Fundamentals of Linear Control

Mauricio de Oliveira

Supplemental material for Chapter 1

Before You Start

In this script you will mostly be plotting data and performing basic calculations. The following MATLAB commands will be used:

load, to import data saved in a file;
plot, to produce a two-dimensional graph of the data;
fit, to fit a model to the data;
/, to solve a least-squares problem;
fsolve, to numerically solve a nonlinear algebraic equation.

You will also use the following auxiliary commands to make plots look prettier:

figure,title, xlabel, ylabel, xlim, ylim, grid, legend

and

round, to round up numbers;
linspace, to create a vector with evenly spaced numbers in a interval;
@, to declare a function on the fly;
num2str, to convert numbers to strings;
printmat, to print a matrix with row and columns labels.

In some calculations you will use a basic for loop;

1.1 Models and Experiments

Start by reading a file containing data of the terminal velocity of a car travelling in a straight line in response to changes in the pedal excursion:

% load data for static car model

load CarModel

then plot using MATLAB's plot:

% Fig. 1.4:

figure()

plot(us, vs, 'x')

xlabel('pedal excursion (in)')

ylabel('velocity (mph)')

ylim([0 120])

grid

Perform a simple least-square nonlinear model fitting using MATLAB's fit:

% perform a nonlinear least-square fit

f = fit(us', vs', 'alpha*atan(beta*x)', 'StartPoint', [80,1])

f =

     General model:
     f(x) = alpha*atan(beta*x)
     Coefficients (with 95% confidence bounds):
       alpha =       82.84  (75.98, 89.7)
       beta =       1.158  (0.9506, 1.365)

% round to one digit

alpha = round(f.alpha,1)

alpha = 82.8000

beta = round(f.beta,1)

beta = 1.2000

and plot the resulting curve plot along with the data:

% plot fitted data

u = linspace(0,3,100);

v = alpha*atan(beta*u);

% Fig. 1.5:

figure()

plot(u, v, '-', us, vs, 'x');

xlabel('pedal excursion (in)')

ylabel('velocity (mph)')

title(['y = ' num2str(alpha,3) ' atan(' num2str(beta,2) ' u)'])

ylim([0 120])

grid

Next perform a linear least-square fit using the slash operator (/):

% perform a linear fit

g1 = vs / us;

g1 = round(g1,1) % round to one digit

g1 = 47.8000

One could also have used MATLAB's fit for that:

f = fit(us', vs', 'gamma*x', 'StartPoint', 10)

f =

     General model:
     f(x) = gamma*x
     Coefficients (with 95% confidence bounds):
       gamma =       47.84  (39.83, 55.85)

round(f.gamma,1)

ans = 47.8000

Calculate also a linear model by linearizing the model obtained above about the origin:

% linearized nonlinear model

g2 = alpha*beta;

g2 = round(g2,1) % round to one digit

g2 = 99.4000

We plot both models along with the data and the nonlinear model:

% Fig. 1.6:

figure()

plot(u, v, '-', u, g1*u, '--', u, g2*u, '--', us, vs, 'x');

xlabel('pedal excursion (in)')

ylabel('velocity (mph)')

ylim([0 120])

title(['y = ' num2str(alpha,3) ' atan(' num2str(beta,2) ' u), ' ...

'y = ' num2str(g1,3) ' u, ' ...

'y = ' num2str(g2,4) ' u ' ...

])

grid

1.5 Solution with Feedback

Calculate the closed-loop gains:

for the two linear models obtained above, g1 and g2, and also by picking up a third linear coefficient between those two values:

% pick extra point in the middle

g3 = 73.3

g3 = 73.3000

Store the closed-loop transfer-functions in an array:

% build table of closed loop gains

ks = [0.02 0.05 0.5 1 3];

ks = [0.02 0.05 1 3];

mat = [

g1 * ks ./ ( 1 + g1 * ks);

g3 * ks ./ ( 1 + g3 * ks);

g2 * ks ./ ( 1 + g2 * ks);

];

% Table 1.1:

printmat(mat, 'Table 1.1', num2str([g1,g3,g2]), num2str(ks))

Table 1.1 =
                      0.02         0.05            1            3
         47.8      0.48875      0.70501      0.97951      0.99307
         73.3      0.59448      0.78564      0.98654      0.99547
         99.4      0.66533      0.83250      0.99004      0.99666

Calculate the nonlinear closed-loop gains using MATLAB's fsolve to solve the nonlinear equation:

for the value of y. For one value of this can be done as follows:

k = 0.05;

ybar = 60;

f = @(y) y - alpha*atan(beta*k*(ybar-y)); % @ creates function on the fly

y0 = ybar; % initial guess

fsolve(f, y0)

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.

<stopping criteria details>

ans = 48.8433

Repeat for various values of :

% closed loop nonlinear gains

u = linspace(0,3,100);

ybars = 40 : 5 : 100;

i = 1;

H = [];

for k = ks

l = 1;

for ybar = ybars

f = @(y) y - alpha*atan(beta*k*(ybar-y));

H(i,l) = fsolve(f, ybar, optimoptions('fsolve','Display','off')) / ybar;

l = l + 1;

end

i = i + 1;

end

and plot the result as a function of the reference :

% Fig. 1.9:

figure()

plot(ybars, H, [40 100], [1 1], 'k--')

xlabel('velocity (mph)')

ylabel('velocity/target')

leg = [char(ones(4, 1) * double('K = ')) num2str(ks','%3.2f') char(ones(4, 1)) * ' '];

legend(leg, 'Location', 'SouthWest');

ylim([0.5 1.02])

grid