# Bounding the Output Error in a Buck Power Converter Using Perturbation Theory

Authors: Fabiola Angulo; Enric Fossas; Tere M. Seara; Gerard Olivar
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

We show the main results obtained
when applying the average theory to Zero Average Dynamic
control technique in a buck power converter with pulse-width
modulation (PWM). In particular, we have obtained the bound
values for output error and sliding surface. The PWM with
centered and lateral pulse configurations were analyzed. The
analytical results have confirmed the numerical and experimental
results already obtained in previous publications. Moreover,
through an important lemma, we have generalized the theory
for any stable second-order system with relative degree 2,
using properties related to transformations and stability of linear
systems.


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## Body

## 1. Introduction

Switching sources are devices used in the
implementation of power converters. As a consequence of the switching action,
chattering, high-order harmonic distortion, and nonlinear phenomena appear. The
latter can be dealt with control techniques [1], while chattering and harmonic distortion, inherent to
switching, can be reduced, but not avoided, using fixed switching frequency. To
achieve this reduction, some techniques have been reported: adaptive hysteresis
band [2, 3], signal injection with a
selected frequency [3–7], zero average current in each iteration (ZACE)
[8], and recently zero
average error dynamics in each iteration (ZAD) [9].ZAD control scheme, proposed in [9], adds the advantages of
fixed frequency implementations and the inherent robustness of sliding control
modes. It is based on an appropriate design of the duty cycle in such a way
that the sliding surface average in each PWM period is zero and the output
voltage tracks the reference with a very low error. A comparative study of this
algorithm with regards to some other ones previously reported in the literature
can be found in [10],
while in [11], this
ZAD technique was applied to a linear converter, showing good numeric and
experimental results. In [12], a stability analysis and the chaos transition
were explained and proved.There are many studies in the averaging theory, mainly
used for skipping the high frequency phenomenon. In fact, in many papers, the
1-periodic orbit is calculated invoking average theory. However, there are
no studies devoted to bound the error in the
system. The aim of this paper is to use the average theory [13, 14] for finding the maximum error of the system, taking
into account the evolution of the trajectory in the continuous time, not only
in the sampled time. This is important in the applications since usually only
the sampled-time evolution is known. In our paper, we explicitly compute the
error bounds along the whole continuous-time trajectory, giving a correct
insight into the error dynamics. Then, centered and lateral ZAD-PWM schemes
will be considered and steady-state maximum values for the error and the
sliding surface in a sampling period will be computed.At the end of the paper, an important lemma is proved.
It relies on properties of transformations and stability of linear systems.
With this lemma, it is possible to find the maximum error displayed for any
ZAD-controlled stable second-order system with relative degree 2 and unitary
gain, using the procedure shown in the paper.The existence and stability of the 1-periodic orbit is
assumed. This assumption is essential since the procedure is based on the
1-periodic orbit bound, then it cannot be applied to higher-period periodic
orbits, as it can be checked with the results shown at the end of Section4.The paper is organized as follows: Section2 is
devoted to show the ZAD-PWM schemes for a buck converter and to give some
generalities on average theory; bounds for the error and the sliding surface
are computed in Section 3, for lateral and centered PWM; in Section 4 the
obtained results are applied to a particular converter widely studied in the
literature; in Section 5, these results are generalized, in order to be applied
to any second-order linear system; finally, conclusions are collected in the
Section 6.


