PaperlayerView exact revision on arXiv

Source-linked AI summary

Receding-horizon Stochastic Model Predictive Control with Hard Input Constraints and Joint State Chance Constraints

Joel A. Paulson, Edward A. Buehler, Richard D. Braatz, Ali Mesbah

arXiv:1506.08471v1math.OCeess.SY

TL;DR

The paper addresses stochastic control of discrete-time linear systems with unbounded disturbances, hard input constraints, and joint state chance constraints. It derives a feasible and stable convex SOCP using saturated affine disturbance feedback and conservative chance-constraint approximations. In the fermentation case study, the SMPC approach always satisfies state constraints, whereas standard MPC violates them in approximately 20% of closed-loop simulations.

  • Problem

    The paper addresses stochastic linear control with unbounded disturbances, hard input constraints, and joint state chance constraints while requiring tractable optimization.

  • Method

    The method uses a saturated affine disturbance-feedback policy, Cantelli-Chebyshev chance-constraint approximations, and softened constraints within a convex SOCP.

  • Results

    Approximately 20% of standard MPC closed-loop simulations violate state constraints, whereas the SMPC approach always satisfies them in the reported case study.

  • Takeaways & Limitations

    The approach combines hard input-constraint handling with joint chance constraints, guaranteed feasibility, and stochastic stability in the studied setting.

  • Takeaways & Limitations

    Earlier stochastic tube approaches cannot handle hard input constraints, and the paper identifies future extensions beyond its current scope.

Abstract

from arXiv · show

This article considers the stochastic optimal control of discrete-time linear systems subject to (possibly) unbounded stochastic disturbances, hard constraints on the manipulated variables, and joint chance constraints on the states. A tractable convex second-order cone program (SOCP) is derived for calculating the receding-horizon control law at each time step. Feedback is incorporated during prediction by parametrizing the control law as an affine function of the disturbances. Hard input constraints are guaranteed by saturating the disturbances that appear in the control law parametrization. The joint state chance constraints are conservatively approximated as a collection of individual chance constraints that are subsequently relaxed via the Cantelli-Chebyshev inequality. Feasibility of the SOCP is guaranteed by softening the approximated chance constraints using the exact penalty function method. Closed-loop stability in a stochastic sense is established by establishing that the states satisfy a geometric drift condition outside of a compact set such that their variance is bounded at all times. The SMPC approach is demonstrated using a continuous acetone-butanol-ethanol fermentation process, which is used for production of high-value-added drop-in biofuels.

1. Introduction

The paper addresses stochastic linear control with unbounded disturbances, hard input constraints, and joint state chance constraints by deriving a tractable SMPC formulation with feasibility and stability guarantees.

  • The paper targets stochastic linear systems with unbounded disturbances, hard input constraints, and joint state chance constraints.
  • The main challenge is obtaining tractable control policies while handling unbounded disturbances, nonconvex chance constraints, feasibility, and stability.
  • The proposed approach uses saturated affine disturbance feedback, individual chance-constraint approximations, and the Cantelli-Chebyshev inequality.
  • A convex second-order cone program is derived with guaranteed feasibility and stability while retaining hard input constraints and joint chance constraints.
  • The approach is demonstrated on an experimentally validated 12-state acetone-butanol-ethanol fermentation process for producing drop-in biofuels.

2. Problem Statement

The paper formulates receding-horizon SMPC for a discrete-time stochastic linear system with hard input constraints and joint state chance constraints, then approximates the problem into a tractable convex optimization program.

  • The problem concerns receding-horizon control of a discrete-time stochastic linear system with hard input constraints and joint state chance constraints.
  • The disturbances are zero-mean, independent, identically distributed, and known through their covariance and probability density, while their support may be unbounded.
  • The desired controller must enforce hard input constraints, satisfy state chance constraints, and guarantee closed-loop stochastic stability.
  • General feedback policies make the optimization intractable, while unbounded disturbances complicate hard input satisfaction and chance constraints are generally nonconvex.
  • The proposed approximation uses affine disturbance feedback, saturated disturbance terms, and Cantelli-Chebyshev surrogates based on predicted means and variances.

3. Methods for Tractable SMPC Problem

The method stacks predicted states and controls over the horizon, then uses affine disturbance feedback to obtain a tractable convex formulation while preserving hard input constraints under unbounded disturbances.

