CN105629733A - Fractional order cell neural network adaptive synchronization control and circuit design method - Google Patents

Abstract

The invention discloses a fractional order cell neural network adaptive synchronization control and circuit design method. A fractional differential algorithm is selected, and the definition of a cell neural network equation is combined, thereby constructing a three-dimensional cell neural network system. The method comprises the steps: designing a drive-response system with a known drive system nonlinear parameter and an unknown response system nonlinear parameter based on the above system, constructing a new adaptive synchronizing controller and a parameter adaptive adjustment rate, and achieving the synchronization of the drive and response systems in numerical simulation; designing the circuit diagrams of the drive and response systems of the fractional order cell neural network, and achieving the circuit simulation of the controller and the adaptive adjustment rate. A simulation result indicates that the circuit simulation and the numerical simulation have the similar synchronous phase diagram, and the method verifies the accuracy of system theoretical analysis and the actual physical realizability. Therefore, the method is practical in the field of engineering.

Description

A kind of fractional order cell neural network Self-adaptive synchronization control and circuit design method

Technical field

The invention belongs to non-thread line chaotic circuit system field, relate to cell neural network, Synchronization Control theory and fractional order circuit thought.

Background technology

Chaos be the discovery that one of the greatest discovery in physics after the Theory of Relativity and quantum mechanics of 20th century, chaotic motion is the forms of motion that a kind of deterministic nonlinear system is exclusive, and showing as and being seen as finite motion in terms of local angle from global scope is then irregular motion. The feature of chaos is its extreme sensitivity to disturbance, and namely two chaos systems are from the initial condition of minute differences, through regular hour meeting rapid divergence, ultimately results in movement locus entirely different. Just because of the height random of chaotic signal, unpredictability, high complexity, and the being easily achieved property of definitiveness equation so that it is have bigger researching value and tempting application prospect thereof in engineering. Cell neural network (CellularNeuralNetworks, CNN) it is a kind of parallel computation phantom similar to human nerve's network, local connectivity matter is simply prone to ultra-large circuit (VLSI) and realizes, and can produce nonlinear kinetics chaos phenomenon even hyperchaos complex behavior. Cell neural network prediction science, image procossing, pattern recognition, secret communication, logic array computer structure etc. in have been achieved for developing widely. And Control of Chaotic Synchronization has become as one of current study hotspot as the key link of chaos applications, therefore the Synchronization Control research of dynamic stability is had realistic meaning and practical value by dynamic stability no less important by it.

Since nineteen ninety L.M.Pecora and T.L.Carrol proposes the thought of Chaotic Synchronous, the research of Chaotic Synchronous obtains flourish. The synchronous method occurred at present has: drive response method, active-passive method, control observation device method, active control, one way coupled map, adaptive synchronization method etc. Most synchronization scenario is all based on realizing on basis that system structure and parameter are all accurately understood, but in fact, it is difficult to accurately obtain systematic parameter by externally measured, even if some system its structure and parameter already known, interference due to external disturbance and noise, it is also difficult to the parameter making two chaos systems is identical.

Fractional calculus and integer rank calculus almost have same long developing history, and integer rank calculus is the special case of fractional calculus, and integer level system is that the idealization to actual chaos system processes. For in practical engineering application, the feature of new fractional-order system and structure are closer to reality, along with the development of fractional calculus, the Synchronization Control of chaotic systems with fractional order has more prominent using value and development prospect than the Synchronization Control of integer rank chaos system in fields such as secret communication, system controls. Up to now numerous research is compared in the Synchronization Control concentrating on integer rank chaos system, utilizes fractional order to realize cell neural network Self-adaptive synchronization control and parameter identification but rarely has report.

Up to now, although numerous scholars are for the dynamic characteristic of cell neural network, dynamic behavior and various engineer applied etc. have more thesis and work to deliver, such as cell neural network is based on the Equilibrium point of threshold stimulus function, airbound target identification based on cell neural network, the application etc. in License Plate of the memristor cell neural network, but these researchs are substantially based on the cell neural network on integer rank. Since fractional order theory proposes, chaotic systems with fractional order has had bigger development, the research of the such as Complete Synchronization of fractional order Chen chaos system and reverse sync, fractional order Liu chaos system and Experiment of Electrical Circuits thereof and control, fractional order Lorenz hyperchaotic system and circuit simulation thereof etc., but the cell neural network chaos system based on fractional order is studied seldom, utilizes fractional order to realize cell neural network Self-adaptive synchronization control and parameter identification rarely has report especially. But fractional calculus can describe the various dynamicss of real world and the actual physics phenomenon of system more accurately. Therefore the Synchronization Control research of fractional order is had to important theoretical research value and application prospect.

