KR20180093331A - Method for stabilization of nonlinear system including singular perturbation - Google Patents

Abstract

The present invention relates to a stabilizing method of a nonlinear system having singular perturbation, which can more stabilize a nonlinear system by using a sample value controller newly built to converge a state variable of the nonlinear system having singular perturbation towards zero.

Description

[0001] The present invention relates to a method for stabilizing a nonlinear system including a singular perturbation,

The present invention relates to a method of stabilizing a nonlinear system having a singular perturbation, and more particularly, to a method of stabilizing a nonlinear system for asymptotically stabilizing a nonlinear system using a sample value controller for controlling the dynamic characteristics of a nonlinear system exhibiting singular perturbations ≪ / RTI >

The singular perturbation system refers to a dynamic system in which the time scale is relatively fast (ie, fast mode) and the slow state variable (ie, slow mode).

For example, the moment of inertia of a DC motor used to drive an inverted pendulum system is relatively small compared to the moment of inertia of an inverted pendulum and can be ignored. In this case, the fast mode and the non- The dynamic characteristics of the slow mode are simultaneously present. In addition, the tunnel diode circuit including the parasitic inductance and the high gain feedback system also have the dynamic characteristics of the fast mode and the slow mode simultaneously.

In the nonlinear system with the singular perturbation, the dynamic characteristics of the fast mode existing in the nonlinear system, that is, the temporal fast dynamic characteristics, are called as the singular perturbations. In order to stabilize the system as the dynamic characteristics of the fast mode and the slow mode are mixed, Is used.

The sample value controller stabilizes the nonlinear system by controlling the dynamic characteristics of the nonlinear system using the control signals calculated using the sampled state variables in the nonlinear system.

However, since the sample value controller for controlling the nonlinear system in which the dynamic characteristics of the fast mode and the slow mode are mixed is easy to design in the discrete time domain, it is difficult to design the differential equation by integrating the dynamics of the nonlinear system represented by the differential equation It is common to design a sample value controller.

However, since the discretization modeling by integration of nonlinear differential equations representing the dynamic characteristics of a nonlinear system is generally impossible, a discrete-time model including the approximation error is constructed using the approximation equation of the integral equation. The sample value controller There is a problem that the stabilization performance of the nonlinear system is deteriorated.

Japanese Patent Application Laid-Open No. 7-287602

SUMMARY OF THE INVENTION The present invention has been made in view of the above problems, and it is an object of the present invention to provide a nonlinear system capable of stabilizing the nonlinear system by using a new sample value controller constructed to converge a state variable of a non- And a method of stabilizing a nonlinear system having a specific perturbation.

Accordingly, in the present invention, a first step of constructing a dynamic characteristic model of a nonlinear system having a specific perturbation; A second step of constructing a sample value controller for controlling the dynamic characteristics of the nonlinear system depending on the dynamic characteristics model; A third step of periodically calculating a control signal for dynamic characteristic control by the sample value controller based on a state variable of the dynamic characteristic model; And a fourth step of using the control signal as a control input of the nonlinear system, wherein the sample value controller is able to converge the state variable to '0'. .

Specifically, the sample value controller

(Z _kT ) is a fuzzy rule utterance of the dynamic characteristic model, and x _kT and z _kT are values sampled at a time point kT of the state variable x and the front variable z of the dynamic characteristic model, respectively, U _kT is the control signal u calculated by the sample value controller at time kT, and Ki is the control gain of the sample value controller.

And Mi is a solution of a matrix Mj satisfying the linear matrix inequality of Equation 6 constructed in dependence on the dynamic characteristic model of the nonlinear system, Is the inverse of the solution of the matrix S satisfying the linear matrix inequality.

In addition, the dynamic characteristics model of the nonlinear system

이고,

ego,

이고, 상기 I는 항등행렬이고, 상기 ε은 특이섭동 파라미터이고, 상기 x는 상태변수이고, 상기 Ai와 Bi는 각각 시스템행렬과 입력행렬이고, 상기 u는 제어입력이고, 상기 θi(z)는 퍼지규칙발화도이다.remind

, Where I is an identity matrix, the epsilon is a singular perturbation parameter, x is a state variable, Ai and Bi are respectively a system matrix and an input matrix, u is a control input, Fuzzy rule.

