Analysis and experimental investigation into a finite time current observer based adaptive backstepping control of buck converters

Accepted Manuscript

Analysis and Experimental Investigation into a Finite Time Current Observer Based Adaptive Backstepping Control of Buck Converters Tousif Khan Nizami, Arghya Chakravarty, Chitralekha Mahanta PII: DOI: Reference:

S0016-0032(18)30338-7 10.1016/j.jfranklin.2018.05.026 FI 3460

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

22 January 2016 20 August 2017 8 May 2018

Please cite this article as: Tousif Khan Nizami, Arghya Chakravarty, Chitralekha Mahanta, Analysis and Experimental Investigation into a Finite Time Current Observer Based Adaptive Backstepping Control of Buck Converters, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.05.026

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Analysis and Experimental Investigation into a Finite Time Current Observer Based Adaptive Backstepping Control of Buck Converters

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Tousif Khan Nizamia,1,∗, Arghya Chakravartya , Chitralekha Mahantaa a

Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati, Guwahati, India 781039

Abstract

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In this paper, the issue of output voltage regulation in buck type dc-dc converters is addressed using a current sensorless control technique. The proposed strategy integrates a finite time current observer with an adaptive backstepping control scheme to yield a cost-effective and robust control mechanism. The overall controller stability in the sense of Lyapunov is proved. Applicability of the proposed control is verified experimentally on a buck converter in the laboratory. The control scheme is implemented on dSPACE DS1103 platform based on DSP TM320F240 processor. To examine the efficacy of the proposed method, the buck converter is subjected to a wide change in input voltage, load resistance and reference voltage. For comparison purpose, a conventional adaptive backstepping control scheme is evaluated under identical conditions of experimental study to examine the merit of the proposed control. The results obtained reveal that the proposed control is prompt in rejecting perturbations and achieves a smooth, reliable and satisfactory output voltage regulation with faithful and time bound estimation of inductor current. Thereby, this investigation demonstrates the validity of the proposed control in maintaining a stringent output voltage regulation in buck converters.

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Keywords: buck converter, adaptive backstepping, current observer 1. Introduction

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Dc-dc power converters are employed to provide a regulated voltage supply to various static and dynamic loads. Their extensive use can be found in communication networks, automobiles, robotics, servomechanism, switched mode power supplies, medical instruments and uninterrupted power supplies. Dc-dc power converters are complex, non-smooth, nonlinear and time varying systems. They suffer from parametric and non-parametric uncertainties arising out of modeling errors, stray inductance, stray capacitance and unmodeled sensor dynamics. In addition, they are significantly susceptible to source voltage perturbations and load changes. Hence for counteracting such challenges, an efficient and robust control mechanism is needed to make adaptive, reliable, high-quality, lightweight and compact size power supplies. Control methods initially proposed were designed to linearize the converter around a specific operating point ∗

Corresponding author Email addresses: [email protected] (Tousif Khan Nizami), [email protected] (Arghya Chakravarty), [email protected] (Chitralekha Mahanta) 1 Tel: +91-7663864124, Fax:+91-361-2582542 Preprint submitted to Journal of The Franklin Institute

May 19, 2018

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using small-signal analysis, thereafter designing the controller by well matured linear control theory [1; 2]. Such conventional methods pose a serious problem of converter instability and inadequate performance under widely varying load and input voltage. Later, the research was focussed mainly on the design of nonlinear control techniques such as passive control [3; 4], flatness based control [5], predictive control [6], optimal control [7; 8] fuzzy logic control [9; 10], neural networks control [11], neuro-fuzzy control [12; 13], sliding mode control [14; 15] and switched control [16; 17]. In general the output voltage tracking problem in dc-dc converters can be classified under two types: voltage-mode control (VMC) and current-mode control (CMC) [18], based on the type of feedback error loops. The VMC dc-dc converter consists of a single feedback loop driven directly by the output voltage error. Although the VMC offers simplicity in implementation, yet disturbance at the input voltage cannot be detected unless it is reflected on the output. On the contrary, the CMC is a multiple loop feedback structure utilizing the inductor current in addition to the voltage error to produce a fast dynamical response. With respect to dc-dc buck converters, disturbances arising at the input voltage will initially result in change in the inductor current slope prior to its impact on the output. However, when this real-time information is fed to the controller through a current sensing mechanism, an almost constant volt-second balance is obtained by appropriate rectification of the pulse width modulation (PWM) signal operating the power semiconductor switch. This makes the dc-dc buck converter less sensitive to the input voltage perturbations [19]. In this way, the CMC inherently prevents the use of an extra input voltage sensor to curb the input fluctuations and subsequently the additional loop delay appearing due to the input voltage sensor is avoided. Furthermore, the CMC reduces the order of buck converter by one and is instrumental in providing automatic protection against over-current and short-circuit arising in the converter due to abrupt eventualities, thereby enhancing the converter’s reliability. Hence, it can be anticipated that for dc-dc buck converters, the control methodology involving a two loop feedback mechanism of output voltage and inductor current sensing has better suitability and offers multiple benefits while attaining the objective of output voltage regulation. In this direction, given the inherent features of backstepping control (BSC) [20; 21; 22], it is found to be suitable for dc-dc buck converter control operation. Since the state-space averaged model of the dc-dc buck converter can be represented in strict-feedback form, the BSC method can be directly applied to it. The systematic, step-by-step and recursive control Lyapunov function (CLF) based controller design framework guaranteeing asymptotic error convergence in each of the sub-systems, ease in physical realization and successful rejection of linearly parameterized matched and mismatched uncertainties make backstepping an appealing control technique. Moreover, the BSC design requires a full-state feedback involving both voltage and current sensing loops and hence meets the requirements of a reliable converter operation. However, for an accurate computation of the control law, it necessarily requires the exact state-space model of the converter. Furthermore, the unavailability of precise parametric knowledge leads to a stable, yet unsatisfactory transient response. Additionally, the load current disturbance in the BSC yields steady-state error in the output voltage. Motivated by such challenges present in the backstepping control method for dc-dc buck converter control, the first attempt made in this paper is to develop an appropriate robust control methodology based on backstepping technique with online adaptation mechanism for estimating uncertainties affecting the system.

