## Code for Figure 8.11

### main.m

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78% Solves the minimization problem with T=10
%  minimize       1/2*integral{t=0 until t=T}(x1^2 + x2^2 + u^2) dt
%  subject to     dot(x1) = (1-x2^2)*x1 - x2 + u,   x1(0)=0, x1(T)=0
%                 dot(x2) = x1,                     x2(0)=1, x2(T)=0
%                 x1(t) >= -0.25, 0<=t<=T
%
% States and control

% Model equations
x1_dot = (1-x2^2)*x1 - x2 + u;
x2_dot = x1;

% Least squares objective terms
lsq =[x1; x2; u];

% Define the problem structure
ocp = struct('x',[x1; x2],'u',u, 'ode', [x1_dot; x2_dot], 'lsq',lsq);

% Specify problem data
data = struct('T', 10,...
'x0', [0;1],...
'xN', [ 0; 0],...
'x_min', [-0.25; -inf],...
'x_max', [ inf;  inf],...
'x_guess',  [0; 0],...
'u_min', -1,...
'u_max', 1,...
'u_guess', 0);

% Specify solver options
opts = struct('N', 20,...
'verbose', false);

% Create an OCP solver instance
s = dms_gn(ocp, data, opts);

% Display sparsities
figure();
subplot(1,2,1);
title('J sparsity pattern:')
subplot(1,2,2);
title('H sparsity pattern:')

% Initializing figure
figure;
clf;
hold on;
grid on;

% Plot solution
x1_plot = plot(s.t, s.sol.x(1,:), 'r--');
x2_plot = plot(s.t, s.sol.x(2,:), 'b-');
u_plot = stairs(s.t, [s.sol.u(1,:) nan], 'g-.');

xlabel('t')
legend('x1','x2','u')
pause(2);

iter=0;
while s.sol.norm_dw > 1e-8
% SQP iteration
s.sqpstep();
iter = iter + 1;

% Update plots
set(x1_plot, 'Ydata', s.sol.x(1, :));
set(x2_plot, 'Ydata', s.sol.x(2, :));
set(u_plot, 'Ydata', s.sol.u(1,:));
title(sprintf('Iteration %d, |dw| = %g', iter, s.sol.norm_dw))
pause(2);
end

```

### dms_gn.m

```
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232classdef dms_gn < handle
properties
% Symbolic representation of the OCP
ocp
% OCP data
data
% Solver options
opts
% Verbose output
verbose
% Dimensions
N
nx
nu
% Solution time grid
t
% Control interval length
dt
fun
% Nonlinear program
nlp
% Current iteration
n_iter
% Current solution
sol
end
methods
function self = dms_gn(ocp, data, opts)
% Constructor
self.ocp = ocp;
self.data = data;
self.opts = opts;

% Verbosity?
if (isfield(opts, 'verbose'))
self.verbose = opts.verbose;
else
self.verbose = true;
end

% Problem dimensions
self.N = self.opts.N;
self.nx = numel(self.ocp.x);
self.nu = numel(self.ocp.u);

% Time grid
self.t = linspace(0, data.T, self.N+1);

% Interval length
self.dt = data.T / self.N;

% Get discrete-time dynamics
self.rk4();

% Transcribe to NLP
self.transcribe();

% Initialize x
self.sol.w = self.nlp.w0;

% Extract x,u from w
[x_traj,u_traj] = self.fun.traj(self.sol.w);
self.sol.x = full(x_traj);
self.sol.u = full(u_traj);

% Initialize Gauss-Newton method
self.init_gauss_newton();
end

function init_gauss_newton(self)
% Linearize the problem w.r.t. w
self.nlp.J = jacobian(self.nlp.g, self.nlp.w);
self.nlp.JM = jacobian(self.nlp.M, self.nlp.w);

self.nlp.H = self.nlp.JM' * self.nlp.JM;
self.nlp.c = self.nlp.JM' * self.nlp.M;

% Functions for calculating g, J, H
{'w'}, {'g'});
{'w'}, {'J'});
self.fun.H = casadi.Function('H', {self.nlp.w}, {self.nlp.H, self.nlp.c},...
{'w'}, {'H', 'c'});

% Out new step is determined by solving the quadratic program for
% dw = w_new-w
%        minimize    1/2 dw'*H*dw + c'*dw
%        subject to  g + A*dw = 0
%                    lbw-w <= dw <= ubw-w
qp = struct('h', self.nlp.H.sparsity(), 'a', self.nlp.J.sparsity());
qp_options = struct();
if ~self.verbose
qp_options.printLevel = 'none';
end
self.fun.qp_solver = casadi.conic('qp_solver', 'qpoases', qp, qp_options);