## 2. Preliminaries

Figure1 depicts a simplified model of a buck power converter.Figure 1
A schematic diagram of a buck power converter driven with PWM.This system can be modelled by the switched linear system(2.1)ddr(vi)=(−1RC1C−1L0)(vi)+(0VL)W,Y=v, where the
voltage in the capacitor v(r) and the current in the inductor i(r) are the state variables. The control signal W takes discrete values in the set {−1,1} depending on the switch position. The
independent time variable is noted as r and the sampling time as Tc.
Let us define the dimensionless variables z1(τ)=v/V, z2(τ)=(1/V)(L/C)i and τ=r/LC and a new parameter γ=(1/R)L/C.
The sampling time in the new variables is ε=Tc/LC.
The dimensionless dynamics can be written as(2.2)ddτ(z1z2)=(−γ1−10)(z1z2)+(01)u,y=z1,where u∈{−1,1} is the input control which is implemented
through a Pulse Width Modulator (PWM). The output of the system, corresponding
to the controlled variable is y=z1.
This system will be controlled through a ZAD-PWM, guaranteeing Zero Average
Dynamics (ZAD) for an auxiliary variable s(z).
This makes the output z1 track a dimensionless reference
voltage vref=Vref/V.
Since the output y is relative degree 2, and for robustness
purposes, s(z) is defined by [15](2.3)s(z)=(z1−vref)+ks(z˙1−v˙ref),where z1 is the variable to be controlled and ks is the time-constant associated to the error
dynamics. As the work of Carpita lies in the frame of sliding mode control, s(z) is also named sliding surface. Strictly
speaking, the discrete dynamics should be denoted as quasisliding, since the s(z) samples lie on s(z)>0 and on s(z)<0,
consecutively.The ZAD-PWM strategy is based on computing a duty
cycle,D.
In this case, the duty cycle is the interval of time while the signal control
is u=1.
The ZAD-PWM strategy is obtaining D such that(2.4)∫kε(k+1)εs(z(τ))dτ=0,where ε is small and it is known as the dimensionless
sampling (PWM) period. Then, the sliding surface s(z) has zero average in each PWM period. The exact
calculation of D requires solving a transcendental equation.
This is a problem for online implementations which is easily overcome
approximating the original sliding surface by a piecewise linear one, which
will be denoted also s(z).
For this reason, in [9–11, 16],
a simplification was made and s(z) was considered as piecewise straight line.
However, no analytical results had shown the kindness of the technique. In
[16], a first
approximation to the proof of the advantages was done. In this paper, we use
averaging techniques for proving that the approximation for s(z) is good, and we find a bound for the maximum
error in the system.Let us considervref as a constant, and we rewrite the system
dynamics in the more suitable variables: e=z1−vref (the output error) and s=e+kse˙ (the sliding surface). In the following, we
note this variables as x1 and x2,
respectively. As we said previously, the period of the PWM is ε.
Doing the change of variable t=τ/ε,
the duty cycle is now d=D/ε and the equations of the system
are(2.5)(x˙1x˙2)=ε(−1ks1ksγ−1ks−ks1ks−γ)(x1x2)+ksε(0u−vref),y=x1+vref.Now, the new independent time is t,
and the period of the PWM is 1. The system reads as x˙=εf(x,t) as usual in averaging (perturbation) methods.
The PWM provides the system with a width pulse, which is, roughly speaking,
used to control the system. The pulse can be centered (CPWM) or lateral (LPWM),
as it is illustrated in Figure 2. For the full-bridge buck converter and
centered PMW, u is defined as (2.6)u={1ifk⩽t⩽k+d2,−1ifk+d2<t<k+1−d2,1ifk+(1−d2)⩽t⩽k+1, and for lateral pulse,
width modulation u is defined as (2.7)u={1ifk⩽t⩽k+d,−1ifk+d<t<k+1.Equation (5) can be
written in compact form as x˙=εAx+ksε𝕌,
where(2.8)𝕌=(0u−vref),A=(−1ks1ksγ−1ks−ks1ks−γ)being u defined by (6) or (7), depending on the pulse
generation scheme (CPWM or LPWM).𝕌2 and two first integrals. In (a) the CPWM case and in (b) the LPWM case.
(a)(b)Remark 2.1.
When the system operates in a 1-periodic orbit the following statements are
fulfilled.
(i)
Sincex2=s independently of the scheme of the pulse
(centered or lateral), the chosen control technique according to (4) implies 〈x2〉=0 in each iteration. 〈⋅〉 means average value, that is, ∫kk+1x2(t)dt=0.(ii)
As〈x2〉=0,
then 〈x˙2〉=0.(iii)
The periodicity ofx1 implies 〈x˙1〉=0.
With this consideration and 〈x2〉=0 and taking into account the first equation of
the (5), then 〈x1〉 is forced to be zero (〈x1〉=0).(iv)
Replacing these averaged values in the second
equation of (5), it is possible to obtain 〈u−vref〉=0.Corollary 2.2.
The duty cycle for the 1-periodic solution of system (5) is given by d*=0.5(1+vref).Proof.
It
does not matter which pulse generation scheme (centered or lateral) we use,
this value ofd guarantees 〈𝕌〉=0.
This value is unique and it is obtained from the solution of the following
equations, depending on the generation pulse scheme:(2.9)∫0d/2(1−vref)dt+∫d/21−d/2(−1−vref)dt+∫1−d/21(1−vref)dt=0or(2.10)∫0d(1−vref)dt+∫d1(−1−vref)dt=0.Remark 2.3.
Before we define some changes of variables, let us consider a generic systemx˙=εAx+ε𝕌,
where 𝕌 has jump discontinuities. First, assume that 𝕌 is 𝒞∞ in [0,1],
except at a finite number of points pi∈[0,1].
Second, let x(t) be a solution of the differential equation x˙=εAx+ε𝕌.
It is continuous in all [0,1] and 𝒞∞ in [0,1] except at the points pi.
Third, if we define u¯=∫0t𝕌,
this is also 𝒞∞ in all [0,1] except at the points pi,
where it is only continuous. Let us define y=x−εu¯, y is 𝒞∞ everywhere except at the points pi,
where, in principle, it is continuous. This change of variables can be done in
each subinterval [pi,pi+1] and extended everywhere due to the continuity
of u¯ at pi.
Finally, one obtains(2.11)y˙=εAy+ε2Au¯.The function f(t,y)=εAy+ε2u¯ is continuous with regards to t and 𝒞1 with regards to y.
The ODE existence theorem states that all the solutions of equation y˙=εAy+ε2Au¯ are 𝒞1.
This allows us to do the proposed change of variables, which is common in
averaging theory.Subsequently, in order to average the system, let us
define the change of variablesy=x−ε𝕌¯,
where(2.12)𝕌¯=ks∫0t𝕌⁢dτ.Since 〈𝕌〉=0,
then 𝕌¯ is periodic. It is straightforward to obtain(2.13)y˙=εAy+ε2A𝕌¯.For the next change of variables, let us note
that we cannot assume that the function∫0tA𝕌¯ is periodic. The behavior for the CPWM and
LPWM cases are shown in Figure 2. To solve this problem, let us define the mean of A𝕌¯ as(2.14)a=∫01A𝕌¯⁢dt,(2.15)𝒰¯=∫0t(A𝕌¯−a)dτ.Note that 𝒰¯ is periodic. Now, let us define a new change
of variables, namely,(2.16)z=y−ε2𝒰¯.Then,(2.17)z˙=y˙−ε2𝒰¯˙holds.This equation is also well defined in the switching
instant because𝒰¯ is 𝒞1.
Hence, the following equation holds(2.18)z˙=y˙−ε2(A𝕌¯−a).Replacing (9) and (12) in (13)
yields to(2.19)z˙=εAz+ε2a+ε3A𝒰¯.Assuming 𝒰¯=0,
we find the equilibrium point of equation (14) as z*=−εA−1a.
The invertibility of the matrix A is assured by the eigenvalues of the systems
(they are different from zero). Then, (14) can be expressed as(2.20)w˙=εAw+ε3𝒲,where w=z−z* and 𝒲=A𝒰¯.
The general solution for this equation is(2.21)w(t)=eεAtw(0)+ε3∫0teεA(t−σ)𝒲(σ)dσ.A steady-state 1-periodic
solution satisfies w(1)=w(0).
Hence,(2.22)w(0)=(I−eεA)−1ε3∫01eεA(1−σ)𝒲(σ)dσ.Finally, the general solution in
the original variables x of the system is(2.23)x(t)=ε𝕌¯(t)+ε2𝒰¯(t)+w(t)−εA−1a,where w(t) is given by (15) and (16). Then, for bounding x1(t)(e(t)) and x2(t)(s(t)),
it is necessary to bound all terms of (17). The terms 𝕌¯ and 𝒰¯ depend only on the input control signal; and
their integrals are well known. The term A−1a depends on the parameters of the system and on
the input signal. For this reason, these three terms are easily bounded. Then,
in order to bound x(t) (defined by (17)), we will proceed by
obtaining bounds for each component of the variables w(t) and w(0).
This implies to bound eεAt, (I−eεA)−1 and ∫01eεA(1−σ)𝒲(σ)dσ.


## 3. Bounds

This section is
devoted to obtain bounds of the 1-periodic solution in terms of the input
control signal and the constant matrix.Since the expression for𝒲 differs depending on the lateral or centered
pulses, the analysis is done separately. However, the common terms admit a
unique analysis. In addition, since the differences in magnitude between
variables x1 and x2 have to be taken into account, it is better to
analyze each one separately. This will be done in the following subsections.


### 3.1. State Transition Matrix

The state
transition matrix can be written in compact form as(3.1)eεAt=(e11(t)e12(t)e21(t)e22(t)),
where(3.2)e11(t)=e(−γ/2)εt(γ/2−1/ksαsin⁡(αεt)+cos⁡(αεt)),e12(t)=e(−γ/2)εt1αkssin⁡(αεt),e21(t)=e(−γ/2)εt(γ−ks−1/ks)αsin⁡(αεt),e22(t)=e(−γ/2)εt(−γ/2+1/ksαsin⁡(αεt)+cos⁡(αεt)),and α=1−γ2/4.
Expanding these coefficients in a Taylor series up to first order and evaluating
its maximum in the interval [01] yields to(3.3)max⁡|e11(t)|⩽|γ2−1ks|ε+1:=|e11|,max⁡|e12(t)|⩽1ksε:=|e12|,max⁡|e21(t)|⩽|γ−ks−1ks|ε:=|e21|,max⁡|e22(t)|⩽|γ2−1ks|ε+1:=|e22|.


### 3.2. Inverse of(I−eεA)

In this section, and in order to get accurate bounds,
we expand all the functions by Taylor up to second order.(I−eεAt)−1 can be expressed as(3.4)(I−eεAt)−1=1det⁡(I−eεAt)(m11(t)m12(t)m21(t)m22(t)),where m11(t)=1−e22(t),m12(t)=e12(t),m21(t)=e21(t),m22(t)=1−e11(t),
and det⁡(I−eεAt)=1−2e(−γ/2)εtcos⁡(αεt)+e−γεt.
Since order 1 (O(1)) and order εt(O(εt)) terms are cancelled, second-order Taylor
series expansion is required. Hence,(3.5)cos⁡(αεt)=1−12(αεt)2+O((αεt)3)=1−12(αεt)2+ϵ1,e(−γ/2)εt=1−γεt2+12(γεt2)2+O((γεt2)3)=1−γεt2+12(γεt2)2+ϵ2,e−γεt=1−γεt+12(γεt)2+O((γεt)3)=1−γεt+12(γεt)2+ϵ3,where ϵ1,ϵ2,
and ϵ3 correspond to higher-order error terms. From
Taylor theorem, |ϵ1|⩽(αϵt)3/6, |ϵ2|⩽(1/6)(γϵt/2)3,
and |ϵ3|⩽(γϵt)3/6,
and the determinant is reduced to(3.6)det⁡(I−eεAt)=(ε2t2+E),where E=−2e(−γ/2)εtϵ1−2cos⁡(αεt)ϵ2+ϵ3+(1/8)γ2α2ε4t4−(1/2)γα2ε3t3.The leading terms of the adjoint matrix coefficients
are of orderO(εt).
Hence, in the Taylor series expansion, they must be developed until order 1.
The following expansions are used:(3.7)cos⁡(αεt)=1+O((αεt)2)=1+ϵ4,sin⁡(αεt)=αεt+O((αεt)2)=αεt+ϵ5,e(−γ/2)εt=1−γεt2+O((γεt2)2)=1−γεt2+ϵ6,where ϵ4⩽(αεt)2/2, ϵ5⩽(αεt)2/2,
and ϵ6⩽(1/2)(γεt/2)2.
Also, the coefficients mij can be written as(3.8)m11=(γ−1ks)tε+ϵm11,m12=1kstε+ϵm12,m21=(γ−1ks−ks)tε+ϵm21,m22=1kstε+ϵm22,where(3.9)ϵm11=−c1e(−γ/2)εtϵ5−c1sin⁡(αεt)ϵ6−e(−γ/2)εtϵ4−cos⁡(αεt)ϵ6+c1αγ2ε2t2,ϵm12=−1αkse(−γ/2)εtϵ5−1αkssin⁡(αεt)ϵ6+γ2ksε2t2,ϵm21=c2e(−γ/2)εtϵ5+c2sin⁡(αεt)ϵ6+c2αγ2ε2t2,ϵm22=c1e(−γ/2)εtϵ5+c1sin⁡(αεt)ϵ6−e(−γ/2)εtϵ4−cos⁡(αεt)ϵ6−c1αγ2ε2t2,where c1=1/αks−γ/2α and c2=(ks+1/ks−γ)/α.
Finally, let us take t=1 and bound the errors as an addition of the
absolute values of the addends. Exponential and cosine functions are bounded by
1, and sine function is bounded by the angle;
hence, the expressions for each error term
are(3.10)|E|⩽2|α3ε36|+124|γ3ε3|+16|γ3ε3|+|(18γ2α2ε−12γα2)ε3|:=ε3|E*|,|ϵm11|⩽|c1α2ε22|+|c1αγ2ε38|+|α2ε22|+|γ2ε28|+|c1αγ2ε2|,|ϵm12|⩽|12ksε2|+|γ28ksε3|+|γ2ksε2|,|ϵm21|⩽|c2α22ε2|+|c2αγ28ε3|+|c2αγ2ε2|,|ϵm22|⩽|c1α2ε22|+|c1αγ2ε38|+|α2ε22|+|γ2ε28|+|c1αγ2ε2|,where |E*|⩽α3/3+(5/24)γ3+(α2γ/2)|((γ/4)ε−1)|.
Then, assuming ε|E*|<1 and t=1, we obtain a lower bound for
each term as(3.11)|det⁡(𝕀−eεA)|⩾Δ=ε2(1−ε|E*|),|m11|⩽|γ−1ks|ε+|ϵm11|,|m12|⩽|1ks|ε+|ϵm12|,|m21|⩽|γ−ks−1ks|ε+|ϵm21|,|m22|⩽|1ks|ε+|ϵm22|.The aim of the following sections is to evaluatew(0) in (16) and w(t) in(15) in order to obtain the maximum of the
error x1(t) and of the sliding surface x2(t) of (17). As (15) and (16) depend on the pulse
generation scheme (centered pulse or lateral pulse), the analysis are done
independently.


