Auxiliary Minimization Problem Research Articles

Structured Static Output Feedback Mixed ${H}_2/{H}_\infty $ Control for Linear Control Systems

This note considers the design of static output feedback mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${H}_2/{H}_\infty $</tex-math></inline-formula> controllers for linear control systems with certain equality and inequality constraints imposed directly on the feedback matrix. Based on the barrier method, we solve an auxiliary minimization problem to obtain an approximate solution to the original nonconvex constrained optimization problem. Necessary conditions for the optimal solution of the auxiliary minimization problem are derived using the Lagrange multiplier method. Subsequently, an iterative steepest descent algorithm is developed to find an approximate optimal solution. Finally, an example is provided to validate the proposed approach.

On local quantum Gibbs states

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of Rd and look for a density operator over L2(Rd), minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.

Design of Optimal Static Output Feedback Controllers for Linear Control Systems Subject to General Structural Constraints

This work considers the design of optimal static output feedback controllers with priori structural constraints for linear systems. The structural constraints impose individual elements of the feedback matrix to satisfy certain equality or inequality constraints. The barrier method is used to address an auxiliary minimization problem to attain an approximate solution to the original nonconvex constrained optimization problem. The Lagrange multiplier method is applied to derive necessary conditions for the optimal solution. An easy-to-implement algorithm based on the gradient descent method is developed to solve the auxiliary minimization problem, and the convergence of the algorithm is proven. The effectiveness of the proposed methodology is validated using two numerical examples.

Optimal archgrids: a variational setting

The paper deals with the variational setting of the optimal archgrid construction. The archgrids, discovered by William Prager and George Rozvany in 1970s, are viewed here as tension-free and bending-free, uniformly stressed grid-shells forming vaults unevenly supported along the closed contour of the basis domain. The optimal archgrids are characterized by the least volume. The optimization problem of volume minimization is reduced to the pair of two auxiliary mutually dual problems, having mathematical structure similar to that known from the theory of optimal layout: the integrand of the auxiliary minimization problem is of linear growth, while the auxiliary maximization problem involves test functions subjected to mean-square slope conditions. The noted features of the variational setting governs the main properties of the archgrid shapes: they are vaults over a subregion of the basis domain being the effective domain of the minimizer of the auxiliary problem. Thus, the method is capable of cutting out the material domain from the design domain; this process is built in within the theory. Moreover, the present paper puts forward new methods of numerical construction of optimal archgrids and discusses their applicability ranges.

Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

The present paper concerns two methods: IMD (isotropic material design) and YMD (Young modulus material design) concerning Pareto optimal distributions of the material and its isotropic properties. The merit function is the weighted sum of compliances corresponding to n independent loading conditions. The unit cost of the design is assumed as equal to the trace of the elastic moduli tensor C. In the IMD method, the design variables are the bulk and shear moduli. In the YMD method, the Poisson ratio is assumed as given, possibly non‐uniformly distributed in the design domain, the Young modulus being the only design variable. Both the methods reduce to the auxiliary minimization problem involving statically admissible stress fields corresponding to the subsequent loading conditions. The integrand is expressed by a norm of the collection of these stresses, hence has linear growth. The numerical method proposed refers to the primal formulation: it is based on piece‐wise polynomial approximation of the set of statically admissible stresses. The exemplary Pareto optimal solutions show that admitting negative values of Poisson's ratio may contribute to a decrease of the total compliances.

On material design by the optimal choice of Young's modulus distribution

This paper concerns the problem of optimal distribution of Young's modulus within an elastic and isotropic body with prescribed, not necessarily uniform, distribution of Poisson's ratio. The merit function is the total compliance corresponding to a given single load case. The unit cost of the design is assumed as the trace of Hooke's tensor, which is proportional to Young's modulus here. The problem thus formulated is a constrained version of the isotropic material design problem, in which both bulk and shear elastic moduli of isotropy are design variables. In contrast to the latter approach, the optimal material is either nonsingular or degenerates to a void. The optimal distribution of Young's modulus is determined by a solution to an auxiliary minimization problem with the integrand of linear growth. The problem dual to the latter assumes the form similar to that known from the theory of Michell trusses. The proposed numerical approach focuses on solving the former auxiliary problem. The case studies comprise selected optimal designs corresponding to the isotropic material optimization and Young's modulus optimization developed in this paper. The exemplary optimal solutions show that admitting negative values of Poisson's ratio may contribute to a decrease of the total compliance, thus leading to better designs.

