Design of Robust Guaranteed Margin Stability Region PI Controller for Automotive LED Lighting with Parameter Uncertainty

In the modern era, automotive light emitting diode (LED) lighting technologies are booming as a result of high luminous intensity, long life, compact size, high energy efficiency, high reliability etc. As the load is sensitive for input parametric variations, it is essential to regulate the load voltage and current with respect to system specifications. Hence, there is a need for fast response robust controller to feed the required voltage and current during abnormal conditions. In this paper, a robust controller is designed for single switch control buck boost converter for automotive LED lighting. A small signal modelling of the converter is developed with perturbed state variables to analyze the dynamic behavior of the system. The system with dynamic controller has a zero in the right-hand side of the s-plane, which results in poor closed loop stability. Hence, a robust controller is designed with Kharitonov’s theorem by considering the uncertain variations in the converter parameters. A robust performance region is obtained by considering guaranteed gain margin and phase margin benchmark. In this region, the optimal PI controller gain values are obtained using Kharitonov’s sixteen plants theorem. The obtained gain values make the system with minimum oscillation, fast settling time, right hand side zero compensation and stability of the system. The performance analysis is compared with the conventional PI controller and to ensure the peak value of sensitivity for robust controller. The simulation of the system is performed in PSIM and validated in real time using dSPACE1103 controller with experimental results.


I. INTRODUCTION
Lighting technology has significant role in indoor and outdoor lightings, automotive applications etc. In recent days, fluorescent and incandescent lamps are being replaced by LED lamps by its advantages like high luminous efficiency, dimming control, high reliability, small size, robustness and color rendering index [1]. In the automotive LED lighting applications, exterior lights were used as head lights, fog lights, stop/tail lights and turn indicators. LED front light could provide high brightness and less heat dissipation when compared with traditional halogen and xenon lamp in vehicles [2]. Recently, LED lighting is used in environmental friendly electric vehicles in which power is supplied from a battery. In recent years, 48 V battery is used in BLDC motor controller, telecommunication equipment and mild hybrid electric vehicles [3]. Automotive mild EV system uses 48 V battery and the voltage varies between 52 V to 38 V during driving period and voltage dips to 28 V during cold crank. Also, voltage varies dynamically during sudden stop and unclamped load dump condition [4]. The voltage and current variations damage the LED driver when exceeds nominal values. Hence, an efficient converter with appropriate closed loop controller is required to regulate the load voltage by maintaining constant current.
The converter selection plays a vital role for LED driver applications. A DC-DC converter is operated in buck [5], boost [6] and buck-boost (BB) modes of operation based on the available input voltage. For automotive lighting applications, BB converter is generally preferred because of wide range of variations in input voltage. BB converters were investigated for automotive applications with 12 V [7], 24 V [8] and 48 V [9]. Yaxiao Q developed a fourswitch noninverting BB converter with 96% maximum This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. efficiency. Though, the system has higher efficiency, the voltage gain of the converter is less than unity [8]. A twostage resonant BB LED driver was developed in [7] and full bridge series resonant converter was proposed in [9]. In those topologies, efficiency is maintained constant (93%) for wide range of input variations. In [10], a single switchcontrolled LED driver with parasitic model with minimum phase system was developed. The efficiency of the converter was maintained between 91% and 95% for wide input variations with less number of active and passive components. In addition, stability of the system was investigated with derived open loop model of the converter. There exists a zero in the right half of s-plane (RHP) for duty cycle to output voltage transfer function which results in stability issue [11]. To overcome this, a compensator was introduced in the system using current/voltage mode controllers. In [12], a current mode control (CMC) was used in DC-DC converter to compensate the RHP zeros. It results in instability issues when the converter was operated with duty cycle greater than 0.5. In addition, sub harmonics oscillation occurs with poor load regulation. It is not preferred for low power applications. Voltage mode control (VMC) is preferred for the system which requires wide range of duty cycle control with dynamic variations. In [13], modified diode-capacitor DC-DC converter with PI controller for regulating the output voltage was developed. The control scheme results in good dynamic response for high voltage applications and poor response for low voltage applications. This problem was overcome using causal path of bond graph method [14]. The detailed design guidelines for eliminating RHP zero were specified by identifying the polarity of zeros of the causal path. The transfer function was obtained by considering the feed forward path which results is computational problem for the system with more number of forward paths. Tri state BB converter with PI and type 3 partial swam optimization was proposed [15]. The controller was implemented using analog components, but the efficiency of the system was reduced when compared with conventional BB converter with non-minimal phase problem. This issue was addressed in [16], with robust PID controller. The proposed controller reduces the complex tuning effect of the controller parameters and allows the converter to operate close to the performance limit with zero in RHP. "Jinke Zhang et.al" [17] investigated the second order non-linear uncertain dynamical system and pointed out the importance of PID tuning. In [18], a modern PI(D) controller was developed for parameter variations using Kharitonov's theorem. It was used for robust controller design for possible system plant variations, right hand side zero (RHZ) compensation and guaranteed margin performance. The open loop system of a single switch modified H bridge converter (MHBC) was investigated for wide range of input variations [10]. The closed loop stability of the system was investigated with loop model. As an extension, in this work, a controller is designed for wide range of input variations and parameter variations. These variations are uncertain in nature which is called as structure uncertainty, which requires a robust controller. In the suggested work, a robust controller is designed for parametric uncertainty model of the converter. The interval plant stability analysis and controller gain regions using Kharitonov's theorem under guaranteed gain and phase margin were carried out. Sixteen Kharitonv's plants theorem is used to find the common controller gain region in all possible plant uncertainty. The controller gain values for this area are evaluated in terms of time domain specifications, pole zero mapping and sensitivity. The performance analysis was compared with conventional PI controller design by using MATLAB -SISO software tool.
Organization of this paper is as follows: Section II explains the steady state and small signal modeling of MHBC. Section III explains the robust controller design including performance analysis of the controller gain, and Section IV explains the simulation and experimental results. The conclusion is summarized in section V. Fig. 1 shows the circuit diagram of a MHBC, in which switch S1 is triggered with fixed duty cycle and switch S2 is controlled by robust PI controller. The virtual resistance (rL) is used for limiting the inner current. The filter capacitor (C0) is used to reduce the ripple content in DC and maintains constant DC voltage. In the load, ten LEDs are connected in series. Each LED has 2.5 V minimum breakdown voltage and an internal equivalent series resistance of Rs. The PI controller generates the error voltage by monitoring present voltage using a potential divider circuit. The triggering pulses are generated using SR flip flop and duty cycle is varied to control the output voltage.

