Optimal Control Theory

Policy optimization methods (like TRPO and PPO) define a surrogate objective function to estimate the policy gradient, enabling continuous updates. PPO uses a clipped ratio ($ ext{clip}(r_{ heta}, 1- ext{eps}, 1+ ext{eps})$) to constrain the new policy's divergence from the old one, stabilizing training without requiring complex second-order optimization.

The basic model-based recipe (collect data $ ightarrow$ fit dynamics $P(s'|s, a)$ $ ightarrow$ plan) fails when dealing with complex or nonlinear dynamics because extrapolation outside the observed state distribution is unreliable. This issue of generalization and distribution shift must be addressed.

To improve model-based planning, uncertainty quantification is necessary. The distinction between Aleatoric (inherent noise) and Epistemic (model uncertainty) is crucial. Bayesian approaches treat this by modeling the posterior distribution over parameters ($ heta$), allowing for prediction averaging across all plausible models.

A practical approach to estimate model uncertainty is using ensembles: training multiple independent neural network copies. The average of their predictions approximates the predictive posterior distribution, effectively exploring multimodal solution landscapes without requiring complex analytical derivations.

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].

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].

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].

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].

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].

RL addresses sequential decision-making where actions have long-term ramifications. It requires balancing the trade-off between exploitation (using current best knowledge) and exploration (gathering information). Learning relies on maximizing a scalar reward signal rather than explicit labels or supervision.

An MDP formally describes an environment using five components: State Set ($S$), Action Set ($A$), Transition Dynamics ($P(s'|s, a)$), Reward Function ($R$), and Discount Factor ($\gamma$). The Markov property ensures that the future state transition depends only on the current state and action, not on history.

The value function $V^{\pi}(s)$ estimates the expected total payoff starting at state $s$ under policy $\pi$. The optimal value, $V^*(s)$, is the maximum possible return. These values are solved recursively using the Bellman equation, which relates the current state's value to the expected discounted future rewards.

The Policy Gradient algorithm optimizes a stochastic policy $\pi_{\theta}(a|s)$ by maximizing the expected return $E[R]$. The gradient is computed using the log-probability trick, allowing the calculation of $\nabla_{\theta} E[R]$ through sampling, even when the dependency on $\theta$ only affects the sampling distribution.