AA203 Optimal and Learning-Based Control Course Info

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.

Monte Carlo methods estimate the expected return ($G_t$) by rolling out an episode until a terminal state, requiring full episodes. Temporal Difference (TD) learning improves upon this by using bootstrapping—defining the target as the instantaneous reward plus the discounted future value ($ ext{Reward} + ext{Discounted Future Value}$), allowing for online updates and handling non-terminal environments.

SARSA is an on-policy algorithm, meaning it improves the policy ($ ext{e.g., } ext{epsilon-greedy}$) that is actively used to generate data in the environment. Q-learning is off-policy; it learns about a target optimal policy (the greedy policy) while using data generated by a different behavior policy (also $ ext{epsilon-greedy}$), which is crucial for utilizing historical or simulated data.

To overcome the curse of dimensionality inherent in tabular value function representations, RL methods transition to parametric functions (e.g., neural networks) that approximate $V(s)$ or $Q(s, a)$. This allows generalization across states and controls.

Deep Q Networks (DQN) stabilize learning using two methods: Experience Replay (storing transitions in a buffer to decorrelate samples, satisfying the IID assumption required for regression) and Fixed Q Targets (using a delayed copy of the network parameters ($ ext{Q}_{ ext{target}}$) to prevent the target from being a moving variable during optimization).

The state update is modeled as $x_{k+1} = f(x_k, u_k, w_k)$, where $w_k$ is a random disturbance. The system must adhere to the Markovian assumption: the probability distribution of $w_k$ can only depend on the current state ($x_k$) and control ($u_k$), not on the history of previous states or disturbances.

The notion of optimality is defined by minimizing the expected cost, $\mathbb{E}[ ext{Cost}]$, over all possible disturbance realizations. The solution uses a backward dynamic programming recursion (Bellman equation) to find the optimal closed-loop policy $\pi^*$.

To solve problems over an infinite number of stages, a discount factor ($\gamma \in [0, 1]$) is introduced to ensure the convergence of the expected cumulative reward. The optimal value function $V^*$ satisfies a fixed-point equation: $V^*(x) = \max_{u} \{ R(x, u) + \gamma \mathbb{E}[V^*(x')]\}$.

For computational tractability in learning settings where the transition kernel is unknown, the problem can be reformulated using the Q-function ($Q^*$), which represents the expected cumulative reward starting at state $x$ and taking action $u$, followed by optimal actions: $Q^*(x, u) = R(x, u) + \gamma \sum_{x'} T(x'|x, u) V^*(x')$.

While LQR is fundamentally designed to drive a state to the origin (regulation), it can be generalized to perform trajectory tracking by defining an auxiliary problem based on deviation variables. The optimal control law structure remains consistent: $u = u_{nominal} + ext{feedback term}$.

For nonlinear dynamics ($x_{k+1} = f(x_k, u_k)$), the tracking problem can be linearized by performing a Taylor expansion around the nominal trajectory ($\bar{x}, \bar{u}$), allowing the use of LQR techniques on the deviation variables.

Both iLQR and DDP are methods for solving nonlinear optimal control problems iteratively. iLQR linearizes dynamics and quadratizes costs, while DDP directly approximates the Bellman equation by quadratizing the Q-function, making it a second-order algorithm in terms of dynamic derivatives.