According to the stabilization method of the nonlinear system according to the present invention, it is possible to effectively stabilize the nonlinear system as a result by controlling the state variable of the nonlinear system having the singular perturbation to converge to '0'.

1 shows an example of a tunnel diode circuit
Fig. 2 is a circuit diagram modeling the actual driving characteristics of the tunnel diode circuit shown in Fig. 1
3 is a flow chart illustrating a method of stabilizing a nonlinear system having a singular perturbation according to the present invention
4 is a block diagram showing a closed loop sample value control system using a sample value controller according to the present invention.
5 is a graph showing simulation results for illustrating that the state variable (x1, x2) of a nonlinear system having a specific perturbation converges to zero according to the present invention.

Hereinafter, the present invention will be described with reference to the accompanying drawings.

The dynamic characteristics of a nonlinear system with singular perturbations can be expressed by the ordinary differential equation of the form:

Equation 1: x1 '= f1 (x, u)

     ? x2 '= f2 (x, u)

Wherein x1 is the dynamic characteristic of the slow mode of the nonlinear system, x2 'is the fast mode dynamic characteristic of the nonlinear system, x is the state variable of the nonlinear system, u is the dynamic characteristic of the non- (The control signal of the sample value controller).

The nonlinear system having dynamic characteristics expressed by the above-mentioned Equation 1 is, for example, a closed loop system including a mechanical device driven by a motor and a sensor for sensing the output of the mechanical device.

The dynamic characteristics of the sensor for sensing the output of the mechanical device are faster than the dynamic characteristics of the motor, so that the closed loop system exhibits a combination of fast dynamic characteristics with time and slow dynamic characteristics.

FIG. 1 is a schematic diagram of a tunnel diode circuit, and FIG. 2 is a circuit diagram modeling the tunnel diode circuit in consideration of actual driving characteristics of the tunnel diode circuit.

As shown in FIG. 1, a tunnel diode circuit composed of a power source 1, a resistor 2 and a capacitor 3 connected in series and a tunnel diode 4 connected in parallel to the capacitor 3, And a very small parasitic inductance (5) of size ε _L is present.

The dynamic characteristics of the tunnel diode circuit shown in FIG. 2 are modeled as shown in Equation 1 above.

A nonlinear system having a dynamic characteristic represented by an ordinary differential equation of the form as shown in Equation (1) has a dynamic characteristic (x1 ') of a slow mode and a dynamic characteristic (x2') of a fast mode in a time- It is called perturbation.

In the case of the nonlinear system with the above-mentioned singular perturbation, a sample value controller is used to stabilize the system. Since the conventional sample value controller is easy to design in the discrete time domain, the differential equation [x '= f (x)], it is necessary to discretize the dynamic characteristics of the nonlinear system and express it in a form including an integral equation such as x (t + 1) = x (t) + f (x) f (x)] can not be expressed in general form.

Therefore, the conventional sample value controller is designed in accordance with the dynamic characteristic model of the nonlinear system which is discretized using the approximation equation of? F (x).

However, when the sample value controller designed based on the dynamic model of the discretized nonlinear system using the approximation equation of ∫ f (x) is used, it is impossible to converge the state variable of the nonlinear system to '0' As a result, there is a limit to stabilizing the nonlinear system.

In the present invention, the dynamic characteristics of a nonlinear system with a specific perturbation are modeled in the form of a Takagi - Sugeno (T - S) fuzzy model for a continuous time without discretization, By constructing a new sample value controller for dynamics control, nonlinear systems can be more effectively stabilized.

To this end, according to the present invention, as shown in FIG. 3, there are a first step of constructing a global dynamic characteristic model of a nonlinear system having a singular perturbation, a first step of constructing a global dynamic characteristic model of a non- A second step of constructing a sample value controller, a second step of periodically calculating a control signal for controlling the dynamic characteristics of the nonlinear system based on sampled state variables of the dynamic characteristics model (i.e., state variables of the nonlinear system) And a fourth step of using the control signal as a control input of the nonlinear system in step 3, and a fourth step of stabilizing the nonlinear system.

