Stabilizing solutions of the H∞ algebraic Riccati equation

AbstractThe algebraic Riccati equation studied in this paper is related to the suboptimal state feedback H∞ control problem. It is parametrized by the H∞-norm bound γ we want to achieve. The objective of this paper is to study the behavior of the solution to the Riccati equation as a function of γ. It turns out that a stabilizing solution exists for all but finitely many values of γ larger than some a priori determined bound γ−. On the other hand, for values smaller than γ− there does not exist a stabilizing solution. The finite number of exception points can be characterized as switching points where eigenvalues of the stabilizing (symmetric) solution can switch from negative to positive with increasing γ. After the final switching point the solution will be positive semidefinite. We obtain the following interpretation: The Riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exists a static feedback such that the closed-loop transfer matrix has k unstable poles and an L∞ norm strictly less than γ

Stochastic disturbance rejection in model predictive control by randomized algorithms

In this paper we consider model predictive control with stochastic disturbances and input constraints. We present an algorithm which can solve this problem approximately but with arbitrary high accuracy. The optimization at each time step is a closed loop optimization and therefore takes into account the effect of disturbances over the horizon in the optimization. Via an example it is shown that this gives a clear improvement of performance although at the expense of a large computational effort

State synchronization of linear and nonlinear agents in time-varying networks

This paper studies state synchronization of homogeneous time-varying networks with diffusive full-statecoupling or partial-state coupling. In the case of full-state coupling, linear agents as well as a class of nonlineartime-varying agents are considered. In the case of partial-state coupling, we only consider linear agents,but, in contrast with the literature, we do not require the agents in the network to be minimum phase or atmost weakly unstable. In both cases, the network is time-varying in the sense that the network graph switcheswithin an infinite set of graphs with arbitrarily small dwell time. A purely decentralized linear static protocolis designed for agents in the network with full-state coupling. For partial-state coupling, a linear dynamicprotocol is designed for agents in the network while using additional communication among controller variablesusing the same network. In both cases, the design is based on a high-gain methodology

Properties of recoverable region and semi-global stabilization in recoverable region for linear systems subject to constraints

This paper investigates time-invariant linear systems subject to input and state constraints. It is shown that the recoverable region (which is the largest domain of attraction that is theoretically achievable) can be semiglobally stabilized by continuous nonlinear feedbacks while satisfying the constraints. Moreover, a reduction technique is presented which shows, when trying to compute the recoverable region, that we only need to compute the recoverable region for a system of lower dimension which generally leads to a considerable simplification in the computational effort

Optimal control of linear, stochastic systems with state and input constraints

In this paper we extend the work presented in our previous papers (2001) where we considered optimal control of a linear, discrete time system subject to input constraints and stochastic disturbances. Here we basically look at the same problem but we additionally consider state constraints. We discuss several approaches for incorporating state constraints in a stochastic optimal control problem. We consider in particular a soft-constraint on the state constraints where constraint violation is punished by a hefty penalty in the cost function. Because of the stochastic nature of the problem, the penalty on the state constraint violation can not be made arbitrary high. We derive a condition on the growth of the state violation cost that has to be satisfied for the optimization problem to be solvable. This condition gives a link between the problem that we consider and the well known $H_\infty$ control problem

Decentralized control with input saturation: a first step toward design

This article summarizes important observations about control of decentralized systems with input saturation and provides a few examples that give insight into the structure of such systems

Internal stabilization and external LpL_p stabilization of linear systems subject to constraints

Having studied during the last decade several aspects of several control design problems for linear systems subject to magnitude and rate constraints on control variables, during the last two years the research has broadened to include magnitude constraints on control variables as well as state variables. Recent work by Han et al. (2000), Hou et al. (1998) and Saberi et al. (2002) considered linear systems in a general framework for constraints including both input magnitude constraints as well as state magnitude constraints. In particular, Saberi et al. consider internal stabilization while Han et al. consider output regulation in different frameworks, namely a global, semiglobal, and regional framework. These problems require very strong solvability conditions. Therefore, a main focus for future research should focus on finding a controller with a large domain of attraction and some good rejection properties for disturbances restricted to some bounded se

A behavioral approach to H2H_2 optimal control

In this paper we consider a formulation of the $H_2$ optimal control problem in which the controller is not viewed as an operator that transforms measurements to controls. Instead, the controller is assumed to be a device that constrains a set of a-priori specified interconnection variables so as to achieve a controlled system whose manifest variables have a free component and satisfy an upperbound on the $L_2$ norm of its impulse response. We formalize and solve this problem in a behavioral setting, that is, without reference to system representations and input-output structures. The equivalance between the $H_2$ control problem, the generalized $H_2$ control problem and the LQ optimal control problem are established