CC BY
12
Robert. C Dixon ®
Department of Industrial Electronics Faculty of Electronics Engineering, Tomsk State University of Control Systems and Radioelectronics robbyculture@gmail.com, dikson@tpu.ru
STABILITY ANALYSIS THROUGH MONODROMY MATRIX OF A SOFT-SWITCHING TWO - PHASE VOLTAGE CONVERTER WITH FILIPPOV METHOD
Abstract
This article proposes the use of monodromy matrix in stability and analysis of a two- phase boost converter (TPBC) with soft - switching technology. Monodromy matrix has been a useful tool for analysing eigenvalues and eigenvectors of various voltage converters. The calculated eigenvalues are the Floquet multipliers of the system and with Filippov inclusion for analysing the dynamics of discontinuous regime, where jump value occurs. Vector matrices are inherent state transition matrices through smooth intervals and across the switching manifolds, known to be saltation matrices of the system over the state transition matrix for one complete cycle (periodic orbit). The findings of instabilities in system can occur in smooth transition as well as nonsmooth quasi-periodic (or The Hopf curve) bifurcations which mostly occur in boost voltage converters. The Perturbation analysis of unstable limit cycle has been analysed of the TPBC through PWM. Therefore, the occurrence undergoes a discontinuous jump or change in the
fundamental solution matrix of the system.
Keywords: filippov method, TPBC, monodromy matrix, Floquet multipliers, and saltation matrix
Introduction
In recent years, the first qualitative studies regarding nonlinear power electronics models were presented in the 1980s by Brockett and Wood [1]. This were later continued by Hamill and Jeffries [2], who effectively started the study of what was referred to until then as 'unknown' or 'unwanted' instabilities. But Hamill and Jeffries [2] modelled a switched mode power converter with an iterative map and mentioned instability is shown to exist for a particular set of parameters. The loci of these parameters were determined by the complex number Argand diagram mapping. Bifurcation and chaotic phenomena may appear in much simpler circuits, like a series RLC (resistor-inductor-capacitor) circuit with a nonlinear diode. A boost converter under current-controlled has also been shown to exhibit chaotic patterns [3]. These bifurcation phenomena, referred to as border collision, are not common to other smooth systems and can only be found in nonlinear, non-smooth systems and in the power electronics case controlled by a pulse width modulator (PWM - saw-tooth waveform) or pulse frequency modulator (PFM). In fig. 1(a), the patented (see Appendix) two-phase boost converter with soft switching technology is achieved through a small resonant circuitry consists of small capacitance C1, C2 and inductor L3. The stable periodic orbit - 1 of the system, shows the effect of the filter capacitors C1 and C2 with L3 on transistor drain currents of fig. 1(c). In fig. 5(b), the PWM-TPBC duty cycle (D) characteristic is show with varying inductor resistance RL1and RL2. Hence, soft switching occurring from the alternating current in inductor L3, which charges and discharges C1 and C2 with polarity changes of the capacitors throughout its cycles of the two-phase boost voltage converter. Therefore, the system has dynamic properties, exhibited over a range of parameter sweep of the power converter in which direct Newton-Raphson method is used to find the roots of (11).
One way to study the bifurcation, quasi-periodic subharmonics phenomena of periodic systems; for example the boost converter; is to retrieve the Poincare map and calculate its
® Robert. C Dixon, 2016 r.
eigenvalues that are in line with the Floquet multipliers values of the systems, which are the eigenvalues total solution matrix. The periodic orbit has to be noted before the analysing issues of its stability. In power electronics practice, one can use the technique of Middlebrook's averaging [4], [5] for obtaining some information about the system dynamic behaviour and stability of the voltage converters for slow-scale low-frequency system. Hence, using the state equation averaging method has shown that, the averaging method can only capture the some instability that occurs in slow time scales. Therefore, this is effectively acting as a low-pass filter which in turn ignores all phenomena occurring at clock frequency [6], [7]. However, the fastscale high-frequency system instabilities that may occur or develop in the current and voltage waveforms at clock frequency will result in sub-harmonic and chaotic behaviour [8]-[9]. These issues are a matter of concern with the importance for analysing and prediction of such instabilities. The sampled-data method of modeling was introduced and developed in the early 1980s as well and was published by Verghese et al. in [10] and [11].
