--- title: Info-Theoretic MPC with Stochastic Diffusion url: https://www.emergentmind.com/topics/information-theoretic-mpc-with-stochastic-diffusion type: topic --- # Info-Theoretic MPC with Stochastic Diffusion Information-theoretic model predictive control (MPC) with stochastic diffusion extends the path-integral (PI) approach to optimal control by integrating both Gaussian (diffusion) and non-Gaussian (jump, e.g., compound Poisson) noise within a receding-horizon framework. This methodology incorporates information-theoretic cost functionals, importance sampling, and GPU-parallelizable iterative updates to enable real-time stochastic optimal control for nonlinear systems subject to general stochastic disturbances, including rare but significant jump events [1807.06108]. Foundational work on path-integral control for diffusion processes provides the basis for the approach, while recent developments generalize MPC design to handle jump-diffusion systems [1405.7392]. ## 1. Stochastic System Dynamics with Jump-Diffusion The controlled state dynamics are formalized as a continuous-time stochastic differential equation (SDE) on $X_t \in \mathbb{R}^n$: \[ \mathrm{d}X_t = f(X_t, t) \,\mathrm{d}t + G(X_t, t)\,u_t\,\mathrm{d}t + B(X_t, t)\,\mathrm{d}W_t + H(X_t, t)\,\mathrm{d}N_t \] where: - $u_t\in\mathbb{R}^m$ is the control input, - $W_t$ is a standard Brownian motion accounting for Gaussian noise, - $N_t$ is a scalar compound Poisson process representing jumps, with rate parameter $\nu$ and i.i.d. zero-mean Gaussian marks $Q_k \sim \mathcal{N}(0, \Sigma_J)$, - $B$ and $H$ map diffusion and jump noises into state space, - $f$ determines deterministic drift, and $G$ the control channel. This model captures both continuous perturbations and discontinuous events (jumps), generalizing the standard SDE setting [1807.06108]. ## 2. Information-Theoretic Cost Functional The control problem minimizes the expected finite-horizon cost: \[ J(u) = \mathbb{E}^{\mathbb{Q}}\Bigl[\;\phi(X_T) + \int_{t_0}^T \left[q(X_t, t) + \frac12\,u_t^\top R\,u_t\right]\mathrm{d}t\Bigr] \] $\mathbb{Q}$ denotes the path measure under the controlled SDE; $\phi$ and $q$ are terminal and running cost functions, respectively; $R$ is the control penalty. ### Free Energy and KL Bound Defining the uncontrolled (prior) measure $\mathbb{P}$ by omitting the control term in the dynamics, the free energy is \[ \mathcal{F} = -\lambda\,\log \mathbb{E}_{\mathbb{P}}\left[\exp\left(-\frac{1}{\lambda} S(X)\right)\right] \] with $S(X) = \phi(X_T)+\int_{t_0}^T q(X_t,t)\mathrm{d}t$. Jensen's inequality yields \[ \lambda\,\mathcal{F} \leq \mathbb{E}_{\mathbb{Q}}[S(X)] + \lambda\,D_{KL}(\mathbb{Q}\|\mathbb{P}) \] By proper choice of $R$ (via Girsanov's theorem), the stochastic optimal control objective becomes equivalent to minimizing this upper bound, which balances expected cost and relative entropy between controlled and uncontrolled trajectory distributions [1807.06108][1405.7392]. ## 3. Path-Integral Formulation and Importance Sampling The optimal path distribution $\mathbb{Q}^*$ that minimizes cost and relative entropy is \[ \frac{\mathrm{d}\mathbb{Q}^*}{\mathrm{d}\mathbb{P}} = \frac{\exp\left(-\frac{1}{\lambda} S(X)\right)}{\mathbb{E}_{\mathbb{P}}\left[\exp\left(-\frac{1}{\lambda} S(X)\right)\right]} \] Control is obtained by projecting $\mathbb{Q}^*$ onto admissible (parameterized) controls, minimizing $D_{KL}(\mathbb{Q}^* \| \mathbb{Q})$. After time discretization, the critical control update is: \[ u_j^{\text{new}} = u_j + G(x_{t_j})^{-1}B(x_{t_j}) \frac{\sum_{m=1}^M w_m \epsilon_j^m/\sqrt{\Delta t}}{\sum_{m=1}^M w_m} \] with importance weights $w_m = \exp\left(-\frac{1}{\lambda} \tilde S(X^{(m)})\right)$, where $X^{(m)}$ are sampled trajectories under the current control policy with stochastic perturbations $\epsilon_j^m$ reflecting both diffusion and jump events [1807.06108]. The same form appears in the pure diffusion setting, where optimal control is computed as a weighted sum over sampled noise increments [1405.7392]. ## 4. Iterative Model Predictive Control Algorithm Information-theoretic MPC with jump-diffusion is implemented as a receding horizon loop: 1. **Initialization**: Set control sequence $\{u_0,\dots,u_{N-1}\}$. 