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International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ Reference Signal Limiter in the Control of Buck Power Converters with Improved Stability Prasanth Sai1, M. Narendra Kumar2, S. Sunisith3 [email protected], [email protected], [email protected] Abstract—This paper presents the solution for controlling the dc-dc switched power of Buck type converter. This technique states that the control signal can take the values in the range (0, 1) only. Such a control limitation is generally not considered into account when designing the converter regulators. This is only dealt with in the control implementation stage, placing a limiter (controls the range of input signal) between the controller and the controlled system. In this, the control signal limitation is provided by using a non-linear regulator that consists of an internal limiter. The final closed-loop control system is nothing but the non-linear feedback involving a linear dynamics block in closed-loop with a non-linear static element. If these conditions are satisfied when choosing the control design parameters then the regulator improves closed-loop stability and output reference tracking. tracking. However, in these studies it was never accounted for the limitation of the control signal. As a matter of fact, the control is not allowed to be outside the interval (0, 1), due to the technological nature of the controlled circuits. Therefore, the usual practice consists, when it comes to implementing the controllers, in simply placing an isolated limiter between the designed controller and the controlled system (Fig. 1.1). Keywords: Buck Converters, Reference Signal Limiter, Switched Power Converter. Fig 1.1 Basic control setup of Buck converters I. INTRODUCTION There are three main types of switched power converters, namely boost, buck, and buck–boost. These have recently received an increasing deal of interest both in power electronics works and in automatic control applications. This is due to their wide applicability domain, e.g., domestic equipments, communication systems, computers, industrial electronics, battery-operating, embedded equipments, uninterruptible power sources. From an automatic control viewpoint, a switched power converter constitutes a challenging case study as it is variable-structure and nonlinear. Its rapid structure variation is generally coped with using averaged models. Based on these average models, different nonlinear controllers have been developed using Passivity techniques, feedback linearization, flatness methods, sliding mode control and back stepping control technique. In all works the proposed controllers are designed to achieve closed-loop global stability and voltage reference Unfortunately, the nonlinear effect of such a limiter is never taken into account when analyzing the closed-loop system. As a consequence, the aforementioned stability results may lose their global nature, the controller transient performances may deteriorate and the output-reference tracking objective may not be achieved. On the other hand, the problem of controlling linear systems subject to input saturation constraint has received a great deal of interest over the last two decades. During the time intervals of control saturation, the closed-loop system is no longer linear and the controlled system is steered in open loop. Then, global asymptotic stabilization is only possible for stable systems. The problem of output-reference (in presence of control input limitation) has not been so deeply investigated. When the controlled system is of type-1 and the reference signal is step-like, then the tracking problem can be transformed into a disturbance-free regulation problem and existing solutions can be applied. For non-type-1 systems, an integrator should be incorporated in the regulator to make the tracking objective achievable. But, the presence of an integrator 1815 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ generally results in large control actions which, due to the control limitation, may lead to undesirable oscillatory behavior. To avoid such behavior, many authors have proposed linear regulators together with ad-hoc “anti-windup” devices. Our approach consists in first designing a linear control law that achieves the tracking objective in the absence of control limitation. Then, an adequate anti-windup device is incorporated, in the above control law, leading to a nonlinear regulator. It is shown that the resulting closed-loop control system is equivalent to a nonlinear feedback consisting of a linear dynamics block in closed-loop with a nonlinear static element. Sufficient conditions for -stability of this feedback are then established using tools from the absolute stability theory (circle criterion, Barbalat’s lemma). The main condition is the real positivity (RP) of a specific transfer function that involves, on one hand, the poles of the controlled system and, on the other hand, those of the underlying linear closed-loop system. In fact, the RP condition defines a neighborhood of the controlled system poles in which should be assigned those of the closed loop. Such a requirement can always be satisfied through an adequate choice of the controller parameters. Finally, it is shown that if the reference signal is slowly varying then the proposed nonlinear controller stops saturating after finite transient-periods and the system output tracks asymptotically its reference. consists of two parts that is D1TS and D2TS (D1+D2=1). During D1TS, inductor current increases linearly and then in D2TS it ramps down that is decreases linearly. 