Modern Control Design With MATLAB and SIMULINK - Ebook download as PDF File .pdf), Text File .txt) or read book online. Download as PDF, TXT or read online from Scribd MATLAB, SIMULINK, and the Control System Toolbox 11 Advanced Topics in Modern Control MATLAB/SIMULINK combination has become the single most common - and industry-wide standard - software in the analysis and design of modern control.

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The idea of computer-aided design and analysis using MATLAB with the Ogata, K., Modern Control Engineering, 3rd ed., Prentice Hall, Englewood Cliffs. Request PDF on ResearchGate | Modern Control Systems Analysis and Design Using MATLAB and SIMULINK | From the Publisher:Modern Control Systems. Written for students and practicing engineers, this book presents the theory and applications of classical and state-space control system design. Topics covered.

Of particular importance are the protocol stacks that drive industry-standard interfaces like USB. The hardware blocks are put together using computer-aided design tools, specifically electronic design automation tools; the software modules are integrated using a software integrated development environment. Once the architecture of the SoC has been defined, any new hardware elements are written in an abstract hardware description language termed register transfer level RTL which defines the circuit behavior, or synthesized into RTL from a high level language through high-level synthesis.

These elements are connected together in a hardware description language to create the full SoC design. The logic specified to connect these components and convert between possibly different interfaces provided by different vendors is called glue logic.

Further information: Functional verification and Signoff electronic design automation Chips are verified for logical correctness before being sent to a semiconductor foundry. Bugs found in the verification stage are reported to the designer. Traditionally, engineers have employed simulation acceleration, emulation or prototyping on reprogrammable hardware to verify and debug hardware and software for SoC designs prior to the finalization of the design, known as tape-out. Both technologies, however, operate slowly, on the order of MHz, which may be significantly slower — up to times slower — than the SoC's operating frequency.

This is used to debug hardware, firmware and software interactions across multiple FPGAs with capabilities similar to a logic analyzer. In parallel, the hardware elements are grouped and passed through a process of logic synthesis , during which performance constraints, such as operational frequency and expected signal delays, are applied.

This generates an output known as a netlist describing the design as a physical circuit and its interconnections. As an alternative to singularity inputs which are often difficult to apply in practical cases. This fact is illustrated in Figure 2. While the singularity functions and their relatives are useful as test inputs for studying the behavior of control systems.

An unstable system will have a transient response shooting to infinite magnitudes. We shall study next how such a model can be obtained. We will see how this is done when we discuss the response to singularity functions in Section 2. The steady-state. Of course. The complex space representation of the harmonic input given by Eq.

Studying a linear system's characteristics based upon the steady-state response to harmonic inputs constitutes a range of classical control methods called the frequency response methods. Such methods formed the backbone of the classical control theory developed between 1 For these reasons. Qlwt 2. For these powerful reasons.

Modern control techniques still employ frequency response methods to shed light on some important characteristics of an unknown control system. This is an advantage. A simple choice of the harmonic input. If we choose to write the input and output of a linear system as complex functions. You will see that the equation is satisfied in each case. The particular solution is of the same form as the input.

We will shortly see the implications of a complex response amplitude. Consider a linear. While the transient response of a linear. By obtaining a steady-state response to the complex input given by Eq. In the complex space. We can also express the frequency response. You can easily show that if the harmonic input has a non-zero phase. From Eq. If we excite the system at various frequencies.

Instead of the real and imaginary parts. The phasor representation of the steady-state response amplitude is depicted in Figure 2. G ico. The length of the phasor in the complex space is called its magnitude. Equations 2. Representation of a complex quantity as a vector in the complex space is called a phasor. The magnitude of a phasor represents the amplitude of a harmonic function.

Bode plots can be plotted quite easily. Polar plots have an advantage over the frequency plots of magnitude and phase in that both magnitude and phase can be seen in one rather than two plots.

Bode plots are cumbersome to construct by hand. In Bode plots. When talking about stability and robustness properties. With the availability of personal computers and software with mathematical functions and graphics capability.