In the early 1990s, the method of Poincare section was used in the study of nonlinear dynamics, Deane and Hamill [12] on the equivalent concept of the iterated map as a model of power converters (for details, see [6] and [14]). The method use for obtaining sampled-data of the TPBC is made to sample in synchronism with the clock (called stroboscopic sampling) to obtain the discrete-time map (fast-scale high-frequency). The iteration of the map represents the evolution of the state in discrete time, and the fixed point of the map represents the periodic orbit in continuous time.
Once the nonlinear map is obtained linearising it at the fixed point, and the eigenvalues of the Jacobian matrix will determine the stability of the fixed point to small perturbations. This method has been successful for analysing the stability of periodic orbits in systems where linearisation of the map in closed form needs to be obtained. Therefore, in [15] and [16], the expressions of a nonlinear map in current-mode-controlled converters were obtained—whose local linearization at the fixed point showed the stability of the orbit obtained.
However, in many other control schemes—notably in the common voltage-mode-controlled converters—the map cannot be obtained in closed form because of the transcendental form of the equations involved. In such systems, though it is possible to obtain the map numerically and thus it is possible to obtain various bifurcation diagrams by iterating the map, studying the stability of periodic orbits which are a problem in the system. Therefore, a range of bifurcation diagrams have been observed, such as Hopf bifurcation [17], border collision [18], quasi-periodicity [19], flip bifurcation [20] and chaos [21].
Over the past decades, methods have been developed to study the stability of such systems. In the Soviet Union, Aizerman, Gantmakher, and Filippov [22] - [23] developed mathematical theorems applicable to systems with a discontinuous right-hand side equation in obtaining solutions for complex nonlinear systems. The method was used in mechanical systems for stick-slip vibrations or impacting motion [24] - [25], with successful results. Hence, power electronic circuits come under the general class of systems with discontinuous right-hand side. Therefore, Filippov's theory provides any new insight into analysing the stability of power electronic circuits, which has been published in power electronics journals.
Mathematical model and stability analysis
A brief review on existing methods of stability analysis of limit cycles in power electronic circuits such as Fig. 1, of TPBC and it continuous conduction mode time invariant steady-state characteristics [26] - [27]. Most of the analytical tools used for this study stem from similar methods used for smooth systems; therefore, outlining such systems for the analysis of the voltage converter, has followed through. Smooth systems - A system is said to be smooth if its mathematical model can be described by a set of differential equations, where such case align with (1).
Fig. 1 a) TPBC Schematic diagram and b) inductor currents L1, L2 and L3 with commutation switches KF1 and KF2 time-domain and c) zero-cuurent/ zero-voltage switching across VT1 and VT2
Poincare mapping of voltage converters
For power electronic circuits, the sampled-data model developed by Verghese et al., [11] and the equivalent Poincare map approach proposed by Deane and Hamill [13] essentially
samples the state variables discretely at the clock instants if the system is nonautonomous (like dc-dc converters under voltage or current mode control) or at the points of intersection of the trajectory with a Poincare surface in case of autonomous systems (like dc-dc converters under hysteresis control). The sampled-data model is obtained as follows. In this exposition, we assume the system to be nonautonomous, with stroboscopic sampling in synchronism with the clock of period. It is a reasonable assumption in dc-dc converters that the subsystems are linear time-invariant (LTI), and the evolution in each subsystem is defined by a differential equation of the form
dXtt) = ^F1,Kf2) • X (t) + BUn (1)
for i = 1,2,3,4. Assuming operation in the nominal period-1 (t = ma, m = 1,2...,k; xk (T) = xk_1) steady state, in which the first switching in a clock cycle occurring at the time interval z1t where z is the duty ratio. The second switching of the clock cycle has time interval occurring at z2t . Third switching of the time interval at ZsT and the fourth time interval is (1 _ z3)T . Therefore, deriving the following system of equations, before
the first switching in a clock cycle, the state evolves as (see fig. 1.)
