A novel explicit analytical mathematical modeling method for hybrid automaton systems

Abstract

The present invention provides a novel explicit analytical mathematical modeling method for hybrid automata systems. First, the switching mode of a continuous-time switched system model with differential form is transformed into a constant switching signal that depends on an index at the switching moment. Then, the discrete state nodes and discrete input events in the logical evolution graph of a deterministic finite automaton are represented by their own logical vector forms respectively. In addition, by using the semi-tensor product tool of matrix calculation, an algebraic characterization form of the dynamics of the switching signal based on the automaton is given, that is, a restricted automaton-dependent switching mechanism based on the evolution mode of a deterministic finite automaton is designed. Finally, by organically integrating a continuous switched system and a discrete deterministic finite automaton, an explicit mathematical model of a hybrid automaton with a hybrid form is given. The model constructed by the present invention can directly represent and describe the hybrid characteristic problems related to discrete event-driven and continuous dynamic processes in the fields of computers and control.

Description

Technical Field

[0001] The present invention belongs to the field of hybrid dynamic systems between computer science and control science, and specifically designs an explicit analytical model under a novel mathematical representation framework for hybrid automata with evolution characteristics such as discrete jumps and continuous flows simultaneously. Background Art

[0002] The evolution processes in dynamics can generally be divided into two cases: one is the continuous-time dynamic flow evolution, such as the continuous evolution processes of classical mechanical systems and analog circuit systems, etc.; the other is the discrete-time dynamic jump evolution, such as the discrete logical evolution processes of Boolean network systems and digital systems, etc. For the former, it is mainly described by differential equations (or differential inclusions); for the latter, it is mainly characterized by difference equations (or difference inclusions). And the evolution process of hybrid systems can just be regarded as an organic integration of the above two dynamic characteristics, and it has become one of the hot research directions in the fields of computer and control.

[0003] A hybrid dynamic system is a complex dynamic system that has both continuous flow dynamic characteristics and discrete jump dynamic characteristics. Such a system can provide a unified mathematical paradigm for the behavioral characterization of many hybrid evolution phenomena related to physics, events, information, etc. For example: current coupling, event triggering, and nonlinear networks, etc. all have the behavioral characteristics of hybrid evolution. According to the types of interaction between discrete events and continuous variables, hybrid systems can be divided into switching type, water tank type, centralized control type, traveling salesman type, hierarchical type, simulation language type, hybrid Petri net, hybrid automata, and so on.

[0004] In fact, hybrid automata mainly describe an event-driven mechanism, which places discrete events and continuous variables within a single framework in a class of hybrid system models. Here, the automata are a class of discrete dynamic processes that can transfer from one state to another. If each state of the automata is regarded as a set of continuously changing micro-subsystems, then the transfer between states can be regarded as an event-driven process. Hybrid automata can clearly represent the operating mechanism and function of the entire system in terms of structure, and are suitable for a class of hybrid systems that emphasize the logical behavior of the system. Therefore, they have been used to describe and calculate the hybrid behavior of various analog computer systems. In addition, the air conditioning system in a room is a typical hybrid automata model, where the indoor temperature is a continuously changing continuous dynamic process, and the temperature regulator of the air conditioning system is a discrete event system that can discretely switch to adjust the indoor temperature, such as cold, normal, and hot. Therefore, it is precisely because of the general application background and scientific significance of this kind of dynamic system with a hybrid evolution method in real life and theoretical research that it has gradually attracted the extensive attention and interest of researchers. Currently, there are several methods for studying hybrid automata as follows: 1) By using the mode-dependent average dwell time switching law, the stability problem of a class of switching systems based on deterministic finite automata is studied. Here, the deterministic finite automata is a dynamic process that transfers from one discrete state to the next through a given transfer function, and it belongs to a special class of discrete event systems. However, the obtained results are only presented based on the original logical evolution graph of the finite automata. 2) By using a set of norms existing at each node of the automata, the stability problem of a class of discrete-time switched linear systems with automata-constrained switching sequences is studied. However, the developed system is not a truly hybrid automata system. In fact, the evolution process of hybrid systems is an organic integration of the common discrete dynamic jump evolution and continuous dynamic flow evolution in nature. Therefore, how to design a restricted switching scheme based on the automata evolution mode to further establish a new hybrid automata system model and extend the time-dependent switching method or state-dependent switching method? It should be noted that, from the perspective of deterministic finite automata and matrix semi-tensor product, the mathematical representation of a new hybrid automata with automata-dependent switching is still blank.