  • 3.1. Compact notation: Predicted states and controls are stacked over the horizon to express the system dynamics, cost, and constraints compactly.The stacked state vector has dimension R^nx(N+1), while the stacked control vector has dimension R^nuN.
  • 3.2. Tractable feedback control policy: The control policy uses affine functions of past disturbances rather than affine state feedback, enabling a tractable convex problem despite feedback during prediction.The disturbance-feedback and state-feedback parametrizations are equivalent under perfect state observation, but the disturbance-feedback formulation can yield suboptimal solutions.
  • 3.3. Saturation of stochastic disturbances for handling hard input constraints: Saturating disturbance terms enables hard input constraints to be handled when disturbances are unbounded.The saturated policy is converted into convex inequalities, and the input constraint representation is exact under the stated construction.
  • 3.4. Joint chance constraints: Joint state chance constraints are relaxed by decomposing each joint constraint into individual chance constraints and applying the Cantelli-Chebyshev inequality.The resulting deterministic constraints depend on predicted-state means and covariances, with risk allocations summing to the joint violation bound.

finite covariance Σx ∈Sn

The chance-constraint surrogate is convex for fixed auxiliary parameters, but its covariance dependence is nonlinear and can require conservative linearization with an added design choice.

  • Covariance dependence: The covariance-dependent constraints are nonconvex because their expressions depend nonlinearly on Σ_x.The paper notes that linearization around selected state values can convexify the expression, but this tightens constraints and adds a design parameter.
  • Chance-constraint reformulation: The individual chance-constraint conditions are convex in the decision variables when the auxiliary parameter δ_j is fixed.A single deterministic inequality can instead be formed by eliminating δ_j, but this introduces nonlinear covariance dependence.
  • Covariance dependence: The auxiliary parameters relate directly to state covariance size, giving their values a physical interpretation.This interpretation connects the relaxation parameters to the dispersion of predicted states.
  • Feedback-policy choice: Affine state-feedback policies can introduce additional suboptimality because they do not incorporate feedback during prediction in the same way as the disturbance-feedback formulation.The supplied discussion contrasts this limitation with the tractable disturbance-feedback approach.

4. Proposed Approach for Stochastic Model Predictive Control

The proposed SMPC formulation converts the stochastic control problem into a tractable convex program, while softening state constraints to guarantee feasibility and minimize violations.

  • 4. Proposed Approach for Stochastic Model Predictive Control: The methods are combined to derive a tractable convex formulation for the SMPC problem and establish feasibility and stability properties.The formulation uses affine disturbance feedback and chance-constraint approximations as its main ingredients.
  • 4.1. Convex SMPC formulation: Predicted means and covariances define the admissible decision variables because the state-constraint approximations depend only on these quantities.The dynamics of the predicted state mean and variance are used to construct the optimization constraints.
  • 4.2. Feasibility of the Convex SMPC Formulation: Unbounded disturbances prevent feasibility from being guaranteed at every time for compact state constraints under bounded control actions.The paper states that the true states can violate any given compact set infinitely often over an infinite horizon.
  • 4.2. Feasibility of the Convex SMPC Formulation: Soft slack variables are added to the approximated state constraints, and their penalization minimizes constraint violation while retaining the original control objective.The exact penalty method is intended to recover the hard-constrained solution whenever that solution is feasible.
  • 4.2. Feasibility of the Convex SMPC Formulation: With a sufficiently large penalty weight, violations occur only when no solution satisfying the hard constraints exists.This is the exact penalty function property used in the softened formulation.

ϵ ∈R2rN denotes the vector of all slack variables ϵm

The softened formulation has an unrestricted feasible initial-condition domain, and its closed-loop behavior is analyzed through a geometric drift condition.

  • Feasibility domain: Because sufficiently large slack variables are always available, the softened problem has feasible initial conditions throughout R^n_x.The construction sets the policy variables to zero and compensates with slack variables when needed.
  • Penalty selection: The exact penalty weight is selected so that constraint violations are minimized and the original objective is preserved whenever hard feasibility is possible.The paper discusses both conservative lower bounds and practical numerical selection of the penalty weight ρ.
  • Stochastic stability: Stochastic stability is established by verifying a geometric drift condition outside a compact set for the Markov process of closed-loop states.The drift lemma uses a contraction factor λ in [0,1) outside the compact set and a bounded one-step expectation inside it.
  • Stochastic stability: The always-feasible control law yields a stochastically stable closed loop with bounded state variance.This conclusion is stated under the assumptions used for the feasibility theorem.

5. Case Study: Stochastic Optimal Control of a Continuous ABE Fermentation Process

The case study applies SMPC to continuous ABE fermentation, modeling a 12-state, 2-input stochastic process and solving an SOCP for constrained setpoint tracking. Across simulations, SMPC tracks product setpoints while maintaining acidic-species constraints under stochasticity, though conservatism causes slight control-performance loss.