Such a plot of G ico in the complex space is called a polar plot since it represents G ico in terms of the polar coordinates. Referring to Figure 2. Since the range of frequencies required to study a linear system is usually very large.

In general. Despite this. G is the name given to the frequency response of the linear. If you don't specify w. The guitar player makes each string vibrate at a particular frequency. Each string of the guitar is capable of being excited at many frequencies. Before we do that.

These coefficients should be be specified as follows. The example given below will illustrate what Bode plots look like. Musical notes produced by a guitar are related to its frequency response.. Instead of plotting the Bode plot. In the bode command. When the switch.

Just like the guitar. High pitched voice of many a diva has shattered the opera-house window panes while accidently singing at one of the natural frequencies of the window! If a system contains energy dissipative processes called damping. A practical limitation of Bode plots is that they show only an inter- polation of the gain and phase through selected frequency points.

The frequencies at which a system can be excited are called its natural or resonant frequencies. One could determine from the peaks the approximate values of the natural frequencies.

The input to the system is the applied voltage.

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An undamped system. A natural frequency is indicated by a peak in the gain plot. When we use the word excited. To verify whether this is the exact natural frequency. We are assuming. The Bode plots shown in Figure 2. The frequency response is used to define a linear system's property called bandwidth defined as the range of frequencies from zero up to the frequency.

The plot is in polar coordinates. Linear systems with G ico having a higher degree denominator polynomial than the numerator polynomial in Eq. For a general system. In the present plot. Let us now draw a polar plot of G ico as follows note that we need more frequency points close to the natural frequency for a smooth polar plot.

The resulting polar plot is shown in Figure 2. It can be shown rigourously that the Laplace integral converges only if u t is piecewise continuous i. The polar curve is seen in Figure 2. Most of the commonly used input functions are Laplace transformable. Here we would like to consider the total response both transient and steady-state of a linear. The term bounded implies that a function's value lies between two finite limits.

The direction of increasing CD is shown by arrows on the polar curve. The Laplace transform of a function u t is defined only if the infinite integral in Eq. We saw how the representation of a harmonic input by a complex function transformed the governing differential equations into a complex algebraic expression for the frequency response.

For a general input. The convergence of the Laplace integral depends solely upon the shape of the function. Some important properties of the Laplace transform are stated below. If we apply the real differentiation property successively to the higher order time derivatives of f t assuming they are Laplace transformable. For such unknown systems.

Modern Control Design: With MATLAB and SIMULINK

By applying known inputs such as the singularity functions or harmonic signals and measuring the output. Fourier transform is widely used as a method of calculating the Input. The transfer function representation of a system is widely used in block diagrams. It is easy to see that if the input.

A special transform. The latter relationship is easily obtained by comparing Eq. To do so. G itw. U s Output. Note that inverse Laplace transform. The Laplace variable. We can grasp this fact by applying the inverse Laplace transform. U ico as follows: The transfer function. However, use of G s involves interpreting system characteristics from complex rather than purely imaginary numbers.

The roots of the numerator and denominator polynomials of the transfer function, G s , given by Eq. The denominator polynomial of the transfer function, G s , equated to zero is called the characteristic equation of the system, given by. The roots of the characteristic equation are called the poles of the system.

In terms of its poles and zeros, a transfer function can be represented as a ratio of factorized numerator and denominator polynomials, given by the following rational expression:. As in Eq. Such systems are said to be proper. Also, note that some zeros, z,, and poles, PJ, may be repeated i. Such a pole or zero is said to be multiple, and its degree of multiplicity is defined as the number of times it occurs.

Finally, it may happen for some systems that a pole has the same value as a zero i. Then the transfer function representation of Eq.

Pole-zero cancelations have a great impact on a system's controllabilty or observability which will be studied in Chapter 5. To get a better insight into the characteristics of a system, we can express each quadratic factor such as that on the left-hand side of Eq. The damping ratio, g, governs how rapidly the magnitude of the response of an unforced system decays with time.