[0005] Based on the above discussion, the mathematical model described in the present invention is an explicit analytical modeling method that can accurately represent and describe hybrid automata systems. This patent is funded by the China Postdoctoral Science Foundation (2022TQ0179), the National Natural Science Foundation of China (61890920, 61890921), and the National Key Research and Development Program of China (2018YFB1700102). Summary of the Invention

[0006] The present invention addresses the limitation problems brought about by the design of hybrid automata based on logical evolution graphs in computer science and control science, and provides a mathematical representation modeling method. Since a hybrid automaton is a complex non-smooth dynamic system with both continuous flow dynamic characteristics and discrete jump dynamic characteristics, how to design an explicit analytical model for mathematical characterization and description within the framework of hybrid systems has always been a challenging problem.

[0007] To achieve the above object, the technical solution of the present invention is as follows:

[0008] A mathematical expression model for hybrid system motors is proposed, which can directly represent and describe the hybrid characteristic problems related to discrete event-driven and continuous dynamic processes in the fields of computer and control. First, the switching mode of the continuous-time switched system model with differential form is transformed into a constant switching signal depending on the index at the switching moment. Then, the discrete state nodes and discrete input events in the logical evolution graph of the deterministic finite automaton are respectively represented by their own logical vector forms. In addition, using the semi-tensor product tool of matrix calculation, an algebraic characterization form of the switching signal dynamics based on the automaton is given, that is, a restricted automaton-dependent switching mechanism based on the evolution mode of the deterministic finite automaton is designed. Finally, by organically integrating the continuous switched system and the discrete deterministic finite automaton, an explicit mathematical model of the hybrid automaton in hybrid form is given.

[0009] A new explicit analytical mathematical modeling method for hybrid automata includes the following steps:

[0010] Step 1: By piecewise constantizing the switching signal function, the switched system with a switching signal depending on the continuous time is equivalently transformed into a switched system with a switching signal depending on the index at the switching moment. That is,

[0011] 1.1) First, design a class of continuous-time switched systems with the following differential form:

[0012]

[0013] where t is the continuous time, t0 is the initial moment, x(t) is the continuous state vector, is the derivative of x(t) with respect to time,

[0014] x(t0) = x0 is the initial continuous state vector, f is a smooth function, σ(t) represents a switching signal in the form of a piecewise constant function taking values in the set {1, 2,..., n}, 1, 2,..., n respectively represent subsystem 1, subsystem 2,..., subsystem n, and n represents the total number of switching subsystems.

[0015] 1.2) Secondly, let the switching signal system (1) can be transformed into the following form:

[0016]

[0017] where the time points …, t k , t k+1 , … represent the switching moments when the subsystems switch, k is the lower index of the switching moment t k , δ(k) ∈ {1, 2, …, n} represents a constant function depending on the switching moment t k and the lower index k, represents a constant indicator function taking values of 0 or 1. When t ∈ [t k , t k+1 ), the subsystem σ(t k ) = δ(k) is activated.

[0018] Step 2: Use the logical vectorization method and the semi-tensor product of matrices to characterize a discrete-time switching signal dynamics algebraic equation based on an automaton. That is,

[0019] 2.1) First, design a class of deterministic finite automata with the following concepts:

[0020] Let a triple be A = (W = (ω1, ω2, …, ω n ), V = (v1, v2, …, v m ), g). If for any discrete state node ω ∈ W and discrete input event v ∈ V, there does not exist more than one discrete state node ω′ ∈ W satisfying ω′ = g(ω, v), where W is a finite set of discrete state nodes, ω1, ω2, …, ω n are n discrete state nodes, V is a finite set of discrete input events, v1, v2, …, v m are m discrete input events, and g is a transition function. Then the triple A is called a deterministic finite automaton.

