CN110929215B - Multi-stability reconstruction method of memristor system based on mixed state incremental integral transformation - Google Patents
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Abstract
The invention provides a multi-stability reconstruction method of a memristor system based on mixed state incremental integral transformation, which adopts a mixed state incremental integral transformation method combining state variable incremental integral transformation and linear transformation to perform dimensionality reduction reconstruction on the memristor system containing high-order nonlinear terms, solves the problems that the high-order nonlinear terms are difficult to analyze and express time integral and integral mapping state variables are diverged due to asymmetry of the memristor system, obtains a dimensionality reduction reconstruction model for keeping the dynamic behavior of the original memristor system in an integral domain, simultaneously maps initial state values of the original memristor system into independent system parameters, reconstructs the multi-stability phenomenon depending on the initial state values in the original system into complex dynamic behavior depending on the system parameters, and can realize detection and control on multi-stability states in a hardware circuit.
Description
Technical Field
The invention relates to the field of electrical theory and new technology, in particular to a multi-stability reconstruction method of a memristor system based on mixed state incremental integral transformation.
Background
The chaotic oscillating circuit and system based on ideal memristors usually show a multi-stability phenomenon depending on initial values of states, parameters of the memristor circuit and the system are kept unchanged, the initial values of the states are changed, and the running tracks of the chaotic oscillating circuit and the system can be gradually stabilized in different running states. These coexisting stable states may provide more flexibility for engineering applications of memristive circuits and systems, while also presenting new challenges to traditional kinetic analysis and multi-stability control strategies. The super multi-stability of the memristor circuit can be reconstructed by adopting the incremental Wikstroy modeling method with the incremental magnetic flux and the incremental charge as the state variables, and the method can be successfully applied to super multi-stability analysis of the memristor circuit with a line balance point and a surface balance point.
The incremental magnetic flux and the incremental charge are considered to be integrals of voltages and currents in a [0, t ] time period, a state variable incremental transformation method is adopted to map state variables of a general memristor system to an integral domain, dimension reduction modeling and multi-stability reconstruction of the original memristor system are achieved in the integral domain, and then multi-stability point-to-point testing is achieved in a hardware circuit. However, this method is only applicable to memristive systems that have no other complex nonlinear terms than the memristive nonlinear term. The conventional nonlinear system usually has some complex high-order nonlinear terms, and when a state variable incremental integral transformation method is adopted to carry out mathematical model reconstruction, an analytical expression of the high-order nonlinear terms to time integral is difficult to obtain. In addition, the asymmetry of the original system also causes divergence of new state variables obtained after incremental integral transformation, which brings difficulties for theoretical analysis and hardware equivalent test of a reconstructed system. Aiming at the problems, the invention provides a multi-stability reconstruction method of a memristor system based on mixed state incremental integral transformation, and the implementation method and the effectiveness of the multi-stability reconstruction method are illustrated by taking the memristor system containing a single high-order nonlinear term (except the memristor nonlinear term) as an example.
Disclosure of Invention
The method mainly solves the technical problems of dimension reduction modeling and super multi-stability reconstruction of the memristor system with complex nonlinear terms (except the memristor nonlinear terms), and control and detection of multi-stability states are realized in a hardware circuit.
The technical scheme adopted for solving the technical problems is as follows: a multi-stability reconstruction method of a memristor system based on mixed state incremental integral transformation comprises the following steps:
step 1: performing state variable increment integral transformation on the memristor system, mapping the state variable of the memristor system to an integral domain, and establishing an integral domain model of the memristor system;
step 2: replacing a high-order nonlinear integral term which cannot obtain an analytical expression with a newly introduced state variable, and constructing an integral domain model of an initial state value of the explicit expression system in an integral domain;
and step 3: solving the problem of divergence of mapping state variables by utilizing linear transformation, obtaining a dimension reduction reconstruction model for maintaining the dynamic behavior of the original system, and reconstructing super multi-stability sensitive to the initial value of the original system into relatively stable complex dynamic behavior depending on the initial value related system parameters;
and 4, step 4: based on a dimension reduction reconstruction model, a state-controllable super multi-stability hardware test circuit is constructed, an external direct-current voltage source is adopted to assign values to initial value-related constant system parameters, and the control of the working state of the circuit is realized by adjusting external direct-current voltage control signals and initial value-related circuit element parameters.
