CN103645646B - Include complicated circuit system modeling and the control method of small inductance or electric capacity - Google Patents

Abstract

A kind of complicated circuit system modeling including small inductance or electric capacity and control method, complicated circuit system controls technical field.The method, based on UDTFSPM, bind profile norm and LMI approach, designs RFSOFC for controlled CCSWSIOCs, it is achieved the high accuracy of CCSWSIOCs controls.Kinetic model according to CCSWSIOCs, set up and obscure Singular Perturbation Model its uncertain continuous time, select the suitable sampling period, use zero-order holder method, obtained continuous time is obscured Singular Perturbation Model and carries out discretization, obtain the UDTFSPM of CCSWSIOCs, design RFSOFC on this basis.Advantage is, solves existing modeling and control method cannot eliminate the inside small inductance of CCSWSIOCs or electric capacity causes instability or the big problem of steady-state error, the control performance of CCSWSIOCs is greatly improved.The high accuracy applying the present invention to Van der Pol Circuits System controls, and carries out simulating, verifying, shows the effectiveness of proposition method.

Description

Include complicated circuit system modeling and the control method of small inductance or electric capacity

Technical field

The invention belongs to complicated circuit system and control technical field, for the complicated circuit system including small inductance or electric capacity System (Complex Circuit Systems with Small Inductances or Capacitances, brief note CCSWSIOCs), it is provided that a kind of uncertain discrete time obscures singular perturbation modeling and robust control method, it is adaptable to multiple The high accuracy of miscellaneous Circuits System controls it can also be used to system model contains other complex system modelings of little time constant with high-precision Degree controls.

Background technology

Along with developing rapidly of science and technology, electronic circuit system structure is increasingly sophisticated, the control accuracy requirement to Circuits System More and more higher, but, perturbation, parameter that the intrinsic small parameter of complicated circuit system especially CCSWSIOCs causes are the most true Qualitative and extraneous random disturbances etc. causes the steady-state error of system so that designed controller cannot meet high accuracy and control to want Ask.Than custom circuit system, the modeling of CCSWSIOCs is more complicated with control, how to overcome the little electricity that CCSWSIOCs is intrinsic Sense or small capacitances impact, parameter uncertainty and external interference, reduce steady-state error, it is ensured that the high accuracy of CCSWSIOCs controls It it is Chinese scholars study hotspot.

For CCSWSIOCs, use custom circuit system modeling method to set up model, the mould of the highest exponent number will be obtained Type, this will improve controller design difficulty, morbid state numerical characteristics will even occur, it is impossible to design controller.To this end, Existing methods is general Ignore the small inductor of CCSWSIOCs or the impact of electric capacity all over using or they are regarded as external disturbance, reaching model reduction, also It is exactly to retain low frequency part, and ignores HFS or seen as external disturbance.But, this kind of method is to control by sacrificing Precision reaches model reduction, it is impossible to the high accuracy meeting CCSWSIOCs controls requirement, in the urgent need to new Modeling Theory and controlling party Method.

Singular perturbation method is to process Multi-time Scale problem or solve the effective tool including the small parameter differential equation, from 60 years After generation is proposed by Klimushev et al., obtain large development.As the effective tool of process Multi-time Scale problem, singular perturbation side Method is widely used in the modeling of Multi-time Scale system and controls with high accuracy, as complex flexible spacecraft, flexible mechanical arm and chemistry steam The high accuracy evaporating tower etc. controls, and obtains preferable control performance.But, as processing small parameter differential equation instrument, Singular perturbation theory still stays in the aspect that pure mathematics is analyzed, and less is applied to real system.By analyzing the dynamic of CCSWSIOCs Mechanics understands, and CCSWSIOCs is the system typically including small parameter, can use its high accuracy control of singular perturbation technical finesse Problem processed, and research in this respect be meet mathematics, electronic circuit and automatic control technology are intersected, merge and develop urgent It is essential and asks, there is important theory significance and actual application value.

In sum, the kinetic model of CCSWSIOCs is a differential equation comprising little time constant, existing modeling side Method is commonly used to be ignored small parameter or is seen as the outer low accuracy control method disturbed, because in the situation considering small parameter Under, use modeling and the controller design method of conventional system, morbid state numerical solution will be obtained.Under unified model framework, it is considered to The perturbation sex chromosome mosaicism that non-linear, the parameter uncertainty of CCSWSIOCs and small inductance or electric capacity cause, describes its morbid state power Learn, and based on obtained model, designing and the outer control rate of disturbing the steady-state error that with small parameter cause can be reduced or eliminated is to solve Certainly CCSWSIOCs the key of high accuracy control problem, the present invention is made that substantial breakthrough for this.