X (zxT) = (zxT,0)• X0 + h (Z!);
01 (z1T,0)° e4(Z1T); (2)
I (zx)° Af1 (,A1Z1T _ E) B (° BVin).
The second switching in a clock cycle (time interval) is constant. Therefore, the state evolves as
x (z2t) = 02 (z2t,z1t )■ x (z1t)+ 12 (z2 _ z1 ) (3)
The third switching in a clock cycle, the state evolves as
x (z3T ) = 03 (z3T, z2t ) ■ x (z2t ) + 13 (z3 _ z2 ) (4)
Therefore, after the switching and until the end of the clock cycle, the state evolves as
X (T ) = 04 (T, z3T ) ■ X (z3T ) + 14 (1 _ z3 ) (5)
Assuming that there is no discontinuity in the state, one can take the final state before a switching instant to be equal to the initial state after the switching. By applying the method of substitution for reducing the size of the state evolve equation gives
qa ° 04 (T,z3t) 03 (z3T, z2T)02 (z2T, zt) 0x (zt,0);
q3 ° 04 (T, z3t)03 (z3t, z2t)02 (z2T, z1T);
q2 ° 04 (T,z3t) 03 (z3T, z2T); [q1 ° 04 (T, z3t);
Therefore, for one complete cycle (periodic orbit), the equation can be written as
X(T)= qa ■ X0 +03-11( zx)+£2 ■ I2 (z2 _zi)+£}-!,(z, _z2)+iA{\_z3). Where, this gives the sampled-data model of the system
' X (T ) = f (X0, zx, z2, z3);
X(T) = O4 ■ X0 + O3 ■ i1 (zl) + O2 ■ I2(z2 _zl)+ ( )
+°1 ■ I3 (z3 _z2) +14 (1 _z3).
The system periodic orbit, X (T ) = X 0, of (6) yields
X =( E-Q4)
-1 fa • h( zi)+ • /2 (^2 - ^1) +
[+£}• /3 ( Z3 - Z2)+14 (l-Z3)
(7)
Hence, the two-phase boost voltage converter switching events occur when the algebraic equation
h (X0, Z1, Z2, Z3 ) = 0. (8)
is satisfied. Substituting (7) into (6), an equation involving three unknown duty cycle vectors that are z1, z2 and z3. But if z2 is a constant, then only two unknowns is necessary, if and only if, the pulse generator is of saw-tooth form. If the pulse generator is triangular waveform, z2 * const. This can be solved using Gauss-Seidel or Newton-Raphson numerical iterative method routine. This procedure yields the location of the periodic orbit. But summing of all the terms in (6) forms the formula of (9), where k = m for last term of the series, therefore
n a.
/=1
4(m-/)+1 ,
(9)
+1 n ^
j=1V/=j+1
(^ / )4--3+(a / )mm+1
For k = m, of a periodic orbit Xm (T) = X0 , gives
N-1
e-n(^44(m)+1)) >
/=1 j
(10)
Z n £4
j=1 V/=j+1
4(m-/ )-9
(^ / )4--3 + (^/)
4m+1 m
Substituting it into (9) and (10), define m and the values of the state vector variables condition X (t) at the time of switching over the periods T = ka, k = 1,..., m . Thereby, defining m periodic orbits can be expressed as the stroboscopic display of the discrete-time model as
n (^44(m-,)+1 )• f E -n (^44(m-,)+1 )'
(11)
m+1 ,
n 1
.'=j +1
n.