2.1.2 Discontinuous Conduction Mode: When the inductor current has an interval of time staying at zero with no charge and discharge then it is said to be working in Discontinuous Conduction Mode (DCM) operation and the waveform of inductor current is illustrated. At lighter load currents, converter operates in DCM. The regulated output voltage in DCM does not have a linear relationship with the input voltage as in CCM. In DCM, each switching cycle is divided into of three parts that is D1TS , D2TS and D3TS (D1+D2+D3=1). During the third mode i.e. in D3TS, inductor current stays at zero. II. CONVERTER MODELING A Buck converter is constituted of power electronic components connected together, as shown in Fig. 2.1. Figure 2.2: Inductor current waveform of PWM converter (a) CCM (b) boundary of CCM &DCM (c) DCM Fig 2.1 Buck converter circuit 2.1. Modes of Operation: The operation of dc-dc converters can be classified by the continuity of inductor current flow. So dc-dc converter has two different modes of operation that are (a) Continuous conduction mode (CCM) and (b) Discontinuous conduction mode (DCM). A converter can be design in any mode of operation according to the requirement. 2.1.1 Continuous Conduction Mode: When the inductor current flow is continuous of charge and discharge during a switching period, it is called Continuous Conduction Mode (CCM) of operation shown in figure 2.1 The converter operating in CCM delivers larger current than in DCM. In CCM, each switching cycle T S The switching converters convert one level of electrical voltage into another level by switching action. They are popular because of their smaller size and efficiency compared to the linear regulators. Dc-dc converters have a very large application area. These are used extensively in personal computers, computer peripherals, and adapters of consumer electronic devices to provide dc voltages. The wide variety of circuit topology ranges from single transistor buck, boost and buck-boost converters to complex configurations comprising two or four devices and employing soft-switching or resonant techniques to control the switching losses. The non-isolated dc/dc converters can be classified as follows: • Buck converter (step down dc-dc converter), • Boost converter (step up dc-dc converter), •Buck-Boost converter (step up-down dc-dc converter, opposite polarity), and •Cuk converter (step up-down dc-dc converter). 1816 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ 2.2 A Pulse-Width-Modulation Based Sliding Mode Controller for Buck Converters: Sliding mode (SM) controllers are well known for their robustness and stability. Most of the previously proposed SM controllers for switching power converters are hysteresis modulation (HM) based. Naturally, they inherit the typical disadvantages of having variable switching- frequency operation and being highly control sensitive to noise. Possible solutions are to incorporate constant timer circuits into the hysteretic SM controller to ensure constant switching frequency operation or to use adaptive hysteresis hand that varies with parameter changes to control and fixate the switching frequency. However, these solutions require additional components and are unattractive for low cost voltage conversion applications. An alternative solution to this is to change the modulation method of the SM controllers from HM to pulse-width modulation (PWM). The idea is based on the assumption that at a high switching frequency, the control action of a sliding mode controller is equivalent to the duty cycle control action of a PWM controller. Hence, the migration of a sliding mode controller from being HM based to PWM based is made possible. III. DC-DC BUCK CONVERTER The operation of the tri-state dc-dc buck converter with hysteretic current-mode control scheme is discussed in this subsection. Figure 3.6 shows the tri-state buck converter topology. It consists of two controlled switches S1 and S2, an uncontrolled switch D , an inductor L and a capacitor C, a load resistance R . Switch S2 is the additional switch which is connected across the inductor. The operation of the tri-state converter includes three different configuration or structures that are show in figure. At the start of the clock period, the switch S1 is turned on and the switch S2 is turned off. During this interval (mode 1), inductor current increases with a slope of (vin-v0/L) When iL reaches a peak value (upper bound), S1 turns off. Then iL starts falling with a slope of (-v0/L) until it reaches some lower threshold. This interval is denoted as mode 2. During this interval, diode is forward biased and both