Summary of the invention

It is an object of the invention to provide a kind of based on uncertain discrete-time fuzzy Singular Perturbation Model (Uncertain Discrete time Fuzzy Singularly Perturbed Model, be abbreviated UDTFSPM) CCSWSIOCs high accuracy Robust control method, solves existing CCSWSIOCs control method and cannot eliminate small inductance or electric capacity, disturb with systematic parameter not outward The steady-state error problem that definitiveness causes, greatly improves the overall control performance of CCSWSIOCs.

The technical scheme is that

The UDTFSPM of CCSWSIOCs sets up and High Precision Robust control method, and the method is based on UDTFSPM, bind profile model Number and LMI (Linear Matrix Inequalities is abbreviated LMIs) method, set for controlled CCSWSIOCs Meter robust fuzzy static output feedback controller (Robust Fuzzy Static Output Feedback Controller, Brief note RFSOFC), it is achieved the high accuracy of CCSWSIOCs controls.According to the kinetic model of CCSWSIOCs, set up it uncertain Property continuous time obscure Singular Perturbation Model, then according to real system require select the sampling period, use zero-order holder side Method, carries out discretization to obtained continuous model, it is thus achieved that the UDTFSPM of CCSWSIOCs, designs RFSOFC on this basis.

Specifically include:

As in figure 2 it is shown, the present invention implements on CCSWSIOCs, the hardware components of described control system specifically includes that controlled CCSWSIOCs, sensor, controller and executor, wherein executor includes buffer and zero-order holder.

Step one, kinetics equation according to CCSWSIOCs, set up the uncertain continuous time of controlled CCSWSIOCs Fuzzy Singular Perturbation Model

Slowly or state variable measured directly can see the change of CCSWSIOCs as slow variable, small parameter is correlated with Or change state variable faster and see fast variable as, use sector nonlinear method, set up CCSWSIOCs uncertain even Continuous time ambiguity Singular Perturbation Model.

Rule i: if ξ₁T () is φ_i1..., ξ_qT () is φ_ig, then

$\begin{array}{c}{E}_{\mathrm{\ϵ}}\stackrel{.}{x}\left(t\right)=(A_{\mathrm{ci}}+\mathrm{\Δ}{Z}_{\mathrm{ci}})x\left(t\right)+B_{\mathrm{ci}}\left(t\right)+\mathrm{Dw}\left(t\right)\\ y\left(t\right)=\mathrm{Hx}\left(t\right)\end{array}----\left(1\right)$

Wherein,

$E_{\mathrm{\ϵ}}\left[\begin{array}{cc}{I}_{n\×n}& 0\\ 0& {\mathrm{\ϵI}}_{m\×m}\end{array}\right],x\left(t\right)=\left[\begin{array}{c}{x}_s\left(t\right)\\ x_f\left(t\right)\end{array}\right],H=\left[\begin{array}{cc}{I}_{n\×n}& 0_{n\×m}\end{array}\right],$

x_s(t)∈RⁿFor slow variable, xf (t) ∈ R^mFor fast variable, u (t) ∈ R^qFor controlling input, y (t) ∈ R^tDefeated for system Go out, w (t) ∈ R^qFor disturbing outward, φ_i1..., φ_ig(i=1,2 ..., r) it is fuzzy set, ξ₁(t) .., ξ_gT () is measurable System variable, A_ci, B_di, D is suitable dimension matrix, Δ A_ciFor the uncertain matrix of suitable dimension, ε is singular perturbation parameter.

Step 2, set up the UDTFSPM of controlled CCSWSIOCs

Sensor and executor in control system all use time type of drive, and the two uses the identical sampling time T_s, under the effect of zero-order holder, by above continuous time model (1), the discrete UDTFSPM that turns to:

Rule i: if ξ₁K () is φ_i1..., ξ_gK () is φ_ig, then

$\stackrel{.}{x}\left(k\right)=E_{\mathrm{\ϵ}}(A_i+\mathrm{\Δ}{A}_i)x\left(k\right)+E_{\mathrm{\ϵ}}{B}_{i}u\left(k\right)+E_{\mathrm{\ϵ}}\mathrm{Dw}\left(k\right)$

