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.     

Robust control via concave minimization local and global algorithms

This paper is concerned with the robust control problem of linear fractional representation (LFT) uncertain systems depending on a time-varying parameter uncertainty. Our main result exploits a linear matrix inequality (LMI) characterization involving scalings and Lyapunov variables subject to an additional essentially nonconvex algebraic constraint. The nonconvexity enters the problem in the form of a rank deficiency condition or matrix inverse relation on the scalings only. It is shown that such problems, but also more generally rank inequalities and bilinear constraints, can be formulated as the minimization of a concave functional subject to LMI constraints. First of all, a local Frank and Wolfe (1956) feasible direction algorithm is introduced in this context to tackle this hard optimization problem. Exploiting the attractive concavity structure of the problem, several efficient global concave programming methods are then introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. Computational experiments indicate the viability of our algorithms, and in the worst case, they require the solution of a few LMI programs     

Nonsmooth optimization for multiband frequency domain control design

Multiband frequency domain synthesis consists in the minimization of a finite family of closed-loop transfer functions on prescribed frequency intervals. This is an algorithmically difficult problem due to its inherent nonsmoothness and nonconvexity. We extend our previous work on nonsmooth H_∞ synthesis to develop a nonsmooth optimization technique to compute local solutions to multiband synthesis problems. The proposed method is shown to perform well on illustrative examples.     

Robust eigenstructure clustering by non-smooth optimisation

We extend classical eigenstructure assignment to more realistic problems, where additional performance and robustness specifications arise. Our aim is to combine time-domain constraints, as reflected by pole location and eigenvector structure, with frequency-domain objectives such as the H₂, H_∞ or Hankel norms. Using pole clustering, we allow poles to move in polydisks of prescribed size around their nominal values, driven by optimisation. Eigenelements, that is poles and eigenvectors, are allowed to move simultaneously and serve as decision variables in a specialised non-smooth optimisation technique. Two aerospace applications illustrate the power of the new method.     

Nonsmooth H/sub /spl infin// synthesis

We develop nonsmooth optimization techniques to solve H/sub /spl infin// synthesis problems under additional structural constraints on the controller. Our approach avoids the use of Lyapunov variables and therefore leads to moderate size optimization programs even for very large systems. The proposed framework is versatile and can accommodate a number of challenging design problems including static, fixed-order, fixed-structure, decentralized control, design of PID controllers and simultaneous design and stabilization problems. Our algorithmic strategy uses generalized gradients and bundling techniques suited for the H/sub /spl infin// norm and other nonsmooth performance criteria. We compute descent directions by solving quadratic programs and generate steps via line search. Convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods. The proposed approach proves to be efficient even for systems with several hundreds of states.     

D.C. optimization approach to robust control: Feasibility problems

The feasibility problem for constant scaling in output feedback control is considered. This is an inherently difficult problem since the set of feasible solutions is non-convex and may be disconnected. Nevertheless, we show that this problem can be reduced to the global minimization of a concave function over a convex set, or alternatively, to the global minimization of a convex program with an additional reverse convex constraint. Thus this feasibility problem belongs to the realm of d.c. optimization, a new field which has recently emerged as an active promising research direction in nonconvex global optimization. By exploiting the specific d.c. structure of the problem, several algorithms are proposed which at every iteration require solving only either convex or linear subproblems. Analogous algorithms with new characterizations are proposed for the bilinear matrix inequality (BMI) feasibility problem.     

Thermoplastic Elastomer Hydrogels via Self‐Assembly of an Elastin‐Mimetic Triblock Polypeptide

Protein‐based analogues of conventional thermoplastic elastomers can be designed with enhanced properties as a consequence of the precise control of primary structure. Protein 1 undergoes a reversible sol–gel transition, which results in the formation of a well‐defined elastomeric network above a lower critical solution temperature. The morphology of the network is consistent with selective microscopic phase separation of the endblock domains. This genetic engineering approach provides a method for specification of the critical architectural parameters, such as block length and sequence, which define macromolecular properties that are important for downstream applications.     

Relaxations of parameterized LMIs with control applications

A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. However, in contrast to LMIs, parameterized LMI (PLMIs) feasibility problems involve infinitely many LMIs hence are very hard to solve. In this paper, we propose several effective relaxation techniques to replace PLMIs by a finite set of LMIs. The resulting relaxed feasibility problems thus become convex and hence can be solved by very efficient interior point methods. Applications of these techniques to different problems such as robustness analysis, or Linear Parameter-Varying (LPV) control are then thoroughly discussed and illustrated by examples. Copyright © 1999 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.