Feedback Control Systems (FCS) Dr. Imtiaz Hussain URL : Lecture Closed Loop Frequency Response

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(jω 1 ), where ω 1 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(jω 1 ). 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=ω 1 is the ratio of magnitudes of vector OA to vector PA. The phase of the closed loop transfer function at ω=ω 1 is the angle formed by OA to PA (i.e Φ-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 M 2 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

Closed Loop Frequency Response

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