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This paper presents a trajectory generation method that optimizes a quadratic cost functional with respect to linear system dynamics and to linear input and state constraints. The method is based on continuous-time flatness-based trajectory generation, and the outputs are parameterized using a polynomial basis. A method to parameterize the constraints is introduced using a result on polynomial nonpositivity. The resulting parameterized problem remains linear-quadratic and can be solved using quadratic programming. The problem can be further simplified to a linear programming problem by linearization around the unconstrained optimum. The method promises to be computationally efficient for constrained systems with a high optimization horizon. As application, a predictive torque controller for a permanent magnet synchronous motor which is based on real-time optimization is presented. I. INTRODUCTIONTrajectory optimization in real-time is an important part of many modern control systems, for instance, predictive control. The trajectory optimization problem must be solved in the sampling interval, determined by the plant dynamics. A computationally efficient solution, however, encounters two major obstacles: first, the optimization horizon has a minimum length, either to avoid suboptimality or as stability criterion, and second, the trajectory must satisfy input and state constraints to be feasible. For fast systems, many of the existing constrained predictive control schemes, which are based on numerical iterations, are not applicable.Concerning the horizon length problem, the continuous approach to predictive control is of interest [1]. Using differential flatness, the optimal control problem can be rewritten as output optimization problem. The basis function approach is applied to obtain a finite-parameter optimization problem [2] [3] [4] [5], analogeously to discrete-time optimization. A long optimization horizon is, however, obtained with comparably few optimization parameters.A remaining issue regarding computational efficiency is the inclusion of constraints. The classical method is the application of penalty functions [3]. As an alternative, a coordinate transformation was proposed to modify the constrained problem to a nonlinear unconstrained problem [6], solvable with nonlinear calculus of variations methods. The existing optimization methods with penalty functions or nonlinear coordinate transformations lead to a nonlinear or non-linear-quadratic problem, increasing the computational burden. If only feasibility is of interest, numerical procedures applying time scaling to slow down some variables can be used [1]. This paper treats the special case of the linearly constrained linear-quadratic problem. The linear structure of the system is exploited in the transformation of the problem to a finite-parameter optimization problem. The linear-quadratic structure of the problem is maintained. The unconstrained problem is solved algebraically as it is convex. A quadratic

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