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Next, in order to provide satisfactory tracking performance in dc-dc converters, most of the aforementioned modern control methods, including the backstepping control technique rely on the knowledge of inductor current. The current is generally measured using Hall-effect based current sensor. Nevertheless, such sensors used for measuring current, suffer from the following issues: 1) the remnant flux introduces a time varying dc bias into the control loop; 2) a high degree of sensitivity to the measurement noise results in an imprecise current measurement (although low-pass filters can be used, they add time delay and subsequently deteriorate the stability margin); 3) additional circuitry of the current sensor contributes to power loss; 4) high cost of current sensor substantially increases the overall cost of the controller; 5) current sensor when exposed to high temperatures and high current surges, faces potential threat of overheating eventually leading to the its failure. Other devices available to measure the current include resistive current sensor and current transformer (CT). The resistive current sensor demands a precise and noise-free differential amplifier for the effective action. Moreover, it increases power loss in the circuit. Hence, such methods are not viable for use in high power applications. In the contrary, the CT method of current sensing is popular in the industry, but the placement of CT in series increases the inductive path leading to high voltage spikes during switching transients. In addition, CTs need to be reset for the magnetizing inductance under every switching cycle, which may impose a restriction on the upper limit of the duty ratio in dc-dc converters. Taking into account, the aforementioned challenges, current sensorless control techniques are promising indeed in the sense that they provide a cleaner, inexpensive and noise-free current estimation, besides facilitating the compactness of the device and at same time avoiding problems of component aging and regular maintenance/calibration in regular current sensors. A brief background on current sensorless control schemes for dc-dc power converters, proposed in literature are discussed hereafter. A integral state reconstructor based methodology embedded in sliding mode control design technique for dc-dc converter systems is proposed in [23]. Although, the performance obtained is near satisfactory however the physical implementation is difficult [24]. Further in the same context, a duty cycle perturbation based method for improving the steady state errors influenced by parasitic elements are presented in [25]. The proposed control mechanism requires sensing of input voltage at regular intervals and hence requires additional voltage loop with a voltage sensing devices, hence not viable. Furthermore, a methodology involving sensorless control scheme with predictive current control (PCC) has been proposed in [26]. The method eliminates the disturbances arising in inductor current within one switching cycle. Nonetheless, the design hinders the real-time flexibility and faces difficulty in particular combination of PWM driven by current control mode. The work in [27] proposed an observer based control mechanism, taking into account, a more accurate model of the system. Implementation of such controllers involved high degree of computational complexity. In addition, to arrive at an accurate model describing converter dynamics, for better estimation of inductor current, parasitic parameters have been considered in modelling. However, an extra voltage sensor is required to measure the switch node voltage for finding the values of parasitic parameters of the system. It must be emphasized that estimation of parasitics depends on load conditions, which happens to be usually uncertain and time varying. Comparatively a less complex sensorless control strategy involving input voltage and feedforward current estimator is presented in [28]. The method reduces the influence of output voltage perturbations on the performance of current observer. Nevertheless, the error in current estimation is found to be comparatively large. In [29], adaptive voltage positioning with sensorless mechanism for dc-dc 3

