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Controller design of DC/DC buck converter based on feedback linearization with integrator Miklós Csizmadia Miklós Kuczmann Department of Automation, Research Centre for Vehicle Industry Széchenyi István University Győr, Hungary Email: [email protected] Department of Automation, Research Centre for Vehicle Industry Széchenyi István University Győr, Hungary Email: [email protected] Abstract—The paper presents the feedback linearization technique of voltage mode DC/DC buck converter. The nonlinear state-space model is presented and has been stabilized by pole placement technique and finally the voltage control of output has been realized by integral control. The designed controller has been simulated by MATLAB. Results are shown that the developed controller has good dynamic and static behavior. I. II. S TATE S PACE M ODEL OF B UCK C ONVERTER The simplest form of SMPS (Switched-mode power supply) is the buck converter which can step down the voltage. The circuit diagram of a buck converter is shown in Figure 1, where SW and D are representing the power electronics switch and the diode, respectively. The input DC voltage is presented by Vi , the load resistance is shown by R, and the output voltage is UR . I NTRODUCTION An integral part of electric systems are DC/DC converters which and widely used in lot of applications. These circuits are provides regulated DC voltage in the output(s) at different levels. Nowadays most power electronics system are based on GaN (Gallium nitride) or SiC (Silicon carbide) semiconductors. Their positive advantages and properties: lower switching power loss, lower on-state resistance, greater bandwidth, which to allow to apply the high switching frequency . At the same time the high switching frequency delivers many positive advantages (smaller switching loss, reducing size and volume of circuits (through the reactive elements) etc.) but influence the stability of the system (bifurcation and chaos), therefore it is need to take into account the nonlinear behavior. The current research are focused on the low-frequency region (some hundred kilohertz) [2, 3]. Due to the switching frequency of system are increasing (up to megahertz region) the investigation between switching frequency and systems properties (e.g stability) is needed. The converters are nonlinear and time-variant system which can be derived from variation of load value and input voltage [4, 5]: For control purposes, many linearized models have been developed that facilitate the designing procedure of DC/DC converter. Widely used for this aim the State Space Averaging method (SSA). The goal of this method obtain the small signal function of the DC/DC converter (linearization around equilibrium point). There are some limitations of this technique: (i) the linearized model is valid just for the half of the switching period, (ii) the chaotic behavior or subharmonic behavior can not handle . At the same time the nonlinear controllers shows good results: sliding mode [7, 8] or backstepping  etc. The presented controller in  use feedback linerization (small overshot in output voltage and inductor current) but not contain an integrator. Fig. 1: Circuit diagram of buck converter without parasitic elements In this case, the state variables of the system are the voltage across the capacitor (vC ) and the current through the inductor (iL ). In CCM (Continuous Conduction Mode), it can be distinguished two operating modes of DC/DC buck converter depending on the SW operation: • on state (0 < t < DT ) and • off state (DT < t < T). The first mode equivalent circuit is shown in Fig. 2: Accordingly the state space equations can be express as: 1 u̇C = − RC uC + i̇L = − L1 uC + In matrix form: " # " iL 0 d = 1 u C dt C − L1 1 − RC 1 C iL , 1 L Vi . # "1# iL + L Vi . uC 0 (1) #" (2) The another operating mode during the switch (SW) is not conducting (off), the equivalent circuit (Fig. 3): III. E XACT L INEARIZATION AND CONTROLLER DESIGN APPLYING EXACT FEEDBACK LINEARIZATION The control is said a Static State Feedback Control if it is depending on the state of x (state variable) and external reference signal (v). The input of SISO (Single Input Single Output) system in case of static state feedback can be written : u = α(x) + β(x)v, (9) where v is the external reference input (see Fig. 5). Furthermore, the non-linear system can be represented in the following form : Fig. 2: Buck converter equivalent circuit - on state ẋ = f (x) + g(x)u, y = h(x). (10) As indicated above, the closed loop function : ẋ = f (x) + g(x)α(x) + g(x)β(x)v, y = h(x). Fig. 3: Buck converter equivalent circuit - off state