### 3.3.w(0) and w(t) for Centered Pulse Width
Modulator (CPWM)

Taking into account thatw(t) is given by(3.12)w(t)=eεAtw(0)+ε3∫0teεA(t−σ)𝒲(σ)dσ,where(3.13)w(0)=(I−eεA)−1ε3∫01eεA(1−σ)𝒲(σ)dσ,in order to establish the
maximum value of w(t),
we need to compute the maximum values taken for each component of vector w(0).
For doing this, first we compute 𝒲.
From (5) and (6), we have(3.14)𝕌=(0u−vref):=(𝕌1𝕌2),(3.15)𝕌2={1−vrefif⁢0⩽t⩽d2,−1−vrefif⁢d2<t⩽1−d2,1−vrefif⁢1−d2<t⩽1,d is the time the switch takes value +1. As we
said previously, the duty cycle in the steady state is given by d=0.5(1+vref).
From the definition of 𝕌¯,
we have(3.16)𝕌¯=ks∫0t𝕌⁢dτ:=(𝕌¯1𝕌¯2)T,where 𝕌¯1=0 and 𝕌¯2 are periodic
signals given by(3.17)𝕌¯2={ks(1−vref)tif⁢0⩽t⩽d2,ks(d−(1+vref)t)if⁢d2<t⩽1−d2,ks(2d−2+(1−vref)t)if⁢1−d2<t<1.From (10) and using (24), we
have a=A∫01𝕌¯dt=0.
Then, 〈𝕌¯〉=0 and 𝒰¯=∫0t(A𝕌¯−a)dτ=ksA(b1b2)T,
where b1=0 and(3.18)b2={12(1−vref)t2if⁢0⩽t⩽d2,−14d2+dt−12(1+vref)t2if⁢d2<t⩽1−d2,(1−d)−2(1−d)t+12(1−vref)t2if⁢1−d2<t⩽1,which is also a periodic signal.
Finally, 𝒰¯ can be expressed in compact form
as(3.19)𝒰¯=(b2(1−ksγ)b2):=(𝒰1𝒰2),𝒲=A𝒰¯=(−γb2(ksγ2−γ−ks)b2):=(𝒲1𝒲2).Let us define(3.20)I=∫01eεA(1−σ)𝒲(σ)dσ,whose components Ii fulfill(3.21)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|∫01b2(σ)dσ|.By integrating the input b2(σ) on the interval [0,1],
we have(3.22)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|112(1+vref)−116(1+vref)2+196(1+vref)3|.For the coefficients|mij(t)| defined in (18), let |mij|=max⁡|mij(t)|forall0⩽t⩽1.
Then, (16) and (15) yield to(3.23)|wi(0)|⩽ε1|1−ε|E*||(|mi1I1|+|mi2I2|),|wi(t)|⩽|ei1w1(0)|+|ei2w2(0)|+ε3|Ii|.Finally, using the estimated
bound of w(t),
it is easy to find the maximum values for the error e=x1 and for the sliding surface s=x2,
using the following equation:(3.24)x(t)=ε𝕌¯(t)+ε2𝒰¯(t)+w(t).


### 3.4.w(0) and w(t) for Lateral Pulse (LPWM)

In this case,
the pulse generation order is+1−1.
As it was stated before, d is the time the switch is +1 and it is given
by d=0.5(1+vref).
Proceeding as in the previous subsection, let us define 𝕌=[𝕌1𝕌2]T,
where 𝕌1=0 and(3.25)𝕌2={1−vrefif⁢0⩽t⩽d,−1−vrefif⁢d<t<1the evaluation of(3.26)𝕌¯=ks∫0t𝕌⁢dτ:=[𝕌¯1𝕌¯2]Tyields to 𝕌¯1=0 and(3.27)𝕌¯2={ks(1−vref)tif⁢0⩽t⩽d,ks(2d−(1+vref)t)if⁢d<t<1.For this case, 〈𝕌¯〉≠0.
In order to evaluate the mean of 𝕌¯,
let us compute(3.28)a=A∫01𝕌¯⁢dt=ksA(0−d2+2d−12−vref2)=ksA(0β)being β=−d2+2d−1/2−vref/2.However,𝒰¯=∫0t(A𝕌(σ)−a)dσ can be evaluated as(3.29)𝒰¯=ksA(b1b2)T,where b1=0 and(3.30)b2={12(1−vref)t2−βtif⁢0⩽t⩽d,−d2+(2d−β)t−12(1+vref)t2if⁢d<t⩽1.Finally, 𝒰¯ is given in compact form as(3.31)𝒰¯=(b2(1−ksγ)b2):=(𝒰¯1𝒰¯2),𝒲=A𝒲=(−γb2(ksγ2−γ−ks)b2):=(𝒲1𝒲2).As in the preceding case, the components ofI function(3.32)I=∫01eεA(1−σ)𝒲(σ)dσcan be bounded
by(3.33)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|∫01b2(σ)dσ|.Integrating the input b2(σ) on the interval [0,1] yields to(3.34)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|112(1+vref)−18(1+vref)2+124(1+vref)3|.Finally,(3.35)|wi(0)|⩽ε1|1−ε|E*||(|mi1I1|+|mi2I2|),|wi(t)|⩽(|ei1w1(0)|+|ei2w2(0)|)+ε3|Ii|.Using this bound for w(t),
we find the bound for e(t) and s(t) using(3.36)x(t)=ε𝕌¯(t)+ε2𝒰¯(t)+w(t)−εA−1a,where e=x1 and s=x2.
It is worth to note that in both cases, the output error is defined
by(3.37)e(t)=w1(t)+ε2u¯12,which implies a very low error.


## 3.1. State Transition Matrix

The state
transition matrix can be written in compact form as(3.1)eεAt=(e11(t)e12(t)e21(t)e22(t)),
where(3.2)e11(t)=e(−γ/2)εt(γ/2−1/ksαsin⁡(αεt)+cos⁡(αεt)),e12(t)=e(−γ/2)εt1αkssin⁡(αεt),e21(t)=e(−γ/2)εt(γ−ks−1/ks)αsin⁡(αεt),e22(t)=e(−γ/2)εt(−γ/2+1/ksαsin⁡(αεt)+cos⁡(αεt)),and α=1−γ2/4.
Expanding these coefficients in a Taylor series up to first order and evaluating
its maximum in the interval [01] yields to(3.3)max⁡|e11(t)|⩽|γ2−1ks|ε+1:=|e11|,max⁡|e12(t)|⩽1ksε:=|e12|,max⁡|e21(t)|⩽|γ−ks−1ks|ε:=|e21|,max⁡|e22(t)|⩽|γ2−1ks|ε+1:=|e22|.