2. **Forward Simulation**: For each of $M$ parallel rollouts: - Set the initial state to the current state. - For $j=0$ to $N-1$: - Sample Gaussian noise $\epsilon_j^m$. - With probability $\nu \Delta t$, sample jump noise $\delta_j^m$ and add to $\epsilon_j^m$. - Propagate dynamics using the current control and both noise types. - Accumulate running cost. - Add terminal cost at horizon. 3. **Weighting and Update**: - Compute exponential weights $w_m$ for each trajectory based on total cost. - Update each $u_j$ using the weighted average of noise perturbations. 4. **Apply and Shift**: - Apply $u_0$ to the real system for one interval. - Shift control sequence forward, re-initialize last element. 5. **Repeat** at the next time step. This parallel sampling structure makes the algorithm highly amenable to GPU implementation, supporting high-frequency receding-horizon replanning [1807.06108]. ## 5. GPU Parallelization In the proposed schema, each rollout trajectory is simulated independently and can be assigned to a separate GPU thread or warp. All core steps—noise sampling, propagation through SDE (including both diffusion and jump noise), cost accumulation, and exponential weight computation—are completely thread-local. Reduction operations are then used to aggregate the necessary statistics for control updates. Typical configurations (e.g., $M=3{,}000$, $N=20$) require 10–20 ms for planning, enabling 50 Hz control rates. This efficiency underpins real-time MPC for nonlinear systems subject to compound stochasticity [1807.06108]. ## 6. Empirical Performance and Benchmark Tasks Simulation studies evaluate information-theoretic MPC with jump-diffusion on two canonical nonlinear control tasks: - **Cart-Pole Swing-Up and Balance**: Standard 4-state system, with diffusion ($\Sigma_D = 0.1\,I$) and varying jump covariance ($\Sigma_J \in \{1,1.5,2,3\}$), and jump rates ($\nu \in \{0.1,0.25,0.5\}$). New MPC with jump modeling achieves 96–100% success for moderate jumps over 100 trials; Gaussian-only MPC drops to 61–81% as jumps intensify. - **3D Quadrotor Waypoint Tracking**: 12-state quadrotor model, full attitude kinematics, diffusion ($\Sigma_D = 0.05\,I$) and heavier jump noise ($\Sigma_J \in \{5,10,20,30\}$, $\nu=0.2$). New MPC maintains 100% success at highest jump intensities, while the diffusion-only baseline fails to do so, dropping to 87%. Increasing rollout count ($M$) reduces trajectory variance, but only explicit modeling of jump statistics yields robustness to large disturbances. In both domains, explicit incorporation of jump events in importance sampling yields superior performance, especially as jump magnitude or rate increases. When jumps are negligible, the method matches diffusion-only approaches [1807.06108]. ## 7. Broader Context and Methodological Extensions The information-theoretic MPC framework for jump-diffusions generalizes earlier PI-based optimal control developed for pure diffusion SDEs [1405.7392]. In those earlier settings, the methodology leverages the HJB equation, Cole–Hopf log transformation, and a path-integral Feynman–Kac representation to express control optimality conditions in expectation form over unforced dynamics. Efficient implementation is closely tied to importance sampling, the free energy/relative entropy duality, and numerical strategies such as rapid-exploring random trees (RRT) to bias trajectory proposals toward promising regions. The current jump-diffusion extension preserves the core importance sampling and sampling-based update structure while accounting for discontinuous state transitions and their statistics. This yields a practical algorithm for stochastic receding-horizon control encompassing a much broader class of disturbance models [1807.06108][1405.7392]. Source: https://www.emergentmind.com/topics/information-theoretic-mpc-with-stochastic-diffusion