Trajectory-based data-driven predictive control and the state-space predictor

TL;DR

The paper introduces Trajectory Predictive Control (TPC) as a unifying indirect DDPC framework that expresses future outputs as a linear function of recent input/output history and planned inputs, unifying DeePC, SPC, γ-DDPC, and related methods through trajectory predictors. It then presents a novel state-space predictor that realizes the predictor as an LTI state-space model, making TPC a special case of model predictive control (MPC) and allowing the application of mature MPC theory, including stability and recursive feasibility. Numerical experiments show that TPC with the state-space predictor achieves performance close to an oracle LQG controller even with small training datasets, and that this predictor is especially data-efficient. The paper also analyzes predictor-specific data requirements and finds that the state-space predictor uses far fewer training examples and provides robust, well-generalizing performance, while relaxation strategies can introduce optimism bias. Overall, the work positions TPC as a versatile, theoretically grounded framework that leverages MPC principles for data-driven control and highlights the state-space predictor as the most data-efficient and reliable option among the studied predictors.

Abstract

We define trajectory predictive control (TPC) as a family of output-feedback indirect data-driven predictive control (DDPC) methods that represent the output trajectory of a discrete-time system as a linear function of the recent input/output history and the planned input trajectory. This paper shows that for different choices of the trajectory predictor, TPC encompasses a wide variety of DDPC methods, including subspace predictive control (SPC), closed-loop SPC, $γ$-DDPC, causal-$γ$-DDPC, transient predictive control, and others. This paper introduces a trajectory predictor that corresponds to a linear state-space model with the recent input/output history as the state. With this state-space predictor, TPC is a special case of linear model predictive control and therefore inherits its mature theory. In numerical experiments, TPC performance approaches the limit of oracle $H_2$-optimal control with perfect knowledge of the underlying system model. For TPC with small training datasets, the state-space predictor outperforms other predictors because it has fewer parameters.

Figures (3)

• Figure 1: Number of estimated parameters in each trajectory predictor. Formulas are general; curves use memory $m = 20$, prediction horizon $h = 15$, and input dimension $n_u = n_y / 2$.
• Figure 2: Mean prediction RMSEs for each predictor on test data gathered in open (left column) and closed loop (center) with training data gathered in open (top row) and closed loop (bottom) over 1,000 Monte Carlo runs. Mean control costs (right) are normalized by the oracle LQG mean cost.
• Figure 3: TPC output $y_1$ (top row) and input $u$ (bottom) with (left column) and without (right) relaxing the equality constraint in \ref{['delta0tpc']} and regularizing the slack variable $e_f(t)$. While the relax-and-regularize approach uses less control effort, it performs worse for tracking and worse overall.

Theorems & Definitions (9)

• Definition 1
• Proposition 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Corollary 4
• proof