# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 1 Introduction
Classical methods for determining the steady state to determine the start of the model of the multibody system are described by differential equations [35], state space, or transfer matrices [36]. However, the methodology proposed in this paper allows modeling and determining the structural properties of the multibody system without requiring its mathematical model. Furthermore, the inversion of the state matrix for multiport systems is carried out with the change of causality. The main contribution of this paper is to obtain the steady state response of an LTI stable system when the state matrix (\(\mathbf{A}\)) is invertible in a multibond graph approach. When  
Figure 1: Comparison between bond graphs and multibond graphs. this system is modeled by an MBGI, the representation in state space \(\left(\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}\right)\) can be calculated, and the dynamic and steady state responses during a simulation process are shown. The inverse of the state matrix \(\left(\mathbf{A}^{-1}\right)\) to get the mathematical description of the steady state using an MBGI is required. However, if the corresponding MBGD of this system is obtained, the steady state in a direct way is determined. This result requires calculating the matrix that relates the inputs of the system with the state variables \(\left(\mathbf{B}^{*}\right)\) and the outputs of the system \(\left(\mathbf{D}^{*}\right)\) where \(\left(\mathbf{B}^{*}=-\mathbf{A}^{-1}\mathbf{B}\right)\) and \(\left(\mathbf{D}^{*}=\mathbf{D}-\mathbf{C}\mathbf{A}^{-1}\mathbf{B}\right)\). Hence, the steady state for the state \(\left(\underline{x}_{\text{ss}}\right)\) and output \(\left(\underline{y}_{\text{ss}}\right)\) variables is defined by \(\underline{x}_{\text{ss}}=B^{*}\underline{u}_{\text{ss}}\) and \(y_{\text{ss}}=D^{*}\underline{u}_{\text{ss}}\), respectively, where \(\underline{u}_{\text{ss}}\) is the steady state for the system inputs. Therefore, the change from integral to derivative causality determines the inversion of the state matrix. The great advantage of this paper with respect to the traditional algebraic approach is that the steady state can be obtained without the need for the mathematical model in the state space of the system. The matrices \(B^{*}\) and \(D^{*}\) that determine the steady state are obtained in a direct way from the MBGD. Furthermore, the inverse of the state matrix is not required, which is shown in Figure 2.  
References [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31] used multibond graphs for the modeling and simulation of systems without requiring the calculation of the steady state response. Furthermore, References [32; 33] modeled and simulated multibody systems using multibond graphs. However, a comparison of this paper with these references cannot be made. Moreover, the systems of these references can be case studies for the determination of the steady state response, as long as these systems have the conditions required by this paper. Given a system, it is desired that all the storage elements have integral causality because derivative causality determines linear dependence. However, derivative causality to obtain the properties of structural controllability and structural observability has been used. In this paper, determining the steady state behavior is applied. Derivative causality allows changing the inputs and outputs of the storage elements that result in interesting properties of the systems. For example, in [37; 38], the invertibility of the state matrix of an LTI system was obtained. In [39], the mixture of the integral and derivative causality for a singularly perturbed LTI system determined the quasi-steady state model. The advantage of this approach is the symbolic determination of the steady state response, and this result can be useful for the analysis or synthesis of systems. Mainly multibond graphs have been used for modeling, and the motivation to develop this paper is to link many procedures and tools for single bond graphs to multibond graphs.