Summary of the invention

The present invention, on the basis of comprehensive cell neural network and the respective advantage of fractional order circuit, have devised a new fractional order cell neural network Circuits System. And utilize one drive system nonlinear parameter of this system constructing known and the drive response system of responding system nonlinear parameter value the unknown, realize this drive response system synchronization by Adaptive synchronization controller and regulation. Owing to its fractional order characteristic is closer to reality physical significance, therefore it has important actual value.

The present invention is achieved through the following technical solutions.

Step (1): according to cytocidal action lattice basic model, design integer rank three-dimensional cell nerve network system, and make system have chaotic characteristic by adjusting parameter.

Step (2): select fractional order differential definition and algorithm.

Step (3): build fractional order synchronous control system model.

Step (4): the integer rank three-dimensional cell neutral net built based on step (1), theoretical in combination with step (2) fractional calculus, design corresponding fractional order dynamic stability. The drive system of the fractional order cell neural network in difference construction step (3) and responding system.

Step (5): design isochronous controller and parameter adaptive regulation, realizes driving the synchronization with responding system in numerical simulation.

Step (6): fractional order cell neural network drive system in design procedure (4) and responding system circuit theory diagrams, controller and self-adaptative adjustment rate to step (5) realize circuit simulation simultaneously.

Further, a kind of fractional order cell neural network Self-adaptive synchronization control method of the present invention, it specifically comprises the following steps that

(S1): according to cytocidal action lattice basic model, integer rank three-dimensional cell nerve network system, the parameters a in adjustment state equation are first designed_j,a_jk,S_jk,(j=1,2,3, k=1,2,3), make system output chaos phenomenon;

$\frac{{\mathrm{dx}}_j}{dt}=-x_j+a_{j}f\left(x_j\right)+\underset{k=1,k\≠j}{\overset{3}{\Σ}}{a}_{jk}f\left(x_k\right)+\underset{k=1}{\overset{3}{\Σ}}{S}_{jk}{x}_k+{\tilde{I}}_j,(j=1,2,3)---\left(1\right)$

And apply MATLAB software designed system is carried out numerical simulation, and observe its chaotic characteristic and attractor phasor;

(S2): definition and algorithm for fractional order differential have the definition of Cauchy integral formula, Grunwald-Letnikov fractional order integration, the definition of Riemann-Liouville fractional order differential, Caputo definition etc., the present invention selects to adopt the fractional calculus of Caputo definition, and its mathematic(al) representation is as follows:

$\frac{d^{q}f\left(t\right)}{{\mathrm{dt}}^q}=\frac{1}{\mathrm{\Γ}(n-q)}{\∫}_0^t\frac{f^{\left(n\right)}\left(\mathrm{\τ}\right)}{{(t-\mathrm{\τ})}^{q-n+1}}d\mathrm{\τ}---\left(2\right)$

�� () in formula is Gamma function, and n-1��q��n, q is mark, and n is integer;

Fractional Differential Equation corresponding to dynamic system can be expressed as:

$\begin{array}{c}{a}_n{D}^{v_n}F(x,y)+a_{n-1}{D}^{v_{n-1}}F(x,y)+\mathrm{...}+a_0{D}^{v_0}F(x,y)\\ =b_m{D}^{{\mathrm{\α}}_m}G(x,y)+b_{m-1}{D}^{{\mathrm{\α}}_{m-1}}G(x,y)+\mathrm{...}+b_0{D}^{{\mathrm{\α}}_0}G(x,y)\end{array}---\left(3\right)$

Wherein v_n,v_n-1��v₀And ��_m,��_m-1����₀Represent corresponding fractional order rank value, a respectively_n,��,a₀And b_m,��,b₀For real number; In formula, (x, y) inputs F for system, and (x, y) exports G for system;

(S3): build fractional order synchronous control system model:

$D_t^{q}X=h\left(x\right(t),t),0\≤t\≤T---\left(4\right)$

Wherein X �� RⁿIt is a n dimension state vector of drive system, h:Rⁿ��Rⁿ, h is split as linear processes two parts, then drive system I is:

$D_t^{q}X=g\left(x\right)+G\left(x\right)A^T---\left(5\right)$

In formula: X �� R is the state variable of drive system, g:Rⁿ��RⁿFor comprising the continuous vector function of linear term, G (x) A^TFor non-linear partial, G:Rⁿ��R^n��nFor parameter vector function, A is the parameter matrix of the nonlinear function of drive system;

Corresponding responding system II is:

$D_t^{q}Y=g\left(y\right)+G\left(y\right){\tilde{A}}^T+U---\left(6\right)$

Y �� R in formula is in response to the state variable of system, U �� RⁿFor controller,It it is the parameter matrix of the nonlinear function of drive system;