B. MODES OF OPERATION
In first mode, S1 and S2 are turned on. The LED load is isolated from the input source. The inductor voltage is same as the source voltage and the capacitor voltage appears across the load. In second mode, switch S1 is still on and switch S2 is turned off. The inductor current increases linearly and reaches the maximum value. In mode 3, both switches are turned off. The inductor energy is discharged through the capacitor and load. In mode 4, S2 is turned on and the inductor current is maintained as constant.
DA and DB denotes the duty cycle of S1 and S2, R0 represents the equivalent LED string resistance. In the mathematical design, a parasitic model was used to This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
By applying voltage second balance equation to the converter, the LED voltage (VLED) is Equation (3) is the open loop transfer function of the converter. It has no zeros and all poles are on the left half of the s plane. The system is stable for wide range of input variations with minimum phase.

C. SMALL SIGNAL CONTROLLER MODELLING
The small signal model is essential for controller design. The state equations for different modes of operation with duty cycles are represented as follows: In this system, it is decided to control the duty cycle of switch S2. The different dynamic transfer functions are derived from (5) -(7) such as duty cycle control to output voltage transfer function (GA), input voltage to output voltage transfer function (GB), duty cycle control to inductor current transfer function (GC) and inductor current to output voltage transfer function (GD).
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

III. DYNAMICS OF A CONVERTER UNCERTAINTY
The Kharitonov's theorem which is the extension of Routh Hurtitz stability criterion is used to analyze the system with interval polynomial analysis. It helps in frequency analysis for guaranteed margin and PI controller stability region [19]

A. EFFECT OF PARAMETER UNCERTANITY
The dynamic behavior of the converter for input voltage is discussed. In addition, the parameter variations of the converter are considered for controller design under normal and abnormal conditions. The parameters nominal value are listed in Table I. The input variation is based on usage of 48 V in EV applications [3] - [4] and parameter variation is referred in robust control of DC to DC converter [20]. The converter performance is analyzed using different transfer functions as shown in Fig. 2. The duty cycle is varied to maintain constant load voltage for input and converter parameter variations. LED lighting is a sensitive load and it may fail whenever there is a sudden change in the driver parameters. Hence, variation in rS1, rS2, C0, L, rL, VD1, VD2, rD1, rD2 and R0 are considered. Based on the parameter variation in Table I, the interval range for the transfer function coefficients are listed in Table II.
The frequency domain analysis (bode plot) approach is used to analyze the relative stability of the system. The frequency response plots for various frequencies are shown in Fig. 3. In Fig. 3 (a), there is a nonminimum phase in which all poles and zeros lie in the lefthand side of s-plane. In Fig. 3 (b) and (c), there is a minimum phase and one zero lies on the right-hand side of s-plane which results in unstable system because of duty cycle variation. Hence, a negative integral gain value is required to compensate the zeros in the positive side which may result in stability issues. A proper design in zero compensator is required with stable and good transient response.