Y (k)=Hx (k) (2) wherein, Δ A_iFor the uncertain matrix of suitable dimension,

$A_i=E_{\mathrm{\ϵ}}^{-1}{e}^{E_{\mathrm{\ϵ}}^{-1}{A}_{\mathrm{ci}}{T}_s},B_i=E_{\mathrm{\ϵ}}^{-1}{\∫}_0^h{E}_{\mathrm{\ϵ}}^{-1}{e}^{E_{\mathrm{\ϵ}}^{-1}{A}_{\mathrm{ci}}\mathrm{\τ}}\mathrm{d\τ}{B}_{\mathrm{ci}}$

Given [x (k)；u(k)；W (k)], apply standard fuzzy reasoning method, obtain the overall situation UDTFSPM:

$\stackrel{\·}{x}\left(k\right)=E_{\mathrm{\ϵ}}(A\left(\mathrm{\μ}\right)+\mathrm{\ΔA}\left(\mathrm{\μ}\right))x\left(k\right)+E_{\mathrm{\ϵ}}B\left(\mathrm{\μ}\right)u\left(k\right)+E_{\mathrm{\ϵ}}\mathrm{Dw}\left(k\right)$

Y (k)=Hx (k) (3) wherein, membership function ${\mathrm{\μ}}_i\left(\mathrm{\ξ}\left(k\right)\right)=\frac{w_i\left(\mathrm{\ξ}\left(k\right)\right)}{\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{w}_i\left(\mathrm{\ξ}\left(k\right)\right)},w_i\left(\mathrm{\ξ}\left(k\right)\right)=\underset{j=1}{\overset{g}{\mathrm{\Π}}}{\mathrm{\φ}}_{\mathrm{ij}}\left({\mathrm{\ξ}}_j\left(k\right)\right),,$ φ_ij(ξ_j (k)) it is ξ_jK () is at φ_ijIn degree of membership, if w_i(ξ (k)) >=0, for i=1,2 ..., r, r are rule number, μ_i(ξ (k)) >=0,μ is made for the ease of recording us_i=μ_i(ξ (k)),

$A\left(\mathrm{\μ}\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\μ}}_i{A}_i,B\left(\mathrm{\μ}\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\μ}}_i{B}_i,\mathrm{\ΔA}\left(\mathrm{\μ}\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{\mathrm{\μ}}_i\mathrm{\Δ}{A}_i.$

Step 3, based on UDTFSPM (3), controlled device is designed RFSOFC

Designing following RFSOFC, its fuzzy rule former piece is identical with the fuzzy rule former piece of system (3).

u(k)=G(μ)y(k) (4)

Wherein,G_iFor controller gain.

Step 4, set up closed-loop model

For controlled system model (3), application controls rate (4), it is thus achieved that closed-loop model:

X sympathizes+1)=E_ε(A(μ)+B(μ)G(μ)H+ΔA(μ))x(k)+E_εDw(k) (5)

Step 5, employing spectral norm method and LMI approach, derive the sufficient condition that RFSOFC exists, Derivation is not sought knowledge the supremum of systematic uncertainty parameter.The linear matrix inequality technique that solve RFSOFC gain is presented herein below Formula group:

Θ_ii< 0 (i=1,2 ... r) (6)

${\mathrm{\&epsiv;}}^2\left[\begin{array}{ccc}\mathrm{\&Omega;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\mathrm{\&epsiv;}\left[\begin{array}{ccc}\mathrm{\&Xi;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{\mathrm{\&Theta;}}_{\mathrm{ii}}<0(i=\mathrm{1,2},...r)---\left(7\right)$

Θ_ij+Θ_ji<0 (1≤i<j≤r) (8)

${\mathrm{\&epsiv;}}^2\left[\begin{array}{ccc}\mathrm{\&Omega;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\mathrm{\&epsiv;}\left[\begin{array}{ccc}\mathrm{\&Xi;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{\mathrm{\&Theta;}}_{\mathrm{ij}}+{\mathrm{\&Theta;}}_{\mathrm{ji}}<0(1\&le;i<j\&le;r)---\left(9\right)$

Wherein,

${\mathrm{\&Gamma;}}_{\mathrm{ij}}=\left[\begin{array}{cc}\mathrm{\&Lambda;}-\mathrm{\&beta;I}& *\\ A_{i}Y+B_i{W}_j& -Z\end{array}\right],$

