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Feedback Control Systems ( FCS ). Lecture-32-33 Closed Loop Frequency Response. Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL : http://imtiazhussainkalwar.weebly.com/. Introduction.
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Feedback Control Systems (FCS) Lecture-32-33 Closed Loop Frequency Response Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/
Introduction • One of the important problems in analyzing a control system is to find all closed-loop poles or at least those closes to the jω axis (or the dominant pair of closed-loop poles). • If the open-loop frequency-response characteristics of a system are known, it may be possible to estimate the closed-loop poles closest to the jω axis.
Closed Loop Frequency Response • For a stable, unity-feedback closed-loop system, the closed-loop frequency response can be obtained easily from that of the open loop frequency response. • Consider the unity-feedback system shown in following figure. The closed-loop transfer function is
Closed Loop Frequency Response • Following figure shows the polar plot of G(s). • The vector OA represents G(jω1), where ω1 is the frequency at point A. • The length of the vector OA is • And the angle is
Closed Loop Frequency Response • The vector PA, the vector from -1+j0 point to Nyquist locus represents 1+G(jω1). • Therefore, the ratio of OA, to PA represents the closed loop frequency response.
Closed Loop Frequency Response • The magnitude of the closed loop transfer function at ω=ω1 is the ratio of magnitudes of vector OA to vector PA. • The phase of the closed loop transfer function at ω=ω1 is the angle formed by OA to PA (i.eΦ-θ). • By measuring the magnitude and phase angle at different frequency points, the closed-loop frequency-response curve can be obtained.
Closed Loop Frequency Response • Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or
Closed Loop Frequency Response • Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or • From above equation we can find the constant-magnitude loci and constant-phase-angle loci. • Such loci are convenient in determining the closed-loop frequency response from the polar plot or Nyquist plot.
Constant Magnitude Loci (M circles) • To obtain the constant-magnitude loci, let us first note that G(jω) is a complex quantity and can be written as follows: • Then the closed loop magnitude M is given as • And M2 is
Constant Magnitude Loci (M circles) • Hence • If M=1 then, • This is the equation of straight line parallel to y-axis and passing through (-0.5,0) point.
Constant Magnitude Loci (M circles) • If M≠1 then, • Add to both sides
Constant Magnitude Loci (M circles) • This is the equation of a circle with
Constant Magnitude Loci (M circles) • The constant M loci on the G(s) plane are thus a family of circles. • The centre and radius of the circle for a given value of M can be easily calculated. • For example, for M=1.3, the centre is at (–2.45, 0) and the radius is 1.88.
Constant Phase Loci (N circles) • The phase angle of closed loop transfer function is • The phase angle α is
Constant Phase Loci (N circles) • If we define • then • We obtain
Constant Phase Loci (N circles) Adding to both sides This is an equation of circle with
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