(S4): the integer rank three-dimensional cell neutral net built based on step (S1), corresponding fractional order dynamic stability is gone out in combination with step (S2) fractional calculus Design Theory, formula (5) drive system I definition in integrating step (S3), constructs the drive system equation of this three-dimensional fractional order cell neural network:

$\left\{\begin{array}{c}\frac{d^{q_1}{x}_1}{{\mathrm{dt}}^{q_1}}=-x_1+s_{11}{x}_1+s_{12}{x}_2+a_{11}f\left(x_1\right)+a_{12}f\left(x_2\right)\\ \frac{d^{q_2}{x}_2}{{\mathrm{dt}}^{q_2}}=-x_2+s_{21}{x}_1+s_{22}{x}_2+s_{23}{x}_3+a_{22}f\left(x_2\right)\\ \frac{d^{q_3}{x}_3}{{\mathrm{dt}}^{q_3}}=-x_3+s_{31}{x}_1+s_{32}{x}_2+s_{33}{x}_3\end{array}\right.---\left(7\right)$

Nonlinear parameter a in formula₁₁,a₁₂,a₂₂It is given value, formula (6) the responding system II definition in integrating step (S3), the responding system equation of the three-dimensional fractional order cell neural network of structure:

$\left\{\begin{array}{c}\frac{d^{q_1}{y}_1}{{\mathrm{dt}}^{q_1}}=-y_1+s_{11}{y}_1+s_{12}{y}_2+{\tilde{a}}_{11}f\left(y_1\right)+{\tilde{a}}_{12}f\left(y_2\right)+u_1\\ \frac{d^{q_2}{y}_2}{{\mathrm{dt}}^{q_2}}=-y_2+s_{21}{y}_1+s_{22}{y}_2+s_{23}{y}_3+{\tilde{a}}_{22}f\left(y_2\right)+u_2\\ \frac{d^{q_3}{y}_3}{{\mathrm{dt}}^{q_3}}=-y_3+s_{31}{y}_1+s_{32}{y}_2+s_{33}{y}_3+u_3\end{array}\right.---\left(8\right)$

Nonlinear parameter in formulaIt is unknown-value.

(S5): the isochronous controller U of (6) formula and parameter adaptive regulation in design procedure (S3), utilize mathematical theory to carry out proving to synchronize character, and undertaken emulate by MATLAB program and synchronize verify;

Drive and the error of response be:

$D_t^{q}e=D_t^{q}Y-D_t^{q}X=g\left(y\right)-g\left(x\right)+G\left(y\right){\tilde{A}}^T-G\left(x\right)A^T+U---\left(9\right)$

Isochronous controller U:

U=[u₁,u₂,u₃](10)

Wherein: $\left\{\begin{array}{c}{u}_1=a_{11}f\left(x_1\right)-{\tilde{a}}_{11}f\left(y_1\right)+a_{12}f\left(x_2\right)-{\tilde{a}}_{12}f\left(y_2\right)-s_{12}{e}_2-{\tilde{e}}_{11}f\left(x_1\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_2=-s_{21}{e}_1-s_{23}{e}_3+a_{22}f\left(x_2\right)-{\tilde{a}}_{22}f\left(y_2\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_3=-s_{31}{e}_1-s_{32}{e}_2\end{array}\right.---\left(11\right)$

Selecting system self-adaptative adjustment rule is simultaneously:

$\left[\begin{array}{c}{\stackrel{\·}{\tilde{a}}}_{11}\\ {\stackrel{\·}{\tilde{a}}}_{12}\\ {\stackrel{\·}{\tilde{a}}}_{22}\end{array}\right]=G{\left(x\right)}^T{\left[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right]}^T=\left[\begin{array}{ccc}f\left(x_1\right)& 0& 0\\ f\left(x_2\right)& f\left(x_2\right)& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]=\left[\begin{array}{c}{e}_{1}f\left(x_1\right)\\ e_{1}f\left(x_2\right)+e_{2}f\left(x_2\right)\\ 0\end{array}\right]---\left(12\right)$

Wherein e_i(i=1,2,3) for the error of drive system Yu responding system;

(S6): design fractional order dynamic stability drive system (7) and responding system (8) circuit theory diagrams, and realize circuit simulation by Multisim design con-trol device (10) and self-adaptative adjustment rate (12).

