MATLAB Introduction and Review

Size: px

Start display at page:

Download "MATLAB Introduction and Review"

Sharleen Mills
6 years ago
Views:

1 MATLAB Introduction and Review Alireza Abouhossein Ph.D. 1

2 Introduction MATLAB: MATrix LABoratory System Analysis: SISO (single input -single output) MIMO (multiple input multiple output) Linear and non-linear Time domain and frequency domain (Bode, Nyquist) Stability (Lyapunov) Controller development PID Pole placement Optimal Control regulators (LQR LQG) Control System Toolbox: Provides tools for modeling, analysis of a system 2

3 Matlab arithmetic i. Definition of variables and evaluation of numerical expressions >> a= 3; b=5; >> a*b Ans = 15 ii. Vectors: >>A = [ ] >> a = >> a= [1:2:10] a = iii. Matrices >> A = [1 2; 3 4] A = ; operator supresses display of results on the monitor 3

4 >>A=[ ]*[4;3;2;1;0] A= 23 >> A=[ ].*[ ] A = Matrix arithmetic

5 Built-in functions Matlab has series of predefined functions >> sin(pi/3) ans = Functions "Element-wise" on Vectors >> a=[0:pi/3:pi]; sin(a) ans= >>t= 0:0.1:1 t =

6 Matrices and Vectors operators Vectors Simple arthmatic operators are multiplication, sumation and subtraction: a*b, a+b, a-b Vector transpose: transpose(a) or a Dimension: length(a) Identifying part of the vector: a(1), a(2:2:length(a)) Accessing matrix elements: A (:, 2) the second column A (1,2) element of the first row / second column matrix operators: A+A, A-A, A*A, A^2 matrix traspose (conjucate): A matrxi inverse: inv(a) determinante: det(a) eigenvalues: eig(a) rank(a) trace(a) norm(a) Finding dimention of a matrix: size(a) 6

7 Linear Control (LTI) systems Specify Linear time-invariant (LTI) systems as Transfer function sys=tf(num,den) Zero/pole/gain models sys=zpk(z,p,k) State/space models sys=ss(a,b,c,d) e.g. h = tf(1,[1 1]) % creates transfer function 1/(s+1) Type 1 + h (s+2)/s+1 To create discrete-time systems, append the sample time Ts to the previous calling sequences. sys=tf(num,den,ts) e.g. sys= zpk(0.5,[ ],1,0.05) (z-0.5)/(z+0.1)(z-0.3) : sampling time

8 System transfer functions G = tf(numg, deng) Input the transfer function in the form of numerator and denomerator. Or you can enter a system function based on zero, pole and gain. G = zpk(z, P, K) Create the continuous-time SISO transfer function: H(s)=-2s/(s-1+j)(s-1-j)(s-2) Create h(s) as a zpk object using: h = zpk(0, [1-i 1+i 2], -2); 8

9 More matrix commands Matrices: Identity matrix nxn: eye(n) Zero matrix nxm: zeros(n,m) Create array of all ones nxm: ones(n,m) Random matrix nxm random values between 0 and 1: rand(n,m) Polynomial characteristic = poly(a) where A is an n-by-n matrix returns an n+1 element row vector whose elements are the coefficients of the characteristic polynomial, (λi-a): poly(a) Clear all: removes all saved variables from the working space whos: lists the variables defined and specifies the type and size 9

10 Polynomials Assume the polynomial: s 4 +3s 3-15s 2-2s+9=0 >> pol=[ ] pol = Finding the roots: roots(pol) ans =

11 Plots Plot >> t=0:0.25:7 >> y =sin(t) >> plot(t,y) A few important commands axis[xminxmax YminYmax] grid on / grid off title, xlabel/ ylabel, Legend hold on, hold off print -depsc nomefile.eps set(gca, )