  • Process model: The ABE fermentation model is linearized around the solventogenesis steady state and contains 12 states, with dilution rate and inlet glucose concentration as inputs.All system states are perturbed by zero-mean white noise with variance 10^-4 mM.
  • Control formulation: The SOCP formulates ABE product setpoint tracking alongside hard input constraints and individual chance constraints on acetate and butyrate.The receding-horizon SOCP is solved at every sampling instant using observed system states.
  • Results: The controller tracked a 5% increase in acetone, butanol, and ethanol setpoints while minimizing concentration variance around the setpoints.The setpoint change was applied at time 10 hours across 100 closed-loop simulation runs.
  • Limitations: Product tracking shows a slight offset, particularly for ethanol, because the controller trades control objectives against state-constraint satisfaction under process stochasticity.The case study reports this tradeoff as a slight drop in control performance.
  • Results: Acetate and butyrate constraints were never violated, attributed to conservative Cantelli-Chebyshev approximations that impose tighter effective bounds.The resulting hard bound is more conservative than the bounds specified by the state chance constraints.
  • Results: SMPC maintained the state constraints in all simulations, whereas standard MPC violated them in approximately 20% of simulations.The comparison concerns closed-loop simulations under stochastic uncertainties.

6. Conclusions

The paper derives a tractable convex SMPC formulation for constrained linear systems with possibly unbounded disturbances and demonstrates it on continuous ABE fermentation. The approach maintains state constraints in simulations, supports stochastic stability, and identifies conservatism reduction as future work.

  • A tractable convex optimization program is derived for receding-horizon stochastic control of linear systems with possibly unbounded disturbances.The formulation addresses hard input constraints and softened state chance constraints to preserve feasibility.
  • The SMPC approach is demonstrated on a continuous ABE fermentation process using probability distributions from 100 closed-loop simulation runs.The simulations examine acetate and butyrate concentrations under closed-loop control.
  • State constraints are always satisfied with SMPC, whereas standard MPC violates them in approximately 20% of closed-loop simulations.This comparison is reported for the fermentation-process simulations.
  • The exact penalty formulation recovers the hard-constrained solution when the latter is feasible, while Cantelli-Chebyshev relaxation introduces conservatism.Future work includes output feedback with measurement noise and reducing this conservatism.
  • The receding-horizon policy yields stochastic stability by satisfying a geometric drift condition outside a compact set, keeping state variance bounded for all time.The stability result is stated for the closed-loop system.

Appendix A

Appendix A provides proofs establishing the Cantelli-Chebyshev bound, the convex SOCP reformulation, constraint handling, and stochastic boundedness results.

  • Lemma 2: The Cantelli-Chebyshev bound follows by constructing a quadratic upper bound and minimizing it at b⋆=Var[Y]/a.Substitution of the minimizing value yields the stated probability upper bound.
  • Theorem 1: The appendix proves the main theorem through value-function derivation, hard-input constraint guarantees, softened chance-constraint inequalities, and SOCP conversion.The proof is organized into four derivations supporting the soft-constrained optimal control problem.
  • Theorem 1: Hard input constraints become linear in the disturbance-feedback decision variables through the dual formulation.The resulting constraints involve M, v, and Z.
  • Theorem 1: The convex quadratic constraints are converted to standard second-order cone constraints by introducing a variable that bounds the quadratic objective terms.This produces the standard SOCP form for the control problem.
  • Lemma 4: A geometric drift argument and a convergent geometric series establish that the expected value function remains uniformly bounded over time.The proof gives sup_t∈N0 E[V(x_t)] ≤ V(x_0) + b(1 − λ)^−1 < ∞.
  • Theorem 2: The stability proof selects a compact set and a decay parameter satisfying 1−λ_max(P) < θ < 1, thereby meeting the drift-condition premises.The corresponding λ is then defined from θ and λ_max(P).

Appendix B

Appendix B describes the state-space model used for the continuous acetone-butanol-ethanol fermentation process, including its state, input, and system-matrix definitions.

  • State-space model: The fermentation model defines a state vector containing concentrations and gene-related variables for the acetone-butanol-ethanol process.The state components include AC, acetate, enzyme-related quantities, acetone, butanol, and other process variables.
  • State-space model: The input vector comprises the dilution rate D and inlet glucose concentration G0.D is measured in hr−1 and G0 in mM.
  • State-space model: The appendix supplies the system matrices and specifies G as a diagonal matrix with twelve unit entries.The matrix is given as G = diag(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1).
Loading 1506.08471v1…