Stability and synthesis problems for nonlinear systems

under uncertainty with application to spacecrafts

V.A.Afanasyev, G.L.Degtyarev, A.S.Meshchanov, T.K.Sirazetdinov

For controlled systems with nonlinear non-stationary objects under nominal (certain) and uncertain limited external and parametric perturbations we submit the methods for analysis of stability of unperturbed motion and synthesis of continuous vector nonlinear controls, which provide exponential stability with zero overshoot and zero stationary deviation from a program motion by a norm of a state vector. The classical statements for stability problems and parametric stability were developed by A.M.Lyapunov and P.A.Kuzmin. Here the submitted methods are based on development of a consequence from Wazewski inequality and Sylvester criterion for negative definite quadratic forms and also on efficient application of numerical simulation of control systems. Compensation of the mentioned perturbations on the basis of identification of uncertain perturbations is considered together with its application for efficient control of soft vertical touchdown of a return space vehicle.

Problem setting:

a controlled system of equations with nonlinear non-stationary object is considered in deflections љfrom some known model (program) motion, and all uncertain elements of system (1) are reduced to one reduced vector of uncertain limited disturbances.

Stabilization control is presented in system as a sum with terms that will be compensating in initial system the effects of vector nominal and uncertain limited disturbances and on desired stabilization processes in system.

Problems

1. To find a stability criterion of unperturbed motion in considered system.

2. To find the control that assures in system with control the exponential decrease of norm of state vector during required time, and zero steady-state error.

3. To give a method to identify the reduced vector of uncertain disturbances and find in the control u, that will be ensuring the compensation of generally unfavorable influence of limited nominal and uncertain disturbances on the controlled system.

4. To apply the obtained results to resolve a problem of the soft vertical touchdown of a return space vehicle (RSV) under conditions of nominal and uncertain disturbances.

From this research we obtain that -

1. As it follows from analytical and numerical results the developed nonlinear control has the following advantages:

1) Presence of the feedback together with the identification of uncertain and compensation of nominal and uncertain inequalities under specified conditions ensures the required high quality of controlled processes in systemsљ with a nonlinear non-stationary object under nominal and uncertain inequalities;

2) further to the above cited advantage 1) in comparison with linear stabilization controls determined by equations linearized relatively model (program) motions the deflections are allowed be not small but finite and sufficiently large (limited by only owing to limitations on the full control);

3) in comparison with the controls optimal by propellant consumption at some more consumption the proposed control is applied under action of limited nominal and uncertain parametric disturbances without recalculation of the control for various initial deflections from the program motion in a finite or sufficiently large area of initial and current deflections;

4) in comparison with a multistep terminal control by the exact model it is not required to do comparatively big calculations of controls during each of long steps for construction of the control by using anew a terminal point;

5) in comparison with a multistep terminal control by the simplified model there is also an advantage that it is no necessary to recalculate the controls although to a lesser extent then in the multistep control by the exact model.

2. When the initial deflections change within the bounds of the permissible area including zero values when RSV accurately hits in the model initial state, the propellant consumption has insignificant changes of about + 5 %.

3. To further expand the RSV initial permissible deflections area relatively the model initial point and increase thereby its maneuverability there is an additional opportunity to use the thrust reversal at the expense of additional braking engines installed along the opposite generating lines of two cones, or quick roll banks of the two-cone RSV by 180 degrees round its longitudinal axis that has stable orientation perpendicular to the velocity vector by using auxiliary propulsion engines of small thrust.

In contrast of Lyapunov's well-known methods during synthesis of controls the decrease of a state vector norm is considered as a control quality coefficient, uncertain disturbances are taken into account and the Silvestre criterion is used for nonlinear non-stationary objects (not only linear stationary ones). Particularly using the Matlab programming system.