## 3.2. Inverse of(I−eεA)

In this section, and in order to get accurate bounds,
we expand all the functions by Taylor up to second order.(I−eεAt)−1 can be expressed as(3.4)(I−eεAt)−1=1det⁡(I−eεAt)(m11(t)m12(t)m21(t)m22(t)),where m11(t)=1−e22(t),m12(t)=e12(t),m21(t)=e21(t),m22(t)=1−e11(t),
and det⁡(I−eεAt)=1−2e(−γ/2)εtcos⁡(αεt)+e−γεt.
Since order 1 (O(1)) and order εt(O(εt)) terms are cancelled, second-order Taylor
series expansion is required. Hence,(3.5)cos⁡(αεt)=1−12(αεt)2+O((αεt)3)=1−12(αεt)2+ϵ1,e(−γ/2)εt=1−γεt2+12(γεt2)2+O((γεt2)3)=1−γεt2+12(γεt2)2+ϵ2,e−γεt=1−γεt+12(γεt)2+O((γεt)3)=1−γεt+12(γεt)2+ϵ3,where ϵ1,ϵ2,
and ϵ3 correspond to higher-order error terms. From
Taylor theorem, |ϵ1|⩽(αϵt)3/6, |ϵ2|⩽(1/6)(γϵt/2)3,
and |ϵ3|⩽(γϵt)3/6,
and the determinant is reduced to(3.6)det⁡(I−eεAt)=(ε2t2+E),where E=−2e(−γ/2)εtϵ1−2cos⁡(αεt)ϵ2+ϵ3+(1/8)γ2α2ε4t4−(1/2)γα2ε3t3.The leading terms of the adjoint matrix coefficients
are of orderO(εt).
Hence, in the Taylor series expansion, they must be developed until order 1.
The following expansions are used:(3.7)cos⁡(αεt)=1+O((αεt)2)=1+ϵ4,sin⁡(αεt)=αεt+O((αεt)2)=αεt+ϵ5,e(−γ/2)εt=1−γεt2+O((γεt2)2)=1−γεt2+ϵ6,where ϵ4⩽(αεt)2/2, ϵ5⩽(αεt)2/2,
and ϵ6⩽(1/2)(γεt/2)2.
Also, the coefficients mij can be written as(3.8)m11=(γ−1ks)tε+ϵm11,m12=1kstε+ϵm12,m21=(γ−1ks−ks)tε+ϵm21,m22=1kstε+ϵm22,where(3.9)ϵm11=−c1e(−γ/2)εtϵ5−c1sin⁡(αεt)ϵ6−e(−γ/2)εtϵ4−cos⁡(αεt)ϵ6+c1αγ2ε2t2,ϵm12=−1αkse(−γ/2)εtϵ5−1αkssin⁡(αεt)ϵ6+γ2ksε2t2,ϵm21=c2e(−γ/2)εtϵ5+c2sin⁡(αεt)ϵ6+c2αγ2ε2t2,ϵm22=c1e(−γ/2)εtϵ5+c1sin⁡(αεt)ϵ6−e(−γ/2)εtϵ4−cos⁡(αεt)ϵ6−c1αγ2ε2t2,where c1=1/αks−γ/2α and c2=(ks+1/ks−γ)/α.
Finally, let us take t=1 and bound the errors as an addition of the
absolute values of the addends. Exponential and cosine functions are bounded by
1, and sine function is bounded by the angle;
hence, the expressions for each error term
are(3.10)|E|⩽2|α3ε36|+124|γ3ε3|+16|γ3ε3|+|(18γ2α2ε−12γα2)ε3|:=ε3|E*|,|ϵm11|⩽|c1α2ε22|+|c1αγ2ε38|+|α2ε22|+|γ2ε28|+|c1αγ2ε2|,|ϵm12|⩽|12ksε2|+|γ28ksε3|+|γ2ksε2|,|ϵm21|⩽|c2α22ε2|+|c2αγ28ε3|+|c2αγ2ε2|,|ϵm22|⩽|c1α2ε22|+|c1αγ2ε38|+|α2ε22|+|γ2ε28|+|c1αγ2ε2|,where |E*|⩽α3/3+(5/24)γ3+(α2γ/2)|((γ/4)ε−1)|.
Then, assuming ε|E*|<1 and t=1, we obtain a lower bound for
each term as(3.11)|det⁡(𝕀−eεA)|⩾Δ=ε2(1−ε|E*|),|m11|⩽|γ−1ks|ε+|ϵm11|,|m12|⩽|1ks|ε+|ϵm12|,|m21|⩽|γ−ks−1ks|ε+|ϵm21|,|m22|⩽|1ks|ε+|ϵm22|.The aim of the following sections is to evaluatew(0) in (16) and w(t) in(15) in order to obtain the maximum of the
error x1(t) and of the sliding surface x2(t) of (17). As (15) and (16) depend on the pulse
generation scheme (centered pulse or lateral pulse), the analysis are done
independently.


## 3.3.w(0) and w(t) for Centered Pulse Width
Modulator (CPWM)

Taking into account thatw(t) is given by(3.12)w(t)=eεAtw(0)+ε3∫0teεA(t−σ)𝒲(σ)dσ,where(3.13)w(0)=(I−eεA)−1ε3∫01eεA(1−σ)𝒲(σ)dσ,in order to establish the
maximum value of w(t),
we need to compute the maximum values taken for each component of vector w(0).
For doing this, first we compute 𝒲.
From (5) and (6), we have(3.14)𝕌=(0u−vref):=(𝕌1𝕌2),(3.15)𝕌2={1−vrefif⁢0⩽t⩽d2,−1−vrefif⁢d2<t⩽1−d2,1−vrefif⁢1−d2<t⩽1,d is the time the switch takes value +1. As we
said previously, the duty cycle in the steady state is given by d=0.5(1+vref).
From the definition of 𝕌¯,
we have(3.16)𝕌¯=ks∫0t𝕌⁢dτ:=(𝕌¯1𝕌¯2)T,where 𝕌¯1=0 and 𝕌¯2 are periodic
signals given by(3.17)𝕌¯2={ks(1−vref)tif⁢0⩽t⩽d2,ks(d−(1+vref)t)if⁢d2<t⩽1−d2,ks(2d−2+(1−vref)t)if⁢1−d2<t<1.From (10) and using (24), we
have a=A∫01𝕌¯dt=0.
Then, 〈𝕌¯〉=0 and 𝒰¯=∫0t(A𝕌¯−a)dτ=ksA(b1b2)T,
where b1=0 and(3.18)b2={12(1−vref)t2if⁢0⩽t⩽d2,−14d2+dt−12(1+vref)t2if⁢d2<t⩽1−d2,(1−d)−2(1−d)t+12(1−vref)t2if⁢1−d2<t⩽1,which is also a periodic signal.
Finally, 𝒰¯ can be expressed in compact form
as(3.19)𝒰¯=(b2(1−ksγ)b2):=(𝒰1𝒰2),𝒲=A𝒰¯=(−γb2(ksγ2−γ−ks)b2):=(𝒲1𝒲2).Let us define(3.20)I=∫01eεA(1−σ)𝒲(σ)dσ,whose components Ii fulfill(3.21)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|∫01b2(σ)dσ|.By integrating the input b2(σ) on the interval [0,1],
we have(3.22)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|112(1+vref)−116(1+vref)2+196(1+vref)3|.For the coefficients|mij(t)| defined in (18), let |mij|=max⁡|mij(t)|forall0⩽t⩽1.
Then, (16) and (15) yield to(3.23)|wi(0)|⩽ε1|1−ε|E*||(|mi1I1|+|mi2I2|),|wi(t)|⩽|ei1w1(0)|+|ei2w2(0)|+ε3|Ii|.Finally, using the estimated
bound of w(t),
it is easy to find the maximum values for the error e=x1 and for the sliding surface s=x2,
using the following equation:(3.24)x(t)=ε𝕌¯(t)+ε2𝒰¯(t)+w(t).


