Solving Convex Optimization Research Articles

Output feedback model predictive control for LPV systems using parameter-dependent Lyapunov function

In this paper, we consider a robust dynamic output feedback model predictive controller (MPC) design for linear parameter varying (LPV) systems. According to the proposed MPC algorithm, the control law is computed based on linear matrix inequality (LMI) at each sampling time by solving convex optimization problem. Also, a new parameter-dependent Lyapunov function is proposed to get a less conservative condition for stability of the system. The effectiveness of the result proposed is verified via a numerical example.

Robust model predictive control for discrete uncertain nonlinear systems with time-delay via fuzzy model

An extended robust model predictive control approach for input constrained discrete uncertain nonlinear systems with time-delay based on a class of uncertain T-S fuzzy models that satisfy sector bound condition is presented. In this approach, the minimization problem of the “worst-case” objective function is converted into the linear objective minimization problem involving linear matrix inequalities (LMIs) constraints. The state feedback control law is obtained by solving convex optimization of a set of LMIs. Sufficient condition for stability and a new upper bound on robust performance index are given for these kinds of uncertain fuzzy systems with state time-delay. Simulation results of CSTR process show that the proposed robust predictive control approach is effective and feasible.

Parametric Uncertainty Bounds for Stabilizing Receding Horizon H Controls

This letter presents parametric uncertainty bounds (PUBs) for stabilizing receding horizon ∞ control (RHHC). The proposed PUBs are obtained easily by solving convex optimization problems represented by linear matrix inequalities (LMIs). We show, by numerical example, that the RHHC can guarantee a ∞ norm bound for a larger class of uncertain systems than conventional infinite horizon ∞ control (IHHC).

Global asymptotic stability analysis for cellular neural networks with time delays

Abstract This paper provides a new sufficient condition for the global asymptotic stability and uniqueness of the equilibrium point of cellular neural networks with time delays. This condition is expressed in terms of linear matrix inequalities, which can be easily checked by various recently developed algorithms in solving convex optimization problems. Numerical examples are provided to show that the proposed stability result is less conservative than some previously established ones in the literature.

04/02822 An alternative method for the reliability differentiated transmission pricing: Hur, D. et al. Electric Power Systems Research, 2004, 68, (1), 11–17

This paper presents a mixed-integer linear programming (MILP) model for the optimal energy management of residential microgrids, modeled as unbalanced, three-phase, electrical distribution system (EDS). Initially, the problem is formulated as a mixed-integer nonlinear programming (MINLP) problem. Then, a set of linear approximations and equivalent mathematical representations are used to obtain a precise MILP model. The proposed formulation considers three-phase generation units (GU), single-phase photovoltaic (PV) resources, and single-phase energy storage systems (ESS), as well as load management. The aim of the proposed model is to minimize the final operational costs of the microgrid while considering operational constraints of the EDS and an unexpected outage of the main grid through a security-constrained set of equations. The optimal solution of the MILP model is found using commercial convex optimization solvers. The proposed model was tested in a residential, three-phase EDS. Results show that the proposed linearizations and approximations produce accurate solutions when compared with a nonlinear three-phase OPF formulation, with an error in the objective function near to 2% and a maximum error in the voltage near to 1%. Efficiency and flexibility of the proposed methodology are also discussed.

On the Complexity of Computing Estimates of Condition Measures of a Conic Linear System

Condition numbers based on the “distance to ill-posedness” ρ(d) have been shown to play a crucial role in the theoretical complexity of solving convex optimization models. In this paper, we present two algorithms and corresponding complexity analysis for computing estimates of ρ(d) for a finite-dimensional convex feasibility problem P(d) in standard primal form: find x that satisfies Ax = b, x ∈ CX, where d = (A, b) is the data for the problem P(d). Under one choice of norms for the m- and n-dimensional spaces, the problem of estimating ρ(d) is hard (co-NP complete even when CX = ℜn+). However, when the norms are suitably chosen, the problem becomes much easier: We can estimate ρ(d) to within a constant factor of its true value with complexity bounds that are linear in ln(C(d)) (where C(d) is the condition number of the data d for P(d)), plus other quantities that arise naturally in consideration of the problem P(d). The first algorithm is an interior-point algorithm, and the second algorithm is a variant of the ellipsoid algorithm. The main conclusion of this work is that when the norms are suitably chosen, computing an estimate of the condition measures of P(d) is essentially not much harder than computing a solution of P(d) itself.

Polynomial Interior Point Cutting Plane Methods

Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programing relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Long-step versions of the algorithms for solving convex optimization problems are presented.

Mirror descent and nonlinear projected subgradient methods for convex optimization

The mirror descent algorithm (MDA) was introduced by Nemirovsky and Yudin for solving convex optimization problems. This method exhibits an efficiency estimate that is mildly dependent in the decision variables dimension, and thus suitable for solving very large scale optimization problems. We present a new derivation and analysis of this algorithm. We show that the MDA can be viewed as a nonlinear projected-subgradient type method, derived from using a general distance-like function instead of the usual Euclidean squared distance. Within this interpretation, we derive in a simple way convergence and efficiency estimates. We then propose an Entropic mirror descent algorithm for convex minimization over the unit simplex, with a global efficiency estimate proven to be mildly dependent in the dimension of the problem.