B. INTERVAL PLANT STABILIZATION KHARITONOV'S THEOREM
The Kharitonov's theorem is used for stability analysis and to identify boundary region of controller design. It generates different possible transfer functions with all possible combinations including the uncertainty range of system parameters. Hurwitz polynomial characteristic equation shows the stability information about the interval plant transfer function. Its polynomial range is between qnmin and qnmax, where n represents the order of the characteristic equation and q represents the minimum and maximum values of coefficient [18] - [19]. The polynomial Hurwitz characteristic equation is as follows: The polynomial equation is solved by Lyapunov's second method to reduce Hermite matrix of the four Kharitonov's polynomials (12) and create the Kharitonov's rectangular box. In this box, the stability of the interval plant is identified.
The polynomial equations are developed based on the converter input variations and parameter uncertainty. The characteristic equation of single loop control with input and duty cycle variations for unity feedback system is given in (13) and the obtained Kharitonov's polynomials are presented in (14). The response of the system is analyzed for a maximum switching frequency of 20 kHz. The real and imaginary plot of Kharitonov's polynomials is rectangular in shape for variation in frequency as shown in Fig. 4. As all rectangles not encircles the origin point, Hurwitz polynomial is stable for all uncertainty condition with the boundary limit [21].

A GUARANTEED MARGIN FOR NOMINAL CONVERTER
The SLVMC should meet the required gain margin (GM) and phase margin (PM) with satisfied time domain specifications. Fig. 5 shows the MHBC SLVM with added guaranteed margin function (GMF) in (15) where A and θ represents the required gain margin and phase margin respectively. It is required to choose the marginal values for stability and robustness checking. Here, C(s) represents the PI controller transfer function and GA(s) represents the converter transfer function.
The odd and even coefficients in the numerator and denominator components are segregated for obtaining Hurwitz polynomial which includes GMF and PI controller. Its polynomial characteristic equation is The controller gain is obtained by segregating the real part and imaginary part and equating it to zero.
The gain and phase margin of the system defines the relative stability and desired closed loop system transient response. As a result of different margins, various real root boundaries, complex root boundaries and infinite root boundaries are obtained. The stability region of the controller for various values of KP and KI based on the variation of ω is obtained. The selection of maximum frequency [20] is important to get proper finite region with condition Im (GA(jω)) = 0 and the maximum frequency is 1.6 kHz. Fig. 6 shows the region for GMF to generate the PI gain and transient response. The values in the shaded region is fully stable and has minimum peak overshoot. The different gain margin with zero phase margin curve is shown in Fig.  6(a). The shaded region shows the guaranteed GM of

B SELECTION OF CONTROLLER REGION WITH GMF
with stable points and minimum peak over shoot. The values of KP and KI obtained from the shaded area are KP = -0.015 to 0.015 and KI = 0 to 22. Based on this, the transient response for fixed gain margin and different phase margin is shown in Fig. 6(b). The shaded portion of the image depicts a stability area for nominal plants with KP = -0.015 to 0.008 and KI = 0 to 22.

C SELECTION OF CONTROLLER REGION WITH SIXTEEN PLANT GMF
The Kharitonov's sixteen plant [18] is generated based on MHBC uncertain parameters. In a continuous time nonlinear system, the stability is analyzed for both normal and abnormal conditions. This approach is extended to robustness criteria for more number of possible uncertain combinations. The guaranteed margin controller gain derivations are obtained from plant (18)- (21) and those expressions are as follows: Where (i,j = 1,2,3,4) represents sixteen Kharitonov's nonlinear plant based on parameter uncertainty of numerator and denominator coefficients for multi linear interval systems. Fig. 7(a)-(c) show the sixteen-plant frequency response gain and phase margin of the plant, nyquist plots and pole zero mapping. On account of the non-minimum system, single zero is in RHP. The entire poles in each plant is in LHP and it is nearer to origin plane when compared to zeros. All plants get negative gain and phase margin in open loop system for perturbations of input with respect to duty cycle. Hence, a superlative controller is required for all sixteen-plant transfer function to make it a stable and for good transient response.
The variation in frequency and controller gain for sixteen plants is shown in Fig. 8

D DISCRETE TIME DOMAIN MODEL
The continuous time domain model of the system described in equation (25) is converted into discrete domain system using z transform. The system is sampled at sampling time period Ts(ij) >> 1/2ωgc(ij), where i, j are varied from one to four for various sixteen plant system. By considering the different values of controller gain in stability region, minimum value of ωgc and maximum bandwidth is selected to achieve fast closed loop response. The discrete time domain transfer function for minimum Ts controller gain and sixteen plant discrete coefficient range is achieved using zero order hold and is expressed as  Fig. 9(g). The step response of the nominal plant within stabilizing gain region is shown in Fig. 10 (a). It is noted that all responses get the minimum overshoot and smooth response at all operating conditions. The step response for variation in the input voltage with positive integrator gain (KP= 2e -3 ; KI = 8) is shown in Fig. 10 (b). It is inferred that the controller works for wide input variations in MHBC.