The circuit design method of fractional order cell neural network Self-adaptive synchronization control method of the present invention, is characterized in that comprising the following steps:

(SS1) its corresponding integer rank circuit is designed according to integer rank three-dimensional cell nerve network system formula (1) built in claim 1 step (S1);

(SS2) design relates to the nonlinear object function in claim 1 step (S1) cell neural network in the three-dimensional cell nerve network system circuit design of step (SS1) integer rankModule;

(SS3) the fractional order value (q1=q2=q3=0.95) that selection is determined, and design the fractional order element circuit of this fractional order value correspondence rank value, including chain, tree-shaped, mixed type or novel;

(SS4) select the electric capacity in the circuit of designed integer rank in suitable fractional order element circuit replacement step (SS1), obtain the corresponding fractional order driving system circuit of system;

(SS5) formula (10) formula controller U and formula (12) formula self-adaptative adjustment rate being updated to the responding system formula (8) in claim 1 step (S4), the system of meeting with a response is:

$\left\{\begin{array}{c}\frac{d^{q_1}{y}_1}{{\mathrm{dt}}^{q_1}}=(s_{11}-1)y_1+s_{12}{x}_2+2a_{11}f\left(x_1\right)-{\tilde{a}}_{11}f\left(x_1\right)+2a_{12}f\left(x_2\right)-{\tilde{a}}_{12}f\left(x_2\right)\\ \frac{d^{q_2}{y}_2}{{\mathrm{dt}}^{q_2}}=s_{21}{x}_1+(s_{22}-1)y_2+s_{23}{x}_3+(a_{12}+a_{22})f\left(x_2\right)-{\tilde{a}}_{12}f\left(x_2\right)\\ \frac{d^{q_3}{y}_3}{{\mathrm{dt}}^{q_3}}=s_{31}{x}_1+s_{32}{x}_2+(s_{33}-1)y_3\end{array}\right.---\left(13\right)$

Design nonlinear factorWithIntegrating circuit, design responding system circuit further in accordance with responding system equation (13);

(SS6) to drive circuit in step (SS4) and in step (SS5) response circuit carry out circuit integrated emulation, the synchronization character of checking design system.

Present invention is characterized in that this system is fractional order dynamic stability compared with traditional cell neural network, in designed drive response synchro system, the nonlinear parameter of drive system is it is known that and the nonlinear parameter of responding system is unknown. But still make this drive response system realize Synchronization Control by designing isochronous controller and parameter adaptive regulation. In conjunction with fractional order circuit, theoretical and multiplexed combination circuit thought, have devised corresponding synchronization control circuit schematic diagram. Simulation result shows that circuit simulation has similar synchronization phasor with numerical simulation, demonstrate this Systems Theory analyze correctness and actual physics on realizability.

Accompanying drawing explanation

The chaos attractor phasor that Fig. 1 fractional order dynamic stability numerical computations produces. A () is variable x₁-x₂, (b) is variable x₂-x₃, (c) is variable x₁-x₃��

Fig. 2 fractional order CNN adaptive synchronicity system model variable and error curve diagram. Wherein (a) is for driving and response variable x_i-y_i(i=1,2,3) variation track curve chart, (b) is system model error e_i(i=1,2,3) Asymptotic Synchronization figure.

Fig. 3 fractional order CNN drive system (driver) circuit theory diagrams. Wherein (a) fractional order CNN drive system (driver) circuit theory diagrams, (b) is drive system (driver) equivalent circuit diagram.

Fig. 4 fractional order CNN driving system circuit simulation result phasor. A () is variable x₁-x₂, (b) is variable x₂-x₃, (c) is variable x₁-x₃��

Fig. 5 fractional order CNN responding system circuit theory diagrams.

Fig. 6 fractional order CNN drive response system x_i-y_i(i=1,2,3) Simulation results. A () is variable x₁-y₁, (b) is variable x₂-y₂, (c) is variable x₃-y₃��

Fig. 7 f (x) module FX circuit theory diagrams and simulation waveform thereof. A () is circuit theory diagrams, (b) is circuit simulation waveform, and (c) is equivalent circuit diagram.

Fig. 8 fractional order each unit circuit diagram. Wherein, (a) is chain element circuit; (b) tree-shaped element circuit; C () is mixed type element circuit; D () is Novel unit circuit.

Detailed description of the invention

Below with reference to accompanying drawing, the present invention is described in further detail.

Embodiment:

1, adopting the fractional calculus of Caputo definition, its mathematic(al) representation is as follows:

$\frac{d^{q}f\left(t\right)}{{\mathrm{dt}}^q}=\frac{1}{\mathrm{\Γ}(n-q)}{\∫}_0^t\frac{f^{\left(n\right)}\left(\mathrm{\τ}\right)}{{(t-\mathrm{\τ})}^{q-n+1}}d\mathrm{\τ}---\left(13\right)$

�� () in formula is Gamma function, and n-1��q��n, q is mark, and n is integer, and the Laplace of this formula converts expression formula and is:

$L\left\{\frac{d^{q}f\left(t\right)}{{\mathrm{dt}}^q}\right\}=s^{q}L\left\{f\left(t\right)\right\}-\underset{k=0}{\overset{n-1}{\Σ}}{s}^k\[\frac{d^{q-1-k}f\left(t\right)}{{\mathrm{dt}}^{q-1-k}}\]---\left(14\right)$

