structural properties of minimax controllers for a class of differential games arising in nonlinear h control abstract this paper introduces in precise mathematical terms two properties named certainty equivalence and generalized certainty equivalence that nonlinear minimax controller problems might possess the certainty equivalence is a generalization of the one introduced earlier by basar and bernhard 1991 and bernhard 1990 which applies to problems where the worst case disturbance may not be unique but the worst case state trajectory is the generalized certainty equivalance on the other hand extends this to accomodate nonunique worst case state trajectories and leads to the constrollers that guarantee a bounded upper value for the underlying game the paper also shows that for a large class of games and under certain conditions certainty equivalent as well as generalized certainty equivalent controllers admit infinite dimensional estimator kalman filter structures where the estimator gain depends on the state of the estimator

a multilevel technique for the approximate solution of operator lyapunov and algebraic riccati equations we consider multigrid or more appropriately multilevel techniques for the numerical solution of operator lyapunov and algebraic riccati equations the riccati equation which is quadratic plays an essential role in the solution of linear quadratic optimal control problems the linear lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the newton kleinman nk algorithm for the solution of algebraic riccati equations both equations are operator equations when the underlying linear system is infinite dimensional in this case finite dimensional discretization is required however as the level of discretization increases the convergence rate of the standard iterative techniques for solving high order matrix lyapunov and riccati equations decreases to deal with this multileveling is introduced into the iterative nk method for solving the algebraic riccati equation and smith s method for solving matrix lyapunov equations theoretical result

a strict error bound with separated contributions of the discretization and of the iterative solver in non overlapping domain decomposition methods this paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non overlapping domain decomposition method feti or bdd we propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

adaptive controller design for tracking and disturbance attenuation in parametric strict feedback nonlinear systems the authors develop a systematic procedure for obtaining robust adaptive controllers that achieve asymptotic tracking and disturbance attenuation for a class of nonlinear systems which are described in the parametric strict feedback form and are subject to additional exogenous disturbance inputs their approach to adaptive control is performance based where the objective for the controller design is not only to find an adaptive controller but also to construct an appropriate cost functional compatible with desired asymptotic tracking and disturbance attenuation specifications with respect to which the adaptive controller is worst case optimal three main issues of the paper are the backstepping methodology worst case identification schemes and singular perturbations analysis closed form expressions have been obtained for an adaptive controller and the corresponding value function a numerical example involving a third order system is given

on the certainty equivalence principle and the optimal control of partially observed dynamic games the purpose of this paper is to explain that while in general the certainty equivalence controller does not achieve the optimal value of a dynamic min max game with partial information the optimal controller may be determined using the recently developed information state approach examples are provided illustrating when the certainty equivalence controller is or is not optimal

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

analysis of preconditioners for saddle point problems mixed finite element formulations give rise to large sparse block linear systems of equations the solution of which is often sought via a preconditioned iterative technique in this work we present a general analysis of block preconditioners based on the stability conditions inherited from the formulation of the finite element method the babuska brezzi or inf sup conditions the analysis is motivated by the notions of norm equivalence and field of values equivalence of matrices in particular we give sufficient conditions for diagonal and triangular block preconditioners to be norm and field of values equivalent to the system matrix

solving hermitian positive definite systems using indefinite incomplete factorizations incomplete ldl factorizations sometimes produce an indefinite preconditioner even when the input matrix is hermitian positive definite the two most popular iterative solvers for symmetric systems cg and minres cannot use such preconditioners they require a positive definite preconditioner one approach that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process we explore a different approach use the incomplete factorization with a krylov method that can accept an indefinite preconditioner the conventional wisdom has been that long recurrence methods like gmres or alternatively non optimal short recurrence methods like symmetric qmr and bicgstab must be used if the preconditioner is indefinite we explore the performance of these methods when used with an incomplete factorization but also explore a less known krylov method called pcg odir that is both optimal and uses a short recurrence and can use an indefinite preconditioner furthermore we propose another optimal short recurrence method called ip minres that can use an indefinite preconditioner and a variant of pcg odir which we call ip cg that is more numerically stable and usually requires fewer iterations

a stopping criterion for the conjugate gradient algorithm in a finite element method framework the conjugate gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self adjoint elliptic partial differential equations taking into account recent results 13 19 20 22 which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process we adapt the stopping criterion introduced in 3 moreover we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion finally we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion

mesh independence of kleinman newton iterations for riccati equations in hilbert space in this paper we consider the convergence of the infinite dimensional version of the kleinman newton algorithm for solving the algebraic riccati operator equation associated with the linear quadratic regulator problem in a hilbert space we establish mesh independence for this algorithm and apply the result to systems governed by delay equations numerical examples are presented to illustrate the results

optimal control of discrete time uncertain systems with imperfect measurement we consider a discrete time optimal control problem in the presence of uncertainties in the dynamics and in the measurement an optimal in min max sense output feedback controller is derived from a discrete bellman equation we propose a formulation of the last equation in the estimation space which allows approximation by a reduced order equation corresponding to a prescribed collection of estimation sets the approach is illustrated up to numerical experiments by a pursuit evasion game on the plane with incomplete information of the current state of the evader

