In Figure 13.18, which shows performance characteristics on the z-plane, what do radial lines emanating from the origin represent?
Explanation
The geometry of the s-plane, where horizontal lines represent constant peak time, is transformed by z = e^(sT) into the z-plane. This mapping results in these horizontal lines becoming radial lines that emanate from the origin, as depicted in Figure 13.18.
Other questions
What device is used to convert analog signals to digital signals within a control loop containing a digital computer?
What is the stability criterion for a digital control system in the z-plane?
According to the derivation in Example 13.1, what is the closed-form z-transform, F(z), for a sampled unit ramp f(kT) = kT?
The transfer function of a zero-order hold, which holds the last sampled value over the sampling interval, is given by which Laplace transform?
In the context of analog-to-digital conversion, what is quantization error?
What is the primary purpose of the Tustin transformation in the design of digital compensators?
In Example 13.12, a digital lead compensator Gc(z) is designed for a plant Gp(s) = 1/(s(s+6)(s+10)). What is the chosen sampling time, T, for this design?
For a digital compensator to be physically realizable, what must be true about the relationship between the order of its numerator and denominator polynomials?
What is the z-transform, R(z), for a unit step input R(s) = 1/s?
In the Antenna Control Case Study, the design gain K is calculated to achieve a 0.5 damping ratio. What is the value of K?
How is the static velocity error constant, Kv, for a digital system defined?
In Example 13.6, a missile control system is analyzed. For a gain of K = 100, where are the poles of the closed-loop system located, and is the system stable?
A bilinear transformation s = (z-1)/(z+1) is used for stability analysis. Where does this transformation map points from the left half of the s-plane (where alpha < 0)?
In gain design on the z-plane, how are lines of constant settling time, Ts, represented?
What is the steady-state error e*(infinity) for a unity feedback digital system with a unit step input?
What is the result of sampling a time waveform f(t) with an ideal sampler?
In Example 13.9, what is the steady-state error for a ramp input for the system with G1(s) = 10/(s(s+1))?
What is one of the main advantages of using digital computers for control over analog controllers?
When reducing block diagrams for sampled-data systems, the z-transform of a product of two continuous-time transfer functions, z{G1(s)G2(s)}, is generally not equal to what?
In Example 13.8, the denominator of a digital transfer function is given as D(z) = z^3 - z^2 - 0.2z + 0.1. After applying the bilinear transformation, what is the resulting polynomial in s?
What is the minimum sampling frequency required to avoid distortion when sampling a signal, according to the principle mentioned in the text?
What does the z-transform theorem f(infinity) = lim(z->1) (1-z^-1)F(z) represent?
In Example 13.7, a digital system becomes unstable for sampling intervals T > 0.2 seconds. What sampling frequency does this correspond to?
When using the power series method to find the inverse z-transform of F(z), what do the coefficients of the resulting power series in z^-k represent?
In the Antenna Control Digital Cascade Compensator Design case study, what is the design requirement for the settling time of the compensated system?
How does placing a pole at z=1 in the open-loop pulse transfer function G(z) affect the steady-state error of a digital system?
In Example 13.10, the root locus for the system G(z) = K(z+1)/((z-1)(z-0.5)) is analyzed for stability. What is the approximate gain K at which the system becomes unstable?
In the digital compensator implementation flowchart shown in Figure 13.28, what do the 'Delay T seconds' blocks represent?
How are curves of constant damping ratio, zeta, represented on the z-plane?
In the Antenna Control Case Study for digital cascade compensator design, a lead compensator zero is placed at s = -1.71. What is the reason for this placement?
For the system in Example 13.11 with a designed gain of K=0.0627, what is the approximate overshoot of the sampled step response?
What is the maximum quantization error for an analog-to-digital converter using n binary bits and a maximum analog voltage of M?
What is the inverse Tustin transformation used for?
When deriving the pulse transfer function, C(z) = G(z)R(z), what conceptual component is assumed to be at the system output?
In the digital compensator implementation shown by the difference equation in (13.99), the current output x*(t) depends on what combination of signals?
Which of the following z-transforms from Table 13.1 corresponds to the time function f(t) = sin(omega*t)?
According to the guideline from Astrom and Wittenmark mentioned in Section 13.10, the sampling interval T should be in what range, relative to the zero dB frequency (omega_phi_M) of the compensated analog system?
In the inverse z-transform method via partial-fraction expansion, why is F(z)/z expanded instead of F(z) directly?
For the digital system in Figure 13.9(c), with two cascaded subsystems G1(z) and G2(z) each with a sampler at its input, what is the overall pulse transfer function for the output C(z)?
In the context of the z-transform, what is the significance of the variable T?
A continuous system pole at s = -a is transformed into the z-plane using z = e^(sT). What is the location of the corresponding pole in the z-plane?
What is the primary effect of decreasing the sampling interval, T, when using the Tustin transformation to design a digital compensator?
In the digital control system of Figure 13.25(a), what components does the 'Digital controller' block represent?
For the cascade of G1(s) and a zero-order hold in Example 13.4, the pulse transfer function G(z) is found using the relation G(z) = (1 - z^-1) * Z{G1(s)/s}. What is G1(s)/s in this example?
In the final step of Example 13.4, with T=0.5, what is the final expression for the pulse transfer function G(z)?
What is the primary trade-off involved in choosing the sampling rate for a digital control system?
If a digital system's closed-loop poles are found to be at 0.5 + j0.5 and 0.5 - j0.5, is the system stable?
In the digital compensator flowchart of Figure 13.29, which implements Gc(z) = (z+0.5)/(z^2 - 0.5z + 0.7), what value is multiplied by the e*(t-T) sample?
What is the key difference in plotting a root locus for a digital system on the z-plane compared to a continuous system on the s-plane?