W_j=[W_1j 0_q×m],

$\mathrm{\&Lambda;}=\left[\begin{array}{cc}{Z}_{11}& 0\\ 0& 0\end{array}\right]-\mathrm{\&gamma;Y}-{\mathrm{\&gamma;Y}}^T,$

$\mathrm{\&Omega;}=\left[\begin{array}{cc}0& 0\\ 0& Z_{22}\end{array}\right],\mathrm{\&Xi;}=\left[\begin{array}{cc}0& *\\ Z_{21}& 0\end{array}\right],$

γ (0 < γ≤1), β are the constant more than zero, and the value of γ, β can be selected by designer that (designer is suitable by selecting γ, β value obtains optimal controller gain), $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{21}^T\\ Z_{21}& Z_{22}\end{array}\right](Z_{11}\&Element;R^{n\&times;n},Z_{22}\&Element;R^{m\&times;m})$ For symmetric positive definite matrix,

$Y=\left[\begin{array}{cc}{Y}_{11}& 0\\ 0& Y_{22}\end{array}\right](Y_{11}\&Element;R^{n\&times;n},Y_{22}\&Element;R^{m\&times;m}),$

Controller gain:

$G_i=W_{1i}*Y_{11}^{-1}$ For i=1,2 ..., r. (10)

Step 6, gained controller Matlab code is transferred to C language code, implant controller.Controller uses thing Part type of drive, when sampled data arrives controller, controller calculates at once, and control signal is passed to executor, Executor reads control signal according to the fixing sampling period, generates and controls input, acts on controlled CCSWSIOCs, thus real The high accuracy of existing CCSWSIOCs controls.

Advantages of the present invention:

(1), use UDTFSPM to describe the parameter uncertainty of CCSWSIOCs, non-linear and small inductance draws with electric capacity The perturbation risen, solving existing CCSWSIOCs model cannot the ill kinetics of accurate description CCSWSIOCs and the control that causes The low problem of performance；

(2), propose RFSOFC method for designing based on UDTFSPM, solve existing control method and be difficult to eliminate CCSWSIOCs The steady-state error difficult problem that intrinsic small inductance and electric capacity cause, the high accuracy reaching CCSWSIOCs controls.

(3), the present invention proposes in controller gain solution procedure without knowing the new of the supremum of uncertain parameters Method, solves existing uncertain system control method and is difficult to the probabilistic upper bound of precompensation parameter asks when processing real system Topic, provides new way for uncertain control theory.

(4), the RFSOFC that proposes of the present invention, can not only the perturbation that causes of the internal small parameter of suppression system, it is also possible to effectively Overcome outer disturbing, thus be substantially improved the control accuracy of system.

(5), the RFSOFC gain that the present invention proposes can avoid existing Static output anti-by solving one group of LMI acquisition Solve iteration LMI in feedback control method selects initial value difficulty problem.

Accompanying drawing explanation

The flow chart of Fig. 1 the inventive method.

Fig. 2 CCSWSIOCs structure chart.

Fig. 3 Van der Pol circuit diagram.

Fig. 4 closed loop system condition responsive curve.

Fig. 5 system output i_L(t) response curve.

Fig. 6 system output v_c1(t) response curve.

Fig. 7 system output v_c2(t) response curve.

Fig. 8 system enters the system output i after stable state_L(t) response curve.

Fig. 9 system enters the system output v after stable state_c1(t) response curve.

Figure 10 system enters the system output v after stable state_c2(t) response curve.

Detailed description of the invention

Van der Pol Circuits System is typical CCSWSIOCs, applies the inventive method to Van der below Pol Circuits System, and combine Fig. 1 and Fig. 2, its implementation is described, detailed process is as follows:

Step one, analyze the kinetics of Van der Pol Circuits System, set up its of uncertainty continuous time and obscure unusual Perturbation model.

Considering Van der Pol circuit as shown in Figure 3, wherein R3 is nonlinear impedance, and its voltage is

${\mathrm{vR}}_3\left(t\right)=-{\mathrm{ai}}_L\left(t\right)-{\mathrm{bi}}_L^3\left(t\right)---\left(11\right)$

|i_L(t) |≤1, R₁, R₂For resistance, C₁, C₂For electric capacity, L is inductance, C_εFor small capacitance.

