Logical Dynamical System Research Articles

Output feedback stabilization of logical dynamical systems with state‐dependent control constraints

Output feedback stabilization of logical dynamical systems with state‐dependent control constraints

Asymptotical Stabilization of Logic Dynamical Systems via Output-Based Random Control

Asymptotical Stabilization of Logic Dynamical Systems via Output-Based Random Control

Hybrid model predictive control for vehicle trajectory tracking with integrated attitude adjustment

This paper presents a vehicle trajectory tracking method that integrates attitude adjustment. The method utilizes the asymmetric adjustment of the damping of the adjustable damping shock absorber to lift the left and right vehicle body and carry out the attitude adjustment of the vehicle. Additionally, it actively tilts toward the inside of the curve during turning. In order to address the nonlinear constraints of damping asymmetry regulation and the mixed logic relationship between suspension asymmetry damping switching and left and right body lifting during body attitude regulation, a mixed logic dynamic system (MLD) for vehicle attitude adjustment is established. Subsequently, the vehicle trajectory tracking controller is designed using the hybrid model predictive control theory (HMPC), which integrates active steering control and attitude adjustment. The optimal control problem of hybrid systems is transformed into a hybrid integer quadratic programming problem and solved through finite time domain rolling optimization. The effectiveness of the proposed integrated controller is confirmed through Simulink simulation. Compared with pure tracking control and roll tracking control with zero roll angle as the control target, this controller demonstrates its ability to effectively track the reference trajectory and enhance the vehicle’s dynamic performance.

Iterative Optimization-based Control of a Class of Mixed Logical Dynamical Systems with STL Specifications: A Case Study on Microgrids

Motivated by the interest in modeling real-world systems as mixed logical dynamical (MLD) systems with signal temporal logic (STL) specifications, this paper introduces an efficient iterative optimization scheme for this class of systems. We focus on a microgrid system with energy losses, providing two formulations: a mixed-integer-based model and a model tailored to the proposed technique. A numerical case study using model predictive control (MPC) demonstrates that the proposed method significantly improves computation time compared to the classical mixed-integer linear programming (MILP) approach for longer time horizons, while also addressing discontinuity and volatility issues sometimes observed in the MILP approach.

Multimethod computing technology for searching energy-optimal motion modes of electric rolling stock of railways

The article is devoted to the problem of finding energy-optimal modes of movement of electric rolling stock of railways, considered as a set of problems of finding a global extremum of a functional defined on a set of logical-dynamic system (LDS) movement trajectories that are acceptable under operating conditions, and a structural-parametric synthesis of the LDS control strategy and optimization of its parameters based on random search methods, swarm intelligence, evolutionary algorithms and multi-agent models. A multi-method technology for optimizing the modes of train movement according to the criterion of minimum energy consumption, and software and algorithmic support based on the original multi-agent evolutionary-swarm method for synthesizing the optimal-terminal control of multi-mode moving objects developed by the author are proposed. The specified computing technology includes the following stages: formation of a set of promising control strategies; their optimization and recombination based on the evolutionary genetic algorithm; finding the global extremum of the functional using parametric optimization of the solutions identified at the previous stages using the modified particle swarm method. The computational experiments performed with the simulation model of train movement showed that when a sufficiently large number of options for control strategies are generated at the first stage, the algorithm converges to the global extremum of the functional.

Stability Analysis of Block Logical Dynamical Systems and Its Application in Logical Networks with Time Delay

Stability Analysis of Block Logical Dynamical Systems and Its Application in Logical Networks with Time Delay

Lyapunov-based stability of time-triggered impulsive logical dynamic networks

In this paper, we consider the stability of impulsive logical dynamic systems (ILDNs) from two aspects: impulsive disturbance and impulsive control. The existing results on the stability of ILDNs only consider the stability of ILDNs under a given impulsive instant sequence (IIS), which has some limitations and motivates us to consider the stability of ILDNs under any IIS. By constructing a merged ILDN, under the assumption that the non-impulsive process and impulsive process are all globally stable, a necessary and sufficient condition on the stability of ILDNs under any IIS is proposed at first. Unfortunately, the obtained result is too restrictive, and in practice, it is not common that a stable LDN is still stable under any IIS. The fact is that for any stable LDN, there exist some IISs such that its stability is destroyed, and these IISs are called impulsive disturbances; for any unstable LDN, there exist some IISs such that the stability can be achieved, and these IISs are called impulsive control. To investigate the stability of ILDNs, the concept of average impulsive interval is first introduced to ILDNs, and several sufficient conditions are proposed to ensure the stability of LDNs under the time-triggered IISs with the property of average impulsive interval. Moreover, the obtained results are applied to the set stability of ILDNs.

Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems

This article presents a new set representation named the hybrid zonotope that is equivalent to the union of 2N constrained zonotopes – convex polytopes – through the addition of N binary zonotope factors. The major contribution of this manuscript is a closed-form solution for exact forward reachable sets of discrete-time, linear hybrid systems modeled as mixed logical dynamical systems. The proposed approach captures the worst-case exponential growth in the number of convex sets required to represent the nonconvex reachable set while exhibiting only linear growth in the complexity of the hybrid zonotope set representation. Redundancy removal techniques are provided that leverage binary trees to store the combinations of binary factors of the hybrid zonotope that map to nonempty convex subsets. Numerical examples show the hybrid zonotope’s ability to compactly represent nonconvex reachable sets with an exponential number of features. Furthermore, the hybrid zonotope is shown to be closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections.

Finite-time controllability and stabilization of probabilistic logical systems with state-dependent constraint via subset transition method

Constraints are very common for practical control systems. For logical systems, the existing technique of pre-feedback is an effective way of treating state-dependent constraints in control when the state is measurable. However, it is inapplicable for the case when measurement information is not available. In this situation, in order for the control input to not violate the state-dependent constraint, the control at each step must be selected from the common admissible controls of all possible states. Motivated by this observation, in this study, we propose a novel technique, termed the subset transition method, for finite-time controllability and stabilization of probabilistic logical dynamic control system (PLDCS) with a state-dependent control constraint. The main idea of this method is to construct an unconstrained deterministic logical control system over the power set of the state space, called the subset transition system (SubSTS), characterizing the transitional dynamics between subsets under common admissible controls. We prove that a control sequence is admissible with respect to all states in an initial subset if and only if it does not steer the SubSTS from the initial subset to the empty set. Based on this, necessary and sufficient conditions for set controllability and set stabilizability are obtained. Examples are presented to demonstrate the application of the obtained results.

Feedback Shaping for Boolean Networks

Boolean networks, as logical dynamical systems where the system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. In this paper, we present a framework to evaluate whether and how the closed-loop dynamics of a controlled Boolean network can be shaped into any prescribed form by state-feedback control. We refer to this problem as to the feedback shaping for Boolean networks. First of all, based on the linear representation of Boolean networks, we establish a necessary and sufficient rank condition for a controlled Boolean network to be feedback shapable or not. Next, we design an algorithm for the synthesis of closed-loop dynamics for a feedback shapable Boolean network, such that for any given controlled Boolean network and a desired closed-loop dynamics, one can always find a feedback control law so that the closed-loop dynamics is precisely realized.

Group Cooperation in Intergroup Conflicting Networks: An Evolutionary Game Approach

Most intergroup conflicts arise from rivalry over limited resources, malicious disturbance, or hostile attitudes; therefore, it is critical to investigate individuals’ behaviors involved in intergroup conflicting contexts. Focusing on cooperation issues among individuals, in this study we establish an evolutionary game framework for analyzing cooperation and conflicts that arise within and inter-groups of intergroup conflicting networks. We first model the intergroup conflicting networked evolutionary games (ICNEGs). Then, we analyze the ICNEGs and prove that the evolution of ICNEGs can be expressed as a logical dynamic system. Finally, we apply the obtained results to a simplified Israeli-Palestinian conflict scenario. Our case study demonstrates that only by adopting suitable initial strategy profiles can a certain scale of group cooperation be continuously generated without suffering casualties.

Koopman Representation for Boolean Networks

Boolean networks, as logical dynamical systems whose system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. The representation and control of Boolean networks have attracted a lot of attention in recent years. On a parallel line of research, Koopman developed an operator view of nonlinear dynamical systems in early 1930s, which shows that using observable functions all nonlinear dynamics can be represented as (infinite dimensional) linear systems. In this paper, we present a framework for representing Boolean networks via Koopman approach. First of all, we construct addition and scalar multiplication operations over the set of logical functions over the binary field, defining a linear Boolean function space. Then associated with any Boolean mapping, the induced Koopman operator is defined as an operator over such Boolean function space. Next, we show that if there exists a set of logical functions that are invariant in the Koopman sense, a Boolean network can be represented by a finite-dimensional linear system. This Koopman perspective for Boolean networks is also extended to controlled Boolean networks.

A Simplifying-Matrix Method to Reduce the Storage Complexity in Studying Finite State Machines in the Framework of STP

The storage complexity problem is one of the severe challenges faced by research of finite state machines (FSMs) in the framework of STP, which must be overcome to make theoretical methodologies applicable to real applications. In this brief we propose a simplifying-matrix method to reduce the complexity of storing the dynamic matrix of an FSM when using computers to deal with problems. This method consists of three key components. Firstly, a mapping from logical matrices to logical vectors is proposed to retain effective information and remove large quantities of useless information from dynamic matrices of FSMs. Secondly, an alternative simplified dynamic matrix is constructed. Finally, a mapping from scalars to logical vectors is proposed to optimize the original model of FSMs and preserve the bilinear feature. In addition, we show that the proposed method is also applicable to reduction of the storage complexity of various logic dynamic systems in the framework of STP, such as, game systems, Boolean control networks, fuzzy systems and graphs.

