一种新的最优极点配置方法

本文从LQ逆问题着眼提出了一种新的最优极点配置方法,推导了加权矩阵Q和R与开环特征多项式、最优闭环特征多项式之间的关系。只要给定一组期望的闭环极点,即可确定与之对应的加权矩阵Q和R,从而得到一个具有指定极点的最优控制系统。     

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].     

Some considerations for estimator-based compensator design

In this paper some criteria are presented for dividing the closed-loop system poles of a feedback system between the estimator and the controller when using an estimator-based compensator. The criteria are based on frequency response considerations, but are related to some well-known facts in the time domain. It is well known, for example, that the closed-loop system poles are the poles of the estimator in union with the controller poles. It is also well known that the time response of the closed-loop plant states due to a command depends only on the closed-loop control poles, and the response of the estimate error depends only on the closed-loop estimator poles. In the frequency domain, the following can be said: the open-loop compensator transfer function from the sensor measurement to the control and the closed-loop transfer function from the sensor measurement to the system output are both independent of how the closed-loop poles are distributed between the controller and the estimator, but the closed-loop transfer function from the command to the control or system output is not. Placing the slower closed-loop poles in the controller causes the control gains to be smaller, which in turn causes the effect of the command signal to be attenuated, both at the control and at the closed-loop system output. Making use of these facts allows the closed-loop poles to be divided between the controller and the estimator on a more intelligent basis than ‘the estimator poles should be three times faster than the controller poles’. In an example, these concepts are applied to the design of a platform despin control system for a dual-spin satellite.     

On constraints in pole assignment by static feedback

A design method based on pole assignment considerations for an nth order linear time-invariant single-input p-output system by means of static (constant gain) feedback is presented for the case in which the rank p of an output matrix C is less than the number of system states n. The constraints in pole assignment are described in the form of linear relationships between the coefficients of the closed-loop characteristic polynomial and some open-loop system parameters. It has been shown that the problem of pole assignment is equivalent to the problem of solving a set of n linear equations with p unknowns in such a way that n – p equations are made linearly dependent by proper choice of coefficients of the closed-loop characteristic polynomial. As a result, all n system poles can be shifted to selected locations which satisfy the constraints. Of particular interest are the design methods for the cases when the number of arbitrarily assigned poles are equal to p and p – I.     

二次型最优离散系统的两个必要条件

本文给出了线性定常二次型最优离散系统的两个必要条件,得到了单输入单输出系统开环特征多项式系数a_i和最优闭环特征多项式系数b_i与权阵Q之间的直接关系,所得结论有益于设计一个具有指定闭环极点的最优控制系统。     

LQ regulator design method for system with delay based on spectral decomposition of the hamiltonian

A design method to construct an LQ regulator for a system with time-delay is proposed. It is based on a new technique for composing the solution of an infinite-dimensional Riccati equation using the spectral decomposition of the hamiltonian. This design method has the feature that the closed-loop poles can be calculated a priori from the open-loop poles and one scalar design parameter, so that we can choose the parameter to obtain an optimal control law which results in the desired degree of exponential stability of the closed-loop system.     

Linear quadratic regulators with eigenvalue placement in a horizontal strip

A method for optimally shifting the imaginary parts of the open-loop poles of a multivariable control system to the desirable closed-loop locations is presented. The optimal solution with respect to a quadratic performance index is obtained by solving a linear matrix Lyapunov equation.     

Remote Stabilization Via Communication Networks With a Distributed Control Law

In this note, we investigate the problem of remote stabilization via communication networks involving some time-varying delays of known average dynamics. This problem arises when the control law is remotely implemented and leads to the problem of stabilizing an open-loop unstable system with time-varying delay. We use a time-varying horizon predictor to design a stabilizing control law that sets the poles of the closed-loop system. The computation of the horizon of the predictor is investigated and the proposed control law explicitly takes into account an estimation of the average delay dynamics. The resulting closed loop system robustness with respect to some uncertainties on the delay estimation is also considered. Simulation results are finally presented.     

Compensation of uncertain continuous systems by using the internal model control principle

In this paper the design of compensators for uncertain continuous plants is investigated. The standard derived compensators are based on the application of the internal model control (IMC) method. The required a priori knowledge on the plant is rather weak, namely, an upper bound of the plant relative order, the numbers of the strictly unstable and critically unstable plant poles being integrators and upper and lower bounds of the amplitude-versus-frequency plot over the low frequency band in the case of minimum-phase open-loop systems. If the open-loop system has unstable zeros and/or poles then the above bounds are required to be known for a modified magnitude plot which substitutes the unstable zeros (poles) by stable poles (zeros) which are their complex-conjugate reflections on the left-hand plane. An absolute upper bound of the open-loop phase plot is obtained on a finite frequency interval which allows the closed-loop system to guarantee a prescribed relative stability in many practical situations. The method is dependent on the alternative design of phase lead/lag classical compensators and to indirect adaptive control situations where the adaptive identifier is used for the parametrization of the adaptive controller.     

Assessment of stability robustness for adaptive controllers

The problem of improving the stability characteristics of discrete-time SISO reduced order adaptive controllers is addressed in this note. A method is proposed to adjust the observer poles of an indirect adaptive scheme in order to ensure stability of the closed-loop system in spite of possible unstructured uncertainties (e.g., process-model order mismatch). Assuming that the identified model transfer function module converges within a previously defined uncertainty range, the frequency response-like compensation method is shown to lead to a stable design, provided that the reduced order estimated model is stable and stably invertible and the system is open-loop stable.     