Make x₁(t)=i_L(t), x₂(t)=v_c1(t), x₃(t)=v_c2(t), x₄(t)=v_ε(t), application Kirchoff s voltage and electricity Stream theorem, sets up the state-space model of Van der Pol circuit:

$\begin{array}{c}{\stackrel{.}{x}}_1\left(t\right)=\frac{1}{L}({\mathrm{ax}}_1\left(t\right)+{\mathrm{bx}}_1^3\left(t\right))+\frac{1}{L}{x}_2\left(t\right)-\frac{1}{L}u\left(t\right)\\ {\stackrel{.}{x}}_2\left(t\right)=-\frac{1}{C_1}{x}_1\left(t\right)-{\mathrm{Rx}}_2\left(t\right)+\frac{1}{R_1{C}_1}{x}_3\left(t\right)+\frac{1}{R_2{C}_1}{x}_4\left(t\right)\\ {\stackrel{.}{x}}_3\left(t\right)=\frac{1}{R_1{C}_2}{x}_2\left(t\right)-\frac{1}{R_1{C}_2}{x}_3\left(t\right)\\ {\stackrel{.}{x}}_4\left(t\right)=\frac{1}{R_2{C}_{\mathrm{\&epsiv;}}}{x}_2\left(t\right)-\frac{1}{R_2{C}_{\mathrm{\&epsiv;}}}{x}_4\left(t\right)+w\left(t\right)\end{array}---\left(12\right)$

Wherein,A=0.4, b=0.5, C₁=C₂=0.5F, R₁=0.5 Ω, R₂=0.5 Ω, L=0.1H, C_ε= 0.035F。

Analyzing model (12) to understand, this model includes small capacitance C_εAnd there is nonlinear characteristic, it is individual typical CCSWSIOCs.It addition, some parameters in model (12) have some minor variations in system operation, hence set up Parameter uncertainties sex chromosome mosaicism to be considered during fuzzy Singular Perturbation Model.

Use the nonlinear characteristic of fuzzy model descriptive model (12) when setting up model, use Singular Perturbation Model to describe little Parameter problem, utilizes the uncertain problem of uncertain parameter descriptive system parameter.Particularly as follows:

In the case of considering that systematic parameter exists uncertainty, according to model (12), use sector nonlinear method with strange Different perturbation modeling technique, sets up uncertainty continuous time of Van der Pol Circuits System and obscures Singular Perturbation Model:

Rule i: if x₁T () is φ_i1, then

$E_{\mathrm{\&epsiv;}}\stackrel{.}{x}\left(t\right)=(A_{\mathrm{ci}}+\mathrm{\&Delta;}{A}_{\mathrm{ci}})x\left(t\right)+B_{\mathrm{ci}}u\left(t\right)+\mathrm{Dw}\left(t\right)$

Y (t)=Hx (t) for i=1,2 (13)

Wherein,

$E_{\mathrm{\&epsiv;}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \mathrm{\&epsiv;}\end{array}\right],x\left(t\right)=\left[\begin{array}{c}{x}_s\left(t\right)\\ x_f\left(t\right)\end{array}\right],\mathrm{\&epsiv;}=0.07,$

x_s(t)=[x₁(t) x₂(t) x₃(t)]^TFor slow variable, x_f(t)=x₄T () is fast variable, u (t) is control input,

y(t)∈R³For system export, w (t) for disturb outward, φ_i1(i=1,2) is fuzzy set, Δ A_ci∈R^4×4For uncertain Property matrix,

$A_{c1}=\left[\begin{array}{cccc}4& 10& 0& 0\\ -2& -8& 4& 4\\ 0& 4& -4& 0\\ 0& 4& 0& -4\end{array}\right.0,A_{c2}=\left[\begin{array}{cccc}9& 10& 0& 0\\ -2& -8& 4& 4\\ 0& 4& -4& 0\\ 0& 4& 0& -4\end{array}\right],$

Step 2: according to model (13), set up the UDTFSPM of Van der Pol Circuits System.

Taking the sampling period is T_s=0.1, application zero-order holder method discretization model (13), it is thus achieved that Van der Pol electricity The UDTFSPM of road system,

Rule i: if x₁K () is φ_i1, then

$\stackrel{.}{x}\left(k\right)=E_{\mathrm{\&epsiv;}}(A_i+\mathrm{\&Delta;}{A}_i)x\left(k\right)+E_{\mathrm{\&epsiv;}}{B}_{i}u\left(k\right)+E_{\mathrm{\&epsiv;}}\mathrm{Dw}\left(k\right)$