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

an iterative method with error estimators abstract iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions in many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small this paper presents a new iterative method based on the lanczos process for the solution of linear systems of equations with a symmetric matrix the method is designed to allow the computation of estimates of the euclidean norm of the error in the computed approximate solutions these estimates are determined by evaluating certain gauss anti gauss or gauss radau quadrature rules

a geometric multigrid preconditioning strategy for dpg system matrices abstract the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2010 2011 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes a key question that has not yet been answered in general though there are some results for poisson e g is how best to precondition the dpg system matrix so that iterative solvers may be used to allow solution of large scale problems in this paper we detail a strategy for preconditioning the dpg system matrix using geometric multigrid which we have implemented as part of camellia roberts 2014 2016 and demonstrate through numerical experiments its effectiveness in the context of several variational formulations we observe that in some of our experiments the behavior of the preconditioner is closely tied to the discrete test space enrichment we include experiments involving adaptive meshes with hanging nodes for lid driven cavity flow demonstrating that the preconditioners can be applied in the context of challenging problems we also include a scalability study demonstrating that the approach and our implementation scales well to many mpi ranks

parameter identification for uncertain plants using h methods abstract we demonstrate the effective use of h filtering and cost to come methods for parameter identification in deterministic uncertain plants that are linear in the unknown parameters but nonlinear otherwise the cost to come method is an approach that has been used earlier to solve linear and nonlinear h optimal control and filtering problems it consists of constructing a cost to come function which assists in the design of an optimal observer scheme the method is used here in the design of a parameter identification scheme for uncertain plants where measurements on the state of the system are available but not on its derivative two approaches are adopted in both of which the parameter estimation problem is formulated as an h filtering problem one of the approaches uses a more standard prefiltering of the past states input and disturbance signals the other approach is a novel design method which leads to a new class of identification schemes it involves two subproblems fsdi full state derivative information problem where it is assumed that both the state and its derivative are available to the parameter estimator and npfsi noise perturbed fsi problem where the estimator is assumed to measure a noise perturbed measurement of the state in the latter problem we use singular perturbation methods to prove asymptotic convergence of the performance of the identifier to that of the unperturbed case thus providing an asymptotically optimal solution to the fsi full state measurement problem to illustrate both approaches several simulation studies on a numerical example are provided

convergence analysis for reduced order adaptive controller design of uncertain siso linear systems with noisy output measurements we consider in this article a class of uncertain siso linear systems that are subject to system and measurement noises reduced order adaptive controller designs have been proposed before for such systems by the authors and stability analysis of the closed loop systems has been established here we analyse further the robustness properties for these reduced order adaptive control systems by providing detailed convergence analysis results for the key closed loop signals and parameter estimates we rigorously prove that whenever the exogenous disturbance input is of finite energy and bounded and the reference trajectory and its derivatives up to rth order are bounded r being the relative degree of the transfer function of the true system a set of signals including the tracking error the estimation error between the system output and its estimate the projection signal are of finite energy and converge to zero and the system states and their estimates exhibit asymptotic behaviours with certain form

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

riccati theory and singular estimates for a bolza control problem arising in linearized fluid structure interaction abstract we consider a bolza boundary control problem involving a fluid structure interaction model the aim of this paper is to develop an optimal feedback control synthesis based on riccati theory the model considered consists of a linearized navier stokes equation coupled on the interface with a dynamic wave equation the model incorporates convective terms resulting from the linearization of the navier stokes equation around equilibrium the existence of the optimal control and its feedback characterization via a solution to a riccati equation is established the main mathematical difficulty of the problem is caused by unbounded action of control forces which in turn result in riccati equations with unbounded coefficients and in singular behavior of the gain operator this class of problems has been recently studied via the so called singular estimate control systems secs theory which is based on the validity of the singular estimate se g avalos differential riccati equations for the active control of a problem in structural acoustic j optim theory appl 91 1996 695 728 i lasiecka mathematical control theory of coupled pde s in nsf cmbs lecture notes siam 2002 with unbounded controls i lasiecka a tuffaha riccati equations for the bolza problem arising in boundary point control problems governed by c0 semigroups satisfying a singular estimate j optim theory appl 136 2008 229 246 it is shown that the fluid structure interaction does satisfy the singular estimate se condition this is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface thus the established singular estimate allows for the application of recently developed general theory which in turn implies well posedness of the feedback synthesis and of the associated riccati equation moreover the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

