A unity feedback system has a nominal characteristic equation q(s)=s^3+3 s^2+3 s+6=0 . The coeff

A unity feedback system has a nominal characteristic...

Key Concepts

Unity Feedback System

A unity feedback system is one in which the output is directly fed back as the input without any additional gain or modification. This way, the overall behavior of the closed-loop system can be characterized by its open-loop transfer function and the feedback configuration, simplifying the analysis of system stability and performance.

Characteristic Equation

The characteristic equation is a polynomial derived from the closed-loop transfer function whose roots determine the dynamic behavior of the system. In the context of control systems stability, finding the roots of this equation (also called poles) allows one to infer whether the system will behave in a stable, oscillatory, or unstable manner.

Coefficient Uncertainty

Coefficient uncertainty refers to variations in the parameters of the characteristic equation within specified ranges. Such uncertainty is critical because the stability of the system may be sensitive to these variations. Analyzing the effect of parameter changes helps determine if the system maintains its stability across the entire range of possible parameter values.

Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is a systematic method used to determine the stability of a system by constructing the Routh array based on the coefficients of the characteristic polynomial. By checking the sign changes and positivity of the first column of the Routh array, one can conclude whether all the roots of the polynomial have negative real parts, ensuring system stability.

Robust Stability

Robust stability is concerned with the ability of a system to remain stable under variations or uncertainties in its parameters. It examines whether the stability conditions hold for all possible values within the specified intervals of the coefficients. This concept is particularly important in practical systems where parameters may vary due to modeling errors, component tolerances, or external disturbances.

Transcript

00:01 For roots s equals negative sigma plus or minus i omega d, the time constant tau is 1 over sigma on damp national frequency, omega n, is the square root of sigma squared plus omega d squared.

00:19 And damping ratio zeta is sigma over omega n.

00:26 The characteristic equation is s minus s1 times s minus s2 equals 0.

00:32 So for a, with the roots negative 4 plus or minus 5i, sigma is 4 and omega d is 5, tau then is 1 over 4 .25 seconds.

00:49 This would come from the characteristic equation would be s plus 4 minus 5i times s plus 4 plus 5i.

01:00 Equals 0 and that multiplies to s squared plus 8 s plus 41 equals 0 tau would be 0 .25 seconds omega n would be then the square root of 4 squared plus 5 squared which is the square of 41 or 6 .403 omega d is 5 and zeta would be 4 .4 .0 3.

01:25 Omega d is 5 and zeta would be 4 over 6 .403 which is 0 .625.

01:43 For v with the root to 4 plus or minus 5 i here the real part's positive so it is unstable and so omeg sigma is negative 4 the damped frequency magnitude is 5 then tau is 1 over 4 or 0 .25 seconds the undemped natural frequency omega n would be the square root of negative 4 squared plus 5 squared that's the square to 41 or 6 .403.

02:20 The damping ratio zeta is negative 4 over 6 .403, which is negative 0 .625.

02:30 Negative means unstable.

02:35 And the characteristic equation, s minus 4 minus 5i times s minus 4 plus 5i multiplies to s squared minus 8 s plus 41 equals 0.

02:47 For c, s, the roots are negative 4 and negative 5, it's over dammed.

02:52 So the characteristic equation would be s plus 4 times s plus 5, which is s squared plus 9 s plus 20 equals 0...

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