# THE APPLICATIVE PROBLEMS

Three applicative examples are treated. The first one is a longitudinal missile autopilot. The linearization of a nonlinear missile model at a trim point is considered, with parametric uncertainties in the stability

derivatives and unmodeled high frequency bending modes.

The second applicative example is the lateral flight control system of a civil transport aircraft. Depending on the problem to be solved, we consider either the single rigid aerodynamic model (with parametric uncertainties in the stability derivatives), or a more complete model including a flexible structural model (with damping ratio about 1 % for the bending modes).

The third application is a telescope mock-up used to study high accuracy pointing systems. The mock-up, which is composed of a two axis gimbal system mounted on Bendix flexural pivots, is representative of very flexible plants (with damping ratio about 0.1 % for the bending modes).

The following problems will be solved in parts 3, 4 and 5:

1. Application of classical p tools to the robustness analysis of a rigid aircraft or missile (chapter 6): the idea is to evaluate existing methods for computing bounds of the s. s.v. on realistic examples. The two examples are complementary: because of the small number of parametric uncertainties in the missile model, exponential time methods for computing bounds of the s. s.v. p can be applied. Conversely, since there is a large number of parametric uncertainties in the aircraft model, only polynomial time methods can be applied.

2. Advanced robustness analysis of the missile (chapter 9): the robust stability and performance properties of the autopilot are studied in the presence of uncertain stability derivatives and unmodeled bending modes. This example also emphasizes the usefulness of the skewed p tools for some classes of practical problems.

3. Computation of the robustness margin in the special case of flexible structures (chapter 10): the s. s.v. p(p) is to be computed as a function of ui. The robustness margin is then obtained as the inverse of the maximal s. s.v. over the frequency range. As said above, p(p) is usually computed at each point p of a frequency gridding, and the robustness margin is deduced as the inverse of the maximal s. s.v. over this gridding. In some special cases such as the control of flexible structures, the p plot (corresponding to the value of the s. s.v. p(u>) as a function of ш) may present narrow and high peaks, so that it is possible to miss the critical frequency (i. e. the frequency which corresponds to the maximal s. s.v. over the frequency range), even when using a very fine frequency gridding. If this critical frequency is missed, the robustness margin is overevaluated, i. e. the result is too optimistic. Chapter 10 proposes a method which computes an estimate of the s. s.v. p(p) as a function of w and which gives a reliable value of the robustness margin. The method is first applied to the flexible transport aircraft, and then to the telescope mock-up.

4. Computation of a robust delay margin (chapter 11): the issue is to analyze the robustness properties of a closed loop in the face of classical model uncertainties (uncertain parameters and neglected dynamics) and uncertain time delays. This difficult problem presents a great practical interest. Indeed, when embedding control laws on a real-time computer, delays are to be considered simultaneously at the plant inputs (because of the time needed to compute the value of the plant input signal as a function of the plant output signal) and outputs (because of the sensors measuring the plant output signal). The method is applied to the missile.

5. Detection of limit-cycles in a nonlinear parametrically uncertain closed loop (chapter 12): as an extension of the famous Lur’e problem, consider the interconnection of a Linear Time Invariant (LTI) system (subject to LTI parametric uncertainties) with autonomous separable nonlinearities (e. g. saturations). The first issue is to detect the presence of a limit-cycle inside this closed loop with a necessary condition of oscillation. The second issue is to guarantee the absence of limit – cycles despite parametric uncertainties, with a sufficient condition of non-oscillation. The ц and skewed /.t tools provide a solution to this interesting nonlinear analysis problem.

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