Stanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 11: Introduction to MPC
This lecture provides a deep dive into advanced control theory, transitioning from theoretical concepts like Hamilton-Jacobi-Isaacs (HJI) equations for computing reachable sets to the practical framework of Model Predictive Control (MPC). The discussion emphasizes that MPC achieves closed-loop performance by repeatedly solving an open-loop optimal control problem over a finite horizon (receding horizon optimization). Key theoretical challenges addressed include ensuring persistent feasibility and stability, which requires leveraging concepts from invariant set theory.
Key takeaways
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Reachable Sets via HJI Equation
0:35
Avoidance sets and reachable sets are computed by solving a differential game using the Hamilton-Jacobi-Isaacs (HJI) equation. This involves reframing the Boolean problem of set membership into an optimal control cost function $h(x)$ [0:35].
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Backward Reachable Tube (BRT)
7:40
To ensure safety over the entire trajectory, not just the endpoint, one must compute a Backward Reachable Tube (BRT). This is achieved by modifying the cost function to minimize the minimum value of $h(x)$ across the entire optimization horizon [7:40].
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MPC Receding Horizon Principle
19:30
MPC solves an open-loop optimal control problem over a finite prediction horizon $[t, t+N_p]$ at each time step $t$. It then uses only the first computed input ($u_t$) and discards the rest of the plan, recomputing everything from scratch based on new state measurements (receding horizon) [19:30].
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MPC Design Goals
36:00
The primary goals when designing an MPC controller are ensuring persistent feasibility (the problem remains solvable at all future times) and guaranteeing stability (convergence to the desired state, e.g., the origin) [36:00].
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Invariant Sets for Feasibility
1:08:40
To guarantee persistent feasibility in MPC, one must identify a control invariant set $\mathcal{C}$. This is a set where, if the system starts within it, there exists a control input $u$ that guarantees the next state remains inside $\mathcal{C}$ [38:00].