Gain Margins of Adaptive Control Systems

This paper presents a systematic study of the gain margins (GM) of adaptive control systems: the specification of the parameter range of a control gain matrix in a designed adaptive control system for maintaining the desired closed-loop signal boundedness and asymptotic tracking performance. It is proved that the GM is infinity for continuous-time direct model reference adaptive control (MRAC) schemes applied to multi-input, multi-output (MIMO) linear time-invariant (LTI) systems, while it is finite with an upper bound for the discrete-time case. The derived GM results are applicable to systems with adaptive nonlinear or pole placement control designs. Analytical GM result is also obtained for some indirect multivariable MRAC systems; in particular, the GM is infinity but with a lower bound for such systems. Methods for enlarging the GM by proper choices of design parameters are presented. Moreover, the effect of sample time on the GM of sampled-data adaptive control systems is studied, which establishes the relationship between the continuous-time and discrete-time GM results. A simulation example is provided to illustrate some of the theoretical results.