Model Order Reduction Problem Research Articles

A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction

In this paper, an optimization-based model order reduction (MOR) framework is proposed. The method involves setting up a quasi-convex program that solves a relaxation of the optimal Hinfin norm MOR problem. The method can generate guaranteed stable and passive reduced models and is very flexible in imposing additional constraints such as exact matching of specific frequency response samples. The proposed optimization-based approach is also extended to solve the parameterized model-reduction problem (PMOR). The proposed method is compared to existing moment matching and optimization-based MOR methods in several examples. PMOR models for large RF inductors over substrate and power-distribution grid are also constructed.

Efficient simulation of nonuniform transmission lines using integrated congruence transform

This paper presents a new algorithm based on integrated congruence transform for efficient simulation of nonuniform transmission lines. The proposed algorithm introduces the concept of model-order reduction (MOR) via implicit usage of the Hilbert-space moments in distributed networks. The key idea in the proposed algorithm is the development of an orthogonalization procedure that does not require the explicit computation of the Hilbert-space moments in order to find their spanning orthogonal basis. The proposed orthogonalization procedure can thus be used to compute an orthogonal basis for any set of elements that are related through a differential operator in a generalized Hilbert space, without the need to have these elements in an explicit form. The proposed algorithm also addresses the problem of MOR of nonuniform transmission lines, through defining a weighted inner product and norm mappings over the Hilbert space of the moments. Numerical examples demonstrate more accurate numerical approximation capabilities over using the moments explicitly.

Accurate Low-Dimensional Approximation of the Linear Dynamics of Fluid Flow

Methods for approximating a stable linear autonomous dynamical system by a system of lower order are examined. Reducing the order of a dynamical system is useful theoretically in identifying the irreducible dimension of the dynamics and in isolating the dominant spatial structures supporting the dynamics, and practically in providing tractable lower-dimension statistical models for climate studies and error covariance models for forecast analysis and initialization. Optimal solution of the model order reduction problem requires simultaneous representation of both the growing structures in the system and the structures into which these evolve. For autonomous operators associated with fluid flows a nearly optimal solution of the model order reduction problem with prescribed error bounds is obtained by truncating the dynamics in its Hankel operator representation. Simple model examples including a reduced-order model of Couette flow are used to illustrate the theory. Practical methods for obtaining approximations to the optimal order reduction problem based on finite-time singular vector analysis of the propagator are discussed and the accuracy of the resulting reduced models evaluated.

An application of selective modal analysis to tokamak modeling and control

In this paper, the problem of model order reduction for tokamak devices is considered. Due to the complexity of the machine, the available models of the plasma behavior and its electromagnetic coupling with the active and metallic structures have to be very detailed to be usefully employed for simulation and control design. Such a level of detail consequently yields linearized models of the system of very high order. Having to deal with such high-order models makes the use of many control schemes quite complicated. To cope with this problem, the natural strategy is to derive models of reduced order, which reproduce with sufficient accuracy the full-order model behavior. The use of a reduction scheme which is based on selective modal analysis (SMA) is proposed. Its basic property is that the physical meaning of the state variables is preserved in the reduced model. This fact can be usefully employed both in the analysis of the system properties and for control design. In particular, the design of an LQG controller for the International Thermonuclear Experimental Reactor is discussed.

A comparison of homotopies for alternative formulations of the L2 optimal model order reduction problem

A fundamental problem in control systems theory is finding a reduced order model that is optimal in the L 2 sense to a given (full order) system model. The numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous. A number of homotopy-based approaches have been developed. The primary numerical issues are the number of degrees of freedom in the homotopy parameter vector, the well-posedness of the finite-dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. This paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms. The results show that the numerical well-conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent.

