a trust region spectral bundle method for nonconvex eigenvalue optimization we present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions we prove global convergence of our method in the sense that for an arbitrary starting point every accumulation point of the sequence of iterates is critical the method is tested on several problems in feedback control synthesis

on hopf bifurcations in singularly perturbed systems it has been shown recently that under some generic assumptions there exists a hopf curve spl lambda spl lambda spl epsiv for singularly perturbed systems of the form x spl dot f x y spl lambda spl epsiv y spl dot g x y spl lambda near the singular surface defined by det g sub v 0 in this note we are concerned with the hopf curve and obtain three results 1 we prove that the eigenvalue crossing condition for the hopf curve holds without additional assumption 2 we provide an improved form of an existing derivative formula for the hopf curve which is more suitable for practical computations and 3 we give a quite precise description of the spectrum structure of the linearization along the hopf curve all three results stated in the main theorem are useful for a better understanding of hopf bifurcations in singularly perturbed systems our analysis is based on a factorization of parameter dependent polynomials lemma 2 3

finite band solutions of the coupled dispersionless hierarchy the coupled dispersionless hierarchy is derived with the help of the zero curvature equation based on the lax matrix we introduce an algebraic curve of arithmetic genus n from which we establish the corresponding meromorphic function the baker akhiezer function and dubrovin type equations the straightening out of all the flows is given under the abel jacobi coordinates using the asymptotic properties of and we obtain the explicit theta function representations of the meromorphic function the baker akhiezer function and of solutions for the whole hierarchy

controller design for linear multivariable feedback systems with stable plants using optimization with inequality constraints this paper proposes a two part design method for unity feed back linear multivariable systems with stable plants first a known algebraic technique yields a matrix q with the property that the family of exponentially stable and proper q globally parametrizes the family of exponentially stable closed loop systems with proper controllers second having obtained such a family of qs a non linear programming problem is formulated which when solved will yield the best feasible q with respect to designer specified objective and constraint functions the method developed makes possible the systematic formulation and solution of a class of design problems which represent practical design considerations as inequality constraints in non linear programming problems several new design problem formulations are given and then solved for some specific plants these formulations include for example constraints to avoid plant saturation and the mixing of time and frequency domain objectives and constraints

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

a modified sqp algorithm for minimax problems in this paper a modified nonmonotone line search sqp algorithm for nonlinear minimax problems is presented during each iteration of the proposed algorithm a main search direction is obtained by solving a reduced quadratic program qp in order to avoid the maratos effect a correction direction is generated by solving the reduced system of linear equations under mild conditions the global and superlinear convergence can be achieved finally some preliminary numerical results are reported

design of multivariable feedback systems with simple unstable plant this paper proposes a design methodology for distributed linear multivariable feedback systems with simple unstable plants a simple unstable plant has either first or second order unstable poles the methodology developed provides a global characterization of all realizable compensators which stabilize a given simple unstable plant a design example is given to show that this methodology can be used to generate in an appropriate computer aided design environment controllers which are optimal with respect to designer specified criteria additionally it is shown that the nature of the design methodology gives geometric insight into the dynamics of the process whereby an unstable plant is stabilized

a non linear programming approach to the computer aided design of regulators using a linear quadratic formulation a design technique is proposed for linear regulators in which a feedback controller of fixed structure is chosen to minimize an integral quadratic objective function subject to the satisfaction of integral quadratic constraint functions application of a non linear programming algorithm to this mathematically tractable formulation results in an efficient and useful computer aided design tool particular attention is paid to computational efficiency and various recommendations are made two design examples illustrate the flexibility of the approach and highlight the special insight afforded to the designer

a new problem in control system design and circuit design an optimisation problem arising in the design of circuits and control systems is formulated and an outer approximation algorithm for solving it is presented the problem is characterized by constraints that must be satisfied for all functions in a given subset of a function space it is shown that an outer approximation type algorithm increases the number of variables and constraints at each iteration a practical algorithm will require a constraint dropping scheme

nondifferentiable optimization algorithm for designing control systems having singular value inequalities it has been known for some time that many control system design requirements can be expressed as differentiable inequalities more recently it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix this paper presents an optimization algorithm which permits all these constraints to be considered

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

stability of singular hopf bifurcations abstract a stability formula is given for the singular hopf bifurcation arising in singularly perturbed systems of the form x e f x y y g x y in this paper the derivation of the formula is based on a reduction technique and on an existing stability formula for hopf bifurcation

on the periodic orbit bifurcating from a zero hopf bifurcation in systems with two slow and one fast variables the hopf bifurcation in slow fast systems with two slow variables and one fast variable has been studied recently mainly from a numerical point of view our goal is to provide an analytic proof of the existence of the zero hopf bifurcation exhibited for such systems and to characterize the stability or instability of the periodic orbit which borns in such zero hopf bifurcation our proofs use the averaging theory

brief paper 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

the localized induction hierarchy and the lund regge equation an evolution equation of a curve is constructed by summing up the infinite sequence of commuting vector fields of the integrable hierarchy for the localized induction equation lie it is shown to be equivalent to the lund regge equation the intrinsic equations governing the curvature and torsion are deduced in the form of integrodifferential evolution equations a class of exact solutions which correspond to the permanent forms of a curve evolving by a steady rigid motion are presented the analysis of the solutions reveals that given the shape there are two speeds of motion one of which has no counterpart in the case of the lie

computer aided design via optimization many design problems including control design problems involve infinite dimensional constraints of the form f z a 0 for all a e a where a denotes time or frequency or a parameter vector in other design problems tuning or trimming of certain parameters after manufacture of the system is permitted the corresponding constraint is that for each a in a there exists a value t of the tuning parameter in a permissible set t such that f z a t 0 recent algorithms for solving design problems having such constraints are summarized

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

the relationship between the dispersionless equation and the localized induction hierarchy through the pohlmeyer lund regge equation fukumoto and miyajima fukumoto y and miyajima m 1996 j phys a math gen 29 8025 have discussed that an evolution equation obtained by summing up an infinite number of equations of the localized induction hierarchy lih is equivalent to the pohlmeyer lund regge equation plre on the other hand the dispersionless equation dle is also shown to be equivalent to the plre so we investigate a link between the equation derived from the lih and the dle through the plre we find that their result is not correct however we show that there is a special case where a solution of the lih satisfies the plre

a finitely convergent algorithm for convex inequalities in engineering design problems it is often necessary to find a solution to a system of nonlinear inequalities in which the functions involved are not necessarily differentiable in this paper assuming convexity of the functions an algorithm is presented for solving such systems of inequalities in a finite number of iterations

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h