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Robust Optimal Control with Adjustable Uncertainty Sets

Xiaojing Zhang, Maryam Kamgarpour, Angelos Georghiou, Paul Goulart, John Lygeros

arXiv:1511.04700v2math.OCcs.SYeess.SY

TL;DR

The paper addresses robust control problems in which uncertainty sets are usually fixed, rather than selected alongside a policy to maximize their size or shape while meeting constraints. It develops convex approximations using restricted set representations and affine policies. The framework covers practical convex uncertainty sets and is applied to power-system reserve provision, with reported computational and solution-quality results.

  • Problem

    Standard constrained robust optimal control uses uncertainty sets known in advance, whereas choosing a set's size and shape while maintaining constraints for every disturbance is generally intractable.

    Source: §1 · p. 2

  • Method

    The framework restricts uncertainty sets to affine transformations of primitive convex sets and control policies to affine functions of those primitives, yielding convex reformulations.

    Source: §1 · p. 2

  • Results

    The power-system reserve problem is solvable as a linear optimization problem in 0.3 seconds, with optimal solutions through eight time steps and a relative suboptimality gap bounded by 0.71% for larger horizons.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The framework provides a unified treatment of reserve provision, input tracking, and robustness analysis and covers practical uncertainty-set families including norm balls, ellipsoids, and polytopes.

    Source: §1 · pp. 2–3

    The framework uses affine policies; future work includes studying their optimality for reserve provision, as well as stability, recursive feasibility, external uncertainties, and partial observations.

    Source: §7 · p. 19

Abstract

from arXiv · show

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. We demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.

1. Introduction

The paper makes uncertainty sets decision variables in finite-horizon robust control, seeking their optimal size and shape while maintaining constraints for every disturbance realization. It develops tractable approximations by restricting uncertainty-set representations and control policies, then applies the framework to power-system reserves.

1. Introduction

  • The framework generalizes adjustable uncertainty sets using conic convex optimization and unifies reserve provision, input tracking, and robustness analysis.The objective is to choose set size and shape while ensuring a feasible control policy keeps the system within its constraints for all allowed disturbances.Source: §1 · p. 2
  • Affine transformations of primitive convex sets and affine policies yield efficient convex reformulations whose size grows polynomially in the decision parameters.Any convex set can serve as a primitive set, and higher-dimensional primitive sets allow more flexible uncertainty sets.Source: §1 · pp. 2–3
  • The framework covers norm balls, ellipsoids, and hyper-rectangles, and polytopic primitives yield continuous piecewise-affine controllers.A simplex primitive supports optimization over compact polytopes with a predefined number of vertices.Source: §1 · p. 3
  • The reserve-provision formulation is a linear optimization problem solvable in 0.3 seconds; solutions are optimal through eight time steps, with a relative suboptimality gap bounded by 0.71% for larger horizons.The paper reports these results for its power-system reserve problem.Source: §1 · p. 3

Notation

  • The notation defines A = diag(A1, ..., An) as a block-diagonal matrix with the listed matrices on its diagonal.It also defines cone ordering and convex-hull notation.Source: Notation · p. 3

2. Problem Formulation

  • The problem jointly optimizes causal disturbance-feedback policies and uncertainty sets, maximizing a set-quality metric while minimizing operating cost under state and input constraints.The formulation is generally intractable because it optimizes over arbitrary sets and causal functions while imposing robust constraints.Source: §2 · pp. 4–5

3. Uncertainty Set Approximation via Primitive Sets

The section restricts adjustable uncertainty sets to affine transformations of primitive sets, then reformulates the robust constraints to obtain tractable approximations. It describes volume objectives and shows how common geometric families fit this representation.

3. Uncertainty Set Approximation via Primitive Sets

  • Any compact convex set can serve as a primitive set, and each uncertainty set is shaped as an affine transformation of one.The shaping matrix can scale, rotate, and project the primitive set; the shaping vector translates it.Source: §3 · pp. 5–6
  • Eliminating the uncertainty variables yields an equivalent formulation that can be dualized into a finite-dimensional convex inner approximation.Its size grows polynomially with problem parameters, and it is tractable when the shaping objective is concave.Source: §3 · pp. 6–7

3.1. Volume Maximization

  • When the primitive and uncertainty sets have equal dimension and the shaping matrix is positive definite, maximizing log det of that matrix maximizes set volume convexly.For projected sets with more primitive dimensions than uncertainty dimensions, volume maximization is generally non-convex, so a convex heuristic objective is suggested.Source: §3.1 · p. 7

3.2. Specific Families of Uncertainty Sets

  • The primitive-set representation can capture practical uncertainty-set families, including norm balls, ellipsoids, hyper-rectangles, and polytopes.These families are obtained by selecting the primitive set and shaping constraints appropriately.Source: §3.2 · p. 7

Ball Uncertainty Set

The paper represents p-norm balls as scaled and translated primitive sets, with a tractable conic optimization formulation.

Ball Uncertainty Set

  • A p-norm ball is represented by a unit p-norm ball scaled by radius r and translated by y.The objective ϱ(W) = r is proportional to log det(Y) and therefore maximizes volume.Source: Ball Uncertainty Set · p. 8
  • The representation uses the p-order cone, whose dual is the q-order cone with 1/p + 1/q = 1.The shaping matrix is restricted to Y = rI.Source: Ball Uncertainty Set · p. 8
  • With linear shaping constraints, the resulting optimization problem is a tractable conic convex program.Source: Ball Uncertainty Set · p. 8

Ellipsoidal Uncertainty Set

Ellipsoidal and axis-aligned rectangular uncertainty sets are represented using primitive sets and constrained shaping parameters. The specified formulations permit convex optimization.

Ellipsoidal Uncertainty Set

  • An ellipsoid is represented by transforming the unit Euclidean ball with a positive-definite symmetric matrix identified with Σ^1/2.Maximizing log det of the shaping matrix maximizes volume, and the resulting problem can be solved efficiently.Source: Ellipsoidal Uncertainty Set · p. 8

Axis-aligned Hyper-Rectangular Uncertainty Sets

  • An axis-aligned rectangle uses the unit infinity-norm ball and a diagonal shaping matrix of half-widths γ.The log-determinant objective maximizes volume; an alternative weighted circumference objective is also convex.Source: Axis-aligned Hyper-Rectangular… · p. 8
  • For either rectangular objective, the dual cone is the first-order cone and the optimization problem is convex.Source: Axis-aligned Hyper-Rectangular… · p. 8

Polyhedral Uncertainty Set

The section represents a polytope through a simplex primitive set and its vertices as shaping parameters. Because generic volume maximization is intractable, it describes alternative tractable vertex objectives.

  • A polytope with m vertices can be represented as the image of an m-vertex simplex, with the shaping matrix columns corresponding to its vertices.The polytope is expressed as W = YS.Source: Polyhedral Uncertainty Set · p. 8
  • Maximizing the volume of a generic polytope with m vertices is computationally intractable.Source: Polyhedral Uncertainty Set · p. 8
  • Pushing vertices in user-specified directions gives a linear optimization problem, while pulling them toward chosen points gives a convex quadratic problem.Source: Polyhedral Uncertainty Set · pp. 8–9

4. Policy Approximation via Affine Decision Rules

Restricting uncertainty sets and policies to affine forms yields a tractable convex approximation with a finite-dimensional formulation. The paper relates these affine policies to policies in the original problem, including continuous piece-wise affine policies when the lifted uncertainty set is a polytope.

4. Policy Approximation via Affine Decision Rules

  • Affine policy restrictions turn the semi-infinite formulation into a finite-dimensional convex optimization problem when the uncertainty-set metric is chosen as specified, with polynomial growth in the problem parameters.The policy is causal and affine in the uncertainty-set representation.Source: §4 · pp. 9–10
  • Every feasible solution of problem (13) is feasible in the original problem (4), so solving (13) provides an upper bound for problem (4).The paper describes problem (13) as an inner approximation of problem (4).Source: §4 · p. 10

4.1. Relationship between the policies π(·) and eπ(·)

  • Affine policies in the lifted formulation can correspond to non-linear policies in the original formulation, and the optimized lifted policy can be used to compute control inputs for measured disturbances.The correspondence uses a lifting operator from W to S.Source: §4.1 · pp. 10–11
  • When S is a polytope, each feasible affine lifted policy has an equal-objective feasible continuous piece-wise affine counterpart in the original formulation.The paper also notes that piece-wise affine lifted policies can yield piece-wise affine original policies for polyhedral S.Source: §4.1 · pp. 10–11
  • Every feasible affine policy in problem (10a) has an equal-objective feasible affine counterpart in problem (10b); the converse holds when Y is invertible.Under invertible Y, the lifted formulation uses convex optimization instead of the bilinear optimization required by the original affine-policy restriction.Source: §4.1 · pp. 10–11
  • More complex piece-wise affine lifted policies are a stated alternative to affine policies, with tractable convex design methods cited for that setting.The stated correspondence to piece-wise affine original policies assumes that S is a polytope.Source: §4.1 · p. 11

5. Example 1: Frequency Reserve Provision

The example models a commercial building’s reserve offer as an adjustable capacity set, with controls required to meet every admissible reserve request while respecting system constraints. It studies how prices and rewards shape reserve provision and electricity consumption.

5.1. Problem Formulation

  • The formulation separates strictly causal nominal controls from causal adjustments that respond to the current reserve request, requiring each adjustment to match that request.The building dynamics include room, wall, floor, and ceiling temperatures, with ambient temperature, solar radiation, and internal gains as uncontrolled inputs.Source: §5.1 · pp. 12–13