For a mechanical or electrical system, damping is the property which converts a part of the unforced system's energy to heat, thereby causing the system's energy - and consequently the output - to dissipate with time. Examples of damping are resis- tances in electrical circuits and friction in mechanical systems. From the discussion following Eq.

For the present example, the poles are found by solving Eq. These numbers could also have been obtained by comparing Eq. The natural frequency agrees with our calculation in Example 2. One can see the dependence of the response, y t , on the damping-ratio, g, in Figure 2. Clearly, the larger the value of the damping-ratio, g, the faster the response decays to zero.

As soon as we see a linear system producing an unbounded response to a bounded input i. A further discussion of stability follows a little later. Locations of poles and zeros in the Laplace domain determine the characteristics of a linear, time-invariant system. Some indication of the locations of a poles and zeros can be obtained from the frequency response, G icu.

Let us go back to Figure 2. Due to the presence of a zero at the origin see Eq. The difference between the number of zeros and poles in a system affects the phase and the slope of the Bode gain plot with frequency in units of dB per decade of frequency , when the frequency is very large i.

Note that the expressions in Eq. For example, the transfer function in Eq. Three different output variables in the Laplace domain are of interest when the aircraft is displaced from the equilibrium point defined by a constant angle of attack, O.

Q, a constant longitudinal velocity, DO, and a constant pitch-angle, OQ: The input variable in the Laplace domain is the elevator angle, d s. The three transfer functions separately defining the relationship between the input, 5 5 , and the three respective outputs, v s , a s , and 0 s , are as follows:. Since we know that the denomi- nator polynomial equated to zero denotes the characteristic equation of the system, we can write the characteristic equation for the aircraft's longitudinal dynamics as.

Comparing the result with that of Example 2. These values are the following: Note that the CST command damp also lists the eigenvalues, which are nothing but the roots of the characteristic polynomial same as the poles of the system.

We will discuss the eigenvalues in Chapter 3. Alternatively, we could have used the intrinsic MATLAB function roots to get the pole locations as the roots of each quadratic factor. As expected, the poles for each quadratic factor in the characteristic equation are complex conjugates. Instead of calculating the roots of each quadratic factor separately, we can multiply the two quadratic factors of Eq. The first mode is highly damped, with a larger natural frequency 1.

The second characteristic mode is very lightly damped with a smaller natural frequency 0. While an arbitrary input will excite a response containing both of these modes, it is sometimes instructive to study the two modes separately. There are special elevator inputs, 8 s , which largely excite either one or the other mode at a time.

You may refer to Blakelock [3] for details of longitudinal dynamics and control of aircraft and missiles. Figures 2.

From the Bode plots Figures 2. The peaks due to complex poles sometimes disappear due to the presence of zeros in the vicinity of the poles. As expected, the natural frequencies agree with the values already. However, Figure 2. Hence, both modes essentially consist of oscillations in the pitch angle, 9 ia. The present example shows how one can obtain an insight into a system's behavior just by analyzing the frequency response of its transfer function s.

Note from Figures 2. Such a decay in the gain at high frequencies is a desirable feature, called roll-off, and provides attenuation of high frequency noise arising due to unmodeled dynamics in the system. We will define sensitivity or robustness of a system to transfer function variations later in this chapter, and formally study the effects of noise in Chapter 7. Using Eq. A system with transfer function having poles or zeros in the right-half s-plane is called a non-minimum phase system, while a system with all the poles and zeros in the left-half s-plane, or on the imaginary axis is called a minimum phase system.

We will see below that systems which have poles in the right-half s-plane are unstable. This usually results in an unacceptable transient response. Popular examples of such systems are aircraft or missiles controlled by forces applied aft of the center of mass.

For this reason, a right-half plane zero in an aircraft or missile transfer function is called 'tail-wags- the-dog zero'. Control of non-minimum phase systems requires special attention. Before we can apply the transfer function approach to a general system, we must know how to derive Laplace transform and inverse Laplace transform of some frequently encountered functions. This information is tabulated in Table 2.

