Optimal Proportional-Integral Guidance Law

Abstract

This chapter proposes a new optimal guidance law based on PI concept to reduce the sensitivity to unknown target maneuvers. By augmenting the integral ZEM as a new system state, a linear quadratic optimization problem is formulated and then the proposed guidance law is analytically derived through optimal control theory. The closed-form solution of the proposed guidance law is presented to provide better insight of its properties. Additionally, the working principle of the integral command is investigated to show why the proposed guidance law can be utilized to reduce the sensitivity to unknown target maneuvers. The analytical results reveal that the proposed optimal guidance law is exactly the same as an instantaneous direct model reference adaptive guidance law with a specified reference model. The potential significance of the obtained results is that it can provide a point of connection between PI guidance laws and adaptive guidance laws. Therefore, it allows us to have better understanding of the physical meaning of both guidance laws and provides the possibility in designing a new guidance law that takes advantages of both approaches. Finally, the performance of the guidance law developed is demonstrated by nonlinear numerical simulations with extensive comparisons.

Appendix. Derivation of Guidance Command (5.55)

This appendix presents a brief derivation of guidance command (5.55) for the completeness of the chapter. From Eq. (2.1), we have

$$\begin{aligned} \ddot{\sigma }= - \frac{{2\dot{r}}}{r}{\dot{\sigma }} - \frac{1}{r}{a_{M_\sigma }} + \frac{1}{r}{a_{T_\sigma }} \end{aligned}$$

(5.56)

By considering the unknown target maneuver as external disturbance, we ignore the unknown term $a_{T_\sigma }$ in Eq. (5.56) as

$$\begin{aligned} \ddot{\sigma }= - \frac{{2\dot{r}}}{r}{\dot{\sigma }} - \frac{1}{r}{a_{M_\sigma }} \end{aligned}$$

(5.57)

For notational simplicity, define ${\varvec{x}} = \left[ x_1,x_2\right] ^T$ with $x_1 = \sigma$ and $x_2 = \dot{\sigma }$. Then, the LOS rate dynamics can be rewritten as

$$\begin{aligned} \begin{aligned}&{{\dot{x}}_1} = {x_2} \\&{{\dot{x}}_2} = - \frac{{2\dot{r}}}{r}{x_2} - \frac{1}{r}{a_{{M_\sigma }}} \\ \end{aligned} \end{aligned}$$

(5.58)

Based on Eq. (5.57), we can formulate the following optimization problem.

Problem 5.2

Find guidance command $a_{M_\sigma }$ that minimizes cost function

$$\begin{aligned} J = \int _0^\infty {\left( {{{\varvec{x}}^T}{\varvec{Q}}{\varvec{x}} + {R}a_{M_\sigma }^2} \right) dt} \end{aligned}$$

(5.59)

subject to

$$ \text {Eq.\,(5.58)} $$

where $\varvec{Q}$ and R are the user-defined positive definite weighting matrices.

The guidance command (5.55) can then be obtained by solving optimization Problem 5.2 at each step with proper weighting matrices $\varvec{Q}$ and R.