The Model and Quadratic Stability Problem of Buck Converter in DCM

Quadratic stability is an important performance for control systems. At first, the model of Buck Converter in DCM is built based on the theories of hybrid systems and switched linear systems primarily. Then quadratic stability of SLS and hybrid feedback switching rule are introduced. The problem of Buck Converter’s quadratic stability is researched afterwards. In the end, the simulation analysis and verification are provided. Both experimental verification and theoretical analysis results indicate that the output of Buck Converter in DCM has an excellent performance via quadratic stability control and switching rules.


Introduction
Hybrid systems (HS) is defined as a unitized dynamic system interacted by discrete and continuous parts.DC-DC converters are typical HS because the operation of each mode can be regard as the continuous dynamic subsystems and the turn-on or turn-off of power switch as the discrete dynamic subsystems.
Switched linear systems (SLS), an important type of HS, have attracted considerable attention in modeling, analysis and design.Quadratic stability is one of the important problems of SLS.This problem is more complex since it depends on the switching rules as the stability of all the subsystems.
Looking from the existing literatures [1]- [5], Lyapunov theories are the dominant approaches used in study of stability.For example, the stability of DC-DC Converters in CCM (Continuous Current Mode) was analyzed in [6].The aim of this paper is to study the quadratic stability of DC-DC converters via using Lyapunov function based on the model in DCM (Discontinuous Current Mode).

Buck converter in DCM
The topology structure of Buck converter is shown in Fig. 1.Assume all of the components are perfect.Three work modes of it in DCM are shown in Fig. 2.

SLS Model of Buck Converter in DCM
Supposing all of the components are ideal and the state vector is x( t) = [ i L u C ] T ,the output vector is y( t) = u C , consequently, the state equations of Buck Converter in SLS model are [7] °°® The parameters matrixes of Buck Converter in DCM are expressed as follows according to Fig. 2.
3 Quadratic stability of SLS

Propaedeutics
Given a stable equilibrium point x .Since any other equilibrium point can be shifted to the origin via a change of variable x ~ =x -x , we can assume the switched equilibrium is the origin x =0 without loss of generality.
Definition: if and only if there exist a matrix P = P T > 0 and a constant ε > 0 such that for the quadratic function system trajectories, the switched equilibrium x = 0 can be said quadratically stable [8].(a) when subsystem i is active, , a convex combination of the subsystems is defined as

Quadratic stability
Theorem 1: As for system (l), the point x = 0 is a quadratic stable switched equilibrium if there exist Since the convex combination A eq in Eq. 4 is stable, there exist two positive definite symmetric matrices P and Q such that According to Eq. 4, Eq. 6 can be rewritten as From Eq. 5, we can also get the null term 0 Now, a new equation is obtained as follows where 0<ε d λ min and λ min is the smallest positive real eigenvalue of Q.Then Eq. 9 is equivalent to Eq. 10 as follows.

Hybrid feedback switching rule
In this rule, a lower bound on the decay rate )) ( ( min with ε > 0 is presented primarily, then the active subsystem is switched off only when it no more satisfies the required constraint.
The procedures of hybrid feedback switching rule is listed as follows.
(initialization) at time t = 0 activate the subsystem

Switching control of buck converter
Buck Converter in DCM includes three subsystem, consequently there exist three coefficients α 1 , α 2 and α 3 in the convex combination of Eq.3 .According to Eq.The load resistor R has two salutations during the process of simulation, one is from 10Ω to 20Ω when t=0.2s, the other is from 20Ω to 10Ω when t=0.4s.According to Fig. 4,we can see the output voltage reach its steady state with some overshoots and it spends more time to stabilize compared with the converter in Fig. 3 when the load resistor changed.So the hybrid feedback switching rule appropriates good performance of transient and steady state dynamic responses.

Conclusion
In this paper, the model of Buck Converter in DCM, which is based on the concept and theory of SLS, is built first and foremost.Then the quadratic stability and switching rule of it are studied.Afterwards, simulation results are presented.This approach can also be extended to other converters and port controlled Hamilton linear switched system.The research results are beneficial to the development of nonlinear control strategy and the practical applications of power electronic systems.

2 ¦ 3 ¦
:w=1, v=0 (Switch tube is off and diode is on) :w=0, v=0 (Switch tube and diode are all off) 3, we have α 3 =1-(α 1 +α 2 ).Then the convex combination can be written as semi-definite positive.Q can be chosen as a semi-definite matrix because ) ( x V is not identically vanishing along all system trajectories for DC-DC converters.Let λ min =2/R is the smallest positive real eigenvalue of Q and 0<ε d λ min .So the regions of subsystems can be divided by hybrid feedback rule.

Figure 3 .
Figure 3. Simulation results of buck converter on hybrid feedback switching control rule.

Figure 4 .
Figure 4. Simulation results of buck converter in openloop control.

stability of buck converter
on x U t (ρ off < ρ on ) .5 Quadratic

Coordinate Transformation of Buck Converter
The state equations after coordinate transformation is given by Eq. 11 and Table1.

Table 1 .
Parameters matrixes of buck converter after coordinate transformation