Note that Table 2. It is interesting to see in Table 2. Let us assume that the system has the. The output. Let us first derive the system's governing differential equation by applying inverse Laplace transform to the transfer function with zero initial conditions. To understand why this is so. Here we will apply a similar approach to find out a linear system's response to singularity functions.

Then the residue command is used as follows to give the terms of the partial fraction expansion: Laplace transform. All one has to do is to specify the numerator and denominator polynomials of the rational function in s. For a system with complex poles such as Example 2. If a pole. In terms of the elements of p and k. We had ended Section 2. One can thus obtain G s from g t by applying the Laplace transform. Then from Eqs. Since the transfer function contains information about a linear system's characteristics.

We know from Table 2. In a manner similar to the impulse response. Using the partial fractions ex- pansion of G s. Since the poles can also be represented in terms of their natural frequency. The same computation in a low-level language. Note that both t and g are vectors of the same size.

This is a characteristic of a underdamped. Substituting these numerical values in Eq. Taking the inverse Laplace transform of Eq. Such a program is the M-file named impresp. We can also evaluate the step response.

If the system is strictly proper i. We know from Example 2. Instead of having to do inverse Laplace transformation by hand. A i For increased accuracy. We postpone the discussion of the CST command impulse until Chapter 4. The M-file impresp.

Usage of CST command impulse yields the same result. Note the programming steps required in impresp to identify the multiplicity of each pole of G s. We must begin with the specification of the transfer function as follows: The M-file stepresp. The GUI tool associated with the command step also lets you get the values of s t and t at any point on the step response curve by merely clicking at that point.

Note that if G s is a proper transfer function i. The CST command step is a quick way of calculating the step response. Note that the plot of the impulse response clearly shows an initial. The time taken by the phugoid mode to decay to zero is an indicator of the sluggishness of the longitudinal dynamics of the airplane.

It is also clear that the impulse response excites both the modes. This behavior meets our expectation from the natural frequencies and damping-ratios of the two modes calculated in Example 2. Since we know how to find the response of a linear. The integral on the right- hand side of Eq. At the end of Section 2. Note that we can apply a change of integration variable.

Then by the superposition principle Eq. Such an Table 2. Substituting Eq. The use of convolution integral of Eq. Examples 2. Such a numerical evaluation of the convolu- tion integral is performed by the M-file. Note that response calls impresp. The M-file response. The simplest numerical integration or quadrature is the assumption that the integrand is constant in each time interval. We first specify the time vector. The smaller the magnitude of the steady-state error.

We briefly discussed the implications of each of these in Chapter 1 using the car-driver example. There are three properties which determine whether a control system is good or bad. The calculated response is plotted in Figure 2. Not all systems have this property. More generally. Note the ease with which response. For such systems. The performance parameters are usually defined for the step response.

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Now we are well equipped to define each of these three properties precisely. There are some performance parameters that indicate the speed of a control system's response. A more general method of evaluating response of even multi-input. If the error. Let us first consider the performance of a control system. Then the settling time can be determined as the time taken when the y t settles to within 2 percent of y oo. For a second order system such as the one considered in Example 2.

The performance of a control system is determined by the locations of its poles in the Laplace domain. Another closed-loop configuration is also possible in which H s is placed in the feedback path of or in parallel with G s.

Many times. Mp are contradictory. We saw in Chapter 1 an example of how a controller performs the task of controlling a plant in a closed-loop system by ensuring that the plant output. The controller applies an input. The steady-state error. As may be clear from examining expressions for step or impulse response.

Such a closed-loop control system is said to have the controller. Tp and those determining the deviation of the response from the desired steady-state value such as peak value.

Consider a general single-input. Note that the performance parameters are intimately related to the damping-ratio. How the control system characteristics are modified to achieve a desired set of performance parameters is an essential part of the control system design. Such poles that dominate the control system's performance are called the dominant poles.

In such cases. We will discuss later what are the precise requirements for stability. Such a system is called a tracking system. Going back to Figure 2. Examples of tracking systems are a telescope tracking a comet. Looking at the block-diagram of Figure 2. An unstable system cannot reach a steady-state. Yd s from the real integration property Eq. Note that the DC gain of the closed-loop transfer function in Figure 2. U s can be increased.

