Absolute stability of systems with controllers that provide given oscillation index

Linear multivariable tracking systems are considered, which controllers provide given or achievable separate oscillation indices, in particular, they minimize $H_{\infty }$ norm of the system closed loop transfer matrix that connects a vector of references with a vector of controlled variables. An aggregate of the separate oscillation indices for the $i$th closed loop connecting the $i$th reference signal with the $i$th controlled variable is considered as a performance index. Such an approach is of great practical interest for engineers that design automatic systems. Based on the multivariable circle criterion of absolute stability, it is proved that the closed loop system is stable in whole if non-stationary sector nonlinearities are entered in the control loop at the plant output. Relation between the oscillation index obtained during controller synthesis and the size of sector which bounds feasible nonlinearities characteristics is found. This result of the paper is sufficient. Relation between the separate oscillation index and the Nyquist plot of system with the break point at the corresponding plant output is proved.