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

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.

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 Section 4.

The paper is organized as follows: Section 2 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

Figure 1 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 system𝑑𝑣𝑖=−1𝑑𝑟1𝑅𝐶𝐶−1𝐿0𝑣𝑖+0𝑉𝐿𝑊,𝑌=𝑣, (2.1) 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 𝑇_𝑐. 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=𝑇_𝑐/√𝐿𝐶. The dimensionless dynamics can be written as𝑑𝑧𝑑𝜏𝑧¹₂=−𝛾1𝑧−10𝑧¹₂+01𝑢,𝑦=𝑧₁, (2.2)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 𝑣_ref=𝑉_ref/𝑉. Since the output 𝑦 is relative degree 2, and for robustness purposes, 𝑠(𝐳) is defined by [15]𝑠(𝐳)=(𝑧₁−𝑣_ref)+𝑘_𝑠(𝑧̇₁−𝑣̇_ref), (2.3)where 𝑧₁ is the variable to be controlled and 𝑘_𝑠 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^(𝑘+1)𝜀_𝑘𝜀𝑠(𝐳(𝜏))𝑑𝜏=0, (2.4)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 consider 𝑣_ref as a constant, and we rewrite the system dynamics in the more suitable variables: 𝑒=𝑧₁−𝑣_ref (the output error) and 𝑠=𝑒+𝑘_𝑠𝑒̇ (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 are𝑥̇₁𝑥̇₂−1=𝜀𝑘_𝑠1𝑘_𝑠1𝛾−𝑘_𝑠−𝑘_𝑠1𝑘_𝑠𝑥−𝛾𝑥¹₂+𝑘_𝑠𝜀0𝑢−𝑣_ref,𝑦=𝑥₁+𝑣_ref. (2.5)Now, the new independent time is 𝑡, and the period of the PWM is 1. The system reads as 𝑥̇=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 𝑑𝑢=1if𝑘⩽𝑡⩽𝑘+2,𝑑−1if𝑘+2𝑑<𝑡<𝑘+1−2,𝑑1if𝑘+(1−2)⩽𝑡⩽𝑘+1, (2.6) and for lateral pulse, width modulation 𝑢 is defined as 𝑢=1if𝑘⩽𝑡⩽𝑘+𝑑,−1if𝑘+𝑑<𝑡<𝑘+1. (2.7)Equation (5) can be written in compact form as 𝐱̇=𝜀𝐀𝐱+𝑘_𝑠𝜀𝕌, where0𝕌=𝑢−𝑣_ref−1,𝐀=𝑘_𝑠1𝑘_𝑠1𝛾−𝑘_𝑠−𝑘_𝑠1𝑘_𝑠−𝛾 (2.8)being 𝑢 defined by (6) or (7), depending on the pulse generation scheme (CPWM or LPWM).

(a)

(b)

Figure 2 

𝕌2 and two first integrals. In (a) the CPWM case and in (b) the LPWM case.

Remark 2.1. When the system operates in a 1-periodic orbit the following statements are fulfilled.
(i) Since 𝑥₂=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, ∫_𝑘^𝑘+1𝑥₂(𝑡)𝑑𝑡=0.(ii) As ⟨𝑥₂⟩=0, then ⟨𝑥̇₂⟩=0.(iii) The periodicity of 𝑥₁ implies ⟨𝑥̇₁⟩=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−𝑣_ref⟩=0.

Corollary 2.2. The duty cycle for the 1-periodic solution of system (5) is given by 𝑑^∗=0.5(1+𝑣_ref).

Proof. It does not matter which pulse generation scheme (centered or lateral) we use, this value of 𝑑 guarantees ⟨𝕌⟩=0. This value is unique and it is obtained from the solution of the following equations, depending on the generation pulse scheme:₀^𝑑/2(1−𝑣_ref)𝑑𝑡+^1−𝑑/2_𝑑/2(−1−𝑣_ref)𝑑𝑡+¹_1−𝑑/2(1−𝑣_ref)𝑑𝑡=0 (2.9)or^𝑑₀(1−𝑣_ref)𝑑𝑡+¹_𝑑(−1−𝑣_ref)𝑑𝑡=0. (2.10)