12 M-file Using Matlab editor you can create.m file Many commands and arithmetic can be written in the file. Variables can be passed between different m files. Input-out can be defined Similar to basic programming high level languages you can use if/then/else, while,end, for/end, etc. New functions may be added to MATLAB's vocabulary if they are expressed in terms of other existing functions. The commands and functions that comprise the new function must be put in a file whose name defines the name of the new function, with a filename extension of '.m'. e.g. function [mean,stdev] = stat(x) %STAT Interesting statistics. n = length(x); mean = sum(x) / n; stdev = sqrt(sum((x - mean).^2)/n); 12

13 Example: mechanical system modeling F(t) K 1 M2 M1 x 1 K 2 D 1 D 2 x 2 Remember x 1, x 2 are function of time x 1 (t) Input to the system: F(t) we show by U(s) by Lapalce transformation to frequency domain Output: displacement of the system/ response of the system to the disturbance force shown by Y(s) in this case it is desired to find x 1? G(s) =Y(s)/U(s) 13

14 Mechanical Example con s F(t) K 2 FBD M1 M2 Force balance K 1 b 1 b 2 Understand the input(s) and output(s) x 1 x 2 m1 = 10 m2 = 5 k1 = 100 Datum k2 = 50 FBD b1 = 10 b2 = 20 F(t) K 1 X 1 M1 b 1 (dx 2 /dt-dx 1 /dt) -b 1 (dx 2 /dt-dx 1 /dt) M2 K 2 ( X 2 ) b 1 (dx 2 /dt) *Don t forget inertia forces 14

15 Mechanical example cont s After combining two equations we have: (1) (2) F(s) = + 15

16 Mechanical example cont s Transfer Function: >>G= tf(num,den) >>numg=[m 2 (b 1 +b 2 ) k 2 ] >>DenG= [m 1 m 2 (m 1 (b 1 +b 2 )+b 1 m 2 ) (b 1 b 2 +k 2 m 1 +k 1 m 2 ) (k 2 b 1 +k 1 b 1 +k 1 b 2 ) k 1 k 2 ] Response of the system to step input: >>impulse(g, t) 16

17 System response to impulse input >>Figure (1) >>t = 10; >>impulse(g, t) >>grid on You can assign the results to matrices. >>[yi,t]=impulse(g, t) >> plot (t,yi) Amplitude Impulse Response Time (seconds) 17

18 System response to step input System response to step input >> figure(2) >> t = 10; % sec >> step(g, t) >> grid on Amplitude Step Response Time (seconds) 18

19 Response to a known signal Response to an input known signal: f(t) >> figure(3) >> t = 0:0.01:50; % signal for the given time interval (sec) >> f = 50*[ones(1,2500), zeros(1,2500) ones(1,1)]; % input signal >> lsim(g, f, t) >> grid on >> axis([ ]) Linear Simulation Results Time (seconds) 19

20 State-Space representation Let and So the state variables are X= [ ] T (1) (2) Isolate acceleration variable and then rewrite: The equations are in the form of, = Ax(t)+Bu(t) y = Cx(t) 20

21 Mechanical example: SS 21

22 Response to the step input Use lsim to find SS equation response to a step input >>A=[ ; ;-k1/m1 0 -b1/m1 b1/m1;0 -k2/m2 b1/m2 -(b1+b2)/m2] >>B=[0; 0; 1/m1; 0] >>C=[ ; ] >>D=0 >>T=0:0.05:10; >>U=0.2*ones(size(T)); >>[Y,X]=lsim(A,B,C,D,U,T); >>plot(t,y) 22

23 Eigenvalues/Eigenvectors [A,B,C,D]=tf2ss(b,a) [EV, EV] = eig (A) Use ss to make state-space model from the transfer function Is there any differences between using ss command and the ss you calculated by hand? Any differences in eigenvalues and eigenvectors of two state-space models? 23