5.2. Simulation Results

  • The study uses a one-room building model with a 24-hour horizon, hourly steps, room-temperature bounds of 21°C–25°C, and weekday or weekend electricity-price patterns.The model has four HVAC actuators and deterministic uncontrolled inputs.Source: §5.2 · p. 13
  • The numerical problem uses symmetric reserve sets W_k = [−Y_k, Y_k] and affine input policies over the box primitive set S = {s : −1 ≤ s ≤ 1}.The input policies are an inner approximation of the general problem.Source: §5.1 · p. 12, §5.2 · p. 13
  • Nighttime periods without temperature constraints are not modeled, although the authors say this feature can be incorporated into the setup.Source: §5.2 · p. 13
  • The building offers reserves only when the reward factor λ exceeds the electricity price c_k, because reserve provision requires increased nominal consumption and can add electricity cost.When λY_k < c_kY_k, it chooses not to provide reserves.Source: §5.2 · pp. 14–15
  • For most reward factors, the building offers more total reserves on weekends than weekdays, while small rewards favor weekdays; Figure 2 supports choosing reserve levels for a given reward.The comparison uses weekday and weekend prices over λ ∈ [15, 50].Source: Figure 2 · p. 15, §5.2 · p. 16
  • As reserve capacity rises with λ, nominal electricity consumption rises in the same pattern, while the maximal-to-minimal consumption interval represents operating slack.The consumption profiles are shown for weekday and weekend prices in Figure 2.Source: §5.2 · pp. 15–16
  • For horizons 9–24, the relative gap between affine policies and the exact solution is below 0.71% for λ ∈ [15, 50]; the 24-hour formulation has roughly 13,000 variables and 18,000 constraints and solves in 0.3 seconds.For N ≤ 8, enumeration verifies affine-policy optimality over the same reward range; enumeration grows exponentially with N.Source: §5.2 · p. 17

6. Example 2: Robustness Analysis

The example compares rectangular, ellipsoidal, and polytopic uncertainty sets for a one-step robustness analysis. The polytopic approximation yields the largest uncertainty set and is verified optimal for this example, while affine policies constrain the ellipsoidal solution.

Example 2: Robustness Analysis

  • The comparison considers rectangular, ellipsoidal, and 30-vertex polytopic uncertainty sets, using different objectives to approximate their volumes.Rectangular and ellipsoidal sets use log det(Y); the polytopic objective is a quadratic cost that approximates volume.Source: §6 · p. 18
  • The rectangular family restricts sets to be axis-aligned, whereas ellipsoidal and polytopic families allow rotated and skewed sets.This geometric difference helps explain why the rectangular approximation is less flexible in the example.Source: §6 · p. 18
  • The polytopic approximation returns the optimal solution to problem (17) and yields the largest uncertainty set in this example.The authors verify optimality by checking all combinations of vertices of the state and input constraints.Source: §6 · p. 18, Table 1 · p. 18
  • The ellipsoidal uncertainty set is not the maximum-volume ellipse contained in the polytopic set because its associated policy is restricted to be affine.The affine ellipsoidal policy hits the input constraints; expanding the set in the 45-degree direction would violate them.Source: §6 · p. 19
  • For polytopic uncertainty, the recovered policy is piece-wise affine, and this policy structure supports the complex shape of the optimal set.The reported set has vertices not captured by either the rectangular or ellipsoidal optimum.Source: §6 · p. 19

7. Conclusion

The conclusion presents a convex framework for optimizing adjustable uncertainty sets under affine representations and policies. It identifies reserve provision stability and recursive feasibility, among other topics, as future work.

Conclusion

  • Affine-transformed primitive uncertainty sets and policies affine in the primitive sets yield a convex approximation whose size grows polynomially with problem data and primitive-set description.The paper reports that this makes the resulting optimization tractable.Source: §7 · p. 19
  • A numerical example demonstrates efficient solution and derivation of optimal bidding and reserve strategies for a power-system reserve capacity problem.The paper presents reserve provision as an application of the framework.Source: §7 · p. 19
  • Future work includes stability and recursive feasibility when reserve capacities are bid in a receding-horizon manner.The authors also identify affine-policy optimality, external uncertainties, and partial state observations as topics for further investigation.Source: §7 · p. 19

Appendix A. Definition, Lemma and Proofs

The appendix defines a lifting operator to relate uncertainty realizations in the adjusted set to points in the primitive set. Its properties support constraint transfer and the proof that the original and transformed optimization formulations correspond.

Appendix A. Definition, Lemma and Proofs

  • A lifting operator maps each uncertainty realization in W to a point in the primitive set S while satisfying the listed structural properties.The operator is written componentwise across stages; its properties preserve causality.Source: Appendix A · p. 20
  • Property (P3) requires that mapping a lifted point back through Y and y recovers the original uncertainty realization.The appendix states this as YL(w) + y = w for all w in W.Source: Appendix A · p. 20
  • Lemma 1 equates robust constraint satisfaction over W with satisfaction after substituting the affine representation W = YS + y.It also gives an implication for transferring constraints through a lifting operator.Source: Appendix A · p. 20
  • The proof of Proposition 2 constructs feasible solutions in either formulation from feasible solutions in the other, preserving causality and objective value.The construction composes a policy with the affine uncertainty-set map in one direction and with the lifting operator in the other.Source: Appendix A · pp. 21–22
  • For affine policies, Proposition 3 establishes correspondence when Y is invertible; for polytopic primitive sets, it constructs a causal continuous piece-wise affine policy.The latter construction uses the lifting operator.Source: Appendix A · pp. 22–23

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