Remark 2.3. Before we define some changes of variables, let us consider a generic system 𝑥̇=𝜀𝐴𝑥+𝜀𝕌, where 𝕌 has jump discontinuities. First, assume that 𝕌 is 𝒞^∞ in [0,1], except at a finite number of points 𝑝_𝑖∈[0,1]. Second, let 𝑥(𝑡) be a solution of the differential equation 𝑥̇=𝜀𝐴𝑥+𝜀𝕌. It is continuous in all [0,1] and 𝒞^∞ in [0,1] except at the points 𝑝_𝑖. Third, if we define 𝑢=∫^𝑡₀𝕌, this is also 𝒞^∞ in all [0,1] except at the points 𝑝_𝑖, where it is only continuous. Let us define 𝑦=𝑥−𝜀𝑢, 𝑦 is 𝒞^∞ everywhere except at the points 𝑝_𝑖, where, in principle, it is continuous. This change of variables can be done in each subinterval [𝑝_𝑖,𝑝_𝑖+1] and extended everywhere due to the continuity of 𝑢 at 𝑝_𝑖. Finally, one obtains𝑦̇=𝜀𝐴𝑦+𝜀²𝐴𝑢. (2.11)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 𝑦̇=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 variables 𝐲=𝐱−𝜀𝕌, where𝕌=𝑘_𝑠^𝑡₀𝕌𝑑𝜏. (2.12)Since ⟨𝕌⟩=0, then 𝕌 is periodic. It is straightforward to obtain𝐲̇=𝜀𝐀𝐲+𝜀²𝐀𝕌. (2.13)

For the next change of variables, let us note that we cannot assume that the function ∫^𝑡₀𝐀𝕌 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 as𝐚=¹₀𝐀𝕌𝑑𝑡,𝒰=^𝑡₀(𝐀𝕌−𝐚)𝑑𝜏. (2.14)(2.15)Note that 𝒰 is periodic. Now, let us define a new change of variables, namely,𝐳=𝐲−𝜀²𝒰. (2.16)Then,𝐳̇=𝐲̇−𝜀²𝒰̇‌ (2.17)holds.

This equation is also well defined in the switching instant because 𝒰 is 𝒞¹. Hence, the following equation holds𝐳̇=𝐲̇−𝜀²(𝐀𝕌−𝐚). (2.18)Replacing (9) and (12) in (13) yields to𝐳̇=𝜀𝐀𝐳+𝜀²𝐚+𝜀³𝐀𝒰. (2.19)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 as𝐰̇=𝜀𝐀𝐰+𝜀³𝒲, (2.20)where 𝐰=𝐳−𝐳^∗ and 𝒲=𝐀𝒰. The general solution for this equation is𝐰(𝑡)=𝑒^𝜀𝐀𝑡𝐰(0)+𝜀^3𝑡₀𝑒^{𝜀𝐀(𝑡−𝜎)}𝒲(𝜎)𝑑𝜎. (2.21)A steady-state 1-periodic solution satisfies 𝐰(1)=𝐰(0). Hence,𝐰(0)=(𝐈−𝑒^𝜀𝐀)⁻¹𝜀³¹₀𝑒^{𝜀𝐀(1−𝜎)}𝒲(𝜎)𝑑𝜎. (2.22)Finally, the general solution in the original variables 𝐱 of the system is𝐱(𝑡)=𝜀𝕌(𝑡)+𝜀²𝒰(𝑡)+𝐰(𝑡)−𝜀𝐀⁻¹𝐚, (2.23)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 𝑒^𝜀𝐀𝑡, (𝐈−𝑒^𝜀𝐀)⁻¹ and ∫¹₀𝑒^{𝜀𝐀(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 for 𝒲 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 𝑒^𝜀𝐀𝑡=𝑒₁₁(𝑡)𝑒₁₂𝑒(𝑡)₂₁(𝑡)𝑒₂₂(𝑡), (3.1) where𝑒₁₁(𝑡)=𝑒^{(−𝛾/2)𝜀𝑡}(𝛾/2−1/𝑘_𝑠𝛼sin(𝛼𝜀𝑡)+cos(𝛼𝜀𝑡)),𝑒₁₂(𝑡)=𝑒^{(−𝛾/2)𝜀𝑡}1𝛼𝑘_𝑠𝑒sin(𝛼𝜀𝑡),₂₁(𝑡)=𝑒^{(−𝛾/2)𝜀𝑡}(𝛾−𝑘_𝑠−1/𝑘_𝑠)𝛼sin(𝛼𝜀𝑡),𝑒₂₂(𝑡)=𝑒^{(−𝛾/2)𝜀𝑡}(−𝛾/2+1/𝑘_𝑠𝛼sin(𝛼𝜀𝑡)+cos(𝛼𝜀𝑡)), (3.2)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 tomax|𝑒₁₁𝛾(𝑡)|⩽|2−1𝑘_𝑠|𝜀+1∶=|𝑒₁₁|,max|𝑒₁₂1(𝑡)|⩽𝑘_𝑠𝜀∶=|𝑒₁₂|,max|𝑒₂₁(𝑡)|⩽|𝛾−𝑘_𝑠−1𝑘_𝑠|𝜀∶=|𝑒₂₁|,max|𝑒₂₂𝛾(𝑡)|⩽|2−1𝑘_𝑠|𝜀+1∶=|𝑒₂₂|. (3.3)