4(m-/)-9 \
) •( )4--3 +( V/ )
4m+1 m
+Z n(^44(m-,)-9) •(m);--3 + (m)
j=1V /=j+1 j
)4k +1 )k .
Where (10) system stability eigenvalues can be calculated using Jacobian method but the monodromy matrix with Filippov's theorem is more straightforward way. For this the Floquet multiplier which are the eigenvalues found on an Argand diagram can determine the dynamic behaviour of (10) with magnitude of 11max| < 1.
Filippov's method
Filippov demonstrated that, a system that possesses a discontinuous right-hand-side, can be ambiguity in the definition of solutions upon the system. To illustrate the method, Fig. 2, a simple one-dimensional (1-D) system is used to show the theorem - (9).
X m =
m I m+1
X 0 =
X
+
k
m)
F
7
At*)
a)
p
?
AI5)
b)
Fig. 2 a) Filippov discontinuous system and b) a set range of value function Filippov concluded that by making f0 (X (t)) not have a single valued function but a range of valued function that satisfies fig. 2b and have upper and lower regime for before and after switching.
T( /+( X (t)), /-(x (t )))Bfc
/0 (x (t )) = ||_( /+(x (t)), /_(x (t )))ma
/-(X(t)) + (1 -z)/+(x(t)),"ze [0,1]
(11) outline a closed convex set regime containing /-(X (t))
/0 (X (t)) = ™ {(/+ (X (t)), /_ (X (t)))} • Therefore, (9) can be written as
'/-( X (t)), / X (t)< 0;
and
/+(x(t)),
(12)
that denotes
dX (t) ( dt
F( X (t))-
c5{(/+(X(t)),/-(X(t)))}, /X(t)= 0; /+(X (t)), / X (t)< 0.
(13)
The extension of (11) to (13) is known as Filippov's convex method and the solution of (13) is known as Filippov solution [23]. The Filippov solution wholes, if it is upper semi-continuous (which is the extension of continuity into set valued functions). The validity of Filippov solution for all initial condition is guaranteed, if the orbit stays almost zero time on the switching manifold or hypersurface. Monodromy matrix analysis
A continuation of the mathematical model as follows, where, Fig. 2(a), where R^ is the resistance in the current loop, R is the resistance of inductors L1 and L2, R3is the resistance of L3 , Uy - reference voltage, the output capacitor is assumed to have no Equivalent Series Resistance (ESR), and feedback amplification factors are b, a1 and a2 respectively. i1 and i2 form the inductor currents loop. Inductors currents L1 and L2 are of anti-phase, which allows for a reduction in capacitance size with low ripple factor on output voltage, where uout = uC .
The mathematical model of the TPBC switching technique feedback equation can be expressed as
Uco (X,t) =a2a1Uy-aRvy (iL1 +iL2) -a2a1bUc (17)
The system starts at new operating period at the time instant of t = (k -1) t , the switch
KFlturns on while KF2 stays off and the trajectories of the state variable x=[iL1,iL2,iL3,UC1,UC2,UC] run in subsystem [A1, B1]. When the switch KFj is turned off at
t=(k-1)T+z{T, KF2 stays off, so the converter runs in subsystem [A2, B1]. The other subsystems
[A3, B1], the switch KFi is off and KF2 turns on at t = (k-1)T+z{T+z2T . Finally, the switch KFi
and KF2 is off for subsystem [A4,B1] at t=(k-1)T+zjT+z2T+z3T+kT. Afterward, the conditional
variable X crosses the switch-off regime and then turns back to x ((k -1)t). At the moment
when the switch is turned off, the switching surface h2 is expressed by (1).
Given that the filter capacity of C is large, UC is kept almost constant at the switching frequency. As iH = UH/R ~ UC/R, the switching surface, where UH is considered the controlled
object, can be rewritten from (1) as follows:
Uo = ioRe + Uc, if Re = 0, which is the Equivalent Series Resistance (ESR) of the output
capacitor Re. Then uo = Uc
aaUy -aR (L+iL2) -a2«iPUc -Upu (t)=0 (15)
The switching surface, where Uo is considered the controlled object, can be rewritten as:
Dn =aPUy -aR (iLi +iL2)-a2aiPUc-UpU (?)=0, (16)
where i = 1,... ,4.
