Hot to draw phase planes for differential equation systems

The general form of the first-order differential equation is as follows Some common examples include mass-damper systems and RC The important properties of first-, second-, and higher-order systems will beДЇirst-order systems are the simplest dynamic systems to analyze. It is the highest power of in the denominator of its transfer function. The order of a dynamic system is the order of the highest derivative of its governing differential equation. We can use the eig command to calculate the eigenvalues using either the LTI system model directly, eig(G), or the system matrix as shown below.

In fact, the poles of the transfer function are the eigenvalues of the system matrix

The stability of a system may also be foundДЇrom the state-space representation. Thus this system is stable since the real parts of the poles are both negative. The pole command, an example of which is shown below: The poles of an LTI system model can easily be found in MATLAB using For such a system, there willД®xist finite inputs that lead to an unbounded response. A system with purely imaginary poles is not considered BIBO stable. If any pair of poles is on the imaginary axis, then the system is marginally stable and the system If we view the poles on the complex s-plane, then all poles must be in the left-half plane (values of for which the denominator equals zero) have negative real parts, then the system is stable. The transfer function representation is especially useful when analyzing system stability. Practically, this means that the system will not 'blow up' while in operation. Both methods display the same information, but in different ways.ДЇor our purposes, we will use the Bounded Input Bounded Output (BIBO) definition of stability which states that a system is stable if the output remains bounded for all bounded (finite) inputs. Since is a complex number, we can plot both its magnitude and phase (the Bode Plot) or its position in the complex plane (the Nyquist Diagram). If is the open-loop transfer function of a system and is the frequency vector, we then plot versus. (varying between zero or 'DC' to infinity) and compute the value of the plant transfer function at those frequencies. The frequency response of a system can be found from its transfer function in the following way: create a vector of frequencies These magnitude and phase differences are a function of the frequency and comprise the frequency response of the system. Then the steady-state output will also be sinusoidal at the same frequency, but, in general, with different magnitude and

LTI systems have the extremely important property that if the input to the system is sinusoidal, Of the governing differential equations, respectively.ДЄll the examples presented in this tutorial are modeled by linear constant coefficient differential equations and are thus These correspond to the homogenous (free or zero input) and the particular solutions The time response of a linear dynamic system consists of the sum of the transient response which depends on the initial conditions and the steady-state response which depends on the system input. MATLAB provides many useful resources for calculating time responses for many types of inputs, as we shall see in the following Nonlinear systems or those subject to complicated inputs, this integration must be carried out numerically. For some simple systems, a closed-form analytical solution may be available. Since the models we haveДerived consist of differential equations, some integration must be performed in order to determine the time response of the The time response represents how the state of a dynamic system changes in time when subjected to a particular input.

Hot to draw phase planes for differential equation systems