The mathematical model of a four-dimensional memristive Jerk system with cubic nonlinear terms is:
equation (1) contains four state variables x 1 (t)、x 2 (t)、x 3 (t) and x 4 (t) wherein x 4 (t) is the internal state variable of the memristor; a and b are constant system parameters set to a =0.7 and b =10, respectively;
the state variable of the system (1) is mapped to an integral domain through state variable increment integral conversion, and the following steps are taken:
integrating four differential equations of the formula (1) from 0 to t, introducing a new state variable W to represent a cubic nonlinear integral term which cannot be expressed analytically, and deriving to obtain a four-dimensional system as follows:
new state variable X obtained by incremental integral transformation due to asymmetry of original system 1 And W gradually diverges; to this end, take X 1 Taking the difference value of the sum W as a new state variable, and carrying out linear state variable mapping on the system (6); three new state variables are introduced, defined as:
Y 1 ＝X 1 -W,Y 2 ＝X 2 ,Y 3 ＝X 3 (7)
therefore, the memristive system (1) is subjected to dimensionality reduction reconstruction in an integral domain to form a three-dimensional system, namely:
wherein, Y 1 ，Y 2 And Y 3 For recalling three state variables x of the system (1) 1 (t)、x 2 (t) and x 3 (t) new state variables obtained after incremental integral transformation of the mixed state, and Y 1 (0)＝Y 2 (0)＝Y 3 (0)＝0；δ 1 ，δ 2 ，δ 3 And delta 4 Corresponding to the initial values x of four states of the memristor system (1) 1 (0)、x 2 (0)、x 3 (0) And x 4 (0) The initial value of (a) is related to the system parameter; the dimensionality reduction system (8) reconstructs the multi-stability phenomenon depending on the initial state value in the memristor system (1) into a complex dynamic behavior depending on the initial value related system parameters in a new system, and is favorable for mechanism analysis and experimental test of the initial value sensitive multi-stability phenomenon;
in order to obtain high-quality experimental results, the state variable of the system (8) is amplified by 4 times, an equivalent realizing circuit is constructed by adopting an analog multiplier, a proportional operational circuit, an integrating circuit and the like based on an operational amplifier, and a corresponding equivalent circuit state equation set can be expressed as follows:
v 1 、v 2 、v 3 three state variables of the system (8) are represented, respectively, RC =10k Ω × 100nf =1ms representing the time constant of the integrator; circuit parameter selection R 1 ＝16R/b＝16kΩ，R 2 ＝8R＝80kΩ，R 3 ＝R/(1–δ 4 )，R 4 = R/a =14.3k Ω; by adjusting R 3 And three applied DC voltage control signals V 1 ＝-4δ 1 V，V 2 ＝4δ 2 V and V 3 ＝4δ 3 V, the experimental circuit can be controlled to work in a designated attraction sub-mode, and multi-stability point-to-point testing is achieved.
The invention has the following beneficial effects: the method comprises the steps of performing dimensionality reduction reconstruction on a memristor system (except for a memristor nonlinear term) containing a high-order nonlinear term through a mixed state variable increment integral conversion method, solving the problems that the high-order nonlinear term is difficult to analyze and express time integral and the memristor system is asymmetric to cause divergence of integral mapping state variables, obtaining a dimensionality reduction reconstruction model for keeping the dynamic behavior of the original memristor system in an integral domain, mapping an initial state value of the original memristor system into an independent system parameter, reconstructing a multi-stability phenomenon depending on the initial state value in the original system into a complex dynamic behavior depending on the system parameter, and accordingly realizing detection and control of multi-stability states in a hardware circuit.
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In order that the present invention may be more readily and clearly understood, reference will now be made in detail to the present invention, examples of which are illustrated in the accompanying drawings.
