![]() ![]() (varying between zero or "DC" to infinity) and compute the value of the plant transfer function at those frequencies. If is the open-loop transfer function of a system and is the frequency vector, we then plot versus. 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). Both methods display the same information, but in different ways.įor 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. Practically, this means that the system will not "blow up" while in operation. The transfer function representation is especially useful when analyzing system stability. 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. A system with purely imaginary poles is not considered BIBO stable. 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 willĮxist finite inputs that lead to an unbounded response. Thus this system is stable since the real parts of the poles are both negative. In fact, the poles of the transfer function are the eigenvalues of the system matrix The stability of a system may also be foundįrom the state-space representation. 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. The order of a dynamic system is the order of the highest derivative of its governing differential equation. It is the highest power of in the denominator of its transfer function. Some common examples include mass-damper systems and RC The important properties of first-, second-, and higher-order systems will beįirst-order systems are the simplest dynamic systems to analyze. The form of a first-order transfer function is The general form of the first-order differential equation is as follows Where the parameters and completely define the character of the first-order system. The DC gain, is the ratio of the magnitude of the steady-state step response to the magnitude of the step input. For stable transferįunctions, the Final Value Theorem demonstrates that the DC gain is the value of the transfer function evaluated at = 0. The time constant of a first-order system is which is equal to the time it takes for the system's response to reach 63% of its steady-state value for a step input (from For first-order systems of the forms shown, the DC gain is. Zero initial conditions) or to decrease to 37% of the initial value for a system's free response. The time scale for which the dynamics of the system are significant.įirst-order systems have a single real pole, in this case at. Therefore, the system is stable if is positive and unstable if is negative. Note: MATLAB also provides a powerful graphical user interface for analyzing LTI systems which can be accessed using the syntax We can calculate the system time response to a step input of magnitude using the following MATLAB commands: Standard first-order system have no zeros. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |