Performance Analysis of Speed Control of Direct Current (DC) Motor using Traditional Tuning Controller Vivek Shrivastva 1, Rameshwar Singh 2 1 M.Tech Control System Deptt. of Electrical Engg. NITM Gwalior 2 Assistance Prof. Deptt. Of Electrical Engg. NITM Gwalior Abstract The control of the speed of a DC motor is an important issue and has been studied since the early decades in the last century. This paper presents a comparison of time response specification between different controller and Linear Quadratic Regulator (LQR) for a speed control of a DC motor. The goal is to determine which control strategy delivers better performance with respect to DC motor s speed. Performance of these controllers has been verified through simulation using MATLAB/SIMULINK software package. According to the simulation results the Comparing with PID- ZN controller, PID-MZN controller and PI-PSO controller, the tuning method was more efficient in improving the step response characteristics such as, reducing the rise time, settling time and maximum overshoot in speed control of DC motor. Linear quadratic regulator method gives the better performance and superiority of liner quadratic regulator method over other controller. Keywords PID-ZN, PID-MZN, PID PSO controller, DC motor, Linear quadratic regulator. I. INTRODUCTION Dc motors are controllable over a wide range with stable and linear characteristics. Therefore, they are the most common choice in the industries for both constant speed and constant load operation [1].The field of electrical energy will be divided into three areas: Electronics, Power and Control. Electronics basically deals with the study of semiconductor devices and circuits at lower power. Power involves generation, transmission and distribution of electrical energy. The electric motors are perhaps the most widely used energy converters in the modern machine tools and robots. These motors require automatic control of their main parameters such as speed, position, acceleration etc. In this paper to control the speed of DC motor, their simplicity, ease of applications such as reliability and favourable cost have long been a backbone of industrial applications and it will have a long tradition of use as adjustable speed machines and a wide range of options have evolved for this purpose. In these applications, the motor should be precisely controlled to give the desired performance [2]. The past decades witnessed many advancing improvements keeping in mind the requirement of the end users. 119 Several approaches have been documented in literatures for determining the PID parameters of such controllers which is first found by Ziegler- Nichols tuning [3,4],Proportional-Integral-Derivative PID controller, due to its simplicity, stability, and robustness, is a type of controller that is most widely applied [11], The fuzzy logic, unlike conventional logic system, is able to model inaccurate or imprecise models. The fuzzy logic approach offers a simpler, quicker and more reliable solution that is clear advantages over conventional techniques [6], Genetic algorithm [7], and PSO [8]. Linear quadratic regulator design technique is well known in modern optimal control theory and has been widely used in many applications. It has a very nice robustness property. This attractive property appeals to the practicing engineers. Thus, the linear quadratic regulator theory has received considerable attention since 1950s. The liner quadratic regulator technique seeks to find the optimal controller that minimizes a given cost function (performance index). This cost function is parameterized by two matrices, Q and R, that weight the state vector and the system input respectively. These weighting matrices regulate the penalties on the excursion of state variables and control signal. One practical method is to Q and R to be diagonal matrix. The value of the elements in Q and R is related to its contribution to the cost function. To find the control law, Algebraic Riccati Equation (ARE) is first solved, and an optimal feedback gain matrix, which will lead to optimal results evaluating from the defined cost function is obtained [9-10]. In this paper to achieve accurate control performance of speed control of dc motor by using of Linear Quadratic Regulator (LQR) technique presented. In this paper compared with PID-ZN controller, PID-MZN controller and PI-PSO controller. And minimizing of overshoot, minimizing of rise time and minimizing setting time. II. SYSTEM MODELING OF DC MOTOR DC motors are most suitable for wide range speed control and are therefore used in many adjustable speed drives.