If the initial condition of function f (t) is zero, then formula (14) can be simplified shown as:

$L\left\{\frac{d^{q}f\left(t\right)}{{\mathrm{dt}}^q}\right\}=s^{q}L\left\{f\left(t\right)\right\}---\left(15\right)$

Fractional Differential Equation for its correspondence of dynamic system can be expressed as:

$\begin{array}{c}{a}_n{D}^{v_n}F(x,y)+a_{n-1}{D}^{v_{n-1}}F(x,y)+\mathrm{...}+a_0{D}^{v_0}F(x,y)\\ =b_m{D}^{{\mathrm{\α}}_m}G(x,y)+b_{m-1}{D}^{{\mathrm{\α}}_{m-1}}G(x,y)+\mathrm{...}+b_0{D}^{{\mathrm{\α}}_0}G(x,y)\end{array}---\left(16\right)$

Wherein v_n,v_n-1��v₀And ��_m,��_m-1����₀Represent corresponding fractional order rank value respectively. In formula, (x, y) inputs F for system, and (x y) exports, it is assumed that they are satisfied by the condition that initial value is 0 G for system. It is done Laplace conversion, it is possible to the transmission function obtaining Fractional Differential Equation is:

$H\left(s\right)=\frac{b_m{s}^{{\mathrm{\α}}_m}+b_{m-1}{s}^{{\mathrm{\α}}_{m-1}}+...+b_0{s}^{{\mathrm{\α}}_0}}{a_n{s}^{v_n}+a_{n-1}{s}^{v_{n-1}}+...+a_0{s}^{v_0}}---\left(17\right)$

It is not difficult to find out from formula (17): transfer function H (s)=1/s can be used in a frequency domain^qRepresent fractional order differential operator q.

2, in order to new fractional order cell neural network can produce stable chaos phenomenon, the matrix parameter of system (7) is made to be chosen as:

$A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]=\left[\begin{array}{ccc}7& -1& 0\\ 0& -2& 0\\ 0& 0& 0\end{array}\right],S=\left[\begin{array}{ccc}{s}_{11}& s_{12}& s_{13}\\ s_{21}& s_{22}& s_{23}\\ s_{31}& s_{32}& s_{33}\end{array}\right]=\left[\begin{array}{ccc}-2& 9& 0\\ 2& 1& 2\\ -3& -15& 1\end{array}\right]$

Fractional order drive system (7) equation is changed to:

$\left\{\begin{array}{c}\frac{d^{q_1}{x}_1}{{\mathrm{dt}}^{q_1}}=-3x_1+9x_2+7f\left(x_1\right)-f\left(x_2\right)\\ \frac{d^{q_2}{x}_2}{{\mathrm{dt}}^{q_2}}=2x_1+2x_3-2f\left(x_2\right)\\ \frac{d^{q_3}{x}_3}{{\mathrm{dt}}^{q_3}}=-3x_1-15x_2\end{array}\right.---\left(18\right)$

(18) three Lyapunov index respectively L in formula₁=5.0529, L₂=-1.2626, L₃=-1.7904, its maximum is more than zero. And the Lyapunov dimension of system is:

$D_L=j+\frac{1}{\left|L_{j+1}\right|}\underset{i=1}{\overset{j}{\Σ}}{L}_i=2+\frac{L_1+L_2}{\left|L_3\right|}=4.1456$

Therefore this system creates chaos phenomenon, and designed system is carried out numerical simulation and observes its chaotic characteristic with attractor phasor as it is shown in figure 1, also indicate that and which create chaos phenomenon by MATLAB simultaneously.

3, according to (6) formulaCorresponding fractional order responding system is:

$\left\{\begin{array}{c}\frac{d^{q_1}{y}_1}{{\mathrm{dt}}^{q_1}}=-y_1+s_{11}{y}_1+s_{12}{y}_2+{\tilde{a}}_{11}f\left(y_1\right)+{\tilde{a}}_{12}f\left(y_2\right)+u_1\\ \frac{d^{q_2}{y}_2}{{\mathrm{dt}}^{q_2}}=-y_2+s_{21}{y}_1+s_{22}{y}_2+s_{23}{y}_3+{\tilde{a}}_{22}f\left(y_2\right)+u_2\\ \frac{d^{q_3}{y}_3}{{\mathrm{dt}}^{q_3}}=-y_3+s_{31}{y}_1+s_{32}{y}_2+s_{33}{y}_3+u_3\end{array}\right.---\left(19\right)$

(19) nonlinear parameter in formulaIt is unknown-value.

3, design isochronous controller and parameter adaptive regulation.

