A pseudospectral method for the optimal control of constrained feedback linearizable systems

• Published in: IEEE transactions on automatic control Vol. 51; no. 7; pp. 1115 - 1129
• Main Authors: Qi Gong, Wei Kang, Ross, I.M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.07.2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Computer science; control theory; systems; Constrained optimal control; Constraints; Control systems; Control theory; Control theory. Systems; Cost function; Dynamical systems; Exact sciences and technology; Feedback; Feedback control; Gradient methods; Large-scale systems; Linear feedback control systems; Mathematical models; Nonlinear equations; nonlinear systems; Optimal control; Optimization; Optimization methods; pseudospectral; State feedback; Constrained optimal control; Spectral method; pseudospectral; Sobolev space; Non linear control; Feedback regulation; Continuous time; Linearizing control; Dynamical system; Non linear system; Constrained optimization; Closed feedback; Control constraint; nonlinear systems; Discretization; Continuous control; Performance requirement; pseudospectral method; Optimal control; Discrete programming; Time optimal control; Numerical convergence; State constraint
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2006.878570

We consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints. In general, a linearizing feedback control does not minimize the cost function. Such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. In this paper, we consider a pseudospectral (PS) method to compute optimal controls. We prove that a sequence of solutions to the PS-discretized constrained problem converges to the optimal solution of the continuous-time optimal control problem under mild and numerically verifiable conditions. The spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution. The proposed ideas are illustrated by several numerical examples.