Computationally Efficient Trajectory Optimization for Linear Control Systems with Input and State Constraints

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.

💡 Research Summary

The paper addresses the problem of generating optimal trajectories for linear control systems that are subject to both input and state constraints, with a particular focus on computational efficiency over long prediction horizons. The authors build their approach on the concept of continuous‑time differential flatness, which guarantees that all system states and inputs can be expressed as functions of a set of flat outputs and a finite number of their derivatives. By selecting appropriate flat outputs, the original high‑dimensional dynamic problem is reduced to a lower‑dimensional one that depends only on the flat output trajectories.

To obtain a tractable representation of these trajectories, the flat outputs are parameterized using a polynomial basis (e.g., Bézier or Lagrange polynomials). This choice has two major advantages. First, the entire trajectory over a fixed time interval can be described by a finite vector of polynomial coefficients, turning an infinite‑dimensional functional optimization into a finite‑dimensional parameter optimization. Second, differentiation of the trajectory becomes a simple matrix multiplication with the derivative matrix of the chosen basis, preserving linearity of the system dynamics with respect to the coefficients.

The most challenging aspect of constrained trajectory optimization is the handling of inequality constraints that must hold for all times in the horizon. The authors exploit a result from polynomial non‑positivity theory: a polynomial that is required to be non‑positive over a compact interval can be enforced by a set of linear inequalities on its coefficients. By expressing input and state limits as polynomial inequalities in the same basis, the authors convert the original semi‑infinite constraint set into a finite set of linear constraints. This transformation avoids the need for sum‑of‑squares (SOS) programming or other semidefinite relaxations, keeping the problem structure purely linear‑quadratic.

Consequently, the overall optimization problem retains a quadratic cost (typically a weighted sum of control effort and tracking error) together with linear equality constraints (the flatness‑derived dynamics) and linear inequality constraints (the polynomial‑based bounds). This is exactly the structure of a standard Quadratic Programming (QP) problem, which can be solved efficiently by modern solvers such as OSQP, Gurobi, or CPLEX. The authors further propose a linearization step around the unconstrained optimum: if the unconstrained solution lies well within the feasible region, the constraints can be approximated by their first‑order Taylor expansion, turning the QP into a Linear Programming (LP) problem. LP solvers are typically faster and can handle even larger horizons, making real‑time implementation feasible for high‑frequency control loops.

To demonstrate practical relevance, the method is applied to a predictive torque controller for a permanent‑magnet synchronous motor (PMSM). The motor’s voltage, current, torque, and speed limits are expressed as polynomial constraints in the chosen basis. The flatness‑based trajectory generation yields a reference torque profile, while the QP (or LP) computes the optimal voltage commands that respect all physical limits. Experimental results on a hardware test‑bench show that the proposed approach reduces computation time by more than an order of magnitude compared with a conventional Model Predictive Control (MPC) implementation, without sacrificing tracking performance or constraint satisfaction. In particular, torque ripple and current overshoot are comparable or improved, and the controller operates reliably at sampling rates well above 10 kHz.

In summary, the paper contributes a systematic framework that combines differential flatness, polynomial parameterization, and polynomial non‑positivity to transform constrained trajectory optimization for linear systems into a standard QP (or LP) problem. This transformation preserves optimality while dramatically lowering computational load, enabling real‑time deployment in applications with long horizons and tight constraints. Future work suggested includes extending the methodology to mildly nonlinear systems, handling multi‑input‑multi‑output (MIMO) configurations, and exploring higher‑order polynomial bases to capture more complex dynamics without sacrificing the linear‑quadratic structure.