The purpose of the work is to provide conditions for machined workpiece surface quality control based on the use of gas-magnetic supports in machine-tool systems (on example of the 3K227A internal grinding machine). To model the trajectory of tool and workpiece mutual movement the methods of nonlinear dynamics are used. The literature data on the problem are analyzed and experimental studies are carried out. Based on the considered issues of controlling the dynamic stability of technological systems under machining by means of contactless controlled gas-magnetic supports of spindle units, a scheme for adaptive control of the machine-tool system is proposed. It allows in many respects to eliminate external mechanical effects on the technological system, internal vibrations caused by drives and moving parts, as well as to compensate temperature deformations of the spindle unit frame and body. An adaptive control system based on gas-magnetic supports of the spindle assembly and the workpiece, as well as the control systems of the position of workpiece and tool (on example of 3K227A internal grinding machine) are demonstrated in action. The gas-magnetic support control system developed at Komsomolsk-na-Amure State Technical University makes it possible to set the position of the rotor axis with the accuracy of 0.1 microns. The study results obtained lead to the conclusion that the control of two adaptive links on the gas-magnetic supports, i.e., the spindle assembly of the tool and the spindle assembly of the workpiece, allows to achieve a rotation accuracy of up to 0.2 microns. The methods of nonlinear dynamics make it possible to construct an attractor (trajectory) of the tool tip movement in the real time, which provides a possibility to affect the input parameters of the machining process and thereby to control the output parameters. In addition to this, the control system of machine-tool system dynamic stability is applicable to other processing types as well, including edge cutting machining.