Methods of Ensuring Invariance with Respect to External Disturbances: Overview and New Advances

Abstract

In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of acting on the control plant. Methods of the synthesis are presented on the example of a third-order nonlinear system with single input and single output (SISO-systems), dynamic feedback synthesis is presented at a descriptive level and the focus is on procedures of static feedback synthesis. For the systems in which the matching conditions are not satisfied, it is concluded that it is expedient to introduce smooth and bounded nonlinear local feedbacks. Within the framework of the block control principle, we developed an iterative procedure of synthesis of S-shaped sigmoid feedbacks for such systems. Nonlinear local feedbacks ensure stabilization of the output variable with the given accuracy and settling time as in a system with traditionally used linear local feedbacks with high gains. However, in contrast to it, sigmoid functions do not lead to a large overshoot of state variables and control actions.

Synthesis of a Control System for a Waste Heat Boiler with Forced Circulation under Restrictions on Control Actions

“…Let a linear controller (15) be found for the extended system (14), the parameters of which satisfy the requirement ( 16) and ( 17) and ensure the fulfillment of the target condition x(∞) = 0. To select the parameters of the controller (23), it is necessary to analyze the closed system (14), similar to that performed in the synthesis of system ( 8) and ( 9), taking into account…”

“…Currently, various approaches to the synthesis of feedback are being intensively developed, taking into account restrictions on controls and phase variables with the formation of controls in the form of linear functions with saturation (sat functions) [19][20][21] and sigma functions [22,23].…”

Robust Stabilization via Super-Stable Systems Techniques

“…. , A r−1 B, one needs to group m i basis rows of matrix P i by transpositions (similar to (7)) and zero out the top linearly dependent lines (similar to (11)). In this case, the leftmost block A r B can be discarded, since its elements do not participate in the formation of the matrix T.…”

“…Classical methods for analyzing open-loop systems include results on interval stability of polynomials [5,6], robust frequency methods [7], the D-partition technique [8], H ∞ optimization methods [9], and others. Direct and very effective methods of robust control include the use of sliding-mode technique [10] and deep feedback [11]. Note that both methods provide the independence of motions in the sliding mode (slow motions) only from the matching uncertainties.…”

“…Finally, based on (17), it is necessary to form a feedback on the variables of the original systems ( 1) and ( 2), since it is these variables that are measured. By substitutions of variables (7), (11), and (15), the resulting transformation matrix and the resulting modal control law (2) are as follows:…”