A graphical approach for controller design with desired stability margins for a DC–DC boost converter

Correspondence Md. Imran Kalim, Department of Electrical Engineering, Indian Institute of Technology Patna, Bihta, Patna-801106, Bihar, India. Email: imran2025kalim@rediffmail.com Abstract A novel graphical tuning method of PID controller for output voltage regulation of a DC– DC boost converter is proposed. The advantage of the presented method is that the desired robustness level in terms of gain and phase margins can be pre-specified. A technique to limit the noise level in the control signal to a pre-specified value by selecting the derivative gain (kd ), is also discussed. Simplicity of the technique is an added advantage. Concept of root crossing boundaries along with the gain phase margin tester is applied to compute constant gain and phase margin boundaries in the controller parameters plane. The overlapping area of constant gain and phase margin boundaries within the all stability region is the desired gain and phase margin region (DGPMR). Controller parameters are obtained by computing the centroid of the triangular convex region of the DGPMR. Effectiveness of the proposed tuning strategy is illustrated by simulation and hardware experimental results. Furthermore, the proposed method yields improved closed-loop performance compared to a recently reported tuning strategy in nominal as well as perturbed scenarios.

design issues related to sliding mode controllers for boost converter were presented in [5]. However, sliding mode controllers have chattering problem and it enhances the ripple content in output voltage. In [6], an optimal linear quadratic regulator (LQR) using convex optimization method was proposed for the boost converter. A model predictive control (MPC) approach has been reported in [7]. A robust control scheme using the concept of time-delayed switching was designed in [8]. A two degree-of-freedom internal model controller for output voltage regulation of boost converter was experimentally evaluated in [9].
Despite many recent theoretical and practical developments in modern control theory, PID controllers are still most widely used in industries. This is because of its simplicity, robustness, acceptable control effort and an outstanding cost/benefit ratio with satisfactory performance for a large class of process models. Furthermore, PID controllers can be easily implemented on analogue as well as digital hardware. In [10], authors have designed a PID controller for boost converter using a novel optimization algorithm named as Bees GA, which combines the evolution of a queen bee in a hive with Genetic Algorithm (GA). Rise time, peak overshoot, settling time and steady-state error of output voltage response of the boost converter were used to formulate the objective function for the optimization process. In [11], authors have used an optimization algorithm based on the foraging behaviour of a colony of honey bees to obtain the optimal values of PID controller for the boost converter. A modified form of particle swarm optimization (PSO), namely probabilistic PSO was applied to compute the PID controller parameters in [12]. Three different evolutionary algorithms, namely genetic algorithm (GA), differential evolution (DE) and artificial immune system (AIS) were used to design the PID controller in [13]. Objective function based on the error between the desired reference voltage and the actual output voltage was used in the optimization algorithms. However, the above-reported approaches do not guarantee robustness to system parameter variations. Tuning rules for a PI controller in terms of boost converter parameters to ensure stability were obtained in [14]. In [15], authors have tuned the PID controller by Ziegler-Nichols method, internal model control (IMC) method, synthesis method and equating coefficient method. It was observed from the results that the synthesis method yields the least ISE value. Complete root contour (CRC) method was used for controller design of boost converter in [16]. PID controller parameters for buck and boost converters were obtained from a switching surface based on a state trajectory in [17].
Since closed-loop stability is the primary requirement, it is desirable to obtain all stabilizing PID gains set before controller tuning. Various methods have been reported in the literature to compute the all stabilizing PI, PD and PID controllers such as Nyquist plot approach [18], D-decomposition method [19], stability boundary locus approach [20], singular frequency method [21], kronecker summation method [22], Hermite-Biehler theorem [23], signature method [24]. Gain and phase margins are the two commonly used relative stability measures. All stabilizing PI/PID controllers with user-defined gain and phase margins have been computed in [19,23].
A robust and non-fragile PI controller was designed by calculating the centroid stable point of the all stability region in [25]. PI/PD controller parameters were obtained by computing weighted geometrical centre (WGC) from the all stability region in [26]. A method for obtaining the PI/PD controller parameters by calculating the centroid from the all stability region has been reported in [27]. However, the above three reported methods do not guarantee a desired degree of robustness. Moreover, procedure to tune the PID controller was not addressed.
A novel and simple graphical tuning method is presented in this paper which ensures desired stability margins and also limits the noise level in the control signal to a pre-specified value. All stability region is obtained by computing different root crossing boundaries in the controller parameters plane. Furthermore, the desired gain and phase margin boundaries are obtained using the gain phase margin tester (GPMT). Overlapping area of the above-said boundaries inside the all stability region is called the desired gain and phase margin region (DGPMR). Required controller parameters are obtained by computing the centroid of the triangular convex region of the DGPMR. The paper is organized as follows: Mathematical model of the boost converter is derived in Section 2. Controller design is addressed in Section 3. Simulation, hardware experimental results and comparison are given in Section 4 whereas Section 5 provides the concluding remarks about the proposed work. Figure 1 shows the equivalent circuit of a DC-DC boost converter, where v in (t ), v o (t ) and i L (t ) are the dc input voltage, output capacitor voltage and inductor current, respectively. R, L, r L and C denote the load resistance, inductance, ESR of the inductor and output capacitance, respectively. Perturbation in load current is denoted byî d (t ). The switch M and the diode D operate complementarily in continuous conduction mode (CCM). Dynamics of the boost converter operating in CCM can be expressed by following state-space equations.

