CN103645646A - Modeling and control method for complex circuit system with small inductances or capacitances - Google Patents

Abstract

Disclosed is a modeling and control method for a complex circuit system with small inductances or capacitances (CCSWSIOCs) and the method belongs to the technical field of complex-circuit-system control. The method is based on an UDTFSPM (uncertain discrete time fuzzy singularly perturbed model) and combines methods of spectral norm and linear matrix inequalities to design an RFSOFC (robust fuzzy static output feedback controller) for controlled CCSWSIOCs so as to realize high-precision control of the CCSWSIOCs. According to a dynamical model of the CCSWSIOCs, an uncertain continuous-time fuzzy singularly perturbed model is established. An appropriate sampling period is selected and a zero-order holder method is employed to perform discretization on the continuous-time fuzzy singularly perturbed model so as to obtain the UDTFSPM of the CCSWSIOCs and thus the RFSOFC is designed on the basis. The advantages of the method are that problems of incapability of elimination of instability or large steady-state errors resulted from the internal minute inductances or capacitances of the CCSWSIOCs in the prior modeling and control methods are solved and control performance of the CCSWSIOCs is improved significantly. The modeling and control method for the complex circuit system with the small inductances or capacitances is applied to precision control of a Van der Pol circuit system to perform simulation verification and validity of the method is shown.

Description

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

Technical field

The invention belongs to complicated circuit system control technology field, for complicated circuit system (the Complex Circuit Systems with Small Inductances or Capacitances that includes small inductance or electric capacity, brief note CCSWSIOCs), the fuzzy singular perturbation modeling of a kind of uncertain discrete time and robust control method are provided, the high precision that is applicable to complicated circuit system is controlled, and also can be used for other complex system modelings and high precision control that system model contains little time constant.

Background technology

Along with scientific and technological develop rapidly, electronic circuit system structure is increasingly sophisticated, to the control accuracy of Circuits System, require also more and more higher, yet, complicated circuit system perturbation, parameter uncertainty and extraneous random disturbance etc. that especially the intrinsic small parameter of CCSWSIOCs causes cause the steady-state error of system, make designed controller cannot meet high precision and control requirement.Than custom circuit system, the modeling and control of CCSWSIOCs is more complicated, how to overcome the intrinsic small inductor of CCSWSIOCs or little capacitive effect, parameter uncertainty and external interference, reduce steady-state error, guarantee that the high precision control of CCSWSIOCs is Chinese scholars study hotspot.

For CCSWSIOCs, adopt custom circuit system modeling method to set up model, will obtain the model of very high exponent number, this will improve controller design difficulty, even occur ill numerical characteristics, cannot CONTROLLER DESIGN.For this reason, existing method generally adopts to be ignored the small inductor of CCSWSIOCs or the impact of electric capacity or they is regarded as to external disturbance, reaches model reduction, namely retains low frequency part, and ignores HFS or seen as external disturbance.Yet this kind of method is to reach model reduction by sacrificing control accuracy, the high precision that cannot meet CCSWSIOCs is controlled requirement, in the urgent need to new Modeling Theory and control method.

Singular perturbation method is process Multi-time Scale problem or solve the effective tool that includes the small parameter differential equation, after being proposed by people such as Klimushev the sixties, obtains large development.As the effective tool of processing Multi-time Scale problem, singular perturbation method is widely used in modeling and the high precision of Multi-time Scale system to be controlled, and as the high precision of complex flexible spacecraft, flexible mechanical arm and chemical distillation tower etc. is controlled, and obtains good control performance.Yet as processing small parameter differential equation instrument, singular perturbation theory still stays in the aspect that pure mathematics is analyzed, the less real system that is applied to.Dynamics by analysis CCSWSIOCs is known, CCSWSIOCs is the system that typically includes small parameter, can adopt its high precision control problem of singular perturbation technical finesse, and research is in this respect to meet mathematics, electronic circuit and automatic control technology is intersected, merged and the active demand of development, there is important theory significance and actual application value.

