Linear quadratic regulators with eigenvalue placement in a vertical strip

A computational method is presented for finding the linear quadratic regulator such that the optimal closed-loop system has eigenvalues lying within a vertical strip in the complex plane. The proposed method is suitable for the two-stage optimal design of two-time scale systems.     

Linear quadratic regulators with eigenvalue placement in a specified region

A linear optimal quadratic regulator is developed for optimally placing the closed-loop poles of multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at ±π/2k (k = 2 or 3) from the negative real axis with a sector angle ≤π/2, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane. Also, a shifted sector method is presented to optimally place the closed-loop poles of a system in any general sector having a sector angle between π/2 and π. The optimal pole placement is achieved without explicitly utilizing the eigenvalues of the open-loop system. The design method is mainly based on the solution of a linear matrix Lyapunov equation and the resultant closed-loop system with its eigenvalues in the desired region is optimal with respect to a quadratic performance index.     

Linear systems with bounded inputs: global stabilization with eigenvalue placement

This work presents a technique for obtaining a bounded continuous feedback control function which stabilizes a linear system in a certain region. If the open-loop system has no eigenvalues with positive real part, the region of attraction of the resulting closed-loop system is all ℝⁿ, i.e., the feedback control is a global stabilizer; otherwise, the region contains an invariant (‘cylindric-like’) set where the controller does not saturate. The proposed control is a linear-like feedback control with state-dependent gains. The gains become implicitly defined in terms of a nonlinear scalar equation. The control function coincides in an ellipsoidal neighbourhood of the origin with a linear feedback law which is a solution of a linear quadratic regulator problem. This design allows eigenvalue placement in a specified region. © 1997 by John Wiley & Sons, Ltd.     

Comments on “Linear quadratic regulators with eigenvalueplacement in a vertical strip”

In the original paper by Shieh, Hani and McInnis (ibid., vol.31, p.241-3, 1986), a method was proposed for the design of linear feedback control such that all the poles of the closed-loop system lie in a vertical strip. There are errors in the conditions for the eigenvalues to lie in the open vertical strip and also in the proof of a theorem. We correct these inaccuracies     

Probabilistic robust design with linear quadratic regulators

In this paper, we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators (LQR). We consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism. The approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters. In particular, we propose an iterative algorithm for designing a controller which is based upon subgradient iterations. At each step of the sequence, we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the LQR problem. The main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps. In addition, at a fixed step, we compute a lower bound of the probability that a quadratically stabilizing controller is found.     

Arbitrary robust eigenvalue placement by a static-state feedback

It is demonstrated that robust eigenvalue placement in the disk of an arbitrary radius r centered at -2r can be achieved by a static-state feedback for systems with so-called matched perturbations of uncertain parameters in the state coefficient matrix A (i.e. with perturbations of A in the range of the input matrix B). This implies, in particular, that such systems can be robustly stabilized with an arbitrarily fixed degree of exponential decay, and thus it extends previously known results on robust stabilization without eigenvalue placement conditions. This result is in sharp contrast with the case of general perturbations in either A or B or both, where there are limits for the degree of exponential stabilizability which depend on the size of perturbations     

The quadratic eigenvalue problem in electric power systems

The application of the quadratic eigenvalue problem in electrical power systems is reviewed. The spectrum and pseudospectrum of an electrical power system are defined.     

Linear quadratic control with stability degree constraint

The problem of static state feedback Linear Quadratic (LQ) optimal control subject to a prescribed degree of stability for the closed-loop system is considered in this paper. A necessary optimality condition is given via the Lagrange multiplier method. A globally convergent algorithm is provided to solve the optimization problem. It is shown that the algorithm recovers the standard LQ feedback gain provided the desired stability degree is small enough to be within the range by the standard LQ design. As for other cases the optimum occurs on the boundary of the α-region. A numerical example shows that the proposed algorithm provides a better design compared to the existing methods.     

Linear quadratic differential games with cheap control

This paper considers a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero. It is shown that if a certain minimum phase condition is satisfied then the value of the game will approach zero as the weighting on the minimizing control approaches zero.     

用改进的差分式Hopfield网络实现线性二次型最优控制

为解决差分式Hopfield网络能量函数的局部极小问题,本文对之改进得到一种具有迭代学习功能的线性差分式Hopfield网络.理论分析表明,该网络具有稳定性,且稳定状态使其能量函数达到唯一极小值.基于线性差分式Hopfield网络稳定性与其能量函数收敛特性的关系,本文将该网络用于求解多变量时变系统的线性二次型最优控制问题.网络的理论设计方法表明,网络的稳态输出就是欲求的最优控制向量.数字仿真取得了与理论分析一致的实验结果.     