## 3.4.w(0) and w(t) for Lateral Pulse (LPWM)

In this case,
the pulse generation order is+1−1.
As it was stated before, d is the time the switch is +1 and it is given
by d=0.5(1+vref).
Proceeding as in the previous subsection, let us define 𝕌=[𝕌1𝕌2]T,
where 𝕌1=0 and(3.25)𝕌2={1−vrefif⁢0⩽t⩽d,−1−vrefif⁢d<t<1the evaluation of(3.26)𝕌¯=ks∫0t𝕌⁢dτ:=[𝕌¯1𝕌¯2]Tyields to 𝕌¯1=0 and(3.27)𝕌¯2={ks(1−vref)tif⁢0⩽t⩽d,ks(2d−(1+vref)t)if⁢d<t<1.For this case, 〈𝕌¯〉≠0.
In order to evaluate the mean of 𝕌¯,
let us compute(3.28)a=A∫01𝕌¯⁢dt=ksA(0−d2+2d−12−vref2)=ksA(0β)being β=−d2+2d−1/2−vref/2.However,𝒰¯=∫0t(A𝕌(σ)−a)dσ can be evaluated as(3.29)𝒰¯=ksA(b1b2)T,where b1=0 and(3.30)b2={12(1−vref)t2−βtif⁢0⩽t⩽d,−d2+(2d−β)t−12(1+vref)t2if⁢d<t⩽1.Finally, 𝒰¯ is given in compact form as(3.31)𝒰¯=(b2(1−ksγ)b2):=(𝒰¯1𝒰¯2),𝒲=A𝒲=(−γb2(ksγ2−γ−ks)b2):=(𝒲1𝒲2).As in the preceding case, the components ofI function(3.32)I=∫01eεA(1−σ)𝒲(σ)dσcan be bounded
by(3.33)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|∫01b2(σ)dσ|.Integrating the input b2(σ) on the interval [0,1] yields to(3.34)|Ii|⩽(γ|ei1|+|ksγ2−ks−γ||ei2|)|112(1+vref)−18(1+vref)2+124(1+vref)3|.Finally,(3.35)|wi(0)|⩽ε1|1−ε|E*||(|mi1I1|+|mi2I2|),|wi(t)|⩽(|ei1w1(0)|+|ei2w2(0)|)+ε3|Ii|.Using this bound for w(t),
we find the bound for e(t) and s(t) using(3.36)x(t)=ε𝕌¯(t)+ε2𝒰¯(t)+w(t)−εA−1a,where e=x1 and s=x2.
It is worth to note that in both cases, the output error is defined
by(3.37)e(t)=w1(t)+ε2u¯12,which implies a very low error.


## 4. Estimation of the Maximum Values for the Output Error and the Sliding Surface

Let us now
compute all of these calculus for the parameters values used in [11, 12, 16, 17]. These parameters are γ=0.35,vref=0.8,
and ε=T=0.1767,
and they correspond to R=20⁢Ω,C=40⁢μF,L=2⁢mH,⁢Vref=32⁢V,V=40⁢V, and Tc=50⁢μs in a physical system. With the value of vref,
we obtain d=0.9.


### 4.1. CPWM

In [12], it was proved that the
limit of the stability is close to ks=3.24.
Values below this point lead the system to other
operation point. Then, for obtaining a 1-periodic orbit, we use ks=4.5.
The following bounds were found  det⁡(I−eεA)⩾0.0285,|e11|⩽1.0083,|e12|⩽0.0393,|e21|⩽0.7726,|e22|⩽1.0083,I1⩽0.0043,I2⩽0.0379,|m11|⩽0.0392,|m12|⩽0.0440,|m21|⩽0.8640, and |m22|⩽0.0559.
Hence, the following hold(4.1)w1(0)⩽0.00036,w2(0)⩽0.0011,w1(t)⩽0.00043,w2(t)⩽0.0016.


#### 4.1.1. Output Error Estimation

Using (27) and
taking into account that the first component of the input vector 𝕌 is zero, error dynamics is defined by the
first components of w(t) and ε2𝒰¯,
thus,(4.2)e(t)=w1(t)+ε2𝒰¯1.The maximum of |𝒰¯1|=|b2| holds at t=0.5 and it is 0.0225, then(4.3)max⁡|e(t)|⩽0.0011.This is equivalent to a maximum
error value of 0.14% in steady state when the reference reaches the value 0.8.
The error behavior obtained from a numerical simulation is depicted in Figure 3. Note that the real error value is lower than the estimated one. However, the
bound is really close.Figure 3
Behavior of the error in a sampling interval for CPWM scheme.


#### 4.1.2. Sliding Surface Error Estimation

In this case,
the expression is adjusted to analyze the second component as(4.4)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t).The maximum associated to |𝕌¯2| holds at t=0.45 and it is 0.4050; while the maximum of |𝒰¯2|=|(1−ksγ)b2| holds at t=0.5 and it is 0.0129. Then, the
following holds(4.5)max⁡|s(t)|⩽0.0728.Note the agreement between the
maximum of the piecewise sliding surface approximation and our result. The
leading term in the later inequality is ε𝕌¯2 and it takes the value 0.0716. Simulation
results are depicted in Figure 4.Figure 4
Behavior of the sliding surface in a sampling interval for CPWM scheme.


### 4.2. LPWM

In [12] it was proved that the
limit of the stability is close to ks=0.182.
Values below this point lead the system to a
chaotic behavior. Then, for obtaining a 1-periodic orbit, we use ks=0.7068.
With this value and the previous parameters, we obtain det⁡(I−eεA)⩾0.0285,|e11|⩽1.2191,|e12|⩽0.2500,|e21|⩽0.3130, and |e22|⩽1.2191.
Therefore I1⩽0.008,I2⩽0.0155,|m11|⩽0.2297,|m12|⩽0.2799,|m21|⩽0.3501,
and |m22|⩽0.2915.
Thus, the bounds for w(0) and w(t) are(4.6)w1(0)⩽0.0012,w2(0)⩽0.0014,w1(t)⩽0.0019,w2(t)⩽0.0022.


#### 4.2.1. Output Error Estimation

Using (29), and
computing h=A−1a,
we obtain h=(0ksβ)T:=(0h2)T.
As 𝕌¯1=0,
the error e(t) fulfills(4.7)e(t)=w1(t)+ε2𝒰¯1.The maximum value for excitation 𝒰¯1 holds at t=0.95 and it is 0.0023, hence,(4.8)max⁡|e(t)|⩽0.0019.This is equivalent to a maximum
error of 0.24% in steady state for a reference equal to 0.8. The output error
behavior is depicted in Figure 5. Simulations took 500 points per sampling
period.Figure 5
Behavior of the error in a sampling interval for LPWM scheme.