3.2. Inverse of (𝐈−𝑒^𝜀𝐀)

In this section, and in order to get accurate bounds, we expand all the functions by Taylor up to second order.

(𝐈−𝑒^𝜀𝐀𝑡)⁻¹ can be expressed as(𝐈−𝑒^𝜀𝐀𝑡)⁻¹=1det(𝐈−𝑒^𝜀𝐀𝑡)𝑚₁₁(𝑡)𝑚₁₂𝑚(𝑡)₂₁(𝑡)𝑚₂₂(𝑡), (3.4)where 𝑚₁₁(𝑡)=1−𝑒₂₂(𝑡),𝑚₁₂(𝑡)=𝑒₁₂(𝑡),𝑚₂₁(𝑡)=𝑒₂₁(𝑡),𝑚₂₂(𝑡)=1−𝑒₁₁(𝑡), and det(𝐈−𝑒^𝜀𝐀𝑡)=1−2𝑒^{(−𝛾/2)𝜀𝑡}cos(𝛼𝜀𝑡)+𝑒^{−𝛾𝜀𝑡}. Since order 1 (𝑂(1)) and order 𝜀𝑡(𝑂(𝜀𝑡)) terms are cancelled, second-order Taylor series expansion is required. Hence,1cos(𝛼𝜀𝑡)=1−2(𝛼𝜀𝑡)²+𝑂((𝛼𝜀𝑡)³1)=1−2(𝛼𝜀𝑡)²+𝜖₁,𝑒^{(−𝛾/2)𝜀𝑡}=1−𝛾𝜀𝑡2+12(𝛾𝜀𝑡2)²+𝑂((𝛾𝜀𝑡2)³)=1−𝛾𝜀𝑡2+12(𝛾𝜀𝑡2)²+𝜖₂,𝑒^{−𝛾𝜀𝑡}1=1−𝛾𝜀𝑡+2(𝛾𝜀𝑡)²+𝑂((𝛾𝜀𝑡)³1)=1−𝛾𝜀𝑡+2(𝛾𝜀𝑡)²+𝜖₃, (3.5)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 todet(𝐈−𝑒^𝜀𝐀𝑡)=(𝜀²𝑡²+𝐸), (3.6)where 𝐸=−2𝑒^{(−𝛾/2)𝜀𝑡}𝜖₁−2cos(𝛼𝜀𝑡)𝜖₂+𝜖₃+(1/8)𝛾²𝛼²𝜀⁴𝑡⁴−(1/2)𝛾𝛼²𝜀³𝑡³.

