We present an algorithm for robust model predictive control with consideration of uncertainty and safety constraints. Our framework considers a nonlinear dynamical system subject to disturbances from an unknown but bounded uncertainty set. By viewing the system as a fixed point of an operator acting over trajectories, we propose a convex condition on control actions that guarantee safety against the uncertainty set. The proposed condition guarantees that all realizations of the state trajectories satisfy safety constraints. Our algorithm solves a sequence of convex quadratic constrained optimization problems of size n*N, where n is the number of states, and N is the prediction horizon in the model predictive control problem. Compared to existing methods, our approach solves convex problems while guaranteeing that all realizations of uncertainty set do not violate safety constraints. Moreover, we consider the implicit time-discretization of system dynamics to increase the prediction horizon and enhance computational accuracy. Numerical simulations for vehicle navigation demonstrate the effectiveness of our approach.
The convex restriction of the power flow feasible sets identifies the convex subset of power injections where the solution for power flow is guaranteed to exist and satisfy the operational constraints. In contrast to convex relaxations, the convex restriction provides a sufficient condition for power flow feasibility and is particularly useful for problems involving uncertainty in the power generation and demand. In this paper, we present a general framework of constructing convex restriction of an algebraic set defined by equality and inequality constraints and apply the framework to power fl…
Electric power grids regularly experience uncertain fluctuations from load demands and renewables, which poses a risk of violating operational limits designed to safeguard the system. In this paper, we consider the robust AC OPF problem that minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. The robust AC OPF problem requires that the system satisfy operational limits for all uncertainty realizations within a specified uncertainty set. Guaranteeing robustness is particularly challenging due to the non-convex, nonlinear AC power flow…
We consider strongly convex optimization problems with affine-type restrictions. We build dual problem and solve dual problem by Fast Gradient Method. We use primal-dual structure of this method to construct the solution of the primal problem. The paper contain a lot of different tricks that allows to generalize mentioned above results for almost all methods we would like to choose to solve the dual problem.
This paper presents a convex sufficient condition for solving a system of nonlinear equations under parametric changes and proposes a sequential convex optimization method for solving robust optimization problems with nonlinear equality constraints. By bounding the nonlinearity with concave envelopes and using Brouwer's fixed point theorem, the sufficient condition is expressed in terms of closed-form convex inequality constraints. We extend the result to provide a convex sufficient condition for feasibility under bounded uncertainty. Using these conditions, a non-convex optimization problem c…
We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost over a finite horizon. Hard constraints are introduced first, and then reformulated in terms of probabilistic constraints. It is shown that, for a suitable parametrization of the control policy, a wide class of the resulting optimization problems are convex, or admit reasonable convex approximations.
This paper details an investigation into the computational performance of algorithms used for solving a convex formulation of the optimization problem associated with model predictive control for energy management in hybrid electric vehicles with nonlinear losses. A projected interior point method is proposed, where the size and complexity of the Newton step matrix inversion is reduced by applying inequality constraints on the control input as a projection, and its properties are demonstrated through simulation in comparison with an alternating direction method of multipliers (ADMM) algorithm,…
We consider the Chance Constrained Model Predictive Control problem for polynomial systems subject to disturbances. In this problem, we aim at finding optimal control input for given disturbed dynamical system to minimize a given cost function subject to probabilistic constraints, over a finite horizon. The control laws provided have a predefined (low) risk of not reaching the desired target set. Building on the theory of measures and moments, a sequence of finite semidefinite programmings are provided, whose solution is shown to converge to the optimal solution of the original problem. Numeri…
Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path fr…
We present a heuristic policy and performance bound for risk-sensitive convex stochastic control that generalizes linear-exponential-quadratic regulator (LEQR) theory. Our heuristic policy extends standard, risk-neutral model predictive control (MPC); however, instead of ignoring uncertain noise terms, our policy assumes these noise terms turns out either favorably or unfavorably, depending on a risk aversion parameter. In the risk-seeking case, this modified planning problem is convex. In the risk-averse case, it requires minimizing a difference of convex functions, which is done (approximate…
This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an affine subspace.
University of California, Berkeley（加州大学伯克利分校）；University of Melbourne（墨尔本大学）；University of Toronto（多伦多大学）
We incorporate future information in the form of the estimated value of future gradients in online convex optimization. This is motivated by demand response in power systems, where forecasts about the current round, e.g., the weather or the loads' behavior, can be used to improve on predictions made with only past observations. Specifically, we introduce an additional predictive step that follows the standard online convex optimization step when certain conditions on the estimated gradient and descent direction are met. We show that under these conditions and without any assumptions on the pre…
We study the problem of networked online convex optimization, where each agent individually decides on an action at every time step and agents cooperatively seek to minimize the total global cost over a finite horizon. The global cost is made up of three types of local costs: convex node costs, temporal interaction costs, and spatial interaction costs. In deciding their individual action at each time, an agent has access to predictions of local cost functions for the next $k$ time steps in an $r$-hop neighborhood. Our work proposes a novel online algorithm, Localized Predictive Control (LPC), …
Recently there has been an increasing interest in primal-dual methods for model predictive control (MPC), which require minimizing the (augmented) Lagrangian at each iteration. We propose a novel first order primal-dual method, termed \emph{proportional-integral projected gradient method}, for MPC where the underlying finite horizon optimal control problem has both state and input constraints. Instead of minimizing the (augmented) Lagrangian, each iteration of our method only computes a single projection onto the state and input constraint set. Our method ensures that, along a sequence of aver…
One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geometry of the objective function. The abstract results of this article are applied in other papers to the Monge-Kantorovich optimal transport problem and the minimization of entropy functionals.
Input constrained Model predictive control (MPC) includes an optimization problem which should iteratively be solved at each time-instance. The well-known drawback of model predictive control is the computational cost of the optimization problem. This results in restriction of the application of MPC to systems with slow dynamics, e.g., process control systems and small-scale problems. Therefore, implementing fast numerical optimization algorithms has been a point of interest. Interior-point methods are proved to be appropriate algorithms, from computational cost point-of-vie, to solve input-co…
For multiparametric mixed-integer convex programming problems such as those encountered in hybrid model predictive control, we propose an algorithm for generating a feasible partition of a subset of the parameter space. The result is a static map from the current parameter to a suboptimal integer solution such that the remaining convex program is feasible. Convergence is proven with a new insight that the overlap among the feasible parameter sets of each integer solution governs the partition complexity. The partition is stored as a tree which makes querying the feasible solution efficient. Th…
We present a Model Predictive Control (MPC) algorithm for energy management in aircraft with hybrid electric propulsion systems consisting of gas turbine and electric motor components. Series and parallel configurations are considered. By combining a point-mass aircraft dynamical model with models of electrical losses and losses in the gas turbine, the fuel consumed over a given future flight path is minimised subject to constraints on the battery, electric motor and gas turbine. The optimization is formulated as a convex problem under mild assumptions and its solution is used to define a pred…
For constrained linear systems with bounded disturbances and parametric uncertainty, we propose a robust adaptive model predictive control strategy with online parameter estimation. Constraints enforcing persistently exciting closed loop control actions are introduced for a set-membership parameter identification scheme. The algorithm requires the online solution of a convex program, satisfies constraints robustly, and ensures recursive feasibility and input-to-state stability. Almost sure convergence to the actual system parameters is demonstrated under assumptions on stabilizability, reachab…
We describe general heuristics to approximately solve a wide variety of problems with convex objective and decision variables from a nonconvex set. The heuristics, which employ convex relaxations, convex restrictions, local neighbor search methods, and the alternating direction method of multipliers (ADMM), require the solution of a modest number of convex problems, and are meant to apply to general problems, without much tuning. We describe an implementation of these methods in a package called NCVX, as an extension of CVXPY, a Python package for formulating and solving convex optimization pr…
Non-convex optimal control problems occurring in, e.g., water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to find a globally optimal solution, and numerical experience indicates that algorithms aiming for Karush-Kuhn-Tucker points often find (near-)optimal solutions. In our paper, we provide a theoretical underpinning for this phenomenon, showing that on a broad class of problems the objective can be shown to be an invariantly convex function (invex function) of the control decision variables when state varia…
Regularized Model Predictive Control via Contractivity and Implicit Lur'e Analysis
Toyota Central Research and Development Laboratories, Incorporated（丰田中央研究所）；Center for Control, Dynamical Systems, and Computation, UC Santa Barbara（加州大学圣塔芭芭拉分校控制、动力学与计算中心）；Department of Mechanical Engineering, Rice University（莱斯大学机械工程系）
We propose a model predictive control (MPC) approach for minimising the social distancing and quarantine measures during a pandemic while maintaining a hard infection cap. To this end, we study the admissible and the maximal robust positively invariant set (MRPI) of the standard SEIR compartmental model with control inputs. Exploiting the fact that in the MRPI all restrictions can be lifted without violating the infection cap, we choose a suitable subset of the MRPI to define terminal constraints in our MPC routine and show that the number of infected people decays exponentially within this se…