[0021] 2.2) Secondly, for a deterministic finite automaton A, let ω j ∈ W and v i ∈ V be respectively identified by their own logical vector forms (that is, ) and (that is, ). Therefore, according to the semi-tensor product method of matrices, the dynamic properties of the switching signal based on the automaton can be described by an algebraic equation in difference form as follows:

[0022]

[0023] Among them, and respectively represent two logical vectors. "~" represents an equivalent meaning between different forms. represents the semi-tensor product operation, k is the discrete time, v(k) is the discrete logical input vector, δ(k) is the discrete logical state vector, and L is a transition structure matrix.

[0024] Step 3: By integrating and embedding the discrete-time switching signal dynamics equation based on automata as shown in formula (3) into the continuous-time switching system as shown in formula (2), a hybrid automaton system with a mathematical hybrid form can be represented as follows:

[0025]

[0026] Among them, x(t) is the continuous state, x0 is the initial continuous state, δ(k) is the discrete state, δ0 is the initial discrete state, and v(k) is the discrete input.

[0027] Therefore, through formula (4), a mathematical explicit analytical model for the hybrid system motor is obtained. This model can directly represent and describe the hybrid characteristic problems related to discrete event-driven and continuous dynamic processes in the fields of computer and control.

[0028] The beneficial effects of the present invention are as follows:

[0029] The hybrid automaton system established by the present invention not only considers the continuous-time switching system with differential form, but also considers the discrete-time deterministic finite automaton with difference form, as well as their interactions. That is to say, compared with the time-dependent or state-dependent switching system, the proposed model is a new type of hybrid dynamic system. That is, the discrete state of the system undergoes discrete jumps based on the logical evolution mode of the deterministic finite automaton, and the continuous state of the system undergoes continuous dynamic evolution on the corresponding activated subsystems. Therefore, the switching mechanism of the hybrid automaton system is a constrained logic switching scheme that simultaneously considers the switching index sequence and the switching time sequence. Among them, the switching index sequence is obtained through the predetermined logical jump evolution of the deterministic finite automaton, and the switching time sequence has a certain pre-allocated duration interval. Description of the Drawings

[0030] Figure 1 is a schematic diagram of a certain switching converter circuit to be studied in the present invention. Note: Figure 1 k, L, C, R, e in c , i l , e srespectively refer to the switch, inductor, capacitor, load resistor, capacitor voltage, inductor current, and power supply voltage of the converter.

[0031] Figure 2 is the schematic diagram of the hybrid automaton logic evolution designed by the present invention for the switching converter circuit. Note: Figure 2 v1 and v2 in [[ ]] respectively refer to two discrete input events.

[0032] Figure 3 is the discrete input evolution trend diagram of the established hybrid automaton system model. Note: Figure 3 1 and 2 in the vertical coordinate in [[ ]] respectively refer to the discrete input events v1 and v2, and the horizontal coordinate represents the discrete time k.

[0033] Figure 4 is the discrete switching law evolution trend diagram of the established hybrid automaton system model. Note: Figure 4 the vertical coordinate in [[ ]] refers to the switching signal dependent on the automaton (i.e., Figure 4 1 and 2 in the vertical coordinate respectively refer to Figure 2 subsystem 1 and subsystem 2 in [[ ]]), and the horizontal coordinate represents the discrete time k.

[0034] Figure 5 is the continuous state response evolution trend diagram of the established hybrid automaton system model. Note: Figure 5 the vertical coordinate in [[ ]] refers to two continuous sub-states of the hybrid automaton system (i.e., Figure 5 x1 and x2 in [[ ]] respectively represent the capacitor voltage and inductor current of the converter circuit), and the horizontal coordinate represents the continuous time t. Detailed implementation manners

[0035] The present invention will be further described below in conjunction with specific embodiments.