Y (k)=Hx (k) for i=1,2 (14) wherein, x (k)=[x_s(k) x_f(k)] T, x_s(k)=[x₁(k) x₂ (k) x₃(k)]^T, x_f(k)=x₄(k), Δ A_i∈R^4×4For uncertain matrix,

Application equation below

$A_i=E_{\mathrm{\&epsiv;}}^{-1}{e}^{E_{\mathrm{\&epsiv;}}^{-1}{A}_{\mathrm{ci}}{T}_s},B_i=E_{\mathrm{\&epsiv;}}^{-1}{\&Integral;}_0^h{E}_{\mathrm{\&epsiv;}}^{-1}{e}^{E_{\mathrm{\&epsiv;}}^{-1}{A}_{\mathrm{ci}}\mathrm{\&tau;}}\mathrm{d\&tau;}{B}_{\mathrm{ci}}$

:

$\begin{array}{cc}{A}_1=\left[\begin{array}{cccc}1.3810& 0.9683& 0.1697& 0.0558\\ -0.1937& 0.6058& 0.2516& 0.0461\\ -0.0339& 0.2516& 0.7213& 0.0155\\ -2.2781& 9.4044& 3.1567& 0.7442\end{array}\right],B_1=\left[\begin{array}{c}-1.5749\\ 0.1168\\ 0.0135\\ 1.1579\end{array}\right],& \\ A_2=\left[\begin{array}{cccc}2.3006& 1.2967& 0.2067& 0.0716\\ -0.2593& 0.5879& 0.2500& 0.0453\\ -0.0413& 0.2500& 0.7211& 0.0154\\ -2.9211& 9.2464& 3.1438& 0.7376\end{array}\right]{,B}_2=\left[\begin{array}{c}-1.5749\\ 0.1168\\ 0.0135\\ 1.1579\end{array}\right],& \end{array}$

Given [x (k)；u(k)；W (k)], apply standard fuzzy reasoning method, obtain the overall situation UDTFSPM:

$\stackrel{.}{x}\left(k\right)=E_{\mathrm{\&epsiv;}}(A_i+\mathrm{\&Delta;}{A}_i)x\left(k\right)+E_{\mathrm{\&epsiv;}}{B}_{i}u\left(k\right)+E_{\mathrm{\&epsiv;}}\mathrm{Dw}\left(k\right)$

y(k)=Hx(k) (15)

Membership function μ_i(x₁(k)) (i=1,2) take, μ₂(x₁(k))=1-μ₁(x₁(k)),

$A\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i\left(x_1\left(k\right)\right)A_i,B\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i\left(x_1\left(k\right)\right)B_i,\mathrm{\&Delta;A}\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i\left(x_1\left(k\right)\right)\mathrm{\&Delta;}{A}_i$

Step 3, on the basis of setting up overall situation UDTFSPM (15), to Van der Pol design of circuit system RFSOFC.

Controller rule i: if x₁K () is φ_i1, then

u(k)=G(μ)y(k) (16)

Wherein,G_iFor controller gain.

Step 4, set up closed-loop model

For controlled system model (15), application controls rate (16), it is thus achieved that closed-loop model:

x(k+1)=E_ε(A(μ)+B(μ)G(μ)H+ΔA(μ))x(k)+E_εDw(k) (17)

Step 5, employing spectral norm method and LMI approach, derive the sufficient condition that RFSOFC exists, Derivation is not sought knowledge the supremum of systematic uncertainty parameter.The linear matrix inequality technique that solve RFSOFC gain is presented herein below Formula group:

Θ_ii< 0 (i=1,2 ... r) (18)

${\mathrm{\&epsiv;}}^2\left[\begin{array}{ccc}\mathrm{\&Omega;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\mathrm{\&epsiv;}\left[\begin{array}{ccc}\mathrm{\&Xi;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{\mathrm{\&Theta;}}_{\mathrm{ii}}<0(i=\mathrm{1,2},...r)---\left(19\right)$

Θ_ij+Θ_ji<0 (1≤i<j≤r) (20)

${\mathrm{\&epsiv;}}^2\left[\begin{array}{ccc}\mathrm{\&Omega;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\mathrm{\&epsiv;}\left[\begin{array}{ccc}\mathrm{\&Xi;}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{\mathrm{\&Theta;}}_{\mathrm{ij}}+{\mathrm{\&Theta;}}_{\mathrm{ji}}<0(1\&le;i<j\&le;r)---\left(21\right)$