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buck converter is proposed. The methodology offers ease in physical realization and depends merely on output voltage signal. Nonetheless, the works takes into account the assumption that equivalent resistance from input end to output end is a constant. Such an assumption is not very realistic since the equivalent resistance varies with the load disturbances which in turn adversely effects the output performance in real-time. Extended-Kalman-filter based observers [30] have also been used to estimate the inductor current in dc-dc converters to improve the estimation accuracy. In [31], the current estimation is found to be closer to the actual profile of inductor current due to consideration of possible parasitic parameters. However, the proposed method requires an auxiliary circuitry for switching voltage sampling. Further, the compensation of such parametric variation depends largely on the load conditions, which are otherwise unknown or difficult to estimate. Now, in the context of adaptive backstepping control design presented in this paper, the output voltage of buck converter is controlled by an appropriate current variable corresponding to an appropriate control law. The robustness of the output voltage under load uncertainties is ensured by choice of an appropriate inductor current variable acting as the virtual control input for the stabilization of load affected voltage dynamics, while generating the desired control signal. Therefore, an enhanced transient performance in the voltage profile at the instances of load changes can be ensured if the desired current variable converges to its appropriate current value, corresponding to the load change, in least possible time. Further, the output voltage performance depends on how promptly the controller responds to such uncertainties with the generation of a desired current profile, which thereby controls the voltage, to guarantee an asymptotically stable output voltage tracking. Hence, resorting to a current sensorless control necessitates the development of an observer which is accurate or rather exact and fast enough in estimating the inductor current signal. This would surely help in rendering a prompt controller action consequently yielding an efficient and robust voltage tracking with satisfactory transient and steady state behaviour. Therefore, the accuracy and fast convergence of current observer error are the two important aspects to be dealt with. This is due to the fact that if the observer does not estimate the current profile fast enough and is rather not accurate, it will subsequently result in degraded post load change transient performance of the output voltage with persisting steady state error. With this motivation, high gain observers with asymptotic stability emerges to be a good choice to achieve a close to accurate estimation within very short transients. However, there are some issues encountered in high gain observers with asymptotic stability which restrict their application to the concerned problem as enlisted below. 1. Peaking effect which may lead to overall closed loop system instability 2. No guarantee on exactness of estimation, that is the estimation accuracy is not significantly appreciable. 3. Noise amplification with high observer gains, which is highly undesirable as the signal to be estimated gets lost due to amplification of noise. Further, the current signal to be observed must be a slow varying signal, so as to allow the observer to converge asymptotically to yield a more closer value near to actual current state in sufficient time. This way, an estimation of a high frequency current signal can lead to steady state error, due to rise in the L2 bounds of the current error in the estimation. This leads to a an inaccurate control signal generation, which will subsequently result in output performance degradation. 4

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The existing works on current sensorless control design focus on the accuracy of current estimation and thereafter the design of a suitable controller. However, there exists no analytical proof justifying the correctness of estimation. Since most of the works are centred around the design of asymptotically stable (exponentially stable) current observers, there are no measures or closed form analytical expression to quantify the rapidity of estimation. In addition, very few works address the effect of parametric uncertainties in the estimation of inductor current. Design of a current observer guaranteeing an analytically exact estimation is promising in view of its potential in exactly recovering the output performance obtained using a full state feedback controller. Moreover, the finite time convergence of estimation would provide the designer with a extra design freedom over the fastness of current estimation opposed to asymptotically stable observers. Therefore, design and usage of a finite time current observer is attributed to exactness in the estimation, design freedom over convergence time of estimation, boundedness of estimation in presence of noise (very small bound) and no peaking effect. Based on a new ideology utilizing the structure of system dynamics and to mitigate the ill effects of measurement noise, a finite time current observer based adaptive backstepping control is proposed for output voltage regulation in DC-DC buck converters. As opposed to the design of a full state observer, herein the inductor current is considered to be a perturbation to the output voltage dynamics of the dc-dc buck converter. Thereafter, utilizing the idea behind disturbance observers, the inductor current is estimated using an observer designed from the first order output voltage dynamics only. Further, the observer is so designed that the resultant error dynamics converge to zero in finite time. Apart from addressing all the open issues of current sensorless based control design mentioned earlier, the proposed controller designed for dc-dc buck converter also ensures a robust exact and quick estimation of unknown inductor current irrespective of rapid unknown fluctuations in load current and input voltage. The theoretical results obtained are further analyzed and verified experimentally in this paper. Further, the superiority of the proposed current sensorless controller in achieving a good output transient, steady state performance and more importantly exact performance recovery is also reflected in simulation and the hardware results. Rest of the paper is organized as follows. Section 2 presents the buck converter modeling and the problem statement. The proposed adaptive backstepping control scheme and its overall stability analysis are discussed in Section 3. Experimental results and discussion are presented in Section 4. Finally, the conclusions are drawn in Section 5.

Remark 1. The importance of finite time current observer in the context of proposed control for buck converter is explained as follows. At the instances of load changes the voltage error is almost in a small vicinity of the origin and hence x1 and ξ1 are sufficiently close. On account of the same, due to negative homogeneity of the finite time observer the estimation occurs at a significantly faster time scale. That is to say, that if the degree of homogeneity of the observer is -1, the actual time scale, t of the voltage dynamics, is related to the observer time scale τ by the relation t = t0 + τ , where  is a very small number (0 ≤  ≤ 1) which characterizes the distance of the initial condition of the observer error variables from the origin. Therefore, we notice that if the voltage error is sufficiently close to the origin then  → 0 implies that the time scale t freezes at t0 , while the observer estimates in time scale τ . Hence, we achieve a very fast estimation of the desired current profile without peaking, due to negative homogeneity. Further, in the sequel, we have proved in the paper that the estimation is exact and negative 5

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homogeneity combined with the asymptotic stability (not exponential stability) implies a finite time stable origin.