The switches KF1 and KF2 are controlled by comparators which compare the control signal uco (X, t) with a suitable periodic saw-tooth waveform up1 (t) and u 2 (t) with
commutation occurring when KF1 is open - uco (X, t) < Up1 (t) and KF2 closed for
Uco (X, t) > Up2 (t) otherwise.
The state variable trajectory of the closed-loop regulator forms a period-1 limit cycle in the phase space over a complete switching period. Based on theory of monodromy matrix [24], [25], if the Floquet multipliers of the monodromy matrix are all within the unit circle, then the system is stable.
In Fig. 3, an example of the steady-state phase portrait observed for the continuous conduction mode (CCM) for a period one orbit of the system for m = 1 with varying periodic time T.
zTT)
X ZjTT)
X,
^ ((z2- z )T)
z-p^T
*Mt ,o)
X44 ((X-z3-r) = X(0)
■fl \
[T,z,T
3 hn; X-((z3-z2)
( ( z- - z2 ) T )
Fig. 3 The phase-portrait for one periodic orbit of the system for m = 1
Therefore, if the maximum magnitude Floquet multiplier calculated equals 1, then quasi -bifurcation and subharmonics occurs; otherwise, it is unstable. Based on (16), the normal vector and derivative of hn with respect to time t are expressed as follows
n = [
= I -a2Rxy -a2Rxy 0 0 0 -a2a
1b ]j
(18)
\S 2
The status matrices state vector four topologies of the subsystem that belong to the converter are shown in Table 1.
Table 1
With respect to that of the current conduction mode boost converter, the iterated equation of the switching point can be as follows [23], from that of (6) and (7)
Wv Y(T\ n\= i^4X0 + ^ ( Z1) + ^2 ( Z2 - Z1) + f (x0, x (t) ,0) = 1+^3 (z3 - ^2) + /4 (1-^3 )-x (t )= 0
(19)
Numerically solving (18) and previously presented equations using MATLAB with the Newton-Raphson method, the values of X and vector duty cycle z = {[z1, z2, z3 j1,..., [z1, z2, z3 ]k}
can be obtained for the periodic orbit [28]. Thus, based on (16), normal vector n and the manifold (see Fig. 3) derivative Dhn of time t are given as
( + f(Z2-)T -- f()T ) nT
f (Z •T ) = 1 ( + )T--)T ) nT (20)
(+f(
(1-Z3 )T -- f(Z3 - z2 )T )n
ump matrix can be written as follows:
Based on (20), the j
Firstly, at interval (k-1) T,t\ has |^(k-1)T,kJ^T, the switching condition is defined by (16) and the possible normal n is given by (18). The two smooth vector fields are /-T (X(zxT)) = AjX + BUm and (x((z2-Z1)t))= A2x+BUln. Sincef+^T */-T, the system has a discontinuous vector field at the switching surface Fig. 3 - X6 on the output voltage UC. The first jump matrix for interval + /^-Z)T and -/ziT gives
' /(+2-z1)T (x ((Z2 - z )t), (Z2 - Z1 )t )-'
s^1 = E+-
- f-T (X (ZiT), ZiT)
(21)
dh
f-T (X (Z1T). ZiT) + ^
t=( Z2-Zi)T
where E in this article represent an identity matrix (6x6).