First, as in the first step, the dynamic characteristics model is constructed by modeling the global dynamic characteristics of the nonlinear system having the singular perturbation.

If the dynamics of the nonlinear system expressed as Equation 1 is fuzzy using the sector nonlinearity technique,

Equation 2:

Then, Equation 2 can be expressed as Takagi-Sugeno (T-S) fuzzy model by singleton fuzzification, multiplication inference, and center value mean non-fuzzy, . That is, the global dynamic characteristic model of the nonlinear system is constructed as shown in Equation 3 below.

Equation 3:

Here,

을 만족한다.ego,

.

(X1, x2) is a state variable, and Ai and Bi are a state variable, and I is an identity matrix, and is a singular perturbation parameter representing a degree of separating a dynamic characteristic of a fast mode from a dynamic characteristic of a slow mode, Where u is the control input (or control signal of the sample value controller) for controlling the dynamic characteristics, and θi (z) and z are respectively the nonlinear system and the system matrix determined by the dynamic characteristics of the nonlinear system It is the fuzzy rule utterance used in the Takagi-Sueno fuzzy model form and the first half variable.

는 x를 미분한 것으로서 x'과 동일한 표현이다.For reference,

Is the derivative of x and is the same expression as x '.

In order to control the nonlinear system having the same dynamic characteristic as Equation (3), a sample value controller of the following Equation (4) is constructed by modeling the time interval [kT, kT + T] depending on the dynamic characteristic model of Equation (3).

Equation 4:

Here, the θi (z _kT) is a value sampled at kT time (t = kT) of a fuzzy rule firing also and specifically, fuzzy rule firing Fig θi (z) of the dynamic characteristic model, the x _kT, z _kT is (T = kT) of the state variable x and the front-end variable z of the dynamic behavior model, and u _kT is a value sampled at a time t (k) ) u, and Ki is a control gain of the sample value controller.

A sample value control system of a nonlinear system having a slow mode and a fast mode dynamic characteristic expressed by Equations (1) and (3) using a sample value controller constructed as shown in Equation (4) can be constructed as shown in FIG.

The closed-loop type sample value control system shown in FIG. 4 is for asymptotically stabilizing the nonlinear system 10 using the sample value controller 12, and includes an output terminal of the nonlinear system 10 and a sample value controller And a Zero Order Holder (ZOH) 16 connected between the output of the sample value controller 12 and the input of the nonlinear system 10.

The switching unit 14 is configured to be turned on at predetermined time intervals to allow the value of the state variable x of the nonlinear system 10 to be input to the sample value controller 12, The control signal u calculated by the sample value controller 12 is supplied as the control input of the nonlinear system 10 based on the value of the state variable x sampled at time intervals so as to be input to the nonlinear system 10 .

In other words, the sample value controller 12 receives the value of the state variable x of the nonlinear system 10 that is periodically sampled (for example, sampled at the time kT) and controls the control for the dynamic characteristics of the nonlinear system 10 And the control signal u is used as a control input for the dynamic characteristics and state control of the nonlinear system 10 via the ZOH 16. [ At this time, the ZOH 16 keeps the periodically calculated control signal u until the next control signal is generated so as to be continuously inputted to the nonlinear system 10 as a control input for stabilization of the nonlinear system 10 .

In addition, the sample value controller 12 receives the state variable x of the nonlinear system 10 to be periodically sampled, and also receives the first-half variable z.

Here, the model value of the sample value control system of the nonlinear system is expressed as Equation 5 below.

Equation 5:

Further, the expression (5) controls the gain of the control sample values constituting the sample value control system expressed as Ki = Kj = Mi * N ( i, j = 1,2, ... r) , and wherein N = S ^{- 1} .

In this case, the control gain K i constituting Equation 4 is expressed as " Kj " in order to distinguish subscripts from other variables in Equation 5, and means that i and j are limited to the same value in Equation 5 It is not.

Where Mi is a solution of a matrix Mj determined by the linear matrix inequality of Equation 6 below and N is a predetermined nonspecific matrix as an inverse matrix for a solution of a matrix S determined by the linear matrix inequality .