In sum, the kinetic model of CCSWSIOCs is a differential equation that comprises little time constant, existing modeling method generally adopts to be ignored small parameter or is seen as outer low accuracy control method of disturbing, because in the situation that considering small parameter, the modeling and control device method for designing that adopts conventional system, will obtain ill numerical solution.Under unified model framework, consider the perturbation problem that non-linear, the parameter uncertainty of CCSWSIOCs and small inductance or electric capacity cause, its ill dynamics is described, and based on obtained model, design can reduce or eliminate outer control rate of disturbing the steady-state error causing with small parameter be solve CCSWSIOCs the key of high precision control problem, the present invention has made substantial breakthrough for this reason.

Summary of the invention

The object of the present invention is to provide a kind of based on uncertain discrete-time fuzzy Singular Perturbation Model (Uncertain Discrete time Fuzzy Singularly Perturbed Model, brief note UDTFSPM) CCSWSIOCs High Precision Robust control method, solve existing CCSWSIOCs control method and cannot eliminate small inductance or electric capacity, disturb the steady-state error problem causing with systematic parameter uncertainty outward, significantly improve the whole control performance of CCSWSIOCs.

Technical scheme of the present invention is:

The UDTFSPM of CCSWSIOCs sets up and High Precision Robust control method, the method is based on UDTFSPM, bind profile norm and LMI (Linear Matrix Inequalities, brief note LMIs) method, for controlled CCSWSIOCs design robust fuzzy static output feedback controller (Robust Fuzzy Static Output Feedback Controller, brief note RFSOFC), the high precision that realizes CCSWSIOCs is controlled.According to the kinetic model of CCSWSIOCs, set up its uncertain continuous time of fuzzy Singular Perturbation Model, then according to real system, require to select the sampling period, adopt zero-order holder method, obtained continuous model is carried out to discretize, the UDTFSPM that obtains CCSWSIOCs, designs RFSOFC on this basis.

Specifically comprise:

As shown in Figure 2, the present invention implements on CCSWSIOCs, and the hardware components of described control system mainly comprises: controlled CCSWSIOCs, and sensor, controller and actuator, wherein actuator comprises impact damper and zero-order holder.

Step 1, according to the kinetics equation of CCSWSIOCs, set up uncertain continuous time of the fuzzy Singular Perturbation Model of controlled CCSWSIOCs

The state variable variation of CCSWSIOCs is slow or that can directly measure is seen slow variable as, small parameter is correlated with or is changed state variable faster and sees fast variable as, adopt sector nonlinear method, set up uncertain continuous time of the fuzzy Singular Perturbation Model of CCSWSIOCs.

Rule i: if ξ ₁(t) be φ _i1..., ξ _q(t) be φ _ig, so

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

Wherein,

$E_{\mathrm{\&epsiv;}}\left[\begin{array}{cc}{I}_{n\&times;\mathrm{<figure-callout\; id="0"\; label="n"\; filenames="HDA0000430077050000022.png,HDA0000430077050000031.png"\; state="\{\{state\}\}">n</figure-callout>}}& 0\\ 0& {\mathrm{\&epsiv;I}}_{m\&times;m}\end{array}\right],x\left(t\right)=\left[\begin{array}{c}{x}_s\left(t\right)\\ x_f\left(t\right)\end{array}\right],H=\left[\begin{array}{cc}{I}_{n\&times;n}& 0_{n\&times;m}\end{array}\right],$

X _s(t) ∈ R ⁿfor slow variable, xf (t) ∈ R ^mfor fast variable, u (t) ∈ R ^qfor control inputs, y (t) ∈ R ^tfor system output, w (t) ∈ R ^qfor disturbing outward, φ _i1..., φ _ig(i=1,2 ..., r) be fuzzy set, ξ ₁(t) .., ξ _g(t) be 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 in control system and actuator all adopt time type of drive, and the two adopts 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 ξ ₁(k) be φ _i1..., ξ _g(k) be φ _ig, so

