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PID controllers Fig: 1 Fig: 2

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System compensation Setting the gain is the first step in adjusting the system for satisfactory performance. In many practical cases, however, the adjustment of the gain alone may not provide sufficient alteration of the system behavior to meet the given specifications. As is frequently the case, increasing the gain value will improve the steady-state behavior but will result in poor stability or even instability. It is then necessary to redesign the system (by modifying the structure or by incorporating additional devices or components) to alter the overall behavior so that the system will behave as desired. Such a redesign or addition of a suitable device is called compensation. A device inserted into the system for the purpose of satisfying the specifications is called a compensator. The compensator compensates for deficit performance of the original system.

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Proportional-plus-derivative control of second-order systems Fig: 1 Control system Fig: 2 Unit-step response curves of the second-order system

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Design procedures for lag compensation by the root-locus method

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Lead compensators There are many ways to realize continuous-time (or analog) lead compensators, such as electronic networks using operational amplifiers, electrical RC networks, and mechanical spring-dashpot systems. Compensators using operational amplifiers are frequently used in practice. Figure 1 shows an electronic circuit using operational amplifiers. The transfer function for this circuit is given by

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Root-locus approach to control system design In building a control system, we know that proper modification of the plant dynamics may be a simple way to meet the performance specifications. This, however, may not be possible in many practical situations because the plant may be fixed and may not be modified. Then we must adjust parameters other than those in the fixed plant. We assume that the plant is given and unalterable. The design problems, therefore, become those of improving system performance by insertion of a compensator. Compensation of a control system is reduced to the design of a filter whose characteristics tend to compensate for the undesirable and unalterable characteristics of the plant. The root-locus method is a graphical method for determining the locations of all closed-loop poles from knowledge of the locations of the open-loop poles and zeros as some parameter (usually the gain) is varied from zero to infinity. The method yields a clear indication of the effects of parameter adjustment.

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Ziegler-Nichols first method for tuning PID controllers Ziegler and Nichols proposed rules for determining values of the proportional gain K_p , integral time T_i, and derivative time T_d based on the transient response characteristics of a given plant. Such determination of the parameters of PID controllers or tuning of PID controllers can be made by engineers on site by experiments on the plant. (Numerous tuning rules for PID controllers have been proposed since the Ziegler-Nichols proposal. They are available in the literature. Here, however, we introduce only the Ziegler-Nichols tuning rules.) There are two methods called Ziegler-Nichols tuning rules. In both methods, they aimed at obtaining 25% maximum overshoot in step response. (Fig 1) First method:

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Lag compensator using operational amplifiers

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Ziegler-Nichols second method for tuning PID controllers In the second method, we first set T_i = ∞ and T_d = 0. Using the proportional control action only (see Figure 1), increase K_p from 0 to a critical value K_cr where the output first exhibits sustained oscillations. (If the output does not exhibit sustained oscillations for whatever value K_p may take, then this method does not apply.) Thus, the critical gain K_cr and the corresponding period P_cr are experimentally determined (see Figure 2). Ziegler and Nichols suggested that we set the values of the parameters K_p , T_i, and T_d according to the formula shown in Table 1.

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Tuning rules for PID controllers PID control of plants:

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Compensators: If a compensator is needed to meet the performance specifications, the designer must realize a physical device that has the prescribed transfer function of the compensator. Numerous physical devices have been used for such purposes. In fact, many noble and useful ideas for physically constructing compensators may be found in the literature. Among the many kinds of compensators, widely employed compensators are the lead compensators, lag compensators, lag-lead compensators, and velocity-feedback (tachometer) compensators. In this chapter we shall limit our discussions mostly to these types. Lead, lag, and lag-lead compensators may be electronic devices (such as circuits using operational amplifiers) or RC networks (electrical, mechanical, pneumatic, hydraulic, or combinations thereof) and amplifiers. In the actual design of a control system, whether to use an electronic, pneumatic, or hydraulic compensator is a matter that must be decided partially based on the nature of the controlled plant.

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Lead compensation techniques based on the root-locus approach The root-locus approach to design is very powerful when the specifications are given in terms of time-domain quantities, such as the damping ratio and undamped natural frequency of the desired dominant closed-loop poles, maximum overshoot, rise time, and settling time. Consider a design problem in which the original system either is unstable for all values of gain or is stable but has undesirable transient-response characteristics. In such a case, the reshaping of the root locus is necessary in the broad neighborhood of the jω axis and the origin in order that the dominant closed-loop poles be at desired locations in the complex plane. This problem may be solved by inserting an appropriate lead compensator in cascade with the feedforward transfer function.

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Lag-lead compensator using operational amplifiers Fig: 1Lag-lead compensator

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Lag-lead compensator using operational amplifiers and design Fig 1: Control system

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Lag compensation techniques based on the root-locus approach Consider the problem of finding a suitable compensation network for the case where the system exhibits satisfactory transient-response characteristics but unsatisfactory steady-state characteristics. Compensation in this case essentially consists of increasing the open loop gain without appreciably changing the transient-response characteristics. This means that the root locus in the neighborhood of the dominant closed-loop poles should not be changed appreciably, but the open-loop gain should be increased as much as needed. This can be accomplished if a lag compensator is put in cascade with the given feedforward transfer function. To avoid an appreciable change in the root loci, the angle contribution of the lag network should be limited to a small amount, say 5°. To assure this, we place the pole and zero of the lag network relatively close together and near the origin of the s plane. Then the closed-loop poles of the compensated system will be shifted only slightly from their original locations. Hence, the transient-response characteristics will be changed only slightly.

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PI controllers Fig 1: Pneumatic proportional controller

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PD controllers Proportional-plus-derivative control of a system with inertia load:

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Root-locus plots:

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Root-locus method: The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. If the system has a variable loop gain, then the location of the closed-loop poles depends on the value of the loop gain chosen. It is important, therefore, that the designer know how the closed-loop poles move in the s plane as the loop gain is varied. From the design viewpoint, in some systems simple gain adjustment may move the closed-loop poles to desired locations. Then the design problem may become the selection of an appropriate gain value. If the gain adjustment alone does not yield a desired result, addition of a compensator to the system will become necessary. The closed-loop poles are the roots of the characteristic equation.

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The relative stability of feedback control systems

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Sensitivity and the root locus One of the prime reasons for the use of negative feedback in control systems is the reduction of the effect of parameter variations. The effect of parameter variations can be described by a measure of the sensitivity of the system performance to specific parameter changes. The logarithmic sensitivity originally suggested by Bode is given as:

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Root loci for positive-feedback systems Fig: 1 Control system

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Application of Routh's stability criterion to control system analysis

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General rules for constructing root loci For a complicated system with many open-loop poles and zeros, constructing a root locus plot may seem complicated, but actually it is not difficult if the rules for constructing the root loci are applied.

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Parameter design by the root locus method Initially, the root locus method was developed to determine the locus of roots of the characteristic equation as the system gain, K, is varied from zero to infinity. However, as we have seen, the effect of other system parameters may be readily investigated by using the root locus method. Fundamentally, the root locus method is concerned with a characteristic equation, which may be written as

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Stability of a second-order system: A second-order system is described by the two first-order differential equations

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Root sensitivity of a control system

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Orthogonality of root loci and constant-gain loci

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Comparison between pneumatic systems and hydraulic systems: The fluid generally found in pneumatic systems is air; in hydraulic systems it is oil. And it is primarily the different properties of the fluids involved that characterize the differences between the two systems. These differences can be listed as follows:

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Root-contour plots In the above equation, the parameter a is written as a multiplying factor. For a given value of K, the effect of a on the closed-loop poles can be investigated from the equation. The root contours for this system can be constructed by following the usual procedure for constructing root loci. We shall now construct the root contours as K and a vary, respectively, from zero to infinity. The root contours start from the poles (at s = ±j√K) and terminate at the zeros (at s = 0 and infinity).

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Root-Locus Plots of Negative-Feedback and Positive Feedback Systems Table 1 shows various root-locus plots of negative-feedback and positive feedback systems. The closed-loop transfer functions are given by

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Root loci for systems with transport lag Fig: 1 Thermal system Fig: 2 Block diagram of the system

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Conditionally stable systems:

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Cancellation of poles G(s) with zeros of H(s): Fig: 1

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The stability of state variable systems: The stability of a system modeled by a state variable flow graph model can be readily ascertained. The stability of a system with an input-output transfer function T(s) can be determined by examining the denominator polynomial of T(s). Therefore, if the transfer function is written as

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Approximation of transport lag or dead time If the dead time T is very small, then e^{- Ts} is frequently approximated by

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Comments on the root-locus plots:

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Nonminimum-phase systems Fig: 1 (a) Nonminimum phase system; (b) root-locus plot.

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Typical pole-zero configurations and corresponding root loci Several open-loop pole-zero configurations and their corresponding root loci in Table 1

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Concept of stability: When considering the design and analysis of feedback control systems, stability is of the utmost importance. From a practical point of view, a closed-loop feedback system that is unstable is of little value. As with all such general statements, there are exceptions; but for our purposes, we will declare that all our control designs must result in a closed-loop stable system. Many physical systems are inherently open-loop unstable, and some systems are even designed to be open-loop unstable. Most modern fighter aircraft are open-loop unstable by design, and without active feedback control assisting the pilot, they cannot fly. Active control is introduced by engineers to stabilize the unstable system—that is, the aircraft—so that other considerations, such as transient performance, can be addressed. Using feedback, we can stabilize unstable systems and then with a judicious selection of controller parameters, we can adjust the transient performance. For open-loop stable systems, we still use feedback to adjust the closed-loop performance to meet the design specifications.

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Transient response of higher-order systems

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Routh's stability criterion 3. If all coefficients are positive, arrange the coefficients of the polynomial in rows and columns according to the following pattern:

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Dominant closed-loop poles The relative dominance of closed-loop poles is determined by the ratio of the real parts of the closed-loop poles, as well as by the relative magnitudes of the residues evaluated at the closed-loop poles. The magnitudes of the residues depend on both the closed-loop poles and zeros. If the ratios of the real parts exceed 5 and there are no zeros nearby, then the closed loop poles nearest the jw axis will dominate in the transient-response behavior because these poles correspond to transient-response terms that decay slowly. Those closed-loop poles that have dominant effects on the transient-response behavior are called dominant closed-loop poles. Quite often the dominant closed-loop poles occur in the form of a complex-conjugate pair. The dominant closed-loop poles are most important among all closed-loop poles.

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Axis Shift: Let us consider the simple third-order characteristic equation

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Stability analysis in the complex plane Fig: 1 Region in the complex plane

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Nyquist stability criterion

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The resonant frequency & the resonant peak value The magnitude of

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Phase and gain margins

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Nichols chart

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Constant phase-angle loci (N-circles): Fig 1: A family of constant N circles

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Determination of static acceleration error constants Fig: 1 Unity-feedback control system

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uadratic factors of polar plots

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Special case when G(s)H(s) involves poles and/or zeros on the jωaxis

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Polar plot:

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Nyquist stability criterion The foregoing analysis, utilizing the encirclement of the -1 j0 point by the G(jω)H(jω) locus, is summarized in the following Nyquist stability criterion: Nyquist stability criterion [for a special case when G(s)H(s) has neither poles nor zeros on the jω axis.]: In the system shown in Figure 1, if the open-loop transfer function G(s)H(s) has k poles in the right-half s plane and lim_s→∞G(s)H(s) = constant, then for stability the G(jω)H(jω) locus, as ω varies from -∞ to ∞, must encircle the -1 j0 point k times in the counterclockwise direction.

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Minimum-phase systems and non-minimum-phase systems Transfer functions having neither poles nor zeros in the right-half s plane are minimum-phase transfer functions, whereas those having poles and/or zeros in the right-half s plane are nonminimum-phase transfer functions. Systems with minimum-phase transfer functions are called minimum-phase systems, whereas those with nonminimum-phase transfer functions are called nonminimum-phase systems. For systems with the same magnitude characteristic, the range in phase angle of the minimum-phase transfer function is minimum among all such systems, while the range in phase angle of any nonminimum-phase transfer function is greater than this minimum.

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Comments on phase and gain margins It is also important to point out that conditionally stable systems will have two or more phase crossover frequencies, and some higher-order systems with complicated numerator dynamics may also have two or more gain crossover frequencies, as shown in Figure 1. For stable systems having two or more gain crossover frequencies, the phase margin is measured at the highest gain crossover frequency.

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Determination of static position error constants

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Relative stability analysis through modified Nyquist plots The Nyquist path for stability tests can be modified in order that we may investigate the relative stability of closed-loop systems. For the following second-order characteristic equation,

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Phase angle of quadratic factor

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Stability analysis using Nyquist stability criterion If the Nyquist path in the s plane encircles Z zeros and P poles of 1 G(s)H(s) and does not pass through any poles or zeros of 1 G(s)H(s) as a representative point s moves in the clockwise direction along the Nyquist path, then the corresponding contour in the G(s)H(s) plane encircles the -1 j0 point N = Z - P times in the clockwise direction. (Negative values of N imply counterclockwise encirclements.)

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Determination of static velocity error constants Consider the unity-feedback control system shown in Figure 8-15. Figure 8-17 shows an example of the log-magnitude plot of a type 1 system. The intersection of the initial - 20 dB/decade segment (or its extension) with the line ω = 1 has the magnitude 20 log Kv. This may be seen as follows: In a type 1 system

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Polar Plots of Simple Transfer Functions The polar plots of simple transfer functions are given below:

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Relative stability analysis by conformal mapping

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General shapes of polar plots Fig: 1 Polar plots of type 0, type 1, and type 2 systems Fig: 2 Polar plots in the high-frequency range

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Transfer function of a first order factor in Bode diagram The transfer function 1/(1 jωT) has the characteristics of a low-pass filter. For frequencies above ω = 1/T, the log magnitude falls off rapidly toward -∞. This is essentially due to the presence of the time constant. In the low-pass filter, the output can follow a sinusoidal input faithfully at low frequencies. But as the input frequency is increased, the output cannot follow the input because a certain amount of time is required for the system to build up in magnitude. Thus, at high frequencies, the amplitude of the output approaches zero and the phase angle of the output approaches -90°. Therefore, if the input function contains many harmonics, then the low-frequency components are reproduced faithfully at the output, while the high-frequency components are attenuated in amplitude and shifted in phase. Thus, a first-order element yields exact, or almost exact, duplication only for constant or slowly varying phenomena. An advantage of the Bode diagram is that for reciprocal factors, for example, the factor 1 jωT, the log-magnitude and the phase-angle curves need only be changed in sign. Since

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Multiple-loop system of a polar plot

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Log-magnitude versus phase plots Another approach to graphically portraying the frequency-response characteristics is to use the log-magnitude versus phase plot, which is a plot of the logarithmic magnitude in decibels versus the phase angle or phase margin for a frequency range of interest. [The phase margin is the difference between the actual phase angle Φ and -180°; that is, Φ - (-180°) = 180° Φ.]