#### 4.2.2. Sliding Surface Error Estimation

In this case,
we have(4.9)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t)−εh2⩽ε|𝕌¯2−h2|+|ε2𝒰¯2|+|w2(t)|.The maximum associated to |𝕌¯2| holds at t=0.9 and it is 0.1272, while the maximum of |𝒰¯2| holds at t=0.95 and it is 0.0017. In addition, h2 =0.0636;
hence,(4.10)max⁡|s(t)|⩽0.0135.Simulation results are depicted
in Figure 6.Figure 6
Behavior of the sliding surface in a sampling interval for LPWM scheme.Previous results must be carefully applied. This is
becausew(0) is calculated assuming the existence and
stability of 1-periodic orbit in (16) and all the procedure is based on the
bound for w(0).
Results obtained for other operation conditions may not agree. For example, the
limit of the stability for this system is close to ks=3.24 [12]. In particular, for ks=3.1,
the system works in 2-periodic mode. The results obtained when this technique
is applied are shown below. In Figure 7, the behavior of the error is shown,
and the maximum calculated with our approximations is presented too. Note that
the predicted maximum does not agree with the numerical results. The reason is
found in the duty cycle (see Figure 8). Now the system is 2-periodic and the
steady state of the duty cycle is different from d=0.9.
The same situation modifies the results obtained for the sliding surface
(Figure 9).Figure 7
Behavior of the error for CPWM working in 2-periodic mode. In the same figure the bound calculated with the previous process is shown.Figure 8
Duty cycle for CPWM working in 2-periodic mode. Note that the steady state is notd=0.9.Figure 9
Behavior of the
sliding surface for CPWM working in 2-periodic mode. In the same figure, the
bound for the sliding surface.Application of this technique in other operation
points can lead to much higher or lower bounds compared
to numerical results, and the approximation of the
sliding surface to a piecewise straight line may not be valid.


## 4.1. CPWM

In [12], it was proved that the
limit of the stability is close to ks=3.24.
Values below this point lead the system to other
operation point. Then, for obtaining a 1-periodic orbit, we use ks=4.5.
The following bounds were found  det⁡(I−eεA)⩾0.0285,|e11|⩽1.0083,|e12|⩽0.0393,|e21|⩽0.7726,|e22|⩽1.0083,I1⩽0.0043,I2⩽0.0379,|m11|⩽0.0392,|m12|⩽0.0440,|m21|⩽0.8640, and |m22|⩽0.0559.
Hence, the following hold(4.1)w1(0)⩽0.00036,w2(0)⩽0.0011,w1(t)⩽0.00043,w2(t)⩽0.0016.


### 4.1.1. Output Error Estimation

Using (27) and
taking into account that the first component of the input vector 𝕌 is zero, error dynamics is defined by the
first components of w(t) and ε2𝒰¯,
thus,(4.2)e(t)=w1(t)+ε2𝒰¯1.The maximum of |𝒰¯1|=|b2| holds at t=0.5 and it is 0.0225, then(4.3)max⁡|e(t)|⩽0.0011.This is equivalent to a maximum
error value of 0.14% in steady state when the reference reaches the value 0.8.
The error behavior obtained from a numerical simulation is depicted in Figure 3. Note that the real error value is lower than the estimated one. However, the
bound is really close.Figure 3
Behavior of the error in a sampling interval for CPWM scheme.


### 4.1.2. Sliding Surface Error Estimation

In this case,
the expression is adjusted to analyze the second component as(4.4)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t).The maximum associated to |𝕌¯2| holds at t=0.45 and it is 0.4050; while the maximum of |𝒰¯2|=|(1−ksγ)b2| holds at t=0.5 and it is 0.0129. Then, the
following holds(4.5)max⁡|s(t)|⩽0.0728.Note the agreement between the
maximum of the piecewise sliding surface approximation and our result. The
leading term in the later inequality is ε𝕌¯2 and it takes the value 0.0716. Simulation
results are depicted in Figure 4.Figure 4
Behavior of the sliding surface in a sampling interval for CPWM scheme.


## 4.1.1. Output Error Estimation

Using (27) and
taking into account that the first component of the input vector 𝕌 is zero, error dynamics is defined by the
first components of w(t) and ε2𝒰¯,
thus,(4.2)e(t)=w1(t)+ε2𝒰¯1.The maximum of |𝒰¯1|=|b2| holds at t=0.5 and it is 0.0225, then(4.3)max⁡|e(t)|⩽0.0011.This is equivalent to a maximum
error value of 0.14% in steady state when the reference reaches the value 0.8.
The error behavior obtained from a numerical simulation is depicted in Figure 3. Note that the real error value is lower than the estimated one. However, the
bound is really close.Figure 3
Behavior of the error in a sampling interval for CPWM scheme.


## 4.1.2. Sliding Surface Error Estimation

In this case,
the expression is adjusted to analyze the second component as(4.4)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t).The maximum associated to |𝕌¯2| holds at t=0.45 and it is 0.4050; while the maximum of |𝒰¯2|=|(1−ksγ)b2| holds at t=0.5 and it is 0.0129. Then, the
following holds(4.5)max⁡|s(t)|⩽0.0728.Note the agreement between the
maximum of the piecewise sliding surface approximation and our result. The
leading term in the later inequality is ε𝕌¯2 and it takes the value 0.0716. Simulation
results are depicted in Figure 4.Figure 4
Behavior of the sliding surface in a sampling interval for CPWM scheme.


## 4.2. LPWM

In [12] it was proved that the
limit of the stability is close to ks=0.182.
Values below this point lead the system to a
chaotic behavior. Then, for obtaining a 1-periodic orbit, we use ks=0.7068.
With this value and the previous parameters, we obtain det⁡(I−eεA)⩾0.0285,|e11|⩽1.2191,|e12|⩽0.2500,|e21|⩽0.3130, and |e22|⩽1.2191.
Therefore I1⩽0.008,I2⩽0.0155,|m11|⩽0.2297,|m12|⩽0.2799,|m21|⩽0.3501,
and |m22|⩽0.2915.
Thus, the bounds for w(0) and w(t) are(4.6)w1(0)⩽0.0012,w2(0)⩽0.0014,w1(t)⩽0.0019,w2(t)⩽0.0022.


### 4.2.1. Output Error Estimation

Using (29), and
computing h=A−1a,
we obtain h=(0ksβ)T:=(0h2)T.
As 𝕌¯1=0,
the error e(t) fulfills(4.7)e(t)=w1(t)+ε2𝒰¯1.The maximum value for excitation 𝒰¯1 holds at t=0.95 and it is 0.0023, hence,(4.8)max⁡|e(t)|⩽0.0019.This is equivalent to a maximum
error of 0.24% in steady state for a reference equal to 0.8. The output error
behavior is depicted in Figure 5. Simulations took 500 points per sampling
period.Figure 5
Behavior of the error in a sampling interval for LPWM scheme.