Isochronous controller U is deployable is:

$\left\{\begin{array}{c}{u}_1=a_{11}f\left(x_1\right)-{\tilde{a}}_{11}f\left(y_1\right)+a_{12}f\left(x_2\right)-{\tilde{a}}_{12}f\left(y_2\right)-s_{12}{e}_2-{\tilde{e}}_{11}f\left(x_1\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_2=-s_{21}{e}_1-s_{23}{e}_3+a_{22}f\left(x_2\right)-{\tilde{a}}_{22}f\left(y_2\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_3=-s_{31}{e}_1-s_{32}{e}_2\end{array}\right.---\left(19\right)$

Wherein e_i=y_i-x_i,

Building a Lyapunov-krasovskii functional function is:

$V\left(t\right)=\frac{1}{2}{e}^{T}e+\frac{1}{2}{\tilde{e}}^T\tilde{e}=\frac{1}{2}(e_1^2+e_2^2+e_3^2)+\frac{1}{2}({\tilde{e}}_{11}^2+{\tilde{e}}_{12}^2+{\tilde{e}}_{22}^2)---\left(20\right)$

Selecting system self-adaptative adjustment rule is simultaneously:

$\begin{array}{c}\left[\begin{array}{c}{\stackrel{\·}{\tilde{a}}}_{11}\\ {\stackrel{\·}{\tilde{a}}}_{12}\\ {\stackrel{\·}{\tilde{a}}}_{22}\end{array}\right]=G{\left(x\right)}^T{\left[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right]}^T=\left[\begin{array}{ccc}f\left(x_1\right)& 0& 0\\ f\left(x_2\right)& f\left(x_2\right)& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]\\ =\left[\begin{array}{c}{e}_{1}f\left(x_1\right)\\ e_{1}f\left(x_2\right)+e_{2}f\left(x_2\right)\\ 0\end{array}\right]\end{array}---\left(21\right)$

Conclusion can be obtained: work as s by theoretical derivation₁₁�� 1, s₂₂�� 1, s₃₃When��1, V'(t in formula (20))��0, namely have:Set up, and obviously have V (t) >=0, therefore error

Therefore responding system Y and drive system X tends to Tong Bu, namely has during t �� ��, Y-X �� 0,

With MATLAB Numerical Simulation Results as shown in Figure 2.

4, linear resistance, linear capacitance, operational amplifier LM741 and fractional order element circuit are utilized, design fractional order cell neural network driving system circuit schematic diagram is as shown in Figure 3, and carrying out circuit simulation, simulation result is similar to Numerical Simulation Results Fig. 1, and simulation result is as shown in Figure 4.

5, the equation according to self-adaptative adjustment rate designs nonlinear factorWithIntegrating circuit figure, responding system circuit can be designed further in accordance with responding system equation; Can designing circuit theory Fig. 5 institute of overall Self-adaptive synchronization control system in conjunction with drive system Fig. 3, and utilize Multisim to be driven-respond synchronization simulation, simulation result is as shown in Figure 6

6, drive and response circuit relates to the nonlinear object function in cell neural network (1)Module, it uses amplifier TL082CD design when �� 18V to realize; The circuit diagram of design is as shown in Figure 7.

7, reality of the present invention can realize 4096 kinds of multiplexed combination fractional order circuits, for brevity, chooses fractional order q_i(i=1,2,3) identical value (i.e. q₁=q₂=q₃=0.95) chain, tree-shaped, mixed type, novel four kinds of combinations. Its complex frequency domain expression formula is respectively as follows:

(a) chain $H\left(s\right)=\frac{R_1}{{\mathrm{sR}}_1{C}_1+1}+\frac{R_2}{{\mathrm{sR}}_2{C}_2+1}+\frac{R_3}{{\mathrm{sR}}_3{C}_3+1};$

(b) tree-shaped $H\left(s\right)=\[R_1+(R_2//\frac{1}{{\mathrm{sC}}_2})\]//\[\frac{1}{{\mathrm{sC}}_1}+(R_3//\frac{1}{{\mathrm{sC}}_3})\];$

(c) mixed type $H\left(s\right)=\{\[\left(\right(R_1//\frac{1}{{\mathrm{sC}}_1})+R_2)//\frac{1}{{\mathrm{sC}}_2}\]+R_3\}//\frac{1}{{\mathrm{sC}}_3};$

D () is novel $H\left(s\right)=R_1//\frac{1}{{\mathrm{sC}}_1}//\[R_2+\frac{1}{{\mathrm{sC}}_2}\]//\[R_3+\frac{1}{{\mathrm{sC}}_3}\].$

The component parameters of each unit circuit is as shown in table 1, and corresponding circuit diagram is as shown in Figure 8.