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

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

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

Given [x (k); U (k); W (k)], application standard fuzzy reasoning method, obtains overall UDTFSPM:

$\stackrel{\&CenterDot;}{x}\left(k\right)=E_{\mathrm{\&epsiv;}}(A\left(\mathrm{\&mu;}\right)+\mathrm{\&Delta;A}\left(\mathrm{\&mu;}\right))x\left(k\right)+E_{\mathrm{\&epsiv;}}B\left(\mathrm{\&mu;}\right)u\left(k\right)+E_{\mathrm{\&epsiv;}}\mathrm{Dw}\left(k\right)$

Y (k)=Hx (k) (3) wherein, membership function ${\mathrm{\&mu;}}_i\left(\mathrm{\&xi;}\left(k\right)\right)=\frac{w_i\left(\mathrm{\&xi;}\left(k\right)\right)}{\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{w}_i\left(\mathrm{\&xi;}\left(k\right)\right)},w_i\left(\mathrm{\&xi;}\left(k\right)\right)=\underset{j=1}{\overset{g}{\mathrm{\&Pi;}}}{\mathrm{\&phi;}}_{\mathrm{ij}}\left({\mathrm{\&xi;}}_j\left(k\right)\right),,$ φ _ij(ξ _j(k)) be ξ _j(k) at φ _ijin degree of membership, establish w _i(ξ (k))>=0, for i=1,2 ..., r, r is regular number, μ _i(ξ (k))>=0,

for the ease of recording us, make μ _i=μ _i(ξ (k)),

$A\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i{A}_i,B\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i{B}_i,\mathrm{\&Delta;A}\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i\mathrm{\&Delta;}{A}_i.$

Step 3, based on UDTFSPM (3), to controlled device design RFSOFC

Design 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), obtains closed-loop model:

X sympathizes+and 1)=E _ε(A (μ)+B (μ) G (μ) H+ Δ A (μ)) x (k)+E _εdw (k) (5)

Step 5, employing spectral norm method and LMI method, derive the adequate condition that RFSOFC exists, and derivation does not require the supremum of knowing systematic uncertainty parameter.The LMI group that solves RFSOFC gain below:

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]，

$\mathrm{\&Lambda;}=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="11"\; label="Z"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{11}& 0\\ 0& 0\end{array}\right]-\mathrm{\&gamma;Y}-{\mathrm{\&gamma;Y}}^T,$

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

γ (0< γ≤1), β is greater than zero constant, γ, the value of β can be selected by deviser (deviser is by selecting suitable γ, and β value obtains optimal controller gain), $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{21}^T\\ Z_{21}& Z_{22}\end{array}\right]({\mathrm{<figure-callout\; id="11"\; label="Z"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{11}\&Element;R^{n\&times;n},Z_{22}\&Element;R^{m\&times;m})$ For symmetric positive definite matrix,

$Y=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="11"\; label="Y"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{11}& 0\\ 0& Y_{22}\end{array}\right]({\mathrm{<figure-callout\; id="11"\; label="Y"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{11}\&Element;R^{n\&times;n},Y_{22}\&Element;R^{m\&times;m}),$

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 codes, implant controller.Controller adopts event driven manner, when sampled data arrives controller, controller calculates at once, and control signal is passed to actuator, actuator reads control signal according to the fixing sampling period, generate control inputs, act on controlled CCSWSIOCs, thereby realize the high precision control of CCSWSIOCs.

Advantage of the present invention:

(1), adopt UDTFSPM to describe the perturbation that the parameter uncertainty of CCSWSIOCs, non-linear and small inductance and electric capacity cause, solve the ill dynamics that existing CCSWSIOCs model cannot accurate description CCSWSIOCs and the low problem of control performance causing;

(2), the RFSOFC method for designing based on UDTFSPM is proposed, solve existing control method and be difficult to eliminate the intrinsic small inductance of CCSWSIOCs and the steady-state error difficult problem that electric capacity causes, reach the high precision control of CCSWSIOCs.

