Adaptive suboptimal control of a linear system with bounded disturbance (Q1061677)

An adaptive suboptimal control of a single-input single-output linear discrete plant with unknown parameters and unmeasurable disturbance is concerned. Let \(u_ t\), \(y_ t\), \(v_ t\), and \(\tau\) be the plant input, the output, the disturbance (at moment t), and the plant parameter vector respectively. The upper bound of \(| v_ t|\) is a known positive constant, \(\rho\). In the case where \(\tau\) is known, the optimal control problem is to design the output feedback controller which minimizes a certain functional \(I(y_ 1,y_ 2,...,u_ 1,u_ 2,...)=I(c(\tau),\tau,v_ t)),\) where c(\(\tau)\) is the feedback controller parameter vector. Due to the disturbance, this problem is restated as the following: find \(c_ 0(\tau)\) minimizing Q(c(\(\tau)\),\(\tau\),\(\rho)\)) such that \(I(c_ 0(\tau),\tau,v_ t))\leq Q(c_ 0(\tau),\tau,\rho))=\sup_{v_ t} I(c_ 0(\tau),\tau,v_ t))\). In this paper, \(\tau\) is unknown. Since the disturbance has an uncertainty, the ordinary parameter estimation techniques are not applicable. So, the estimated parameter \({\hat \tau}\) and the controller parameter \(\hat c(\)\({\hat \tau}\)) should be determined as solutions of the suboptimal control problem. The author presents an algorithm to obtain \({\hat \tau}\) and \(\hat c(\)\({\hat \tau}\)), which satisfies \(I(\hat c({\hat \tau}),{\hat \tau},v_ t))\leq Q(c_ 0(\tau),\tau,\rho))+\epsilon\) for any arbitrary positive constant \(\epsilon\). The main purpose of this paper is to prove theoretically that this algorithm has a solution under adequate conditions. Some simple examples are also presented.