The location of the continuous-time stationary Kalman filter poles

It is shown that the closed-loop poles of the continuous-time Kalman filter reside in a region in the left half of the complex plane that is confined by two concentric circles whose radii depend on the system matrices and the signal-to-noise ratio. This region includes the system open-loop poles and excludes the imaginary axis. In the case where the system dynamic matrix has a simple eigenstructure, this region possesses an additional boundary, that is parallel to the imaginary axis at a distance that varies with the signal-to-noise ratio.     

Constrained pole placement for linear systems using low-order output feedback controllers

The paper presents a simple approach to the problem of designing low-order output feedback controllers for linear continuous systems. The controller can place all of the closed-loop poles within a circle, C(- f , 1/ g ) , with centre at - f and radius of 1/ g in the left half s-plane. The design method is based on transformation of the original system and then applying the bounded-real-lemma to the transformed system. It is shown that subjected to the solvability of an algebraic Riccati equation (ARE), output feedback controllers can then be systematically derived. Furthermore, the order of the controller is low and equals only the number of the open-loop poles lying outside the circle. A step-by-step design algorithm is given. Numerical examples are given to illustrate the design method.     

On pole assignment in multi-input controllable linear systems

It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.     

Right half plane poles and zeros and design tradeoffs in feedback systems

This paper expresses limitations imposed by right half plane poles and zeros of the open-loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed-loop system. The limitations are determined by integral relationships which must be satisfied by these functions. The integral relationships are interpreted in the context of feedback design.     

Feedback control system synthesis for plants with large parameter variations

A synthesis procedure is developed for a dominant third-order system (as opposed to the usual dominant second-order system) where the size of the complex closed-loop pole region is not specifically restricted but is contained within a circular boundary. The plant considered is of third order, with a real pole at the origin and a pair of complex poles with negative real parts. The added real pole affords an additional degree of freedom with which to meet the system-time domain specifications. The compensation prescribed is a biquadratic network with complex zeros, and compensation poles are constrained to exhibit only second-order effects. The procedure is based on the association of the desired closed-loop pole parameter values with open-loop parameter values in the (Omega^{2},Sigma) plane, which is in actuality the natural-frequency-squared damping-constant plane. A constant gain factor is assumed, but this constraint may be removed if a tandem gain control system is employed. This assures that the gain variations are much slower than the system response. Relations between the desired nominal closed-loop poles and the plant region and compensation locations are derived for a prescribed circular closed-loop pole region. The nominal plant pole location is a design specification, while the allowable plant region depends on the closed-loop pole location and region, i.e., a change in size of one region is reflected as a proportional change in size of the plant closed-loop pole region. The remote compensation pole location is primarily determined by minimizing transient distortion while ensuring a specified degree of stability.     

LQ逆问题研究

本文研究了线性定常系统LQ逆问题解的存在性问题,给出了逆问题解的参数化公式,得 到了加权矩阵Q与开环、闭环特征多项式系数之间的解析关系.只要给定一组稳定的闭环极 点,即可确定与之对应的Q阵.     

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.     

Design of cascade controllers for zero assignment in multivariable systems†

The paper describes a method for the design of a constant cascade controller to assign some zeros and numerator coefficients of the (open or closed-loop) transfer function matrix of a linear multivariable system. The cascade controller does not affect the poles of the system which can be positioned prior to the zero assignment. The requirements on the transfer function matrix for arbitrary assignment of some zeros and numerator coefficients are discussed. The cascade controller is calculated from sets of simple linear equations and thus the method is computationally very efficient. An interactive computer programme is described for the computer-aided design of cascade controllers. Finally, two simple numerical examples of zero assignment in open-loop and closed-loop systems are worked out in detail to illustrate the method.     

H∞-almost disturbance decoupling with internalstability for linear systems subject to input saturation

For a linear system subject to input saturation and input-additive disturbances, we show that: (1) the H_∞-almost disturbance decoupling problem with local asymptotic stability is always solvable via state feedback as long as the system in the absence of saturation is stabilizable, no matter where the open-loop poles are; and (2) the H_∞-almost disturbance decoupling problem with semiglobal asymptotic stability is solvable via state feedback as long as the system in the absence of saturation is stabilizable with all its open-loop poles located in the closed left-half plane. The results generalize those in Lin et al. (1996) by not requiring the disturbance to be bounded by a known bound, or even bounded     

Closed-Loop Dynamics of Cooperative Vehicle Formations With Parallel Estimators and Communication

The control of cooperative formations of vehicles can be based on parallel estimation, where each vehicle determines its control action from a locally maintained estimate of the entire observable formation state. Vehicles may communicate with one another allowing the local estimates to incorporate information from other estimators in the formation. This paper studies the dynamics that arise in this situation and provides a complete analysis of the formation stability for this class of decentralized control problems. In the absence of communication, the local estimator-controllers' open-loop dynamics necessarily appear in the closed-loop system dynamics, giving a more stringent closed-loop stability condition than in the single controller case. The estimators achieve consensus if and only if the controllers' open-loop dynamics are stable. Communication amongst the estimators can be used to specify the complete system dynamics and we present a framework for the analysis and design of communicated information links in the formation. We relate the complete closed-loop system poles to the transmitter and receiver gains, and the spectral properties of the Laplacian of the graph describing the communication links within the formation. These results also apply to parallel estimation problems in other applications including power system control and redundant channel control architectures.