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Error in the First-order factors in Bode diagrams The error in the magnitude curve caused by the use of asymptotes can be calculated. The maximum error occurs at the corner frequency and is approximately equal to - 3 dB since

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First-order factors of polar plots

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Closed-loop frequency response for non-unity-feedback systems The constant M and N loci and the Nichols chart cannot be directly applied to control systems with nonunity feedback, but rather require a slight modification. If the closed-loop system involves a nonunity-feedback transfer function, then the closed-loop transfer function may be written

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Conditionally stable systems of a polar plot

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Constant-magnitude loci (M-circles) Fig: 1 A family of constant M circles

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Mapping theorem Let F(s) be a ratio of two polynomials in s. Let P be the number of poles and Z be the number of zeros of F(s) that lie inside some closed contour in the s plane, with multiplicity of poles and zeros accounted for. Let this contour be such that it does not pass through any poles or zeros of F(s). This closed contour in the s plane is then mapped into the F(s) plane as a closed curve. The total number N of clockwise encirclements of the origin of the F(s) plane, as a representative point s traces out the entire contour in the clockwise direction, is equal to Z - P. (Note that by this mapping theorem the numbers of zeros and of poles cannot be found, only their difference.) We shall not present a formal proof of this theorem here but leave the proof to Problem A-8-10. Note that a positive number N indicates an excess of zeros over poles of the function F(s) and a negative N indicates an excess of poles over zeros. In control system applications, the number P can be readily determined for F(s) = 1 G(s)H(s) from the function G(s)H(s). Therefore, if N is determined from the plot of F(s), the number of zeros in the closed contour in the s plane can be determined readily. Note that the exact shapes of the s-plane contour and F(s) locus are immaterial so far as encirclements of the origin are concerned, since encirclements depend only on the enclosure of poles and/or zeros of F(s) by the s-plane contour.

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Gain adjustments Fig 1: M-circle

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Transport lag in Bode Diagrams

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Nyquist stability criterion applied to inverse polar plots In analyzing multiple-loop systems, the inverse transfer function may sometimes be used in order to permit graphical analysis; this avoids much of the numerical calculation. (The Nyquist stability criterion can be applied equally well to inverse polar plots. The mathematical derivation of the Nyquist stability criterion for inverse polar plots is the same as that for direct polar plots.) The inverse polar plot of G(jω)H(jω) is a graph of 1/[G(jω)H(jω)] as a function of ω. For example, if G(jω)H(jω) is

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Jury's stability test For the Jury Test for stability of a discrete-time system, let us consider a transfer function

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Cutoff frequency and bandwidth

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Transport lag of polar plot The transport lag

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Introduction to Frequency-Response Analysis The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal. The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle. For example, consider the system Y(s) = T(s)R(s) with r(t) = A sin ωt. We have

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Unit-ramp response of first-order systems:

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Second-order systems Rise Time: Here the rise time is obtained in terms of . The system is assumed to be underdamped. Considering the equation,

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Bode diagrams or logarithmic plots A sinusoidal transfer function may be represented by two separate plots, one giving the magnitude versus frequency and the other the phase angle (in degrees) versus frequency. A Bode diagram consists of two graphs: One is a plot of the logarithm of the magnitude of a sinusoidal transfer function; the other is a plot of the phase angle; both are plotted against the frequency in logarithmic scale. The standard representation of the logarithmic magnitude of G(jω) is 20log |G(jω)|, where the base of the logarithm is 10. The unit used in this representation of the magnitude is the decibel, usually abbreviated dB. In the logarithmic representation, the curves are drawn on semilog paper, using the log scale for frequency and the linear scale for either magnitude (but in decibels) or phase angle (in degrees). (The frequency range of interest determines the number of logarithmic cycles required on the abscissa.)

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Resonant peak magnitude M_r and resonant peak frequency ω_r

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DC servomotors: There are many types of dc motors in use in industries. DC motors that are used in servo systems are called dc servomotors. In dc servomotors, the rotor inertias have been made very small, with the result that motors with very high torque-to-inertia ratios are commercially available. Some dc servomotors have extremely small time constants. DC servomotors with relatively small power ratings are used in instruments and computer-related equipment such as disk drives, tape drives, printers, and word processors. DC servomotors with medium and large power ratings are used in robot systems, numerically controlled milling machines, and so on. In dc servomotors, the field windings may be connected in series with the armature or the field windings may be separate from the armature. (That is, the magnetic field is produced by a separate circuit.) In the latter case, where the field is excited separately, the magnetic flux is independent of the armature current. In some dc servomotors, the magnetic field is produced by a permanent magnet and, therefore, the magnetic flux is constant. Such dc servomotors are called permanent magnet dc servomotors.

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he Gain in Bode diagrams A number greater than unity has a positive value in decibels, while a number smaller than unity has a negative value. The log-magnitude curve for a constant gain K is a horizontal straight line at the magnitude of 20 log K decibels. The phase angle of the gain K is zero. The effect of varying the gain K in the transfer function is that it raises or lowers the log-magnitude curve of the transfer function by the corresponding constant amount, but it has no effect on the phase curve.

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Overdamped system : In this case, the two poles of C(s)/R(s) are negative real and unequal. For a unit-step input, R(s) = 1/s and C(s) can be written

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Underdamped systems : In this case, C(s)/R(s) can be written

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First-order factors in Bode diagrams The log magnitude of the first-order factor 1/(1 jωT) is

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General procedure for plotting Bode diagrams First rewrite the sinusoidal transfer function G(jω)H(jω) as a product of basic factors. Then identify the corner frequencies associated with these basic factors. Finally, draw the asymptotic log-magnitude curves with proper slopes between the corner frequencies. The exact curve, which lies close to the asymptotic curve, can be obtained by adding proper corrections. The phase-angle curve of G(jω)H(jω) can be drawn by adding the phase-angle curves of individual factors. The use of Bode diagrams employing asymptotic approximations requires much less time than other methods that may be used for computing the frequency response of a transfer function. The ease of plotting the frequency-response curves for a given transfer function and the ease of modification of the frequency-response curve as compensation is added are the main reasons why Bode diagrams are very frequently used in practice.

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Quadratic factors in Bode diagram Control systems often possess quadratic factors of the form

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Step response of second-order systems Fig: 1 Fig: 2 Second-order system.

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The logarithmic magnitude of 1/jω in decibels is

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Steady-state Errors

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Servo system with velocity feedback: The derivative of the output signal can be used to improve system performance. In obtaining the derivative of the output position signal, it is desirable to use a tachometer instead of physically differentiating the output signal. (Note that the differentiation amplifies noise effects. In fact, if discontinuous noises are present, differentiation amplifies the discontinuous noises more than the useful signal. For example, the output of a potentiometer is a discontinuous voltage signal because, as the potentiometer brush is moving on the windings, voltages are induced in the switchover turns and thus generate transients. The output of the potentiometer therefore should not be followed by a differentiating element.)

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Closed-loop frequency response of unity-feedback systems

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Static acceleration error constant Fig: 1 Response of a type 2 unity-feedback system to a parabolic input.

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Second-order systems settling Time:

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Second-order systems Peak Time: Here the peak time is obtained in terms of . The system is assumed to be underdamped.

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Static position error constant K_v: The steady-state error of the system with a unit-ramp input is given by

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Definitions of transient-response specifications: 5. Settling time, t_s:The settling time is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%). The settling time is related to the largest time constant of the control system. Which percentage error criterion to use may be determined from the objectives of the system design in question.

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Typical test signals: The commonly used test input signals are those of step functions, ramp functions, acceleration functions, impulse functions, sinusoidal functions and the like. With these test signals, mathematical and experimental analyses of control systems can be carried out easily since the signals are very simple functions of time. Which of these typical input signals to use for analyzing system characteristics may be determined by the form of the input that the system will be subjected to most frequently under normal operation. If the inputs to a control system are gradually changing functions of time, then a ramp function of time may be a good test signal.

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Unit-step response of first-order systems

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Second-order systems Maximum overshoot: Here the maximum overshoot is obtained in terms of . The system is assumed to be underdamped.

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Critically damped system: If the two poles of C(s)/R(s) are nearly equal, the system may be approximated by a critically damped one.