[0036] A method for explicit analytical hybrid automaton mathematical modeling is given below for a class of pulse width modulation driven boost converter circuits with switch switching, which specifically includes the following steps:

[0037] Step 1: Consider a class of switching converter circuits as shown in Figure 1 and its converter can be modeled as a continuous time switching system similar to the formula (1) shown:

[0038]

[0039] Wherein, is the system state matrix of subsystem 1, is the system state matrix of subsystem 2, is the system input matrix of subsystem 1, is the system input matrix of subsystem 2, and s = t / T is the continuous time similar to t. The switch k can be switched at most once in each period T, and L, C, R, e c , i l , e s respectively refer to the inductor, capacitor, load resistor, capacitor voltage, inductor current, and power supply voltage of the converter. is the continuous state vector, and u = e s is the control input.

[0040] Step 2: For the switched converter as shown in Figure 1 and the switching system in Step 1, design a hybrid automaton evolution process as shown in Figure 2 . Among them, the deterministic finite automaton with two discrete state nodes and two discrete input events can be represented as A = (W = (ω1, ω2) = {1, 2}, V = (v1, v2), g). Then, by using the logical vector form of discrete state nodes and discrete input events, it can be represented that: subsystem Subsystem Then, through the schematic diagram of the logical evolution of the hybrid automaton Figure 2 , we can obtain Therefore, the transition structure matrix L in formula (3) can be obtained, that is,

[0041]

[0042] Step 3: By substituting the transition structure matrix L into formula (3) and letting the switching signal Then, integrate the formula (3) with L into the continuous switching system in Step 1. Therefore, a hybrid automaton system with a hybrid form can be represented as follows:

[0043]

[0044] Implementation results

[0045] 1) From the discrete evolution of Figure 3 and Figure 4 , it can be seen that the discrete state of the hybrid automaton is finally globally stabilized within a limit cycle {1, 2} under the drive of discrete input events (that is, Figure 3 ). That is, the evolution order of the discrete state of the hybrid automaton model is Mode 1 → Mode 2 → Mode 1 → Mode 2 → Mode 1 → Mode 2 → ···. Figure 4 ). That is, the evolution order of the discrete state of the hybrid automaton model is Mode 1 → Mode 2 → Mode 1 → Mode 2 → Mode 1 → Mode 2 → ···.

[0046] 2) From the continuous evolution of Figure 5It can be seen that the two consecutive states of the hybrid automaton model are driven by the discrete switching law (i.e., Figure 4 ) and finally globally converge exponentially and uniformly to the equilibrium point 0 within the cycle {1, 2}.

[0047] Therefore, such a result conforms to the essential characteristics of the hybrid automaton, and at the same time proves that the proposed model has a new mathematical characterization ability with explicit analysis for the design problem of the hybrid automaton. That is, the designed automaton-dependent switching scheme is feasible and effective.

[0048] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are only used to illustrate the technical solutions of the present invention and cannot be construed as limiting the present invention. Those of ordinary skill in the art can modify and replace the above embodiments within the scope of the present invention without departing from the principle and purpose of the present invention.

Claims

1. An explicit analytical mathematical modeling method for hybrid automata systems, for modeling a pulse-width modulation driven boost converter circuit with switching, characterized in that The steps are as follows: Step 1: For the switched converter circuit, and its converter is modeled as a continuous-time switched system shown in Equation (1): Among them, is the system state matrix of subsystem 1, is the system state matrix of subsystem 2, is the system input matrix of subsystem 1, is the system input matrix of subsystem 2, s = t / T is the continuous time of t, The switch k can be switched at most once in each period T, L, C, R, e c , i l , e s respectively refer to the inductor, capacitor, load resistor, capacitor voltage, inductor current and power supply voltage of the converter, is the continuous state vector, u = e s is the control input; Step 2: For the switching converter and the switching system in Step 1, design the evolution process of the hybrid automaton, where the deterministic finite automaton with two discrete state nodes and two discrete input events is represented as A = (W = (ω1, ω2) = {1, 2}, V = (v1, v2), g); then, by using the logical vector form of the discrete state nodes and discrete input events, represent: the subsystem Subsystem Then, through the logical evolution of the hybrid automaton, obtain Obtain the transition structure matrix L in formula (3), that is,[[]] Step 3: By substituting the transfer structure matrix L into formula (3) and setting the switching signal Then, the formula (3) with L is integrated and embedded into the continuous switching system in Step 1. Thus, a hybrid automaton system with a hybrid form is represented as follows:

Images