Risk Aware Mpc

Definition

Risk-aware model predictive control (MPC) is a receding-horizon optimal-control scheme in which a
coherent tail-risk measure $\rho$ (typically Conditional Value-at-Risk, CVaR) replaces the plain
expectation in the cost and/or the chance constraints, so the controller penalizes the costly tail of
the disturbance distribution rather than only its mean. It sits between the robust (worst-case) and
risk-neutral (expected-value) paradigms: less conservative than worst-case, yet — unlike risk-neutral
MPC — it explicitly bounds rare, high-consequence outcomes (akella2024risk).
A central practical advantage is that coherent measures such as CVaR give convex inner
approximations of chance constraints for any disturbance distribution, so the stochastic program
admits a tractable convex reformulation without a Gaussian assumption. The survey states this for
generic discrete-time robotic systems (self-driving vehicles, bipedal/ground robots, subterranean
exploration); it is regime-agnostic and makes no space-manipulator dynamics assumption — neither
free-flying nor free-floating.

Key Equations

Symbols per notation.md.
For a linear discrete-time plant ${\mathbf{x}}_{k+1}=A{\mathbf{x}}_k+B{\mathbf{u}}_k+D{\mathbf{d}}_k$
with process noise $\mathbf{d}$, weights $Q,R,P$, and a coherent risk measure $\rho$, the
risk-aware MPC at time $t$ over an $N$-step horizon is (predicted state ${\mathbf{x}}_{k|t}$,
inputs $U_t=\{{\mathbf{u}}_{k|t}\}$):

$$J_t^{\ast }(\mathbf{x}(t))=\underset{U_t}{\min }\rho \left[{\mathbf{x}}_{t+N|t}^{\top }P{\mathbf{x}}_{t+N|t}+\sum _{k=t}^{t+N-1}({\mathbf{x}}_{k|t}^{\top }Q{\mathbf{x}}_{k|t}+{\mathbf{u}}_{k|t}^{\top }R{\mathbf{u}}_{k|t})\right]$$

subject to the dynamics, the input set ${\mathbf{u}}_{k|t}\in \mathcal{U}$, and risk-tightened state /
terminal constraints in place of probabilistic ones:

$${\rho }_{\alpha }(F_x{\mathbf{x}}_{k|t}-g_x)\le 0,{\rho }_{\alpha }(F_f{\mathbf{x}}_{t+N|t}-g_f)\le 0.$$

When ${\rho }_{\alpha }={\mathrm{CVaR}}_{\alpha }$ this is a convex inner approximation of the chance constraint
$\mathrm{Prob}({\mathbf{x}}_{k|t}\notin {\mathcal{X}}_S)\le \alpha$; the same chance constraint reduces to a
deterministic QP only in the Gaussian case (via $F_x{\hat{\mathbf{x}}}_{k|t}+F_x{\Phi }^{-1}(1-\alpha ){\Sigma }_{k|t}\le g_x$).

Notation note. The source writes a single risk level as $\beta \in (0,1)$ (with $\beta \to 1$
risk-neutral and $\beta \to 0$ risk-averse) and uses it for both the CVaR subscript ${\rho }_{\beta }$
and the chance-constraint bound $\mathrm{Prob}(\cdot )\le \beta$. notation.md already
reserves $\beta$ for the isotropic-floor damping scalar in the singularity layer, so to avoid a glyph
clash this page maps the source’s risk level to $\alpha$ — notation.md’s “CVaR confidence level (risk
layer)” — everywhere it appears (CVaR subscript and chance bound alike); $\beta$ is not used
on this page. The coherent-risk operator $\rho$ is not yet in notation.md (introduced source-faithfully here).

Source Support

• akella2024risk — survey “Risk-Aware Robotics: Tail Risk Measures in
  Planning, Control, and Verification.” Its “Risk-Aware MPC” subsection and the MPC sidebar give the
  stochastic-MPC formulation, the $\rho$-cost/constraint reformulation, the Gaussian-chance-constraint
  reduction, and the data-driven / obstacle-avoidance MPC literature (CVaR, EVaR, distributionally-robust
  variants). Primary and only cited source for this page.

Related Topics

• model_predictive_control — the underlying receding-horizon scheme;
  risk-aware MPC is this with the expectation/chance constraints swapped for a coherent risk measure.
• conditional_value_at_risk — the canonical tail-risk measure $\rho$
  used in the cost and as the convex inner approximation of the chance constraints.
• chance_constraints — the probabilistic safety constraints
  $\mathrm{Prob}(\mathbf{x}\notin {\mathcal{X}}_S)\le \alpha$ that the CVaR reformulation tightens into a
  tractable convex form for non-Gaussian noise.
• coherent_risk_measures — the axioms (subadditivity, monotonicity,
  translational invariance, positive homogeneity ⇒ convexity) that make the $\rho$-constrained program
  convex and well-posed.
• nonlinear_mpc — for nonlinear plants the convex QP reformulation no longer holds;
  the survey points to sampling-based solvers (e.g. model-predictive path integral) with CVaR constraints.

Open Questions

• The source’s formulation is a linear discrete-time plant; our free-flying space manipulator has
  strongly nonlinear, dynamically-coupled base–arm dynamics (ffsm_dynamics,
  dynamic_coupling). Which CVaR-MPC reformulation survives linearization about
  the circumcentroidal nominal trajectory, and does the convex-inner-approximation guarantee carry over?
• The survey is regime-agnostic (terrestrial/automotive/ground-mobile case studies). Does any cited
  result assume the actuation authority of a fully-actuated base, or only the generic
  ${\mathbf{x}}_{k+1}=A{\mathbf{x}}_k+B{\mathbf{u}}_k+D{\mathbf{d}}_k$ plant — i.e. is anything
  here specific to (or invalidated by) the free-flying vs. free-floating distinction?
• The risk constraints here are pointwise-in-time; the survey notes a trajectory-wise (time-supremum)
  CVaR alternative. For an inspection pass, should the keep-out / pointing-error risk be enforced
  pointwise or over the whole standoff trajectory?