Problem statement. The problem stated by Israel Military Industries (Ta’as) was to get the pitch angle q of a flying platform (Delilah) to track some signal q_c given in real time when undergoing unmeasured external disturbances. The actual nonlinear dynamic system is given by its linear 5-dimensional approximations calculated for 42 values of the pair “height - Mach number”. The coordinates are two velocity components, height, q and . The corresponding parameters of the linear systems as well as the current coordinates are not available in real time. Only q, the height and the Mach number are measured. The system has second relative degree, which means that the control (horizontal stabilizer angle) appears explicitly only in the second time derivative of q. Some delay and noise are also present in measurements, and the system contains an actuator whose reaction involves both delay and discretization. The difficulty of the problem is seen from the fact that good tracking of q_c by q and boundedness of _c, imply good tracking of _c by as well. Real time differentiation being a well-known extremely complicated problem, no wonder that the linear controller developed by Ta’as was not successful.

Solution method. The controller is based on a branch of the variable structure system (VSS) theory - sliding modes of higher orders - that was developed at the Institute and is now rapidly expanding. The Delilah’s pitch control problem is the first practical implementation of that approach.

The main advantage of VSS is extremely low sensitivity to interior and exterior disturbances. They don't need parameter identificators, transient process may be made arbitrarily fast, precision is very high. These systems are based on so-called sliding modes. However, regular sliding modes feature high frequency switching of the control influence, which might cause dangerous vibrations of the system. On the other hand, second order sliding modes used here have the same positive qualities, provide for higher accuracy, and are achieved by means of continuous (Lipshitz) bounded control. The proposed algorithm demonstrates adaptive properties: the control influence automatically takes on the unique value, providing for accurate tracking under conditions of uncertainty, noises and disturbances.

It is seen from the computer simulation results (Figs. 1, 2) that the transient time is shortened with a significant accuracy improvement. Measurement errors, delays and noises are taken into account here. At the last stage the problem was simplified: a special data unit was installed measuring _c and . As a result practically ideal tracking was attained (Fig. 3). The corresponding actuator output (horizontal stabilizer angle) is shown in Fig. 4.

Fig.1 Original tracking algorithm performance Fig.2 Our algorithm performance

Fig. 3 Resulting tracking with and_c measured Fig. 4 Resulting actuator output

During the last two years Ta’as engineers have been performing a series of computer simulation experiments, based on the algorithm developed at the Institute. Due to the most encouraging results they have received, a flight experiment was performed in December 1997. We have been lately informed by Ta’as executives that the flight experiments demonstrated an excellent compliance with the computer simulation results.