Robust Control Applied to Improve the Performance of a Buck-Boost Converter

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1 Robust Control Applied to Improve the Performance of a Buck-Boost Converter WILMAR HERNANDEZ Universidad Politecnica de Madrid EUIT de Telecomunicacion Department of Circuits and Systems Ctra. Valencia km 7, Madrid SPAIN whernan@ics.upm.es Abstract: In this paper, H loop-shaping control is applied to improve the performance of a buckboost dc-dc power converter based on pulse-width-modulation (PWM) techniques. Here, classical control techniques (i.e., proportional-integral-derivative (PID) control) and post-modern control techniques (i.e., H control), are used to design the feedback loop of a buck-boost dc-dc power converter. The results of the experiment are satisfactory and show that robust controllers do not depend strongly on the operating point, and that H loop-shaping control performs better than PID control. Key Words: buck-boost converter, state-space averaging procedure, robust stability, PID control, H loop-shaping control, disturbance rejection 1 Introduction In the last two decades the classical approach to designing power converters has been transformed into the one based on the application of linear and nonlinear control techniques to improve the performance of these converters. In 1, a nonlinear control algorithm was used to reduce the sensitivity of the control-to-output transfer function of a boost converter to the nature and magnitude of resistive loads. Also, in 2 the most simple form of the general H algorithm 3 5 was applied to reduce the sensitivity of a boost converter to disturbances in the input voltage of the power converter and in its output load. What is more, Naim et al. 2 showed that dc-dc power converters whose transfer functions have right-half-plane (RHP) zeros can be controlled satisfactorily by using H optimal control. In 6, μ synthesis with DK-iteration was applied to design a robust voltage controller for a buck-boost converter with current mode control; and the nonlinear H control theory 7 has also been applied satisfactorily to improve the performance of a Ćuk converter, see 8. Another application of μ synthesis with DKiteration to improve the performance of a boost converter can be found in 9, and another application of the standard H control problem to improve the performance of both a boost converter and a buck-boost converter can be found in 10. In 11 a linear quadratic regulator (LQR) combined with a linear state estimator was used to improve the time domain performance of a Ćuk converter, and an application of the linearquadratic-gaussian control/loop-transfer-recovery procedure (LQG/LTR) to designing a robust controller for a series parallel resonant dc-dc converter can be found in 12. In the present paper a robust controller for a buck-boost dc-dc power converter by using H loop-shaping control was designed 3 5, 13, 14. Moreover, in order to carry out a comparison between classical control and post-modern control, here the H loop-shaping controller is compared with a robust proportional-integralderivative (PID) controller. The results are satisfactory and show the importance of using robust controllers when designing the feedback-control loop of power converters. 2 The Buck-Boost Converter In this paper, the state-space averaging procedure 15, 16 has been employed for the determination of the transfer function from the duty cycle to the output voltage. Fig. 1 shows the low-power buck-boost converter used in this paper. Here, ISBN: ISSN

2 the switching frequency is 250 khz, the nominal input voltage is 12 V and the output voltage is -12 V. It is assumed for simplicity that the converter is operated so that either the transistor or the diode is always conducting; this assumption rules out the discontinuous conduction mode, in which both devices are off during some part of each cycle. D = Rr C DV in r L (R + r C )+Rr C (1 D)+R 2 (1 D) 2 Here, D represents the DC component of the duty cycle, i L (t) is the inductor current, v C (t) is the capacitor voltage, and I L and V C represent the DC components of the inductor current and the capacitor voltage, respectively. Also, r L and r C represent the effect of the equivalent series resistance of the inductor and the capacitor, respectively, and V in is the DC component of the input voltage. Therefore, in accordance with 2, if it is assumed for a moment that r L =0,then(4) shows the transfer function from the duty cycle to the output voltage. v o (s) d(s) = V in LC (s LD R(1 D) 2 1)(sCr C +1) s 2 + s L+Rr CC(1 D) LCR + (1 D)2 LC (4) Figure 1: The buck-boost converter of this paper Then, with the assumption of real components, (1) and (2) show the finite dimensional linear time invariant dynamical model of the system, where x(t) is the system state vector, d(t) isthe system input (the duty cycle), and v o (t) isthe system output (the voltage across the output capacitor). In (3), the corresponding transfer function from d(s) tov o (s) is defined, where v o (s) andd(s) are the Laplace transforms of v o (t) andd(t) with zero initial conditions (x(0) = 0). where A = B = ẋ(t) = Ax(t)+Bd(t) (1) v o (t) = Cx(t)+Dd(t) (2) v o (s) = C(sI A) 1 B + Dd(s) (3) x(t) = il (t) v C (t) (R+r C )r L +Rr C (1 D) (R+r C )L R(1 D) (R+r C )L R(1 D) (R+r C )C 1 (R+r C )C (R 2 (1 D)+R(r C +r L )+r C r L )V in L(r L (R+r C )+Rr C (1 D)+R 2 (1 D) 2 ) RDV in (r L (R+r C )+Rr C (1 D)+R 2 (1 D) 2 )C C = RrC (1 D) R R+r C R+r C For the purpose of this paper, the equivalent series resistance of the power transistor and the diode have been neglected. However, according to 2, r L and r C have not been neglected. Therefore, if we take into account that for this paper r L = 0.2 Ω, r C =0.1ΩandD = 0.53, then for the values shown in Fig. 1, the transfer function from the duty cycle to the output voltage is given by (5). G p (s) = v o(s) d(s) = (s )(s 38696) s s (5) 3 Robust Controller Design For a single-input-single-output (SISO) plant, if the designer guarantees that the system has a gain margin equal to infinity, a gain reduction margin equal to 0.5 and a (minimum) phase margin of 60 o, the robustness of the controlled system is guaranteed 4, 5. This paper s buck-boost dc-dc power converter is based on pulse-width-modulation (PWM) techniques and the voltage-mode PWM mode of operation was used. Moreover, in order to know whether the feedback-controlled system contains hidden unstable modes, an internal stability analysis of the closed-loop system was made. ISBN: ISSN

3 3.1 Internal Stability In this paper, in order to carry out the internal stability analysis of the feedback-controlled system, the block diagram shown in Fig. 2 was used. It consists of the plant G p (s), the controller C(s), the plant output disturbance ω 1, the plant input disturbance ω 2, the controller input signal z 1 and the plant input signal z 2. Assume that the feedback-controlled system is well-posed and that neither G p (s)norc(s)have hidden unstable modes 4. The system shown in Fig. 2 can be given by z1 ω1 = T (s) (6) where T (s) = z 2 ω 2 (I + G p C) 1 G p (I + CG p ) 1 C(I + G p C) 1 (I + CG p ) 1 Figure 2: Diagram used to carry out the internal stability analysis. Theorem 1 The feedback-controlled system in Fig. 2 is internally stable if the internal signals (i.e., z 1 and z 2 ) are bounded for all bounded inputs (i.e., ω 1 and ω 2 ). (7) Proof: An excellent proof of this theorem can be found in Chapter 5 of Zhou et al. 4. According to Skogestad and Postlethwaite 5, the feedback-controlled system shown in Fig. 2 is internally stable if T (s) in (7) is stable. According to Youla et al. 17, all the stabilizing controllers for this paper s power converter can be parameterized. Lemma 2 is due to the Youla-parameterization or Q-parameterization. Lemma 2 For the case under analysis, G p (s) is a stable plant, the closed-loop system in Fig. 2 is internally stable provided that Q = C (I + G p C) 1 is stable. Proof: An excellent proof of this lemma can be found in Skogestad and Postlethwaite PID Controller Design Among all the classical control techniques developed over the years, PID control stands out as one of the most effective control technique when meeting the performance specifications of feedbackcontrolled systems. The PID-controller is the most widely used controller in the process industry and its transfer function is given by C(s) =Π p +Π d s + Π ( i s = Πd s 2 ) +Π p s +Π i (8) s where the gains Π p,π d and Π i are the PID coefficients 18. In (8), it can be seen that C(s) involvesdifferentiation of the input, it is an improper transfer function. Experience tells us that (8) can be transformed into a proper transfer function by letting the derivative action be effective only over a limited frequency range 5. Thus, in order to limit the derivative action, the transfer function given by (8) can be changed into the following transfer function C(s) =Π p + Π d s (ɛθ d s +1) + Π i s (9) where ɛ 0.1 andθ d is the derivative time constant. Therefore, (9) can be re-written as C(s) = Π d + ɛθ d Π p ɛθ d Π i Π d +ɛθ d Π p s2 + Πp+ɛΘ dπ i Π d +ɛθ ( d Π s + p ) (10) s s + 1 ɛθ d For the problem at hand, a PID-controller that gave a closed-loop system with robust stability margins (i.e., phase margin = 60.4 o and gain margin = db) was (s )2 C(s) = s (s ) The loop transfer function is given by L(s) =G p (s)c(s) (11) ISBN: ISSN

