In this case, the system of first-order differential equations can be represented as a matrix equation, that is. Fortunately, over a sufficiently small operating range (think tangent line near a curve), the dynamics of most The most common in control systems being 'saturation' in which an element of the system reaches a hard physical limit to its These nonlinearities arise in many different ways, one of

In other words, is typically some complicated function of the state and inputs.

In reality, nearly every physical system is nonlinear. The second common assumption concerns the linearity of the system. The state variables,, and control inputs,, however, may still be time dependent. This is often a very reasonable assumption because the underlying physical laws themselves do not typically depend on time.įor time-invariant systems, the parameters or coefficients of the function are constant. , then the system is said to be time invariant. The function does not depend explicitly on time, i.e. There are two common simplifications which make the problem more tractable. The relationship given in Equation (1) is very general and can be used to describe a wide variety of different systems unfortunately, The system order usually corresponds to the number of independent energy storage elements in the system. is referred to as the system order and determines the dimensionality of the state-space. , required in order to capture the 'state' of a given system and to be able to predict the system's future behavior (solve Though the state variables themselves are not unique, there is a minimum number of state variables, The state at any future time,, may be determined exactly given knowledge of the initial state,, and the time history of the inputs,, between and by integrating Equation (1). is the vector of external inputs to the system at time, and is a (possibly nonlinear) function producing the time derivative (rate of change) of the state vector,, for a particular instant of time. For instance, in a simple mechanical mass-spring-damper system, the two state variables could be the position and velocity In the above equation, is the state vector, a set of variables representing the configuration of the system at time. For many physical systems, this rule can be stated asĪ set of first-order differential equations: Entering Transfer Function Models into MATLABĭynamic systems are systems that change or evolve in time according to a fixed rule.

Entering State-Space Models into MATLAB.

Scilab state space to transfer function