Polytopic Invariant Sets Research Articles

Model predictive real-time architecture for secondary voltage control of microgrids

The integration of new elements and technologies to the power grid, driven by the expansion of electric microgrids, brings various challenges requiring a suitable control architecture for systems to operate ensuring economic and reliable access to electricity. This paper presents the development of a time-varying constraint real-time model predictive secondary voltage control (MPVC) strategy for microgrids built up in a multi-class Python environment with InfluxDB and SQLExpress databases storing the controller and microgrid data, and its experimental implementation with Modbus communication to physical devices at the Universidad Pontificia Bolivariana (UPB) campus microgrid. The microgrid characteristics are described and the experimental setup is presented. The mathematical model of the resource used as case study is obtained through experimental identification, and it is integrated to the MPVC scheme. Polytopic invariant sets are used as terminal sets in the MPVC scheme to ensure stability of the solution, and time-varying constraints of reactive power availability are considered. Validation studies of the proposed control system in computational simulation are presented prior to real application in the UPB campus microgrid. Controller performance results are described for three different implementation scenarios of low, medium and high variability of external disturbances, presenting adequate voltage regulation around the reference value and considerable dependence on the meteorological forecast quality. The model predictive voltage controller is finally benchmarked against a commercial microgrid controller to evaluate its performance, standing out as a more comprehensive and tailor-made solution.

Campus Microgrid Data-Driven Model Identification and Secondary Voltage Control

Microgrids deal with challenges presented by intermittent distributed generation, electrical faults and mode transition. To address these issues, to understand their static and dynamic behavior, and to develop control systems, it is necessary to reproduce their stable operation and transient response through mathematical models. This paper presents a data-driven low-order model identification methodology applied to voltage characterization in a photovoltaic system of a real campus microgrid for secondary voltage regulation. First, a literature review is presented focusing on secondary voltage modeling strategies and control. Then, experimental data is used to estimate and validate a low-order MIMO (multiple input–multiple output) model of the microgrid, considering reactive power, solar irradiance, and power demand inputs and the voltage output of the grid node. The obtained model reproduced the real system response with an accuracy of 88.4%. This model is used for dynamical analysis of the microgrid and the development of a secondary voltage control system based on model predictive control (MPC). The MPC strategy uses polytopic invariant sets as terminal sets to guarantee stability. Simulations are carried out to evaluate the controller performance using experimental data from solar irradiance and power demand as the system disturbances. Successful regulation of the secondary voltage output is obtained with a fast response despite the wide range of disturbance values.

A Polytopic Invariant Set Based Iterative Learning Model Predictive Control

Model predictive control (MPC) is often combined with iterative learning control (ILC), which results in the so-called iterative learning model predictive control (ILMPC), to control batch processes with constraints. It is a long standing and challenging problem that how to simultaneously guarantee system stability and constraint satisfaction in ILMPC design. Several invariant set-based methods, such as the zero-terminal state and the ellipsoidal invariant set, have been proposed to solve this problem. However, these methods are often restrictive with conservative control performance and limited applicability. In this paper, we propose a polytopic invariant set based ILMPC method to reduce conservativeness. Specifically, a polytopic invariant set is designed based on geometric computation and proved to be convex and compact. An iterative algorithm is proposed to compute the maximal one. Numerical simulations are provided to demonstrate its effectiveness.

Model predictive control synthesis algorithm based on polytopic terminal region for Hammerstein-Wiener nonlinear systems

An improved model predictive control algorithm is proposed for Hammerstein-Wiener nonlinear systems. The proposed synthesis algorithm contains two parts: offline design the polytopic invariant sets, and online solve the min-max optimization problem. The polytopic invariant set is adopted to replace the traditional ellipsoid invariant set. And the parameter-correlation nonlinear control law is designed to replace the traditional linear control law. Consequently, the terminal region is enlarged and the control effect is improved. Simulation and experiment are used to verify the validity of the wind tunnel flow field control algorithm.

MPC‐based tracking for real‐time systems subject to time‐varying polytopic constraints

SummaryThis paper presents a real‐time MPC‐based tracking strategy for linear systems subject to time‐varying constraints. The framework is quite general because the time‐varying constraints can apply both to the state and to the input. To handle the problem, a polytopic invariant set computed off‐line is homogeneously dilated or contracted on‐line to fit the polytopic time‐varying constraints. The invariant set is used as an admissible terminal constraint set so that it guarantees stability and convergence in the tracking task. The on‐line cost of the homothetic invariant set computation is low enough to cope with systems subject to stringent real‐time constraints. Copyright © 2015 John Wiley & Sons, Ltd.

