Linear quadratic optimal control system design by Chebyshev-based state parameterization

Abstract: "A Chebyshev-based representation of the state vector is proposed for designing optimal control trajectories of unconstrained, linear, dynamic systems with quadratic performance indices. By approximating each state variable by a finite-term, shifted Chebyshev series, the linear quadratic (LQ) optimal control problem can be cast as a quadratic programming (QP) problem. In solving this QP problem, one approach is to treat the state initial conditions as constraints that are included in the performance index using a Lagrange multiplier technique.A computationally more efficient approach is to tailor the state representation such that the boundary conditions (which include the known initial conditions) are decoupled from the coefficients of the Chebyshev series. In both approaches the necessary condition of optimality isderived as a system of linear algebraic equations from which near optimal trajectories can be designed."