Interactive Control Systems Tools

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Analysis of Control Systems

The first step of analysis of a control system is to derive a mathematical model of the system. Once such a model is obtained, various methods for analysis of the system are available.

Transient Response Analysis

When analyzing and designing control systems it is convenient to have a basis for comparison of performance of various systems. This basis can be set up by specifying a number of test input signals and by comparing the response of different systems to these signals.

The commonly used test input signals are step functions, ramp functions, acceleration functions, impulse functions, sinusoidal functions and the like. Since these test input signals are simple functions of time it is easy to perform mathematical and experimental analysis of control systems. How do we choose which test signal to use? If the inputs to the real control system are suddenly changing then a good approximation might be the step function. Similarly, if the inputs to the control system are gradually changing with the time then the ramp function might be a good candidate. The use of such test signals allows to compare the responses of different systems or allows to compare the response of different designs.

Important system behavior to which we must give careful consideration includes not only transient response, but steady state response, absolute stability, relative stability, steady state error and like.

Root Locus Analysis

The transient response of a closed-loop system is closely related to the location of the closed loop poles. From the design viewpoint, in some systems simple gain adjustment can move the closed loop poles to the desired locations. The closed loop poles are the roots of the characteristic equation of the system.

A simple method for finding the roots of the characteristic equation and how they vary depending on the change of the system gain has been developed by W. R. Evans and is called root locus method.

The locus of the roots of the characteristic equation of the closed loop system as the gain varies from zero to infinity gives the name of the method. Such a plot shows clearly the contribution of each open loop pole or zero to the locations of the closed loop poles. Tis method is very powerful graphical technique for investigating the effects of the variation of a system parameter on the locations of the closed loop poles. General rules for constructing the root locus exist and if the designer follows them, sketching of the root loci becomes a simple matter.

Frequency Response Analysis

The frequency response means the steady state response of the system to a sinusoidal input. In frequency response we vary the frequency of the input signal and study the resulting response. An advantage of the frequency response is that the tests are simple and can be made accurately by using a sinusoidal generator and measurement equipment.

The Nyquist stability criterion enables to investigate both the absolute and relative stability of closed loop system from the knowledge of its open loop frequency response characteristics.

Bode plots based analysis and design are simple and allow even to construct the transfer function of unknown system easily from its frequency response.

Analysis in State Space

A modern complex system may have many inputs and outputs, and these may be related in a complicated manner. The state space approach is best suited to system analysis of such systems because it reduces the complexity of the mathematical expressions. The state space approach describes the system equations in terms of a system of first order differential equations which may be combined into first order vector-matrix notation. This greatly simplifies the mathematical representation and allows the analysis to be carried out by procedures that are similar to those required for analysis of first order scalar differential equations even for multi-input multi-output systems.

Lyapunov stability analysis and the quadratic optimal control are based on the state space approach.