Set-Valued Estimation of State and Parameter Vectors within Adaptive Control Systems

The problem under consideration is that of obtaining simultaneously set-valued estimates for state and parameter vectors of linear (in parameters and in phase coordinates) discrete-time systems under uncontrollable bounded disturbances and given bounded noise in measurements. There is no other a priori information on disturbances and noise except for they are bounded. It is shown that in the absence of noise in measurements and in the presence only of uncontrollable additive disturbances having an effect on stationary plants being investigated, the problem of obtaining set-valued parameter estimates is equivalent to the problem of determining a set-valued solution of a set of linear algebraic equations under uncertainty in their right-hand sides. With additive measurement noise, set-valued estimation procedure should be changed considerably since in this case one has to determine the whole set of solutions of a set of algebraic equations under uncertainty in coefficients as well as in right-hand sides. The problem of simultaneous estimation of state and parameter vectors can be reduced in the long run to the last-mentioned algebraic one. The problem of set-valued estimation for nonstationary systems with restricted parameter drift rate is also considered.