SYNTHESIZING SLIDING MODES IN CASE OF NON-INVARIANCE AND OF THE DIFFERENCE OF THE DOUBLE CONTROL DIMENSION FROM THE DIMENSION OF LINEAR NON-STATIONARY OBJECTS
The methods were developed to synthesize moving sliding diversities and discontinuous vector control in order to set systems in the sliding mode with a high quality of transients, in case of failure of the known conditions of its disturbance non-invariance. Since the methods depend on the system dimensions and on its vector control, the methods of synthesizing the moving sliding diversities and discontinuous controls are proposed for the cases where the system dimension is not equal to the double dimension of the control vector. The methods of synthesizing the moving sliding diversities and discontinuous vector control, aimed at setting systems in the sliding mode with a high quality of transients, in case of failure of the known conditions of its disturbance non-invariance and in case of the equality of the system dimension to the double dimension of the control vector,  , were first obtained for linear stationary and non-stationary objects in works [1]. Cases  and  for systems with linear stationary objects were studied in work [2], and a complete mathematical proof was obtained that the linear dependency of the vector of uncertainties with the columns of the nominal (known) part of the matrix of entering the control into the system was the necessary and sufficient condition for the analytically exact rejection of the known or identified in one or another manner disturbances in the normally represented systems for both continuous and discontinuous control. This condition, as known from [3, 4], is the invariance condition of sliding modes towards the effect provided by nominal and uncertain disturbances. Without complying with this condition, it could not be managed previously to ensure the high quality of control processes at disturbances without increasing the control values and energy consumption. This paper develops the results of initial work [2] at  and  for linear non-stationary objects in the context of the high topicality of the problem related to ensuring the high quality of transients in sliding modes and in case of failure of the disturbance invariance condition.