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Predictive Linear Online Tracking for Unknown Targets
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Proposes PLOT, a predictive online tracking controller that learns an unknown time-varying AR target model with multi-step RLS predictors (forgetting factor, delayed-feedback learners) and applies receding-horizon certainty-equivalent control, with O(sqrt(T V_T)) dynamic regret and a Crazyflie quadrotor demo.
Problem
Online tracking of an unknown, non-stationary target in known linear control systems with quadratic costs, where target states are revealed sequentially; the goal is low dynamic regret against a non-causal optimal controller.
Approach
Decouple tracking into online prediction and control: W separate k-step-ahead RLS predictors with exponential forgetting (improper learning, delayed feedback handled by k non-overlapping learners) feed a certainty-equivalent receding-horizon feedforward law; regret is measured through the total variation of target dynamics V_T rather than of target states.
Key results
- PLOT achieves dynamic regret bound O(sqrt(T V_T)) and logarithmic regret when target dynamics are static (V_T=0).
- In simulation with a time-varying target, PLOT outperforms DAP, Riccatitron, SS-OGD, FTL and Naive LQR in accumulated dynamic regret.
- On a real Crazyflie quadrotor tracking an infinity-shaped reference (T=40 s, W=5, gamma=0.8), PLOT achieves smaller tracking error than Naive LQR.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PLOT's learned, time-varying target dynamics and forgetting factor instead of the compared online control methods | Online tracking controllers · Non-stationary target simulation, T=200 seconds | dynamic regret | Helped |
| Use PLOT instead of Naive LQR's state-error feedback without an affine term | Crazyflie 2.1 · Infinity-shaped target, T=40 seconds | tracking error | Helped |
| Increase the prediction horizon from too-small values (ablation) | PLOT · Circular target with zero dynamics path length | tracking performance | Helped |
| Increase the prediction horizon beyond the effective horizon (ablation) | PLOT · Circular target with zero dynamics path length | dynamic regret | No clear effect |
| Set the forgetting-factor exponent to a=0.25 instead of a=0.1, 0.5, or 1 (ablation) | PLOT · Spiral target with V_T = \mathcal{O}(\sqrt{T}) | dynamic regret | Helped |
Surprising findings
- The indirect PLOT approach can perform worse than direct baselines when references are unstructured and independent across time steps, as the model fits arbitrary data.
Limitations
- Guarantees assume realizable AR target dynamics; non-realizable targets are left open.
- Forgetting-factor tuning requires knowledge of V_T unless a FLH meta-algorithm is used, which is not explored.
- Assumes known linear system dynamics; hardware results use linearized model.
Experimental setup
Datasets: simulated target trajectories (circular, spiraling, time-varying AR dynamics)
Benchmarks: simulated quadrotor tracking of circular and spiral targets hardware Crazyflie infinity-shaped reference tracking
Metrics: dynamic regret tracking error accumulated regret
Baselines: disturbance-action policy (DAP) with memory Riccatitron SS-OGD Follow the Leader Naive LQR
Builds on: online non-stochastic control recursive least squares with forgetting factors delayed feedback learning reduction performance difference lemma
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