Classical Controller Design Strategy - Lab Report Example

Controller design Student number Lecturer Abstract This paper presents a report about the controller design. The paper on the basis of literature describes various types of controllers as well as why controllers are needed. As an experimental paper, the information herein focuses on design of controllers and mainly dwells on two controllers: the I-PD base on the standard principles of controller design and an experimental design of the classical controller based on the Ziegler-Nichols approaches. The design uses Matlab for the Mathematical workouts and Simulink (Matlab program) for the simulation of the designed controller to attain the required state of stability. The design controller in this experiment uses the single axis camera controller for the purpose of experimentation of the controller design. As such, the experimentation and details hereinafter refer to/apply to the single-axis controller. Contents Abstract 2 1.Introduction 3 1.1Background 3 1.2Objective 5 2Experimentation 6 2.1Controller Design 7 2.2.1 System closed loop block design 7 2.2.2Controller Requirements 10 2.2.2.1Simulink Results 12 2.4 Ziegler-Nichols approaches 14 3Discussion 15 4Conclusion 15 5References 15 1. Introduction 1.1 Background As Dube and Gulati (2005) describe controllers are a very essential part of any system. Controllers are essential in improving steady state accuracy by decreasing steady state errors which in turn improves the stability of the system. The controllers help in reducing the offsets produced in the system. The maximum overshoot is also controlled and noise signals produced in the system greatly reduced. Slow response of the over damped system is also made faster with the help of these controllers. For a single axis camera, the transfer function using the differential is driven by a servomotor. The input torque that rotates the camera to rotate at an output angle can relate as illustrated below. where: Τ is the input torque provided by the servomotor (Nm) (T=moving part’s combined inertia = 0.2 (kg*m2) • J is the combined Inertia of the moving parts (Kgm2 ) J = moving part’s combined inertia = 0.2 (kg * m2) • D is the combined viscous damping (Kg m2/s) D = combined viscous damping = 0.7 ((kg* m2)/s) • θ Camera directional angle (rad) θ = directional angle (rad) • ω is the camera angular velocity (rad/s) ω = dθ/dt thus relating the torque and ω, Laplace transform is very useful in working out on differential equation in order to make the mathematical process easy to handle. Using the Laplace transform, the Laplace transform is given by: This implies T(s)=J*sw(s) + dw(s)=w(Js+D) A PID controller was used to continuously calculate an error value as the difference between a desired set point and a measured process variable. For a camera tracking an object, the high performance is measured by a significant reduction in settling time and peak overshoot and shorter rise time (Zone.ni.com, 2016). To address this issue well, the I-PD controller is more effective compared to the common PID (J.SujiPrasad, Varghese and A. Balakrishnan, 2012). This is what the study resorted to in order to come up with controller design as illustrated bellow in the controller blocks. 1.2 Objective A single axis camera is expected to have smaller range of response characteristics. It should meet/exceed the following response characteristics (KUAN et al., 2016). overshoot ≤ 10% 95% settling time ≤ 2:5 seconds rise time ≤1 seconds The main objective of the project therefore is to design a controller that meets/exceeds the response characteristics above 2 Experimentation For the above description, the I-PD block design for the controller was as shown below. The system specification for this experimentation was as indicated below; J = 0.2 Kgm2 D =0.7 Kgm2/s Kt = 1.6 V/rad According to the transfer equation, G(s)=w(s)/T(s)=1/Js+D Thus substituting, the new block diagram is. Standard rules were applied in the block minimization process. The standard rules can be found in Msubbu.in (2016) for reference. The transfer function rules. In this case, reducing all to one block in series, the associative and commutative property rule was applied and the resultant as illustrated. 2.1 Controller Design As indicated earlier, the I-PD controller design is the most appropriate design for the single axis camera which for example the time response related requirements are a big necessity of concern. 2.2.1 System closed loop block design The task is for the experiment is to solve for the block design shown below into a single closed loop transfer function block diagram step by step following the relevant rules. Thus incorporating the already simplified above for the system, the diagram below presents the diagram including the controller for an I-PD controller presentation. First, the part below is solved by changing the upper par part to G(s) and the lower part to H(s) for computational simplicity. For the negative feedback loop rule (blocks in parallel), Output/input=feed forward transfer function/(1+feedforward X feedback). The transfer function=G(s)/1+GH which solves to the simplest form using matlab to This thus results to the new block design as shown below The next step is to solve the series part below using the commutativity and associative rule By multiplying the two blocks Ki/s and the one on the right which simplifies the block to the one shown below including the resultant of multiplication Then, the next step is to solve the right side to one block Taking and , the parallel solution of Which then form the block to be Multiplying this with the series block, then the resulting block is as given below From the eventual transfer function from the whole process of block reduction using standard rules. , which is a single transfer function with one block. The following section discusses how the given specifications were used to derive the values of the unknown. 