## 70  Nagurka Problem

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

6.4 Further example

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

n’th-order linear time-invariant system.

### 70.1  Problem description

Find u over t in [0; 1 ] to minimize

J =
 1 0
x′*x + u′*u dt + 10*x1(tF)2

subject to:

 dx dt
= A*x + u

```A = [0 1 0 ... 0
0 0 1 ... 0
... ... ...
0 0 0 ... 1
1 -2 3 ... (-1)^(n+1)*n]```

The initial condition are:

 x(0) = [ 1  2  ...  n ] ,
 −∞ <= u(1:n) <= ∞ .

Reference: [25]

### 70.2  Problem setup

```toms t
n  = 6;
t_F = 1;

p = tomPhase('p', t, 0, t_F, 25);
setPhase(p);

x = tomState('x', n, 1);
u = tomState('u', n, 1);

nvec = (1:n);
A = [sparse(n-1,1), speye(n-1); ...
sparse(nvec.*(-1).^(nvec+1))];

% Initial guess
guess = icollocate(x == nvec');

% Initial conditions
cinit = (initial(x) == nvec');

% ODEs and path constraints
ceq = collocate(dot(x) == A*x+u);

% Objective
objective = 10*final(x(1))^2 + integrate(x'*x + u'*u);
```

### 70.3  Solve the problem

```options = struct;
options.name = 'Nagurka Problem';
solution = ezsolve(objective, {ceq, cinit}, guess, options);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);
```
```Problem type appears to be: qp
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem:  1: Nagurka Problem                    f_k     109.074347751904550000
sum(|constr|)      0.000000000002256926
f(x_k) + sum(|constr|)    109.074347751906810000
f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Barrier QP solver
Optimal solution found

FuncEv    3 GradEv    3 ConstrEv    3 Iter    3
CPU time: 0.046875 sec. Elapsed time: 0.032000 sec.
```

### 70.4  Plot result

```subplot(2,1,1)
x1 = x(:,1);
plot(t,x1,'*-');
legend('x1');
title('Nagurka Problem - First state variable');

subplot(2,1,2)
u1 = u(:,1);
plot(t,u1,'+-');
legend('u1');
title('Nagurka Problem - First control variable');

figure(2)
surf(t, 1:n, x')
xlabel('t'); ylabel('i'); zlabel('x');
title('Nagurka Problem - All state variables');

figure(3)
surf(t, 1:n, u')
xlabel('t'); ylabel('i'); zlabel('u');
title('Nagurka Problem - All control variables');
```