Frequency response design

A recent addition to the model-based tuning correlations is Internal Model Control (Rivera, Morari, and Skogestad, Internal Model Control 4 PID Controller Design, lEC Proc. Des. Dev., 25, 252, 1986), which offers some advantages over the other methods described here. However, the correlations are similar to the ones discussed above. Other plant testing and controller design approaches such as frequency response can be used for more complicated models. 

Fig. 6.39 Closed-loop frequency response for both lead compensator designs.

Properly designed, they effectively can eliminate the transmission of high-frequency response. The filter should prevent the transmission of all acoustic response with as little restriction to steady flow as possible. Filters can be designed for almost any desired degree of acoustic response control. 

Accurate control of potential, stability, frequency response and uniform current distribution required the following low resistance of the cell and reference electrode small stray capacitances small working electrode area small solution resistance between specimen and point at which potential is measured and a symmetrical electrode arrangement. Their design appears to have eliminated the need for the usual Luggin capillary probe. 

We have given up the pretense that we can cover controller design and still have time to do all the plots manually. We rely on MATLAB to construct the plots. For example, we take a unique approach to root locus plots. We do not ignore it like some texts do, but we also do not go into the hand sketching details. The same can be said with frequency response analysis. On the whole, we use root locus and Bode plots as computational and pedagogical tools in ways that can help to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. 

On the other hand, frequency response analysis cannot reveal information on dynamic response easily—something root locus does very well. Hence controller design is always an iterative procedure. There is no one-stop-shopping. There is never a unique answer. 

Frequency response analysis allows us to derive a general relative stability criterion that can easily handle systems with time delay. This property is used in controller design. 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

The pulse experiment is not crucial for our understanding of frequency response analysis and is provided on our Web Support, but we will do the design calculations in Section 8.3. 

The gain and phase margins are used in the next section for controller design. Before that, let s plot different controller transfer functions and infer their properties in frequency response analysis. Generally speaking, any function that introduces additional phase lag or magnitude tends to be destabilizing, and the effect is frequency dependent. 

If we want to design a PI controller, how should we proceed with frequency response methods Let s presume that the unit of the time constants is in minutes. 

The LTI Viewer was designed to do comparative plots, either comparing different transfer functions, or comparing the time domain and (later in Chapter 8) frequency response properties of a transfer function. So a more likely (and quicker) scenario is to enter, for example,... 

We will simply state that the SISO system design tool sisotool, as explained in Session 6, can be used to do frequency response plots. Now, we want to use the default view, so we just need to enter ... 

Sampled-data controllers can be designed in the same way continuous controllers are designed. Root locus plots in the z plane or frequency-response plots are made with various types of >(z) s (different orders of M and N and different values of the a, and 6, coefficients). This is the same as using different combinations of lead-lag elements in continuous systems. 

Fig. 11.1. Two basic types of current ampliflers. (a) Feedback picoammeter. It consists of two components, an operational amplifier (op-amp) A, and a feedback resistor 1 fb- a typical value of the feedback resistor used in STM is 10 fl. The stray capacitance Cfb is an inevitable parasitic element in the circuit. In a careful design, Cfb 0.5 pF. The input capacitance Cm is also an inevitable parasitic element in the circuit. Those parasitic capacitors, the thermal noise of the feedback resistor, and the characteristics of the op-amp are the limiting factors to the performance of the picoammeter. (b) An electrometer used as a current amplifier (the shunt current amplifier). The voltage at the input resistance is amplified by the circuit, which consists of an op-amp and a pair of resistors R, and R2. The parasitic input capacitance Cm limits the frequency response, and the Johnson noise on Rm is the major source of noise. Also, the input resistance for this arrangement is large.

The existence of the stray capacitance results in a time delay RmCvn- The frequency response of the amplifier is reduced. In a practical design, a compromise between bandwidth and noise is often to be made, as explained in the following subsection. 

If xj/ < 0 the output is said to lag the input, and if xj/ > 0 it is said to lead the input. The relationship between input and output for a sinusoidal forcing function as t -> oo constitutes an important tool in the analysis and design of control systems termed frequency response analysis. Of particular importance are the amplitude ratio (AR) and the phase shift xfr. The AR represents the relationship between the output and the input amplitudes... 

For linear, time-invariant systems a complete characterization is given by the impulse or complex frequency response [Papoulis, 1977], With perceptual interpretation of this characterization one can determine the audio quality of the system under test. If the design goal of the system under test is to be transparent (no audible differences between input and output) then quality evaluation is simple and brakes down to the... 

Thus, in order to simulate a perceptually convincing room reverberation, it is necessary to simulate both the pattern of early echoes, with particular concern for lateral echoes, and the late energy decay relief. The latter can be parameterized as the frequency response envelope and the reverberation time, both of which are functions of frequency. The challenge is to design an artificial reverberator which has sufficient echo density in the time domain, sufficient density of maxima in the frequency domain, and a natural colorless timbre. 

The filter bank used in the MPEG/Audio coding system will be used as an example. A 511 tap prototype filter is used. Figure 2.7 shows the prototype filter (window function). It has been optimized for a very steep filter response and a stop band attenuation of better than 96 dB. Figure 2.8 shows the frequency response of the filter bank. In addition to the attenuation requirements it was designed as a reasonable tradeoff between time behavior and frequency localization [Dehery, 1991],... 

Process design modifications usually have a bigger impact on operability (dynamic resilience). Dynamic resilience depends on controller structure, choice of measurements, and manipulated variables. Multivariable frequency-response techniques have been used to determine resilience properties. A primary result is that closed-loop control quality is limited by system invertability (nonmin-imum phase elements). Additionally, it has been shown that steady-state optimal designs are not necessarily optimal in dynamic operation. 

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