Secondly, at interval
1 2 h > t2
has tS
1 2 lk , lk
= (Z2 - Z1 )T • For interval +fz3-z2)t
y A
f Z2-z)T for S232 which is similar in expression as (21) likewise the third interval
12 13
lk, lk
and
has
ts ( [ tk2, t3k ] )=( Z 3 - Z 2 ) T
and final interval for
'sll tk, kT I ) =
] )=( k - Z3 )T, the switching condition
n
k = 1 is defined by (17). Periodic orbit of interval (1 - z3) t has saltation matrix S3 43 . Solutions of the (19) for the inequalities of S^21, S242 and S3A43
nT f +
T
T f-
(Z2 - Z1 )T X n f ( Z1 )T
> 0;
nT f+( w X nT f / „ )7. > 0;
J (z3- 2 )T J (Z2-z1 )T
(22)
nT f + (,-
(1-Z3 )T
X nT f
( z
2 )T
> 0.
Thus, only three Filippov solution switching surface manifolds of the TPBC were obtained and the monodromy matrix for one completed switching period [29]. This can be expressed as follows:
Me ((k-1)T,X((k-1)T),kT) =
&off (T,Z3T)XS^on (Z3T,Z2T)X
xS^o/f (Z2T, Z1T) X SA^on (Z1T, 0);
A
(23)
For k = 1, of the periodic orbit, gives
&off (T, Z3T ) X sf4 0 on ( Z3T, Z2T ) X
Me (0, X (0), T ) =
off
XSA320off (Z2T,Z1T)XSA210on (Z1T,0)•
(24)
Fig. 3, phase portrait voltage-current trajectory periodic orbit and how the monodromy matrix Me can be derived from the diagram for the Floquet multiplier stability. In fig. 4, the time invariant continuous conduction mode (CCM) characteristics of iL12 3, io, ¿0, U0 and hn are shown in (a), and (6) Poincare mapping bifurcation diagram for u0 (KFhKF2) and coefficient of proportionality «2.
Simulation results and analysis
The TPBC system pulse width modulation frequency has been kept constant. The varying parameter here is the input voltage and the effect of the changes on the output voltage characteristics. In Fig. 4, Floquet multipliers were obtained and plotted. The TPBC circuit parameters are constant for the following values: un =1,..,250V; rl,2 = 0.01W; R3 = 0.02 W ;
Rxy = 0.0255 W ; RH = 20 W; ; ax = 14; a2 = 10 L12 = 450 L3 = 43 \iH; CVT12 = 4nF; C = 80^F ;
U y = 0 75V ; P= 0.0048; transistor model - MOSFET N-chan by STMicroelectronics Vds=800V,
Rds(on)=0.35W, Qgate = 44nF and fp = 75 kHz .
Fig. 4 Argand diagram consists of two circles, with blue as unit circle, that is, |Xmax| =1. The coordinates within the red circle are stable eigenvalues of the system including those that are between the red and blue circle. Border collision and periodic double are those X =1 and for outside unstable regime.
Fig. 4 Loci of the eigenvalues of Me
Fig. 5a), b), and c), shows the hypersurface relation of the output voltage, the bifurcation diagram of the swept input voltage verses the output voltage, where u0 = f (uin ).
Fig. 5 a) - The time invariant CCM of the hn manifold, b) the duty cycle of voltage mode control at R0 = 0W, current-mode control R0 > 0W with c) and d) - Poincare mapping bifurcation diagram for stable and unstable regimes
Conclusion
Although monodromy matrix with the inclusion of Filippov's method is a convincing way of analysing voltage converters dynamics over a particular parameter sweep of the system. This approach reduces the need for using Jacobian method, which can be challenging in deriving Jacobian solution. The proposed TPBC topologies state matrices are considered to be large. This system through simulation of Jacobian solution consumes a considerable amount of time before reporting the results, which are not the eigenvalue solution. Hence, the monodromy solution to the TPBC can produce accurate dynamic results and analysis of the system behaviour through Argand diagram characteristics of the voltage converter operational regime(s). The experimental verification for finding the monodromy matrix has been done through mathematical simulation algorithms with MATLAB and not through physical experiment. But experimental verification for soft switching has been achieved resulting in a patented technology of the two-phase boost voltage converter.
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