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## Chapter 5 Frequency-Domain Analysis

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**Chapter 5 Frequency-Domain Analysis**NUAA-Control System Engineering**Content in Chapter 5**5-1 Frequency Response (or Frequency Characteristics) 5-2 Nyquist plot and Nyquist stability criterion 5-3 Bode plot and Bode stability criterion**A Perspective on the Frequency-Response Design Method**The design of feedback control systems in industry is probably accomplished using frequency-response methods more than any other. Advantages of frequency-response design: -It provides good designs in the face of uncertainty in the plant model -Experimental information can be used for design purposes. Raw measurements of the output amplitude and phase of a plant undergoing a sinusoidal input excitation are sufficient to design a suitable feedback control. -No intermediate processing of the data (such as finding poles and zeros) is required to arrive at the system model.**Frequency response**G(s) H(s) The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. For a LTI system, when the input to it is a sinusoid signal, the resulting output , as well as signals throughout the system, is sinusoidal in the steady-state; The output differs from the input waveform only in amplitude and phase.**magnitude**phase The closed-loop transfer function of the LTI system: For frequency-domain analysis, we replace s by jω: The frequency-domain transfer function M(jω) may be expressed in terms of its magnitude and phase:**Gain characteristic**Phase characteristic The magnitude of M(jω) is The phase of M(jω) is Gain-phase characteristics of an ideal low-pass filter**Example. Frequency response of a Capacitor**Consider the capacitor described by the equation where v is the input and i is the output. Determine the sinusoidal steady-state response of the capacitor. Solution. The transfer function of the capacitor is So Computing the magnitude and phase, we find that**Gain characteristic:**Phase characteristic: Output: For a unit-amplitude sinusoidal input v, the output i will be a sinusoid with magnitude Cω, and the phase of the output will lead the input by 90°. Note that for this example the magnitude is proportional to the input frequency while the phase is independent of frequency.**Resonant peak**Cutoff rate Resonant frequency Bandwidth Frequency-Domain Specifications Typical gain-phase characteristic of a control system**Frequency response of a prototype second-order system**Closed-loop transfer function: Its frequency-domain transfer function: Define**With , we have**The magnitude of M(ju) is The phase of M(ju) is Resonant peak The resonant frequency of M(ju) is Since frequency is a real quantity, it requires So**With , we have**According to the definition of Bandwidth**For a prototype second-order system ( )**Resonant peak Resonant frequency Bandwidth**Correlation between pole locations, unit-step response and**the magnitude of the frequency response**Example. The specifications on a second-order unity-feedback**control system with the closed-loop transfer function are that the maximum overshoot must not exceed 10 percent, and the rise time be less than 0.1 sec. Find the corresponding limiting values of Mr and BW analytically. Solution. Maximum overshoot: Rise time:**Resonant peak**Bandwidth Based on time-domain analysis, we obtain and Frequency-domain specifications:**Adding a zero at**Open-loop TF： The additional zero changes both numerator and denominator. Effects of adding a zero to the OL TF Open-loop TF： Closed-loop TF： Closed-loop TF：**For fixed ωn and ζ, we analyze the effect of .**As analyzing the prototype second-order system, using similar but more complicate calculation, we obtain Bandwidth where**The general effect of adding a zero the open-loop transfer**function isto increase the bandwidth of the closed-loop system.**Open-loop TF：**Closed-loop TF： Adding a pole at Open-loop TF： Closed-loop TF： Effects of adding a pole to the OL TF**The effect of adding a pole the open-loop transfer function**isto make the closed-loop system less stable, while decreasing the bandwidth.**G(s)**H(s) Nyquist Criterion What is Nyquist criterion used for? Nyquist criterion is a semigraphical method that determines the stability of a closed-loop system; Nyquist criterion allows us to determine the stability of a closed-loop system from the frequency-response of the loop function G(jw)H(j(w)**Review about stability**Closed-loop TF: Characteristic equation (CE): Stability conditions: Open-loop stability: poles of the loop TF G(s)H(s) are all in the left-half s-plane. Closed-loop stability: poles of the closed-loop TF or roots of the CE are all in the left-half s-plane.**Definition of Encircled and Enclosed**Encircled: A point or region in a complex function plane is said to be encircled by a closed path if it is found inside the path. Enclosed: A point or region in a complex function plane is said to be encircled by a closed path if it is encircled in the countclockwise(CCW) direction. Point A is encircled in the closed path; Point A is also enclosed in the closed path;**Number of Encirclements and Enclosures**Point A is encircled once; Point B is encircled twice. Point C is enclosed once; Point D is enclosed twice.**Δ( s)-plane**s-plane Mapping Mapping from the complex s-plane to the Δ(s) -plane Exercise 1: Consider a function Δ(s) =s-1, please map a circle with a radius 1 centered at 1 from s-plane to the Δ(s)-plane .