24 Discrete-time/ Continuous-time Converting continuous time to discrete time >> G(z)= c2d (G, sampling_time, method ) Converting discrete time to continuous time: >> G = d2c(g_discrete, method ) Methods: 'zoh : Zero-order hold on the inputs. The control inputs are assumed piecewise constant over the sampling period. tustin : Bilinear (Tustin) approximation to the derivative. 'matched : Zero-pole matching method for SISO systems only 24

25 Bode plot is the representation of the magnitude and phase of G(j*w) bode(g), grid on G(s)= ω c 5 Frequency analysis-bode Magnitude (db) Diagram:c2d Bode Diagram System: G Frequency (rad/s): 4.98 Magnitude (db): Phase (deg) Frequency (rad/s) Note: a decibel is defined as 20*log10 ( G(j*w ) 25

26 Discretization Shannon practice αω c ω s 10αω c in this case: 5 α 10; ω s =2 /T ω c = 5, α = 5 25 ω s /T 40 Converting the continuous-time to discrete-time: >> T = 1/ ; %sec. ω s 100 rad/sec 2 / >> G_T1 =c2d(g,t, tustin ); >> G_M1 = c2d(g,t, matched ); >> Bodemag(G, b,g_t1, r,g_m1, g-- ) >> T = 1/3 0.33; %sec. ω s 20 rad/sec 2 / >> G_T1 =c2d(g,t, tustin ); >> G_M1 = c2d(g,t, matched ); >> Bodemag(G, b,g_t1, r,g_m1, g-- ) 26

27 Discretization 0 Bode Diagram 0 Bode Diagram G(s) Tustin Matched G(s) Tustin Matched Magnitude (db) Magnitude (db) Frequency (rad/s) Frequency (rad/s) T = 1/ ; %sec. ω s 100 rad/sec T = 1/3 0.33; %sec. ω s 20 rad/sec 27

28 Discretization T = 1/15 ω s 100 rad/sec T = 1/ ω s 20 rad/sec 28

29 Hard-disk read/write head controller θ J: inertia of the head assembly : 0.01 kgm^2 D: viscous damping in the bearings : Nm.(rad/sec) K: spring back constant: 10 Nm/rad K i =motor torque constant :0.05 Nm/rad Θ: angular position of the head i: input current l Laplace transform from ito θ, Recall, H=tf(num,den) 29

30 Hard disk cont s Design a digital controller that provides accurate positioning of the read/write head. The design is preformed in digital domain. First, discretize the continues plant we have plant equipped with a D/A converter (ZOH) connected to its input Ts= 0.005; %sampling period Hd= c2d(h,ts, ZOH ) Draw bode plot of both systems Hd and H bode(h,'-',hd,'--') To analyze the discrete system, plot its step response, Step(Hd) 30

31 Hard disk cont s The system oscillates quite a bit. This is probably due to very light damping. You can check this by calculating open loop poles. openloop poles of discrete model: Damp(Hd) The poles are very light equivalent damping and near the unit circle. You need to design a compensator that increases the Damping of these poles. Amplitude Step Response Time (seconds) 31

32 Hard disk cont s The simplest compensator is gain. Try root locus to select an appropriate feedback gain. >>rlocus(hd) The poles quickly leave the unit circle and go unstable Root Locus 2 >> rlocfind(oloop) 1.5 To pick a set of poles, pick a point inside the circle Imaginary Axis /T 1 /T 1 /T 0.9 /T 0.8 /T 0.7 /T 0.5 /T 0.6 /T 0.4 /T 0.3 /T /T /T /T /T 0.2 /T /T 0.6 /T 0.4 /T 0.5 /T 0.3 /T Real Axis

33 Hard disk cont s >> ddamp(poles,ts) To analyze this design, form the closed-loop system and plot the closed loop step response: >> cloop=feedback(oloop,k); >>step(cloop) 33

34 Hard disk cont s The response is fast and settles in 14 samples that is 14 * T s = 0.07 sec. You disk drive has seek time of x 10-4 Step Response Amplitude Time (seconds) 34

