微分代数系统的无源性

考察了微分代数系统的无源性的控制问题.提出了微分代数系统无源的定义以及KYP特性的定义.利用类似微分几何理论的方法,通过引入微分代数系统的M导数,推出了微分代数系统无源与KYP特性等价的定理和微分代数系统无源性的充分必要条件.最后给出了无源控制器存在的条件.M导数的方法可以看作是L导数方法的延伸.本文所获得的一系列结果,使来自于物理系统的无源概念与控制理论有机结合起来.     

A new two-stage Rosenbrock scheme for differential-algebraic systems

In this work conditions for the two-stage Rosenbrock system parameters with third-order accuracy for the differential-algebraic systems of index 1 have been obtained. The coefficients of the system are complex numbers. A new system with L2 stability has been constructed with third-order accuracy for differential-algebraic systems and with fourth-order accuracy for differential systems. The convergence of this scheme is proved. This scheme was tested on standard stiff tests and compared with the already known schemes of the same class.     

Implicit-system approach to the robust stability for a class of singularly perturbed linear systems

Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied. The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero. The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only. Our results can be extended to other singular perturbation problems for DAEs of more general form.     

Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems

We consider the design of a feedback control law for control systems described by a class of nonlinear differential-algebraic equations so that certain desired outputs track given reference inputs. The nonlinear differential-algebraic control system being considered is not in state variable form. Assumptions are introduced and a procedure is developed such that an equivalent state realization of the control system described by nonlinear differential-algebraic equations is expressed in a familiar normal form. A nonlinear feedback control law is then proposed which ensures, under appropriate assumptions, that the tracking error in the closed loop differential-algebraic system approaches zero exponentially. Applications to simultaneous contact force and position tracking in constrained robot systems with rigid joints, constrained robot systems with joint flexibility, and constrained robot systems with significant actuator dynamics are discussed.     

Robust stability of differential-algebraic equations with an arbitrary unsolvability index

Consideration was given to the linear stationary systems of differential-algebraic equations with an arbitrarily high unsolvability index. Conditions were established guaranteeing the internal structure of the system at hand against the internal structural modifications caused by the perturbations of the matrix coefficients. Under the assumptions of structural persistence, the sufficient conditions for robust stability were obtained, and the values of real stability radii were given.     

The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

Hybrid systems of differential-algebraic equations – Analysis and numerical solution

We consider hybrid systems of differential-algebraic equations and present a general framework for general nonlinear over- and underdetermined hybrid systems that allows the analysis of existence and uniqueness and the application of index reduction methods for hybrid differential-algebraic systems. A particular difficulty in the numerical simulation of hybrid systems is (numerical) chattering, i.e., fast oscillations between modes of operations. A regularization technique using sliding modes allows us to regularize the system behavior in the case of chattering. Further, we show how chattering behavior during the numerical solution can be prevented using sliding mode simulation. The advantage of the sliding mode simulation is illustrated by numerical examples.     

Combinatorial optimization model and MIP formulation for the structural analysis of conditional differential-algebraic systems

In this paper we consider the structural analysis problem for differential-algebraic systems with conditional equations. This problem consists, given a conditional differential-algebraic system, in verifying if the system is structurally nonsingular for every state, and if not in finding a state in which the system is structurally singular. We give a formulation for this problem as an integer linear program. This is based on a transformation of the problem into a matching problem in an auxiliary graph. We also show that the linear relaxation of that formulation can be solved in polynomial time. Using this, we develop a Branch-and-Cut algorithm for solving the problem and present some experimental results.     

Reprint of “The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions”

The Kalman–Yakubovich–Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution of this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. Consequences of our results are bounded real and positive real lemmas for differential-algebraic systems.     

具有奇异位置的多体系统动力学方程的改进算法

多体系统进行数值仿真时,很多选择了微分代数混合方程作为多体系统动力学数学模型.本文在现有的约束稳定化理论基础上,提出了针对具有奇异位置的多体系统动力学方程的改进算法.算法通过修正速度违约和控制稳定项,讨论了具有奇异位置的微分代数混合方程的数值仿真问题并给出了稳定项中相关系数的建议值,从而有效克服了求解混合方程时因为构型奇异给计算造成的困难.算例分别采用改进算法与ADAMS软件进行仿真,计算结果的比较表明了改进算法的有效性.本文给出的基于能量守恒的能量差曲线也证明了改进算法的有效性.     