FIG. 1 is a flow chart of a multi-stability reconstruction method of a memristor system based on mixed-state incremental integral transformation according to the present invention;
FIG. 2 is a numerical simulation result of an exemplary four-dimensional memristor Jerk system (1) in an original state variable domain and an integral domain in the present invention, wherein a diagram (a) shows that the original memristor system is in x 1 (t)-x 2 (t) a phase trajectory plot of the plane; graph (b) shows the original memristive system at x 3 (t)-x 4 (t) a phase trajectory plot of the plane; the figure (c) is an integral domain dimension reduction reconstruction system
A planar phase rail diagram; the figure (d) is an integral domain dimension reduction reconstruction system
A planar phase rail diagram;
FIG. 3 is an equivalent circuit for implementing the integral domain dimensionality reduction reconstruction system;
FIG. 4 is a diagram of a system parameter δ associated with an initial value obtained by MATLAB simulation in an integral domain dimension reduction reconstruction system 4 Varying typical attractors, where graph (a) is δ 4 = 16 time integral domain reconstruction system of an attractor trajectory diagram; graph (b) is δ 4 =10 time integral domain reconstruction system of an attractor trajectory diagram; graph (c) is δ 4 Inverse transformation to original system state variable in time of = -16An attractor trajectory graph for the domain; diagram (d) is δ 4 An attractor phase trajectory diagram inversely transformed to an original system state variable domain when the = -10 hours;
FIG. 5 shows the measured values of random number R 3 (and δ) 4 Related circuit parameter), where graph (a) is R 3 =614 Ω time integral domain reconstruction system of the attractor trajectory diagram; in the figure (b), R is 3 An attractor trajectory diagram of an integral domain reconstruction system when =1.06k Ω; in the figure (c), R is 3 When the phase is =614 Ω, the phase is inversely transformed to an attraction sub-phase trajectory diagram of the original system state variable domain; in the figure (d), R is 3 And (4) inversely transforming to an attraction sub-phase trajectory diagram of the original system state variable domain when the k omega is 1.06k omega.
Detailed Description
The present invention will now be described in detail with reference to the accompanying drawings.
Fig. 1 shows a multi-stability reconstruction method of a memristor system based on mixed-state incremental integral transformation, which is characterized in that: the method comprises the following steps:
step 1: performing state variable increment integral transformation on the memristor system, mapping the state variable of the memristor system to an integral domain, and establishing an integral domain model of the memristor system;
step 2: replacing a high-order nonlinear integral term which cannot obtain an analytical expression with a newly introduced state variable, and constructing an integral domain model of an initial state value of the explicit expression system in an integral domain;
and step 3: solving the problem of divergence of mapping state variables by utilizing linear transformation, obtaining a dimension reduction reconstruction model for maintaining the dynamic behavior of the original system, and reconstructing super multi-stability sensitive to the initial value of the original system into relatively stable complex dynamic behavior depending on the initial value related system parameters;
and 4, step 4: based on a dimension reduction reconstruction model, a state-controllable super multi-stability hardware test circuit is constructed, an external direct-current voltage source is adopted to assign values to initial value-related constant system parameters, and the control of the working state of the circuit is realized by adjusting external direct-current voltage control signals and initial value-related circuit element parameters.
By using the multi-stability reconstruction method of the memristor system based on the mixed state increment integral transformation, the dimensionality reduction modeling and the super multi-stability reconstruction are carried out on the four-dimensional memristor Jerk system with the cubic nonlinear term. The mathematical model of the four-dimensional memristor Jerk system is
The system comprises four state variables x 1 (t)、x 2 (t)、x 3 (t) and x 4 (t) wherein x 4 (t) is the internal state variable of the memristor. a and b are constant system parameters, whose values are set to a =0.7 and b =10. The system has three line balance points and can show complex super multi-stability depending on the initial value of the state.
Firstly, the state variable of the system (1) is mapped to an integral domain through state variable increment integral conversion, and the state variable is taken
Integrating the four differential equations of the formula (1) from 0 to t to obtain
Dx can be obtained according to the special volt-ampere relation of the memristor, namely the fourth formula of the system (1) 4 (t)＝-x 3 (t) dt, which is substituted into the integral term of the second expression of the expression (3), is easily obtained
However, the cubic non-linear integral term of the third equation of equation (3) cannot be analytically expressed using the mapped state variable defined by equation (2). To solve this problem, the whole integral term is set as a new state variable
Substituting (5) into (3) can derive the following four-dimensional kinetic system
Simulation analysis shows that the state variable X of the mapping 1 And W gradually increases over time, which is detrimental to system analysis and testing. To this end, take X 1 And the difference value of W is used as a new state variable to carry out linear state variable mapping on the system (6). Three new state variables are introduced, defined as
Y 1 ＝X 1 -W,Y 2 ＝X 2 ,Y 3 ＝X 3 (7)
Thus, the memristive system (1) is reduced in dimension and reconstructed into a three-dimensional system in the integral domain, namely
Wherein, Y 1 ，Y 2 And Y 3 For memristive system (1) three state variables x 1 (t)、x 2 (t) and x 3 (t) new state variables obtained after incremental integral transformation of the mixed state, and Y 1 (0)＝Y 2 (0)＝Y 3 (0)＝0；δ 1 ，δ 2 ，δ 3 And delta 4 For initial values x corresponding to four states of the memristive system (1) 1 (0)、x 2 (0)、x 3 (0) And x 4 (0) The initial value of (a) is related to the system parameter. Moreover, the following mapping relation exists between the mapping state variable and the original state variable:
ignoring the small dc offset caused by the initial value of the state, the typical chaotic attractor of the system (8) can be transformed back to its original state variable domain by state variable derivation based on the relationship described in equation (9).
