Computationally efficient trajectory optimization for linear control systems with input and state constraints

Abstract

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

Classical Model Predictive Control of a Permanent Magnet Synchronous Motor

“…Convexity guarantees the existence of a unique global minimum, and convergence of the optimization algorithm in finite time. The proof that convexity is maintained with this transformation is given in [10]. Graphically, for a simple example with two parameters, J can be represented as in Fig.…”

“…A simpler way is to sample the trajectories for i dq (t) and u dq (t) at an interval T n . In [10] it is proven that if an additional interlay is added, the transformation guarantees maintenance of the original constraints. For instance, a constraint i d (t) ≤ 0∀t ∈ [0, T ] is parameterized as…”

“…It is embedded in a flatnessbased predictive control scheme [7], [8], where trajectory optimization and prediction are decoupled, and the trajectory generation can be simplified [9]. A suboptimal trajectory generation algorithm [10] is applied which is based on a continuous approach, the variables are not discretized but represented as a polynomial with undetermined coefficients. The cost function is linearized in the unconstrained optimum in order to use a linear programming (LP) solver, which is amongst the simplest and fastest numerical optimizers.…”

Inversion of linear and nonlinear observable systems with series-defined output trajectories

“…In this paper, the two steps of computing the inverse and defining the output trajectory are interchanged. Given a trajectory defined according to (3), the corresponding state trajectories x x x(t) and control input u(t) are searched. Thus, the state and control input trajectories are determined as algebraic functions of time and the arbitrary parameters α α α…”

Flachheitsbasierte Trajektorienoptimierung entlang von Wegpunkten für ein lineares Ballbot-System

“…Eine flachheitsbasierte Trajektorienplanung für einen einfachen Arbeitspunktwechsel (rest-to-rest) ist jüngst in [15] für eine BallbotPlattform mit zwei entkoppelten Antrieben untersucht worden. Durch die Planung der Trajektorie in flachen Koordinaten erübrigt sich die numerische Integration Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 7/6/15 6:05 PM der Systemdifferentialgleichungen, wodurch sich der Berechnungsaufwand reduziert [13,17,18]. Ein Trajektorienentwurf entlang definierter Wegpunkte wurde allerdings noch nicht betrachtet.…”