Low-Complexity Polytopic Invariant Sets for Linear Systems Subject to Norm-Bounded Uncertainty

We propose a novel algorithm to compute low-complexity polytopic robust control invariant (RCI) sets, along with the corresponding state-feedback gain, for linear discrete-time systems subject to norm-bounded uncertainty, additive disturbances and state/input constraints. Using a slack variable approach, we propose new results to transform the original nonlinear problem into a convex/LMI problem whilst introducing only minor conservatism in the formulation. Through numerical examples, we illustrate that the proposed algorithm can yield improved maximal/minimal volume RCI set approximations in comparison with the schemes given in the literature.

Polytopic invariant and contractive sets for closed-loop discrete fuzzy systems

In this work a procedure for obtaining polytopic λ-contractive sets for Takagi–Sugeno fuzzy systems is presented, adapting well-known algorithms from literature on discrete-time linear difference inclusions (LDI) to multi-dimensional summations. As a complexity parameter increases, these sets tend to the maximal invariant set of the system when no information on the shape of the membership functions is available. λ-contractive sets are naturally associated to level sets of polyhedral Lyapunov functions proving a decay-rate of λ. The paper proves that the proposed algorithm obtains better results than a class of Lyapunov methods for the same complexity degree: if such a Lyapunov function exists, the proposed algorithm converges in a finite number of steps and proves a larger λ-contractive set.

Computation of polytopic invariants for polynomial dynamical systems using linear programming

This paper deals with the computation of polytopic invariant sets for polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given instant, it will remain in the set forever in the future. Polytopic invariants for polynomial systems can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes. Using the blossoming principle together with properties of multi-affine functions on rectangles and Lagrangian duality, we show that certified lower bounds of the optimal values of such optimization problems can be computed effectively using linear programs. This allows us to propose a method based on linear programming for verifying polytopic invariant sets of polynomial dynamical systems. Additionally, using sensitivity analysis of linear programs, one can iteratively compute a polytopic invariant set. Finally, we show using a set of examples borrowed from biological applications, that our approach is effective in practice.

Invariant sets computation for convex difference inclusions systems

In this paper we introduce the Convex Difference Inclusion (CDI) systems as a modeling framework useful to analyze set-theory and invariance-related issues for nonlinear and uncertain systems. The dynamics of a CDI system is given by a set-valued map whose values are convex, compact subsets of the space and are determined by convex bounding functions. Necessary and sufficient boundary-type conditions for invariance and contractiveness, characteristic of the linear systems, are given for the CDI systems. Lyapunov functions are proved to be induced by contractive sets for CDI systems, as in the linear context. A computational procedure for obtaining polytopic invariant and contractive sets for nonlinear systems, based on the properties of the CDI systems, is presented.

A Note to Calculation of Polytopic Invariant and Feasible Sets for Linear Continuous-Time Systems

A common way of guaranteeing stability in model predictive control (MPC) is to add into the open-loop optimization problem a terminal constraint set which is invariant and feasible. This note presents an algorithm for calculation of low-complexity polytopic invariant sets for linear continuous-time systems with complex conjugate closed-loop eigenvalues. Non-symmetrical amplitude and rate input constraints are considered.

Nonlinear model predictive control with polytopic invariant sets

Ellipsoidal invariant sets have been widely used as target sets in model predictive control (MPC). These sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope. The choice of inclusion polytope has a significant effect on the size of the target ellipsoid, but the optimal inclusion polytope cannot in general be computed systematically. This paper shows that use of polytopic invariant sets overcomes this difficulty, allowing larger stabilizable sets without loss of performance. In the interests of online efficiency, consideration is focused on interpolation-based MPC.

NONLINEAR MODEL PREDICTIVE CONTROL WITH POLYTOPIC INVARIANT SETS

Ellipsoidal invariant sets have been widely used as target sets in MPC. These sets can be computed by constructing appropriate Linear Difference Inclusions together with additional constraints to ensure that the ellipsoid lies within a given Inclusion Polytope. The choice of this polytope has a significant effect on the size of the computed ellipsoid, but the optimal inclusion polytope cannot in general be computed systematically. This paper shows that use of polytopic invariant sets overcomes this difficulty, resulting in larger stabilizable sets without loss of closed-loop performance. In the interests of online efficiency, consideration is focused on interpolation-based NMPC.