A Simultaneous Mixed LQR/H∞ Control Approach to the Design of Reliable Active Suspension Controllers

AbstractThis paper is concerned with the synthesis of reliable controllers for quarter‐car active suspension systems. By a simultaneous mixed LQR/H∞ control approach, a static output feedback controller is derived for guaranteeing good suspension performance under possible sensor fault or suspension component breakdown. The considered simultaneous mixed LQR/H∞ control problem is a nonconvex optimization problem; therefore, the linear matrix inequality approach is not applicable. Based on the barrier method, we solve an auxiliary minimization problem to get an approximate solution for the simultaneous mixed LQR/H∞ control problem. Necessary conditions for the local optimum of the auxiliary minimization problem are derived. Moreover, a three‐stage solution algorithm is developed for solving the auxiliary minimization problem. The simulation shows that the obtained static output feedback suspension controllers can improve suspension performance in nominal mode and all considered failure modes.

Optimal regional pole placement for sun tracking control of high-concentration photovoltaic (HCPV) systems: case study

This paper proposes an optimal regional pole placement approach for sun tracking control of high-concentration photovoltaic systems. A static output feedback controller is designed to minimize an LQG cost function with a sector region pole constraint. The problem cannot be solved by LMI approach since it is a non-convex optimization problem. Based on the barrier method, we instead solve an auxiliary minimization problem to obtain an approximate solution. Simulation results show the benefit of our approach. Copyright © 2010 John Wiley & Sons, Ltd.

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.

Optimal Static Output Feedback Simultaneous Regional Pole Placement

The problem of optimal simultaneous regional pole placement for a collection of linear time-invariant systems via a single static output feedback controller is considered. The cost function to be minimized is a weighted sum of quadratic performance indices of the systems. The constrained region for each system can be the intersection of several open half-planes and/or open disks. This problem cannot be solved by the linear matrix inequality (LMI) approach since it is a nonconvex optimization problem. Based on the barrier method, we instead solve an auxiliary minimization problem to obtain an approximate solution to the original constrained optimization problem. Moreover, solution algorithms are provided for finding the optimal solution. Furthermore, a necessary and sufficient condition for the existence of admissible solutions to the simultaneous regional pole placement problem is derived. Finally, two examples are given for illustration.

A new method for mixed H2/H∞ control with regional pole constraints

AbstractIn this paper, the problem of state feedback mixed H2/H∞ control with regional pole constraints is studied. The constraint region is represented by several algebraic inequalities. This constrained optimization problem cannot be solved via the LMI approach. Based on the barrier method, we instead solve an auxiliary minimization problem to get an approximate solution. We shall show that the obtained minimal solution of the auxiliary minimization problem can be arbitrarily close to the infimal solution of the original problem. An example is provided to illustrate the benefits of the approach. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Robust, reduced-order modeling for state-space systems via parameter-dependent bounding functions

One of the most important problems in dynamic systems theory is to approximate a higher-order system model with a low-order, relatively simpler model. However, the nominal high-order model is never an exact representation of the true physical system. In this paper the problem of approximating an uncertain high-order system with constant real parameter uncertainty by a robust reduced-order model is considered. A parameter-dependent quadratic bounding function is developed that bounds the effect of uncertain real parameters on the model-reduction error. An auxiliary minimization problem is formulated that minimizes an upper bound for the model-reduction error. The principal result is a necessary condition for solving the auxiliary minimization problem which effectively provides sufficient conditions for characterizing robust reduced-order models.

Strictly proper projective controllers for disturbance attenuation

A design methodology is presented which applies projective controls to the disturbance attenuation problem for large flexible structures. Through this method, low-order strictly proper controllers are designed which improve the disturbance attenuation properties of the system. A new parameterization of the closed-loop system in terms of free parameters in the projective controller leads to an auxiliary minimization problem which determines the parameters of the controller. A flexible structure example demonstrates the use of this methodology.