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2. Buck Converter Modeling and Problem Statement

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Figure 1 shows the circuit diagram of the buck converter. It provides a reduced level of dc voltage at its load end. Here, vo represents the voltage across the load and iL is the inductor current. The load on the converter is modeled as an effective resistance R. The input dc voltage source is termed as E. A power electronic switch Sw is used for chopping the input supply and a power diode D is used for free-wheeling mechanism. A low pass filter LC smoothen vo and iL profiles. During the on mode of the switch Sw , the diode D does not conduct and input source E acts L

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Figure 1: Circuit Diagram of buck converter

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as a sole source of energy to the buck converter. During off mode of the switch, D resumes conduction ensuring flow of current through the inductor. Hence, the stored energy in the inductor drives the load resistance R during off mode of power switch Sw . Let x1 = vo and x2 = iL be the states of the buck dc-dc converter. Buck converter dynamics can be represented in state-space as x2 x1 + (1) x˙ 1 = − RC C x1 uE x˙ 2 = − + (2) L L where u ∈ Z+ = {0, 1} is the control signal denoting the opening and closing operation of switch Sw . The objective is to obtain a faithful tracking of output voltage vo , besides ensuring a satisfactory transient behavior. 3. Proposed Controller Design The proposed control integrates the finite time current observer with an adaptive backstepping control mechanism. The schematic diagram of the proposed control is shown in Figure 2.

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Figure 2: Proposed control

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3.1. Adaptive backstepping control Adaptive backstepping control design is a step-by-step procedure to derive the final control law with the stability ensured in the sense of Lyapunov [20][32]. The procedure of backstepping involves consideration of certain appropriate functions of system state to act as stabilizing virtual control inputs for lower order subsystems. Such a procedure is carried out recursively till the final control law is arrived at. Briefly, the steps involved are summarized below: Step 1: The term xR1 in (1) is assumed to be an unknown nonlinear function. It is estimated by framing an updation law to yield W ∗T Φ where W ∗ is the optimum weight and Φ is the regressor matrix. Replacing xR1 in (1) with W ∗T Φ, we get x˙ 1 = −

W ∗ T Φ x2 + C C

Therefore, (2) and (3) define the dynamics of buck converter. Step 2: The error variables are defined as  z1 = x1 − vr z2 = xˆC2 − α

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where xˆ2 is the estimated inductor current obtained using finite time current observer discussed in the next subsection and vr is the reference output voltage.

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Step 3: Virtual control input α is selected as α = −c1 z1 +

ˆ TΦ W − z2 + v˙ r C

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ˆ is the adaptive estimate of unknown where c1 > 0 is the controller gain and W ˜ = W∗ − W ˆ . Subsequently, the next error z2 can be rewritten as Moreover, W

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2 1 X X ξ˜˙2 x1 ∂α ∂α ˆ˙ T ∂α + E u − − x˙ k − vr (k+1) − W (k) ˆT C LC k=1 ∂xk ∂vr LC ∂W k=0

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where ξ˜2 = x˜2 = xˆ2 − x2 is the error in current state estimation. The user is guided about appropriate selection of control law u to stabilize z2 . Hence, u is found from (6) as ! 2 1 X X ∂α x1 ∂α ∂α LC ˆ˙ T (7) v (k+1) + W u= − c2 z2 − z1 + + x˙ k + (k) r ˆT E LC k=1 ∂xk ∂v r ∂ W k=0

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where c2 > 0 is also a controller gain. Step 4: Further, for finding the optimal weight required for estimation of the nonlinear uncertain term xR1 , an online Lyapunov based adaptive learning law is formulated to yield a ˆ (t) is given by close approximation. The optimum weight vector estimate W Z t Φ(x1 (ν))z1 (ν) ˆ (t) = W ˆ (t0 ) − γ dν (8) W C t0

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where γ > 0 is the adaption rate. In order to compute the control law u in (7) precisely, the exact knowledge of system states is essential. Hence, the proposed methodology proposes the control strategy with a minimal usage of sensing units. The inductor current state x2 is reconstructed from the finite time current observer. Remark 2: The choice of adaptation gain parameter γ is crucial in order to attain a fast and satisfactory dynamic response of the output voltage state of the converter. A high value of γ results in a faster adaptation and enhanced transient response. However, a very high value of γ also results in a decreased stability margin of dc-dc buck converter system. Therefore, it is recommended to judiciously prescribe the value of γ, in such a way that a desired and satisfactory control response is achieved, besides preserving a safe stability margin. 3.2. Finite time current observer This section presents a finite time convergence based estimation of current x2 . This would enhance output transient performance by reconstructing the inductor current under both nominal and perturbed situations. Hence, its applicability to the control design problem of buck converter is well suited. On the lines of [33], we design the current observer followed by the finite time stability analysis. Prior to the design, we first introduce the definition of finite time stability for a more clear understanding.

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Lemma 1. Let us consider a dynamical system given by x˙ = f (x) with x ∈ Dx ⊆ R. The system has a finite time stable origin if ∃ c > 0, µ ∈]0, 1[ and a Lyapunov function V : Dx → R+ satisfying V˙ (x) ≤ −c(V (x))µ , ∀x ∈ Dx [34].