Wherein,

${\mathrm{\&Gamma;}}_{\mathrm{ij}}=\left[\begin{array}{cc}\mathrm{\&Lambda;}-\mathrm{\&beta;I}& *\\ A_{i}Y+B_i{W}_j& -Z\end{array}\right],$

W_j=[W_1j 0_q×m],

$\mathrm{\&Lambda;}=\left[\begin{array}{cc}{Z}_{11}& 0\\ 0& 0\end{array}\right]-\mathrm{\&gamma;Y}-{\mathrm{\&gamma;Y}}^T,$

$\mathrm{\&Omega;}=\left[\begin{array}{cc}0& 0\\ 0& Z_{22}\end{array}\right],\mathrm{\&Xi;}=\left[\begin{array}{cc}0& *\\ Z_{21}& 0\end{array}\right],$

γ (0 < γ≤1), β are the constant more than zero, and the value of γ, β can be selected by designer that (designer is suitable by selecting γ, β value obtains optimal controller gain), $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{21}^T\\ Z_{21}& Z_{22}\end{array}\right](Z_{11}\&Element;R^{n\&times;n},Z_{22}\&Element;R^{m\&times;m})$ For symmetric positive definite matrix,

$Y=\left[\begin{array}{cc}{Y}_{11}& 0\\ 0& Y_{22}\end{array}\right](Y_{11}\&Element;R^{n\&times;n},Y_{22}\&Element;R^{m\&times;m}).$

Understanding according to model (17), the partial parameters in LMI (18)-(19) is taken as:

R=2, n=3, m=1, ε=0.07

Gather method by examination, can obtain when γ=0.9, optimum RFSOFC during β=15, its gain is:

$\begin{array}{c}{G}_1=W_{11}*Y_{11}^{-1}=\left[\begin{array}{ccc}1.6222& -3.0308& -1.1054\end{array}\right],\\ G_2=W_{12}*Y_{11}^{-1}=\left[\begin{array}{ccc}1.8719& -2.4498& -0.9309\end{array}\right]\end{array}---\left(22\right)$

Step 6, gained controller Matlab code is transferred to C language code, implants Van der Pol Circuits System Control system.

Control program in step 7, operation controller, is controlled Van der Pol Circuits System, overall control System construction drawing is as in figure 2 it is shown, concrete control process is: sensor uses time type of drive, according to the fixing sampling time, Sampled signal and timestamp thereof are packaged into packet (being called for short sampled data bag) and send controller to；Controller uses event to drive Flowing mode, when sampled data bag arrives, controller is controlled signal at once and calculates, and control signal is passed to executor； Executor is made up of buffer and zero-order holder.After controlling data arrival executor, the timestamp that executor is carried Compare with the timestamp of control signal in relief area, and judge newly arrived control packet whether " newly "；"Yes" then will Newly arrived control signal and timestamp thereof preserve in the buffer, and "No" then abandons this and controls packet.Zero-order holder is adopted Use time type of drive, i.e. zero-order holder according to the fixing sampling period, read control signal from relief area, and generate control Input adjusts helicopter attitude, thus realizes the stability contorting of Helicopter System.It should be noted that sensor and executor adopt With the identical sampling period, and the two should keep clock to synchronize.

Simulating, verifying:

Using matlab software to emulate Van der Pol circuit control system, simulated conditions is:

1., initial value takes x (0)=[-0.9 0.5 0.4 1]^T；

2., inductive current i is measured_LT noise that () causes be amplitude be 1 × 10^-6The white noise of ampere, measures capacitance voltage v_c1T noise that () causes be amplitude be 0.5 × 10^-6The white noise of volt, measures capacitance voltage v_c2T noise that () causes is amplitude It is 0.15 × 10^-6The white noise of volt；

3., external disturbance $w\left(k\right)=\left\{\begin{array}{c}\mathrm{0.1,12}\&le;k\&le;15\\ 0,\mathrm{others}\end{array}\right.,$

For Van der Pol Circuits System (14), application has the control rate (16) of controller parameter (22), and gained is imitated True result is as shown in figs. 4 through 10.

Fig. 4 shows closed loop system condition responsive curve, Fig. 5-Fig. 7 show closed loop system export response curve, Fig. 8- Figure 10 shows the output response curve after system entrance stable state.Fig. 4-Fig. 7 shows that designed control can not only make system quick Enter stable state, and external disturbance and system mode can be suppressed to measure and control the noise that input introduces.Easy from Fig. 8-Figure 10 Find out inductive current i_LT the control accuracy of () reaches 6 × 10^-5Ampere, capacitance voltage v_c1T the control accuracy of () reaches 11 × 10^-6Volt, Capacitance voltage v_c1T the control accuracy of () reaches 9 × 10^-6Volt.