The leading terms of the adjoint matrix coefficients are of order 𝑂(𝜀𝑡). Hence, in the Taylor series expansion, they must be developed until order 1. The following expansions are used: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, (3.7)where 𝜖₄⩽(𝛼𝜀𝑡)²/2, 𝜖₅⩽(𝛼𝜀𝑡)²/2, and 𝜖₆⩽(1/2)(𝛾𝜀𝑡/2)². Also, the coefficients 𝑚_𝑖𝑗 can be written as𝑚₁₁1=(𝛾−𝑘_𝑠)𝑡𝜀+𝜖_𝑚11,𝑚₁₂=1𝑘_𝑠𝑡𝜀+𝜖_𝑚12,𝑚₂₁1=(𝛾−𝑘_𝑠−𝑘_𝑠)𝑡𝜀+𝜖_𝑚21,𝑚₂₂=1𝑘_𝑠𝑡𝜀+𝜖_𝑚22, (3.8)where𝜖_𝑚11=−𝑐₁𝑒^{(−𝛾/2)𝜀𝑡}𝜖₅−𝑐₁sin(𝛼𝜀𝑡)𝜖₆−𝑒^{(−𝛾/2)𝜀𝑡}𝜖₄−cos(𝛼𝜀𝑡)𝜖₆+𝑐₁𝛼𝛾2𝜀²𝑡²,𝜖_𝑚121=−𝛼𝑘_𝑠𝑒^{(−𝛾/2)𝜀𝑡}𝜖₅−1𝛼𝑘_𝑠sin(𝛼𝜀𝑡)𝜖₆+𝛾2𝑘_𝑠𝜀²𝑡²,𝜖_𝑚21=𝑐₂𝑒^{(−𝛾/2)𝜀𝑡}𝜖₅+𝑐₂sin(𝛼𝜀𝑡)𝜖₆+𝑐₂𝛼𝛾2𝜀²𝑡²,𝜖_𝑚22=𝑐₁𝑒^{(−𝛾/2)𝜀𝑡}𝜖₅+𝑐₁sin(𝛼𝜀𝑡)𝜖₆−𝑒^{(−𝛾/2)𝜀𝑡}𝜖₄−cos(𝛼𝜀𝑡)𝜖₆−𝑐₁𝛼𝛾2𝜀²𝑡², (3.9)where 𝑐₁=1/𝛼𝑘_𝑠−𝛾/2𝛼 and 𝑐₂=(𝑘_𝑠+1/𝑘_𝑠−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 are𝛼|𝐸|⩽2|³𝜀³61|+24|𝛾³𝜀³1|+6|𝛾³𝜀³1|+|(8𝛾²𝛼²1𝜀−2𝛾𝛼²)𝜀³|∶=𝜀³|𝐸^∗|,|𝜖_𝑚11|⩽|𝑐₁𝛼²𝜀²2|+|𝑐₁𝛼𝛾²𝜀³8𝛼|+|²𝜀²2𝛾|+|²𝜀²8|+|𝑐₁𝛼𝛾2𝜀²|,|𝜖_𝑚121|⩽|2𝑘_𝑠𝜀²𝛾|+|²8𝑘_𝑠𝜀³𝛾|+|2𝑘_𝑠𝜀²|,|𝜖_𝑚21|⩽|𝑐₂𝛼²2𝜀²|+|𝑐₂𝛼𝛾²8𝜀³|+|𝑐₂𝛼𝛾2𝜀²|,|𝜖_𝑚22|⩽|𝑐₁𝛼²𝜀²2|+|𝑐₁𝛼𝛾²𝜀³8𝛼|+|²𝜀²2𝛾|+|²𝜀²8|+|𝑐₁𝛼𝛾2𝜀²|, (3.10)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|det(𝕀−𝑒^𝜀𝐀)|⩾Δ=𝜀²(1−𝜀|𝐸^∗|),|𝑚₁₁1|⩽|𝛾−𝑘_𝑠|𝜀+|𝜖_𝑚11|,|𝑚₁₂1|⩽|𝑘_𝑠|𝜀+|𝜖_𝑚12|,|𝑚₂₁|⩽|𝛾−𝑘_𝑠−1𝑘_𝑠|𝜀+|𝜖_𝑚21|,|𝑚₂₂1|⩽|𝑘_𝑠|𝜀+|𝜖_𝑚22|. (3.11)