Model Predictive Control: Convex Optimization Versus Constrained Dynamic Backpropagation

ABSTRACT In this paper different methods for implementation of a model based controller are compared. The first method is based on solving convex optimization problems in the unknown control inputs and states. An interior point method exploiting the structure of this optimization problem is used. The second method is based on finding a suboptimal and stable controller for a given state feedback control law that takes care of the input constraints, using neural networks methodology of constrained dynamic backpropagation. Both methods are compared in terms of performance, stability, robustness and computational issues

An LMI approach to dencentralized H8 control

We consider the design of a decentralized controller for a linear time invariant (LTI) system. This system is modelled as an interconnection of subsystems. For every subsystem, a linear time invariant controller is sought such that the overall closed loop system is stable and achieves a given H performance level. The main idea is to design every local controller such that the corresponding closed loop subsystem has a certain input-output (dissipative) property. This local property is constrained to be consistent with the overall objective of stability and performance. The local controllers are designed simultaneously, avoiding the traditional iterative process: both objectives (the local one and the global one) are achieved in one shot. Applying this idea leads us to solving the following new problem: given an LTI system, parameterize all the dissipative properties which can be achieved by feedback. The proposed approach leads to solving convex optimization problems that involve linear matrix inequality constraints.

Nonlinear performance of a PI controlled missile: an explanation

The primary aim of this paper is to investigate the practical interest of the incremental norm approach for analysing (realistic) nonlinear dynamical systems. In this framework indeed, incremental stability, a stronger notion than ℒ︁2-gain stability, ensures suitable qualitative and quantitative properties. On the one hand, the qualitative properties essentially correspond to (steady-state) input/output properties, which are not necessarily obtained when ensuring only ℒ︁2-gain stability. On the other hand, it is possible to analyse quantitative robustness performance properties using the notion of (nonlinear) incremental performance, the latter being defined in the continuity of the (linear) H∞ performance (i.e. through the use of a weighting function). As testing incremental properties is a difficult problem, stronger, but computationally more attractive, notions are introduced, namely quadratic incremental stability and performance. Testing these properties reduces indeed to solving convex optimization problems over Linear Matrix Inequalities (LMIs). As an illustration, we consider a classical missile problem, which was already treated using several (linear and nonlinear) approaches. We focus here on the analysis of the nonlinear behavior of this PI controlled missile: using the notions of quadratic incremental stability and performance, the closed loop nonlinear missile is proved to meet desirable control specifications. Copyright © 1999 John Wiley & Sons, Ltd.

Nonlinear performance of a PI controlled missile: an explanation

The primary aim of this paper is to investigate the practical interest of the incremental norm approach for analysing (realistic) nonlinear dynamical systems. In this framework indeed, incremental stability, a stronger notion than ℒ︁2-gain stability, ensures suitable qualitative and quantitative properties. On the one hand, the qualitative properties essentially correspond to (steady-state) input/output properties, which are not necessarily obtained when ensuring only ℒ︁2-gain stability. On the other hand, it is possible to analyse quantitative robustness performance properties using the notion of (nonlinear) incremental performance, the latter being defined in the continuity of the (linear) H∞ performance (i.e. through the use of a weighting function). As testing incremental properties is a difficult problem, stronger, but computationally more attractive, notions are introduced, namely quadratic incremental stability and performance. Testing these properties reduces indeed to solving convex optimization problems over Linear Matrix Inequalities (LMIs). As an illustration, we consider a classical missile problem, which was already treated using several (linear and nonlinear) approaches. We focus here on the analysis of the nonlinear behavior of this PI controlled missile: using the notions of quadratic incremental stability and performance, the closed loop nonlinear missile is proved to meet desirable control specifications. Copyright © 1999 John Wiley & Sons, Ltd.

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.

The Expected–Projection Method: Its Behavior and Applications to Linear Operator Equations and Convex Optimization

It was shown by Butnariu and Flåm [J. Nnmer. Funct. Anal. Optim. 15: 601–636, 1995] that, under some conditions, sequences generated by the expected projection method (EPM) in Hilbert spaces approximate almost common points of measurable families of closed convex subsets provided that such points exist. In this work we study the behavior of the EPM in the more general situation when the involved sets may or may not have almost common points and we give necessary and sufficient conditions for weak and strong convergence. Also, we show how the EPM can be applied to finding solutions of linear operator equations and to solving convex optimization problems.

The perturbed proximal point algorithm and some of its applications

Following the works of R. T. Rockafellar, to search for a zero of a maximal monotone operator, and of B. Lemaire, to solve convex optimization problems, we present a perturbed version of the proximal point algorithm. We apply this new algorithm to convex optimization and to variational inclusions or, more particularly, to variational inequalities.

A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio

We extend Clarkson's randomized algorithm for linear programming to a general scheme for solving convex optimization problems. The scheme can be used to speed up existing algorithms on problems which have many more constraints than variables. In particular, we give a randomized algorithm for solving convex quadratic and linear programs, which uses that scheme together with a variant of Karmarkar's interior point method. For problems withn constraints,d variables, and input lengthL, ifn = Ω(d2), the expected total number of major Karmarkar's iterations is O(d2(logn)L), compared to the best known deterministic bound of O(\(\sqrt n\)L). We also present several other results which follow from the general scheme.

Lösung strukturierter aufgaben der optimalen steuerung mittels zerlegungsverfahren

A method for solving convex optimization problems in Hilbert spaces, based on the principle of goal coordination, is described. This general approach is then applied to a linear quadratic control problem, which is assumed to have a special structure. The arising informal problems of smaller dimension and without dynamics must be solved several times. The method is illustrated by an example.