On Estimation in Interception Endgames

Abstract

The paper presents an overview of the different errors created in an interception endgame by the measurement noise and the need of using an estimator in the guidance loop. Approaches for reducing the effects of the estimation error are described and some directions for further investigations are pointed out.

Appendix. Partial Differential Equation for the Cumulative Distribution Function of the Terminal State

Let z(t) be the state of the scalar continuous-time system

$$ \dot{z} = h_{1}(t)u, $$

(1)

where the control u is given by a saturated linear strategy

$$ u(t,z) = \mathrm{sat}\bigl(K(t)z\bigr) = \begin{cases} 1, & K(t)z > 1, \\ K(t)z, & |K(t)z| \le1, \\ - 1, & K(t)z < - 1, \end{cases} $$

(2)

and since the state z(t) is not known accurately, the actual control is

$$ u = \mathrm{sat}\bigl[K(t) \bigl(z(t) + \eta(t)\bigr)\bigr], $$

(3)

where η(t) is a random estimation error. Let f _z (x,t) denote the probability density function of z(t). In [29], this probability density function was approximated by $\hat{f}_{z(t_{n + 1})}(x)$, where z(t _n+1)=z _n+1 is the state of the discrete-time system

$$ z_{n + 1} = z_{n} + b_{n}u_{n}, $$

(4)

t ₀=0, t _n =t ₀+nΔt, n=1,…,N, u _n =sat(k _n (z _n +η _n )), b _n =Δth ₁(t _n ), k _n =K(t _n ), η _n =η(t _n ).

The approximation yields [30], for n=0,1,…,N−1,

(5)

where $A_{n} = 1 + \frac{1}{b_{n}k_{n}}$, $B_{n}(x) = \frac{x}{b_{n}k_{n}}$, and the probability density function $\hat{f}_{z(0)}(x) = f_{z_{0}}(x)$. It is assumed that the initial value of z and the probability density functions $f_{\eta (t_{n})}(x)$ of the estimation errors are known.

By the change of variables y=−A _n s+B _n (x), (5) can be rewritten for t _n =t, t _n+1=t+Δt as

(6)

where α(t,x,Δt)=−x−1/K(t)−Δth ₁(t), β(t,x,Δt)=−x+1/K(t)+Δth ₁(t), γ(t,Δt)≜Δth ₁(t)K(t)+1, δ ₁(t,x,Δt,y)≜x−Δth ₁(t)K(t)y. By taking the limit as Δt→0 one obtains that $\lim_{\Delta t \to 0}g(x,t,\Delta t) = \hat{f}_{z(t)}(x)$, yielding $\lim_{\Delta t \to 0}[ \hat{f}_{z(t + \Delta t)}x - \hat{f}_{z(t)}(x) ] = 0$. By applying the l’Hôpital’s rule,

$$ \lim_{\Delta t \to 0}\frac{\hat{f}_{z(t + \Delta t)}(x) - \hat{f}_{z(t)}(x)}{\Delta t} = \lim_{\Delta t \to 0}\frac{\partial g(x,t,\Delta t)}{\partial\Delta t}. $$

(7)

By differentiating g(t,x,Δt) w.r.t. Δt,

$$ \lim_{\Delta t \to 0}\frac{\partial g(x,t,\Delta t)}{\partial\Delta t} = a(x,t) \frac{\partial \hat{f}_{z(t)}(x)}{\partial x} + b(x,t)\hat{f}_{z(t)}(x), $$

(8)

where

Note that for Δt→0, the set of collocation points t _n fills the entire interval [0,t _f ]. Thus, based on the results of [30], the discrete probability density function $\hat{f}_{z(t)}(x)$ becomes a continuous probability density function f _z (x,t). In such a case, it is reasonable to set

$$ \lim_{\Delta t \to 0}\frac{\hat{f}_{z(t + \Delta t)}(x) - \hat{f}_{z(t)}(x)}{\Delta t} = \frac{\partial f_{z}(x,t)}{\partial t}. $$

(9)

Thus, due to (7)–(9),

$$ \frac{\partial f_{z}(x,t)}{\partial t} = a(x,t)\frac{\partial f_{z}(x,t)}{\partial x} + b(x,t)f_{z}(x,t), $$

(10)

which is the linear first-order partial differential equation for f _z (x,t). This equation is subject to the initial condition $f_{z}(x,0) = f_{z_{0}}(x)$. By some algebra, the coefficient a(x,t) is simplified as

$$ a(x,t) = h_{1}(t) \biggl( K(t)\int_{ - x - 1/K(t)}^{ - x + 1/K(t)} F_{\eta (t)}(y)\,dy - 1 \biggr). $$

(11)

By direct differentiation, $\frac{\partial a(x,t)}{\partial x} = b(x,t)$, i.e. Eq. (10) can be rewritten as

$$ \frac{\partial f_{z}(x,t)}{\partial t} = \frac{\partial}{\partial x} \bigl[ a(x,t)f_{z}(x,t) \bigr]. $$

(12)

By integrating (12) w.r.t. x from −∞ to x, one obtains

$$ \int_{ - \infty }^{x} \frac{\partial f_{z}(\xi,t)}{\partial t}\,d \xi= \frac{\partial}{\partial t}\int_{ - \infty }^{x} f_{z}(\xi,t)\,d\xi= \bigl[a(\xi,t)f_{z}(\xi,t) \bigr]\big|_{\xi = - \infty }^{\xi = x} = a(x,t)f_{z}(x,t), $$

(13)

or alternatively

$$ \frac{\partial F_{z}(x,t)}{\partial t} = a(x,t)\frac{\partial F_{z}(x,t)}{\partial x}, $$

(14)

where $F_{z}(x,t) = \int_{ - \infty }^{x} f_{z}(\xi,t)\,d\xi$ is the cumulative distribution function of z(t), which satisfies the linear first-order transition-type PDE (14), subject to the initial condition $F_{z}(x,0) = \int_{ - \infty }^{x} f_{z_{0}}(\xi) \,d\xi$.

By solving the PDE (14) the cumulative distribution function of z(t _f ) can be computed based on the probability density functions $f_{\eta (t_{n})}(x)$ of the estimation errors during the interval [0,t _f [.