The aim of the following sections is to evaluate 𝐰(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 that 𝐰(𝑡) is given by𝐰(𝑡)=𝑒^𝜀𝐀𝑡𝐰(0)+𝜀^3𝑡₀𝑒^{𝜀𝐀(𝑡−𝜎)}𝒲(𝜎)𝑑𝜎, (3.12)where𝐰(0)=(𝐈−𝑒^𝜀𝐀)⁻¹𝜀³¹₀𝑒^{𝜀𝐀(1−𝜎)}𝒲(𝜎)𝑑𝜎, (3.13)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 have0𝕌=𝑢−𝑣_ref𝕌∶=¹𝕌₂𝕌₂,=1−𝑣_ref𝑑if0⩽𝑡⩽2,−1−𝑣_ref𝑑if2𝑑<𝑡⩽1−2,1−𝑣_ref𝑑if1−2<𝑡⩽1, (3.14)(3.15)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+𝑣_ref). From the definition of 𝕌, we have𝕌=𝑘_𝑠^𝑡₀𝕌𝑑𝜏∶=𝕌₁𝕌₂^𝑇, (3.16)where 𝕌₁=0 and 𝕌₂ are periodic signals given by𝕌₂=𝑘_𝑠(1−𝑣_ref𝑑)𝑡if0⩽𝑡⩽2,𝑘_𝑠(𝑑−(1+𝑣_ref𝑑)𝑡)if2𝑑<𝑡⩽1−2,𝑘_𝑠(2𝑑−2+(1−𝑣_ref𝑑)𝑡)if1−2<𝑡<1. (3.17)From (10) and using (24), we have 𝐚=𝐀∫¹₀𝕌𝑑𝑡=0. Then, ⟨𝕌⟩=0 and 𝒰=∫^𝑡₀(𝐀𝕌−𝐚)𝑑𝜏=𝑘_𝑠𝐀(𝑏₁𝑏₂)^𝑇, where 𝑏₁=0 and𝑏₂=12(1−𝑣_ref)𝑡²𝑑if0⩽𝑡⩽2,−14𝑑²1+𝑑𝑡−2(1+𝑣_ref)𝑡²𝑑if2𝑑<𝑡⩽1−2,1(1−𝑑)−2(1−𝑑)𝑡+2(1−𝑣_ref)𝑡²𝑑if1−2<𝑡⩽1, (3.18)which is also a periodic signal. Finally, 𝒰 can be expressed in compact form as𝑏𝒰=²(1−𝑘_𝑠𝛾)𝑏₂𝒰∶=¹𝒰₂,𝒲=𝐀𝒰=−𝛾𝑏₂(𝑘_𝑠𝛾²−𝛾−𝑘_𝑠)𝑏₂𝒲∶=¹𝒲₂. (3.19)Let us define𝐈=¹₀𝑒^{𝜀𝐀(1−𝜎)}𝒲(𝜎)𝑑𝜎, (3.20)whose components 𝐼_𝑖 fulfill|𝐼_𝑖|⩽(𝛾|𝑒_𝑖1|+|𝑘_𝑠𝛾²−𝑘_𝑠−𝛾||𝑒_𝑖2|)|¹₀𝑏₂(𝜎)𝑑𝜎|. (3.21)By integrating the input 𝑏₂(𝜎) on the interval [0,1], we have|𝐼_𝑖|⩽(𝛾|𝑒_𝑖1|+|𝑘_𝑠𝛾²−𝑘_𝑠−𝛾||𝑒_𝑖21|)|12(1+𝑣_ref1)−16(1+𝑣_ref)²+196(1+𝑣_ref)³|. (3.22)

For the coefficients |𝑚_𝑖𝑗(𝑡)| defined in (18), let |𝑚_𝑖𝑗|=max|𝑚_𝑖𝑗(𝑡)|forall0⩽𝑡⩽1. Then, (16) and (15) yield to|𝑤_𝑖1(0)|⩽𝜀|1−𝜀|𝐸^∗||(|𝑚_𝑖1𝐼₁|+|𝑚_𝑖2𝐼₂|),|𝑤_𝑖(𝑡)|⩽|𝑒_𝑖1𝑤₁(0)|+|𝑒_𝑖2𝑤₂(0)|+𝜀³|𝐼_𝑖|. (3.23)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:𝐱(𝑡)=𝜀𝕌(𝑡)+𝜀²𝒰(𝑡)+𝐰(𝑡). (3.24)