4 where G p (s) isgivenby(5)andc(s) isgivenby (11). Finally, for this paper s PID-controller, it can also be checked that T (s) (see (7)) is stable. Therefore, the closed-loop system is internally stable. 3.3 H Loop-Shaping Controller Design In the present paper, the one degree-of-freedom (1DOF) H loop-shaping design procedure was also used 14. Excellent information on H loopshaping control can also be found in 3 5. In this control technique, the converter s transfer function G p (s) (see (5)) is represented as the following left coprime factor perturbed plant G p U =(O + U O ) 1 (R + U R ) (12) where the transfer functions U O and U R are stable and unknown. They represent the uncertainty in G P (s). In addition, U R U O <δ (13) whereδ is the stability margin (0 < δ) 3 5. Fig. 3 shows the block diagram of the 1DOF H loopshaping design problem. and optimal controller that guarantees that (14) holds. Finally, according to Glover and McFarlane 13 and Skogestad and Postlethwaite 5, the minimum value of γ min and the maximum stability margin δ max are given by γ min = 1 ( = δ max 1 R O 1 2 H ) 1 2 (15) where H denotes Hankel norm. An efficient procedure to find a controller that robustly stabilizes a given shaped plant with respect to coprime factor uncertainty using H optimization is given by Skogestad and Postlethwaite 5 (Chapter 9, p. 378). In addition, the MATLAB function coprimeunc can be used to obtain the abovementioned controller. In order to carry out the controller design process, the plant G p (s) (5) was shaped by using the pre-compensator F (s) given by s F (s) = 0.02 (16) s Therefore, the shaped plant was given by G p S (s) = G p (s)f (s) (17) and, using the MATLAB function coprimeunc, the H loop-shaping positive-feedback controller given by (18) was obtained. C(s) = s s s s s s (18) In order to guarantee tracking, the gain K 0 given by K 0 = C(0) = Figure 3: Block diagram of the 1DOF H loopshaping design problem. According to Skogestad and Postlethwaite 5, the feedback-controlled system shown in Fig. 3 is robustly stable provided that γ = C (I G I p C) 1 O 1 1 δ (14) where the symbol = denotes equal by definition. Here, the H control problem is to find a robust was placed between at the system input. In this case, the loop transfer function is given by L(s) =G p S (s)c(s) where G S p (s) is the shaped plant given by (17) and C(s) is the controller given by (18). For this paper s H loop-shaping controller (18), the system also had robust stability margins (i.e., phase margin = 88.4 o and gain margin = db). Furthermore, γ min was equal to 1.73, which is a satisfactory value of γ min 5. Moreover, for the controller given by (18), it can also be checked that the feedback-controlled system is internally stable. ISBN: ISSN

5 4 Experimental Results Experimental results of the response of the PIDcontrolled buck-boost converter to a rectangular perturbation in the input voltage of ± 2.4 V at 50 Hz are shown in Fig. 4. Also, experimental results of the response of the PID-controlled buckboost converter to a rectangular perturbation in the load of ± 220 ma at 50 Hz are shown in Fig. 5. Here, the PWM actuator was implemented by using the National Semiconductor Regulating Pulse Width Modulator LM3524D. Output voltage ( v ).2.2 7th WSEAS International Conference on Application of Electrical Engineering (AEE 08), Trondheim, Norway, July 2-4, 2008 Time ( s ) Also, as in 8, the implementation of (18) was carried out by using computer assistance as a digital controller. In addition, for a sampling rate of 4 khz and applying the bilinear transformation, the pulse transfer function of the 1DOF H loopshaping controller given by (18) multiplied by the pre-compensator given by (16) is the one given by