A Control Strategy for Highly Regulated Magnet Power Supplies Using a LQR Approach

A linear quadratic regulator (LQR) based proportionalIntegrator-derivative (PID) controller is proposed for the SMPS based magnet power supply of the high current proton injector operational at VECC. The state weighting matrix Q of the LQR based controller, for a fixed control weighting matrix R, is derived analytically using guaranteed dominant pole placement approach for desired maximum overshoot and rise time. The uniqueness of this scheme is that the controller gives the desired closed loop response with minimum control effort, hence avoiding the actuator saturation by utilizing both optimum approach of LQR technique and simplicity of the conventional PID controller. The perturbation of controller and power supply parameters is studied along with the load disturbance to verify the robustness of proposed control mechanism. INTRODUCTION A PID controller is one of the favorite controllers in the industry. The three controlling parameters of PID controller are proportional (Kp), integral (Ki) and derivative (Kd) [1]. The magnet power supply having rating 10V/200A feeds the solenoid coil, which produces desired magnetic field in the plasma chamber. For efficient performance of the ion source we need a highly stable and reliable power supply. In this paper a LQR based PID controller is proposed for the design of the DC-DC converter pulse width modulator (PWM) controller. The PWM controller will be finally used to regulate the magnet power supply. The simulation is performed in MATLAB software. The design methods of PWM controller have been discussed extensively in the literature [2, 3]. We have used LQR approach for obtaining an optimal control which is finally re-arranged to give the parameters of a PID controller [4]. Buck converter is a DC-DC converter that is widely used in the industry today, due to its high efficiency, low cost and more important small size. This small size results due to the use of high frequency switching, which is accomplished by using high frequency IGBT [2]. Figure 1 shows the closed loop optimal controller where X(t) is the state variable matrix, u(t) is the control vector, A is the state transitions matrix, B is the control matrix, C is the output matrix, R is the control weighting matrix and P is the Riccati coefficient matrix. LQR gives the value of control vector u(t) in an optimum way using Calculus of Variation approach [5] which minimizes the quadratic performance cost function as given by    dt t t t t t u J T T     0 ) ( ) ( ) ( ) ( ) ( Ru u QX X (1) with the plant dynamics given in the state space form as 0 0 ) ( ; ) ( ) ( ) ( ) ( ) ( X X u B X A X    t t t t t t  (2) ) ( ) ( ) ( t t t X C Y  (3) where X0 is the initial values of the state variables. Q is the state variable weighting matrix. The optimal control u(t) is given by [5] ) ( ) ( ) ( t t t KX PX B R u T 1      (4) The value of matrix P can be evaluated by solving the continuous algebraic Riccati equation, 0 P B PBR Q PA P A T 1 T      . (5) After getting the optimal gain vector u(t), the PID parameters can be easily calculated using the procedure discussed in reference [4]. Since in the present work we have studied only linear time invariant system, the matrices A(t), B(t) and C(t) are therefore, independent of time i.e. A(t)=A, B(t)=B and C(t)=C. Figure 1: Schematic of a closed loop optimal controller. MAGNET POWER SUPPLY DESIGN The buck converter is a switch based DC-DC converter [2]. The output current is regulated based on the control of switch element which is generally IGBT. Figure 2 shows the closed loop block diagram of the buck converter based power supply. The PID controller is used to regulate the duty cycle of the PWM signal. e(t) is the error signal, u(t) is the control signal, D(t) is the duty cycle and Y(t) is the final output. ___________________________________________ #saurabh@vecc.gov. in THPPC117 Proceedings of ICALEPCS2013, San Francisco, CA, USA ISBN 978-3-95450-139-7 1334 C op yr ig ht c ○ 20 14 C C -B Y3. 0 an d by th e re sp ec tiv e au th or s