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

(4), the RFSOFC that proposes of the present invention, can not only suppress the perturbation that internal system small parameter causes, also can effectively overcome outer disturbing, thereby greatly improve the control accuracy of system.

(5), the RFSOFC gain that proposes of the present invention can obtain by solving one group of LMI, that can avoid solving in existing static output feedback control method iteration LMI selects the difficult problem of initial value.

Accompanying drawing explanation

The process flow diagram of Fig. 1 the inventive method.

Fig. 2 CCSWSIOCs structural drawing.

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.

Embodiment

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

The dynamics of step 1, analysis Van der Pol Circuits System, sets up its continuous time of uncertain fuzzy Singular Perturbation Model.

Consider 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, R ₁, R ₂for resistance, C ₁, C ₂for electric capacity, L is inductance, C _εfor small capacitance.

Make x ₁(t)=i _l(t), x ₂(t)=v _c1(t), x ₃(t)=v _c2(t), x ₄(t)=v _ε(t), apply kirchhoff voltage and current theorem, set up the state-space model of Van der Pol circuit:

$\begin{array}{c}{\stackrel{.}{x}}_1\left(t\right)=\frac{1}{L}({\mathrm{ax}}_1\left(t\right)+{\mathrm{<figure-callout\; id="1"\; label="bx"\; filenames="HDA0000430077050000022.png,HDA0000430077050000041.png"\; state="\{\{state\}\}">bx</figure-callout>}}_1^3\left(t\right))+\frac{1}{L}{x}_2\left(t\right)-\frac{1}{L}u\left(t\right)\\ {\stackrel{.}{x}}_2\left(t\right)=-\frac{1}{C_1}{x}_1\left(t\right)-{\mathrm{Rx}}_2\left(t\right)+\frac{1}{R_1{C}_1}{x}_3\left(t\right)+\frac{1}{R_2{C}_1}{x}_4\left(t\right)\\ {\stackrel{.}{x}}_3\left(t\right)=\frac{1}{R_1{C}_2}{x}_2\left(t\right)-\frac{1}{R_1{C}_2}{x}_3\left(t\right)\\ {\stackrel{.}{x}}_4\left(t\right)=\frac{1}{R_2{C}_{\mathrm{\&epsiv;}}}{x}_2\left(t\right)-\frac{1}{R_2{C}_{\mathrm{\&epsiv;}}}{x}_4\left(t\right)+w\left(t\right)\end{array}---\left(12\right)$

Wherein, a=0.4, b=0.5, C ₁=C ₂=0.5F, R ₁=0.5 Ω, R ₂=0.5 Ω, L=0.1H, C _ε=0.035F.

Analytical model (12) is known, and this model includes small capacitance C _εand having nonlinear characteristic, is a typical CCSWSIOCs.In addition, some parameters in model in system operational process (12) have some subtle change, while therefore setting up fuzzy Singular Perturbation Model, will consider parameter uncertainty problem.

While setting up model, adopt the nonlinear characteristic of fuzzy model descriptive model (12), adopt Singular Perturbation Model to describe Small Parameter, utilize the uncertain problem of uncertain parametric description systematic parameter.Be specially:

In taking into account system parameter, exist in uncertain situation, according to model (12), adopt sector nonlinear method and singular perturbation modeling technique, set up uncertain fuzzy Singular Perturbation Model continuous time of Van der Pol Circuits System:

Rule i: if x ₁(t) be φ _i1, so

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

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)=[x ₁(t) x ₂(t) x ₃(t)] ^tfor slow variable, x _f(t)=x ₄(t) be fast variable, u (t) is control inputs,