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The above lemma is equivalent to the statement that if the origin of a dynamical system is finite time stable there exists a closed form explicit continuous function for the settling time to the origin. Having introduced the concept of finite time stability, the dynamics of proposed current observer for x2 is given as,  1 ˆT ξ˙1 = − λε1 |ξ1 − x1 | 2 sgn(ξ1 − x1 ) − WC Φ + ξC2   ˙ξ2 = − λ22 sgn(ξ2 − υ1 ) (9) 2ε  1 ξ  λ1 2 υ1 = − ε |ξ1 − x1 | 2 sgn(ξ1 − x1 ) + C

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where ξ1 and ξ2 denote the estimates of x1 and x2 respectively. The terms λ1 , λ2 are the observer gains and ε > 0 is a small number close to zero. In subsequent analysis, we shall find a bound on ε so as to achieve a finite time convergence of the observer error to the origin. Proceeding further, let us define ∆(x1 , t) = x2 , ξ˜1 = ξ1 − x1 and ξ˜2 = ξ2 − ∆(·). Hence, the error dynamics are written as, ) ˙ξ = − λ1 |ξ˜ | 21 sgn(ξ˜ ) + ξ˜ ˜ 1 1 1 2 ε (10) ˙ξ = − λ2 sgn(|ξ˜ | 21 sgn(ξ˜ )) + ∆(·) ˜ ˙ 1 2 1 2 2ε

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Next, the finite time stability of the observer error dynamics described in (9), is stated in Proposition 3.1 below.

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Proposition 3.1. Considering the finite time current observer error dynamics given by (10) and assuming that the disturbance ∆(x, t) is atleast once continuously differentiable, the resulting observer error variables ξ˜1 and ξ˜2 converge to the origin in finite time provided the gain λ2 ˙ ˆ = ξ2 . > sup{∆(.)} = L, L > 0 yielding ∆(·) 2ε2

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Proof: For analyzing the finite time stability of observer error dynamics in (10), the degree of homogeneity of the vector fields associated must be obtained first. Applying a homogeneity transformation Tr : (t, ξ˜1 ) 7−→ (rt, r3−i ξ˜1 ) to the observer error dynamics yields the degree of homogeneity of the associated vector fields to be −1 < 0. Therefore, as a next step, to ensure the finite time stability of the observer, let us consider a Lyapunov function V0 = ζ T P ζ, where, ζ := [ζ1 ζ2 ]T = [dξ˜1 c1/2 ξ˜2 ]T and dξ˜1 cν := |ξ˜1 |ν sgn(ξ˜1 ). The first time derivative of ζ is given by, " # " # λ1 ˜ 12 ˙ 1 ˜ −1/2 1 ˜ −1/2 ˜ ˜2 ) | ξ | (− d ξ c + ξ | ξ | ξ 1 1 1 ε ζ˙ = 2 1 ˙ = 2 λ ˙ − 2ε22 sgn(ξ˜2 − ξ˜˙2 ) + ∆(·) ξ˜2  1 −1/2 λ1  1 |ξ˜1 | (− ε dξ˜1 c 2 + ξ˜2 ) 2 (11) = 1 ˙ − λ22 sgn( λ1 dξ˜1 c 2 ) + ∆(·) 2ε

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− λε1 dξ˜1 c1/2 + ξ˜2 ˙ ξ˜1 ))dξ˜1 c1/2 −( λε22 dξ˜1 c1/2 − ∆(·)sgn( 2λ1 ε

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Thereby the first time derivative of V0 can be written as,

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V˙ 0 = |ξ˜1 |−1/2 ζ T (AT P + P A)ζ = −|ξ˜1 |−1/2 ζ T Qζ < 0

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It can be noted that (14) above is satisfactory because the matrix A is Hurwitz and the matrix P is a positive definite symmetric matrix satisfying the Lyapunov criterion given by AT P + P A = −Q with Q > 0. The matrix A is guaranteed to be Hurwitz if and only if the observer gain λ2 λ1 ˙ > sup(∆(.)) = L and > 0. Using Rayleigh principle, |ξ˜1 |1/2 ≤ dξ˜1 c1/2 ≤ kζk2 < 2 2ε ε −1/2 1/2 βmin (P )V0 and then V˙ 0 can be rewritten as,

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V˙ 0 ≤ −|ξ˜1 |−1/2 βmin (Q)kζk2 1/2

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βmin (P )βmin (Q) is the observer gain parameter and βmax (·) and βmin (·) βmax (P ) denote the maximum and minimum eigen values of a square matrix. Though the transformation ζ is continuous, it follows that ζ reduces to zero in finite time which means that the observer error variables ξ˜1 and ξ˜2 converge to the origin in finite time. From (16) it can be further solved to obtain an explicit expression describing the maximum finite time required for the estimation error to converge to the origin. Therefore, let us consider, V0 (t0 ) = V0 (0), and the final convergence time T satisfying V0 (T ) = 0 due to negative definiteness of V˙ 0 (t) and proceed where the term Γ =

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as follows.

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As per the condition, V0 (T ) = 0. Substituting the same in the above inequality, we get closed form expression for the convergence time T in terms of initial conditions and observer gains as defined below. 2V0 (0)1/2 Γ

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Therefore, equation(3) reflects that the relevant parameter influencing the finite time convergence is the bound on the convergence time given by T . In contrary, such closed form definition of the upper bound on the convergence time for any arbitrary initial condition is not possible in case of asymptotic (exponentially converging) observers.

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Henceforth, using Theorem 2.1 given in [34], the observer error dynamics are inferred to be finite time stable from (16). This completes the proof. 2. Next, to justify the exactness in the estimation of inductor current using the proposed current observer, the results have been summarized as Proposition 3.2.

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Proposition 3.2. The current observer given in (9) yields an exact estimation of the current ˙ exists. This means that provided that it is atleast once continuously differentiable, i.e sup{∆} at steady state, the exact ultimate bound on the observer error variables is given by ξ˜1 = 0 and ξ˜2 = 0.