Summary, the simulation result for Van der Pol circuit control system shows, the employing present invention can not only Effectively process the perturbation that the inside small parameter (small inductance or electric capacity) of CCSWSIOCs causes, and effectively suppress outer and disturb and be System state measurement and the noise controlling input introducing, be substantially reduced steady-state error, and the high accuracy reaching CCSWSIOCs controls to refer to Mark.

Claims (1)

1. the complicated circuit system modeling including small inductance or electric capacity is characterized in that with control method:

Step one, kinetics equation according to CCSWSIOCs, obscure the uncertain continuous time setting up controlled CCSWSIOCs Singular Perturbation Model

Slowly or state variable measured directly can see the change of CCSWSIOCs as slow variable, small parameter is relevant or becomes Change state variable faster and see fast variable as, use sector nonlinear method, set up the uncertain consecutive hours of CCSWSIOCs Between fuzzy Singular Perturbation Model:

Rule i: if ξ₁T () is φ_i1..., ξ_gT () is φ_ig, then

Wherein,

H=[I_n×n O_n×m],

x_s(t)∈RⁿFor slow variable, x_f(t)∈R^mFor fast variable, u (t) ∈ R^qFor controlling input, y (t) ∈ R^lExport for system, w (t)∈R^qFor disturbing outward, φ_i1..., φ_ig, wherein, i=1,2 ..., r is fuzzy set, ξ₁(t) ..., ξ_gT () is for measuring System variable, A_ci, B_ci, D is suitable dimension matrix, Δ A_ciFor the uncertain matrix of suitable dimension, ε is singular perturbation ginseng Number；

Step 2, set up the UDTFSPM of controlled CCSWSIOCs

Sensor and executor in control system all use time type of drive, and the two uses identical sampling time T_s, Under the effect of zero-order holder, by above continuous time model (1), the discrete UDTFSPM that turns to:

Rule i: if ξ₁K () is φ_il..., ξ_gK () is φ_ig, then

Wherein, Δ A_iFor the uncertain matrix of suitable dimension,

Given [x (k)；u(k)；W (k)], apply standard fuzzy reasoning method, obtain the overall situation UDTFSPM:

Wherein, membership functionφ_ij(ξ_j(k)) it is ξ_jK () exists φ_ijIn degree of membership, if ω_i(ξ (k)) >=0, wherein, i=1,2 ..., r, r are rule number, μ_i(ξ (k)) >=0, μ is made for the ease of recording us_i=μ_i(ξ (k)),

Step 3, based on UDTFSPM (3), controlled device is designed RFSOFC

Designing following RFSOFC, its fuzzy rule former piece is identical with the fuzzy rule former piece of UDTFSPM (3),

U (k)=G (μ) y (k) (4)

Wherein,G_iFor controller gain；

Step 4, set up closed-loop model

For controlled system model (3), application controls rate (4), it is thus achieved that closed-loop model:

X (k+1)=E_ε(A(μ)+B(μ)G(μ)H+ΔA(μ))x(k)+E_εDw(k) (5)

Step 6, employing spectral norm method and LMI approach, derive the sufficient condition that RFSOFC exists, and derives Process is not sought knowledge the supremum of systematic uncertainty parameter, and the LMI that solve RFSOFC gain is presented herein below Group:

Θ_ii＜ 0, i=1,2 ... r (6)

Θ_ij+Θ_ji＜ 0, l≤i ＜ j≤r (8)

Wherein,

W_j=[W_1j 0_q×m],

0 ＜ γ≤1, β is the constant more than zero, and the value of γ, β can be selected by designer, and i.e. designer is by selecting suitable γ, β value obtains optimal controller gain,Wherein, Z₁₁∈R^n×n, Z₂₂∈R^m×mFor symmetric positive definite matrix,Wherein, Y₁₁∈R^n×n, Y₂₂∈R^m×m,

Controller gain:

Step 7, gained controller Matlab code being converted into C language code, implant controller, controller uses event to drive Flowing mode, when sampled data arrives controller, controller calculates at once, and control signal is passed to executor, performs Device reads control signal according to the fixing sampling period, generates and controls input, acts on controlled CCSWSIOCs, thus realize The high accuracy of CCSWSIOCs controls.