H∞ Control with Multi-Saddles

This paper discusses feedback control problems for affine nonlinear singular systems. Namely, the problems of regularization (regular singular systems have for a particular initial state and input, a unique solution), non-interaction and exact linearization. First, an algorithm providing necessary and sufficient conditions for the regularizability of the affine nonlinear singular systems is introduced. This algorithm resembles the well-known constrained dynamics algorithm for affine nonlinear systems. For regularizable affine nonlinear singular systems, necessary and sufficient conditions for the solvability of the non-interacting control problem are derived. They are based on another algorithm producing a sequence of integers, the generalization of the well known notion of relative degree for usual affine nonlinear systems. Finally, the problem of feedback exact linearization of the dynamic part of singular systems is briefly addressed. An example is provided to illustrate the main results.

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.

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.

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.

The authors present a state-space solution to the discrete H/sub 2/ control problem and give direct formulas for an H/sub 2/-optimal sampled-data control problem with state feedback and disturbance feedforward using the powerful lifting technique. The discrete-time H/sub 2/-optimal control is first treated via state feedback and disturbance feedforward and then via dynamic output feedback. New direct formulas for a sampled-data H/sub 2/ problem using state measurement are presented. The optimal sampled-data control is applied to a two-motor system and a comparison is made with the optimal analog control. >

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.

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.

This thesis addresses the design of optimization-based control laws for the case where convergence to a desired set-point, minimization of an arbitrary performance index, or a combination of the two, is the desired objective. The results are developed within the sample-data Model Predictive Control (MPC) framework considering constrained nonlinear continuous-time time-varying dynamical systems. For a given time sampling, a sample-data MPC control strategy consists of i) choosing among all future finite horizon predictions of state and input trajectories of the system the one that minimizes the given performance index, ii) applying the optimal input trajectory to the system until a new time sample is reached, and iii) iterating this process. The performance index is chosen to describe the specific control problem under consideration. In a classic Tracking-MPC framework, where the main goal is to steer the state of the system to a desired steady-state, the performance index is properly chosen to penalize the distance from the current state to a desired one. In order to capture more complex control objectives, in recent years a growing attention has been dedicated to a new class of controllers that goes under the name of Economic-MPC. Here, the term economic is used to stress the fact that the performance index is a general index of interest that we wish to minimize, e.g., economic, which does not denote the distance to a desired set point. This setting makes full use of the potentialities of optimization-based control strategies. Although, it comes with disadvantages. In fact, by choosing an arbitrary performance index, it is difficult to predict, and therefore certify, the evolution of the closed-loop system, which could potentially manifest undersirable behaviors. This thesis provides analysis and certification of a variety of closed-loop behaviors stemming from the use Economic-MPC controllers. A set of tools for design of provably correct MPC controllers is provided for the case where the performance index is of the Tracking-MPC type, purely economic, or a combination of the two. The results focus the certification of both closed-loop economic performance and closed-loop state evolution. The proposed strategies are applied to a range of motion control problems for underactuated vehicles. An MPC controller for Trajectory-Tracking and Path-Following with convergence guarantees is first proposed and then extended, using the results presented on Economic-MPC, to address the control problems of distributed formation keeping, energy efficient trajectory-tracking, and target-following through highly observable trajectories.

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.

Method of Moment Restrictions in Robust Control and Filtering

Second order autonomous systems are key systems in the study of non linear systems because their solution trajectories can be represented by curves in the plane (Khalil, 2002), which helps in the development of control strategies through the understanding of their dynamical behaviour. Such autonomous systems are often obtained when considering feedback control strategies, because the closed loop system might be rewritten in terms of the state system and perturbation terms, which are function of the state as well. Thus, analyses of stability properties of second order autonomous systems and their convergence are areas of interest on the control community. Moreover, several applications consider nonlinear second order systems; there are various examples of this: 1. In mechanical systems the pendulum, the inverted pendulum, the translational oscillator with rotational actuator (TORA) and the mass-spring systems; 2. In electrical systems there are examples such as the tunnel diode circuit, some electronic oscillators as the negative-resistance twin-tunnel-diode circuit; and finally 3. Other type of these systems are mechanical-electrical-electronic combinations, for example a two degree of freedom (DOF) robot arm or a mobile planar robot and among every degree of freedom on a robotic structure can be represented by a second order nonlinear system. Therefore, due to the wide applications in second order nonlinear systems, several control laws have been proposed, which comprises from simple ones, like linear controllers, to the more complex, like sliding mode, backstepping approach, output-input feedback linearization, among others (Khalil, 2002). Despite the development of several control strategies for nonlinear second order systems, it is not surprising that for several years and even nowadays the classical PID controllers have been widely used in technical and industrial applications and even on research fields. This is due to the good understanding that engineers have of them. Moreover, the PID controllers have several important functions: provide feedback, has the ability to eliminate steady state offset through integral action, and it can anticipate the future through derivative action. PID controllers are sufficient for many control problems, particularly when system dynamics are favourable and the performance requirements are moderate. These types of

As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.

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