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Proof: Following the procedure of finding the ultimate bounds presented in [36], we now consider the observer error dynamics described in (10) and using a specific diffeomorphism, error dynamics have to be transformed to a more suitable form to make the analysis convenient. Uti1 ¯ = 0D , where, Di = ∆(·), ˙ lizing the diffeomorphism [ψ1 , ψ2 ]T = [|ξ˜1 | 2 sgn(ξ˜1 ), εξ˜2 ]T and D ψ1 (ξ˜1 ) we obtain the transformed dynamics as, (       ) 0 ˜ ˙ ψ ( ξ ) ψ1 −λ1 1 ψ1 0 ¯ 1 = + ε2 D (18) −λ2 0 ψ2 1 ψ˙ 2 ε | {z } | {z } F

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analysis of current estimation error using the auxiliary dynamics in (19). Finally, the original estimation error can be obtained using the equation of ψ1 in a formal change of co-ordinates in (19). Next, using the concept of linear control theory and Lemma 1 mentioned in [36], the ¯ ∞ , where, ultimate bound on the estimation error is derived as |ψk | ≤ ε2 {F}k kDk   Z ∞ F1 F= = |eF τ G|dτ (20) F2 0

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¯ ∞ < K with the change of co-ordinates, the original estimation error can be obtained Since kDk as  1 1 2ε2 |ξ˜1 | 2 F1 K = |ξ˜1 | 2 sgn(ξ˜1 ) (21) 1 |ξ˜1 | 2 (2ε2 F1 K − 1) = 0 By solving the above equation, we find that if ε2 < 21 F1 K, the estimation error ξ˜1 exactly converges to zero. Similarly, the ultimate bound on the current estimation error ξ˜2 can be 1 found using the relation, ψ˜2 ≤ 2ε2 F2 K|ξ˜1 | 2 , implying ξ˜2 = 0. Therefore, it is proved that the finite time current observer (9) achieves an exact estimation of the current state. 2

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3.3. Stability Analysis The stability analysis under the Lyapunov stability criterion for the closed loop signal boundedness of the overall converter system under the action of the proposed control is investigated here. The result has been summarized in Proposition 3.3 as follows.

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Proposition 3.3. For the buck converter (2)-(3) affected by different uncertainties, the control law in (7) achieves an asymptotic output voltage tracking. The closed loop trajectories reside finally within a small maximal set S ∗ in the vicinity of the origin and are ultimately bounded in the set S, where S ∗ ⊆ S and are defined as S ∗ := {z ∈ R2 | kzk2 < (kξ˜˙2 /Ck∞ )/min{c1 , c2 }} ˜ T /Ck2 + kξ˜˙2 /Ck2 )}, where βmax (M ) denotes and S := {(z1 , z2 ) ∈ R2 |kzk22 ≤ βmax (M )(kW ∞ ∞ the maximum eigen value of the user defined matrix M ∈ R2×2 whose elements are given as a11 = 1/c21 , a12 = a21 = 1/c21 c2 , a22 = (c21 + 1)/c21 c22 and c1 , c2 > 0 are the controller gains.

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Proof: The asymptotic stability of the error variables z1 and z2 defined in (4) are proved here. The proof is carried out in a systematic manner by considering a stepwise procedure as discussed below. The controller error dynamics z1 and z2 in (4) can be rewritten by substituting the virtual control law α and the actual control input u from (5) and (7) respectively to yield, 

z˙1 z˙2





    ˜T   ˜˙ z1 1 W Φ 0 ξ2 −c1 1 + + = z2 0 1 C −1 −c2 C | {z } | {z } | {z } A

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Let us define z := [z1 z2 ]T and consider a positive definite continuously differentiable Lyapunov function V : R2 × [0, ∞) → R+ defined as (23)

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1 1 ˜T ˜ V = z T Pz + W W 2 2γ

where P is a positive definite symmetric matrix satisfying the Lyapunov criterion. Now, taking the time derivative of V (·) yields,

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˜ TΦ 1 ˜ T Φ T ˜ TW ˆ˙ 1 T W ξ˜˙2 1 T T 1W W 1 T T T ˙ + z PB2 + z A Pz + B2 Pz − V (z, t) = z PA + z PB1 2 2 C 2 C 2 2 C γ ˙ ˙ T T ˆ ˜ Φ ˜ W W ξ˜2 W 1 + z T PB2 − = (PA + AT P)z + z T PB1 2 C C γ 1 ≤ − z T Qz + kPB2 kkξ˜˙2 /Ck kzk (24) 2 where Q is positive definite symmetric matrix. For V˙ to be negative definite, kzk2 >

