Missile autopilot robustness using the real multiloop stability margin

Abstract

The robustness of a longitudinal missile autopilot to uncertain aerodynamics is determined using the real multiloop stability margin. This method determines control system stability robustness to simultaneous real- parameter variations without conservatism. This paper presents an overview of the theory and a computer program developed to evaluate control system robustness to real-paramete r uncertainties. The program imple- ments a signal flow graph decomposition method to compute the robustness analysis model. A polynomial-time convex-hull algorithm is presented. MISSILE flight control system must guarantee stability and performance in the face of large aerodynamic uncer- tainties and disturbances. This requires the feedback controller to maintain system stability and loop performance for all pos- sible variations in the plant behavior. Determination of the degree of robustness to these aerodynamic uncertainties is the focus of this paper. A recent publication by deGaston and Safonov1 outlines an algorithm that will exactly compute the stability margin km of diagonally perturbed multivariable sys- tems without conservatism. This paper presents an overview of that theory and describes software used to analyze a bank-to- turn missile autopilot. In the early 1980s, several papers described how multivari- able feedback systems can be analyzed in the frequency do- main.2'3 Doyle4 developed several robustness theorems that were fundamental in developing the analysis techniques used to analyze model uncertainties. These methods utilize singular- value theory as a means of measuring the size of multi-input multi-output frequency-dependent matrices. These tests mod- eled control system uncertainties using a full single-block ma- trix structure. If the uncertainties were truly structured in this form, then these tests are not conservative. If the uncertainties did enter the system in a structured way, these tests produced conservative estimates of stability robustness. Doyle5'6 later developed a method of incorporating the structure of the uncertainty into the uncertainty analysis. This capability reduced the conservatism of the previous singular- value techniques by utilizing a multiple-block diagonal struc- tural model of the uncertainties. This test is called the struc- tured singular-value (SSV) ^ test. Morton7'8 applied the \L test to the analysis of real-parameter variations. The SSV /* software generally models the real- parameter variation as a complex variation and produces a slightly conservative bound. Jones9 and Fan and Tits10 have worked on reducing this conservatism, but reliable software implementing the real SSV is not readily available. In contrast to the foregoing SSV-based tests, robustness to uncertainties is also being analyzed using polynomial methods. A myriad of papers have recently been published that utilize Kharitonov's theorem11 and variants of it. Kharitonov's theo- rem analyzes the robustness question by examining the Hur- witz (stability) properties of a family of polynomials whose coefficients are based on the system's characteristic equation and uncertainties. Barmish and DeMarco12 present an excellent review of the literature on these polynomial methods.