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+𝑣_ref). Proceeding as in the previous subsection, let us define 𝕌=[𝕌₁𝕌₂]^𝑇, where 𝕌₁=0 and𝕌₂=1−𝑣_refif0⩽𝑡⩽𝑑,−1−𝑣_refif𝑑<𝑡<1 (3.25)the evaluation of𝕌=𝑘_𝑠^𝑡₀𝕌𝑑𝜏∶=[𝕌₁𝕌₂]^𝑇 (3.26)yields to 𝕌₁=0 and𝕌₂=𝑘_𝑠(1−𝑣_ref𝑘)𝑡if0⩽𝑡⩽𝑑,_𝑠(2𝑑−(1+𝑣_ref)𝑡)if𝑑<𝑡<1. (3.27)For this case, ⟨𝕌⟩≠0. In order to evaluate the mean of 𝕌, let us compute𝐚=𝐀¹₀𝕌𝑑𝑡=𝑘_𝑠𝐀0−𝑑²1+2𝑑−2−𝑣_ref2=𝑘_𝑠𝐀0𝛽 (3.28)being 𝛽=−𝑑²+2𝑑−1/2−𝑣_ref/2.

However, 𝒰=∫^𝑡₀(𝐀𝕌(𝜎)−𝐚)𝑑𝜎 can be evaluated as𝒰=𝑘_𝑠𝐀(𝑏₁𝑏₂)^𝑇, (3.29)where 𝑏₁=0 and𝑏₂=12(1−𝑣_ref)𝑡²−𝛽𝑡if0⩽𝑡⩽𝑑,−𝑑²1+(2𝑑−𝛽)𝑡−2(1+𝑣_ref)𝑡²if𝑑<𝑡⩽1. (3.30)Finally, 𝒰 is given in compact form as𝑏𝒰=²(1−𝑘_𝑠𝛾)𝑏₂∶=𝒰₁𝒰₂,𝒲=𝐀𝒲=−𝛾𝑏₂(𝑘_𝑠𝛾²−𝛾−𝑘_𝑠)𝑏₂𝒲∶=¹𝒲₂. (3.31)

As in the preceding case, the components of 𝐈 function𝐈=¹₀𝑒^{𝜀𝐀(1−𝜎)}𝒲(𝜎)𝑑𝜎 (3.32)can be bounded by|𝐼_𝑖|⩽(𝛾|𝑒_𝑖1|+|𝑘_𝑠𝛾²−𝑘_𝑠−𝛾||𝑒_𝑖2|)|¹₀𝑏₂(𝜎)𝑑𝜎|. (3.33)Integrating the input 𝑏₂(𝜎) on the interval [0,1] yields to|𝐼_𝑖|⩽(𝛾|𝑒_𝑖1|+|𝑘_𝑠𝛾²−𝑘_𝑠−𝛾||𝑒_𝑖21|)|12(1+𝑣_ref1)−8(1+𝑣_ref)²+124(1+𝑣_ref)³|. (3.34)

Finally,|𝑤_𝑖1(0)|⩽𝜀|1−𝜀|𝐸^∗||(|𝑚_𝑖1𝐼₁|+|𝑚_𝑖2𝐼₂|),|𝑤_𝑖(𝑡)|⩽(|𝑒_𝑖1𝑤₁(0)|+|𝑒_𝑖2𝑤₂(0)|)+𝜀³|𝐼_𝑖|. (3.35)Using this bound for 𝐰(𝑡), we find the bound for 𝑒(𝑡) and 𝑠(𝑡) using𝐱(𝑡)=𝜀𝕌(𝑡)+𝜀²𝒰(𝑡)+𝐰(𝑡)−𝜀𝐀⁻¹𝐚, (3.36)where 𝑒=𝑥₁ and 𝑠=𝑥₂. It is worth to note that in both cases, the output error is defined by𝑒(𝑡)=𝑤₁(𝑡)+𝜀²𝑢²₁, (3.37)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,𝑣_ref=0.8, and 𝜀=𝑇=0.1767, and they correspond to 𝑅=20Ω,𝐶=40𝜇F,𝐿=2mH,𝑉_ref=32V,𝑉=40V, and 𝑇_𝑐=50𝜇s in a physical system. With the value of 𝑣_ref, we obtain 𝑑=0.9.

