Adaptive Control Design Based on Adaptive Optimization Principles

Recently, we introduced an adaptive control design for linearly parameterized multi-input nonlinear systems admitting a known Control Lyapunov Function (CLF) that depends on the unknown system parameters. The main advantage of that design is that it overcomes the problem where the estimation model becomes uncontrollable (at regions of the state space where the actual system is controllable). However, the resulted adaptive control design is quite complicated and, moreover, it exhibited poor transient behavior in various applications. In this technical note, we propose and analyze a new computationally efficient adaptive control design that overcomes the aforementioned shortcomings. The proposed design is based on an adaptive optimization algorithm introduced recently by the author, which makes sure that the parameters to be optimized (which correspond to the controller parameters in this technical note) are modified so as to both lead to a decrease of the function to be minimized and satisfy a persistence of excitation condition. The main advantage of the proposed adaptive control design is that it can produce arbitrarily good transient performance outside (a) the regions of the state space where the actual system becomes uncontrollable and (b) a region of the parameter estimates space which shrinks exponentially fast. It is also worth noting that the class of systems where the proposed algorithm is applicable is more general than that of our previous work; however, it has to be emphasized that, due to the fact that the proposed algorithm involves the use of random sequences, all the established stability and convergence results are guaranteed to hold with probability one.