| In this research, we study the reduced-order robust adaptive control design problem for a class of uncertain SISO linear systems, that are subject to system and measurement noise. Two controller order reduction methodologies are obtained with different level of simplification as compared to the full-order design. The key technique for these two order-reduction methodologies both lies in the modification of a particular step of the backstepping design in the controller design part. The first order-reduction methodology reduces the controller structure by n - 1 or n - 2 integrators, depending on the eigen-structure of a particular feedback matrix. The second order-reduction methodology simplifies controller structure by n integrators, the trade-off for this order reduction is that the worst-case estimate for the expanded state vector has to be chosen as a suboptimal choice, rather than the optimal choice. It is shown that the resulting reduced-order adaptive controllers preserve the strong robustness properties of the full-order adaptive controller in disturbance attenuation, boundedness of closed-loop signals, and output tracking. Simulation results corroborate our theoretical findings.;Furthermore, convergence analysis is investigated for the reduced-order adaptive control system achieved using the first order-reduction design methodology. We explore the conditions under which various closed-loop signals converge. We rigorously prove that, whenever the exogenous disturbance inputs is of finite energy and bounded, and the reference trajectory and its derivatives up to rth order are bounded, r being the relative degree of the transfer function of the true system, then a set of closed-loop signals are of finite energy and converge to zero; the system states and their estimates exhibit asymptotic behaviors with certain formats. If the rth order time derivative of the reference trajectory is uniformly continuous, then the rth order noiseless derivative of the output asymptotically tracks the rth order derivative of the reference trajectory. With an additional persistency of excitation condition, the parameter estimates converge to their true values and the state estimates asymptotically track the true state. Two simulation examples are included to illustrate the convergence analysis results. |