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Unit-impulse response of first-order systems

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The transfer function between the motor shaft displacement and the error voltage is obtained from Equations above as follows:

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Effect of load on servomotor dynamics: Most important among the characteristics of the servomotor is the maximum acceleration obtainable. For a given available torque, the rotor moment of inertia must be a minimum. Since the servomotor operates under continuously varying conditions, acceleration and deceleration of the rotor occur from time to time. The servomotor must be able to absorb mechanical energy as well as to generate it. The performance of the servomotor when used as a brake should be satisfactory.

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Impulse response of second-order systems:

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Comparison of steady-state errors in open-loop control system and closed loop control system:

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Introduction to control system Automatic control has played a vital role in the advance of engineering and science. In addition to its extreme importance in space-vehicle systems, missile-guidance systems, robotic systems, and the like, automatic control has become an important and integral part of modern manufacturing and industrial processes. For example, automatic control is essential in the numerical control of machine tools in the manufacturing industries, in the design of autopilot systems in the aerospace industries, and in the design of cars and trucks in the automobile industries. It is also essential in such industrial operations as controlling pressure, temperature, humidity, viscosity, and flow in the process industries. Since advances in the theory and practice of automatic control provide the means for attaining optimal performance of dynamic systems, improving productivity. Relieving the drudgery of many routine repetitive manual operations, and more, most engineers and scientists must now have a good understanding of this field. Some basic terminologies must be defined before the study of control systems.

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DIFFERENTIAL EQUATIONS OF PHYSICAL SYSTEMS

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Complex variable: A complex number has a real part and an imaginary part, both of which are constant. If the real part and/or imaginary part are variables, a complex number is called a complex variable. In the Laplace transformation we use the notation s as a complex variable; that is,

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Block diagram of a closed-loop system:

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Block diagram reduction: It is important to note that blocks can be connected in series only if the output of one block is not affected by the next following block. If there are any loading effects between the components, it is necessary to combine these components into a single block. Any number of cascaded blocks representing non loading components can be replaced by a single block, the transfer function of which is simply the product of the individual transfer functions.

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Transfer functions of cascaded elements:

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Open-loop transfer function and feedforward transfer function:

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Procedures for drawing a block diagram:

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Transfer function of a multiple-loop system

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Modeling in state space Modern control theory: The modern trend in engineering systems is toward greater complexity, due mainly to the requirements of complex tasks and good accuracy. Complex systems may have multiple inputs and multiple outputs and may be time varying. Because of the necessity of meeting increasingly stringent requirements on the performance of control systems, the increase in system complexity, and easy access to large-scale computers, modern control theory, which is a new approach to the analysis and design of complex control systems, has been developed since around 1960. This new approach is based on the concept of state. The concept of state by itself is not new since it has been in existence for a long time in the field of classical dynamics and other fields.

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Transfer function of an interacting system: Fig: 1Two-path interacting system (a) Signal-flow graph, (b) Block diagram.

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Closed-loop system subjected to a disturbance Fig: 1 Closed-loop system subjected to a disturbance.

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State-space representation of nth-order systems of linear differential equations in which the forcing function involves derivative terms: If the differential equation of the system involves derivatives of the forcing function, such as

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State-space model of Electrical systems LCR circuit:

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State-space equations: The equations of x and y can be written as:

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Transfer function of a hydraulic actuator:

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ransfer Functions of Dynamic Elements and Networks:

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Transfer function of an armature-controlled motor

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State-space representation of nth-order systems of linear differential equations in which the forcing function does not involve derivative terms: Consider the following nth-order system:

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Correlation between transfer functions and state-space equations we will study how to derive the transfer function of a single-input-single output system from the state-space equations. Let us consider the system whose transfer function is given by

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Speed control system The basic principle of a Watt's speed governor for an engine is illustrated in the schematic diagram of Figure 1. The amount of fuel admitted to the engine is adjusted according to the difference between the desired and the actual engine speeds.

