Matlab and Simulink for Modeling and Control

Transcription

1 Matlab and Simulink for Modeling and Control 1 Introduction Robert Babuška (r.babuska@tudelft.nl) This text is a handout for the first MATLAB session within the course Systeem- en Regeltechniek 2. A short introduction to MATLAB is included in the beginning for those who are totally inexperienced with this software. The main focus is then on the elementary use of the Control System Toolbox for the implementation and analysis of linear time-invariant (LTI) systems. Three sets of basic tools are available: 1. General MATLAB functions for matrix algebra (*, inv, exp,... ), nonlinear functions (sin, abs,... ), data visualization (plot, stairs,... ), and data storage (save, load). Type help or demo for more details on general-purpose MATLAB functions. 2. Control System Toolbox functions and tools for the definition and analysis of linear time-invariant (LTI) systems and synthesis of controllers (tf, ss, pole, rltool, place,... ). Type help control for a list of all Control Systems Toolbox functions. 3. Simulink for the implementation and simulation of nonlinear (and of course also linear) dynamic systems. Type help simulink for more information or type simulink to open the main Simulink library. 2 Matlab basics Entering commands. Commands in MATLAB can be entered in three ways: via the Command Window, through scripts (programs stored in files with extension *.m, the so called M-files), and functions (also stored in files with extension *.m). When using the Command Window, MATLAB operates in an interpreter mode, executing the commands directly. Scripts and functions are to a certain degree precompiled (automatically, during the first call) in order to speed up the execution. For very simple ad hoc computations, it is possible to use the Command Window directly (use up and down arrows to browse through the history of previously entered commands. In all other cases (e.g., working on a project), it is much more effective to write scripts and functions to execute a desired sequence of commands (create a new file via the File menu or click on the New M-file icon and save the file in your working directory). Data types and variables. MATLAB supports all usual data types like real numbers, characters, strings, structures, cell arrays, etc. However, the most prominent feature of this package is its ability to compute with vectors and matrices. This facilitates very efficient programming of linear algebra computations, including all kinds of applications, such as filtering, signal and image processing, control, etc. Rather than using for and while loops, like in C or Pascal, it is recommended to vectorize computations as much as possible. For instance, to plot a graph of the function exp(-x) for x from 1 to 10, the C-style program would be something like: for i = 1 : 10, y(i) = exp(-i); end; plot(y); while a single MATLAB command can do the job: plot(exp(-(1:10))). To practice this way of programming, interested readers can download from Blackboard a separate document titled Simple exercises to get used to MATLAB vector and matrix notation. However, in some cases the use of for-loops is inevitable. Data handling and visualization. MATLAB has extensive capabilities for importing, processing, storing, and visualizing data. Use the functions load and save to load and save MATLAB data files (*.mat). To create plots, see functions plot, label, title, mesh, etc. 1