The steady-state error given by Eq. This command is quite useful when the transfer function is too complicated to be easily manipulated by hand. Let us determine the steady-state error of this system if the desired output.

From Example 2. This calls for the antenna to move at a constant angular velocity. Let us see what kind of controller transfer function. For both the choices of H s. Let us see what can be done to reduce the steady-state error of system in Example 2. Note that for the closed-loop system of Figure 2. An example of tracking systems whose desired output is a ramp function is an antenna which is required to track an object moving at a constant velocity. If we can make the steady-state error to a ramp function finite by somehow changing the system.

Then the desired output of the antenna is v d r -c-r t. This process is called control system design. The residues of the partial fraction expansion Eq. We know that the poles of the closed-loop transfer function are distinct i. The direct term c is a null vector. We have seen in Examples 2. We know from the real integration property of the Laplace transform Eq. Classical control assigns a type to a closed-loop system of Figure 2.

When yd t is a ramp function. The error. Since G s in Examples 2. Precisely how many poles H s should have to reduce the steady-state error of a closed- loop system to a particular desired output. In Example 2. From Table 2. Based on our experience with Examples 2. From Examples 2.

Modern Control Design: With MATLAB and SIMULINK

We can now define stability and instability more precisely for linear systems. While nonlinear systems can have more than one equilibrium points.

While discussing steady-state error. In Table 2. The pendulum in Example 2. The system of Example 2. Such linear systems are said to be asymptotically stable. Using MATLAB you can easily obtain a location of the poles and zeros of a system in the Laplace domain with either the intrinsic command roots num and roots den.

Tabular methods such as Routh-Hurwitz were indispensible before the availability of digital computers. We can summarize the stability criteria by saying that if either the real part of any one pole is positive. In this manner. The aircraft of Example 2. From such a plot. If all the poles of a system have real parts less than or equal to zero. In physical systems.

There are the following three categories under which all linear control systems fall in terms of stability: If the real parts of all the poles roots of the denominator polynomial of the transfer function are negative. Such a system is said to be unstable. A stable linear system having all poles with negative real parts is asymptotically stable. Such a system is said to be stable but not asymptotically stable because the response does not tend to an infinite magnitude.

If any pole of the linear system has a positive real part. The root- locus method determine stability simply by investigating whether the return difference becomes zero for any value of s in the right half of the Laplace domain i.

If we see the loci of one or more poles crossing into the right-half s-plane. By the same criteria. Y s has returned to itself. The return difference is a property of the feedback loop. Before the advent of state-space methods. We shall further discuss stability from the viewpoint of state-space methods as opposed to the classical frequency and Laplace domain methods in Chapter 3. Design using graphical methods is an instructive process for single-input.

How far away the locus closest to the imaginary axis is from crossing over into the right half 5-plane also indicates how far away the system is from being unstable. Constructing a root-locus plot by hand is difficult. The Bode plot is a graphical method that we have already seen.

The function G s H s is called the return ratio. A pole on the imaginary axis indicates zero stability margin. Other graphical methods are the root-locus. By drawing a locus of the each of the roots of the return difference function. The closed-loop transfer func- tion. The user can also specify a vector k containing all the values of K for which the roots are to be computed by entering rlocus sys.

In the CST command rlocus sys. The design parameter. KG s H s as a design parameter.: The value of K for which both the poles are on the imaginary axis i. The poles with zero real part correspond to a damping-ratio. You can get the value of gain. Note that this system is different from the one shown in Figure 2. For the root-locus plot of more general systems. We can find the values of the design parameter K for specific pole locations along the root-loci using the CST command rlocfind as follows: To do this you must first plot a root-locus.

Let us plot the root-locus of the closed-loop system as the controller design parameter. For the present example. Introduction of poles and zeros in the open-loop transfer function. The closed-loop transfer function. You can show that both the systems have the same closed-loop transfer function. This choice of H s allows introducing a pole of G s H s at a location that can be varied by changing the design parameter. H s cascade with the plant. G ia H ico.