stopping criteria for adaptive finite element solvers we consider a family of practical stopping criteria for linear solvers for adaptive finite element methods for symmetric elliptic problems a contraction property between two consecutive levels of refinement of the adaptive algorithm is shown when a family of smallness criteria for the corresponding linear solver residuals are assumed on each level or refinement more importantly based on known and new results for the estimation of the residuals of the conjugate gradient method we show that the smallness criteria give rise to practical stopping criteria for the iterations of the linear solver which guarantees that the inexact adaptive algorithm converges a series of numerical experiments highlights the practicality of the theoretical developments

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

parameter identification for uncertain linear systems with partial state measurements under an h sup spl infin criterion this paper addresses the worst case parameter identification problem for uncertain single input single output siso and multi input multi output mimo linear systems under partial state measurements and derives worst case identifiers using the cost to come function method in the siso case the worst case identifier obtained subsumes the kreisselmeier observer as part of its structure with parameters set at some optimal values its structure is different from the common least squares ls identifier however in the sense that there is additional dynamics for the state estimate coupled with the dynamics of the parameter estimate in a nontrivial way in the mimo case as well the worst case identifier has additional dynamics for the state estimate which do not appear in the conventional ls based schemes also for both siso and mimo problems approximate identifiers are obtained which are numerically much better conditioned when the disturbances in the measurement equations are small the theoretical results are then illustrated on an extensive numerical example to demonstrate the effectiveness of the identification schemes developed

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

residual and backward error bounds in minimum residual krylov subspace methods minimum residual norm iterative methods for solving linear systems ax b can be viewed as and are often implemented as sequences of least squares problems involving krylov subspaces of increasing dimensions the minimum residual method minres c paige and m saunders siam j numer anal 12 1975 pp 617 629 and generalized minimum residual method gmres y saad and m schultz siam j sci statist comput 7 1986 pp 856 869 represent typical examples in c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear revealing upper and lower bounds on the residual norm of any linear least squares ls problem were derived in terms of the total least squares tls correction of the corresponding scaled tls problem in this paper theoretical results of c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear are extended to the gmres context the bounds that are developed are important in theory but they also have fundamental practical implications for the finite precision behavior of the modified gram schmidt implementation of gmres and perhaps for other minimum norm methods

superlinear convergence of krylov subspace methods for self adjoint problems in hilbert space the conjugate gradient and minimum residual methods for self adjoint problems in hilbert space are considered linear and superlinear convergence results with respect to both q and r rates are reviewed new results on ell step q superlinear and r superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a hilbert space theory

a numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems a hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed the method which combines use of a chandrasekhar type system with an iteration of the newton kleinman form with variable acceleration parameter smith schemes is formulated to efficiently compute directly the feedback gains rather than solutions of an associated riccati equation the hybrid method is particularly appropirate when used with large dimensional systems such as those arising in approximating infinite dimensional distributed parameter control systems e g those governed by delay differential and partial differential equations computational advantages of the proposed algorithm over the standard eigenvector potter laub schur based techniques are discussed and numerical evidence of the efficacy of these ideas is presented

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

minimax control of switching systems under sampling we consider a general class of systems subject to two types of uncertainty a continuous deterministic uncertainty that affects the system dynamics and a discrete stochastic uncertainty that leads to jumps in the system structure at random times with the latter described by a continuous time finite state markov chain when only sampled values of the system state is available to the controller along with perfect measurements on the state of the markov chain we obtain a characterization of minimax controllers which involves the solutions of two finite sets of coupled pdes and a finite dimensional compensator for the linear quadratic case a complete characterization is given in terms of coupled generalized riccati equations which also provides the solution to a particular h optimal control problem with randomly switching system structure and sampled state measurements

homotopy for rational riccati equations arising in stochastic optimal control we consider the numerical solution of the rational algebraic riccati equations in mathbb r n arising from stochastic optimal control in continuous and discrete time applying the homotopy method we continue from the stabilizing solutions of the deterministic algebraic riccati equations which are readily available the associated differential equations require the solutions of some generalized lyapunov or stein equations which can be solved by the generalized smith methods of o n 3 computational complexity and o n 2 memory requirement for large scale problems the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms in comparison the alternative modified newton s methods require a difficult initial stabilization step some illustrative numerical examples are provided