2 3 Entering and analyzing LTI models in Matlab Linear time-invariant systems can be entered in the state-space form (ss) or as transfer functions, either by means of the numerator and denominator coefficients (tf) or by means of the zeros, poles, and the gain (zpk). Coefficients of polynomials are entered in the descending order of the powers of s (continuous time) or z (discrete time). For instance, the polynomial A(s) = 3s 3 +2s+10 is defined as the vector: A = [ ]. Consider the transfer function G(s) = ω 2 s 2 +2ζωs+ω 2 aω s+aω (1) First, define some values for the parameters in MATLAB: w = 3; z = 0.8; a = 2; Now, the transfer function (1) can be entered by using the function tf and the product operator: G1 = tf(wˆ2,[1 2*w*z wˆ2]) G2 = tf(a*w,[1 a*w]) G = G2*G1 Note that complex systems can be defined in terms of simpler subsystems combined in parallel (operator +), series (operator *) or feedback (function feedback). The standard MATLAB operators are overloaded (redefined) for the LTI class of the Control System Toolbox such that they compute the respective combination of the linear systems (instead of addition or multiplication of scalars or matrices). Another possibility is to first define a system representing the Laplace operator s (corresponding to a differentiator) and then enter the transfer function as an algebraic expression ins(in a symbolic way): s = tf([1 0],1) G1 = wˆ2/(sˆ2 + 2*w*z*s + wˆ2) Exercise 1: Enter the above transfer functions in MATLAB. Use functions pole, zero, and damp to get information about the poles and zeros of the system and the corresponding frequencies and relative damping coefficients. Is the system stable? ConvertGto the zero-pole-gain and state-space forms (using the functions zpk and ss). Form a feedback system G c by using the function feedback with a proportional controller K p = 3. Is the relative damping of G c larger or smaller than the damping of G? 4 Time-domain and frequency-domain plots There is a variety of functions for plotting all possible responses, such as step, impulse, bode, etc. The most important ones are integrated in an LTI viewer (ltiview). As a default plot, the step response is shown. By right-clicking on the plot area, a menu pops up that allows us to choose a different plot or to compute some important characteristics, such the settling time, rise time, gain and phase margins, etc. Exercise 2: Use the LTI viewer to get the step response and impulse response for our system G. What is the settling time and the rise time? 5 Control design for a magnetic levitation system Electromagnetic levitation is used for friction-less support of high-speed trains, in bearings of low-energy motors, and other applications. The system to be controlled consists of an electromagnet which is attracted to an object made of a magnetic material. The air gap between this material and the electromagnet must be kept constant despite disturbances, such as forces acting on the train. 2

3 The goal is to design a controller that will keep the air gap at a desired value by adjusting the current through the coil (see the schematic below). This system has one control input u(t), which is the current through the electromagnet and one measured output: y(t), the airgap. A nonlinear model of this process can be derived from elementary physical laws. Through linearization, the following differential equation is obtained: ÿ(t) = ay(t) bu(t) where a > 0 and b > 0 are known positive real parameters. In the sequel, we will represent this model as a transfer function and analyze its properties, both in the open-loop and closed-loop configuration. Please, work out items a) through e) as homework preparation for the lab session. The remaining items f) and g) will be done in the lab, using Matlab and Simulink. a) Represent the above differential equation in a block diagram. 3

4 b) Derive the transfer function of the open-loop process: G(s) = Y(s) U(s) = c) Compute the poles and analyze stability of the open-loop system. p 1 = p 2 = Stability: d) Consider a proportional controller C(s) = K p, using negative feedback. Derive the closed-loop transfer function G c (s) from the reference R(s) to the output Y(s). Can this controller stabilize the system? Motivate your answer. G c (s) = Y(s) R(s) = Motivation: e) Now consider a proportional-derivative (PD) controller: C(s) = K p + K d s. Again, derive the closed-loop transfer function G c (s) from the reference R(s) to the output Y(s). Can the PD controller stabilize the system? Motivate your answer. G c (s) = Y(s) R(s) = Motivation: 4

5 f) Enter the open-loop transfer function G(s) in MATLAB. Use the following parameter values: a = 1962 and b = 0.94; Compute the poles and verify the result by comparing with your solution to item b). Create a vector containing a range of proportional gainsk p = 10, 20, 30,..., You can use the following command: Kp = -[10:10:3000] ; Compute the closed-loop transfer function and its poles for each of the gains in the vector Kp. It is convenient to implement this in a little program (MATLAB script). In this program, you can also define the values of the parameters a and b, and the command for defining the open-loop transfer function. To compute the closed-loop poles for each value of K p, you can use the following forloop (supply appropriate parameters or commands instead of the dots... ): for i = 1 : length(kp), Gc = feedback(...); p(i,:) =...; end; Plot the poles in the complex plane. 1 Note that the function plot directly works also for complex numbers. Sketch the resulting plot in the box below: Does the result obtained correspond to your analysis under item d)? For what values of K p is the closed-loop response oscillatory? K p = 1 Note that the plot of the location of closed-loop poles with respect to the controller gain is called the root-locus. The Control Systems Toolbox has special functions to plot the root locus, such as rlocus and rltool, so we do not need to do this programming ourselves. The basic idea behind, however, is the same. 5

