A Parametric Method To Design Interval Observers

Based on the need to solve this kind of problem that physical state is difficult to measure in control engineering,the research of state reconstruction problem is promoted.Establishing a set of dynamic systems,the interval observer realizes the estimation of the upper and lower bounds of the state respectively,breaks through the characteristics of the observer structure in the traditional sense,and effectively solves the problem of the state reconstruction of the uncertain system.In addition to the structural breakthroughs,the design process of interval observer introduces some information of uncertainty,which makes the observer extremely inclusive and simple in design method for disturbance.Nevertheless,the theory research and application of interval observer is far less rapid and deeper than the development of first-order system observers,and there are still theoretical gaps in many aspects.This paper continues the framework of interval observer design,which is based on the monotone system theory,to extend parametric design method of interval observer,and the possibility of optimization performance about observers is analyzed on the basis of the characteristics of the proposed method.The main research work of this paper is as follows:1.For a class of linear time-invariant systems,full-order interval observer is constructed on the basis of a linear transformation.The characteristic of this method is that the synthesis design of interval observer,that is to construct cooperative and stable error system,is transformed to the mathematical problem of solving a class of the time-invariant Sylvester equations.Finally,the restriction about the existence of observer gain matrix is solved effectively.Meanwhile,in view of the research results already available and by making full use of the rich degree of freedom in the parametric method,the design method of interval observer with H_?-robustness performance index and expected pole configuration is proposed with adjustable observation rate and better interval observation effect.2.Based on the useful information in system output,the design problem of reduced-order interval observers is studied for a class of linear time-invariant systems with the unknown but bounded disturbances in order to reduce the complexity of the observer design.Utilizing both the decoupling matrix and the diagonal matrix,the parametric design method of the interval observer is effectively realized on the premise of ensuring the existence of the observer gain matrix.Meanwhile,a detailed design procedure is provided.Similarly,for the specific design objectives and low design complexity,the concept,the existence conditions and design method of the functional interval observer are considered for the first time in this paper.By a novel dynamic system,the functional interval observer is directly constructed to achieve the interval estimation,and different from the linear transformation method,it provides a new idea of the interval estimation.3.Both advantages and disadvantages of the Lyapunov transformation approach to design interval observers for the time-varying systems are discussed in this paper.Because of the restrictive demanding of the observed systems and transformation matrix,the characteristics of Lyapunov transformation,the design problem about interval observer of time-varying system is equivalent to a mathematical problem of solving a class of differential equations.It avoids complicated Lyapunov transformation,reduces the numerical conservatism,and is suitable for the general linear time-varying systems.4.Consder the complicated systems,a class of generalized time-varying systems and a class of switched systems,as examples,and on the basis of the existing research results,the parametric design methods of interval observer for two types of systems are realized by solving Sylvester matrix equation.Firstly,using the matrix transformation,a unified general form of error dynamic system is obtained,which avoids the stabilization problem of complex analysis and design and reduces the calculation amount effectively for switched systems.Secondly,under the constrained equivalent transformation,the conditions of the positive system of the descriptor time-varying system are provided.Based on this condition,the interval observation of a class of descriptor systems is achieved,which complements the parametric design of the interval observer about complicated systems.