Uncertainty Block Research Articles

A direct-construction approach to multidimensional realization and LFR uncertainty modeling

This article proposes a direct-construction realization procedure that simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall multidimensional (n-D) Roesser model realization or linear fractional representation (LFR) model for a given n-D polynomial or causal rational transfer matrix. It is shown for the first time that the realization problem for an n-D transfer matrix G(z 1, . . . , z n ), which is assumed without loss of generality to be strictly causal and given in the form of G(z 1, . . . , z n )=N r (z 1, . . . , z n )D r ?1 (z 1,..., z n ) with D r (0, . . . , 0)=I and N r (0, . . . , 0) = 0, can be essentially reduced to the construction of an admissible n-D polynomial matrix ? for which there exist real matrices A, B, C such that N r (z 1, . . . , z n ) = CZ? and ? D r ?1 (z 1, . . . , z n ) = (I ? AZ)?1 B with Z being the corresponding variable and/or uncertainty block structure, i.e., $${Z={\rm diag} \{z_1I_{r_1},\ldots,z_nI_{r_n} \}}$$ . This important fact reveals a substantial difference between the 1-D and n-D (n ? 2) realization problems as in the 1-D case ? can only be a monomial matrix and never a polynomial one. Necessary and sufficient conditions for ? to satisfy the above restrictions are given and algorithms are proposed for the construction of such an admissible n-D polynomial matrix ? with low order (for an arbitrary but fixed field of coefficients) and the corresponding realization. Symbolic and numerical examples are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.

Developing thinking in statistics

BLOCK 1 Describing with Words and Numbers Comparing with Words and Numbers Interrelating with Words and Numbers Uncertainty BLOCK 2 Describing with Pictures Comparing with Pictures Interrelating with Pictures Picturing Probability BLOCK 3 Describing with ICT Comparing with ICT Interrelating with ICT Probability with ICT BLOCK 4 Models and Modelling Statistical Investigation Teaching and Learning Statistics

An algorithm for sampling subsets of H/sub /spl infin// with applications to risk-adjusted performance analysis and model (in)validation

In spite of their potential to reduce computational complexity, the use of probabilistic methods in robust control has been mostly limited to parametric uncertainty, since the problem of sampling causal bounded operators is largely open. In this note, we take steps toward removing this limitation by proposing a computationally efficient algorithm aimed at uniformly sampling suitably chosen subsets of H/sub /spl infin//. As we show in the note, samples taken from these sets can be used to carry out model (in)validation and robust performance analysis in the presence of structured dynamic linear time-invariant uncertainty, problems known to be NP-hard in the number of uncertainty blocks.

Enhanced LFR-toolbox for Matlab

We describe recent developments and enhancements of the LFR-toolbox for Matlab for building LFT-based uncertainty models and for LFT-based gain scheduling. A major development is the new LFT-object definition supporting a large class of uncertainty descriptions: continuous- and discrete-time uncertain models, regular and singular parametric expressions, more general uncertainty blocks (nonlinear, time-varying, etc.). By associating names to uncertainty blocks the reusability of generated LFT-representations and the user friendliness of manipulation of LFR-descriptions have been highly increased. Significant enhancements of the computational efficiency and of numerical accuracy have been achieved by employing efficient and numerically robust Fortran implementations of order reduction tools via mex-function interfaces. The new enhancements in conjunction with improved symbolical preprocessing lead generally to a faster generation of LFT-representations with significantly lower orders.

Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness

Robust semidefinite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well established how to systematically construct relaxations on the basis of the full block S-procedure. In general, such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to suggest novel computationally verifiable conditions for when general classes of linear matrix inequality relaxations do not involve any conservatism. If the convex set of uncertainties is finitely generated, we suggest a novel sequence of relaxations which can be proved to be asymptotically exact. Finally, our results are applied to the particularly relevant robustness analysis problem for linear time-invariant dynamical systems affected by uncertainties that are full ellipsoidal or repeated and contained in intersections of disks or circles. This leads to extensions of known results on relaxation exactness for small block structures, including an elementary proof for tightness of standard structured singular value computations for three full complex uncertainty blocks.

Convex Necessary and Sufficient Conditions for Frequency Domain Model (In)Validation Under SLTV Structured Uncertainty

This paper deals with the problem of model (in)validation of discrete time, causal, linear time-invariant (LTI) stable models subject to slowly linear time-varying structured uncertainty, using frequency domain data corrupted by additive noise. It is well known that in the case of structured LTI uncertainty the problem is NP hard in the number of uncertainty blocks. The main contribution of this paper shows that, on the other hand, if one considers arbitrarily slowly time varying uncertainty and noise in L/sub 2/, then tractable, convex necessary and sufficient conditions for (in)validation can be obtained. Additional results include a discussion of the case where the noise is characterized in terms of the L/sub /spl infin// norm.