Variable structure control for three-variable autocatalytic reaction

In this paper, two irreversible exothermic autocatalytic reactions which carry out in continuous stirred tank reactor (CSTR) are considered. A differential-algebraic system is applied to model these chemical reactions. The stability and the dynamic behavior are studied for the differential-algebraic system. The Hopf bifurcation appears when the parameter exceeds a critical value. In order to eliminate this complex behavior, the differential-algebraic system is described by a single-input and single-output system with parameter varying within definite intervals, and then variable structure control with sliding mode based on a special power reaching law is designed to stabilize this chemical system. Numerical simulations are given to illustrate the effectiveness of the method.     

Co-simulation for performance prediction of integrated building and HVAC systems – An analysis of solution characteristics using a two-body system

Integrated performance simulation of buildings and heating, ventilation and air-conditioning (HVAC) systems can help in reducing energy consumption and increasing occupant comfort. However, no single building performance simulation (BPS) tool offers sufficient capabilities and flexibilities to analyze integrated building systems and to enable rapid prototyping of innovative building and system technologies. One way to alleviate this problem is to use co-simulation to integrate different BPS tools. Co-simulation approach represents a particular case of simulation scenario where at least two simulators solve coupled differential-algebraic systems of equations and exchange data that couples these equations during the time integration.This article analyzes how co-simulation influences consistency, stability and accuracy of the numerical approximation to the solution. Consistency and zero-stability are studied for a general class of the problem, while a detailed consistency and absolute stability analysis is given for a simple two-body problem. Since the accuracy of the numerical approximation to the solution is reduced in co-simulation, the article concludes by discussing ways for how to improve accuracy.     

一类非线性微分代数系统的L2-增益分析及状态反馈∞控制

分析了一类非线性微分代数系统的L₂增益, 并讨论了一类状态反馈H_∞控制问题. 在一定条件下给出了控制器的设计, 并保证了闭环指数为 1且零解渐近稳定.     

一类生物经济系统的分析与控制

研究了一类利用微分代数方程描述的生物经济系统的分岔、脉冲及控制问题。基于微分代数系统分岔、矩阵柬指数等理论，研究了该类生物经济系统在参数临界状态下，出现的奇异诱导分岔及脉冲行为，以及构造该系统的广义T-S模糊模型；并利用T-S模糊广义系统控制方法，设计状态反馈控制器，抑制种群的变化，使系统达到稳定状态，从而消除系统中的奇异诱导分岔及脉冲行为。数值仿真实例说明了该控制器的有效性。     

Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems

 We study over- and underdetermined systems of nonlinear differential-algebraic equations. Such equations arise in many applications in circuit and multibody system simulation, in particular when automatic model generation is used, or in the analysis and solution of control problems in the behavior framework.?We give a general (local) existence and uniqueness theory and apply the results to analyze when nonlinear implicit control problems can be made regular by state or output feedback.?The theoretical analysis also leads immediately to numerical methods for the simulation as well as the construction of regularizing feedbacks. Date received: February 21, 2000. Date revised: November 14, 2000.     

Computations with inverse Runge-Kutta schemes

A new subclass of schemes is considered formally reduced to the class of fully implicit Runge-Kutta schemes possessing outstanding accuracy and stability characteristics. The implementation details of the iterative algorithm for solving stiff systems of ODE and differential-algebraic systems of index 1 by means of the proposed schemes are given.     

Bisimulation equivalence of differential-algebraic systems

In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE ? A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.     

Disturbance decoupled estimation for linear differential-algebraic systems

We study the disturbance decoupled estimation problem for linear differential-algebraic systems which are not necessarily regular. We introduce the notion of partial state observers following a recent approach to observer design motivated by considerations for behavioural systems. In our framework, a partial state observer is itself a differential-algebraic system. We derive a characterisation for the existence of (asymptotic) partial state observers. Exploiting the freedom in the proposed observer design, we derive a solution of the disturbance decoupled estimation problem. The characterisation of solvability is obtained via geometric conditions in terms of the generalised Wong sequences.     

Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters.