Optimal feedback solutions to the in nite horizon LQR problem with state and input constraints based on receding horizon real-time quadratic programming are well known. In this paper we develop an explicit solution to the same problem, eliminating the need for realtime optimization. A suboptimal strategy, based on a suboptimal choice of a nite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon, is suggested in order to address the computer memory and processing capacity requirements of the explicit solution. It is shown that the resulting feedback controller is piecewise linear, and the piecewise linear structure is explored and exploited for computational analysis of stability and performance of the suboptimal constrained LQR. The piecewise linear structure can also be exploited for e cient real-time implementation of the controller.

Summary This is a noncritical survey of key literature in the field of real-time production optimization of oil and gas production. The information flow used for optimization of the system is described. The elements in this description include data acquisition, data storage, processing facility model updating, well model updating, reservoir model updating, production planning, reservoir planning, and strategic planning. Methods for well prioritization, gas lift optimization, gas or water injection optimization, and model updating are discussed in the view of the information flow described. Challenges of real-time production optimization are also discussed. Introduction In the daily operation of an oil and gas production system, many decisions (an element of a solution) have to be taken affecting the volumes produced and the cost of production. These decisions are taken at different levels in the organization, but eventually they will reach the physical production system. Fig. 1 gives an overview of a physical production system. For such production systems, the decisions are related to the choke or valve openings, compressor, and pump settings at every instance of time. An objective function is a single-valued and well-defined mathematical function mapping the values of the decision variables into a performance measure. Examples of such performance measures are the total oil production rate, net present value (profit), or the recovery of the reservoir. In the efforts toward better performance of the production system, a question to be answered is which decisions are better to maximize or minimize the objective function. In the process of making good decisions, information about the production system is used. This information may include the physical properties such as pipe diameters and lengths, or it may include measurements from the production system. The environment in which the production of oil and gas is obtained is constantly changing. This will affect the value of the performance measure of the decisions used. For example, if the cooling capacity of the production system is an operational bottleneck, this may no longer be the case if the seawater temperature drops or another pump in the cooling system is started. Incidents in the production system may also affect the value of the performance measure of the decisions. A partial shutdown of the production system because of maintenance will most likely also affect system bottlenecks. Real-time optimization (RTO) is a method for complete or partial automation of the process for making good or optimal decisions. The term "optimal" is defined below. By continuously collecting and analyzing data from the production system, optimal decisions may be found. Either these settings are then implemented directly in the production system or they are presented to an operator or engineer for consideration. If the settings are implemented directly, the RTO is said to be in a closed loop. RTO defined by Saputelli et al. (2003a) reads: "a process of measure-calculate-control cycles at a frequency, which maintains the system's optimal operating conditions within the time-constant constraints of the system". The main aim of RTO is to improve the utilization of the capacity of a production system to obtain higher throughput or net present value. The idea is to operate the production system, at every instant of time, as near to the desired optimum as possible (Sequeira et al. 2002). To achieve this, a model of the production system is optimized to furnish an optimal solution. The model is continuously being updated by measurements from the production system to fit the actual input-output behavior of the processing facilities, wells or network, and reservoir better.

In this paper a method for nonlinear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the state-space called clusters the plant is assumed to be an element in a polytope which vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to be linear systems. The clusters cover the region of interest in the state-space. An affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single nonconvex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed-loop. The feasibility problem is attacked by a branch-and-bound based global approach. If the feasibility check is successful, the Liapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained nonlinear model predictive control is presented. Also, the control design method is applied to a simple example

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