Guaranteeing Input Tracking For Constrained Systems: Theory and Application to Demand Response

A method for certifying exact input trackability for constrained discrete time linear systems is introduced in this paper. A signal is assumed to be drawn from a reference set and the system must track this signal with a linear combination of its inputs. Using methods inspired from robust model predictive control, the proposed approach certifies the ability of a system to track any reference drawn from a polytopic set on a finite time horizon by solving a linear program. Optimization over a parameterization of the set of reference signals is discussed, and particular instances of parameterization of this set that result in a convex program are identified, allowing one to find the largest set of trackable signals of some class. Infinite horizon feasibility of the methods proposed is obtained through use of invariant sets, and an implicit description of such an invariant set is proposed. These results are tailored for the application of power consumption tracking for loads, where the operator of the load needs to certify in advance his ability to fulfill some requirement set by the network operator. An example of a building heating system illustrates the results.Comment: Technical Not

System Level Synthesis

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty -- such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.Comment: To appear in Annual Reviews in Contro

Robust feedback model predictive control of norm-bounded uncertain systems

This thesis is concerned with the Robust Model Predictive Control (RMPC) of linear discrete-time systems subject to norm-bounded model-uncertainty, additive disturbances and hard constraints on the input and state. The aim is to design tractable, feedback RMPC algorithms that are based on linear matrix inequality (LMI) optimizations. The notion of feedback is very important in the RMPC control parameterization since it enables effective disturbance/uncertainty rejection and robust constraint satisfaction. However, treating the state-feedback gain as an optimization variable leads to non-convexity and nonlinearity in the RMPC scheme for norm-bounded uncertain systems. To address this problem, we propose three distinct state-feedback RMPC algorithms which are all based on (convex) LMI optimizations. In the first scheme, the aforementioned non-convexity is avoided by adopting a sequential approach based on the principles of Dynamic Programming. In particular, the feedback RMPC controller minimizes an upper-bound on the cost-to-go at each prediction step and incorporates the state/input constraints in a non-conservative manner. In the second RMPC algorithm, new results, based on slack variables, are proposed which help to obtain convexity at the expense of only minor conservatism. In the third and final approach, convexity is achieved by re-parameterizing, online, the norm-bounded uncertainty as a polytopic (additive) disturbance. All three RMPC schemes drive the uncertain-system state to a terminal invariant set which helps to establish Lyapunov stability and recursive feasibility. Low-complexity robust control invariant (LC-RCI) sets, when used as target sets, yield computational advantages for the associated RMPC schemes. A convex algorithm for the simultaneous computation of LC-RCI sets and the corresponding controller for norm-bounded uncertain systems is also presented. In this regard, two novel results to separate bilinear terms without conservatism are proposed. The results being general in nature also have application in other control areas. The computed LC-RCI sets are shown to have substantially improved volume as compared to other schemes in the literature. Finally, an output-feedback RMPC algorithm is also derived for norm-bounded uncertain systems. The proposed formulation uses a moving window of the past input/output data to generate (tight) bounds on the current state. These bounds are then used to compute an output-feedback RMPC control law using LMI optimizations. An output-feedback LC-RCI set is also designed, and serves as the terminal set in the algorithm.Open Acces

Systematic Sampling and Validation of Machine Learning-Parameterizations in Climate Models

Progress in hybrid physics-machine learning (ML) climate simulations has been limited by the difficulty of obtaining performant coupled (i.e. online) simulations. While evaluating hundreds of ML parameterizations of subgrid closures (here of convection and radiation) offline is straightforward, online evaluation at the same scale is technically challenging. Our software automation achieves an order-of-magnitude larger sampling of online modeling errors than has previously been examined. Using this, we evaluate the hybrid climate model performance and define strategies to improve it. We show that model online performance improves when incorporating memory, a relative humidity input feature transformation, and additional input variables. We also reveal substantial variation in online error and inconsistencies between offline vs. online error statistics. The implication is that hundreds of candidate ML models should be evaluated online to detect the effects of parameterization design choices. This is considerably more sampling than tends to be reported in the current literature.Comment: 13 pages, 4 figure

A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control

Gain-scheduled control based on linear parameter-varying (LPV) models derived from local linearizations is a widespread nonlinear technique for tracking time-varying setpoints. Recently, a nonlinear control scheme based on Control Contraction Metrics (CCMs) has been developed to track arbitrary admissible trajectories. This paper presents a comparison study of these two approaches. We show that the CCM based approach is an extended gain-scheduled control scheme which achieves global reference-independent stability and performance through an exact control realization which integrates a series of local LPV controllers on a particular path between the current and reference states.Comment: IFAC LPVS 201

Robust model predictive control for linear systems subject to norm-bounded model Uncertainties and Disturbances: An Implementation to industrial directional drilling system

Model Predictive Control (MPC) refers to a class of receding horizon algorithms in which the current control action is computed by solving online, at each sampling instant, a constrained optimization problem. MPC has been widely implemented within the industry, due to its ability to deal with multivariable processes and to explicitly consider any physical constraints within the optimal control problem in a straightforward manner. However, the presence of uncertainty, whether in the form of additive disturbances, state estimation error or plant-model mismatch, and the robust constraints satisfaction and stability, remain an active area of research. The family of predictive control algorithms, which explicitly take account of process uncertainties/disturbances whilst guaranteeing robust constraint satisfaction and performance is referred to as Robust MPC (RMPC) schemes. In this thesis, RMPC algorithms based on Linear Matrix Inequality (LMI) optimization are investigated, with the overall aim of improving robustness and control performance, while maintaining conservativeness and computation burden at low levels. Typically, the constrained RMPC problem with state-feedback parameterizations is nonlinear (and nonconvex) with a prohibitively high computational burden for online implementation. To remedy this issue, a novel approach is proposed to linearize the state-feedback RMPC problem, with minimal conservatism, through the use of semidefinite relaxation techniques and the Elimination Lemma. The proposed algorithm computes the state-feedback gain and perturbation online by solving an LMI optimization that, in comparison to other schemes in the literature is shown to have a substantially reduced computational burden without adversely affecting the tracking performance of the controller. In the case that only (noisy) output measurements are available, an output-feedback RMPC algorithm is also derived for norm-bounded uncertain systems. The novelty lies in the fact that, instead of using an offline estimation scheme or a fixed linear observer, the past input/output data is used within a Robust Moving Horizon Estimation (RMHE) scheme to compute (tight) bounds on the current state. These current state bounds are then used within the RMPC control algorithm. To reduce conservatism, the output-feedback control gain and control perturbation are both explicitly considered as decision variables in the online LMI optimization. Finally, the aforementioned robust control strategies are applied in an industrial directional drilling configuration and their performance is illustrated by simulations. A rotary steerable system (RSS) is a drilling technology that has been extensively studied over the last 20 years in hydrocarbon exploration and is used to drill complex curved borehole trajectories. RSSs are commonly treated as dynamic robotic actuator systems, driven by a reference signal and typically controlled by using a feedback loop control law. However, due to spatial delays, parametric uncertainties, and the presence of disturbances in such an unpredictable working environment, designing such control laws is not a straightforward process. Furthermore, due to their inherent delayed feedback, described by delay differential equations (DDE), directional drilling systems have the potential to become unstable given the requisite conditions. To address this problem, a simplified model described by ordinary differential equations (ODE) is first proposed, and then taking into account disturbances and system uncertainties that arise from design approximations, the proposed RMPC algorithm is used to automate the directional drilling system.Open Acces