2 lec Eng
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Advantages of using the Bode diagram
•The main advantage is that multiplication of magnitudes can be converted into addition;
•a simple method for sketching an approximate log-
magnitude curve is available.
It is based on asymptotic approximations. Such approximation by straight-line asymptotes is sufficient if only rough information on the frequency-response characteristics is needed. Should the exact curve be desired, corrections can be made easily to these basic asymptotic plots.
•Expanding the frequency range by use of a logarithmic scale for the frequency is highly advantageous
Stability Analysis
Introduction
When considering the design and analysis of feedback control systems, stability is of the utmost importance.
Among the many forms of performance specifications used in design, the most important requirement is that the system must be stable. An unstable system is generally considered to be useless.
When all types of systems are considered – linear, nonlinear, time-invariant, and time-varying –
the definition of stability can be given in many different forms. We shall deal only with the stability of linear SISO time-invariant systems in the following discussions.
Introduction
For analysis and design purposes, we can classify
stability as absolute stability and relative stability.
Absolute stability refers to whether the system is stable or unstable; it is a yes or no answer.
Once the system is found to be stable, it is of interest to determine how stable it is, and this degree of stability is a measure of relative stability.
Zero-input response approach
The zero-input response is due to the initial conditions only; all the inputs are zero.
Let a linear time-invariant system be described by the state equation:
dX t AX t Bu t dt
For zero input u = 0:
dX t AX t dt
Zero-input response approach
The equilibrium state of a system
The system is not in the motion
dX t AX t 0 dt
For zero input X t 0 is defined as the equilibrium state
of the system
Stability definition
The zero-input stability is defined as follows:
a) |
If the zero-input response |
X t , subject to finite initial |
state |
X t0 , returns to the |
equilibrium state X t 0 as |
t approaches infinity, the system is said to be stable. This type of stability is also known as the asymptotic stability.
Stability definition
b) If the zero-input response |
X t , subject to finite initial |
state X t0 , moves off the |
equilibrium state X t 0 as |
t approaches infinity, the system is said to be unstable.
Stability definition
c) If the zero-input response X t , subject to finite initial
state |
X t0 , |
does not approach and move off the equilibrium |
state |
X t 0 |
as t approaches infinity, the system is said to be |
neutral stable.
Zero-state response approach
The zero-state response is due to the input only; all the initial conditions of the system are zero.
A stable system is defined as a system with a bounded (limited) system response. That is, if the system is subjected to a bounded input or disturbance and the response is bounded in magnitude, the system is said to be stable.
A stable system is a dynamic system with a bounded response to a bounded input.