Y (t) ∈ R ³for system output, w (t) is for disturbing outward, φ _i1(i=1,2) are fuzzy set, Δ A _ci∈ R ^{4 * 4}for uncertain matrix,

$A_{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000430077050000022.png,HDA0000430077050000041.png"\; state="\{\{state\}\}">c</figure-callout>}1}=\left[\begin{array}{cccc}4& 10& 0& 0\\ -2& -8& 4& 4\\ 0& 4& -4& 0\\ 0& 4& 0& -4\end{array}\right.0,A_{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA0000430077050000022.png,HDA0000430077050000033.png"\; state="\{\{state\}\}">c</figure-callout>}2}=\left[\begin{array}{cccc}9& 10& 0& 0\\ -2& -8& 4& 4\\ 0& 4& -4& 0\\ 0& 4& 0& -4\end{array}\right],$

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

Getting the sampling period is T _s=0.1, application zero-order holder method discretization model (13), the UDTFSPM of acquisition Van der Pol Circuits System,

Rule i: if x ₁(k) be φ _i1, so

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

Y (k)=Hx (k) for i=1,2 (14) wherein, x (k)=[x _s(k) x _f(k)] T, x _s(k)=[x ₁(k) x ₂(k) x ₃(k)] ^t, x _f(k)=x ₄(k), Δ A _i∈ R ^{4 * 4}for uncertain matrix,

Apply following formula

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

:

$\begin{array}{cc}{A}_1=\left[\begin{array}{cccc}1.3810& 0.9683& 0.1697& 0.0558\\ -0.1937& 0.6058& 0.2516& 0.0461\\ -0.0339& 0.2516& 0.7213& 0.0155\\ -2.2781& 9.4044& 3.1567& 0.7442\end{array}\right],{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="HDA0000430077050000022.png,HDA0000430077050000041.png"\; state="\{\{state\}\}">B</figure-callout>}}_1=\left[\begin{array}{c}-1.5749\\ 0.1168\\ 0.0135\\ 1.1579\end{array}\right],& \\ A_2=\left[\begin{array}{cccc}2.3006& 1.2967& 0.2067& 0.0716\\ -0.2593& 0.5879& 0.2500& 0.0453\\ -0.0413& 0.2500& 0.7211& 0.0154\\ -2.9211& 9.2464& 3.1438& 0.7376\end{array}\right]{,\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="HDA0000430077050000022.png,HDA0000430077050000033.png"\; state="\{\{state\}\}">B</figure-callout>}}_2=\left[\begin{array}{c}-1.5749\\ 0.1168\\ 0.0135\\ 1.1579\end{array}\right],& \end{array}$

Given [x (k); U (k); W (k)], application standard fuzzy reasoning method, obtains overall 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(x ₁(k)) get (i=1,2)

, μ ₂(x ₁(k))=1-μ ₁(x ₁(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, setting up on the basis of overall UDTFSPM (15), to Van der Pol design of circuit system RFSOFC.

Controller rule i: if x ₁(k) be φ _i1, so

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), obtains 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 method, derive the adequate condition that RFSOFC exists, and derivation does not require the supremum of knowing systematic uncertainty parameter.The LMI group that solves RFSOFC gain below:

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]，

$\mathrm{\&Lambda;}=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="11"\; label="Z"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{11}& 0\\ 0& 0\end{array}\right]-\mathrm{\&gamma;Y}-{\mathrm{\&gamma;Y}}^T,$

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

γ (0< γ≤1), β is greater than zero constant, γ, the value of β can be selected by deviser (deviser is by selecting suitable γ, and β value obtains optimal controller gain), $Z=\left[\begin{array}{cc}{Z}_{11}& Z_{21}^T\\ Z_{21}& Z_{22}\end{array}\right]({\mathrm{<figure-callout\; id="11"\; label="Z"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Z</figure-callout>}}_{11}\&Element;R^{n\&times;n},Z_{22}\&Element;R^{m\&times;m})$ For symmetric positive definite matrix,

