This page contains information on some of the software that is being developed to support the analysis and design techniques described in Feedback Systems. Some specialized scripts have been developed to produce plots that match the style in Feedback Systems. For your convenience, these files are also available as a single zip file or tar file.

Some of these examples make use of the additional scripts below, which you should install in the working directory or MATLAB path:. In this section, we will show how to determine these dynamic properties from the system models. The time response represents how the state of a dynamic system changes in time when subjected to a particular input. Since the models we have derived consist of differential equations, some integration must be performed in order to determine the time response of the system.

## Power System Analysis

For some simple systems, a closed-form analytical solution may be available. However, for most systems, especially nonlinear systems or those subject to complicated inputs, this integration must be carried out numerically.

Fortunately, MATLAB provides many useful resources for calculating time responses for many types of inputs, as we shall see in the following sections. 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. These correspond to the homogenous free or zero input and the particular solutions of the governing differential equations, respectively.

All the examples presented in this tutorial are modeled by linear constant coefficient differential equations and are thus linear time-invariant LTI. LTI systems have the extremely important property that if the input to the system is sinusoidal, then the steady-state output will also be sinusoidal at the same frequency, but, in general, with different magnitude and phase.

## Solutions Manual for Linear Control System Analysis and Design with MATLAB

These magnitude and phase differences are a function of the frequency and comprise the frequency response of the system. The frequency response of a system can be found from its transfer function in the following way: create a vector of frequencies 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. For 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.

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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 all poles of the transfer function values of for which the denominator equals zero have negative real parts, then the system is stable.

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If any pole has a positive real part, then the system is unstable. If we view the poles on the complex s-plane, then all poles must be in the left-half plane LHP to ensure stability. If any pair of poles is on the imaginary axis, then the system is marginally stable and the system will tend to oscillate.

A system with purely imaginary poles is not considered BIBO stable.

## Matlab Nonlinear Dynamics Toolbox

For such a system, there will exist finite inputs that lead to an unbounded response. Thus this system is stable since the real parts of the poles are both negative. The stability of a system may also be found from the state-space representation. In fact, the poles of the transfer function are the eigenvalues of the system matrix.

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. Houpis ; Stuart N. Publisher: CRC Press , This specific ISBN edition is currently not available. View all copies of this ISBN edition:. Synopsis Thoroughly classroom-tested and proven to be a valuable self-study companion, Linear Control System Analysis and Design: Sixth Edition provides an intensive overview of modern control theory and conventional control system design using in-depth explanations, diagrams, calculations, and tables.

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