Fig. 5: Static state feedback control  The state space equations are: 1 u̇C = − RC uC + 1 C iL , (3) i̇L = − L1 uC . In matrix form: " # " 0 d iL = 1 u dt C C − L1 1 − RC The goal is to determine α and β which characterize the control, i.e. convert the nonlinear system into linear one. The output voltage - controlled quantity- can be defined as [11, 12]: y = h(x) = uC . (12) #" # " # iL 0 + Vi . uC 0 (4) The output function need to differentiate by time until the input function appears in expression [11, 12]: ḣ = u̇C = Averaging (1) and (2) by duty cycle (D), can be written as: " # " 0 d iL = 1 u dt C C − L1 1 − RC # "D# iL + L Vi . uC 0 1 1 iL − uC , C RC (5) 1 1 1 1 uC + Vi − + 2 2 uC . (14) LC LC RC 2 R C It can be see that the second derivative (14) contains the input function. Accordingly, the results in Lie algebra framework : ḧ = − ḣ = Lf h, ḧ = L2f h + Lg Lf hu. where f (x) = 1 − RC uC C iL − L1 uC g(x) = (13) #" The state space model can be written into a general matrix form: ẋ = f (x) + g(x)u, (6) "1 (11) 0 Vi L # , (7) , (8) The block diagram of control system are shown in Fig. 4 (see next page). (15) To apply a feedback linearization method to buck converter, firstly need to check the rank of matrix, which can be done easily via Matlab . The rank of matrix is 2 which equal the system order, therefore the system is controllable. Using the following coordinate transformation: " # h(x) uC z = T (x) = = 1 , 1 Lf h(x) C iL − RC uC (16) Fig. 4: Input-output feedback linerization pole placement and integral conrol of buck converter where z = T (x): z1 = h1 , z2 =Lf h1 i.e. For dynamic testing two test has been applied: z˙1 = z2 , z˙2 = L2f h + Lg Lf hu = v. (17) Accordingly, considering new state-variable T , system model will be linear as: 0 1 0 Ṫ = T+ v. (18) 0 0 1 • output load test (R) and • input voltage test. (Vi ). The dynamic behavior of controller was investigated at steady state, which are shown in Fig. 9. and Fig. 10, respectively. The results are clearly shows that the controller is invariant to the changing of the input voltage and has good transient behavior to load variations. In (18), v is a new control input. So, the controller of nonlinear system (3) is the following: u = α(x) + β(x)v, (19) where L2 h α(x) = − Lg Lf f h = − V1i uC − β(x) = 1 Lg Lf h IV. = L Vi RC iL + L V iR2 C uC , LC Vi . (20) L INEAR CONTROLLER DESIGN There are several ways to realize a linear regulator, e.g. PI, PID regulator, pole placement, linear quadratic regulator etc . In this paper the pole placement with integrator has been used. All the original poles have been moved to -0.25, as results the obtained gain matrix is K = [k1 k2 kI ]T = [0.1875 0.75 0.0156]T (in coherent unit system). V. S IMULATION RESULTS To investigate the properties of controller DC/DC buck converter has been designed with following parameters: Vi =52V, Vo =12V, Io =4.8A, R=2.5Ω, L=50µH, C=50µF. The  contains in detail the design steps and procedures of DC/DC buck converter. For better numerical handling coherent unit system is used. The start-up simulation shows that the inductor current (IL ) is very high which may be related to the value of the output capacity. To investigate this, the simulation was run with two different C values which are shown in Fig. 6 and Fig. 7. respectively. Besides that two solvers were compared (Fig.8.). It can be see the type of solver is affects the shape of current. Fig. 6: State variables of the converter in start-up (C=50µF ) VI. C ONCLUSIONS This article is describe the exact feedback linearization with integrator. The simulation results shows the extra integrator means also better transient behavior. At the same time, the inductor current is very high at start-up which can be depend on: the presence of parasitic elements and/or the type of numerical solver. These detailed examination is needed in the future. Another future plan is to realize the controller by microcontroller and investigate the possibility of applying wide bandgap semiconductors (SiC or GaN) in buck converter. ACKNOWLEDGMENT The research presented in this paper was carried out as part of the “Dynamics and Control of Autonomous Vehicles meet- Fig. 7: State variables of the converter in start-up (C=25µF ) Fig. 9: Output resistance test at steady state Fig. 8: Comparison of Runge-Kutta and Euler solver (C=50µF ) Fig. 10: Input voltage test at steady state  ing the Synergy Demands of Automated Transport Systems (EFOP-3.6.2-16-2017-00016)” project in the framework of the New Széchenyi Plan. The completion of this project is funded by the European Union and co-financed by the European Social Fund. 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