Robust Stability and Performance via Fixed-Order Dynamic Compensation

Two robust control-design problems are considered. The Robust Stabilization Problem involves deterministically modeled, bounded but unknown, time-varying parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. For both problems the design goal is a fixed-order (i.e., reduced- or full-order) dynamic (strictly proper) feedback compensator. A sufficient condition for solving the Robust Stabilization Problem is given by means of a quadratic Lyapunov function parameterized by the compensator gains. For the Robust Performance Problem the Lyapunov function provides an upper bound for the closed-loop performance. This leads to consideration of the Auxiliary Minimization Problem: Minimize the performance bound over the class of fixed-order controllers subject to the Lyapunov-function constraint. Necessary conditions for optimality in the auxiliary problem thus serve as sufficient conditions for robust stability and performance in the original problem. Two particular bounds are considered for constructing the quadratic Lyapunov function. The first corresponds to a right shift/ multiplicative white noise model, while the second was suggested by recent work of Petersen and Hollot. The main result is an extended version of the optimal projection equations for fixed-order dynamic compensation whose solutions are guaranteed to provide both robust stability and robust performance.

Unified optimal projection equations for simultaneous reduced-order, robust modelling, estimation and control

An optimal design problem which unifies reduced-order modelling, estimation and control problems is stated. Necessary conditions for optimality are obtained in the form of a coupled system of modified Riccati and Lyapunov equations. The results permit treatment of several new problems, such as reduced-order dynamic compensation with partially known disturbances and unified reduced-order control and estimation. Upon appropriate specialization, results obtained previously for the individual problems of reduced-order modelling, estimation and control are recovered. An additional feature is the inclusion of parameter uncertainty bounds so that the necessary conditions for an auxiliary minimization problem serve as sufficient conditions for simultaneous robust, reduced-order modelling, estimation and control.

Primal and dual formulations of sequential gradient-restoration algorithms for trajectory optimization problems

One of the most effective first-order algorithms for solving trajectory optimization problems is the sequential gradient-restoration algorithm (SGRA). Originally developed in the primal formulation (PSGRA), this algorithm is extended to incorporate a dual formulation (DSGRA). Both the primal formulation and the dual formulation involve a sequence of two-phase cycles, each cycle including a gradient phase and a restoration phase. In turn, each iteration of the gradient phase and the restoration phase requires the solution of an auxiliary minimization problem (AMP). In the primal formulation, the AMP is solved with respect to the variations of the state, the control and the parameter. In the dual formulation, the AMP is solved with respect to the Lagrange multipliers. A characteristic of the dual formulation is that the AMP's associated with the gradient phase and the restoration phase of SGRA can be reduced to mathematical programming problems involving a finite number of parameters as unknowns. A comparison of the primal formulation and the dual formulation is presented. The comparison is done in terms of several trajectory optimization problems having current aerospace interest.

Supplementary optimality properties of the restoration phase of sequential gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the restoration phase. It is shown that the Lagrange multipliers associated with the restoration phase not only solve the auxiliary minimization problem of the restoration phase, but are also endowed with a supplementary optimality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

Supplementary optimality properties of gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.

Optimal initial choice of multipliers in the quasilinearization method for optimal control problems with bounded controls

An algorithm to choose the initial multipliers optimally for quasilinearization solution of optimal control problems with bounded controls and constant multipliers is proposed. It uses diagonal matrices of weighting coefficients in the performance index of an auxiliary minimization problem. This auxiliary performance index comprises the cumulative error in the system constraints and the optimum conditions in the original extremization problem. The auxiliary performance index is quadratically dependent on the multipliers for given state and control functions. The resulting variational problem leads to linear Euler equations. The computational characteristics of the proposed method are demonstrated with two numerical examples.

A computational method for parameter optimization problems arising in control†

A parameter optimization procedure is presented for large-scale problems arising in linear control system design that include equality and inequality constraints. The procedure is based on a novel min—max algorithm for locating a constrained relative minimum without the use of penalty functions or slack variables. This algorithm is constructed from an auxiliary minimization problem with equality constraints. Inequality constraints then are introduced using the notion of an effective constraint. Typical problem formulations are discussed, and an extensive design example is presented.