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Robust Model Predictive Control via Scenario Optimization
Giuseppe C. Calafiore, Lorenzo Fagiano
TL;DR
The paper addresses robust MPC for discrete-time linear systems subject to parametric uncertainty and additive disturbances. It solves convex scenario-based finite-horizon problems using random uncertainty and disturbance samples, then applies a receding-horizon command-selection rule. Under the stated assumptions, constraints are satisfied with chosen reliability p, while the state reaches the terminal set asymptotically or in finite time with probability at least p.
Problem
Robust MPC must provide stability and constraint satisfaction under uncertain parameters and disturbances, including settings where uncertainty and disturbance sets are non-convex.
Method
At each step, the method solves a scenario-based finite-horizon optimal control problem using randomly extracted parameter and disturbance trajectories, with a terminal-law correction structure.
Results
With practical certainty, the closed loop satisfies state and input constraints with probability at least p at every time, and the state converges to or reaches the terminal set with probability at least p.
Takeaways & Limitations
The approach provides a convex randomized robust MPC design for uncertainty and disturbance descriptions that need not be convex or connected.
Takeaways & Limitations
The guarantees require bounded uncertainty and disturbance sets, stochastic independent parameters and disturbances, stabilizability, convex constraints, and an available terminal set and control law.
Abstract
from arXiv · showhide
This paper discusses a novel probabilistic approach for the design of robust model predictive control (MPC) laws for discrete-time linear systems affected by parametric uncertainty and additive disturbances. The proposed technique is based on the iterated solution, at each step, of a finite-horizon optimal control problem (FHOCP) that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances, followed by a specific command selection rule implemented in a receding horizon fashion. The scenario FHOCP is always convex, also when the uncertain parameters and disturbance belong to non-convex sets, and irrespective of how the model uncertainty influences the system's matrices. Moreover, the computational complexity of the proposed approach does not depend on the uncertainty/disturbance dimensions, and scales quadratically with the control horizon. The main result in this paper is related to the analysis of the closed loop system under receding-horizon implementation of the scenario FHOCP, and essentially states that the devised control law guarantees constraint satisfaction at each step with some a-priori assigned probability p, while the system's state reaches the target set either asymptotically, or in finite time with probability at least p. The proposed method may be a valid alternative when other existing techniques, either deterministic or stochastic, are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance-constrained optimization problem.
1 Introduction
Robust MPC must handle uncertainty and disturbances while enforcing constraints and maintaining stability. The paper proposes a randomized robust MPC method that remains convex without requiring convex uncertainty or disturbance sets.
- Robust MPC seeks stability and constraint satisfaction despite uncertainty and disturbances.
- Stochastic MPC uses statistical information but typically requires known system matrices or structures preserving convexity.
- The proposed randomized method handles model uncertainty and additive disturbances without assuming convex or connected uncertainty and disturbance sets.
- The resulting optimization problem is always convex with respect to uncertain parameters and disturbances.
2 Problem formulation
The paper models a discrete-time uncertain linear system with stochastic parameters and disturbances, state and input constraints, and regulation toward a robustly invariant terminal neighborhood.
- The system includes uncertain parameters, unmeasured disturbances, and parameter-dependent matrices A(θ), B(θ), and Bγ(θ).
- The uncertainty sets and system-matrix set are bounded, while parameters and disturbances have probability measures and independent disturbances are i.i.d.
- The control objective is to regulate the state near the origin while satisfying potentially uncertain state and input constraints at every time.
- Constraint sets are convex, contain the origin in their interiors, and are represented by convex functions of states and inputs.
- The parameter may affect constraint functions or thresholds, while no restrictions are imposed on how it affects system matrices beyond stabilizability.
- Because disturbances may be nonzero, regulation targets a neighborhood represented by a robustly positively invariant terminal set.
- The terminal controller u = Kf x keeps the terminal set invariant and satisfies state and input constraints for all modeled parameters and disturbances.
- The control input combines the terminal law Kfxt with a correction vt, and the resulting closed-loop matrix is Acl(θ) = A(θ) + B(θ)Kf.
3 The Scenario-based Finite-Horizon Optimal Control Problem
The scenario-based FHOCP predicts trajectories under randomly sampled uncertainty and disturbances, then selects control corrections through a convex optimization with soft constraints. Scenario results provide probabilistic finite-horizon guarantees for reaching the terminal set and satisfying state and input constraints, while receding-horizon implementation yields closed-loop guarantees.
- Control parameterization: The controller optimizes N control corrections, parameterizing each predicted input as uj|t = Kfxj|t + vj|t.Only the corrective sequence is optimized, giving N m control decision variables.
- Scenario construction: M randomly extracted uncertainty and disturbance scenarios generate predicted state and input trajectories for the finite-horizon control problem.The scenarios are sampled from the product probability measure of uncertain parameters and disturbance sequences.
- Optimization formulation: Soft state and input constraints with slack qt make the scenario FHOCP always feasible, while αqt penalizes constraint violation.The slack is negligible when the corresponding hard-constrained problem is feasible and indicates violation otherwise.
- Optimization formulation: The worst-case cost z∗t upper-bounds sampled scenario costs and the distance between the current state and the terminal set Xf.This upper-bound property is used in the convergence analysis.
- Probabilistic guarantees: For desired reliability p and small β, choosing M according to the scenario bound makes the computed sequence reach Xf and satisfy state and input constraints with probability at least p.The probability β bounds the unfavorable event that the solution reliability falls below p; M grows mildly with β^-1.
- Receding-horizon implementation: Receding-horizon control applies only the first correction, resolves the FHOCP at the next state, and supports probabilistic closed-loop convergence and constraint satisfaction.The paper analyzes repeated scenario FHOCP solutions rather than applying the full finite-horizon sequence once.
4 MPC scheme based on Scenario optimization
The MPCS algorithm repeatedly solves scenario-based finite-horizon problems and selects between newly optimized and previously computed control sequences. Under the stated assumptions, it provides probabilistic constraint satisfaction and convergence to the terminal set.
- Algorithm 4.1: MPCS initializes a scenario optimization problem, then applies the first control correction while retaining the remaining sequence for subsequent steps.At each step it shifts the previous sequence, samples scenarios, resolves the problem, and applies u_t = K_f x_t + v_0|t.
- Algorithm 4.1: The selection rule accepts the new solution when its worst-case cost improves sufficiently; otherwise, it uses the shifted previous solution.The parameter ε controls how much improvement is required before accepting the newly computed sequence.
- Guaranteed properties: With practical certainty, state and input constraints are satisfied at every time step with probability at least p and violation level q_t.This guarantee holds for either the newly computed sequence or an appropriately shifted previously computed sequence.
- Guaranteed properties: The state either converges asymptotically to the terminal set or reaches it in finite time with probability at least p.In the finite-time case, the applied sequence drives the state to X_f within the subsequent N-step window, with the associated constraint violation.
- Convergence mechanism: The running worst-case cost bounds the distance to the terminal set and decreases by at least εd(x_t, X_f) whenever the state is outside X_f.The resulting relations establish convergence, while z_t = 0 is equivalent to x_t belonging to X_f.
5 Numerical example
The numerical example tests MPCS on a nonlinear-in-parameters uncertain system with non-convex disturbances using Monte Carlo simulations. The observed success probabilities exceed the prescribed reliability levels, with receding-horizon performance higher than finite-horizon performance.
- Numerical example: The example uses uniformly and Gaussian-distributed uncertain parameters, with additional uniformly distributed variables defining the disturbance and constraints.The construction includes θ_1, θ_2, θ_3 uniform on [-0.1,0.1] and θ_4, θ_5 Gaussian with zero mean and unit variance.
- Numerical example: The system matrices and constraints depend nonlinearly on uncertain parameters, while the disturbance belongs to a non-convex disconnected set.The authors state that existing robust MPC techniques cannot be directly applied to this example.
- Experimental setup: The experiment uses N = 10, β = 10^-9, and 100,000 Monte Carlo trials from x_0 = [5, 2.75]^T.The initial condition is infeasible for the deterministic counterpart, so some initial scenario solutions have non-negligible constraint violation.
- Results: For p = 0.05, 0.3, 0.6, and 0.95, both finite-horizon and receding-horizon estimated success probabilities exceed the corresponding p values.The results are reported in Table 1 with the corresponding scenario counts M.
- Results: Estimated receding-horizon success probabilities exceed finite-horizon estimates, with good performance already at low numbers of scenarios.The paper attributes the receding-horizon increase to iterative re-optimization and the finite-horizon robustness mainly to the terminal control law.
- Computational complexity: For fixed p and β, scenario count M is independent of state and uncertainty dimensions, while total constraints scale approximately as n · m^2 · N^2.The quadratic scaling is with respect to the control horizon N.