DC motor shown in Figure 1 is the one of most common motors which used in industrial motion control systems and Fig 2 shows a Direct Current (DC) Motor Model. Ra is the armature resistance and La is the armature inductance of dc motor. Rf is the field resistance and Lf is the inductance of the field winding, i a Armature current, J Moment of inertia of motor, K T Torque factor constant, Kb Back emf constant B Viscous Frictional coefficient. [ ] [ ] [ ] [ ] [ ] [ ] [ ]... (4) Obtaining the transfer function of the motor using the state space model by formula G(s)= C (s I - A) -1 B + D [14] in the equation (3) and (4) and obtain the equation 5. Fig 2 show the dc motor armature control system. Fig 1 DC motor [12] Fig. 3- Block Diagram of DC Motor Fig 2 Direct Current (DC) Motor Model [13] A linear model of a simple DC motor consists of an electrical equation and mechanical equation. Using Kirchhoff s Voltage Law (KVL) and Newton s second law, the following equation is obtained: Assuming the above equations, the steady state representation can be obtained as: III. LINEAR-QUADRATIC REGULATOR (LQR) OPTIMAL CONTROL LQR is a method in modern control theory that used state-space approach to analyse such a system. Using state space methods it is relatively simple to work with Multi- Input Multi-Output system. Linear quadratic regulator design technique is well known in modern optimal control theory and has been widely used in many applications, Linear-Quadratic Regulator (LQR) optimal control problems have been widely investigated in the literature. 120
Fig 4 Block diagram of DC motor control system used by LQR Controller. The performance measure is a quadratic function composed of state vector and control input. If the linear time-invariant system is controllable, the optimal control law will be obtained via solving the algebraic Ricci equation optimal control. The function of Linear Quadratic Regulator (LQR) is to minimize the deviation of the speed of the motor. The speed of the motor is specifying that will be the input voltage of the motor and the output will be compare with the input. In general, the system model can be written in state space equation as follows: The diagonal-off elements of these matrices are zero for simplicity. If diagonal matrices are selected, the quadratic performance index is simply a weighted integral of the squared error of the states and inputs. The term in the brackets in equation (8) above are called quadratic forms and are quite common in matrix algebra. Also, the performance index will always be a scalar quantity, whatever the size of Q and R matrices.the conventional linear quadratic regulator problem is to find the optimal control input law u* that minimizes the performance index under the constraints of Q and R matrices. The closed loop of dc motor with linear quadratic regulator show in the fig 1, The closed loop optimal control law is defined as: A is the state matrix of order n n B is the control matrix of order n m. Also, the pair (A, B) is assumed to be such that the system is controllable. The linear quadratic regulator controller design is a method of reducing the performance index to a minimize value. The minimization of it is just the means to the end of achieving acceptable performance of the system. For the design of a linear quadratic regulator controller, the performance index (J) is given by: Where Q is symmetric positive semi-definite state weighting matrix of order and R is symmetric positive definite control weighting matrix of order The choice of the element Q and R allows the relative weighting of individual state variables and individual control inputs as well as relative weighting state vector and control vector against each other. The weighting matrices Q and R are important components of an LQR optimization process. The compositions of Q and R elements have great influences of system performance. The designer is free to choose the matrices Q and R, but the selection of matrices Q and R is normally based on an iterative procedure using experience and physical understanding of the problems involved. Commonly, a trial and error method has been used to construct the matrices Q and R elements. This method is very simple and very familiar in linear quadratic regulator application. However, it takes long time to choose the best values for matrices Q and R. The number of matrices Q and R elements are dependent on the number of state variable (n) and the number of input variable (m), respectively. Fig 4 Block diagram of DC motor control system used by LQR Controller. Where K is the optimal feedback gain matrix, and determines the proper placement of closed loop poles to minimize the performance index in equation (8). The feedback gain matrix K depends on the matrices A, B, Q, and R. There are two main equations which have to be calculated to achieve the feedback gain matrix K. Where P is a symmetric and positive definite matrix obtained by solution of the ARE is Defined as: (10) Then the feedback gain matrix K is given by: K = Substituting the above equation (9) into Equation (7) gives: (12) If the Eigen values of the matrix (A-BK) have negative real parts, such a positive definite solution always exits [9, 10, 14]. 121
IV. RESULTS AND DISCUSSION The performance and tuning using the Linear-Quadratic Regulator (LQR) controller Optimal Control has been compared with several controllers such as PID-ZN controller, PID-MZN controller and PI-PSO controller and its two different rating of dc motor as described below in the table 1and table 4 [5]. Simulations were carried out using MATLAB 7.0.1 on a Pentium IV processor, 2.8 GHz. with 1 GB RAM. Test Case I: in this test system the data of dc motor show in the table 1 and transfer function of dc motor equation 6 and 13 used as a system and find out the response of the system applying the step function as an input. and the tuning of different point such as the LQR parameter Q and R:Q=[0.0000006;0.2], R=[0.000005] and Q=[0.000006;0.7 ],R=[0.000005] used in the motor 1.achieve The best results show in the table 2 and fig 5 and 6.and compare LQR controller than the several controller such as PID-ZN controller, PID-MZN controller and PI- PSO controller for different parameter such as rise time, setting time and overshoot, using the LQR tuning controller Q=[0.00004;0.2], R=[0.000000006] for a dc motor 1and gives the better rise time (0.0466),setting time (0.0833), and overshoot(0.00) show in the fig 7and table 3. Table 1. Dc motor parameters motor 1[5] Table 2 Show Best Two Results Of Dc Motor Using LQR Controller Fig 5 Fig System Response with LQR controller Show the transfer functions for the dc motor 1..(13) Fig 6 System Response with LQR controller 122
Table 3 Comparative analysis of various tuning controller for motor 1 The best results show in the table 5 and fig 8 and 9.and compare LQR controller than the several controller such as PID-ZN controller, PID-MZN controller and PI-PSO controller for different parameter such as rise time, setting time and overshoot, using the LQR tuning controller Q=[.000006 0;0 0.3];R=[0.00000000083]; for a dc motor 2 and gives the better rise time (0.0601),setting time (0.109), and overshoot(0.00) show in the fig 10 and table 6. Table 4. Dc motor parameters motor 4 [5] Show the transfer functions for the dc motor 2. (14) Table 5 Show Best Two Results Of Dc Motor Using Lqr Controller Fig 7 System Response with LQR controller Test Case II: for test system 2 the data of dc motor show in the table 4 and transfer function of dc motor equation 14 used as a system and find out the response of the system applying the step function as an input. and the tuning of different point such as the Linear Quadratic Regulator (LQR) parameter Q and R : Q=[.00000006 0;0 0.3],R=[0.00000008],Q=[.00000020;00.3];R=[0.00000004] ;used in the motor 2.achieve 123
Fig 8 System Response with LQR controller Fig 10 System Response with LQR controller Fig 9 System Response with LQR controller Table 6 Comparative analysis of various tuning controller for motor 1 V. CONCLUSION In this paper the speed of a DC motor is controlled using Linear-Quadratic Regulator (LQR). The simulation results are obtained using MATLAB/SIMULINK. LQR response is compared with that of PID-ZN, PID-MZN,PI-PSO controller. The results show that the overshoot, settling time, peak time and control performance has been improved greatly by using Linear-Quadratic Regulator (LQR) controller. The LQR controller has more advantages, such as higher flexibility, control, better performance compared with ZN, PID-MZN,PI-PSO controller. Hence, the comparison results showed that the Linear-Quadratic Regulator (LQR) controller was the best controller. REFERENCES [1 ] Rishabh abhinav, satya sheel,senior member,ieee, An adaptive,robust control of dc motor using fuzzy-pid controller,ieee international conference on power electronics, drives and energy systems,december 16-19,2012,bengaluru,Indian. [2 ] Dr.Ch.Chengaiah1,K.Venkateswarlu, Comparative Study On Dc Motor Speed Control Using Various Controllers, International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering,, Vol. 3, Issue 1, January 2014. [3 ] J. C. Basilio and S. R. Matos, Design of PI and PID Controllers With Transient Performance Specification,, IEEE Trans. Education, vol. 45, Issue No. 4, 2002, pp. 364-370. [4 ] P.M.Meshram, Rohit G. Kanojiya, Tuning of PID Controller using ziegler-nichols method for Speed Control of DC motor,ieee international confrence on advanced inengineering, science and management (ICAESM-2012) march 30,31,2012, 124
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