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Engineering design: Engineering design is the central task of the engineer. It is a complex process in which both creativity and analysis play major roles. Design is the process of conceiving or inventing the forms, parts, and details of a system to achieve a specified purpose. Design activity can be thought of as planning for the emergence of a particular product or system. Design is an innovative act whereby the engineer creatively uses knowledge and materials to specify the shape, function, and material content of a system. The design steps are

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CONTROL SYSTEM DESIGN The design of control systems is a specific example of engineering design. The goal of control engineering design is to obtain the configuration, specifications, and identification of the key parameters of a proposed system to meet an actual need. The control system design process is illustrated in Figure 1. The design process consists of seven main building blocks, which we arrange into three groups: 1. Establishment of goals and variables to be controlled, and definition of specifications (metrics) against which to measure performance

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Open-loop control systems:

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MECHATRONIC SYSTEMS

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Closed loop systems:

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Mathematical Modeling of Dynamic Systems A mathematical model of a dynamic system is defined as a set of equations that represents the dynamics of the system accurately or, at least, fairly well. Note that a mathematical model is not unique to a given system. A system may be represented in many different ways and, therefore, may have many mathematical models, depending on one's perspective. The dynamics of many systems, whether they are mechanical, electrical, thermal, economic, biological, and so on, may be described in terms of differential equations. Such differential equations may be obtained by using physical laws governing a particular system, for example, Newton's laws for mechanical systems and Kirchhoff's laws for electrical systems. We must always keep in mind that deriving a reasonable mathematical model is the most important part of the entire analysis.

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Mason's Gain Rule: Mason's gain formula is a method for finding the transfer function of a linear signal-flow graph. Let us consider a system,

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Closed-loop control versus open-loop control:

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Block diagrams: A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals. Such a diagram depicts the interrelationships that exist among the various components. Differing from a purely abstract mathematical representation, a block diagram has the advantage of indicating more realistically the signal flows of the actual system. In a block diagram all system variables are linked to each other through functional blocks. The functional block or simply block is a symbol for the mathematical operation on the input signal to the block that produces the output. The transfer functions of the components are usually entered in the corresponding blocks, which are connected by arrows to indicate the direction of the flow of signals. Note that the signal can pass only in the direction of the arrows. Thus a block diagram of a control system explicitly shows a unilateral property.

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Transfer function of an armature controlled DC motor: The armature-controlled DC motor uses the armature current i_a as the control variable. The stator field can be established by a field coil and current or a permanent magnet. When a constant field current is established in a field coil, the motor torque is

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Mechanical system: Consider the satellite attitude control system shown in Figure 1. The diagram shows the control of only the yaw angle e. (In the actual system there are controls about three axes.) Small jets apply reaction forces to rotate the satellite body into the desired attitude. The two skew symmetrically placed jets denoted by A or B operate in pairs. Assume that each jet thrust is F/2 and a torque T = Fl is applied to the system. The jets are applied for a certain time duration and thus the torque can be written as T(t). The moment of inertia about the axis of rotation at the center of mass is J.

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Nonlinear systems: A system is nonlinear if the principle of superposition does not apply. Thus, for a nonlinear system the response to two inputs cannot be calculated by treating one input at a time and adding the results. Examples of nonlinear differential equations are

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Signal-flow graph models

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THE TRANSFER FUNCTION OF LINEAR SYSTEMS The transfer function of a linear system is defined as the ratio of the Laplace transform of the output variable to the Laplace transform of the input variable, with all initial conditions assumed to be zero. The transfer function of a system (or element) represents the relationship describing the dynamics of the system under consideration. A transfer function may be defined only for a linear, stationary (constant parameter) system. A non-stationary system, often called a time-varying system, has one or more time-varying parameters, and the Laplace transformation may not be utilized. Furthermore, a transfer function is an input-output description of the behavior of a system.

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Block Diagram Transformations: The block diagram representation of a given system often can be reduced to a simplified block diagram with fewer blocks than the original diagram. Some of the block diagram transformations are given.

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Transfer function: The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. Consider the linear time-invariant system defined by the following differential equation:

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Transfer function of a complex system

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State-space model of mechanical systems: Referring for equations,

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Transfer functions of nonloading cascaded elements

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Transfer function of the DC motor