Table 1 fractional order each unit circuit element parameter

Claims (2)

1. a fractional order cell neural network Self-adaptive synchronization control method, is characterized in that comprising the following steps:

(S1) according to cytocidal action lattice basic model, first design integer rank three-dimensional cell nerve network system, the parameters in adjustment state equationMake system output chaos phenomenon;

$\frac{{\mathrm{dx}}_j}{dt}=-x_j+a_{j}f\left(x_j\right)+\underset{k=1,k\≠j}{\overset{3}{\Σ}}{a}_{jk}f\left(x_k\right)+\underset{k=1}{\overset{3}{\Σ}}{S}_{jk}{x}_k+{\tilde{I}}_j,(j=1,2,3)---\left(1\right)$

And apply MATLAB software designed system is carried out numerical simulation;

(S2) fractional order differential algorithm is selected: selecting the fractional calculus of Caputo definition, its mathematic(al) representation is as follows:

$\frac{d^{q}f\left(t\right)}{{\mathrm{dt}}^q}=\frac{1}{\mathrm{\Γ}(n-q)}{\∫}_0^t\frac{f^{\left(n\right)}\left(\mathrm{\τ}\right)}{{(t-\mathrm{\τ})}^{q-n+1}}d\mathrm{\τ}---\left(2\right)$

�� () in formula is Gamma function, and n-1��q��n, q is mark, and n is integer;

Fractional Differential Equation corresponding to dynamic system can be expressed as:

$\begin{array}{c}{a}_n{D}^{v_n}F(x,y)+a_{n-1}{D}^{v_{n-1}}F(x,y)+...+a_0{D}^{v_0}F(x,y)\\ =b_m{D}^{{\mathrm{\α}}_m}G(x,y)+b_{m-1}{D}^{{\mathrm{\α}}_{m-1}}G(x,y)+\mathrm{...}+b_0{D}^{{\mathrm{\α}}_0}G(x,y)\end{array}---\left(3\right)$

Wherein v_n,v_n-1��v₀And ��_m,��_m-1����₀Represent corresponding fractional order rank value, a respectively_n,��,a₀And b_m,��,b₀For real number; In formula, (x, y) inputs F for system, and (x, y) exports G for system;

(S3) fractional order synchronous control system model is built:

$D_t^{q}X=h\left(x\right(t),t),0\≤t\≤T---\left(4\right)$

Wherein X �� RⁿIt is a n dimension state vector of drive system, h:Rⁿ��Rⁿ, h is split as linear processes two parts, then drive system I is:

$D_t^{q}X=g\left(x\right)+G\left(x\right)A^T---\left(5\right)$

In formula: X �� R is the state variable of drive system, g:Rⁿ��RⁿFor comprising the continuous vector function of linear term, G (x) A^TFor non-linear partial, G:Rⁿ��R^n��nFor parameter vector function, A is the parameter matrix of the nonlinear function of drive system;

Corresponding responding system II is:

$D_t^{q}Y=g\left(y\right)+G\left(y\right){\tilde{A}}^T+U---\left(6\right)$

Y �� R in formula is in response to the state variable of system, U �� RⁿFor controller,It it is the parameter matrix of the nonlinear function of drive system;

(S4) the integer rank three-dimensional cell nerve network system built based on step (S1), designs corresponding fractional order dynamic stability in combination with step (S2) fractional calculus equation; Define in conjunction with formula (5) the drive system I in (S3), construct the drive system equation of this three-dimensional fractional order cell neural network:

$\left\{\begin{array}{c}\frac{d^{q_1}{x}_1}{{\mathrm{dt}}^{q_1}}=-x_1+s_{11}{x}_1+s_{12}{x}_2+a_{11}f\left(x_1\right)+a_{12}f\left(x_2\right)\\ \frac{d^{q_2}{x}_2}{{\mathrm{dt}}^{q_2}}=-x_2+s_{21}{x}_1+s_{22}{x}_2+s_{23}{x}_3+a_{22}f\left(x_2\right)\\ \frac{d^{q_3}{x}_3}{{\mathrm{dt}}^{q_3}}=-x_3+s_{31}{x}_1+s_{32}{x}_2+s_{33}{x}_3\end{array}\right.---\left(7\right)$

Nonlinear parameter a in formula₁₁,a₁₂,a₂₂It is given value; Define in conjunction with formula (6) the responding system II in (S3), the responding system equation of the three-dimensional fractional order cell neural network of structure:

$\left\{\begin{array}{c}\frac{d^{q_1}{y}_1}{{\mathrm{dt}}^{q_1}}=-y_1+s_{11}{y}_1+s_{12}{y}_2+{\tilde{a}}_{11}f\left(y_1\right)+{\tilde{a}}_{12}f\left(y_2\right)+u_1\\ \frac{d^{q_2}{y}_2}{{\mathrm{dt}}^{q_2}}=-y_2+s_{21}{y}_1+s_{22}{y}_2+s_{23}{y}_3+{\tilde{a}}_{22}f\left(y_2\right)+u_2\\ \frac{d^{q_3}{y}_3}{{\mathrm{dt}}^{q_3}}=-y_3+s_{31}{y}_1+s_{32}{y}_2+s_{33}{y}_3+u_3\end{array}\right.---\left(8\right)$

