Olav Slupphaug

Explicit sub-optimal linear quadratic regulation with state and input constraints

Optimal feedback solutions to the infinite 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 real-time optimization. It is shown that the resulting feedback controller is piecewise linear. This explicit functional structure is exploited for efficient real-time implementation. A suboptimal strategy, based on a suboptimal choice of a finite 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.

Constrained quadratic stabilization of discrete-time uncertain non-linear multi-model systems using piecewise affine state-feedback

In this paper a method for non-linear 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 statespace known as clusters the plant is assumed to be an element in a polytope whose vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to being linear systems. The clusters cover the region of interest in the state-space. A n 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 non-convex 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 Lyapunov matrix and the piecewise affine sta...

On explicit suboptimal LQR with state and input constraints

Optimal feedback solutions to the infinite-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 real-time optimization. A suboptimal strategy, based on a suboptimal choice of a finite 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 exploited for computational analysis of stability and performance as well as efficient real-time implementation.

Brief Linear MPC with optimal prioritized infeasibility handling: application, computational issues and stability

All practical MPC implementations should have a means to recover from infeasibility. We present a recently developed infeasibility handler which computes optimal relaxations of the relaxable constraints subject to a user-defined prioritization, by solving only a single linear program on-line in addition to the standard quadratic programming problem on-line. A stability result for this infeasibility handler combined with the Rawlings-Muske MPC controller is provided, and various practical and computational issues are discussed. From a simulated FCCU main fractionator case study, we conclude that the proposed strategy for designing the proposed infeasibility handler is applicable to the problems of realistic size.

Infeasibility handling in linear MPC subject to prioritized constraints

All practical MPC implementations should have a means to recover from infeasibility. We propose an algorithm designed for linear state-space MPC which optimally relaxes an infeasible prioritized MPC optimization problem into a feasible one by solving only one LP on-line in addition to the standard MPC optimization problem. By optimal it is meant that the violation of a lower prioritized constraint cannot be made less without increasing the violation of a higher prioritized constraint. It is shown how to design the LP off-line such that the computed constraint violations are optimal.

A structured approach to optimizing offshore oil and gas production with uncertain models

Abstract Optimizing offshore production of oil and gas has received comparatively little attention despite the large scale of revenues involved. The complexity of multiphase flow means that any model for use in production optimization must be fitted to production data for accuracy, but the low information content of production data means that the uncertainty in the fitted parameters of any such model will be significant. Due to costs and risk the information content in production data cannot be increased through excitation unless the benefits are documented. A structured approach is suggested which iteratively updates setpoints while documenting the benefits of each proposed setpoint change through excitation planning and result analysis. In simulations on an analog which mimics a real-world oil field and its typical low information content data the approach is able to realize a significant portion of the available profit potential while ensuring feasibility despite large initial model uncertainty.

Uncertainty modelling and robust output feedback control of nonlinear discrete systems: a mathematical programming approach

We present a mathematical programming approach to robust control of nonlinear systems with uncertain, possibly time-varying, parameters. The uncertain system is given by different local affine parameter-dependent models in different parts of the state space. It is shown how this representation can be obtained from a nonlinear uncertain system by solving a set of continuous linear semi-infinite programming problems, and how each of these problems can be solved as a (finite) series of ordinary linear programs. Additionally, the system representation includes control and state constraints. The controller design method is derived from Lyapunov stability arguments and utilizes an affine parameter-dependent quadratic Lyapunov function. The controller has a piecewise affine output feedback structure, and the design amounts to finding a feasible solution to a set of linear matrix inequalities combined with one spectral radius constraint on the product of two positive definite matrices. A local solution approach to this non-convex feasibility problem is proposed. Complexity of the design method and some special cases such as state feedback are discussed. Finally, an application of the results is given by proposing an on-line computationally feasible algorithm for constrained nonlinear state-feedback model predictive control with robust stability. Copyright

Bilinear matrix inequalities and robust stability of nonlinear multi-model MPC

A bilinear matrix inequality (BMI)-based approach to an online computationally efficient robust nonlinear model predictive control (MPC) is proposed. Theoretical results and a simple example accompany the proposed method.

Comment on "Stability issues on Takagi-Sugeno fuzzy model-parametric approach" [with reply]

The stability conditions for Takagi-Sugeno fuzzy systems suggested in the above mentioned article by Lo-Chen (ibid. vol.7 (1999)) are shown by a counterexample not to be sufficient. In reply, Lo-Chen point out that the example also indicates the fuzzy stability problem is related to the premise part of fuzzy rules.

Production optimization; System Identification and Uncertainty Estimation

Real-time optimization of oil and gas production requires a production model, which must be fitted to data for accuracy. A certain amount of uncertainty must typically be expected in production models fitted to data due to the limited information content in data. It is usually not acceptable to introduce additional excitation at will to reduce this uncertainty due to the costs and risks involved. The contribution of this paper is twofold. Firstly, this paper discusses estimation of uncertainty in production optimization resulting from fitting models to production data with low information content, a concept that has previously mainly been applied in reservoir management. Secondly, this paper illustrates how system identification can be used to find production models which can be solved with little computational effort and which are designed to be easily fitted to production data. The method is demonstrated on a synthetic example before being applied to a case study of a North Sea oil and gas field. In offshore oil and gas production, the suggested method is expected to have applications in the development of structured approaches to uncertainty handling, for instance excitation planning and real-time optimization under uncertainty.