4.1. CPWM

In [12], it was proved that the limit of the stability is close to 𝑘_𝑠=3.24. Values below this point lead the system to other operation point. Then, for obtaining a 1-periodic orbit, we use 𝑘_𝑠=4.5. The following bounds were found det(𝐼−𝑒^𝜀𝐀)⩾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 hold𝑤₁(0)⩽0.00036,𝑤₂𝑤(0)⩽0.0011,₁(𝑡)⩽0.00043,𝑤₂(𝑡)⩽0.0016. (4.1)

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,𝑒(𝑡)=𝑤₁(𝑡)+𝜀²𝒰₁. (4.2)The maximum of |𝒰₁|=|𝑏₂| holds at 𝑡=0.5 and it is 0.0225, thenmax|𝑒(𝑡)|⩽0.0011. (4.3)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 as𝑠(𝑡)=𝜀𝕌₂+𝜀²𝒰₂+𝑤₂(𝑡). (4.4)The maximum associated to |𝕌₂| holds at 𝑡=0.45 and it is 0.4050; while the maximum of |𝒰₂|=|(1−𝑘_𝑠𝛾)𝑏₂| holds at 𝑡=0.5 and it is 0.0129. Then, the following holdsmax|𝑠(𝑡)|⩽0.0728. (4.5)Note the agreement between the maximum of the piecewise sliding surface approximation and our result. The leading term in the later inequality is 𝜀𝕌₂ 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 𝑘_𝑠=0.182. Values below this point lead the system to a chaotic behavior. Then, for obtaining a 1-periodic orbit, we use 𝑘_𝑠=0.7068. With this value and the previous parameters, we obtain det(𝐼−𝑒^𝜀𝐀)⩾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) are𝑤₁(0)⩽0.0012,𝑤₂𝑤(0)⩽0.0014,₁(𝑡)⩽0.0019,𝑤₂(𝑡)⩽0.0022. (4.6)

4.2.1. Output Error Estimation

Using (29), and computing 𝐡=𝐀⁻¹𝐚, we obtain 𝐡=(0𝑘_𝑠𝛽)^𝑇∶=(0ℎ₂)^𝑇. As 𝕌₁=0, the error 𝑒(𝑡) fulfills𝑒(𝑡)=𝑤₁(𝑡)+𝜀²𝒰₁. (4.7)The maximum value for excitation 𝒰₁ holds at 𝑡=0.95 and it is 0.0023, hence,max|𝑒(𝑡)|⩽0.0019. (4.8)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 have𝑠(𝑡)=𝜀𝕌₂+𝜀²𝒰₂+𝑤₂(𝑡)−𝜀ℎ₂⩽𝜀|𝕌₂−ℎ₂|+|𝜀²𝒰₂|+|𝑤₂(𝑡)|. (4.9)The maximum associated to |𝕌₂| holds at 𝑡=0.9 and it is 0.1272, while the maximum of |𝒰₂| holds at 𝑡=0.95 and it is 0.0017. In addition, ℎ₂ =0.0636; hence,max|𝑠(𝑡)|⩽0.0135. (4.10)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 because 𝐰(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 𝑘_𝑠=3.24 [12]. In particular, for 𝑘_𝑠=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 not 𝑑=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 function𝛼𝑌(𝑠)=¹𝑠²+𝛼₂𝑠+𝛼₁𝑈(𝑠). (5.1)
The system can be written as𝑥̇₁𝑥̇₂=01−𝛼₁−𝛼₂𝑥₁𝑥₂+0𝛼₁𝑢,𝑦=10𝐱. (5.2)
The output corresponds to the first state of the new state vector. Doing 𝜏=(1/√𝛼₁)𝑡 with 𝜏 the current independent variable and 𝑡 the new independent variable, we have𝑑𝑥₁=1𝑑𝑡√𝛼₁𝑥₂≜𝑤₂, (5.3)hence,𝑑𝑤₂=1𝑑𝑡𝛼₁𝑑𝑥₂𝑑𝜏=−𝑥₁−𝛼₂√𝛼₁𝑤₂+𝑢. (5.4)Now, defining 𝑤₁=𝑥₁, the dynamical system reads as𝑤̇₁𝑤̇₂=𝛼01−1−²√𝛼₁𝑤₁𝑤₂+01𝑢,𝑦=10𝐰. (5.5)Finally, let us define 𝛾=𝛼₂/√𝛼₁,𝐏=(¹⁰_𝛾1) 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 as𝑠(𝑡)=𝜀𝕌₂+𝜀²𝒰₂+𝑤₂(𝑡)−𝜀ℎ₂. (6.1)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 Figures 7, 8, and 9). For example, the 𝑘_𝑠 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 (𝑣_ref). 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 condition 𝜀|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.