**N - number of encirclements of the origin by the**-plane locus Z - number of zeros of encircled by the s-plane locus P - number of poles of encircled by the s-plane locus Principle the Argument Let be a single-valued function that has a finite number of poles in the s-plane. Suppose that an arbitrary closed path is chosen in the s-plane so that the path does not go through any one of the poles or zeros of ; The corresponding locus mapped in the -plane will encircle the origin as many times as the difference between the number of zeros and poles (P) of that are encircled by the s-plane locus . In equation form:**s-plane**Note Nyquist path does not pass through any poles or zeros of Δ(s); if Δ(s) has any pole or zero in the right-half plane, it will be encircled by . Nyquist Path A curve composed of the imaginary axis and an arc of infinite radius such that the curve completely encloses the right half of the s-plane . Nyquist path is in the CCW direction Since in mathematics, CCW is traditionally defined to be the positive sense.**Δ( s)-plane**s-plane G( s)H(s)-plane Nyquist Criterion and Nyquist Diagram Nyquist Path Nyquist Diagram: Plot the loop function to determine the closed-loop stability Critical point: (-1+j0)**G( s)H(s)-plane**s-plane Nyquist Path G(s)H(s) Plot The Nyquist Path is shown in the left figure. This path is mapped through the loop tranfer function G(s)H(S) to the G(s)H(s) plot in the right figure. The Nyquist Creterion follows: Nyquist Criterion and G(s)H(s) Plot**s-plane**N - number of encirclements of (-1,j0) by the G(s)H(s) plot Z - number of zeros of that are inside the right-half plane P - number of poles of that are inside the right-half plane Nyquist Criterion and Nyquist Plot G( s)H(s)-plane Nyquist Path Nyquist Plot The condition of closed-loop stability according to the Nyquist Creterion is:**has the same poles as , so P can be obtained by**counting the number of poles of in the right-half plane.**Question 2: what if**An example Consider the system with the loop function Matlab program for Nyquist plot (G(s)H(s) plot) >>num=5; >>den=[1 3 3 1]; >>nyquist(num,den); Question 1: is the closed-loop system stable? N=0, P=0, N=-P, stable**1. With root locus technique:**>>num=1; >>den=[1 3 3 1]; >>rlocus(num,den); For K* varies from 0 to ∞, we draw the RL When K*=8 (K=1.6), the RL cross the jw-axis, the closed-loop system is marginally stable. For K*>8 (K>1.6), the closed-loop system has two roots in the RHP and is unstable.**>>K=1;**>>num=5*K; >>den=[1 3 3 1]; >>nyquist(num,den); 2. With Nyquist plot and Nyquist criterion: K=1 No pole of G(s)H(s) in RHP, so P=0; Nyquist plot does not encircle (-1,j0), so N=0 Thus N=-P The closed-loop system is stable**>>K=1.6;**>>num=5*K; >>den=[1 3 3 1]; >>nyquist(num,den); 2. With Nyquist plot and Nyquist criterion: K=1.6 No pole of G(s)H(s) in RHP, so P=0; The Nyquist plot just go through (-1,j0) The closed-loop system is marginally stable**>>K=4;**>>num=5*K; >>den=[1 3 3 1]; >>nyquist(num,den); 2. With Nyquist plot and Nyquist criterion: K=4 No pole of G(s)H(s) in RHP, so P=0; Nyquist plot encircles (-1,j0) twice, so N=2 Thus Z=N+P=2 The closed-loop system has two poles in RHP and is unstable**Nyquist Criterion for Systems with Minimum-Phase Transfer**Functions What is called a minimum-phase transfer function? A minimum-phase transfer function does not have poles or zeros in the right-half s-plane or on the jw-axis, except at s=0. Consider the transfer functions Both transfer functions have the same magnitude for all frequencies But the phases of the two transfer functions are drastically different.**A minimum-phase system (all zeros in the LHP) with a given**magnitude curve will produce the smallest change in the associated phase, as shown in G1.**Consider the loop transfer function:**If L(s) is minimum-phase, that is, L(s) does not have any poles or zeros in the right-half plane or on the jw-axis, except at s=0 Then P=0, where P is the number of poles of Δ(s)=1+G(s)H(s), which has the same poles as L(s). Thus, the Nyquist criterion (N=-P) for a system with L(s) being minimum-phase is simplified to**Nyquist criterion for systems with minimum-phase loop**transfer function For a closed-loop system with loop transfer function L(s) that is of minimum-phase type, the system is closed-loop stable , if the Nyquist plot (L(s) plot) that corresponds to the Nyquist path does not enclose (-1,j0) point. If the (-1,j0) is enclosed by the Nyquist plot, the system is unstable. The Nyquist stability can be checked by plotting the segment of L(jw) from w= ∞ to 0.**Example**Consider a single-loop feedback system with the loop transfer function Analyze the stability of the closed-loop system. Solution. Since L(s) is minimum-phase, we can analyze the closed-loop stability by investigating whether the Nyquist plot enclose the critical point (-1,j0) for L(jw)/K first. w=∞: w=0+:**The frequency is positive, so**stable the Nyquist plot does not enclose (-1,jw); the Nyquist plot goes through (-1,jw); marginally stable unstable the Nyquist plot encloses (-1,jw).**>>z=[]**>>p=[0, -2, -10]; >>k=1 >>sys=zpk(z,p,k); >>rlocus(sys); By root locus technique**Relative Stability**Gain Margin and Phase Margin For a stable system, relative stability describes how stable the system is. In time-domain, the relative stability is measured by maximum overshoot and damping ratio. In frequency-domain, the relative stability is measured by resonance peak and how close the Nyquist plot of L(jw) is to the (-1,j0) point. The relative stability of the blue curve is higher than the green curve.**Gain Margin (GM)**(for minimum-phase loop transfer functions) Phase crossover L(jw)-plane Phase crossover frequency ωp For a closed-loop system with L(jw) as its loop transfer function, it gain margin is defined as**Gain margin represents the amount of gain in decibels (dB)**that can be added to the loop before the closed-loop system becomes unstable.**Phase Margin (PM)**(for minimum-phase loop transfer functions) Gain margin alone is inadequate to indicate relative stability when system parameters other the loop gain are subject to variation. With the same gain margin, system represented by plot A is more stable than plot B. Gain crossover frequency ωg Phase margin:

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