35 Hard disk cont s You need to introduce some lead or a compensator with some zeros D(z)=(z+a)/(z+a) With a=-0.85 and b=0 D=zpk(o.85,0,1,Ts) Oloop =Hd*D Bode(Hd, --,oloop, - ) The plot shows the compensator has shifted up the phase plot (added lead) in the fraquncy range of ω >10 rad/sec. Magnitude (db) Phase (deg) Bode Diagram Frequency (rad/s) 35

36 Let s try root locus again: >>rlocus(oloop) >>zgrid The poles stay within the unit circle Root Locus Root Locus Imaginary Axis Imaginary Axis Real Axis Without lead compensator Real Axis With lead compensator 36

37 The Singular Value Decomposition >> A=[1 2;3 4]; >> [U,S,V] = svd(a) U = S = V = Where Would you use SVD? The Singular Value Decomposition (SVD) is a widely used technique to decompose a matrix into several component matrices. E.g. when the matrix is not full rank: that means the row and columns and the matrix are linearly dependent. 37

38 Using SIMULINK First bring up the Simulink Go to continuous icon Click on Transfer function You can either enter the coeff. Of Num/Denum or enter Num/Denum as variable but values of the available should be loaded in the working space. 38

39 Simulink Cont s 39

40 Simulink Cont s 40

41 Simulink Solver Solver type Step of integration: variable or fixed step 41

42 MIMO Creating multi-input/multi-output (MIMO) cell arrays provide an ideal means to specify the resulting arrays of numerators and denominators num = {0.5,[1 1]} %1-by-2 cell array of numerators den = {[1 0],[1 2]} %1-by-2 cell array of denominators h = tf (num,den) h11 = tf(0.5,[1 0]) % 0.5/s h12 = tf ([1 1], [1 2]) % (s+1)/(s+2) H = [h11,h12] 42

43 MIMO Computer low frequncy (DC) gain of linear time invariant system K=dcgain(sys) The continous-time DC gain is the transfer function value at the frequncy s = 0. For the state space models matrices (A,B,C,D), this value is K= D CA -1 B 43

44 MIMO The discrete-time DC gain is the transfer function value at z= 1. For SS models with matrices (A,B,C,D), this value is K= D+C(I-A) -1 B The DC gain is infinite for system with integrators. 44

45 Example: MIMO H=[1 tf([1-1],[1 1 3]); tf(1,[1 1]) tf([1 2], [1-3])] Dcgain(H)? 45

46 ACTIVE Suspension: Homework Active suspension: This example concerns active suspension of a vehicle for passenger comfort. To drive the equation of motions We draw FBD of the entire system m1 = 250kg m2 = 40kg k1 = 1000N/m k2 = 20000N/m D 0 = 500Ns/m r = 0.5 m Road Surface K1 M1 M2 K2 D 0 Road radius r X 1 M1: Mass of chassis and passengers M2: Mass of Wheel carriage X 2 46

47 Active Suspension M 1 = u K 1 (x 1 -x 2 )-M 1 g- D( 1- M1 K 1 (x 1 x 2 ) - u M 1 g M 2 = K 1 (x 1 -x 2 ) + - u M 2 g K 2 (x 2 -r) K 1 (x 1 x 2 ) u M2 M 2 g K 2 (x 2 r) 47

48 Active suspension Define x 3 = 1, x 4 = can be assembled as. Then the equations = Ax + B 1 u + B 2 r + c 1 48

49 Automatic control and system theory Let s download this toolbox Books and Matlab Toolbox of Prof. Marro available at: nnimarro/gm_books.htm 49

50 Solution to the active suspension system The system can be written in form of = Ax + B 1 u + B 2 r + c 1 Define x 3 = 1, x 4 =. Then, the SS can be written as: 50

Outline. Classical Control. Lecture 1

Outline. Classical Control. Lecture 1 Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction