The term “dynamic system” (either natural or man-made) may refer to a system whose response at any given time is a function of its input stimuli, its current state, and the current time. Such systems may range from simple to highly complex systems. Examples of dynamic systems include a falling body, the rotation of the earth, bio-mechanical systems (muscles, joints, etc.), bio-chemical systems (gene expression, protein pathways), weather and climate pattern systems, etc. Examples of man-made or engineered dynamic systems may include a bouncing ball, a spring with a mass tied to its end, automobiles, airplanes, control systems in major appliances, communication networks, audio signal processing, nuclear reactors, stock markets, etc.
Professionals from diverse areas such as engineering, science, education, and economics may build mathematical models of dynamic systems in order to better understand system behavior as it changes with the progression of time. The models may aid in building better systems in terms of a variety of performance measures, such as quality, time-to-market, cost, speed, size, power consumption, robustness, etc. The models may also aid in analyzing, debugging, repairing, and predicting performance of existing systems. The models may also serve an educational purpose of educating others on the basic principles governing physical systems. The models and results may often be used as a tool for communicating information about different systems.
The process of linearization may refer to the approximation of complex dynamic system models (i.e., non-linear models) that yields a linear model, which may be used by various engineering analysis tools. After linearization, the approximation of a complex model may be accurate for regions near an operating point. The operating point of a model may define its total “state” at any given time. For example, for a model of a car engine, the operating points may be typically described by variables such as engine speed, throttle angle, engine temperature, and the surrounding atmospheric condition. The behavior or what is typically known as the “dynamics” of the model may be generally affected by the values of the operating points. For example, in a car, the behavior of an engine can greatly vary if the car is being operated in the high elevations of Colorado or the low elevations of Florida.
A model that includes at least one algebraic equation to represent a dynamic system may have a singular mass matrix. A singular matrix refers to a square matrix that does not have a matrix inverse. A mass matrix refers to a matrix that appears in the system equation M(t,x,u){dot over (x)}=F(t,x,u), where M is a mass matrix, F is a forcing function, t, x and u are the variables of the system, and {dot over (x)} is the derivative of the variable x with respect to time.
The standard approach to linearizing a system having a singular mass matrix may result in the descriptor form of:E{dot over (x)}=Ax+Bu y=Cx+Du where u is an input vector, y is an output vector, x is a state vector and A, B, C, D and E are matrices. In the descriptor form, the matrix E may be singular. System designers are often more comfortable working with the following state-space representation.{dot over (x)}=Ax+Bu y=Cx+Du 
The state-space representation of a system having a singular mass matrix may be obtained by perturbation techniques. The perturbation techniques are a method that is used to find an approximate solution to a problem that cannot be solved exactly. For example, state variables and input variables are first perturbed and then the resulting perturbations to state variable derivatives and output variables are measured in the descriptor form of the system. Based on the measurement, the state-space representation of the system may be obtained. These perturbation techniques, however, are expensive and time-consuming.
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