When the initial value of the system state is set to (0, 10) -6 0, -0.5), the numerical simulation obtains that the memristor system (1) is in x 1 (t)-x 2 (t) and x 3 (t)-x 4 The phase diagram of the (t) plane is shown in fig. 2 (a) and (b), respectively. Correspondingly, in the integral domain reconstruction system, take δ 1 ＝δ 3 ＝0、δ 2 ＝10 -6 、δ 4 Numerical simulation results when the value is not less than-0.5
And
the planar phase diagrams are shown in fig. 2 (c) and (d), respectively, and the two are well matched. It follows that by adjusting the system parameter δ in relation to the initial value of the state 1 、δ 2 、δ 3 And delta 4 The dynamic behavior of the original memristive system (1) related to the state initial value can be reproduced in the integral domain reconstruction system (8).
An equivalent realization circuit is constructed based on the formula (8), and circuit reconstruction and test of super multi-stability can be performed. In order to obtain high-quality experimental results, the state variable of the system (8) is amplified by 4 times, the equivalent realization circuit is shown in FIG. 3, and the state equation system of the circuit can be expressed as
Wherein v is 1 、v 2 、v 3 Three state variables of the system (8) are represented respectively, RC =10k Ω x 100nF =1ms represents the time constant of the integrator, and the circuit parameter is selected to be R 1 ＝16R/b＝16kΩ，R 2 ＝8R＝80kΩ，R 3 ＝R/(1–δ 4 )，R 4 = R/a =14.3k Ω. By adjusting R 3 And three applied DC voltage control signals V 1 ＝-4δ 1 V，V 2 ＝4δ 2 V and V 3 ＝4δ 3 V, the experimental circuit can be controlled to work in a designated attraction sub-mode, and multi-stability point-to-point testing is achieved.
In order to verify the correctness and the effectiveness of the multi-stability reconstruction and test method, delta is selected 1 ＝0、δ 2 ＝0.1、δ 3 ＝0，δ 4 Two exemplary attractors, set at-16 and-10, respectively, were subjected to numerical simulation and experimental testing. FIG. 4 shows the system parameter δ associated with the initial value obtained by MATLAB simulation 4 Varying exemplary attractor phase trajectory plot, wherein plot (a) and plot (b) are δ 4 An attractor phase trajectory diagram for the = -16/-10-time integral domain reconstruction system; in graphs (c) and (d), δ 4 And when the phase is = 16/-10, the variable mapping relation of the formula (9) is inversely transformed to the attraction sub-phase trajectory diagram of the original system state variable domain. FIG. 5 is a graph of the measured time of flight R, captured based on the circuit experiment of FIG. 3 3 (and δ) 4 Related circuit parameters). In the test process, three external direct current voltage control signals are respectively set to be V 1 ＝0V，V 2 =0.4V and V 3 =0V; the three integrating capacitors need to be fully discharged, so that the equivalent realization circuit is forced to start oscillation under the condition of an all-zero state initial value. In FIG. 5, the symbol (a) is R 3 =614 Ω time integral domain reconstruction system of the attractor trajectory diagram; in FIG. 5, the symbol (b) is R 3 An attractor trajectory diagram of an integral domain reconstruction system when =1.06k Ω; in FIG. 5, the symbol (c) is R 3 The inverse transformation is carried out to an attractor phase trajectory diagram of an original system state variable domain when the value is =614 Ω; in FIG. 5, R is in diagram (d) 3 And (4) inversely transforming to an attraction sub-phase trajectory diagram of the original system state variable domain when the k omega is 1.06k omega.
The numerical simulation result shown in fig. 4 is completely consistent with the experimental measurement result shown in fig. 5, which proves that the coexisting attractor depending on the initial value of the state can be stably reproduced in the dimensionality reduction reconstruction system of the integral domain, reflects that the method for reconstructing and testing the super multi-stability of the memristor system based on the mixed state incremental integral transformation, provided by the invention, has scientific theoretical basis and physical realizability, and can play a positive role in promoting the mechanism research and application exploration of the multi-stability.
