Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays

Numerical Methods for Solution of the Extended Linear Quadratic Control Problem

The problem of robust stabilization of a linear system leads to the classical ℋ︁∞ control problem. The same analysis applied to a nonlinear system leads to the problem of ensuring via output feedback that a nonlinear operator be Lipshitz continuous, with a prescribed Lipshitz modulus. We show that, in the same way as the ℋ︁∞ control problem is equivalent to a minimax control problem, the Lipshitz modulus control problem can be approached via a minimax team decision problem. This motivates us to re-visit a class of the so-called ‘static’ team decision problems for nonlinear dynamical control systems. Because of the ‘static’ character, signaling plays no role in that case, which is important for the equivalence with the Lipshitz modulus control problem. We show that under some conditions, a certainty equivalence principle applies that yields a practical solution to the team problem at hand. To reach that conclusion we must first investigate a ‘partial team’ problem where one of the team members has all the information. Copyright © 1999 John Wiley & Sons, Ltd.

There are many tasks which require robotic manipulators interaction with environment. It consists of three control problems, i.e., position control, impact control and force control. The position control means the way of reaching to the environment. The moment of touching to the environment yields the impact control problem and the force control is to maintain the desired force trajectory after the impact with the environment. These three control problems occur in sequence, so each control algorithm can be developed independently. Especially for redundant manipulators, each of these three control problems has been important independent research topic. For example, joint torque minimization and impulse minimization are typical techniques for such control problems. The three control problems are considered as a single task in this paper. The position control strategy is developed to improve the performance of the task, i.e., minimization of the individual joint torques and impulse. Therefore, initial conditions of the impact control problem are optimized at the previous position control algorithm. Such a control strategy yields improved result of the impact control. Similarly, the initial conditions for the force control problem are indirectly optimized by the previous position control and impact control strategies. The force control algorithm uses the individual joint torque minimization concept. It also minimizes the force disturbances. The simulation results show the proposed control strategy works well.

Applying the line method approximation to a parabolic boundary control problem a sequence of ordinary control problems is generated. It is shown that the line method is a consistent and stable discretization. The convergence of the extreme values of the ordinary control problems to the extreme value of the parabolic control problem is proved.Finally, error estimations are given.

This paper focuses on numerical methods for solving time-optimal control problems using discrete-valued controls. A numerical Two-Phase Scheme, which combines admissible optimal control problem formulation with enhanced branch-and-bound algorithms, is introduced to efficiently solve bang-bang control problems in the field of engineering. In Phase I, the discrete restrictions are relaxed, and the resulting continuous problem is solved by an existing optimal control solver. The information on switching times obtained in Phase I is then used in Phase II wherein the discrete-valued control problem is solved using the proposed algorithm. Two numerical examples, including a third-order system and the F-8 fighter aircraft control problem, are presented to demonstrate the use of this proposed scheme. Comparing to STC and CPET methods proposed in the literature, the proposed scheme provides a novel method to find a different switching structure with a better minimum time for the F-8 fighter jet control problem.

We consider a dynamic scheduling problem for parallel server systems. J. M. Harrison has proposed a scheme for using diffusion control problems to approximately solve such control problems for heavily loaded systems. This approach has been very successfully used in the special case when the diffusion control problem can be reduced to an equivalent one for a one-dimensional workload process. However, it remains a challenging open problem to make substantial progress on using Harrison’s scheme when the workload process is more than one-dimensional. Here we present some new structural results concerning the diffusion control problem for parallel server systems with arbitrary workload dimension. Specifically, we prove that a certain server-buffer graph associated with a parallel server system is a forest of trees. We then exploit this graphical structure to prove that there exists a matrix, used to define the workload process, that has a block diagonal-like structure. An important feature of this matrix is that, except when the workload is one-dimensional, this matrix is frequently different from a choice of workload matrix proposed by Harrison. We demonstrate that our workload matrix simplifies the structure of the control problem for the workload process by proving that when the original diffusion control problem has linear holding costs, the equivalent workload formulation also has a linear cost function. We also use this simplification to give sufficient conditions for a certain least control process to be an optimal control for the diffusion control problem with linear holding costs. Under these conditions, we propose a continuous review threshold-type control policy for the original parallel server system that exploits pooling of servers within trees in the server-buffer graph and uses non-basic activities connecting different trees in a critical manner. We call this partial pooling. We conjecture that this threshold policy is asymptotically optimal in the heavy traffic limit. We illustrate the solution of the diffusion control problem and our proposed threshold control policy for a three-buffer, three-server example.

Reduced-order controllers for the H∞ control problem with unstable invariant zeros

The paper considers non-local control and optimal control problems for a linear time-varying dynamic system described by an ordinary differential equation with undivided multipoint conditions and with an integral condition imposed on the components of the state vector in some part of the time interval of the system operation. For the problem of optimal control, it is assumed that the quality criterion is specified over the whole time interval of system operation and has the meaning of the norm of some normed space. Conditions for complete controllability of a dynamic system with imposed non-local conditions are formulated. Conditions are obtained under which a solution to the control problem with such nonlocal conditions exists, and this solution is constructed. A constructive approach to solving control problems and a method for constructing an optimal control function are proposed. Continuity and non-uniqueness of the control function are shown. A solution to a specific control problem is given as an illustration.

This paper deals with various duality results for a new class of constrained robust nonlinear control problems. Concretely, we introduce a minimisation problem associated with a vector of functionals of multiple integral type. In addition, the considered multi-objective controlled optimisation problem is governed by several mixed constraints generated by partial derivatives of second order and uncertain data. For this new class of control problems, we establish and characterise Wolfe, Mond-Weir and mixed type robust dual models. To this aim, by using the notion of convex multiple integral functional, generated by a controlled second-order Lagrangian with uncertain data, and the concept of robust weak efficient solution to the problem under study, we build a new mathematical context to state and prove the duality theorems.

This paper studies the optimum design of beam networks modeled with Timoshenko beams. To account for multiple load cases, an auxiliary optimal control problem is introduced. Optimal distributed control problems for Timoshenko beam networks are solved through the associated optimality system, where the shape functional of the network is defined by the optimal value of the control cost. For control problems exhibiting the turnpike property, the optimum network design is carried out using the steady-state beam model and the corresponding steady-state control problem. A domain decomposition method is adopted to handle topological changes, while the Steklov–Poincaré operator is used to reformulate the beam network model as an interface problem on subdomain boundaries. This approach is applicable under additional assumptions on the network loading. Consequently, the topological derivative of the Steklov–Poincaré operator is incorporated into the optimality system of the control problem, enabling sensitivity analysis with respect to topological changes. The topological derivative of the cost functional with respect to the size of small cycles is derived and computed. Finally, numerical experiments are presented to illustrate and corroborate the analytical results.

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Infinite horizon singular optimal control problems with control taking values in a closed cone U in Rn lead to a dynamic programming equation of the form: maxF2x,v, v',v'' ,F1x,v,v' =0, forall x∈Q, where Q , the state space of the control problem, is some nonempty connected open subset of Rn , and F1, F2 are continuous real-valued functions on QxR2 and QxR3 respectively, with the coercivity assumption for F 2, that is, the function r&rarrr;F2x,u,p,r is nonincreasing on R . A major concern is to determine how smooth the value function v is across the free boundary of the problem. Linear-convex deterministic and stochastic singular control problems in dimension one are considered. We present the analysis of the above equation together with smoothness of the value function across the free boundary. The interest here is the explicitness of the results and the fact that the smooth fit principle depends on the parameters of the control problem. In particular, the value function for the stochastic control problem is founded explicitly, and at the same time, optimal controls are identified using a verification theorem.

5 - Connections with the classical theory

H∞ Control with Multi-Saddles

We consider the design of fixed‐order (or low‐order) linear controllers which meet certain performance and/or robustness specifications. The following three problems are considered; covariance control as a nominal performance problem,𝒬‐stabilization as a robust stabilization problem, and robust L∞ control problem as a robust performance problem. All three control problems are converted to a single linear algebra problem of solving a linear matrix inequality (LMI) of the type BGC + (BGC)T + Q < 0 for the unknown matrix G. Thus this paper addresses the fixed‐order controller design problem in a unified way. Necessary and sufficient conditions for the existence of a fixed‐order controller which satisfies the design specifications for each problem are derived, and an explicit controller formula is given. In any case, the resulting problem is shown to be a search for a (structured) positive definite matrix X such that X ∈ 𝒞1 and X−1 ∈ 𝒞2 where 𝒞1 and 𝒞2 are convex sets defined by LMIs. Computational aspects of the nonconvex LMI problem are discussed.