$Y=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="11"\; label="Y"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{11}& 0\\ 0& Y_{22}\end{array}\right]({\mathrm{<figure-callout\; id="11"\; label="Y"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{11}\&Element;R^{n\&times;n},Y_{22}\&Element;R^{m\&times;m}).$

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

r=2，n=3，m=1，ε=0.07

By examination, gather method, can obtain when γ=0.9, β=15 o'clock optimum RFSOFC, its gain is:

$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA0000430077050000022.png,HDA0000430077050000041.png"\; state="\{\{state\}\}">G</figure-callout>}}_1={\mathrm{<figure-callout\; id="11"\; label="W"\; filenames="HDA0000430077050000041.png"\; state="\{\{state\}\}">W</figure-callout>}}_{11}*Y_{11}^{-1}=\left[\begin{array}{ccc}1.6222& -3.0308& -1.1054\end{array}\right],\\ {\mathrm{<figure-callout\; id="2"\; label="G"\; filenames="HDA0000430077050000022.png,HDA0000430077050000033.png"\; state="\{\{state\}\}">G</figure-callout>}}_2=W_{12}*Y_{11}^{-1}=\left[\begin{array}{ccc}1.8719& -2.4498& -0.9309\end{array}\right]\end{array}---\left(22\right)$

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

Control program in step 7, operation controller, Van der Pol Circuits System is controlled, overall system control structural drawing as shown in Figure 2, concrete control procedure is: sensor adopts 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 adopts event driven manner, and when sampled data bag arrives, controller carries out control signal calculating at once, and control signal is passed to actuator; Actuator is comprised of impact damper and zero-order holder.When controlling data, arrive after actuator, in the timestamp that actuator is carried and buffer zone, the timestamp of control signal compares, and judges newly arrived control packet whether " newly "; "Yes" is kept at newly arrived control signal and timestamp thereof in buffer zone, and "No" abandons this and controls packet.Zero-order holder adopts time type of drive, and zero-order holder, according to the fixing sampling period, reads control signal from buffer zone, and generates control inputs adjustment helicopter attitude, thereby realizes the stable control of Helicopter System.It should be noted that sensor and actuator adopt the identical sampling period, and the two should keep clock synchronous.

Simulating, verifying:

Adopt matlab software to carry out emulation to Van der Pol circuit control system, simulated conditions is:

1., initial value is got x (0)=[0.9 0.5-0.4 1] ^t;

2., inductance measuring current i _l(t) noise causing is that amplitude is 1 * 10 ^-6the white noise of ampere, measures capacitance voltage v _c1(t) noise causing is that amplitude is 0.5 * 10 ^-6the white noise of volt, measures capacitance voltage v _c2(t) noise causing is that amplitude 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 simulation result is as shown in Fig. 4-10.

Fig. 4 has shown closed-loop system condition responsive curve, and Fig. 5-Fig. 7 has shown closed-loop system output response curve, and Fig. 8-Figure 10 has shown that system enters the output response curve after stable state.Fig. 4-Fig. 7 shows that designed control can not only make system enter fast stable state, and can suppress the noise that external disturbance and system state measurement and control inputs are introduced.From Fig. 8-Figure 10, easily find out inductive current i _l(t) control accuracy reaches 6 * 10 ^-5ampere, capacitance voltage v _c1(t) control accuracy reaches 11 * 10 ^-6volt, capacitance voltage v _c1(t) control accuracy reaches 9 * 10 ^-6volt.

Comprehensively above-mentioned, simulation result for Van der Pol circuit control system shows, the perturbation that the inside small parameter (small inductance or electric capacity) that adopts the present invention not only can effectively process CCSWSIOCs causes, and effectively suppress outer disturbing with system state and measure the noise of introducing with control inputs, greatly reduce steady-state error, the high precision that reaches CCSWSIOCs is controlled index.