In the 5-plane. The resulting plot is shown in Figure 2. It employs a polar plot see Section 2. The direction of the Nyquist plot indi- cates the direction of increasing co. Such a polar plot is called a Nyquist plot. The application of Nyquist stability criteria is restricted to linear. The direction of the curves in Figure 2. This fact is depicted in Figure 2. It is quite possible that the two polar plots for positive and negative frequencies of some functions may overlap.

Since G s H s is proper. You may verify that a polar plot of the same frequency response for negative frequencies overlaps the curve shown in Figure 2. Since G s H s of a linear.

G s H s of the system in Figure 2. The Nyquist plot shown in Figure 2. If G s H s has poles on the imaginary axis. Nyquist plot in the G s H s plane. Applying Cauchy's theorem to the contour enclosing the entire right-half s-plane shown in Figure 2. Studying the effect of each detour around imaginary axis poles on the Nyquist plot is necessary.

An anti-clockwise encirclement of G s H s — — 1 is considered positive. Note that Cauchy's theorem does not allow the presence of poles of G s H s any- where on the imaginary axis of. The Nyquist stability criterion is based on a fundamental principle of complex algebra. The system shown in Figure 2. N could be either positive. The proof of Cauchy's theorem is beyond the scope of this book.

The open-loop transfer function G s H s has two poles with positive real parts i. Hence by the Nyquist stability criterion. If a control system meets its performance and stability objectives in the presence of all kinds of expected noises whose mathematical models are uncertain.

The transfer function of the open-loop control-system. Since we have to deal with actual control systems in which it is impossible or difficult to mathematically model all physical processes. We are now in a position to mathematically compare the sensitivities of open and closed-loop systems. We intuitively felt in Chapter 1 that a closed-loop system is more robust than an open-loop system.

In Chapter 1. More specifically. The magnitude of the return difference. The greater the value of the return difference. A circle of unit radius is overlaid on the Nyquist plot. Point A denotes the intersection of G s H s with the negative real axis nearest to the point —7.

We need not confine ourselves to closed-loop systems with a constant controller transfer function when talking about robustness. From Nyquist stability theorem.

The farther away the G s H s locus is from —1. Point A is situated at a distance a from. The Nyquist plot is more intuitive for analyzing stability robustness. It is clear that the gain of G s H s at point B is unity. In the Bode gain plot. From Figure 2. The phase margin is defined as the difference between the phase of G s H s at point B. The gain margin is defined as the factor by which the gain of G s H s can be increased before the locus of G s H s hits the point — 1.

Since the Bode plot is a plot of gain and phase in frequency domain i. The gain present at this frequency is — For a system having a frequency response. The numerical values of gain margin. In Figure 2. The gain and phase margins can be directly read from the resulting plot in which the point —1 of the Nyquist plot corresponds to the point 0 dB.

Another way of obtaining gain and phase margins is from the Nichols plot. Then the command [mag. This procedure is illustrated for the present example by the following commands: Seeing the formidable nature of Eqs.

Here sys denotes the LTI object of the open-loop transfer function. Criteria for achieving stability and performance robustness of multivariable systems are more generally expressed with the use of modern state-space methods rather than the classical single-input. The intersection of the G ia H io Nichols plot with the closed-loop gain contours give the closed-loop gain frequency response. We have seen how the stability robustness of a closed-loop control system is defined by the gain and phase margins determined from the open-loop transfer function.

Single-Output Systems We have seen in the previous sections how the steady-state error. Such a noise is called measurement noise. We have considered variations in the overall transfer function. While designing a closed-loop system. It can be appreciated that for achieving performance robust- ness. Apart from giving the gain and phase margins. To reduce the sensitivity of a closed-loop system to measurement noise or to make the system robust with respect to measurement noise.