Nonlinear parameter in formulaIt is unknown-value;

(S5) the isochronous controller U of (5) formula and parameter adaptive regulation in design procedure (S3), utilizes mathematical theory to carry out proving to synchronize character, and is undertaken emulate by MATLAB program and synchronize verify;

The error of drive system and responding system is:

$D_t^{q}e=D_t^{q}Y-D_t^{q}X=g\left(y\right)-g\left(x\right)+G\left(y\right){\tilde{A}}^T-G\left(x\right)A^T+U---\left(9\right)$

Isochronous controller U:

U=[u₁,u₂,u₃](10)

Wherein: $\left\{\begin{array}{c}{u}_1=a_{11}f\left(x_1\right)-{\tilde{a}}_{11}f\left(y_1\right)+a_{12}f\left(x_2\right)-{\tilde{a}}_{12}f\left(y_2\right)-s_{12}{e}_2-{\tilde{e}}_{11}f\left(x_1\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_2=-s_{21}{e}_1-s_{23}{e}_3+a_{22}f\left(x_2\right)-{\tilde{a}}_{22}f\left(y_2\right)-{\tilde{e}}_{12}f\left(x_2\right)\\ u_3=-s_{31}{e}_1-s_{32}{e}_2\end{array}\right.---\left(11\right)$

Selecting system self-adaptative adjustment rule is simultaneously:

$\left[\begin{array}{c}{\stackrel{\·}{\tilde{a}}}_{11}\\ {\stackrel{\·}{\tilde{a}}}_{12}\\ {\stackrel{\·}{\tilde{a}}}_{22}\end{array}\right]=G{\left(x\right)}^T{\left[\begin{array}{ccc}{e}_1& e_2& e_3\end{array}\right]}^T=\left[\begin{array}{ccc}f\left(x_1\right)& 0& 0\\ f\left(x_2\right)& f\left(x_2\right)& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{e}_1\\ e_2\\ e_3\end{array}\right]=\left[\begin{array}{c}{e}_{1}f\left(x_1\right)\\ e_{1}f\left(x_2\right)+e_{2}f\left(x_2\right)\\ 0\end{array}\right]---\left(12\right)$

Wherein e_i(i=1,2,3) for the error of drive system Yu responding system;

(S6) design fractional order dynamic stability drive system formula (7) and responding system formula (8) circuit theory diagrams, and realize circuit simulation with Multisim design con-trol device formula (10) and self-adaptative adjustment rate formula (12).

2. the circuit design method of the fractional order cell neural network Self-adaptive synchronization control method described in claims 1, is characterized in that comprising the following steps:

(SS1) its corresponding integer rank circuit is designed according to integer rank three-dimensional cell nerve network system formula (1) built in claim 1 step (S1);

(SS2) design relates to the nonlinear object function in claim 1 step (S1) cell neural network in the three-dimensional cell nerve network system circuit design of step (SS1) integer rankModule;

(SS3) the fractional order value (q1=q2=q3=0.95) that selection is determined, and design the fractional order element circuit of this fractional order value correspondence rank value, including chain, tree-shaped, mixed type or novel;

(SS4) select the electric capacity in the circuit of designed integer rank in suitable fractional order element circuit replacement step (SS1), obtain the corresponding fractional order driving system circuit of system;

(SS5) formula (10) formula controller U and formula (12) formula self-adaptative adjustment rate being updated to the responding system formula (8) in claim 1 step (S4), the system of meeting with a response is:

$\left\{\begin{array}{c}\frac{d^{q_1}{y}_1}{{\mathrm{dt}}^{q_1}}=(s_{11}-1)y_1+s_{12}{x}_2+2a_{11}f\left(x_1\right)-{\tilde{a}}_{11}f\left(x_1\right)+2a_{12}f\left(x_2\right)-{\tilde{a}}_{12}f\left(x_2\right)\\ \frac{d^{q_2}{y}_2}{{\mathrm{dt}}^{q_2}}=s_{21}{x}_1+(s_{22}-1)y_2+s_{23}{x}_3+(a_{12}+a_{22})f\left(x_2\right)-{\tilde{a}}_{12}f\left(x_2\right)\\ \frac{d^{q_3}{y}_3}{{\mathrm{dt}}^{q_3}}=s_{31}{x}_1+s_{32}{x}_2+(s_{33}-1)y_3\end{array}\right.---\left(13\right)$

Design nonlinear factorWithIntegrating circuit, design responding system circuit further in accordance with responding system equation (13);

(SS6) to drive circuit in step (SS4) and in step (SS5) response circuit carry out circuit integrated emulation.