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

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.

💡 Research Summary

The paper introduces a unifying framework for indirect data‑driven predictive control (DDPC) called Trajectory Predictive Control (TPC). In TPC the future output trajectory over a prediction horizon is expressed as a linear function of the most recent input‑output history and the planned future inputs:

 y_f(t) = P z_p(t) + F u_f(t) + e_f(t)

where z_p(t) stacks the last m input‑output samples, u_f(t) is the planned input sequence of length h, and e_f(t) captures modeling error or regularization slack. The matrices P and F are identified from a batch of training data organized in Hankel matrices, typically via least‑squares or regularized regression. Because the predictor is linear, if the stage cost and constraints are convex, the resulting optimal control problem remains convex and can be solved efficiently with off‑the‑shelf solvers.

The authors first show that many recent DDPC schemes are special cases of TPC. The subspace predictor (P_sbs, F_sbs) corresponds to the minimum‑norm solution of the Willems fundamental lemma and recovers Subspace Predictive Control (SPC). By applying an LQ‑decomposition to the data matrix, the γ‑DDPC formulation is obtained, and a causal version (causal‑γ‑DDPC) is derived by forcing the F matrix to be block‑lower‑triangular. Thus TPC provides a systematic taxonomy that unifies SPC, γ‑DDPC, causal‑γ‑DDPC, transient predictive control, and related methods.

The main novel contribution is the introduction of a “state‑space predictor”. The recent input‑output window z_p(t) is taken as the state x(t) of an LTI state‑space model

 x(t+1) = A x(t) + B u(t)
 y(t) = C x(t) + D u(t)

The matrices A, B, C, D are identified directly from the same Hankel data. With this identification, the predictor matrices P and F become exactly the output‑prediction matrices of the conventional linear MPC formulation. Consequently, TPC with the state‑space predictor is mathematically identical to linear MPC, inheriting its mature theory: stability conditions, recursive feasibility, robust and stochastic extensions, and well‑established tuning guidelines.

Numerical experiments on a multi‑input multi‑output system illustrate the practical impact. Training datasets ranging from a few dozen to several hundred samples are generated in closed‑loop with a modest stabilizing controller, and additive measurement noise is added. The state‑space‑based TPC achieves performance virtually indistinguishable from an oracle H₂‑optimal controller that knows the true plant, even with as few as 30 training samples. In contrast, the subspace predictor requires substantially more data to avoid over‑parameterization; with small datasets its performance degrades noticeably. Causal‑γ‑DDPC improves over γ‑DDPC for limited data but still lags behind the state‑space approach. The experiments also confirm that the regularization term r(e_f) = λ‖e_f‖² can be used to trade off tracking accuracy against robustness to model mismatch.

Finally, the paper discusses the relationship between TPC and direct DDPC methods such as DeePC. DeePC embeds the data matrix directly into the optimization via a decision variable α, solving a problem equivalent to TPC with the subspace predictor when the minimum‑norm α is enforced. Hence TPC can be viewed as a bridge between DeePC (purely data‑based) and classical MPC (model‑based).

Limitations are acknowledged: the current theory assumes linear time‑invariant dynamics and deterministic behavior; extensions to nonlinear, time‑varying, or stochastic systems are left for future work. The authors suggest integrating robust or stochastic MPC techniques, reducing computational load for real‑time implementation, and validating the approach on real industrial processes.

In summary, the paper provides a comprehensive unifying perspective on indirect DDPC, introduces a data‑efficient state‑space predictor that makes TPC a special case of linear MPC, and demonstrates through simulations that this approach can achieve near‑optimal performance even with modest amounts of training data.