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˙ 2

ξ˜2 which implies that the trajectories firstly enter kzk2 ≤

and finally converge to λmin (Q) C ∞ 2kξ˜˙2 k∞ the set very close to the origin given by kzk2 < . However, the derived invariant λmin (Q)C set does not set the ultimate or actual bounds on the closed loop trajectories. Therefore, using Lemma 1 stated in [36], the actual bounds of z1 and z2 can be found. Formulating the tracking error dynamics as a linear time invariant uncertain system with bounded current estimation error derivative ξ˜˙2 and load estimation error as in (22), the bound can be found as, ˜ /Ck∞ + Fi2 kξ˜˙2 /Ck∞ , where i = 1, 2 and j = 1, 2. Further, Fij can be evaluated |zi | ≤ Fi1RkW ∞ Aτ as Fij = 0 {e Bj }i dτ , where Bj is the distribution matrix of the j th perturbation. Further, the matrix F is calculated as  1 1  c1 c1 c2 F= (25) 0 c12 Therefore, the actual bounds on the closed loop trajectories are found to be ) ˜ T /Ck∞ + 1 kξ˜˙2 /Ck∞ |z1 | ≤ c11 kW c1 c2 ˙ 1 ˜ |z2 | ≤ kξ2 /Ck∞

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c2

Hence, the actual bounding set in which the error variables z1 and z2 reside at steady state is given by S defined as ˜ T /Ck2 + kξ˜˙2 /Ck2 )} S := {(z1 , z2 ) ∈ R2 |kzk22 ≤ βmax (M )(kW ∞ ∞

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where βmax (M ) is the largest eigenvalue of the matrix M which is given by " 1 # 1 c21 1 c21 c2

c21 c2 1+c21 c21 c22

. 4. Results and Discussion 4.1. Simulation Results

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Figure 3: Simulated response curves for output voltage vo in buck converter under proposed control and adaptive backstepping control (ABSC) [38] during start-up (0-10V)

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In order to find the response of proposed sensorless finite time current observer based adaptive backstepping scheme, simulations have been performed on MATLAB/Simulink tool with a step size of 10µs. The obtained response have been compared with conventional adaptive backstepping control (ABSC) [38] scheme on fair comparison grounds with parameters mentioned in Table 1. The following tests are performed. Test 1: Start-up response (0-10V) The voltage tracking performance obtained by proposed control method and with ABSC [38] method during start-up is shown in Figure 3. Since the observer has replaced the current sensor, hence its performance is on par with conventional ABSC method. As it is clearly evident from 14

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Figure 3, both the methods yields almost similar start-up response for output voltage without yielding any peak overshoot. Similarly, during start-up the response obtained by the inductor current (measured through current sensor) is plotted in Figure 4. Test 2: Load resistance change.

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Figure 6: Simulated response curves for output voltage vo in buck converter under proposed control and adaptive backstepping control (ABSC) [38] during a step change in load resistance R from 10Ω to 20Ω.

During the load resistance perturbation from nominal 20Ω to 10Ω, the response obtained by proposed control and ABSC is shown in Figure 5. Similarly, during load resistance change from 10Ω to nominal 20Ω is also plotted in Figure 6. Meanwhile, the efficacy of proposed finite time current sensor is established by comparing the observed current with, measured current using a current sensor. The performance have been recorded in Figure 7. The strength of proposed observer is evident from its closer estimation of inductor current. During this test, the adaptive estimation of load resistance R is shown in Figure 8. Test 3: Input voltage change. In a similar way, the buck converter is subjected to sudden change in input voltage E from nominal 25V to 17V. The response obtained is plotted in Figure 9. Hence, it can be inferred that the proposed observer based control is found to be successful in rendering nominal performance closer to that of ABSC, but without using current sensor. 15

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Figure 8: Simulated response curves for estimated load resistance under proposed control for from 20Ω to 10 Ω and vice-versa.

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4.2. Experimental Results In order to investigate the performance of the proposed control scheme, a prototype of buck converter with a resistive load is made in the laboratory as shown in Figure 10. The experimental parameters are listed in Table 1. The buck converter consists of a power MOSFET IRF-P460 with rating 500V/18A, a power Diode D-6A4 MIC and a LV-25P isolated voltage transducer for measuring the output voltage. The control algorithm is computed on dSPACE Control Desk DS1103 with an embedded TM320F240 Digital Signal Processor. The control Desk DS1103 is comprised of internal data acquisition card CLP1103 with 16-bit resolution operating within ±10V voltage range for sampling of input signals. The Control panel is linked with the buck converter system through a computer of 4 GB RAM and Intel 3.10GHz processor. An external pulse width modulation (PWM) circuit including a gate driver IC IR2110 for signal amplification is used. Moreover, IC HPCL-2611 helps in isolating the control desk DS1103 from the converter. Thereafter the generated pulse pattern is fed to the gate terminal of MOSFET switch for necessary control action. To investigate the performance and robustness, the proposed control scheme is applied to the buck dc-dc converter system under different test conditions as described below. The proposed controller is also evaluated against conventional adaptive backstepping control (ABSC) procedure [38] under identical experimental conditions. The tests conducted are the following: Test 1: Step change in reference voltage from 0 − 10 V The transient performance of output voltage vo and inductor current iL in response to the 16

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Figure 9: Simulated response curves for output voltage vo in buck converter under proposed control and adaptive backstepping control (ABSC) [38] during sudden change in input voltage E from 25V to 17V at t=3s.

