Đang tải... (xem toàn văn)
Tài liệu hướng dẫn xây dựng mô hình bài toán điều khiển hệ con lắc ngược cả về góc và vị trí. Đây là tâm huyết của nhóm trong quá suốt quá trình nghiên cứu và thực hiện. Mục đích tôi chia sẻ bài viết này nhằm giúp cho những bạn sinh viên hay học sinh có niềm đam mê với chuyên ngành điều khiển tự động của tự động hóa có thể tự nghiên cứu, là một kênh tham khảo đã được cô đọng.
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING INVERTED PENDULUM CONTROLLER DESIGN Instructor: Department: Student Name Automatic Control Student ID Class Hanoi, Table of Contents 2|Page Abstract Inverted pendulum system is typical multi-variable, non- linear, strong coupling, and instability naturally, as a typical control target, it has been subjected to many experts and scholars’ concern In this report, single-link inverted pendulum was chosen as the study subject This paper represents apply Genetic Algorithm (GA) for tuning the parameters of PID controller in balancing system in upwardposition Results was operated in MATLAB/Simulink environment Keyword: Inverted pendulum, GA, PID, MATLAB, Simulink Chapter 1: Overview Stability for the inverted pendulum is a familiar problem in automatic control However, most research that control the pendulum balance with the PID controller only stop at finding the parameters for the controller Genetic algorithm is a heuristic search inspired by the natural evolutionary theory of Charles Darwin This algorithm reflects the natural selection process in which 3|Page the healthiest individuals are chosen to breed to produce offspring of the next generation The paper describes a method of applying the GA algorithm to find the optimal value of PID controller for the stability system This means that using a different PID to stabilize the system is then optimized by a PID controller with parameters from the genetic algorithm 1.1 Inverted Pendulum System The problem associated with stabilization of Inverted Pendulum is a very basic and benchmark problem of Control System The design of Inverted Pendulum consists of a DC motor, Cart, Pendulum and Cart driving mechanism The nature of this system is single input and multi output system where control voltage act as input and the output of the system are cart’s position and the angle Here we must stabilize the pendulum angle to inverted position which is a challenging work to as the inverted position is a highly unstable equilibrium The main characteristics of the system are highly unstable as we have to stabilize the Figure Simple Inverted Pendulum Setup pendulum angle to inverted position, it is highly nonlinear system as because the dynamic of inverted pendulum consists non-linear terms, as the system have a pole on its right hand it is a non-minimum phase system and the system is also under actuated because the system has only one actuator (the DC Motor) and two degree of freedom 1.2 Problems setup and design requirements 4|Page For the PID, root locus, and frequency response sections of this problem we will be only interested in the control of the pendulums position This is because the techniques used in these tutorials can only be applied for a single-input-single-output (SISO) system Therefore, none of the design criteria deal with the cart's position For these sections we will assume that the system starts at equilibrium and experiences an impulse force of 1N The pendulum should return to its upright position within seconds, and never move more than 0.05 radians away from the vertical In, summary, the design requirements for this system are: • Setting time less than seconds • Pendulum angle never more than 0.05 radians from the vertical 1.3 Control Methods There are many controller types Like: P, PI, PD, PID, fuzzy controller, LQR controller, distributed order PID… - PD controller is one of a simple controllers in implementation - LQR controller to improve the system response - PID combined with LQR controller was used aiming to enhance the performance - Fuzzy-logic controllers are applicable to an IP system like fuzzy parallel distributed compensation (PDC) controller 5|Page Chapter 2: Inverted pendulum controller of MATLAB 2.1 System modeling (M): mass of the cart [0.5kg] (m): mass of pendulum [0.2kg] (b): coefficient of friction for cart [0.1 N/m/sec] (l): length to pendulum center of mass [0.006kg.] (I): mass moment of inertia of the pendulum (x): cart position coordinate (Ө): angle vertical pendulum from (down) Figure 1: Free-body diagrams From the free-body diagram, our group will analyze the related force on the system on both the cart and the pendulum to get hold of the transfer function for the inverted pendulum First, we summarize the forces of the cart in the horizontal direction, we will obtain: F-M-b–N=0 (1) Next, we calculate the forces of the pendulum in the horizontal direction, we will get the expression for the reaction N: N= m +ml (2) Then we have sum of the forces of the direction perpendicular to the pendulum, we get: P+N -mg = ml +m (3) When we substitute force N from equation (2) into (1), we achieve: F = (M + m) + b + ml (4) Sum the moments I about the centroid of the pendulum, we have: -Pl -Nl= I Combining equation (4) and (5) we will get: 6|Page (5) (I + m)+ mgl=- ml (6) We denoted ϕ be the deviation angle of the pendulum’s position from the equilibrium The equilibrium is vertically upward position, which is ; therefore, + ϕ Because angle ϕ is small, from the equation (4) and (6), we obtain: (I + m)- mgl=ml (7) F = (M + m) + b - ml (8) To get the transfer function of the inverted pendulum, we will take the Laplace transform with initial condition is zero Therefore, after Laplace transform, with L {and L {, we got: (I + m)- mgl=ml (9) F(s) = (M + m) + b - ml (10) The transfer function will be the relation between the input F(s) and the output Y(s) From equation (9), we have relation X(s) and Y(s), then substitute to equation (10), we will have: F(s) = (M + m) + b - ml Denoting q = (M + m) (I + m)-, we will have the required transfer function: =( 2.2 Controllers of MATLAB 2.2.1 PID Controller According to MATLAB, our group designed the schematic with the force F is the input to the output angle ϕ and the controller is set as the response signal