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In missile guidance theory, one of the fundamental guidance algorithms is the so-called "Proportional navigation", which directs a constant-speed missile into a collision course with the target by continously turning the missile velocity vector at a rate proprtional to the rate of rotation of the line of sight to the target and at the same direction (for non-planar scenerios, the last mentioned fact has non trivial importance).

For slow targets (when compared to the missile) and low angle engagements, the closing speed $V_c$ of the missile is constant in time and approximately equal to the missile speed. Therefore, the proportional navigation rule can be stated as:

$$\vec{a} = -N \vec{V_c}\times \vec{\dot\theta}$$

where $N$ is a contant and $\vec{\dot\theta}$ is the angular velocity vector of the line of sight. The logic behind this law is that the greater the rate of rotation of line of sight, the greater acceleration is needed to be applied to set the missile into collision course with the target. If the line of sight is constant, than the missile and the target are already on collision course, and no acceleration is needed to be applied.

However, I read that there are deeper reasons for this law. For non-manuvering target, PN guidance law with $N=3$ minimizes the so called "total control effort" of the missile, a kind of criterion that quantifies the amount of manuvering required for the missile to intercept its target. The control effort is defined as:

$$\int_0^{t_f}|a(t)|^2dt$$

To see why the last statement is right, let us describe the motion of the target from the reference frame of the missile in a planar engagement. Under the assumptions of slow target and low angles engagement, we can write the motion of the target in polar coordinates system $(r,\theta)$ centered at the missile. The acceleration in polar coordinates is:

$$\vec{a}=(\ddot r-r\dot\theta^2)\hat{r}+(2\dot r\dot\theta + r\ddot\theta)\hat{\theta}$$.

Under the previous assumptions we can suppose $\dot r = -V_c, \ddot r=0$. In addition, since all acceleration is applied perpendiculary to the closing velocity vector, we can derive the differential equation:

$$N V_c\dot \theta \hat{\theta} =(2V_c \dot\theta+V_c(t_f-t)\ddot \theta)\hat{\theta}$$

, and if we denote $z=\dot\theta$ than we get the RDE:

$$\dot z = (\frac{N-2}{t_f-t})z$$

whose solution one can get by seperation of variables, which is :

$$\dot\theta (t)=\dot\theta (0)(\frac{t_f-t}{t_f})^{N-2}$$

Now, to connect this result with the variational problem of minimum control effort, one has to impose certain constraints on the small variations of the control function $a(t)$ - the requirement for interception at time $t_f= \frac{r_0}{V_c}$ is equivalent to the the following condition on the variation $\delta a(t)$:

$$\int_0^{t_f}\delta a (t_f-t)dt=0$$

The logic behind this condition is that for interception, small accelerations have to be weighted by the time-to-go $t_{go}=t_f-t$, since earlier velocity changes have greater effect on miss distance.

Now, if we set $N=3$ we get:

$$\dot\theta(t)=\dot\theta (0)(\frac{t_f-t}{t_f})\implies a(t)=3V_cC(t_f-t)$$

where $C$ is a constant.

Now, since the extermal principle requires:

$$\delta \int_0^{t_f}|a(t)|^2dt = 0\implies \int_0^{t_f}2|a(t)|\delta a dt = 0$$

Plugging inside the last requirement the acceleration formula found before, one can see that for every small variation consistent with the constraint mentioned before, the acceleration function sets the control effort integral in an extremum.

Questions

All this long discussion was just intended to illustrate the connection between PN guidance law and optimization of total control effort. However, I have a much more basic misunderstanding: why the quantity that captures the notion of "amount of manuvers" is defined to be:

$$\int_0^{t_f}|a(t)|^2dt$$?

Why not, for example, $\int_0^{t_f}|a(t)|dt$?

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  • $\begingroup$ Because this is the only special case that a analytical solution exists? You can arbitrarily a metric function but the majority of them do not derive to anything useful. $\endgroup$ Mar 11 at 12:13

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I do not think that the total control effort is fixed to be the integral of squares. For example in this technical report, the total control effort seems to be defined as the sum of lateral accelerations in a time-discrete setting. In the context of optimization, the cost function is often squared as it offers a number of advantages:

  • The cost function is smooth, which often makes it possible to analytically derive solutions
  • It discourages large values, which is advantageous as real-life constrains (e.g. lateral acceleration constraints) are less likely to be violated.

A good example (in my oppinion) is the Linear-quadratic Regulator, which features a quadratic cost function resulting in a very nice, linear feedback law.

For these reasons, I do not think that the total control effort is universally defined to be

$$\int_0^{t_f}|a(t)|^2dt$$

but rather often is defined to be in this form to serve as a optimization cost function or serve other purpose in the framework of the paper.

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