Robust Successor and Precursor Sets of Hybrid Systems Using Hybrid Zonotopes

This letter presents identities for calculating robust successor and precursor sets of discrete-time linear hybrid systems. The proposed technique relies on generating a set containing all possible state transitions of a dynamic system over a region of interest, named the state-update set. Forward and backward reachability can then be performed using only projection, intersection, and Minkowski difference set operations with the state-update set. It is shown how state-update sets may be defined as hybrid zonotopes for mixed logical dynamical systems and linear systems in closed loop with model predictive control.

Survey on applications of algebraic state space theory of logical systems to finite state machines

Algebraic state space theory (ASST) of logical systems, developed based on the semi-tensor product (STP) which is a new matrix analysis tool built in recent ten years, provides an algebraic analysis approach for many fields of science, such as logical dynamical systems, finite-valued systems, discrete event dynamic systems, and networked game systems. This study focuses on comprehensively surveying the applications of the ASST method to the field of finite state machines (FSMs). Some necessary preliminaries on the method are first reviewed. Then the applications of the method in the FSM field are reviewed, including deterministic FSMs, nondeterministic FSMs, probabilistic FSMs, networked FSMs, and controlled and combined FSMs. In addition, other applications related to both STP and FSMs are surveyed, such as the application of FSM to Boolean control networks and the application of graph theory to FSMs. Finally, some potential research directions with respect to the ASST method in the FSM field are predicted.

A Kalman filter-based Hybrid Model Predictive Control Algorithm for Mixed Logical Dynamical Systems: Application to Optimized Interventions for Physical Activity.

Hybrid Model Predictive Control (HMPC) is presented as a decision-making tool for novel behavioral interventions to increase physical activity in sedentary adults, such as Just Walk. A broad-based HMPC formulation for mixed logical dynamical (MLD) systems relevant to problems in behavioral medicine is developed and illustrated on a representative participant model arising from the Just Walk study. The MLD model is developed based on the requirement of granting points for meeting daily step goals and categorical input variables. The algorithm features three degrees-of-freedom tuning for setpoint tracking, measured and unmeasured disturbance rejection that facilitates controller robustness; disturbance anticipation further improves performance for upcoming events such as weekends and weather forecasts. To avoid the corresponding mixed-integer quadratic problem (MIQP) from becoming infeasible, slack variables are introduced in the objective function. Simulation results indicate that the proposed HMPC scheme effectively manages hybrid dynamics, setpoint tracking, disturbance rejection, and the transition between the two phases of the intervention (initiation and maintenance) and is suitable for evaluation in clinical trials.

Survey on Mathematical Models and Methods of Complex Logical Dynamical Systems

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.

Stability of Logical Dynamic Systems With a Class of Constrained Switching

In this paper, a novel constrained switching rule, called “time-triggered logical switching” (TTLS) is considered for switched logical dynamic systems (SLDSs). Compared with the general time-triggered switching, the activation mode of TTLS cannot be arbitrary and is pre-allocated according to the logic operation, which is more practical. The TTLS is described as an LDS, and according to the characteristics of LDS, the stability analysis of SLDSs with TTLS is converted into the stability analysis of SLDSs under the logical switching cycle sequences. Firstly, based on the equivalent algebraic form of SLDSs with TTLS, combining the Lyapunov theory of LDS with average dwell-time method, several sufficient conditions are put forward for ensuring the point stability of the considered SLDSs. Then, by defining the switching cycle invariant subset and constructing a new system, the set stability analysis of the original system is transformed into the point stability analysis of the new system, and further, the obtained results for the point stability analysis are applied to the set stability analysis. At last, the validity of obtained results is illustrated by simulation on gene and protein signaling activity patterns.

Output Feedback Observability of Switched Boolean Control Networks

This paper investigates the output feedback observability of switched Boolean control networks. According to time-varying output feedback (TVOF) and time-invariant output feedback (TIOF) controls, six types of observability are proposed and their relation is discussed. A technique called parallel extension and a new system named induced logical dynamic system are developed. Based on them, some necessary and sufficient conditions are obtained for these TVOF and TIOF observabilities. In addition, the relation between output feedback observability and the existing results about observability is analyzed. Furthermore, some examples are given to illustrate the validity of the main results.

Finite-Time Set Reachability of Probabilistic Boolean Multiplex Control Networks

This study focuses on the finite-time set reachability of probabilistic Boolean multiplex control networks (PBMCNs). Firstly, based on the state transfer graph (STG) reconstruction technique, the PBMCNs are extended to random logic dynamical systems. Then, a necessary and sufficient condition for the finite-time set reachability of PBMCNs is obtained. Finally, the obtained results are effectively illustrated by an example.