Figure 10: Experimental setup of buck dc dc converter

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reference voltage vr = 10 V is evaluated in Figure 11. Figure 11 (a) demonstrates the responses of converter state under ABSC control. The output voltage takes 28 ms to reach the desired reference, besides suffering from 2 V ripple in the steady state. The corresponding response of iL can also be observed in Figure 11 (a). In contrary, the proposed control shown in Figure 11 (b) provides a quick start-up within 15 ms and a clean output voltage profile with negligible ripple. However, the inductor current shows a peak in initial phase while reaching the nominal current of 0.5 A. Test 2: Sudden change in input voltage E from 25 V to 17 V and vice-versa. The effectiveness of the proposed control under a matched uncertainty is examined in this test. After the steady state is reached, the buck converter is exposed to a sudden source voltage change scenario. The input voltage E is suddenly perturbed from the nominal 25 V to 17 V and vice-versa, amounting to 32 % input voltage disturbance. The performances of vo and iL under ABSC in response to source voltage change are shown in Figure 11 (c). The result shows Table 1: Experimental parameters. System parameters Power Converter Rating, P Supply DC voltage, E Filter Inductance, L Inductor resistance, rL DC Capacitor, C Nominal Load Resistance, R Reference output voltage, Vr Switching frequency, fs Controller parameters Adaptive gain, γ Observer rate, λ1 Observer rate, λ2 Observer parameter, ε Backstepping gain, c1 Backstepping gain, c2

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Rating 120 W 25 V 59 mH 4.54 Ω 220 µF, 450 V 20 Ω 10V 20 kHz Value 9 × 10−5 0.05 0.5 0.05 6000 20

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Figure 11: Experimental response curves of buck converter system in CCM: (a) ABSC [38]: output voltage vo and inductor current iL during start up, (b) proposed: output voltage vo and inductor current iL during start up, (c) ABSC [38]: output voltage vo and inductor current iL during a step change in input voltage, E from 25 V to 17 V and vice-versa, (d) proposed: output voltage vo and inductor current iL during a step change in input voltage E from 25 V to 17 V and vice-versa.

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that during the input change from 25 V to 17 V , the conventional ABSC yields a large undershoot of 50 % and reaches 5 V level. The time taken to reject input disturbance is observed to be 70 ms. Similarly, the change of input voltage from 17 V to nominal 25 V produces a 50 % overshoot with a settling time of 70 ms while tracking the set 10 V reference voltage. Corresponding inductor current exhibits high undershoot and overshoot. On the other hand, the performance of the proposed control strategy in 11 (d) is satisfactory and yields no undershoot and overshoot. Subsequently, the inductor current exhibits a cleaner profile. Test 3: Sudden change in load resistance R from 20 Ω to 10 Ω and vice-versa. The robustness of the proposed control is next investigated under a widely varying load conditions. Figure 12 (a) and 12 (b) reveal the converter state response for loading test, under which R changes from 20 Ω to 10 Ω, amounting to 50 % change. The ABSC method produces an undershoot of 45 % in vo and reaches the level of 5.5 V . In addition, the time recorded to reject such a mismatched uncertainty is noted to be 85 ms. Similarly, during unloading test, R changes from 10 Ω to 20 Ω. The ABSC shows a high overshoot of 17 V , accounting to a 70 % peak and convergence time of 90 ms. Interestingly, under the proposed control scheme, the output voltage demonstrates a robust and accurate tracking of desired reference voltage. The 18

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Figure 12: Experimental response curves of buck converter system: (a)ABSC [38]: output voltage vo and inductor current iL during a step change in load resistance R from 20 Ω to 10 Ω and vice-versa, (b) proposed: output voltage vo and inductor current iL during a step change in load resistance R from 20 Ω to 10 Ω and vice-versa, (c) ABSC [38]: output voltage vo and inductor current iL during a step change in vr from 10 V to 15 V , (d) proposed: output voltage vo and inductor current iL during a step change in vr from 10 V to 15 V .

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response obtained in the proposed method is devoid of any overshoot and undershoot during both loading and unloading conditions. Subsequently, the inductor current response in also found to be satisfactory. Test 4: Reference voltage change vr from 10 V − 15 V . Lastly, to test the response speed of the proposed control algorithm, vr is suddenly changed from nominal 10 V to 15 . The transient response obtained with ABSC mechanism is shown in Figure 12 (c). Initially the ABSC response is observed to be faster. However, it takes nearly 200 ms time to converge to the new reference voltage of 15 V . In contrary, the proposed control yields a rapid response of vo in 5 ms. The inductor current dynamics show an overshoot during the trajectory change. The experiments investigated have shown that the proposed control strategy is capable to provide a strict output voltage regulation for a wide range of perturbations under both matched and mismatched conditions. The spikes produced in the inductor current profile during reference voltage change are a result of dependency of error variables z1 and z2 on the derivative of current estimation error ξ˜2 and the same may be tolerated by slight increase in the power 19

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handling capability of the converter. 5. Conclusion

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A novel control structure that enables a stable and robust trajectory tracking of the output voltage in buck converters for a wide varying source voltage and load resistance change is proposed in this work. The proposed control utilizes the adaptive backstepping procedure, besides observing the inductor current using a finite time observer. The finite time convergence of inductor current is validated mathematically. Stability analysis of the overall closed loop buck converter system is established. Experimental investigation is conducted under widely varying input, load and reference voltage changes. The results are evaluated against the conventional adaptive backstepping control method. The results shown indicate that the proposed control scheme is successful in estimating the unknown current which promises potential for realizing a current sensorless controller. References

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