The frequencies at the intersection points can be obtained from a Bode gain plot of G ia H ia. Variations in the overall transfer function are called process noise. This requirement results in the magnitude Bode plot of the closed-loop system decaying rapidly or rolling-off with frequency. Classical control techniques of designing single-input. Closed-loop compensation is generally of two types: The derivative term.

We will consider some commonly employed closed-loop compensation techniques. Cascade and feedback compensation represent two alternatives for achieving the same closed-loop characteristics. PID compensators are very common in industrial applications due to their good robustness over a wide frequency range.

Where we insert the compensator in the control system depends largely upon the physical aspects of imple- mentation. We study only cascade compensation techniques here.

Let us consider an example of PID compensation. The poles. Since we can use H s to compensate for the poor characteristics of a plant. The presence of the integral term. Connecting H s and G s in series.

Com aensator Closed-loop sytem S" n 2. To make the steady- state error zero for a step input. Since the plant is of type 0. PI compensator.

How does this closed-loop pole config- uration affect the step response of the system? This question is best answered by comparing the plant's step response with that of the closed-loop system. The step responses are calculated using the M-file stepresp.

Note that while the closed- loop steady-state error is brought to zero. The closed-loop poles are the following: The plant's natural frequency of The gain and phase margins of the plant. There is a reduction in the gain margin of the closed-loop system due to the increased gain at low frequencies. The DC gain of the closed-loop system is brought to 0 dB.

The plant thus has infinite gain and phase margins. In summary. The margins for the closed-loop system are the following: It is desirable to have a well damped closed-loop response. We can write D s as a quadratic factor in s multiplied by a first order polynomial in 5. The general transfer function of the forward path. Although the phase margin is no longer infinite as for the plant. The performance is thus greatly improved by the PID compensation.

Note the settling time of 0. The large 0. PID compensator. Mechani- cally. Note that the PID compensator has a decreasing gain at low frequen- cies.. The Bode plots of the PID compensator and the closed-loop system are compared with those of the plant in Figure 2. The resulting closed-loop gain and phase plots are much flatter compared to the plant. If we neglect the stiffness and inertia of the gimbal and wheel about the jc-axis. H is the angular momentum vector with a constant magnitude.

Operational amplifiers can also be used to represent summing junctions. The noise amplification may interfere with the working of the entire control system.

The chief difficulty in implementing PID compensators is the ideal differentiator i. An operational amplifier has two input voltages. Modern control systems are relatively inexpensively implemented using either passive unpowered or active powered electronic circuits.

An ideal differentiator is difficult to set up. Control systems based upon operational amplifiers and other active circuits. A simpler alternative to operational amplifier is the approximation of the pure integrator by a passive circuit. In the circuit shown in Figure 2. Lead compensation usually requires amplifi- cation of error signals. A lead compensator given by the transfer function of Eq.

Lag compen- sation is relatively simple to use. Phase lag or lead compensation is traditionally employed in a variety of control applications. To combine the desirable properties of the lead and lag compensators.

A lag compensator is useful in reducing the steady-state error of type 0 plants. So that the DC gain of the type 0 open-loop transfer function. A lead compensator is useful for increasing the speed of the closed-loop response i.

Such a compensator is called a lead-lag compensator. The lead circuit in Figure 2. The transfer function of the lag circuit Figure 2. The frequencies. In each circuit. Note that the transfer function suggests that the lead-lag compensator consists of a lag compensator in series with a lead compensator. If we want to achieve a settling time.

The maximum overshoot requirement of 2 percent of steady-state response. C2s] 2. With such a compensator. The maneuverability of an aircraft depends upon the time taken to achieve a desired roll-angle.

The closed-loop transfer function is thus the. Let us now consider a lag-compensator for this task. It is required to design a closed-loop control system to achieve a unit step desired roll-angle in about three seconds.If one knows what output a system will produce when a known input is applied to it. A lead compensator given by the transfer function of Eq. Assume that the satellite is in a circular orbit of radius. The plant's natural frequency of Since the closed-loop gain never crosses the 0 dB line.

A practical modern control system may simultaneously contain several inputs and several outputs. Qlwt 2.

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