switch S1 and S2 are turned off When the inductor current reaches lower threshold, it stays constant at lower boundary, because the switch S2 shorts the inductor L and voltage across the voltage across the inductor is thus equal to zero. During this interval S2 is turned on while S1 and diode are off. This is the additional interval, denoted as mode 3. The inductor current waveform showing the switch conditions for a tri-state buck converter is shown in figure 3.3. Fig. 3.1. A tri-state buck converter configuration Fig. 3.2. Equivalent circuits under different modes of operation These three modes of operation can be described as follows: Mode 1: when S1 is on and, S2 is off, the state space equation of buck converter is derived as ---(3.1) Where x=[v0 iL] , v0 is the output voltage , iL is the inductor current. Mode 2: when S1 and S2 is off , the equation derived as, T ..(3.2) Mode 3: when S1 is off and S2 is on the state-space equation is …(3.3) 1817 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ Where iL represent the inductor current, v0 and vref represent the output voltage and reference voltage respectively. Here the switching state of the switch is either 1 or 0. Then by taking the derivative of (3.4) with respect to time, Fig. 3.3. Inductor current waveform of tri-state buck converter showing the switch conditions 3.1 Mathematical Analysis of Proposed Controller: The operation of a hysteretic current-mode controller for tri-state dc-dc buck converter is proposed and the schematic diagram of proposed controller is shown figure 3.4. …(3.5) Considering the buck converter when the switch S1 is on, S2 is off. …(3.6) …(3.7) Substituting these equations we get the results as shown below: Fig 3.4. Schematic diagram of the hysteretic controller for tri-state buck converter The digital logic blocks generates required switching pulses for controlling the Switches S1 and S2 .This block consists of two SR flip-fops and some logic gates that can be shown in figure 3.5. …(3.8) The dynamics of the converter circuit, when S1 and S2 are off, can be expressed as, …(3.9) Substituting these equations we get the results as: Figure 3.5. Schematic diagram of pulse generator circuit The state-space description of the system in terms of the desired control variables (i.e, voltage, current etc) is developed. The proposed current controller employs both the output voltage error x1 and the inductor current x2 as the controlled state variables, which are expressed as …(3.4) …(3.10) The dynamics of the converter circuit in mode 3, when S2 is on, S1 is off can be expressed as, …(3.11) Since in this mode of operation, inductor current stays at a constant value, so we get the derivative of a constant value is zero. 1818 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ …(3.12) Then, the state-space modeling of the converter circuit with state variables x1 and x2 is given by, (3.13) Where x=[x1 x2] is the state vector and A‟ s and B‟ s are the system matrices. The state matrices and the input vectors for the three periods are, T As studied from the previous discussion that the basic principles of a hysteresis control is based on the two hysteresis bands (upper and lower bands), whereby the controller turns the switch on when the output current falls below the lower band and turns the switch off when output is beyond the upper bound. The switching action can be determined in the following way: 1. If iL< lower bound, u=1(ON) 2. If iL> upper bound, u=0(OFF) , where u is the control input. In this case, the hysteretic current controller shown in figure 3.2 consists of two control loops. One is the current control loop and the other is the voltage control loop. The difference between the actual output voltage and reference voltage generates the error voltage. A controller may a P or PI type use the voltage error signal to provide a control voltage which is consider as the upper boundary or the upper threshold vhys+ Here a simple proportional controller is considered for easy analysis. The control voltage vcon is a function of the output voltage v0 and a reference signal vref in the form, Hence, there are two switching actions u1 and u2 are occurring on the two switching surfaces in the region of phase plane. The control law determines the switching action. This can be represented mathematically as, …(3.15) Where, h1 and h2 are the two switching surfaces (manifolds) in sliding mode phase plane that can be expressed as the combination of state variables, ..