C(z) = b o + b 1 z 1 + b 2 z 2 + b 3 z 3 + b 4 z 4 1+a 1 z 1 + a 2 z 2 + a 3 z 3 (19) + a 4 z 4 where a 1 = a 2 = a 3 = a 4 = b o = b 1 = b 2 = b 3 = b 4 = Here, (19) was implemented by using the National Instruments Data Acquisition Card NI DAQCard-6062E. And experimental results of the response of the 1DOF H loop-shapingcontrolled buck-boost converter to a rectangular perturbation in the input voltage of ± 2.4 V at 50 Hz are shown in Fig. 6, and experimental results of the response of the 1DOF H loop-shapingcontrolled buck-boost converter to a rectangular perturbation in the load of ± 220 ma at 50 Hz Fig. 7. Figure 4: Experimental results of the response of the PID-controlled buck-boost converter to a rectangular perturbation in the input voltage of ± 2.4 V at 50 Hz. Output voltage ( v ).2.2 Output voltage (v).2.2 Time (s) Figure 6: Experimental results of the response of the 1DOF H loop-shaping-controlled buckboost converter to a rectangular perturbation in the input voltage of ± 2.4 V at 50 Hz. Time ( s ) Figure 5: Experimental results of the response of the PID-controlled buck-boost converter to a rectangular perturbation in the load of ± 220 ma at 50 Hz. 5 Conclusion To conclude, the results of the experiments were satisfactory and showed that both controllers allowed the buck-boost converter to reject perturbations in the input voltage and in the load current ISBN: ISSN

6 Output voltage (v).2.2 Time (s) Figure 7: Experimental results of the response of the 1DOF H loop-shaping-controlled buckboost converter to a rectangular perturbation in the load of ± 220 ma at 50 Hz. satisfactorily. However, the results also show that the buck-boost converter implemented by using the 1DOF H controller attenuates both input ripple and perturbations in the load current at 50 Hz better than the buck-boost converter implemented by using the PID-controller. Acknowledgements: The research was supported by the Universidad Politecnica de Madrid, Spain. References: 1 S. Hiti and D. Borojević, Robust nonlinear control for boost converter, IEEE Transactions on Power Electronics, Vol. 10, 1995, No. 6, pp R. Naim, G. Weiss, and S. Ben-Yaakov, H control applied to boost power converters, IEEE Transactions on Power Electronics, Vol. 12, No. 4, 1997, pp M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice Hall, K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, S. Skogestad and I. Postlethwaite, Multivariable Feedback Control, John Wiley & Sons, S. Buso, Design of a robust voltage controller for a buck-boost converter using μ- synthesis, IEEE Transactions on Control Systems Technology, Vol. 7, No. 2, 1999, pp A. Van der Schaft, L 2 -Gain and Passivity Techniques in Nonlinear Control, 2nd ed., Springer Verlag, A. Kugi and K. Schlacher, Nonlinear H - controller design for a dc-to-dc power converter, IEEE Transactions on Control Systems Technology, Vol. 7, No. 2, 1999, pp G. F. Wallis and R. Tymerski, A generalized approach for μ synthesis of robust switching regulators, IEEE Transactions on Aerospace and Electronic Systems, Vol. 36, No. 2, 2000, pp H. Vidal Idiarte, L. Martínez Salamero, H. Valderrama Blavi, F. Guinjoan, and J. Maxé, Analysis and design of a H control of nonminimu phase-switching converters, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 50, No. 10, 2003, pp F. H. F. Leung, P. K. S. Tam, and C. K. Li, An improved LQR-based controller for switching dc-dc converters, IEEE Transactions on Industrial Electronics, Vol. 40, No. 5, 1993, pp O. Ojo, Robust control of series parallel resonant converters, IEE Proceedings - Control Theory and Applications, Vol. 142, 1995, pp K. Glover and D. McFarlane, Robust stabilization of normalized coprime factor plant descriptions with H bounded uncertainty, IEEE Transactions on Automatic Control, Vol. 34, No. 8, 1989, pp D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Springer-Verlag, R. D. Middlebrook and S. Ćuk, A general unified approach to modelling switchingconverter power stages, IEEE Power Electronics Specialist Conference, 1976, pp K. K. Sum, Switch Mode Power Conversion: Basic theory and design, Marcell Dekker, D. C. Youla, H. A. Jabr, and J. J. Bongiorno, Modern Wierne-Hopf design of optimal controllers, part II: The multivariable case, IEEE Transactions on Automatic Control, Vol. 21, No. 3, 1976, pp K. Ogata, Modern Control Engineering, 3rd ed., Prentice-Hall, ISBN: ISSN