Robust Model Predictive Control for Non-Linear Systems with Input and State Constraints Via Feedback Linearization

Robust predictive control of non-linear systems under state estimation errors and input and state constraints is a challenging problem, and solutions to it have generally involved solving computationally hard non-linear optimizations. Feedback linearization has reduced the computational burden, but has not yet been solved for robust model predictive control under estimation errors and constraints. In this paper, we solve this problem of robust control of a non-linear system under bounded state estimation errors and input and state constraints using feedback linearization. We do so by developing robust constraints on the feedback linearized system such that the non-linear system respects its constraints. These constraints are computed at run-time using online reachability, and are linear in the optimization variables, resulting in a Quadratic Program with linear constraints. We also provide robust feasibility, recursive feasibility and stability results for our control algorithm. We evaluate our approach on two systems to show its applicability and performance

Robust Monotonic Optimization Framework for Multicell MISO Systems

The performance of multiuser systems is both difficult to measure fairly and to optimize. Most resource allocation problems are non-convex and NP-hard, even under simplifying assumptions such as perfect channel knowledge, homogeneous channel properties among users, and simple power constraints. We establish a general optimization framework that systematically solves these problems to global optimality. The proposed branch-reduce-and-bound (BRB) algorithm handles general multicell downlink systems with single-antenna users, multiantenna transmitters, arbitrary quadratic power constraints, and robustness to channel uncertainty. A robust fairness-profile optimization (RFO) problem is solved at each iteration, which is a quasi-convex problem and a novel generalization of max-min fairness. The BRB algorithm is computationally costly, but it shows better convergence than the previously proposed outer polyblock approximation algorithm. Our framework is suitable for computing benchmarks in general multicell systems with or without channel uncertainty. We illustrate this by deriving and evaluating a zero-forcing solution to the general problem.Comment: Published in IEEE Transactions on Signal Processing, 16 pages, 9 figures, 2 table

Adjustable Robust Two-Stage Polynomial Optimization with Application to AC Optimal Power Flow

In this work, we consider two-stage polynomial optimization problems under uncertainty. In the first stage, one needs to decide upon the values of a subset of optimization variables (control variables). In the second stage, the uncertainty is revealed and the rest of optimization variables (state variables) are set up as a solution to a known system of possibly non-linear equations. This type of problem occurs, for instance, in optimization for dynamical systems, such as electric power systems. We combine tools from polynomial and robust optimization to provide a framework for general adjustable robust polynomial optimization problems. In particular, we propose an iterative algorithm to build a sequence of (approximately) robustly feasible solutions with an improving objective value and verify robust feasibility or infeasibility of the resulting solution under a semialgebraic uncertainty set. At each iteration, the algorithm optimizes over a subset of the feasible set and uses affine approximations of the second-stage equations while preserving the non-linearity of other constraints. The algorithm allows for additional simplifications in case of possibly non-convex quadratic problems under ellipsoidal uncertainty. We implement our approach for AC Optimal Power Flow and demonstrate the performance of our proposed method on Matpower instances.Comment: 28 pages, 3 table

Distributionally Robust Model Predictive Control: Closed-loop Guarantees and Scalable Algorithms

We establish a collection of closed-loop guarantees and propose a scalable, Newton-type optimization algorithm for distributionally robust model predictive control (DRMPC) applied to linear systems, zero-mean disturbances, convex constraints, and quadratic costs. Via standard assumptions for the terminal cost and constraint, we establish distribtionally robust long-term and stage-wise performance guarantees for the closed-loop system. We further demonstrate that a common choice of the terminal cost, i.e., as the solution to the discrete-algebraic Riccati equation, renders the origin input-to-state stable for the closed-loop system. This choice of the terminal cost also ensures that the exact long-term performance of the closed-loop system is independent of the choice of ambiguity set the for DRMPC formulation. Thus, we establish conditions under which DRMPC does not provide a long-term performance benefit relative to stochastic MPC (SMPC). To solve the proposed DRMPC optimization problem, we propose a Newton-type algorithm that empirically achieves superlinear convergence by solving a quadratic program at each iteration and guarantees the feasibility of each iterate. We demonstrate the implications of the closed-loop guarantees and the scalability of the proposed algorithm via two examples.Comment: 34 pages, 6 figure

A Sequential Quadratic Programming Method for Optimization with Stochastic Objective Functions, Deterministic Inequality Constraints and Robust Subproblems

In this paper, a robust sequential quadratic programming method of [1] for constrained optimization is generalized to problem with stochastic objective function, deterministic equality and inequality constraints. A stochastic line search scheme in [2] is employed to globalize the steps. We show that in the case where the algorithm fails to terminate in finite number of iterations, the sequence of iterates will converge almost surely to a Karush-Kuhn-Tucker point under the assumption of extended Mangasarian-Fromowitz constraint qualification. We also show that, with a specific sampling method, the probability of the penalty parameter approaching infinity is 0. Encouraging numerical results are reported

How to Understand LMMSE Transceiver Design for MIMO Systems From Quadratic Matrix Programming

In this paper, a unified linear minimum mean-square-error (LMMSE) transceiver design framework is investigated, which is suitable for a wide range of wireless systems. The unified design is based on an elegant and powerful mathematical programming technology termed as quadratic matrix programming (QMP). Based on QMP it can be observed that for different wireless systems, there are certain common characteristics which can be exploited to design LMMSE transceivers e.g., the quadratic forms. It is also discovered that evolving from a point-to-point MIMO system to various advanced wireless systems such as multi-cell coordinated systems, multi-user MIMO systems, MIMO cognitive radio systems, amplify-and-forward MIMO relaying systems and so on, the quadratic nature is always kept and the LMMSE transceiver designs can always be carried out via iteratively solving a number of QMP problems. A comprehensive framework on how to solve QMP problems is also given. The work presented in this paper is likely to be the first shoot for the transceiver design for the future ever-changing wireless systems.Comment: 31 pages, 4 figures, Accepted by IET Communication

Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems

Learning-based control algorithms require data collection with abundant supervision for training. Safe exploration algorithms ensure the safety of this data collection process even when only partial knowledge is available. We present a new approach for optimal motion planning with safe exploration that integrates chance-constrained stochastic optimal control with dynamics learning and feedback control. We derive an iterative convex optimization algorithm that solves an \underline{Info}rmation-cost \underline{S}tochastic \underline{N}onlinear \underline{O}ptimal \underline{C}ontrol problem (Info-SNOC). The optimization objective encodes both optimal performance and exploration for learning, and the safety is incorporated as distributionally robust chance constraints. The dynamics are predicted from a robust regression model that is learned from data. The Info-SNOC algorithm is used to compute a sub-optimal pool of safe motion plans that aid in exploration for learning unknown residual dynamics under safety constraints. A stable feedback controller is used to execute the motion plan and collect data for model learning. We prove the safety of rollout from our exploration method and reduction in uncertainty over epochs, thereby guaranteeing the consistency of our learning method. We validate the effectiveness of Info-SNOC by designing and implementing a pool of safe trajectories for a planar robot. We demonstrate that our approach has higher success rate in ensuring safety when compared to a deterministic trajectory optimization approach.Comment: Submitted to RA-L 2020, review-