(3.16) Where, x1 and x2 are the state variables, k is a constant factor that transforms the current signal to voltage signal, kp is the proportional gain, which is known as sliding coefficient. In these equations u1 and u2 can be known as the discrete control inputs to the system and this equations forms the basis of control law for the hysteretic control system. Generally a SM control design approach consists of two components. The first involves the design of a sliding surface and the second is concerned with the selection of a control law that directs the system trajectory towards system stable operating point. If we assume the width of the hysteresis band Δ→0, then there exists only a single switching surface h1 in phase plane. And for sliding mode to be exist on the switching surface, then h1=0. This is said to be as ideal SM operation. Hence for ideal SM operation dh1/dt= h1=0. Also the equivalent control produces trajectory whose motion is exactly equivalent to the trajectory motion in an ideal SM operation. In equivalent control, the discrete control input is substituted by continuous equivalent control signal ueq and is obtained by solving the equations: …(3.17) …(3.14) where, kp is the gain of proportional controller. The lower threshold vhys- , is determined by subtracting a small constant term Δ , which is given as, vhys- = vhys+ - Δ. 1819 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ Then substituting the values, we get converters can be found out by considering both the switching surfaces for analysis. From these equations we can find out the values of control parameters such that it will satisfy the existence condition. IV. RESULTS …(3.18) 4.1 The Tracking behavior of the controller in presence of a time varying output reference signal y*(t) switching between 9 and 15 V: …(3.19) From above equation ueq can be found as …(3.20) Where ueq is a smooth function of the discrete input function u and bounded by 0 and 1. Then by substituting the values into 0< ueq <1, the function can be expressed as, (3.21) Multiplying the above equation by vin, (3.22) The SM existence region provides a range of sliding area where the sliding motion takes place. The condition for the existence of sliding mode is achieved, if the following inequalities are satisfied. Fig. 4.1. Tracking behavior of the controller in presence of a time varying output reference signal 4.2 Comparison between the computed duty ratio v(t) and the applied value μ(t) in presence of a time-varying input reference: ….(3.23) (3.23) By solving the above equations, we get, …(3.24) The above two equations represents two lines in phase plane with same slope. As there are two switching surfaces, so the SM existence region for hysteresis controlled Fig. 4.2. Comparison between the computed duty ratio v(t) and the applied value μ(t) 1820 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability International Electrical Engineering Journal (IEEJ) Vol. 6 (2015) No.3, pp. 1815-1821 ISSN 2078-2365 http://www.ieejournal.com/ 4.3 Robustness of the controller with respect to load resistance changes: Fig. 4.3. Robustness of the controller with respect to load resistance 4.4 Comparison between the computed duty ratio v(t) and the applied value μ(t) in presence of load resistance changes: Fig. 4.4. Comparison between the computed duty ratio v(t) and the applied value μ(t) in presence of load resistance 4.5. Real part of c(jῳ)/A(jῳ) versus: Fig. 4.5. Real part of c(jῳ)/A(jῳ) versus V. CONCLUSION The control limitation is important to all power converters operating according to the PWM principle, whatever the control design approach. In the present paper, such issue has been dealt with, for converters of the Buck type, using the nonlinear controller. It contrasts with previous solutions that consist in first designing a controller as if there was no control limitation and then incorporating an isolated limiter, placed between the controller and the controlled system. The point is that the effect of such a limiter is also ignored, in previous works, when analyzing the closed-loop control system. VI. REFERENCES [1] S. Sunisith, K. S. Mann, K. Janardhan Rao, “Effective algorithm for reducing DC link neutral point voltage and total harmonic distortion” IEEJ, Vol.5 (2014) Issue No.11, ISSN: 2078-2365, pp 1613-1618. [2] S. Sunisith, Lizi Joseph, M. Saritha, “Comparision of fuzzy PID controller with conventional PID controller in controlling the speed of a brushless dc motor” IEEJ, Vol.5 (2014) Issue No. 11, ISSN: 2078-2365, pp 1605-1612. [3] H. Sira-Ramirez and R. Silva-Ortigoza, Control Design Techniques in Power Electronics Devices. New York: Springer, 2006. [4] H. Sira-Ramirez, R. Perez-Moreno, R. Ortega, and M. Garcia-Esteban, “Passivity-based controllers for the stabilization of DC-to-DC power converters,” Automatica, vol. 33, pp. 499–513, 1997. [5] S. C. Tan, Y. M. Lai, M. K. H. Cheung, and C. K. Tse, “A pulse-width modulation based sliding mode controller for buck converters,” in IEEE Power Electron. Specialists Conf. (PESC), 2004, vol. 5, pp. 3647–3653. [6] V. M. Nguyen and C. Q. Lee, “Indirect implementation of sliding mode control law in buck-type converters,” in IEEE Appl. Power Electron. Conf. and Expo. (APEC), 1990, vol. 1, pp. 111–115. [7] A. Gensior, O.Woywode, J. Rudolph, and H. Güldner, “On differential flatness, trajectory planning, observers, and stabilization for DC-DC converters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 53, pp. 2000–2010, 2006. [8] H. El Fadil, F. Giri, F. Z. Chaoui, M. Haloua, and H. Ouadi, “Nonlinear and adaptive control of buck power converters,” in IEEE Conf. on Decision and Control, Maui, HI, 2003, pp. 4475–4480. 1821 Prasanth et. al., Reference Signal Limiter in the control of Buck Power Converters with Improved Stability