Therefore, we have the transfer function T(s) for the closed-loop system: T(s) = = Figure 2: The schematic We can apply P, I, PI, PID controller to stabilize the angle = For uncovering the appropriate set of coefficients of controller, MATLAB used empirical method 7|Page They used the set of both P, I and D to control the system In the beginning, we set Kp= Ki= Kd = 1, the result will be shown as below: - Comment: For the first set of numbers, the system is unstable - Solution: Increasing the coefficient of Kp to stabilize the system Figure 3: Kp=1, Ki=1, Kd=1 Next, we increased coefficient Kp to 100, and the result was: - Comment: By increasing Kp, the system is stable; otherwise, response time up to seconds and the range of overshoot is high (0.2rad) - Solution Increasing the coefficient Kd to reduce both the response time and the overshoot Figure 4: Kp=100, Ki=1, Kd=1 Finally, we adjusted coefficient of Ki to 20 and the outcome was: - Comment: The response time was reduced by half to second and the overshoot is trivial - Result: This coefficient of this PID controller solved the problem 2.2.2 Other controller methods 8|Page Figure 5: Kp=100, Ki=1, Kd=20 Beside using PID controller, there are still other methods can be mentioned to control the inverted pendulum, namely Linear- Quadratic Regulator (LQR) design, Linear- Quadratic-Gaussian(LQG) design, loop shaping method or Internal model control(IMC) method 2.3 Simulation Instead of making a real inverted pendulum system, our group followed the simulation by using Simscape in MATLAB software Firstly, our group set up the inverted pendulum on cart simulation with the input is force F, and the output will be the cart position as well as its velocity (lack of q and w pendulum) Next, our group will design a closed- loop set up We transfer the disturbance in the input, using a manual switch to convert between not using the controller and using PID controller for adjusting the equilibrium Entering the coefficient of Kp= 100, Ki= 1, Kd= 20 The result is shown as below: 9|Page Figure 6: Response of the cart Figure 7: Response of the pendulum The graph illustrates the response of the inverted pendulum when it has an impact When the inverted pendulum has a force applied, the controller will adjust the angle as well as the position The cart will move with constant velocity, in the negative horizontal direction to balance the pendulum It also immediately reduces the angle to zero comparing to positive vertical direction and its velocity to zero 10 | P a g e Chapter 3: Control Inverted Pendulum with Genetic Algorithm 3.1 Introduction of Genetic Algorithm (GA) The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution The genetic algorithm repeatedly modifies a population of individual solutions At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation Over successive generations, the population "evolves" toward an optimal solution You can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear In this project, we will use GA in MATLAB to find the best PID controller possible 3.2 Using GA to design PID controller For the method we are using, the system must be stable So, the stable closed loop system in Chapter will be used but with no disturbance A new closed loop system with the system in Chapter is the control object has been created We have new system of inverted pendulum: Figure 8: Simulation using GA Now, we will use GA to find of new controller First, we create two new scripts in MATLAB The first script will be named as “PID_GA_mod.m”, and the second script is “runPIDGA” For the first script, a function will be created: 11 | P a g e function fitness = PID_GA_mod(x) kp=x(1); ki=x(2); kd=x(3); b0=91; b1=455; b2=4.55; a0=1; a1=91.18; a2=423.82; a3=0.1; S=tf([b0 b1 b2],[a0 a1 a2 a3]); C=tf([kd kp ki],[1 0]); G=feedback(S*C,1); [y, t]=step(G); The second script: clc; [x, fval] = ga(@PID_GA_mod,3,-diag([1 1]),zeros(3,1)); kp=x(1);ki=x(2);kd=x(3); b0=91; b1=455; b2=4.55; a0=1; a1=91.18; a2=423.82; a3=0.1; S=tf([b0 b1 b2],[a0 a1 a2 a3]); Run the second script and we have the result: =54.0299, =37.0575, =0 The system to a step input: Figure 9: Finding controller using GA 3.3 Simulation and Analysis To evaluate new system, we will adjust disturbance applied to pendulum and simulate it in Simscape: 12 | P a g e The disturbance is close to impulse response, and it will be applied until second of simulation Result from the above scope: - Comment: The angle of pendulum almost has no change from vertical position The angle velocity increases a bit when the force is applied but return to zero immediately Figure 10: Response of the pendulum with new PID Result from the below scope: - Comment: The cart position has no change Cart velocity is similar with pendulum velocity 13 | P a g e Figure 11: Response of the cart with new PID Note: “q pendulum” is the angle of pendulum, “w pendulum” is the angle velocity of pendulum, “x cart” is the position of the cart, “v cart” is the velocity of the cart Analysis: - The cart and pendulum almost not move even when the disturbance is applied No steady state errors Settling time approximate zero Overshoot is much smaller than the previous system (Approximate zero rad when applied by disturbance) With new PID controller from GA, the system performs much better than the system with only one PID controller GA is a very powerful tool in optimization 14 | P a g e References Nguyen Doan Phuoc (2019) “Optimization and Applications in control (4th edition)” Nguyen Doan Phuoc (2009) “Lí thuyết điều khiển tuyến tính(4th edition)” University of Detroit Mercy “Control Tutorials for MATLAB & Simulink’ 15 | P a g e ... less than seconds • Pendulum angle never more than 0.05 radians from the vertical 1.3 Control Methods There are many controller types Like: P, PI, PD, PID, fuzzy controller, LQR controller, distributed... 1.1 Inverted Pendulum System The problem associated with stabilization of Inverted Pendulum is a very basic and benchmark problem of Control System The design of Inverted Pendulum consists of... Inverted pendulum, GA, PID, MATLAB, Simulink Chapter 1: Overview Stability for the inverted pendulum is a familiar problem in automatic control However, most research that control the pendulum