Robust MPC Formulation

• Robust MPC is a control approach that explicitly models uncertainties with sets like ellipsoids or polytopes to guarantee constraint satisfaction and closed-loop stability.
• It employs tube-based techniques where uncertainty tubes are propagated using system Jacobians and feedback policies, ensuring robust constraint tightening.
• The formulation utilizes iterative strategies like zoRO for efficient real-time optimization, offering theoretical guarantees such as recursive feasibility and input-to-state stability.

Robust Model Predictive Control (MPC) formulations encompass a broad class of methods designed to provide constraint satisfaction and stability guarantees for control systems subject to process uncertainties, disturbances, parameter variations, and model mismatch. These approaches systematically propagate uncertainty bounds—typically as ellipsoidal, polytopic, or probabilistic sets—through the prediction horizon and employ robustification mechanisms to ensure constraints are enforced for all allowable realizations. The following sections summarize the technical foundations, canonical algorithms, constraint-tightening structures, specific robustification techniques, and the theoretical guarantees associated with robust MPC design.

1. System Modeling and Uncertainty Representation

Robust MPC is predicated on formal representations of plant dynamics and associated uncertainties. Consider a discrete-time system state $x \in \mathbb{R}^n$, input $u \in \mathbb{R}^m$, nominal (possibly nonlinear) plant $f(x_k,u_k)$, and additive process noise $w_k$,

$$x_{k+1} = f(x_k, u_k), \qquad \tilde{x}_{k+1} = f(\tilde{x}_k, u_k) + w_k.$$

Uncertainty sets for $w_k$ are typically specified as either:

• Ellipsoidal: $$W = \{ w \in \mathbb{R}^n\,|\,w^\top S^{-1} w \le 1 \}$$, $S \succ 0$ (Gao et al., 2023),
• Polytopic: $W = \text{conv}\{w^{(1)}, ..., w^{(M)}\}$,
• Zonotopic,
• Bounded box constraints.

Parametric uncertainty may be represented by sets of system matrices: $$A_k = \bar{A} + \Delta_A^{\text{tr}},\;\;B_k = \bar{B} + \Delta_B^{\text{tr}},$$ with $(\Delta_A^{\text{tr}}, \Delta_B^{\text{tr}})$ in specified polytopes (Bujarbaruah et al., 2021).

Uncertainty propagation induces nonlinear or linear evolution of disturbance tubes (ellipsoidal, polytopic, or zonotopic) around the nominal trajectory, which are recursively propagated using system Jacobians, feedback policies, and uncertainty set dynamics.

2. Tube-Based Robust MPC and Constraint-Tightening

Robust MPC constructs auxiliary sets (tubes) that enclose all reachable system states under bounded process noise. For ellipsoidal tubes, the tube at stage $k$ is $x_k \oplus \mathcal{E}(\Sigma_k)$ with $$\mathcal{E}(\Sigma) = \{ e \,|\, e^\top \Sigma^{-1}e \le 1 \}$$ and set-propagation

$$\Sigma_{k+1} = (A_k - B_k K)\Sigma_k (A_k - B_k K)^\top + S,$$

where $(A_k,\,B_k)$ are Jacobians at $(x_k,u_k)$ and $K$ is a disturbance-rejection gain (Gao et al., 2023).

Constraints are robustified by back-off terms $\beta_k$: $$h_k(x_k,u_k) + \beta_k(\Sigma_k,x_k,u_k) \le 0, \qquad \beta_k = \sqrt{ \nabla h_k^\top [I~K] \Sigma_k [I~K]^\top \nabla h_k },$$ ensuring satisfaction for all admissible uncertainty. For polytopic tubes, similar constructions use Minkowski sums/differences, and explicit set propagation via matrix powers and sum over disturbance sets (Alvarado et al., 2021).

Collision avoidance, actuator bounds, and complex state constraints are enforced through constraint tightening using the evolving uncertainty tubes (Gao et al., 2023).

3. Robust MPC Optimization Algorithms and Iterative Strategies

The robust MPC problem is formulated as an optimization over states, inputs, and tube parameters to minimize cumulative stage and terminal costs, subject to robustified system dynamics and tightened constraints. A typical formulation in the tube-based setting is: $$\begin{aligned} \min_{\{x_k, u_k, \Sigma_k\}_{k=0}^N} &~ \ell_N(x_N) + \sum_{k=0}^{N-1}\ell_k(x_k, u_k) \ \text{s.t.}~ &~ x_{k+1} = f(x_k, u_k),~ \Sigma_{k+1} = \Phi(\Sigma_k, x_k, u_k), \ &~ h_k(x_k, u_k) + \beta_k(\Sigma_k, x_k, u_k) \le 0. \end{aligned}$$

Zero-order robust optimization (zoRO) is a computationally efficient strategy (Gao et al., 2023):

• Fix tube parameters $\Sigma_k$ and back-off terms $\beta_k$ from previous iteration,
• Solve nominal nonlinear program for $\{x_k, u_k\}$,
• Propagate tube parameters and recompute back-off terms,
• Iterate a predefined number of times per control step; single iterations suffice for real-time environments.

Algorithmic sketch:

Initialize Σ_k, compute β_k via (6) for i in range(M): Solve subproblem for x_k, u_k with fixed β_k Propagate Σ_k using (5d) Update β_k using (6) Return u_0^*, store Σ_k for next time step

Ellipsoid propagation proceeds via linearization and tube update formulas as above.

4. Robustification Mechanisms and Back-Off Computation

Back-off terms $\beta_k$ serve to guarantee robust satisfaction of constraints under all disturbance realizations. The explicit form is: $$\beta_k = \sqrt{ \nabla h_k^\top [I~K] \Sigma_k [I~K]^\top \nabla h_k },$$ for stage constraints, and

$$\beta_N = \sqrt{ \nabla h_N^\top \Sigma_N \nabla h_N },$$

for terminal constraints. In collision-avoidance, for obstacle $j$,

$$h^{\text{coll}}_{k,j}(x_k) = r + r_j^{\text{obs}} - \sqrt{ (x_k - x_j^{\text{obs}})^2 + (y_k - y_j^{\text{obs}})^2 } \le 0$$

is tightened by $\beta_k$.

This structure generalizes to arbitrary nonlinear state/input constraints provided the gradients $\nabla h_k, \nabla h_N$ are well-defined.

5. Implementation Characteristics and Theoretical Guarantees

Robust MPC formulations based on zoRO and ellipsoidal-tube propagation are compatible with nonlinear systems, collision avoidance, and state/input constraints. Required system ingredients include:

• Horizon $N$, state dimension $n$, input dimension $m$,
• Discrete/continuous-time dynamics $f$,
• Disturbance set $S$,
• Reference trajectories $(x_k^{\text{ref}}, u_k^{\text{ref}})$,
• Stage/terminal cost matrices $Q, R, Q_f$,
• Precomputed stabilizing feedback $K$,
• Number of zoRO iterations $M$ per cycle.

Theoretical properties include:

• Recursive feasibility: feasible initial solution implies feasibility at subsequent steps (Gao et al., 2023),
• Input-to-state stability (ISS) and closed-loop stability: cost functions serve as ISS Lyapunov functions,
• Robust constraint satisfaction: constraints hold for all disturbance realizations within specified sets,
• Suboptimality: zoRO is generally suboptimal but maintains convergence to reference trajectory in absence of obstacles.

Experimental results confirm real-time implementation on industrial mobile robot prototypes, and demonstrate robust collision-free tracking under bounded process noise (Gao et al., 2023).

6. Extensions, Comparisons, and Applicability

The robust MPC formulation generalized above subsumes classical tube-MPC, zonotope-based, polytopic, and min-max MPC strategies. Key distinctions include:

• zoRO algorithmic structure for computational efficiency at scale,
• Explicit propagation of disturbance ellipsoids via system Jacobians and feedback linearization,
• Back-off computation via constraint gradients and tube shape matrices,
• Applicability to complex nonlinear plants and operational constraints (e.g., collision avoidance, differential-drive robots).

Suboptimality of zero-order iterations is offset by practical performance in real-time applications, with convergence and constraint satisfaction guarantees preserved for unconstrained tracking scenarios. In constrained, obstacle-rich domains, iterative improvement of back-off terms and tube updates achieves near-optimal collision-free planning under process uncertainty.

Robust MPC using zoRO and tube-based constraint-tightening constitutes a foundational methodology for real-time control of uncertain nonlinear systems, with proven theoretical properties and scalable practical realization (Gao et al., 2023).