6 g) Implement the block diagram from item a) in Simulink. Close the loop with negative feedback using an ideal PD controller C(s) = K p +K d s. One option is to use the Simulink PID Controller block which implements a realistic approximation of the PID controller. Set the Filter coefficient (N) equal to Another (better) option is to implement the PD controller as a Subsystem block (in the Commonly Used Blocks library) in which you implement the PD controller using the Derivative, Gain and Sum blocks. As a reference, use a step function with an amplitude of m. Start with the following gains: K p = , K i = 0, K d = 200. Sketch the resulting closed-loop step response in the box below: How large is the steady-state error (in % of the reference)? Verify this value by calculations in MATLAB. Hint: compute first the closed-loop transfer function by means of the function feedback, then use the function dcgain to find the stationary gain of the closed loop transfer function. e ss = Increase the proportional gain by factor 5 and simulate the closed-loop system. What is the effect on the transient response and on the steady-state error? What happens if you also increase the derivative gain? Is it possible to completely remove the steady-state error? How? Got so far? Congratulations, you are done for today! It might be a good idea to store your results on a memory stick or another medium for future reference. 6

7 6 Useful Matlab and Control Systems Toolbox Functions Creating and converting linear models tf - Create (or convert to) a transfer function model. zpk - Create (or convert to) a zero/pole/gain model. ss - Create (or convert to) a state-space model. feedback - Feedback connection of two systems. c2d - Continuous to discrete conversion. d2c - Discrete to continuous conversion. Model analysis dcgain bandwidth pole, eig zero pzmap damp ltiview step impulse lsim bode ctrb, obsv - D.C. (low frequency) gain. - System bandwidth. - System poles. - System zeros. - Pole-zero map. - Natural frequency and damping of system poles. - Response analysis GUI (LTI Viewer). - Step response. - Impulse response. - Response to arbitrary inputs. - Bode diagrams of the frequency response. - Controllability and Observability matrix. Design tools place acker sisotool rlocus rltool - MIMO pole placement. - SISO pole placement. - SISO design GUI (root locus and loop shaping techniques). - Evans root locus. - Runs the SISO design GUI set up for root locus. Data visualization and storage figure - Create figure window. clf - Clear current figure. plot - Plot data. stairs - Stair-step graph. save - Save workspace variables to disk. load - Load workspace variables from disk. Simulink Matlab trim linmod - Finds steady state parameters for a Simulink system. - Linearize a Simulink model around an operating point. 7

8 7 Appendix: nonlinear model of the electromagnetic levitation system Physical modeling. The motion of an object in a magnetic field is determined by the forces that act on it. In our case, these forces are: (i) the gravitational force, (ii) the electromagnetic force, and (iii) a disturbance force. The dynamic equation of the system is derived from the Newton s law: ÿ(t) = 1 m (F grav +F dist F R ), (2) wheremis the mass of the train,f grav = mg andf dist is an external disturbance (e.g., due to the vertical components of external forces acting on the train). The electromagnetic force is a nonlinear function of the current i through the coil and the gapy: F R = K mag i 2 (t) y 2 (t). (3) In our example, we use parameters of a small-scale model: K mag = Hm and m = 8 kg. Linearization. For the ease of notation, assume that the disturbance forcef dist = 0. The equilibrium of the nonlinear model (2) is given by: 0 = mg K mag i 2 0 y 2 0 i 0 = mg K mag y 0 Now we can choose some operating point y 0 for the gap, compute the corresponding current i 0 and linearize the right-hand side of (2) around this operating point: Defining ÿ(t) = 2K magi 2 0 my 3 0 y(t) 2K magi 0 my0 2 u(t). a = 2K magi 2 0 my 3 0 and b = 2K magi 0 my0 2, we obtain the final linearized model: ÿ(t) = ay(t) bu(t) wherea = 1962 and b = were obtained for y 0 = 0.01 m. 8