Development of a skew µ lower bound

Exploitation of the NP hard, mixed µ problem structure provides a polynomial time algorithm that approximates µ with usually reasonable answers. When the problem is extended to the skew µ problem an extension of the existing method to the skew µ formulation is required. The focus of this paper is to extend the µ lower bound derivation to the skew µ lower bound and show its direct computation by way of a power algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonsmooth µ‐synthesis

We revisit robust complex‐ and mixed‐ µ‐synthesis problems based on upper bounds and show that they can be recast as specially structured controller design programs. The proposed reformulations suggest a streamlined handling of µ‐synthesis problems using recently developed (local) nonsmooth optimization methods, where both scalings or multipliers and a controller of given structure are obtained simultaneously. A first cut of the nonsmooth programming software for structured H_∞ synthesis is made available through the MATLAB R2010b Prerelease, Robust Control Toolbox Version 3.5 developed by The MathWorks, Inc. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability‐ and performance‐robustness tradeoffs: MIMO mixed‐µ vs complex‐µ design

We present a non‐trivial case study designed to highlight some of the practical issues that arise when using mixed‐µ or complex‐µ robust synthesis methodologies. By considering a multi‐input multi‐output three‐cart mass–spring–dashpot (MSD) with uncertain parameters and dynamics, it is demonstrated that optimized performance (disturbance‐rejection) is reduced as the level of uncertainty in one or two real parameters is increased. Comparisons are made (a) in the frequency domain, (b) by RMS values of key signals and (c) in time‐domain simulations. The mixed‐µ controllers designed are shown to yield superior performance as compared with the classical complex‐µ design. The singular value decomposition analysis shows the directionality changes resulting from different uncertainty levels and from the use of different frequency weights. The nominal and marginal stability regions of the closed‐loop system are studied and discussed, illustrating how stability margins can be extended at the cost of reducing performance. Copyright © 2008 John Wiley & Sons, Ltd.     

A non‐smooth lower bound on ν

A non‐smooth optimization technique to directly compute a lower bound on the skew structured singular value ν is developed. As corroborated by several real‐world challenging applications, the proposed technique can provide tighter lower bounds when compared with currently available techniques. Moreover, in many cases, the determined lower bound equals the true value of ν. Thanks to the efficiency of the non‐smooth technique, the algorithm can be applied to problems involving even a significant number of uncertain parameters. Another appealing feature of the proposed non‐smooth approach is that the dimension of repeated scalar uncertainties in the overall structured uncertainty matrix has little impact on the computational time. The technique can be used to compute a lower bound on the structured singular value μ as well. Copyright © 2012 John Wiley & Sons, Ltd.     

Pointwise in frequency performance weight optimization in µ‐synthesis

A conceptually different approach to the µ‐synthesis robust performance problem is proposed in this article. The optimization problem posed maximizes the performance weights with respect to a suitable cost function that captures the desired closed‐loop performance. This maximization of performance weights is limited by the fact that there must exist some internally stabilizing controller that guarantees robust performance with respect to these maximized weights. Thus, performance weights and a controller that achieves an optimized level of robust performance are synthesized together by one algorithm in a systematic way. The designer is only required to specify the plant set and an optimization directionality. This directionality only appears in the cost function and reflects the desired closed‐loop properties in particular frequency regions. It is pointed out that choosing this directionality is much easier than choosing the performance weights directly. Correspondingly, this approach greatly simplifies the often long and tedious process of designing ‘good’ performance weights directly and gives an indication of what is the achievable performance. A pointwise in frequency solution to the posed optimization problem is also developed in this article. Copyright © 2005 John Wiley & Sons, Ltd.     

A measure of worst-case H∞ performance and of largest acceptable uncertainty

The structured singular value (SSV or μ) is known to be an effective tool for assessing robust performance of linear time-invariant models subject to structured uncertainty. Yet all single μ analysis provides is a bound β on the uncertainty under which stability as well as H_∞ performance level of κ/β are guaranteed, where κ is preselectable. In this paper, we introduce a related quantity ν which provides answers for the following questions: (i) given β, determine the smallest with the property that, for any uncertainty bounded by β, an H∞ performance level of is guaranteed; (ii) conversely, given , determined the largest β with the property that, again, for any uncertainty bounded by β, an H_∞ performance level of is guaranteed. Properties of this quantity are established and approaches to its computation are investigated. Both unstructured uncertainty and structured uncertainty are considered.     

