Minimum Time Problems
The performance index of the seven problems considered in this chapter is simply the time required to transfer the state of the system from a regular variety to a set which is always a regular variety but in the fifth problem. The amplitude of the control
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Minimum Time Problems
Abstract The performance index of the seven problems considered in this chapter is simply the time required to transfer the state of the system from a regular variety to a set which is always a regular variety but in the fifth problem. The amplitude of the control variable is constrained.
The performance index of the seven problems considered in this chapter is the time required to transfer the state of the double integrator from a regular variety S 0 to a set S f which is always a regular variety but in the fifth problem. As we noticed in Sect. 2.2.4, to which reference is extensively made, the amplitude of the control variable is constrained. More specifically, J=
tf
l(x(t), u(t), t) dt, l(x, u, t) = 1
0
u(t) ∈ U ∀t, U = {u| − 1 ≤ u ≤ 1} It is straightforward to verify that Assumption 2.3, presented in Sect. 2.2.4, is satisfied. As a consequence, the optimal control, if it exists, is piecewise constant (Theorem 2.3). Furthermore the number of switchings of any extremal control is not greater than one (Theorem 2.4). In fact the eigenvalues of the dynamic matrix A are zero and the set U is a parallelepiped. Therefore any extremal control can be either constant over all the control interval and equal to ±1 or piecewise constant with a single switching from 1 to −1 or viceversa. The seven problems of this chapter have the following features. (1) The initial state x0 is given in Problem 7.1 where the final state is the origin of the state space. We ascertain that the optimal control exists whatever x0 is. (2) Particular initial and final states are chosen in Problem 7.2, the latter being different from the origin. We verify that two extremal controls exist, one of which is indeed optimal. (3) The final state in Problems 7.3 and 7.4 is not completely specified and we show that the optimal control exists, though not unique.
© Springer International Publishing Switzerland 2017 A. Locatelli, Optimal Control of a Double Integrator, Studies in Systems, Decision and Control 68, DOI 10.1007/978-3-319-42126-1_7
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7 Minimum Time Problems
(4) The final state must belong to a set which is not a regular variety in Problem 7.5. Also in this case we find the optimal control for each initial state. (5) The last two problems (Problems 7.6 and 7.7) are somehow atypical because the initial state is not given: indeed it must belong to a set which is a regular variety. Their discussion is anyhow not complex and we can compute an optimal solution. Problem 7.1 Let x(0) = x0 , x(t f ) =
0 , 0
where x0 is given. First, we consider the state trajectories corresponding to u(·) = ±1. Their equations are x22 + k, if u(·) = 1 2 x2 x1 = − 2 + k, if u(·) = −1 2
x1 =
where k is a constant. Along these trajectories x2 increases when u(·) = 1, whereas x2 decreases when u(·) = −1 (see Fig. 7.1a). It is straightforward to ascertain that, no matter what the initial state x0 is, we have an infinite number of piecewise constant controls which take on the values ±1 only and generate trajectories eventually en
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