Xiaojing Zhang

A Scenario Approach for Non-Convex Control Design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist efficient randomized approaches, this is not the case for non-convex problems. Methods based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in achieving the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control-design problems that can be addressed via randomization, we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control-affine systems.

On the sample size of randomized MPC for chance-constrained systems with application to building climate control

We consider Stochastic Model Predictive Control (SMPC) for constrained linear systems with additive disturbance, under affine disturbance feedback (ADF) policies. One approach to solve the chance-constrained optimization problem associated with the SMPC formulation is randomization, where the chance constraints are replaced by a number of sampled hard constraints, each corresponding to a disturbance realization. The ADF formulation leads to a quadratic growth in the number of decision variables with respect to the prediction horizon, which results in a quadratic growth in the sample size. This leads to computationally expensive problems with solutions that are conservative in terms of both cost and violation probability. We address these limitations by establishing a bound on the sample size which scales linearly in the prediction horizon. The new bound is obtained by explicitly computing the maximum number of active constraints, leading to significant advantages both in terms of computational time and conservatism of the solution. The efficacy of the new bound relative to the existing one is demonstrated on a building climate control case study.

Stochastic Model Predictive Control using a combination of randomized and robust optimization

In this paper, we focus on Stochastic Model Predictive Control (SMPC) problems for systems with linear dynamics and additive uncertainty. One way to address such problems is by means of randomized algorithms. Typically, these algorithms require substituting the chance constraint of the SMPC problem with a finite number of hard constraints corresponding to samples of the uncertainty. Earlier approaches toward this direction lead to computationally expensive problems, whose solutions are typically very conservative in terms of cost. To address these limitations, we follow an alternative methodology based on a combination of randomized and robust optimization. We show that our approach can offer significant advantages in terms of both cost and computational time. Both the open-loop MPC formulation (i.e. optimizing over input sequences), as well as optimization over policies using the affine disturbance feedback formulation are considered. We demonstrate the efficacy of the proposed approach relative to standard randomized techniques on a building control problem.

Selling robustness margins: A framework for optimizing reserve capacities for linear systems

This paper proposes a method for solving robust optimal control problems with modulated uncertainty sets. We consider constrained uncertain linear systems and interpret the uncertainty sets as “robustness margins” or “reserve capacities”. In particular, given a certain reward for offering such a reserve capacity, we address the problem of determining the optimal size and shape of the uncertainty set, i.e. how much reserve capacity our system should offer. By assuming polyhedral constraints, restricting the class of the uncertainty sets and using affine decision rules, we formulate a convex program to solve this problem. We discuss several specific families of uncertainty sets, whose respective constraints can be reformulated as linear constraints, second-order cone constraints, or linear matrix inequalities. A numerical example demonstrates our approach.

Randomized Nonlinear MPC for Uncertain Control-Affine Systems with Bounded Closed-Loop Constraint Violations

Abstract In this paper we consider uncertain nonlinear control-affine systems with probabilistic constraints. In particular, we investigate Stochastic Model Predictive Control (SMPC) strategies for nonlinear systems subject to chance constraints. The resulting non-convex chance constrained Finite Horizon Optimal Control Problems are computationally intractable in general and hence must be approximated. We propose an approximation scheme which is based on randomization and stems from recent theoretical developments on random non-convex programs. Since numerical solvers for non-convex optimization problems can typically only reach local optima, our method is designed to provide probabilistic guarantees for any local optimum inside a set of chosen complexity. Moreover, the proposed method comes with bounds on the (time) average closed-loop constraint violation when SMPC is applied in a receding horizon fashion. Our numerical example shows that the number of constraints of the proposed random non-convex program can be up to ten times smaller than those required by existing methods.

Robust optimal control with adjustable uncertainty sets

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.

Balancing bike sharing systems through customer cooperation - a case study on London's Barclays Cycle Hire

A growing number of cities worldwide have been installing public bike sharing systems, offering citizens a flexible and “green” alternative of mobility. In most bike sharing systems, customers rent and return bikes at different stations, without prior notification of the system operator. As a consequence, bike systems often become unbalanced, leaving some stations either empty or full. In such a case, customers either cannot pick up or return their bikes, resulting in a low service level. Typically, system operators employ staff to manually relocate bikes using trucks, leading to considerable operational cost. In this paper, we describe various methods to balance bike sharing systems by actively engaging customers in the balancing process. In particular, we show that by appropriately sending “control signals” to customers requesting them to slightly change their intended journeys, bike sharing systems can be balanced without using staffed trucks. Through extensive simulations based on historical data from Londons Barclays Cycle Hire scheme, we show that simple control signals are sufficient to effectively balance the bike sharing system and offer service rates close to 100%.

Stochastic frequency reserve provision by chance-constrained control of commercial buildings

Robust programs with modulated uncertainty sets are problems in which the uncertainty sets are treated as optimization variables. In many applications, these uncertainty sets can be interpreted as reserve capacities for which rewards are offered. One example is the provision of frequency reserve capacities to the power grid by demand-side resources. This paper studies the case in which the reserves are offered by commercial buildings. We determine the optimal size of the reserve capacity set such that the buildings can follow any reserve demanded within the set, while guaranteeing their comfort constraints. Since the actual reserve demands are uncertain, we interpret the reserve demands as random variables and formulate a chance-constrained program to ensure comfort constraints with high probabilities. By restricting the class of policies to affine decision rules and applying the scenario approach to approximate the chance constraints, we obtain a tractable convex problem. We compare our chance-constrained framework to previous work with robust comfort constraints and demonstrate its efficacy based on a numerical case study.

A scenario approach to non-convex control design: Preliminary probabilistic guarantees

Randomized optimization is a recently established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable for a certain class of non-convex problems, but they usually are conservative in terms of performance and are computationally demanding. In this paper, we present a novel scenario approach for a wide class of random non-convex programs. We provide a sample complexity similar to the one for uncertain convex programs, but valid for all feasible solutions inside a set of a-priori chosen complexity. Our scenario approach applies to many non-convex control-design problems, for instance control synthesis based on uncertain bilinear matrix inequalities.

Convex approximation of chance-constrained MPC through piecewise affine policies using randomized and robust optimization

In this paper, we consider chance-constrained Stochastic Model Predictive Control problems for uncertain linear systems subject to additive disturbance. A popular method for solving the associated chance-constrained optimization problem is by means of randomization, in which the chance constraints are replaced by a finite number of sampled constraints, each corresponding to a disturbance realization. Earlier approaches in this direction lead to computationally expensive problems, whose solutions are typically very conservative both in terms of cost and violation probabilities. One way of overcoming this conservatism is to use piecewise affine (PWA) policies, which offer more flexibility than conventional open-loop and affine policies. Unfortunately, the straight-forward application of randomized methods towards PWA policies will lead to computationally demanding problems, that can only be solved for problems of small sizes. To address this issue, we propose an alternative method based on a combination of randomized and robust optimization. We show that the resulting approximation can greatly reduce conservatism of the solution while exhibiting favorable scaling properties with respect to the prediction horizon.