Adaptive Linear Parameter Varying Control Synthesis for Actuator Failure

A robust linear parameter varying (LPV) control synthesis is carried out for a Highly Maneuverable Aircraft Technology (HiMAT) vehicle subject to loss of control effectiveness. The scheduling parameter is selected to be a function of the estimates of the control effectiveness factors. The estimates are provided online by a two-stage adaptive Kalman filter estimator. The inherent conservatism of the LPV design is reduced through the use of a scaling factor on the uncertainty block that represents the estimation errors of the effectiveness factors. Simulations of the controlled system with the online estimator show that a superior fault tolerance can be achieved.

Properties of Optimal Solutions in L1 Identification Problem

This paper presents primal-dual formulation of a convex optimization problem related to l1 interpolatory-projection algorithm. The use of duality theory permits to obtain information on the structure of optimal solutions for l1 identi£cation problem and gives extra analytical tools for the analysis of identi£cation error and the estimation of a lowerbound for noise amplitude. The proposed analysis applies also to the case of l1 model validation problem and characterizes properties of uncertainty block and disturbance signal. A simulation example illustrates the results.

LFT REPRESENTATIONS FOR NONLINEAR MECHANICAL SYSTEMS

In this paper, we present a systematic methodology for constructing LFT representations for general mechanical systems derived via Lagrange's equations. The LFT representation allows for any nonlinear matrix second-order mechanical system to be transformed into an interconnection of an LTI system with a diagonal “uncertainty” block. This uncertainty block is, in fact, state-dependent. Sufficient conditions that ensure well-posedness of the LFT interconnection are given. Using such LFT representations, the stability properties of the system can then be analyzed using Linear Matrix Inequalities (LMIs).

Complex Uncertainties in Feedback Loop: The Gain and Phase Margins

The dynamic uncertainties offer a great flexibility for modeling dynamic systems. The dynamic uncertainties are bounded by some norm, that is usually normalized by unity. One useful application of dynamic uncertainties is to provide means for assuring amplitude and phase margins in a design. A simple way to include the amplitude and phase margins in a design is to include a dynamic uncertainty block in configuration. This block contains one parameter (km), that is a measure of the stability margins to be imposed on the closed loop system. The overall gain from input to output depends on the uncertainties (Δ) and the parameter (km), and it is placed where the margin is need. The advantages of this modeling of unstructured uncertainties are illustrated on a double integrator system.

Robust control of flexible structures against structural damage

This paper deals with the vibration suppression control problem of flexible structures against potential damage. The controller is required not only to robustly stabilize the structure but to satisfactorily suppress the vibration under all the predicted damages that might occur in the 20-bay truss structure under study. The damage to the flexible structure we consider here is represented by the various stiffness losses of predetermined elements. The structure with the predicted damages is modeled as an interval transfer function. The controller design is based on the /spl mu/-synthesis in conjunction with the extremal properties of interval systems. Unlike conventional /spl mu/-synthesis, the technique enables us to use a fixed-size uncertainty block [four real blocks for the single input-single output (SISO) case] regardless of the number of parameters, resulting in a fast, reliable, and simple synthesis procedure.

On computational complexity of invalidating structured uncertainty models

In this paper we study a class of uncertainty model validation/invalidation problems for linear discrete-time systems. We consider models described by a linear fractional transformation (LFT) of modeling uncertainties, which are structured, time invariant or time varying, and are bounded in H ∞ or ℓ 1 norms. The experimental data available for use in invalidation are either input–output time series observations or frequency response measurements. We analyze the computational complexity associated with these problems. While earlier work showed that for LFT models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities, and hence may be tackled readily using well-developed numerical methods, in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all NP -hard with respect to the number of uncertainty blocks, suggesting that the problems are inherently intractable from a computational standpoint. Additionally, we also demonstrate that the problems do become tractable when the LFT model description is changed to an additive one, leading to LMI-based invalidation tests similar to those obtained elsewhere. These results indicate that the main source of the computational difficulty lies in the LFT description together with the structured nature of the uncertainties.

A block-based MAP segmentation for image compressions

A novel block-based image segmentation algorithm using the maximum a posteriori (MAP) criterion is proposed. The conditional probability in the MAP criterion, which is formulated by the Bayesian framework, is in charge of classifying image blocks into edge, monotone, and textured blocks. On the other hand, the a priori probability is responsible for edge connectivity and homogeneous region continuity. After a few iterations to achieve a deterministic MAP optimization, we can obtain a block-based segmented image in terms of edge, monotone, or textured blocks. Then, using a connected block-labeling algorithm, we can assign a number to all connected homogeneous blocks to define an interior of a region. Finally, uncertainty blocks, which are not given any region number yet, are assigned to one of the neighboring homogeneous regions by a block-based region-growing method. During this process, we can also check the balance between the accuracy and the cost of the contour coding by adjusting the size of the uncertainty blocks. Experimental results show that the proposed algorithm yields larger homogeneous regions which are suitable for the object-based image compression.

