# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 1 Introduction  
In the analysis and design of control systems, the most important specifications are: stability, transient response, and steady state response, considering robust designs, as well as economic and social aspects [1]. The dynamics of a physical system are linked to the energy storage elements; when the dynamic performance is over, the steady state is reached. The steady state response is an important characteristic of a system; for example, some equipment in electrical machines or in power electrical systems requires to know the steady state values for calibration. An interesting steady state reference to simulate groundwater flow in unconfined water can be found in [2]. Furthermore, a modified polynomial expansion algorithm for solving the steady state Allen-Cahn equation for heat transfer processes was proposed in [3]. Some of the methodologies in system modeling such as lumped modeling with circuit elements, finite elements, and bond graphs can be found. The lumped modeling has been used to represent microsystems, which are very small systems. The characteristics of circuit analogies also permit efficient modeling of the interaction between the electronic and non-electronic components of a microsystem. A further advantage of circuit models is that they are intrinsically correct from an energy point of view [4]. A bond graph model of a system determines the power interactions with connecting lines, \"bonds\", which carry both power variables and causalities between variables. Bond graph modeling has been applied to various areas, for example: in [5], an uncertain bond graph based fault detection and isolation and an adaptive enhanced unscented Kalman filter based fault estimation and sequential prognosis were developed for an electric scooter with parameter uncertainties. A versatile approach to the synthesis and design of a bond graph model and a Kalman filter observer for an industrial back-support exoskeleton was presented in [6]. The modeling of bond graph buck converter systems wasanalyzed in [7]. Bond graph modeling and the kinematic and dynamic characteristics of a piezoelectric-actuated micro-/nano-compliant platform system were investigated in [8]. In [9], the development of a bond graph model for the simulation of a multi-axis low cost accelerometer was successful in the forecast of the accuracy of velocity and displacement reconstruction from imperfect acceleration measurements, as has been corroborated by experimental results. Lumped modeling with circuit elements and the bond graph have similar characteristics such as the use of generalized variables and can model systems with different energy domains. However, the models in the bond graph determine the static and dynamic relationships in a graphical form that can be linked to the causality of the elements and that in this paper are used to invert the state matrix and obtain the steady state response. On the other hand, in the finite element method, a structure is divided into finite elements of a simple geometry by means of suitable sections. This method can perform the dynamic analysis as the determination of the motion of the mechanism as a function of time [10]. The finite element approach to mechanism analysis has shown many advantages over the others in the aspects of the generality and simplicity in the formulation the governing equations in [11]. Rather than describe the differences and advantages between the finite element method and bond graphs, these modeling methodologies can be complementary. A bond graph modeling approach that is equivalent to a finite element method was formulated in [12]. This formulation led to a new definition of the generalized displacements for a continuous system. In [13], it was shown how variables, specifically the temperature at each point in the deformation zone, can be modeled using a multi-element bond graph approach. Lumped parameter models cannot sufficiently describe the dynamics of many distributed systems, such as a continuous rotating shaft with bending.