2.2.2 Controller Requirements In finding the values of Kp, Ki and Kd in this experimentation, the transfer function method was used to compute. First, the equation to find the settling time for 95% is given by , …….i The equation below is used in deriving the second order transfer function. , Substituting ts=2.5 in the first part ,i, we get , then making wn the subject, we get, , Taking the above as points of note and considering that the equation of rise time is as shown below, the process continues which when substituted, the equation for wd becomes In order to solve for the , wd in C above is substituted and the resulting equation D noted as below . Considering B and substituting into D Then both the sides are squared to give equation E as shown below Which when substituted into equation D and then squared, equation F is generated as shown below both of which the E and F are substituted into A to give G as shown below . From this the unknown values can be illustrated as below Gettting Ki and KD involves getting (G) into a third order transfer function as the first order transfer function. With a value of A multiplied by G it leads to which solves the values of of Kp and Ki by comparing with 2.2.2.1 Simulink Results In order to test if our solution meets/exceeds the requirement, Matlab and Simulink were used to simulate the block diagram shown below Figure 1: The Simulink block diagram. The results were as shown below When run, the Simulink waveform was as shown below. From the diagram, however, it shows that the settling time is exceeding the requirements specified. However the rise time is less than anticipated as well as the rise time thus meeting the two desired response requirement. From there was some tradeoffs when varying the various characteristics. As would be expected from (Ogata, 2010), exceeding the the peak overshoot time (i.e reducing the time it takes to reach for peak overshoot), it results in increased time for settling. It also results in longer rise time. Shorter settling time also results from longer rise time. As such, it can be concluded that exceeding (improved) a given response characteristic came at an expense of another response charactestic. In order strike a balance, the threshold set as shown in the objective section of the paper would help guide in attaining some state of balance for the three response characteristics. A classical method is also one of the methods for controller design. One of the models is as discussed below with the relevant PID 2.4 Ziegler-Nichols approaches The ziegler-Nichols rules, have that the Ki and Kd are zero and thus the system proportional term is adjusted until a critical point Kc. The table below helps in computing the system gains. Using this and the matlab code procedure on this basis, the following terms were found. The simulink was run for this model to determine the validity of the procedure and the results showed the same results. 3 Discussion As shown by a comparison of the two simulations, it can be seen that there is difference. There is a larger overshoot when a classical method such as the Zieglaer-Nichols method is used. The I-PD method is more accurate, hence a better option copared to the classical one. 4 Conclusion As a conclusion, it can be said that the objective was satisfied in adjusting the I-PD controller so that to control the single-axis camera in meeting the desired design in terms time-related response characteristics. The values of the gains Kp, Ki and Kd were successfully coomputed. A comparison of the classical PID rules and the standard rule I-PD controller that the standard rules are more accurate. 5 References Anon, (2016). [online] Available at: http://bolvan.ph.utexas.edu/~vadim/Classes/14s/linang.pdf [Accessed 20 Apr. 2016]. Dann, J. (2016). [online] Available at: http://www.dsif.fee.unicamp.br/~manera/EE640/files/Tutorial_Stability_and_the_s-plane.pdf [Accessed 20 Apr. 2016]. J.SujiPrasad, S., Varghese, S. and A. Balakrishnan, P. (2012). Optimization of I-PD Controller for a FOLIPD Model using Particle Swarm Intelligence. International Journal of Computer Applications, 43(9), pp.23-26. KUAN, T., OON, T., JIAR, Y. and SUPRIYANTO, E. (2016). Facial Tracking based Camera Motion Control System. In: TELECOMMUNICATIONS and INFORMATICS. [online] UTM Skudai: Progressive Health Care and Human Development Research Group. Available at: http://www.wseas.us/e-library/conferences/2010/Catania/TELE-INFO/TELE-INFO-37.pdf [Accessed 20 Apr. 2016]. Michiels, W. and Niculescu, S. (2007). Stability and stabilization of time-delay systems. Philadelphia: Society for Industrial and Applied Mathematics. Read More