and methods of investigating LDSs. Firstly, some preliminary results on LDSs and semi-tensor product (STP) method are presented. Secondly, some new developments on modeling complex LDSs are summarized, including switched LDSs, probabilistic LDSs, delayed LDSs, LDSs with impulsive effects, asynchronous LDSs, constrained LDSs, and implicit LDSs. Finally, the control design techniques of LDSs are reviewed, including reachable set approach, sampled-data control, event-triggered control, and control Lyapunov function method.

Semi-tensor product of matrices (STP of matrices) is a new matrix product and has been successfully applied to many fields, especially to logical dynamic systems. This paper considers how to represent a Boolean matrix as an exponentiation of another one in the framework of the STP of matrices. To this end, Boolean matrices (BMs) are classified into three types, single block, full block and non-full block BMs. For each kind of BMs, a sufficient and necessary condition for the exponentiation-representability is proposed, and an algebraic algorithm of finding all the exponentiation representations of a Boolean matrix is designed. As potential applications, a discussion of how to use the obtained results to analyze and synthesize logical systems mathematically, especially to finite state machines, is finally presented. The results of this paper may provide a theory basis for studying finite-valued systems, logical systems, logical networks and networked evolutionary games, etc.

Consider a networked evolutionary game (NEG). According to its strategy updating rule, a fundamental evolutionary equation (FEE) for each node is proposed, which is based on local information. Using FEEs, the network strategy profile dynamics (SPD) is expressed as a $k$ -valued (deterministic or probabilistic) logical dynamic system. The SPD is then used to analyze the network dynamic behaviors, such as the fixed points, the cycles, and the basins of attractions, etc. Particularly, when the homogeneous networked games are considered, a necessary and sufficient condition is presented to verify when a stationary stable profile exists. Then the equivalence of two NEGs is investigated. Finally, after a rigorous definition of controlled NEGs, some control problems, including controllability, stabilization, and network consensus, are considered, and some verifiable conditions are presented. Examples with various games are presented to illustrate the theoretical results. The basic tool for this approach is the semi-tensor product (STP) of matrices, which is a generalization of the conventional matrix product.

As a kind of commutative and associative four‐dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi‐tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure‐preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high‐level operations, relation of operations between commutative quaternion matrices and their ‐representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure‐preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure‐preserving algorithm for strict image authentication.

In this paper, we present a comprehensive modeling technique for bounded Petri net systems (BPNSs) in the framework of the semi‐tensor product (STP) of matrices. The two dynamic properties of BPNSs, namely, reachability and controllability, are investigated systematacially. First, the dynamics of a bounded Petri net system (BPNS), by resorting to the STP of matrices, are expressed in the form of a discrete‐time bilinear equation, which is called the marking evolution equation (MEE) of BPNSs. Second, controllability and transition‐marking adjacency matrix (TMAM) of BPNSs are defined, respectively. Further, several necessary and sufficient conditions for reachability and controllability of BPNSs are given in terms of the MEE and TMAM. Third, an efficient algorithm to verify reachability property of BPNSs, in this paper, is provided, as well as its computational complexity. Finally, an example is presented to illustrate the theoretical results in this paper. The main contribution of this paper is the presentation of a precise mathematical model for BPNSs. The main advantage of the proposed approach is that not only it can be applied to verify whether or not any given marking is reachable from the other in state space, but also it is very convenient to find all firing sequences between any two reachable markings.

Using semi-tensor product and the vector form of multi-valued logical variables, the mix-valued logical system is expressed as a linear discrete time system and the mix-valued logical control system is expressed as a bilinear discrete time system with respect to the state and control variables. In this paper, the topological property of mix-valued logical network and the reachability and controllability of mix-valued logical control systems are discussed. From the algebraic form of mix-valued logical network, we obtain the number of the logical system. The formula we obtained is based on the rank of structure matrix. Also, the necessary and sufficient conditions are obtained for testing the reachability and controllability.

This study proposes a novel method for the set stability analysis of switched Boolean networks(SBNs) under restricted successors based on the semi-tensor product(STP) of matrices. We demonstrate that all allowable switching signals can be generated by a logic dynamical system (LDS) with free inputs, termed as the generator of switching signals(GSS). Thus, we regard the GSS as a part of the system through series interconnection and transfer the original restricted SBN into a logic system with free inputs. Then, we merge the GSS and the SBN using the STP and prove that the set stability of the original restricted SBN is equivalent to that of the merged LDS under arbitrary switching. Based on this, a necessary and sufficient condition of set stability is obtained. Further, a numerical example is provided to illustrate the use of the proposed result.

In this brief, we investigate the robustness of outputs with respect to disturbances for Boolean control networks (BCNs) by semi-tensor product (STP) of matrices. First, BCNs are converted into the corresponding algebraic forms by STP, then two sufficient conditions for the robustness are derived. Moreover, the corresponding permutation system and permutation graph are constructed. It is proven that if there exist controllers such that the outputs of permutation systems are robust with respect to disturbances, then there must also exist controllers such that the outputs of the corresponding original systems achieve robustness with respect to disturbances. One effective method is proposed to construct controllers to achieve robustness. Examples are also provided to illustrate the correctness of the obtained results.

In this paper we consider the solvability of the matrix equations , with respect to the semi-tensor product. Using the concept of the semi-tensor product which was initially proposed by Cheng et al. [An introduction to Semi-Tensor product of matrices and its applications, World Scientific, 2012], firstly we investigate the matrix–vector equations , under the semi-tensor product, establish some compatible conditions for the matrices, and propose a necessary and sufficient condition for the solvability of the matrix–vector equation. We then explore the solvability of the matrix equations , under the semi-tensor product. Some elementary examples are given to show the efficiency of the results.

This paper investigates the output controllability problem of temporal Boolean networks with inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network is a logical dynamic system describing cellular networks with time delays. Using semi-tensor product of matrices, the temporal Boolean networks can be converted into discrete time linear dynamic systems. Some necessary and sufficient conditions on the output controllability via two kinds of inputs are obtained by providing corresponding reachable sets. Two examples are given to illustrate the obtained results.