### 4.2.2. Sliding Surface Error Estimation

In this case,
we have(4.9)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t)−εh2⩽ε|𝕌¯2−h2|+|ε2𝒰¯2|+|w2(t)|.The maximum associated to |𝕌¯2| holds at t=0.9 and it is 0.1272, while the maximum of |𝒰¯2| holds at t=0.95 and it is 0.0017. In addition, h2 =0.0636;
hence,(4.10)max⁡|s(t)|⩽0.0135.Simulation results are depicted
in Figure 6.Figure 6
Behavior of the sliding surface in a sampling interval for LPWM scheme.Previous results must be carefully applied. This is
becausew(0) is calculated assuming the existence and
stability of 1-periodic orbit in (16) and all the procedure is based on the
bound for w(0).
Results obtained for other operation conditions may not agree. For example, the
limit of the stability for this system is close to ks=3.24 [12]. In particular, for ks=3.1,
the system works in 2-periodic mode. The results obtained when this technique
is applied are shown below. In Figure 7, the behavior of the error is shown,
and the maximum calculated with our approximations is presented too. Note that
the predicted maximum does not agree with the numerical results. The reason is
found in the duty cycle (see Figure 8). Now the system is 2-periodic and the
steady state of the duty cycle is different from d=0.9.
The same situation modifies the results obtained for the sliding surface
(Figure 9).Figure 7
Behavior of the error for CPWM working in 2-periodic mode. In the same figure the bound calculated with the previous process is shown.Figure 8
Duty cycle for CPWM working in 2-periodic mode. Note that the steady state is notd=0.9.Figure 9
Behavior of the
sliding surface for CPWM working in 2-periodic mode. In the same figure, the
bound for the sliding surface.Application of this technique in other operation
points can lead to much higher or lower bounds compared
to numerical results, and the approximation of the
sliding surface to a piecewise straight line may not be valid.


## 4.2.1. Output Error Estimation

Using (29), and
computing h=A−1a,
we obtain h=(0ksβ)T:=(0h2)T.
As 𝕌¯1=0,
the error e(t) fulfills(4.7)e(t)=w1(t)+ε2𝒰¯1.The maximum value for excitation 𝒰¯1 holds at t=0.95 and it is 0.0023, hence,(4.8)max⁡|e(t)|⩽0.0019.This is equivalent to a maximum
error of 0.24% in steady state for a reference equal to 0.8. The output error
behavior is depicted in Figure 5. Simulations took 500 points per sampling
period.Figure 5
Behavior of the error in a sampling interval for LPWM scheme.


## 4.2.2. Sliding Surface Error Estimation

In this case,
we have(4.9)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t)−εh2⩽ε|𝕌¯2−h2|+|ε2𝒰¯2|+|w2(t)|.The maximum associated to |𝕌¯2| holds at t=0.9 and it is 0.1272, while the maximum of |𝒰¯2| holds at t=0.95 and it is 0.0017. In addition, h2 =0.0636;
hence,(4.10)max⁡|s(t)|⩽0.0135.Simulation results are depicted
in Figure 6.Figure 6
Behavior of the sliding surface in a sampling interval for LPWM scheme.Previous results must be carefully applied. This is
becausew(0) is calculated assuming the existence and
stability of 1-periodic orbit in (16) and all the procedure is based on the
bound for w(0).
Results obtained for other operation conditions may not agree. For example, the
limit of the stability for this system is close to ks=3.24 [12]. In particular, for ks=3.1,
the system works in 2-periodic mode. The results obtained when this technique
is applied are shown below. In Figure 7, the behavior of the error is shown,
and the maximum calculated with our approximations is presented too. Note that
the predicted maximum does not agree with the numerical results. The reason is
found in the duty cycle (see Figure 8). Now the system is 2-periodic and the
steady state of the duty cycle is different from d=0.9.
The same situation modifies the results obtained for the sliding surface
(Figure 9).Figure 7
Behavior of the error for CPWM working in 2-periodic mode. In the same figure the bound calculated with the previous process is shown.Figure 8
Duty cycle for CPWM working in 2-periodic mode. Note that the steady state is notd=0.9.Figure 9
Behavior of the
sliding surface for CPWM working in 2-periodic mode. In the same figure, the
bound for the sliding surface.Application of this technique in other operation
points can lead to much higher or lower bounds compared
to numerical results, and the approximation of the
sliding surface to a piecewise straight line may not be valid.


## 5. Generalizing the Theory for Second-Order Systems

The objective
of this section is to show that the procedure developed up to now is not only
for a particular example, but is also correct for general second-order systems,
as the following lemma shows.Lemma 5.1.
Any stable second-order system with relative degree 2
and unitary gain can be put in the form of (2).Proof.
Any second linear order system with relative degree 2
and unitary gain is defined by the following transfer function(5.1)Y(s)=α1s2+α2s+α1U(s).
The system can be written as(5.2)[x˙1x˙2]=[01−α1−α2][x1x2]+[0α1]u,y=[10]x.
The output corresponds to the first state of the new
state vector. Doingτ=(1/α1)t with τ the current independent variable and t the new independent variable, we
have(5.3)dx1dt=1α1x2≜w2,hence,(5.4)dw2dt=1α1dx2dτ=−x1−α2α1w2+u.Now, defining w1=x1,
the dynamical system reads as(5.5)[w˙1w˙2]=[01−1−α2α1][w1w2]+[01]u,y=[10]w.Finally, let us define γ=α2/α1,P=(10γ1) and z=Pw,  then (30) exactly matches to the system considered in (2). Even more, for the
change of coordinates, it is obvious that α1⁢≠0 guaranteeing the invertibility of the matrix A.
Therefore, the methodology developed along the paper is a general tool.


## 6. Conclusions

From the previous sections we can conclude the
following.(i)
Approximations done in [10–12] have been supported from averaging theory. In those
papers, the sliding surface s is assumed to be piecewise linear w.r.t. time.
Our final expression reads as(6.1)s(t)=ε𝕌¯2+ε2𝒰¯2+w2(t)−εh2.In this expression, the leading
term is of order 1. This is a straight line and it corresponds to the primitive
of the input of the system. The other terms can be deleted, as
being explained in Sections 4.1.2 and 4.2.2.(ii)
Due to average theory, it has been
analytically found that the maximum error is always less or equal to 0.3% for
the parameter values in use. This confirms the effectiveness of the technique.(iii)
These results are strongly dependent on the
constants values. A bad selection of the constants values can lead to
wrong-predicted results (see Figures7, 8, and 9). For example, the ks constant value considered in the LPWM case
makes the system with CPWM unstable. For the general second-order system
considered in the last section, the dynamics are
defined by parameter γ.
The results reported in the paper assume a
1-periodic orbit as steady-state dynamics.(iv)
The bounds depend explicitly on the reference
value (vref). As it can be seen, the error increases when
reference decreases. These results are coherent with other numerical studies.(v)
A strong assumption of this procedure is given
by the conditionε|E*|<1 which is stated for avoiding zero division.
However it is also clear that if this term is close to 1, then the calculated
bound will be very high. Thus, these considerations must be taken into account
for applying these computed bounds.(vi)
CPWM and LPWM ZAD theory are generalized to
any second-order system with unitary gain and relative degree 2, and all the
results obtained can be applied to another ZAD-controlled systems.



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*Source: 732039-2008-09-04.xml*