An input normal form homotopy for the L/sup 2/ optimal model order reduction problem

In control system analysis and design, finding a reduced-order model, optimal in the L/sup 2/ sense, to a given system model is a fundamental problem. The problem is very difficult without the global convergence of homotopy methods, and a homotopy based approach has been proposed. The issues are the number of degrees of freedom, the well posedness of the finite dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. A homotopy algorithm based on the input normal form characterization of the reduced-order model is developed here and is compared with the homotopy algorithms based on Hyland and Bernstein's optimal projection equations. The main conclusions are that the input normal form algorithm can be very efficient, but can also be very ill conditioned or even fail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Contragredient transformations applied to the optimal projection equations

The optimal projection approach to solving the H 2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. It is not obvious from their original form how they can be differentiated and how an algorithm for solving nonlinear equations can be applied to them. A contragredient transformation, a transformation which simultaneously diagonalizes two symmetric positive semidefinite matrices, is used to transform the equations into forms suitable for algorithms for solving nonlinear problems. Three different forms of the equations obtained using contragredient transformations are given. An SVD-based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given, together with a numerical example.

On the Hankel-norm approximation of upper-triangular operators and matrices

A matrix T = Tij ∞=-∞ , which consists of a doubly indexed collection Tij of operators, is said to be upper when Tij = 0 for i > j. We consider the case where the Tij are finite matrices and the operator T is bounded, and such that the Tij are generated by a strictly stable, non-stationary but linear dynamical state space model or colligation. For such a model, we consider model reduction, which is a procedure to obtain optimal approximating models of lower system order. Our approximation theory uses a norm which generalizes the Hankel norm of classical stationary linear dynamical systems. We obtain a parametrization of all solutions of the model order reduction problem in terms of a fractional representation based on a non-stationary J-unitary operator constructed from the data. In addition, we derive a state space model for the so-called maximum entropy approximant. In the stationary case, the problem was solved by Adamyan, Arov and Krein in their paper on Schur-Takagi interpolation. Our approach extends that theory to cover general, non-Toeplitz upper operators.

Homotopy methods for solving the optimal projection equations for theH2reduced order model problem

The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. Due to the resemblance of these equations to standard matrix Lyapunov equations, they are called modified Lyapunov equations. The algorithms proposed herein utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial problem a theorem guarantees that all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. It is shown that the appropriate inverse is a differentiable function. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Another class of methods considers the equations in a modified form, using a decomposition of the pseudogramians based on a contragredient transformation. Some freedom is left in making an exact match between the number of equations and the number of unknowns, thus effectively generating a family of methods.

On the relationship between the model order reduction problem and the simultaneous stabilization problem

This note points out the fact that, under certain conditions, the model order reduction problem is very closely tied to the simultaneous stabilization problem. These conditions occur when the reduced order model is to be used to design a suboptimal controller for a high-order plant. Under these conditions the controller must not only stabilize the low-order model, but must also stabilize the high-order plant. The result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high-order plant in the low-order model will guarantee the existence of such a controller.

Mathematical Models of Flexible Spacecraft Dynamics : A Survey of Order Reduction Approaches

Increases in size and mechanical complexity of spacecraft result in increased complexity of the mathematical model of the spacecraft dynamics. In turn, this results in increased computational effort, increased difficulties in understanding the characteristics of the spacecraft dynamics, and increased difficulties in designing as well as implementing suitable algorithms for the control of the spacecraft dynamic motions. Reduction of the order (i.e. the complexity) of the mathematical open loop model with minimal loss of model accuracy, is therefore of prime importance. Literature contains descriptions of a large number of approaches towards open loop model order reduction. These have been surveyed from the point of view of usefulness for application to flexible spacecraft dynamics models. Six basic approaches have been identified, involving: (i) parameter optimization, (ii) aggregation, (iii) singular perturbation, (iv) modal dominance, (v) component cost analysis, and (vi) internal balancing, respectively. The latter three approaches appear to be most meaningful, and convenient in applications. The problem of model order reduction is reviewed, and each of the six approaches is discussed. The latter three approaches are applied to the case of a long, flexible beam in space, controlled with two line torquers.