Improvement of the linear quadratic regulator control applied to a DC-DC boost converter driving a permanent magnet direct current motor

ABSTRACT


INTRODUCTION
DC-DC converters play a very important role in several technical areas such as: manufacturing industry, renewable energies, especially solar photovoltaic and hybrid systems (PV-wind).These converters facilitate the management of the direct current (DC) energy; therefore, it can supply DC electrical systems accurately.Several research have been carried out to control boost, Buck and mixed converters, including a robust-optimal control strategy allows to track and control DC-DC boost converter output voltage errors.The strategy used is based on the design of an optimized fractional-order proportional-integral (FoPI) controller that removes oscillations, overshoots, undershoots and steady-state fluctuations.In order to improve the error convergence rate during a transient response, the FoPI controller is supplemented by a pre-stage nonlinear error modulator, the latter combines the variations of the error and the error derivative via the signed distance method, the control system realized can reject external disturbances such as load transients and input fluctuations, which gives adequate operation to the DC-DC boost converter [1].A linear quadratic integral (LQI) controller is used for DC motors optimum speed regulation.It allows the motor to run without disturbances and fluctuations in steady-state for different responses, the LQI controller is complemented by a Lyapunov-based model reference adaptation system (MRAS) that modulates the controller gains adaptively while maintaining the asymptotic stability of the controller [2].An adaptive and collaborative speed controller used to control a permanent magnet direct current (PMDC) motor; this controller is designed by a proportionalintegral (PI) regulator and a linear-quadratic regulator (LQR), the adaptive combination of the two controllers brings an improvement on the transient and steady-state responses and the removal of the disturbances due to load-torque variations [3].
A theoretical study based on a mathematical model of the boost converter where the Lyapunov function has made closed loop current and the voltage control possible, nevertheless Lyapunov function does not give the boost converter an overall stability, the monodromy matrix which interfere on systems nonlinear dynamic behavior allows stability of the boost converter [4].An adaptive controller proposed to improve the performance of the buck converter which presents a time-bound estimate of the unknown system uncertainties and the exogenous disturbances, moreover an online estimator is carried out to reconstruct the uncertainty incurred, the additive uncertainty estimated then passes to the nominal backstepping controller for further finite-time compensation [5].An estimation based on the sliding mode controller, which is able to provide reference voltage tracking in the presence of uncertainties due to input voltage and reference voltage variations, this closed loop control gives a system stability [6].A design of a robust control law based on pulse-width modulated (PWM) conditions; the strategy applied to a parallel interconnection of DC/DC converters [7].A design of a dynamic sliding mode control based on an internal model for a DC-DC boost converter is proposed in order to reject system disturbances [8].
A design of a continuous signal generator used to control electrocardiographic (ECG) signals using a buck converter was proposed, it consists of the application of sliding mode controller on both current and voltage loops to impose the tracking of ECG voltage reference, this proposed controller is able to perform robustly regarding voltage input dynamics disturbances [9].A novel DC-DC buck converter control technique based on generalized proportional integral observer (GPIO) is designed to estimate localized disturbance, improve anti-disturbance buck converter and transient performances.The PWM command is managed by the dynamic prescribed performance sliding mode control-GPIO (DPPSMC-GPIO) controller [10].
A finite control set model predictive control (FCS-MPC) method combined with a Kalman observer is designed to obtain the load variation and model correction, which can decrease the disturbance, moreover, it can estimate voltage error and its integration [11].In order to solve the problem of regulating the buck converter output voltage under current constraint, a nonlinear algorithm is proposed so that the output voltage follows the reference voltage quickly in a finite time [12].The speed control of PMDC motor by a buck converter using the adaptive backstepping controller and Legendre neural network allows to estimate the uncertainties online then to compensate them effectively during the robust control action.The stability of the closed-loop system under the action of the proposed controller and online adaptive learning laws are proved using Lyapunov's stability criterion [13].A new fast learning neural network used for PMDC motor load torque estimation, angular rate control is generated by DC-DC buck converter which is linked to adaptive backstepping controller, the approach used in this system gives great stability using the Lyapunov criterion [14].To increase the boost converter output voltage, a new improvement technique is based on output voltage closed-loop control which is designed by Simulink automatic code generation, the technology that makes the output voltage 170 V is the TMS320F2833 32-bit floating-point processor [15].A new control technique highlights how nonlinear controllers based on reinforcement learning can improve the boost DC-DC converter dynamic performance compared to standard controllers [16].
The research work carried out in this article is based on the design of three LQR control techniques, which are LQR, LQR-PI and  −