Claims (1)

1. a complicated circuit system modeling and control method that includes small inductance or electric capacity, is characterized in that:

Step 1, according to the kinetics equation of CCSWSIOCs, set up uncertain continuous time of the fuzzy Singular Perturbation Model of controlled CCSWSIOCs;

The state variable variation of CCSWSIOCs is slow or that can directly measure is seen slow variable as, small parameter is correlated with or is changed state variable faster and sees fast variable as, adopt sector nonlinear method, set up uncertain continuous time of the fuzzy Singular Perturbation Model of CCSWSIOCs:

Rule i: if ξ ₁(t) be φ _i1..., ξ _g(t) be φ _ig, so

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

y(t)=Hx(t) (1)

Wherein,

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

X _s(t) ∈ R ⁿfor slow variable, x _f(t) ∈ R ^mfor fast variable, u (t) ∈ R ^qfor control inputs, y (t) ∈ R ^lfor system output, w (t) ∈ R ^qfor disturbing outward, φ _i1..., φ _ig(i=1,2 ..., r) be fuzzy set, ξ ₁(t) .., ξ _g(t) be measurable system variable, A _ci, B _ci, 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 in control system and actuator all adopt time type of drive, and the two adopts 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 ξ ₁(k) be φ _i1..., ξ _g(k) be φ _ig, so

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

y(k)=Hx(k) (2)

Wherein, Δ A _ifor the uncertain matrix of suitable dimension,

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

Given [x (k); U (k); W (k)], application standard fuzzy reasoning method, obtains overall UDTFSPM:

$\stackrel{\&CenterDot;}{x}\left(k\right)=E_{\mathrm{\&epsiv;}}(A\left(\mathrm{\&mu;}\right)+\mathrm{\&Delta;A}\left(\mathrm{\&mu;}\right))x\left(k\right)+E_{\mathrm{\&epsiv;}}B\left(\mathrm{\&mu;}\right)u\left(k\right)+E_{\mathrm{\&epsiv;}}\mathrm{Dw}\left(k\right)$

Y (k)=Hx (k) (3) wherein, membership function ${\mathrm{\&mu;}}_i\left(\mathrm{\&xi;}\left(k\right)\right)=\frac{w_i\left(\mathrm{\&xi;}\left(k\right)\right)}{\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{w}_i\left(\mathrm{\&xi;}\left(k\right)\right)},w_i\left(\mathrm{\&xi;}\left(k\right)\right)=\underset{j=1}{\overset{g}{\mathrm{\&Pi;}}}{\mathrm{\&phi;}}_{\mathrm{ij}}\left({\mathrm{\&xi;}}_j\left(k\right)\right),,$ φ _ij(ξ _j(k)) be ξ _j(k) at φ _ijin degree of membership, establish w _i(ξ (k))>=0, for i=1,2 ..., r, r is regular number, μ _i(ξ (k))>=0,

for the ease of recording us, make μ _i=μ _i(ξ (k)),

$A\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i{A}_i,B\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i{B}_i,\mathrm{\&Delta;A}\left(\mathrm{\&mu;}\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_i\mathrm{\&Delta;}{A}_i$

Step 3, based on UDTFSPM (3), to controlled device design RFSOFC

Design 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), obtains closed-loop model:

x(k+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 method, derive the adequate condition that RFSOFC exists,

Derivation does not require the supremum of knowing systematic uncertainty parameter.The linear matrix that solves RFSOFC gain below

Inequality group:

Θii<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 _ij 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 is greater than zero constant, γ, the value of β can be selected by deviser that (deviser is by selecting suitable γ, and β value obtains

Excellent 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:

for i=1，2，…，r. (10)

Step 6, gained controller Matlab code is transferred to C language codes, implant controller.Controller adopts event driven manner, when sampled data arrives controller, controller calculates at once, and control signal is passed to actuator, actuator reads control signal according to the fixing sampling period, generate control inputs, act on controlled CCSWSIOCs, thereby realize the high precision control of CCSWSIOCs.

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