Observability Of Second-order Time-Varying Linear Non-autonomous Systems

Lv Xianrui and Huang Qingdao [17] put forward the definition of the first-order linear time-invariant systems and their observability.In this paper,we give the definition of the first-order linear time-varying systems and their observability.In [25],Shi Haibin proposed a necessary and sufficient condition for the observability of the first-order linear non-autonomous time-varying systems ?Gram Matrix invertibility condition and provided the corresponding proof.Since the complex calculation and limited application scope of this condition,it mainly focuses on theoretical research.In order to overcome these difficulties,we present another necessary and sufficient condition for the observability of the first-order linear non-autonomous time-varying systems.Define a sequence of matrices (?) as follows,(?)where(?),and make its rank of column reach full,i.e(?)This condition can perfectly avoid the tedious calculation of the integral and the state transition matrix,so that we can better study the observability of the firstorder linear non-autonomous time-varying systems.Then,in order to clearly show the advantages and disadvantages of these two sufficient and necessary conditions for the observability of the first-order linear non-autonomous time-varying systems,this paper gives some examples to show the differences between the two methods above.In addition,most of the current researchers on observability are about the first-order linear time-varying systems.With the continuous improvement of scientific level,we find more and more systems of the second-order are widely used in various fields.The second-order systems can describe the dynamic characteristics of many natural phenomena and have many applications in engineering fields such as mechanics [2,7,11,26,40],aerospace[5,24,34,46] control and communication [18,23,39].For the study of the secondorder linear systems,many researchers are devoted to transforming them into the constant cases.However,many real systems are clearly time-varying,and then serious errors will occur if we continue to describe them with the constant model.Duan and Hu [8] give the general form and the definition of observability of the second-order linear non-autonomous time-varying systems as follows(?)Under some certain assumptions,they proposed that if let(?)then we can transform the second-order systems into the first-order systems,that is(?)where(?)As a result,we can study the observability of the second order linear non-autonomous time-varying systems by the necessary and sufficient conditions of the existing observability of the first order systems.Based on this,we consider whether we can apply the Gram Matrix invertibility condition and the condition of the rank of the matrix sequence given by the first-order systems to the above form of the secondorder systems,so as to obtain the necessary and sufficient conditions for the observability of a second-order linear non-autonomous system.The necessary and sufficient conditions for the observability of the secondorder systems are : there is finite time (?),so that the following Gram Matrix invertibility(?)where(?)denotes the state transition matrix of the second-order systems.Define the following matrix sequence(?) as(?)where(?)The necessary and sufficient conditions for the observability of the second-order systems are rank(?)In [8],the application of two sufficient conditions for the observability of the second-order linear non-autonomous system in satellite orbit system were mentioned.In this paper,we use two sufficient conditions proposed by Duan and Hu as well as the sufficient and necessary conditions proposed in this paper respectively to study the observability of T-H equation,which is introduced by Yang Leping [44] in a combination with related systems,and then get a satisfying conclusion.