Linear quadratic performance with worst case disturbance rejection

The method of the calculus of variations and the maximum principle are preposed for the design of 'LQR' controllers with the worst case disturbance rejection for a linear time-varying (LTV) plant on finite horizon. The disturbance is bounded by either the windowed L     

Linear quadratic suboptimal control with static output feedback

This paper considers the design of a stabilizing static output feedback gain which keeps linear quadratic (LQ) performance index less than a specified number (we call this an ‘LQ suboptimal controller’). Existence of such a controller is shown to be equivalent to the existence of a positive-definite matrix P such that P satisfies two linear matrix inequalities (LMIs) while P⁻¹ satisfies another LMI. All LQ suboptimal controllers are explicitly parametrized by the freedom in the choice of the positive-definite matrix P satisfying the LMIs, and an arbitrary positive scalar and an arbitrary matrix of fixed dimension with a norm bound. A modified version of the min/max algorithm is given to find a positive-definite solution P to the LMIs.     

Optimal regulators using time-weighted quadratic performance index with prespecified closed-loop eigenvalues

Optimization of linear time-invariant systems, using a given time-weighted quadratic performance index, is investigated for the prespecified closed-loop eigenvalues. The necessary conditions have simple forms and this algorithm gives a considerable amount of computational time savings compared to the previous algorithms [1]-[2].     

Linear quadratic regulation of systems with stochastic parameter uncertainties

In this paper, we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency, in a statistical sense, with the results obtained from Monte-Carlo based designs.     

General eigenvalue placement in linear control systems by output feedback

Consider the time-invariant system E[xdot] = Ax + Bu, y = Cx where E is a square matrix that may be singular. The problem is to find constant matrices K and L, such that the feedback law u = Ky+L[ydot] yields x = exp (λt)v_i (where v_i is some constant vector) for some preassigned λ_i (i=l, 2, [tdot], r). This problem is equivalent to that of finding K and L which makes a preassigned λ _i an eigenvalue corresponding to the general eigenvalue problem {λ(E ? BLC) ? (A + BKC)}v=0. Using matrix generalized inverses, a method is developed for the construction of a linear system of equations from which the elements of K and L may be computed.     

Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time

This work develops asymptotically optimal controls for a class of discrete-time hybrid systems involving singularly perturbed Markov chains having weak and strong interactions. The state space of the underlying Markov chain is decomposed into a number of recurrent classes and a group of transient states. Using a hierarchical control approach, by aggregating the states in each recurrent class into a single state, a continuous-time quadratic limit control problem in which the resulting limit Markov chain has much smaller state space is derived. Using the optimal control for the limit problem, a control for the original problem is constructed, which is shown to be nearly optimal. Finally, a numerical example is given to demonstrate the effectiveness of the approximation scheme.     

Linear quadratic regulation for linear time-varying systems with multiple input delays

This paper studies the classic linear quadratic regulation (LQR) problem for both continuous-time and discrete-time systems with multiple input delays. For discrete-time systems, the LQR problem for systems with single input delay has been studied in existing literature, whereas a solution to the multiple input delay case is not known to our knowledge. For continuous-time systems with multiple input delays, the LQR problem has been tackled via an infinite dimensional system theory approach and a frequency/time domain approach. The objective of the present paper is to give an explicit solution to the LQR problem via a simple and intuitive approach. The main contributions of the paper include a fundamental result of duality between the LQR problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system. The duality allows us to obtain a solution to the LQR problem via standard projection in linear space. The LQR controller is simply constructed by the solution of one backward Riccati difference (for the discrete-time case) or differential (for the continuous-time case) equation of the same order as the plant (ignoring the delays).     

Linear quadratic regulation problem for discrete-time systems with multi-channel multiplicative noise

This paper investigates the linear quadratic regulation (LQR) problem for discrete-time systems with multiplicative noise. Multiplicative noise is usually assumed to be a scalar in existing literature works. Motivated by recent applications of networked control systems and MIMO communication technology, we consider multi-channel multiplicative noise represented by a diagonal matrix. We first show that the finite horizon LQR problem can be solved using a generalized Riccati equation. We then prove the convergence of the generalized Riccati equation under the conditions of stabilization and exact observability, and obtain the solution to the infinite horizon LQR problem. Finally, we provide a numerical example to demonstrate the proposed approach.     

Linear quadratic regulation for systems with time‐varying delay

In this paper we study the linear quadratic regulation (LQR) problem for discrete‐time systems with time‐varying delay in the control input channel. We assume that the time‐varying delay is of a known upper bound, then the LQR problem is transformed into the optimal control problem for systems with multiple input channels, each of which has single constant delay. The optimal controller is derived by establishing a duality between the LQR and a smoothing estimation for an associated system with a multiple delayed measurement. Copyright © 2009 John Wiley & Sons, Ltd.     

Linear eigenvalue code for edge plasma in full tokamak X-point geometry

A new code is presented for solving linear eigenvalue problems from fluid models of the edge plasma of tokamaks. The 2DX code solves linearized fluid equations in a 2D cross-section of the plasma, with toroidal mode number resolving the third dimension. Geometry capabilities include both closed and open field lines, allowing solution of X-point problems as well as a variety of other toroidal and cylindrical systems. The code generates a pair of sparse matrices forming a generalized eigenvalue problem which is then solved using a standard sparse eigensolver package. Use of a specialized equation parser permits a high degree of flexibility in both equations and coordinate systems. Both analytic and full geometry benchmark cases are presented.