% Iteration counter
self.n_iter = 0;

% Residual
self.sol.norm_dw = inf;
end

function rk4(self)
% Continuous-time dynamics
x = self.ocp.x;
u = self.ocp.u;
ode = self.ocp.ode;
f = casadi.Function('f', {x, u}, {ode}, {'x','p'}, {'ode'});

% Implement RK4 integrator that takes a single step
k1 = f(x, u);
k2 = f(x+0.5*self.dt*k1, u);
k3 = f(x+0.5*self.dt*k2, u);
k4 = f(x+self.dt*k3, u);
xk = x+self.dt/6.0*(k1+2*k2+2*k3+k4);

% Return as a function
self.fun.F = casadi.Function('RK4', {x,u}, {xk}, {'x0','p'}, {'xf'});

% Least squares objective function
lsq = self.ocp.lsq;
self.fun.Lsq = casadi.Function('Lsq', {x, u}, {lsq}, {'x','p'}, {'lsq'});
end

function transcribe(self)
w = {}; % Variables
g = {}; % Equality constraints
M = {}; % Least squares objective function
lbw = {}; % Lower bound on w
ubw = {}; % Upper bound on w
w0 = {}; % Initial guess for w

% Expressions corresponding to the trajectories we want to plot
x_plot = {};
u_plot = {};

% Initial conditions
w{end+1} = xk;
lbw{end+1} = self.data.x0;
ubw{end+1} = self.data.x0;
w0{end+1} = self.data.x_guess;
x_plot{end+1} = xk;

% Loop over all times
for k=0:self.N-1
% Define local control
w{end+1} = uk;
lbw{end+1} = self.data.u_min;
ubw{end+1} = self.data.u_max;
w0{end+1} = self.data.u_guess;
u_plot{end+1} = uk;

% Simulate the system forward in time
Fk = self.fun.F('x0', xk, 'p', uk);
x_next = Fk.xf;

% Add least squares term to the objective
M{end+1} = self.fun.Lsq(xk, uk);

% Define state at the end of the interval
w{end+1} = xk;
if k==self.N-1
lbw{end+1} = self.data.xN;
ubw{end+1} = self.data.xN;
else
lbw{end+1} = self.data.x_min;
ubw{end+1} = self.data.x_max;
end
w0{end+1} = self.data.x_guess;
x_plot{end+1} = xk;

% Impose continuity
g{end+1} = xk - x_next;
end

% Concatenate variables and constraints
self.nlp.w = vertcat(w{:});
self.nlp.g = vertcat(g{:});
self.nlp.M = vertcat(M{:});
self.nlp.lbw = vertcat(lbw{:});
self.nlp.ubw = vertcat(ubw{:});
self.nlp.w0 = vertcat(w0{:});

% Create a function that maps the NLP decision variable to the x and u trajectories
self.fun.traj = casadi.Function('traj', {self.nlp.w}, {horzcat(x_plot{:}), horzcat(u_plot{:})},...
{'w'}, {'x', 'u'});
end

function sqpstep(self)
% Update iteration counter
self.n_iter = self.n_iter + 1;

% Calculate the QP matrices
self.sol.g = self.fun.g(self.sol.w);
self.sol.J = self.fun.J(self.sol.w);
[self.sol.H, self.sol.c] = self.fun.H(self.sol.w);

% Solve the QP to get the step in in w
qp_solution = self.fun.qp_solver('a', self.sol.J, 'h', self.sol.H, 'g', self.sol.c,...
'lbx', self.nlp.lbw-self.sol.w,...
'ubx', self.nlp.ubw-self.sol.w,...
'lba', -self.sol.g, 'uba', -self.sol.g, 'x0', 0);
dw = full(qp_solution.x);

% Check convergence criteria
self.sol.norm_dw = norm(dw);

% Take (full) step
self.sol.w = self.sol.w + dw;

% Extract x,u from w
[x_traj,u_traj] = self.fun.traj(self.sol.w);
self.sol.x = full(x_traj);
self.sol.u = full(u_traj);

% Print progress
if self.verbose || mod(self.n_iter,10)==1
disp(repmat('-', 1, 70))
fprintf('%15s %15s\n', 'SQP iteration', 'norm(dw)');
end
fprintf('%15d %15g\n', self.n_iter, self.sol.norm_dw);
end
end
end

```