𝑇𝑠+1
. These techniques are applied to the DC-DC boost converter driving a PMDC.The controllers designed gave very efficient results in terms of voltage, current, power, angular speed and motor torque.Our contribution to the research works mentioned above is the design of a robust control system based on the linear quadratic regulator control theory  −

𝑇𝑠+1
; this type of controllers removes oscillations, disturbances and fluctuations from the DC-DC boost converter outputs (voltage, current and power) therefore, this gives the PMDC motor a steady-state stable drive under load variation.The results obtained are simulated by MATLAB Simulink software.

SYSTEM DESCRIPTION
The PMDC motor is controlled by the boost converter where the variation of the speed is obtained by the output voltage of the boost converter using a closed servo loop, the circuit of Figure 1   (1) with   () armature voltage of PMDC motor,   () armature current,   () motor torque,   () load torque,   torque constant,   electromotive force (EMF) constant and   () the EMF.
The equations governing the boost converter operation are as follows [22]- [24].The switch Sw is closed, the diode D is blocked therefore,  0 ≤  ≤  0 + , we will have the (5) to (7), +   = 0 The switch Sw is open, then the diode D is on therefore,  0 +  ≤  ≤  0 + , we obtain the ( 8) to (10), +   =   (10) The dynamic equations of the boost converter in continuous conduction regime are given as (11) and (12), At the equilibrium point the derivatives of the average states are zero and the average order of um is equal to a constant value  ̅  , from ( 11) and ( 12) we can get the average states at equilibrium of the boost converter: The normalized static characteristic of the boost converter is given as (19), The linearized model of the boost converter based on the average equilibrium voltage which is (20) to (22), The average model of the boost converter as a state is as (23), The linearization of the average model around the equilibrium point is (24), with: with  is the state vector,  the input vector,  the output vector, A the state matrix, B the control matrix, and C the output matrix.
To control the boost converter with high static and dynamic performances, we represent the system of state (23) in the form of a closed loop.From this state equation we designed a control system based on the linear quadratic regulator technique which allows the DC-DC converter to operate without disturbances and oscillations and which gives great stability for different characteristics (current and voltage).The Figure 2 represents an LQR type corrector applied to the DC-DC boost converter: where  is the gain matrix, so the control vector equals (28).

𝑢(𝑡) = −𝐾. 𝑥(𝑡) (28)
To improve the performance of the boost converter control, we apply the infinite horizon LQR theory where the criterion used is as (29) [25]- [28]: If the control is linear with invariant time, it becomes a constant state feedback control where the gain matrix K is expressed by (24) and P expressed in the algebraic Riccati (30).() +   () − () −1   () +  = 0 (30) with  ≥ 0 and  > 0, symmetric matrices To stabilize the system, we apply the theory of the state feedback looped system induced by the optimal control (31).

𝑢(𝑡
where P is the solution of Riccati's algebraic equation, it is asymptotically stable if: The {, } pair is controllable and the {, √} pair is detectable.Thus, the matrix P is positive definite if and only if {, √} is completely observable.The eigenvalues of the matrix [ −  −1   ] are then all with negative real part.The transfer function of the PI corrector is (32), From ( 11), ( 12) and ( 24) we will design the transfer function of the boost converter: (33) The closed loop transfer function of the full system is (34).

SIMULATION RESULTS
To obtain different results, the parameters of the DC-DC boost converter and of the PMDC motor are given in Tables 1 and 2. Table 1 gives an estimated parameter of PMDC motor like armature resistance and inductance, back electromotive force constant, torque constant, moment of inertia and viscous friction coefficient.Whereas Table 2 gives an estimated parameters of DC-DC boost converter like input DC voltage, switching frequency, resistor load, boost inductor, boost capacitor, output DC voltage, Kc constant, time constant, proportional gain and integral gain.From these parameters the simulation results using MATLAB are as shown in Figures 4 to 7.   ), it is observed that the voltage obtained by the LQR method (19 V) equals that obtained by the LQR-PI controller in steady state but the performances (response time and oscillations) of the LQR-PI are better than the LQR control.For driving the PMDC motor, the  −   +1 controller gives a continuous output voltage equal to 24 Volts which is sufficient for the operation of the motor in steady state without disturbances and oscillations, but the response time is less than those LQR and LQR-PI control methods.Figure 5 represents the currents at the output of the inductor, we note that the current obtained by the LQR and LQR-PI method are lower than that obtained by the  −   +1 method nevertheless, for the static and dynamic performance we note that the overshoot obtained by the LQR-PI method is smaller compared to the LQR and  −   +1 methods.
Figure 6 represents the angular speed of the PMDC motor driven by the DC-DC boost converter.The speed follows well its applied reference, at the instant [0.05 to 0.2] second the angular speed has decreased during the introduction of the resistive torque which equals 1.3 Nm but at the instant of [0.2 to 0.3] second the angular speed increases up to the rated speed (no-load running) and for the instant of [0.3 to 0.4] second the angular speed has decreased down to 400 rad/s because of the load torque applied and stabilizes at this value in steady state until the end of the 0.4 second period.The motor torque of Figure 7 follows the variation of the load torque applied, we notice at the instant [0.05 to 0.2] second that the motor torque takes the value of the resistive torque which is equal to 1.3 Nm and stabilizes in this value in steady state then this motor torque decreases in the period of [0.2 to 0.3] second during no-load running and in the period of [0.3 to 0.4] the motor torque increases up to the value of the load torque applied (1.3 Nm) and stabilizes at this value until the end of the period.The various simulation tests applied to the system (DC-DC boost converter-PMDC motor) make it possible to confirm the robustness of the LQR, LQR-PI control and the  −   +1 hybrid control.These three control techniques allowed the system to operate at high dynamic and static performance during the introduction of disturbances.

CONCLUSION
The work presented in this article is to carry out three control techniques: the linear quadratic regulator (LQR) control technique, the LQR-PI control as well as the hybrid control  −   +1 , these controls are applied to the boost DC-DC converter.To test the efficiency and robustness of the controls used, MATLAB environment gives different curves.In order to improve the dynamic performance of the system, the LQR, LQR-PI and  −   +1 correctors participate in a considerable way to the elimination of oscillations and fluctuations on different characteristics.The most effective control strategy for driving the PMDC motor is the hybrid control which allows DC-DC boost converter output voltage to increase up to 24 V, this allows the PMDC motor to operate under rated conditions without oscillations, disturbances, overshoot and fluctuations.The simulation results of the applied tests demonstrated the efficiency and robustness of the control techniques applied to the boost DC-DC converter driving the PMDC motor.


ISSN: 2088-8708 Int J Elec & Comp Eng, Vol. 13, No. 6, December 2023: 6131-6140 6132 represents an association of two assemblies, the first consists of a fixed DC source Ve, inductor L, load resistor R, diode D, a capacitor C, ISSN: 2088-8708  Improvement of the linear quadratic regulator control applied to a DC-DC boost … (Adel Bouchahed) 6133 the current flowing through the coil L iL, an output current Idc, an output voltage Vdc and power electronics switch SW , the second assembly represents the circuit of the PMDC motor which includes an armature inductance Lar, armature resistance Rar, motor inertia J, a viscous friction f and an angular velocity ω.

Figure 1 .
Figure 1.PMDC motor drive by boost converter & Comp Eng, Vol. 13, No. 6, December 2023: 6131-6140 6134 with  is the Mosfet control signal.We put:  1 =   et  2 =   the (11) and (12) become: are the inductance of the coil L in [H], the capacitance of the capacitor (C) in [F] and the load resistance R in [Ω].The state variables are the current in the coil and the voltage across the capacitor.The control signal u is between {0;1} and it indicates the state of the switch Sw: open for 0 and closed for 1.It can be replaced by its average value over a chopping period d which represents the duty cycle  =   /  where Ton is the conduction time and Ts is the chopping period.

Figure 3 .
Figure 3.Control of the boost converter driving a PMDC motor

6137Figure 4 .Figure 5 .Figure 6 .Figure 7 . 4 .
Figure 4. DC-DC converter output voltage for a reference of 20 to 24 volts From the boost converter equations, we can design the control law based on LQR, linear quadratic regulator-proportional integral (LQR-PI) and the hybrid control  − Improvement of the linear quadratic regulator control applied to a DC-DC boost … (Adel Bouchahed) 61352.1.LQR control of boost converter

Table 2 .
Estimated parameters of DC-DC boost converter Improvement of the linear quadratic regulator control applied to a DC-DC boost … (Adel Bouchahed)