Energy-Gain Control of Time-Varying Systems: Receding Horizon Approximation
Abstract
Standard formulations of prescribed worstcase disturbance energy-gain control policies for linear time-varying systems depend on all forward model data. In a discrete-time setting, this dependence arises through a backward Riccati recursion. The aim herein is to consider the infinite-horizon $\ell_2$ gain performance of state feedback policies with only finite receding-horizon preview of the model parameters. The proposed synthesis of controllers subject to such a constraint leverages the strict contraction of lifted Riccati operators under uniform controllability and observability. The main approximation result establishes a sufficient number of preview steps for the performance loss to remain below any set tolerance, relative to the baseline gain bound of the associated infinite-preview controller. Aspects of the main result are explored in the context of a numerical example.
Summary
This paper addresses the problem of designing energy-gain controllers for linear time-varying (LTV) systems when only a finite preview of the system's parameters is available. The standard approach requires complete knowledge of future system parameters, which is often impractical. The authors propose a receding horizon approach that approximates the infinite-horizon control policy using a finite number of preview steps. This is achieved by leveraging the strict contraction property of lifted Riccati operators under uniform controllability and observability assumptions. The key result establishes a sufficient number of preview steps needed to ensure that the performance loss, relative to the ideal infinite-preview controller, remains below a specified tolerance. The paper includes a numerical example demonstrating the applicability and exploring aspects of the theoretical findings. This work is significant because it provides a practical method for designing robust controllers for LTV systems when complete future knowledge is unavailable, addressing a key limitation of traditional LTV control design techniques.
Key Insights
- •The paper introduces a novel receding-horizon control design for LTV systems with finite preview, approximating the ideal infinite-horizon controller.
- •The controller synthesis leverages the strict contraction property of lifted Riccati operators, a key theoretical contribution.
- •The paper derives a sufficient condition (equation 36) for the number of preview steps *T* required to achieve a desired performance loss tolerance *β*. The number of preview steps grows logarithmically with `log(1/β)`.
- •The analysis relies on uniform controllability and observability assumptions (Assumption 2) and assumes non-singular *A_t* which may limit its applicability to certain LTV systems, as noted in the paper, generalization to singular *A_t* is beyond the current scope.
- •The numerical example focuses on a periodic system (linearized unicycle kinematics), allowing for exact computation of constants and assessment of the conservativeness of the theoretical bounds.
- •The paper connects the receding horizon approximation to Riccati recursions and Schur decompositions, providing insights into the underlying mathematical structure.
- •The paper clarifies the distinction between their approach (finite preview of model data) and related work on optimal regret (finite preview of the disturbance).
Practical Implications
- •The research has direct applications in robotics, aerospace, and other domains where LTV systems are common but future model data is uncertain or computationally expensive to obtain.
- •Engineers can use the derived equations (specifically Theorem 2) to design controllers for LTV systems with guaranteed performance bounds, even with limited model preview.
- •The receding horizon framework can be implemented in real-time, adapting to changing system dynamics and online roll-out of finite-horizon plans.
- •Future research directions include extending the approach to handle singular *A_t*, exploring adaptive preview horizons, and investigating the impact of noise and model uncertainties.
- •The analysis of periodic systems provides a benchmark for evaluating the performance of the receding horizon controller and assessing the conservativeness of the theoretical bounds.