That is, the control gain Ki can be determined by obtaining matrices S and Mj satisfying the linear matrix inequality as shown in Equation 6 below. The linear matrix inequality in Eq. (6) is constructed using matrices Ai, Bi, and E, which form the dynamic model of the nonlinear system based on the Lyapunov stability theory. In other words, the linear matrix inequality of Equation 6 below is built on the dynamic model of the nonlinear system based on the Lyapunov stability theory.

Equation 6:

Wherein S, Mj, X, Y the matrix corresponding to the different variables to be determined dependent on the dynamic characteristics of the non-linear system, wherein T is a period for sampling the state variable (x) of the nonlinear system, the S ^T, Ai ^T , Bi ^T , and Mj ^T are transpose matrices of S, Ai, Bi, and Mj, respectively.

When with respect to c1, c2 is set as an appropriate constant value, numerical optimization by Method S, Mi = Mj, X = satisfying the linear matrix inequality of formula ^{6 X T> 0, Y =} Y T> 0 is present the The sample value controller constructed as shown in Equation (4) is able to stabilize the system asymptotically by converging the state variable (x) of the nonlinear system having the dynamic characteristics as Equations (1) and (3) to '0'.

Accordingly, the sample value controller can asymptotically stabilize the nonlinear system having the dynamic characteristics as expressed by Equations (1) and (3) by applying the control gain (Ki) calculated using S and Mi satisfying Equation (6) .

In other words, by determining the control gain (Ki) of the sample value controller to satisfy Equation (6), the sample value controller has a control characteristic capable of converging the state variable x of the nonlinear system to '0' By determining the control characteristics of the sample value controller, the sample value control system using the sample value controller controls the state variable x of the nonlinear system having the singular perturbation to converge to '0', so that the nonlinear system is asymptotic It becomes possible to stabilize it.

Hereinafter, a simulation and a result of demonstrating the utility of the present invention will be described.

First, we assume a nonlinear system with the same dynamic characteristics as Equation 3, which consists of the following parameters.

The sampling period T of the sample value controller constructed as shown in Equation 4 is set to 0.02, and c1 and c2 of the linear matrix inequality as shown in Equation 6 are set to 0.05, and solving the above linear matrix inequality results in the following solution.

The control gains of the sample value controller are determined as K1 and K2 by using S, M1 and M2.

Next, the initial state variable x (0) = (1,1) of the nonlinear system is selected and simulation results are shown in FIG.

As shown in FIG. 5, it can be seen that the state variables (x1, x2) of the nonlinear system gradually converge to '0' over time.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the scope of the present invention is not limited to the disclosed exemplary embodiments. Modifications are also included in the scope of the present invention.

10: Nonlinear system
12: Sample value controller
14:
16: Zero Order Holder (ZOH)

Claims (4)

A first step of constructing a dynamic characteristic model of a nonlinear system having a specific perturbation;
A second step of constructing a sample value controller for controlling the dynamic characteristics of the nonlinear system depending on the dynamic characteristics model;
A third step of periodically calculating a control signal for dynamic characteristic control by the sample value controller based on a state variable of the dynamic characteristic model;
And a fourth step of using the control signal as a control input of the nonlinear system,
Wherein the sample value controller is adapted to converge the state variable to '0'.
The method according to claim 1,
The sample value controller

Wherein the nonlinear system is configured in the form of a nonlinear system.
Wherein θi (z _kT) is a fuzzy rule of dynamic characteristics model ignition also, the x _kT, z _kT are each a value sampled at kT time of the state variables of the dynamic characteristic model x, the first half of the variable z, said u _kT is in kT point The control signal u calculated by the sample value controller, and Ki is the control gain of the sample value controller.
The method of claim 2,
Wherein the control gain of the sample value controller is Ki = Mi * N.
Where Mi is a solution of a matrix Mj that satisfies the linear matrix inequality constructed in dependence on the dynamic model of the nonlinear system and N is an inverse matrix for the solution of the matrix S satisfying the linear matrix inequality.
The method according to claim 1,
The dynamic model of the nonlinear system

Wherein the nonlinear system has a specific perturbation.
remind

, Where I is an identity matrix, the epsilon is a singular perturbation parameter, x is a state variable, Ai and Bi are respectively a system matrix and an input matrix, u is a control input, Fuzzy rule.

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