Global optimization for H/sub ∞/ control with constant diagonal scaling

This paper considers the H/sub /spl infin//-control problem with constant diagonal scaling related to the robust control synthesis for systems with structured time-varying uncertainties. The problem is not convex in general, and hence it is difficult to find a global solution. The purpose of this paper is to provide an algorithm to find a suboptimal solution with any specified small tolerance from the globally optimal solution for the optimization problem. The algorithm based on the triangle covering method is proposed. The computational complexity analysis shows that its worst case order is polynomial in the inverse of the tolerance and the size of an a priori given interval of scaling with a fixed number of uncertainty blocks.

Robustness analysis with full-structured uncertainties

This paper deals with μ problems involving full-structured (rather than block-diagonal) uncertainty, i.e. uncertainty blocks where each sub-block (or element) may be an independent uncertainty. Rearranging this problem into standard (block-diagonal) form results in an explosion in the required computation, and so a number of researchers have proposed more efficient bounds, in particular those based on non-similarity scaling, for such problems. Here we show how to map such problems into reduced size standard μ problems. The standard bounds applied to the reduced size problem are then shown to be at least as accurate, and require the same computational effort, as earlier techniques, with the added bonus that the standard μ upper bound is convex. Moreover this new approach is applicable in a much more general setting, allowing one to efficiently compute both robust stability and robust performance for problems involving multiple real and complex uncertainty blocks, any number of which may be full-structured.

State Feedback Minimax and Guaranteed Cost Controllers for Linear Systems with Integral Quadratic Constraint Uncertainty Descriptions

This paper considers the design of minimax and optimal guaranteed cost controllers for uncertain linear systems. The uncertainty is permitted to have a structured description and is bounded by an integral quadratic constraint. For known initial conditions on the system state, it is shown that minimax control can be realised by static full state feedback controllers. A new proof of minimax optimality is given using standard methods for linear systems. The design of such controllers requires the optimal solution of a single Riccati equation dependent upon a set of parameters whose dimension is equal to the number of uncertainty blocks. It is shown that this multivariable optimisation is convex and thus numerically tractable. No explicit criteria to determine the existence of a solution are known; however, if solutions exist, results pertaining to their location are given to assist numerical optimisation. The extension of the approach to arbitrary unknown initial conditions is considered but is shown, in general, not to be possible. A tractable approach, considering the initial condition to be a random variable whose occurrence satisfies a certain probability density function, is presented. The resulting optimisation problem is shown to be convex and thus numerically tractable.

OUTPUT FEEDBACK GUARANTEED COST CONTROL OF UNCERTAIN SYSTEMS ON AN INFINITE TIME INTERVAL

This paper considers the problem of optimal guaranteed cost control of an uncertain system via output feedback. The uncertain system under consideration contains a single uncertainty block subject to an integral quadratic constraint. The cost function considered is a quadratic cost function defined over an infinite time interval. The main result of the paper gives a necessary and sufficient condition for the existence of a guaranteed cost controller guaranteeing a specified level of performance. This condition is given in terms of the existence of suitable solutions to an algebraic Riccati equation and a Riccati differential equation. The resulting guaranteed cost controller is in general time-varying. >

An Uncertainty Augmentation Approach to μ-Analysis

In this paper the concept of uncertainty augmentation is presented as a tool for treating various kinds of uncertainty structures within the μ framework. The original uncertainty block is replaced by a linear fractional transformation (LFT) of an augmented uncertainty block, which may include the time-derivative of the original uncertainties. The LFT is matched to the structure of the uncertainty including its dynamic and time-varying description. The approach is in many respects similar to the integral quadratic constraints (IQCs).

Robust control design strategy with parameter-dominated uncertainty

A robust controller design strategy is presented for systems, the parameters of which are affected dominantly by a few unknown physical quantities. The use of the functional dependence of the model parameters on these physical quantities results in an accurate description of the model uncertainty space. This potentially may lead to a better tradeoff between performance and robustness. The proposed procedure for modeling such uncertain systems relies on the linear-fractional-transformation algebra and reduction of the uncertainty block dimension. The resulting model can be used for control synthesis using any robust control design technique. In this work fi synthesis was adopted since it best fits the structured uncertainty model obtained. An example presents a controller design for an unmanned flight vehicle, the model parameters of which depend on the unknown mass of its payload. The aircraft equations are derived, and a p, controller is synthesized. The controller is compared to other control methods, and its advantages are demonstrated.

Optimal Popov controller analysis and synthesis for systems with real parameter uncertainties

Robust performance analysis plays an important role in the design of controllers for uncertain multivariable systems. Recent research has investigated the use of absolute stability criteria to develop less conservative analysis tests for systems with linear and nonlinear real parameter uncertainties. This note extends previous work on optimal /spl Hscr/2 performance analysis using the Popov criterion by presenting a numerical homotopy algorithm that can be used to analyze systems with less restrictive assumptions on the structure of the uncertainty block. The technique is used to compare relative robustness capabilities of the various control algorithms that have been designed for the Middeck active control experiment (MACE). The analysis is combined with the previously presented Popov controller synthesis to yield compensators that guarantee robust performance for systems with real parameter uncertainty.