In this paper we present a Learning Model Predictive Control (LMPC) strategy for linear and nonlinear time optimal control problems. Our work builds on existing LMPC methodologies and it guarantees finite time convergence properties for the closed-loop system. We show how to construct a time varying safe set and terminal cost function using closed-loop data. The resulting LMPC policy is time varying and it guarantees recursive constraint satisfaction and non-decreasing performance. Computational efficiency is obtained by convexifing the safe set and terminal cost function. We demonstrate that,…
The design of reliable path-following controllers is a key ingredient for successful deployment of self-driving vehicles. This controller-design problem is especially challenging for a general 2-trailer with a car-like tractor due to the vehicle's structurally unstable joint-angle kinematics in backward motion and the car-like tractor's curvature limitations which can cause the vehicle segments to fold and enter a jackknife state. Furthermore, advanced sensors with a limited field of view have been proposed to solve the joint-angle estimation problem online, which introduce additional restrict…
Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably stronger design constraints that arise when the applications of interest, such as aerospace systems, are safety-critical. Formal verification is the process of proving or disproving the ''correctness'' of an algorithm with respect to a certain mathematical description of it by means of a computer. This article discusses the formal verification of the Ellipso…
A Learning Model Predictive Controller (LMPC) for linear system in presented. The proposed controller is an extension of the LMPC [1] and it aims to decrease the computational burden. The control scheme is reference-free and is able to improve its performance by learning from previous iterations. A convex safe set and a terminal cost function are used in order to guarantee recursive feasibility and non-increasing performance at each iteration. The paper presents the control design approach, and shows how to recursively construct the convex terminal set and the terminal cost from state and inpu…
Beyond Pure Sampling: Hybrid Optimization Mechanisms for Non-Convex Model Predictive Control
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA（航空航天工程学院，佐治亚理工学院，美国亚特兰大，GA州）；Development Division, Komatsu Ltd., Tokyo, Japan（Komatsu Ltd.开发部门，日本东京）；Department of Electrical and Computer Engineering, Inha University, Incheon, Republic of Korea（电气与计算机工程系，inha大学，韩国仁川，大韩民国）
This paper aims to improve the reliability of optimal control using models constructed by machine learning methods. Optimal control problems based on such models are generally non-convex and difficult to solve online. In this paper, we propose a model that combines the Hammerstein-Wiener model with input convex neural networks, which have recently been proposed in the field of machine learning. An important feature of the proposed model is that resulting optimal control problems are effectively solvable exploiting their convexity and partial linearity while retaining flexible modeling ability.…
We propose a computationally efficient Learning Model Predictive Control (LMPC) scheme for constrained optimal control of a class of nonlinear systems where the state and input can be reconstructed using lifted outputs. For the considered class of systems, we show how to use historical trajectory data collected during iterative tasks to construct a convex value function approximation along with a convex safe set in a lifted space of virtual outputs. These constructions are iteratively updated with historical data and used to synthesize predictive control policies. We show that the proposed str…
In the framework of Model Predictive Control (MPC), the control input is typically computed by solving optimization problems repeatedly online. For general nonlinear systems, the online optimization problems are non-convex and computationally expensive or even intractable. In this paper, we propose to circumvent this issue by computing a high-dimensional linear embedding of discrete-time nonlinear systems. The computation relies on an algebraic condition related to the immersibility property of nonlinear systems and can be implemented offline. With the high-dimensional linear model, we then de…
We propose a fully distributed control system architecture, amenable to in-vehicle implementation, that aims to safely coordinate connected and automated vehicles (CAVs) at road intersections. For control purposes, we build upon a fully distributed model predictive control approach, in which the agents solve a nonconvex optimal control problem (OCP) locally and synchronously, and exchange their optimized trajectories via vehicle-to-vehicle (V2V) communication. To accommodate a fast solution of the nonconvex OCPs, we apply the penalty convex-concave procedure which solves a convexified version …
We address the problem of designing optimal linear time-invariant (LTI) sparse controllers for LTI systems, which corresponds to minimizing a norm of the closed-loop system subject to sparsity constraints on the controller structure. This problem is NP-hard in general and motivates the development of tractable approximations. We characterize a class of convex restrictions based on a new notion of Sparsity Invariance (SI). The underlying idea of SI is to design sparsity patterns for transfer matrices Y(s) and X(s) such that any corresponding controller K(s)=Y(s)X(s)^-1 exhibits the desired spar…
This paper presents a novel distributed active set method for model predictive control of linear systems. The method combines a primal active set strategy with a decentralized conjugate gradient method to solve convex quadratic programs. An advantage of the proposed method compared to existing distributed model predictive algorithms is the primal feasibility of the iterates. Numerical results show that the proposed method can compete with the alternating direction method of multipliers in terms of communication requirements for a chain of masses example.
We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in $\mathbb{R}^3$, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for …
