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Research on stability of time-delay systems based on Lambert function

2022-07-24 15:23:00 【studyer_ domi】

1、 Content abstract

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426- Can communicate 、 consulting 、 Answering question

2、 Content description

With the development of network technology in today's world 、 The rapid development of communication technology and computer technology and their wide application in various fields . Especially in the research of power system , With the continuous development and innovation of Technology , Gradually applied to power production and power supply system , It provides a stable way of power transmission , Move towards intellectualization . These technologies also make the monitoring of power system 、 The analysis and control mode is towards wide area 、 Networking and intelligence , There is an important characteristic in the power system, that is, the time-delay characteristic , What we want to study is in the power system Lambert Stability of time-delay systems under function . For now , When domestic and foreign scholars analyze the power system , The influence of time-delay characteristics is ignored . But the composition of the power system is very complex 、 Gigantic , When high power needs long-distance transmission , The possibility and harm of power oscillation on the transmission line will become greater , It is difficult to ensure the stability of power system only by locally designed controller . And the power system is often a nonlinear 、 Time delay dynamical systems with randomness , The existence of time delay is the source of system instability , Even a small time delay will cause controller failure , Even cause catastrophic accidents , therefore , in many instances , The power system must be modeled as a time-delay system for research , Make good use of the stability of time-delay systems , The implementation of wide area optimal control from the perspective of the whole system can ensure the safe and stable operation of the system .

Time delay system is the inherent time delay of system components , Or the time delay system introduced for control purposes . Delay systems can be expressed by delay differential equations , It belongs to a class of universal differential equations , And a lot of research has been done in the past decades . From power and automation engineering to biology 、 chemical 、 Physical systems and ecology , Will quote time delay . Such a time delay can limit and reduce the achievable performance of the control system , It may even cause system instability . In various time-delay systems , The pure delay time of the object has great harm to the control performance of the control system , It not only reduces the stability of the system , The characteristics of the transition process deteriorate , And the longer the pure lag takes up the whole dynamic process , The more difficult it is to control , Therefore, the control problem of time-delay systems has become a major problem in the field of automatic control . For time-delay systems , The most important thing is to control , The control of time-delay characteristics is very common . In the development of time-delay system control , There is Smith Predictive control 、 Dahlin algorithm 、 Adaptive control 、 Predictive control 、 Application of various control methods such as variable structure control and intelligent controller in time-delay systems , The advantages and disadvantages of these methods are summarized , It is concluded that compound control is an effective and practical method to control time-delay systems , And it has great application prospects . thus it can be seen , The importance of time-delay systems is self-evident , What we are going to study is how to solve the stability problem of time-delay systems , How to achieve good control effect .

Lambert W Function is also called Omega function or multiplier logarithm , It is a multivalued function , There are infinite branches , And every branch is a single valued function , In the range of real numbers, there are W(x0) and W（x1） Two , The former is called Lambert W The main branch of the function .Lambert W The function is also f(W)=Wexp(W) The inverse function of ,exp(W) It's an exponential function ,W Is any plural . all the time , Scholars at home and abroad have also made some research on this function . In the first , Just use it to solve some basic mathematical problems , For example, the equation with exponent , Delay differential equations , Calculation of trees in Combinatorial Mathematics . and , With the in-depth study of this function , Found more uses for it , From mathematics to physics 、 chemical 、 In biology and other fields , Such as in Quantum Chemistry , Some problems in statistical mechanics use W Function to solve . Especially some complicated problems , application Lamberr W Function representation is much more convenient , Some equation solutions can be expressed by it , Coupled with the current development of related mathematical software, such as Matlab,Maple etc. , Provide corresponding function calculation Lambert W function , It is possible to calculate the solutions of these equations with high accuracy , It also provides convenience for studying the stability of time-delay systems .

3、 Simulation analysis

clc
close all
clear
%%  First order 
tao = 1;
Kp = -2;
k = -4:4;
alpha = 1;
for i = 1:length(k)
    x = -Kp*tao*exp(tao*alpha);
    y = lambertw(k(i),x);
    s(i) = y/tao-alpha;
end
figure
plot(real(s),imag(s),'marker','.','markersize',18,'color','black')
xlabel  Real component 
ylabel  Imaginary part 
set(gcf,'color','w');

%%  Second order 
% -------------- Kp = -2 ------------------------------
n = 2;
Kp = -2;
alpha = 1;
tao = 1;
k = -4:4;

l = 0;
for i = 1:length(k)
    if Kp<0
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*2*l*pi/n);
        y = lambertw(k(i),x);
        s0(i) = n/tao*y-alpha;
    else
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*(2*l+1)*pi/n);
        y = lambertw(k(i),x);
        s0(i) = n/tao*y-alpha;
    end
end
l = 1;
for i = 1:length(k)
    if Kp<0
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*2*l*pi/n);
        y = lambertw(k(i),x);
        s1(i) = n/tao*y-alpha;
    else
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*(2*l+1)*pi/n);
        y = lambertw(k(i),x);
        s1(i) = n/tao*y-alpha;
    end
end

figure
plot(real(s0),imag(s0),'marker','.','markersize',18,'color','red')
hold on
plot(real(s1),imag(s1),'marker','.','markersize',18,'color','black')
hold off 
xlabel  Real component 
ylabel  Imaginary part 
set(gcf,'color','w');

% -------------- Kp = 2 ------------------------------
n = 2;
Kp = 2;
alpha = 1;
tao = 1;
k = -4:4;

l = 0;
for i = 1:length(k)
    if Kp<0
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*2*l*pi/n);
        y = lambertw(k(i),x);
        s0(i) = n/tao*y-alpha;
    else
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*(2*l+1)*pi/n);
        y = lambertw(k(i),x);
        s0(i) = n/tao*y-alpha;
    end
end
l = 1;
for i = 1:length(k)
    if Kp<0
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*2*l*pi/n);
        y = lambertw(k(i),x);
        s1(i) = n/tao*y-alpha;
    else
        x = tao/n*exp(tao/n*alpha)*abs(Kp)^(1/n)*exp(1i*(2*l+1)*pi/n);
        y = lambertw(k(i),x);
        s1(i) = n/tao*y-alpha;
    end
end

figure
plot(real(s0),imag(s0),'marker','.','markersize',18,'color','red')
hold on
plot(real(s1),imag(s1),'marker','.','markersize',18,'color','black')
hold off 
xlabel  Real component 
ylabel  Imaginary part 
set(gcf,'color','w');