MATHEMATICAL MODEL OF DC-DC BOOST CONVERTER
[̇i where, the value of m(t ) is '0/1' when 'M' is 'ON/OFF'. Presence of switch in the circuit makes the above state-space equation discontinuous with respect to time. Controller design generally requires averaged model of the switching power converters [1]. Control input (duty cycle), u(t ) is the average value of m(t ) over a complete switching cycle. Perturbation signals about the nominal steady-state values are introduced to obtain the small-signal model of the boost converter as follows: where steady-state values are represented by upper case letters, while '̂' denotes corresponding perturbed values. From the control point of view,v in (t ) andî d (t ) denote the disturbances in input voltage and load current, respectively. Steady-state values of inductor current and output voltage are obtained by equating the derivatives in (1) to zero and are obtained as follows: where U , V in and R represent nominal values of duty ratio, input voltage and load resistance, respectively. Substituting (2) in (1), we get Considering only the perturbed values and the first-order terms, the small-signal model of the system defined by (1) is obtained as Taking Laplace transform of (5), the small-signal transfer functions from control-to-output voltage, line-to-output voltage and load current-to-output voltage are obtained as follows: Control input to output voltage transfer function: Input voltage to output voltage transfer function: Load current to output voltage transfer function: From (6), it is observed that control input to output voltage transfer function contains one RHP zero, whose location in splane is given by Effect of the RHP zero becomes more pronounced when it moves towards the origin. As can be seen by (9), duty ratio,

CONTROLLER DESIGN
where, Parameters of the DC-DC boost converter considered in the present work are given in Table 1. Substituting these values in (6), (7) and (8), we get Transfer function of the PID controller is assumed as follows: where k p , k i and k d are the proportional, integral and derivative gains, respectively.
From Figure 2, the closed-loop transfer function is obtained as From (15), it is observed that zeros of controller appear in the numerator of T (s). Presence of these zeros deteriorates the closed-loop performance. A pre-filter with the following transfer function is therefore used to cancel the zeros.
Transfer function from set-point to output is obtained as N f = k i ensures zero steady-state error for a step change in reference voltage. The characteristic equation is given by Substituting the values of N p (s), D p (s), N c (s) and D c (s) in (18), we get Substituting s = j in (19), we get Where Re( P ) and Im( P ) denote the real and imaginary parts of P ( j ) and are obtained as where, Roots of any stable polynomial can become unstable by crossing the imaginary axis. So the imaginary axis is mapped on the controller parameters plane to obtain stable and unstable regions. Boundaries of the two regions (stable and unstable) are the root crossing boundaries, which can be finite (Real Root Boundary, Complex Root Boundary) and infinite (Infinite Root Boundary). All stability region in controller parameters plane is the area bounded by the above-said boundaries which contains all values of the controller parameters to ensure stability. Procedure to obtain the root crossing boundaries is as follows: (i) Real Root Boundary (RRB): A real root can cross the imaginary axis at s = 0. RRB line is obtained as k i = 0 by substituting s = 0 in (19). by substituting s = ∞ in (19). Substituting the value of a 1 gives k d = 5.988 × 10 −4 as the IRB line. (iii) Complex Root Boundary (CRB): Complex roots on the imaginary axis can become unstable which implies that both real and imaginary parts of P ( j ) will become zero simultaneously.
By equating Re( P ) and Im( P ) to zero, two equations with three unknowns are obtained. Assuming a suitable value of one of the controller parameter, the root crossing boundaries are drawn in the plane of other two parameters. Pre-specification of one of the controller parameters needs the stable range of the same. Intersection of RRB line, IRB line and CRB curve in the (k p , k i )-plane and (k p , k d )-plane provides the stable ranges of controller parameters [28]. Furthermore, the procedure to calculate the stable ranges of controller parameters is given in Sections 3.1 and 3.2.

3.1
All stability region in (k p , k i )-plane for a fixed value of k d By substituting s = 0 in (19), k i = 0 is obtained as the RRB line. On the other hand, substituting s = ∞ in (19) gives no solution for k p and k i for any value of k d and therefore IRB line does not exist in (k p , k i )-plane. Expressions for CRB curve are obtained by equating (21) and (22) to zero and solving for k p and k i yields (24) and (25) are plotted to obtain CRB curve by varying from 0 to c , where c is known as critical frequency. Furthermore, c is the lowest value of frequency other than = 0 at which k i = 0, the value of k i on the RRB line. c is obtained by Using the above-discussed procedure, all stability region in (k p , is obtained as the IRB line by substituting s = ∞ in (19). Expressions for CRB curve are obtained as follows: CRB curve in (k p , k d )-plane is obtained by plotting (27)  . On similar lines as discussed in the previous subsection, all stability region in (k p , k d )-plane for k i = 0 is obtained as shown in Figure 3b.
From Figure 3a,b, stable ranges of controller parameters are obtained by intersection of RRB line, IRB line and CRB curve as −0.0123 ≤ k p ≤ 21.514, 0 ≤ k i ≤ 7.6135 × 10 5 and −7.428 × 10 −6 ≤ k d ≤ 5.988 × 10 −4 . In this paper, value of kd is pre-specified and the root crossing boundaries are drawn in (k p , k i )-plane. A suitable value of k d is chosen from stable range to have a trade-off between transient performance and noise.

Desired gain and phase margin region
The DGPMR is obtained by placing a GPMT (G pm = Be − j ) in feed-forward path of the closed-loop shown in Figure 2. B and denote the values of gain and phase margins for constant GM and PM boundaries, respectively. Steps to compute different root crossing boundaries in Section 3.1 is repeated with GPMT in the loop. RRB line remains same, that is, k i = 0. There will not be any IRB line in the (k p , k i )-plane for any value of k d . Corresponding k p and k i expressions for the CRB curve are given below.
CRB curve for Constant GM boundary is computed by substituting the desired value of GM and = 0 in (29) and (30), whereas we put B = 1 and the desired value of PM in (29) and (30) to obtain CRB curve for constant PM boundary. c is obtained by equating (30) to zero for desired GM and PM values. Common area of the above-said boundaries inside the all stability region is called the DGPMR. Controller parameters in the DGPMR yield stability margins greater than or equal to the pre-specified values.
A triangular convex region of the DGPMR is obtained by identifying peak point on the CRB curve and two corner points on the RRB line of DGPMR. Coordinates of the peak point is assumed as (k pp , k ip ) and that of corner points are (k pc1 , k ic1 ) and (k pc2 , k ic2 ). Values of k p and k i are obtained by finding the centroid of the triangular convex region of the DGPMR as follows:

Selection of k d based on noise constraint
A new approach for selecting the value of k d to limit the noise level in the control signal to a pre-specified value is presented in this section. Noise in the control signal degrades the performance and shortens the actuator life of a closed-loop system [29]. Power converters are prone to noise due to the presence of multiple noise sources. Standard deviation, which is one of the most commonly used measures of noise level, is calculated as where k is the number of samples of the control signal in steady-state. u av is the average value of k samples of the control signal. In the present work, 20,000 samples of the control signal in steady-state are considered to increase the accuracy level. An arbitrary value of k d (say 1 × 10 −5 ) is chosen from the stable range. Figure 4a shows the all stability region in (k p , k i )plane for k d = 1 × 10 −5 , which is obtained using the procedure given in Section 3.1. It is mentioned in [30] that robust stability is ensured by simultaneous satisfaction of GM > 6dB and PM > 30 deg. Constant GM (9 dB) and PM (60 deg) boundaries for k d = 1 × 10 −5 are obtained using the procedure given in Section 3.3 and shown in Figure 4b,c respectively. Figure 4d shows the all stability region, Constant GM (9 dB) and PM (60 deg) boundaries. The shaded region in Figure 4d is the DGPMR. Centroid of the triangular convex region of the DGPMR is obtained as (0.0068, 3.254).
A band-limited white noise of standard deviation 0.1 with sampling period equal to 5 × 10 −7 is considered here. Closedloop simulation for a unit step reference voltage change is performed by using the above controller parameters with and without considered white noise present in the loop to obtain u and integral of absolute error (IAE = ∫ ∞ 0 |e(t )|dt ), respectively. Instantaneous error, e(t ) is the difference between reference voltage change,v 0re f (t ) and output voltage change, v 0 (t ) at time 't '. Values of u and IAE are obtained as 0.0076 and 5.87 × 10 −3 , respectively. Above-discussed procedure is repeated for several values of k d to obtain values of u and IAE . Curve fitting toolbox of MATLAB is then used to obtain a graphical relationship between u and IAE as shown in Figure 5 and a mathematical relationship between k d and u as follows: where a = 0.0005002, b = 0.8232 and c = 2.153e − 06. It is observed from Figure 5 that improved transient response (i.e. lower IAE value) is obtained at the cost of higher value of u (i.e. increased level of noise content in the control signal) and vice versa. Using (34), value of k d is selected for a specified u .

SIMULATION AND HARDWARE EXPERIMENTAL RESULTS
Following commonly used parallel form of PID controller is considered for simulations and hardware experiments.
In the present work, u = 0.03 from Figure 5 is considered for controller design which yields IAE = 2.66 × 10 −3 . Using (34), corresponding value of k d is calculated as 3.005 × 10 −5 . All stability region, constant GM (9 dB) and PM (60 deg.) boundaries for k d = 3.005 × 10 −5 in (k p , k i )-plane are obtained as shown in Figure 6. The DGPMR is shown by the shaded region. Centroid point is calculated as (k p = 0.0256, k i = 13.82), which is the required controller setting for k d = 3.005 × 10 −5 . Proposed method is compared with that of Kobaku et al. [9]. In [9], authors have proposed a two degree-of-freedom internal model controller for boost converter. Two filters namely setpoint filter (F r (s)) and disturbance filter (F (s)) were designed. Tuning parameter, r for F r (s) was calculated to have a desired set-point tracking performance, whereas d for F (s) was computed to achieve a desired degree of robustness based on maximum value of sensitivity function (M s ). M s is defined as inverse of the shortest distance from the Nyquist curve of open-loop transfer function to the critical point (−1, j 0) and is expressed Value of M s less than 2 ensures robust stability [30]. The proposed method yields M s = 1.246. For a fair comparison, the tuning parameters of the two filters in [9] are selected as r = 1.14ms and d = 0.61ms to achieve the same values of settling time in set-point tracking response and M s as obtained by the proposed method. Different performance measures like settling time (T s ), overshoot (O.S.)/ undershoot (U.S.) and IAE are used for comparison. As reported in [9], 'T s ' is considered as the time required to reach and stay within ±1% band of the steady-state value. MAT-LAB simulations are performed on the linear as well as nonlinear model of the boost converter. The switching frequency is considered as 20 kHz. Figure 7 shows the hardware set-up for experimental evaluation. Control schemes are implemented using a dSPACE 1104 microcontroller. Programmable DC power supply is used to provide variable input voltage to the boost converter, whereas DC programmable load is connected in parallel with the nominal load to provide a step change in connected load. Three different cases are considered to demonstrate the effectiveness of the proposed method.

Case 1: Servo performance
Servo performance is evaluated by applying step changes in reference voltage. First, a step-up change in reference voltage from 50 to 60V at t = 0.1s is considered. Input voltage and load resistance are kept constant at their nominal values of 30V and 50Ω, respectively. Figure 8 shows the simulation results of output voltage, control effort (duty ratio) and inductor current. Values of performance measures are given in Table 2. It is observed from Figure 8 and Table 2 that the two methods yield same settling time ('T s ') as considered in the design. Hardware experimental results are shown in Figure 14. Furthermore, the benefit of the proposed control scheme is illustrated by applying a step-down change in reference voltage from 60 to 50V at t = 0.15 s. Corresponding simulation and hardware   Table 2. For the step-up case, controllers of the two schemes first increase the inductor current to a high value (approximately 9 A) in transient phase to quickly ramp up the output voltage  Control effort c. Inductor current go below 0 A due to unidirectional capability of the boost converter. Consequently, the converter temporarily enters and stays in discontinuous conduction mode (DCM) until average value of inductor current is greater than half of the peak-to-peak ripple in inductor current. However, since the two control schemes are designed with sufficient robustness, they ensure closed-loop stability and output voltage regulation in non-linear simulation as well as hardware experiments. Furthermore, the proposed control scheme outperforms the method of Kobaku et.al. [9]  in non-linear simulation and hardware experimental results for the step-down case as can be seen by Figures 9 and 15 and Table 2.

Case 2: Regulatory performance for input voltage disturbance
Input voltage is changed from 30 to 45V at t = 0.3s and 45 to 30V at t = 0.35s while the load resistance is kept constant at 50 Ω. Corresponding simulation results of input voltage, output voltage, control effort and inductor current are shown in Figures 10 and 11, respectively. Performance measures are given in Table 2. Figures 10 and 11 and Table 2 show that the proposed method yields improved regulatory performance for disturbance in input voltage with lower values of settling time, overshoot and IAE as compared to the method of Kobaku et al. [9]. Corresponding hardware experimental results in Figures 16 and 17 closely agree with the simulation results.

Case 3: Regulatory performance for load current disturbance
Regulatory performance for disturbance in load current is considered for a step change in load from 50 to 125W (i.e. 50 to 20 Ω) at t = 0.5 s and 125 to 50W (i.e. 20 to 50 Ω) at t = 0.55 s.  Table 2. Figures 18 and 19 show the corresponding hardware experimental results.
From Figures 8 and 19 and Table 2, it is observed that hardware experimental results are qualitatively matching with corresponding simulation results in all the above three cases.

Simulation results for the perturbed model
To illustrate the robustness of the proposed controller against perturbation in plant parameters, values of L and C are increased by 20%. MATLAB simulations are performed on the linear and non-linear perturbed models. Corresponding results for the above-said three cases are shown in Figure 20. It is observed from the obtained results that the proposed controller is robust to the assumed perturbations in L and C . Furthermore, the proposed method also outperforms the strategy reported in [9] in the perturbed scenario.
From the above simulation and hardware experimental results, it can be observed that the proposed method yields improved closed-loop performance compared to the strategy reported in [9] under both nominal and perturbed scenarios. Also, it is to be noted that the number of controller parameters is more in the design strategy reported in [9], which can be easily seen from the transfer functions of the two controllers of [9] given below.

CONCLUSION
A novel graphical-based controller design method for output voltage control of a DC-DC boost converter is presented in this paper. A linearised model in the neighbourhood of the equilibrium point is used to design the controller. Frequencydomain performance measures such as gain and phase margins can be pre-specified to achieve a desired degree of robustness. Furthermore, k d is selected to have a trade-off between closedloop transient performance and noise level in the control signal. Simplicity of the technique is an added advantage for the practising engineers in industries. The presented technique has an extra benefit over mathematical-based tuning methods as the controller parameters are obtained graphically whereas later needs tedious calculations. Simulation and hardware results show that the proposed method yields satisfactory closed-loop responses. Robustness of the proposed controller is illustrated by simulation and hardware results. Closed-loop responses of simulation as well as hardware experiments are found to be superior as compared to a recently reported tuning strategy for the boost converter in nominal as well as perturbed scenarios.