Computation of the real structured singular value via pole migration

The paper introduces a new computationally efficient algorithm to determine a lower bound on the real structured singular value μ. The algorithm is based on a pole migration approach where an optimization solver is used to compute a lower bound on real μ independent of a frequency sweep. A distinguishing feature of this algorithm from other frequency independent one‐shot tests is that multiple localized optima (if they exist) are identified and returned from the search. This is achieved by using a number of alternative methods to generate different initial conditions from which the optimization solver can initiate its search from. The pole migration algorithm presented has also been extended to determine lower bounds for complex parametric uncertainties as well as full complex blocks. However, the results presented are for strictly real and repeated parametric uncertainty problems as this class of problem is the focus of this paper and are in general the most difficult to solve. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust performance analysis for uncertain negative‐imaginary systems

Negative‐imaginary systems are important in engineering practice as this class of systems appears quite often in practical problems, for example, lightly damped flexible structures with collocated position sensors and force actuators. In this paper, an analytical framework for robust performance of uncertain negative‐imaginary systems is proposed. The results are obtained by transforming negative‐imaginary systems into a bounded‐real framework via the positive‐real property. This paper deals with all the significant technical difficulties that appear due to the transformation and the punctured j ω‐axis frequency condition of negative‐imaginary systems. The problem is equivalently cast into a structured singular value condition that gives a quantitative performance test for this class of systems. This result also gives an analytical framework for robust stability when the perturbations are mixture of bounded‐real and negative‐imaginary uncertainties. A numerical example is presented to show the usefulness of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Robust multiple model adaptive control: Modified using ν‐gap metric

A new robust adaptive control method is proposed, which removes the deficiencies of the classic robust multiple model adaptive control (RMMAC) using benefits of the ν‐gap metric. First, the classic RMMAC design procedure cannot be used for systematic design for unstable plants because it uses the Baram Proximity Measure, which cannot be calculated for open‐loop unstable plants. Next, the %FNARC method which is used as a systematic approach for subdividing the uncertainty set makes the RMMAC structure being always companion with the µ‐synthesis design method. Then in case of two or more uncertain parameters, the model set definition in the classic RMMAC is based on cumbersome ad hoc methods. Several methods based on ν‐gap metric for working out the mentioned problems are presented in this paper. To demonstrate the benefits of the proposed RMMAC method, two benchmark problems subject to unmodeled dynamics, stochastic disturbance input and sensor noise are considered as case studies. The first case‐study is a non‐minimum‐phase (NMP) system, which has an uncertain NMP zero; the second case‐study is a mass‐spring‐dashpot system that has three uncertain real parameters. In the first case‐study, five robust controller design methods (H₂, H_∞, QFT, H_∞ loop‐shaping and µ‐synthesis) are implemented and it is shown via extensive simulations that RMMAC/ν/QFT method improves disturbance‐rejection, when compared with the classic RMMAC. In the second case‐study, two robust controller design methods (QFT and mixed µ‐synthesis) are applied and it is shown that the RMMAC/ν/QFT method improves disturbance‐rejection, when compared with RMMAC/ν/mixed?µ. Copyright © 2010 John Wiley & Sons, Ltd.     

Fixed‐order robust H∞ control and control‐oriented uncertainty set shaping for systems with ellipsoidal parametric uncertainty

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities. A polynomial method is employed to design a fixed‐order controller that assigns closed‐loop poles within a given region of the complex plane and that satisfies an H_∞ performance specification. The main feature of the proposed method is that it can be extended easily for control‐oriented uncertainty set shaping using a standard input design approach. Consequently, the results can be extended to joint robust control/input design procedure whose controller structure and performance specifications are translated into the requirements on the input signal spectrum used in system identification. This way, model uncertainty set can be tuned for the robust control design procedure. The simulation results show the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Design of an RMPC with a time‐varying terminal constraint set for tracking problem

This paper presents a robust model predictive control algorithm with a time‐varying terminal constraint set for systems with model uncertainty and input constraints. In this algorithm, the nonlinear system is approximated by a linear model where the approximation error is considered as an unstructured uncertainty that can be represented by a Lipschitz nonlinear function. A continuum of terminal constraint sets is constructed off‐line, and robust stability is achieved on‐line by using a variable control horizon. This approach significantly reduces the computational complexity. The proposed robust model predictive controller with a terminal constraint set is used in tracking set‐points for nonlinear systems. The effectiveness of the proposed method is illustrated with a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐time scale singular perturbation control and stability analysis for an autonomous helicopter on a platform

A three‐time scale singular perturbation control law is designed for a nonlinear helicopter model in vertical flight. The proposed control law is based on time scale decomposition and is able to achieve the desired altitude by selecting a desired angular velocity and the associated collective pitch angle of the blades. The stability of the system is performed by presenting a stability analysis for generic three‐time scale singularly perturbed systems, which allows to construct a composite Lyapunov function for the resultant closed‐loop system by using time scale separation and also providing mathematical expressions for the upper bounds of the singularly perturbed parameters that define the three‐time scale. Numerical results on both, the singular perturbation control strategy and the stability analysis, are also presented for the studied nonlinear highly coupled helicopter model. Copyright © 2012 John Wiley & Sons, Ltd.