F CONTROLLER DESIGN PROCEDURE
Design procedure for single loop voltage mode robust controller are: 1) Obtain s domain continuous time transfer function using (8). 2) Select minimum and maximum values for uncertain parameters as represented in Table I. 3) Calculate minimum and maximum values of coefficient of the controller as represented based on Table II.

V. SIMULATION AND EXPERIMENTAL RESULTS
The simulation of closed loop system is performed in PSIM software as per the design specification in Table I. To  evaluate the performance of the developed controller, the  gains KP = 1e -3 and KI = 8 are chosen and the system is  tested with input voltage variations of 15 V to 54 V. Similarly, prototype is developed using dSPACE1103 controller with MATLAB software interface.

A EXPERIMENTAL SETUP
The converter is made up of two IRF540 MOSFET switches and two LT6A02 diodes. The control pulses are generated by DAC pin 1 in dspace controller with sampling frequency of 200 kHz and driven by TLP250 driver IC. The experimental setup of the system is shown in Fig. 11. SIGLENT CP4060 current probe is used to measure the current and MDO3024 mixed domain oscilloscope is used to record the waveforms.
The control scheme implementation is shown in Fig. 12. Switch S1 is controlled with fixed 55% duty cycle. The switching pulse for S2 is generated from the controller to maintain constant output voltage for variations in input voltage. A potential divider is used to measure the output voltage and fed as input to ADC pin 1. To maintain a steady output voltage, the robust PI controller creates control pulses based on the error signal and feeds them as input to the driver circuit through DAC pin 2.

B STEADY STATE AND DYNAMIC RESPONSE
The converter with SLVM controller is tested in both simulation and hardware for steady state input voltage of 48 V and 24 V as represented in Fig. 13. It shows the input voltage, output voltage, output current and inductor current of LED driver. Fig. 13 (a) shows the steady state input voltage at 48 V to get 30 V output voltage and 700 mA output current. The LED output voltage and output current ripple is minimum in simulation and experimental results. The MHBC is operated in continuous conduction mode and the inductor current is varying between 500 mA and 1 A. Similarly, input voltage, output voltage, output current and inductor current for 24 V are shown in Fig. 13(b). It is inferred that, there is some ripple in the inductor current and output current, since input voltage is reduced from 48 V to 24 V. The simulation and experimental waveforms for dynamic response of the MHBC for variations in input voltage (∆Vin) of 48 V, 34 V and 24 V are shown in Fig. 14. In simulation, the input voltage is varied at t = 0.12 s and t = 0.325 s. As a result of the non-minimum phase of the converter, there is a negative dip in all waveforms during dynamic variations and maintained constant after short duration. The corresponding input voltage, output voltage, output current and inductor current are shown in Fig. 14

C PERFORMANCE COMPARISIONS
The conventional PID controller is tuned using PID tuner app in MATLAB -SISO tool [22]. For turning the values of controller gains, various methods say Ziegler -Nicholos frequency response (ZNF), approximate MIGO frequency response (AMIG), robust response time (RR) are available. Fig. 15 shows the step response of the different controller and it is observed that proposed controller model gives fast response and zero peak overshoot. The pole zero mapping plane is shown in Fig. 16 and it is inferred that in the proposed controller, all pole zero mapping lies inside the unit circle In ZNF controller, though all poles and zeros lie inside unit circle, the system has more oscillations and slow response. The peak value of sensitivity is represented as in the characterization of performance and robustness of the feedback system. The peak sensitivity value is 1.34 in more aggressive robust controller [22]. It is inferred from Table III that RRT controller response is closer to the proposed controller but with only integral gain. Hence, it is difficult to process the instant data which creates lagging in the robustness.  The performance comparison of RR, ZNF, AMIG and proposed controller for MHBC is illustrated in Fig. 16. It is inferred that, ZNF and AMIG controllers have high peak overshoot, slow response and high steady state error. The RR controller possesses better response compared to ZNF and AMIG controller. Because of the absence of proposal gain in RR controller, the system possesses slow response compared with proposed controller. The simulation results for transient response for various controllers are shown in Fig. 17. The input voltage is varied from 48 V to 24 V at 0.2 s. The proposed controller has fast response with fast settling time. The corresponding load and inductor currents are illustrated in Fig. 17(b) and Fig. 17(c) respectively.