The above examples are merely illustrative for clearly illustrating the present invention and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments.
Claims (1)
1. A multi-stability reconstruction method of a memristor system based on mixed state incremental integral transformation is characterized by comprising the following steps: the method comprises the following steps:
step 1: performing state variable increment integral transformation on the memristor system, mapping the state variable of the memristor system to an integral domain, and establishing an integral domain model of the memristor system;
and 2, step: replacing a high-order nonlinear integral term which cannot obtain an analytical expression with a newly introduced state variable, and constructing an integral domain model of an initial state value of the explicit expression system in an integral domain;
and step 3: solving the problem of divergence of mapping state variables by utilizing linear transformation, obtaining a dimension reduction reconstruction model for maintaining the dynamic behavior of the original system, and reconstructing super multi-stability sensitive to the initial value of the original system into relatively stable complex dynamic behavior depending on the initial value related system parameters;
and 4, step 4: constructing a state-controllable super multi-stability hardware test circuit based on a dimension reduction reconstruction model, assigning values to initial value-related constant system parameters by adopting an external direct current voltage source, and realizing the control of the working state of the circuit by adjusting an external direct current voltage control signal and initial value-related circuit element parameters;
the mathematical model of a four-dimensional memristive Jerk system with cubic nonlinear terms is:
the system (1) comprises four state variables x 1 (t)、x 2 (t)、x 3 (t) and x 4 (t) wherein x 4 (t) is the internal state variable of the memristor; a and b are constant systemsParameters set as a =0.7 and b =10, respectively;
the state variable of the system (1) is mapped to an integral domain through state variable increment integral conversion, and the following steps are taken:
integrating four differential equations of the formula (1) from 0 to t, introducing a new state variable W to represent a cubic nonlinear integral term which cannot be expressed analytically, and deriving to obtain a four-dimensional system as follows:
new state variable X obtained by incremental integral transformation due to asymmetry of original system 1 And W gradually diverges; to this end, take X 1 Taking the difference value of the sum W as a new state variable, and carrying out linear state variable mapping on the system (6); three new state variables are introduced, defined as:
Y 1 ＝X 1 -W,Y 2 ＝X 2 ,Y 3 ＝X 3 (7)
therefore, the memristive system (1) is subjected to dimensionality reduction reconstruction in an integral domain to form a three-dimensional system, namely:
wherein, Y 1 ，Y 2 And Y 3 For recalling three state variables x of the system (1) 1 (t)、x 2 (t) and x 3 (t) new state variables obtained after incremental integral transformation of the mixed state, and Y 1 (0)＝Y 2 (0)＝Y 3 (0)＝0；δ 1 ，δ 2 ，δ 3 And delta 4 Corresponding to the initial values x of four states of the memristor system (1) 1 (0)、x 2 (0)、x 3 (0) And x 4 (0) The initial value of (a) is related to the system parameter;the dimensionality reduction system (8) reconstructs the multi-stability phenomenon depending on the initial state value in the memristor system (1) into a complex dynamic behavior depending on the initial value related system parameters in a new system, and is favorable for mechanism analysis and experimental test of the initial value sensitive multi-stability phenomenon;
in order to obtain high-quality experimental results, the state variable of the system (8) is amplified by 4 times, an equivalent realizing circuit is constructed by adopting an analog multiplier, a proportional operational circuit, an integrating circuit and the like based on an operational amplifier, and a corresponding equivalent circuit state equation set can be expressed as follows:
v 1 、v 2 、v 3 three state variables of the system (8) are represented, respectively, RC =10k Ω × 100nf =1ms representing the time constant of the integrator; circuit parameter selection R 1 ＝16R/b＝16kΩ，R 2 ＝8R＝80kΩ，R 3 ＝R/(1–δ 4 )，R 4 = R/a =14.3k Ω; by adjusting R 3 And three applied DC voltage control signals V 1 ＝-4δ 1 V，V 2 ＝4δ 2 V and V 3 ＝4δ 3 V, controlling the experiment circuit to work in a designated attraction sub-mode, and realizing multi-stability point-to-point test.
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Application publication date: 20200327 Assignee: Changzhou Ruixinteng Microelectronics Co.,Ltd. Assignor: CHANGZHOU University Contract record no.: X2023980054127 Denomination of invention: Multi stability reconstruction method for memristive systems based on mixed state incremental integral transformation Granted publication date: 20230321 License type: Common License Record date: 20231227 |