Feedback Control Systems FCS Lecture32 33 Closed Loop

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+j 0 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. e Φ -θ). • By measuring the magnitude and phase angle at different frequency points, the closedloop 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.