Chapter 2 MATHEMATICAL FOUNDATION (a) Poles: s = 0, 0, Zeros: s = −1, − 10; (b) Poles: s = −2, ∞, ∞, ∞. −2, −2; Zeros: s = 0. The pole and zero at s = (c). Automatic Control of Bioprocesses (Control Systems, Robotics and Manufacturing) · Read more Automatic Control Systems, 8th ed. (Solutions Manual) · Read more · Automatic Control Systems, 9th Edition - Solutions Manual . Read more. Systems, 8E, by Kuo,Jun 30, - Shop Systems 8Ed - Kuo and Golnaraghi - Solutions. chuntistsicentcha.gq 9-Fundamentals. Automatic Control Systems 8th Ed.
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SYSTEM BY BENJAMIN C KUO 8TH EDITION Automatic Control Systems, 9th Edition by Farid. Golnaraghi, Benjamin C. chuntistsicentcha.gq - ma, . Welcome to the Web site for Automatic Control Systems, 8e by Benjamin C. Kuo and Farid Golnaraghi. This Web site gives you access to the rich tools and. BENJAMIN C. KUO. Automatic. Control Systems. THIRD EDITION. 2 Control Systems. Automatic EHER bo-. CO-O. EDITION. THIRD. PRENTICE. HALL.
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Dynamical systems and control. Systems and Components vol. Control Systems. Coplanar waveguide circuits, components, and systems. Control Theory and Systems Biology. Define the outputs of integrators as state variables. Transfer functions: Define the state variables as: Transfer function: Forward-path transfer function: The same results as above are obtained.
Output equation: A nonsingular DF transformation matrix T cannot be found. Thus, the DF transformation matrix T is the identity matrix I. Rewrite the differential equations as: Thus, the purpose of R is s This improves the time constant of the system. The equations are in the form of CCF with v as the input. State equation: It depends on how the input and the output branches are allocated. Thus, we should create a new node as shown in the following state diagram.
These are not functions of K. Roots of characteristic equation: The system is uncontrollable. The system is controllable. The system is observable. In this case, the state variables are already defined, and the system is uncontrollable as found out in part a. The system is unobservable. AC a Characteristic equation: State diagram: The system with feedback is unobservable.
System is controllable. V is nonsingular. System is observable. Marginally stable. Marginally stabl e. Two poles in RHP. All poles in the LHP. Two roots in RHP. No roots in RHP. Four roots in RHP. Since K is always positive, the last condition cannot be met by any real value of K. Thus, the system is unstable for all values of K. The auxiliary equation is A s The frequency of oscillation is 1.
The solution of the auxiliary The frequency of oscillation is 2. K 2 Thus, 0 From the s 2. Thus, the final condition for stability is: Characteristic equation: Thus, there is no auxiliary equation.
There is no oscillation. The system response would increase monotonically. The second row of B is zero; thus, the second state variable, x is uncontrollable. The system transfer function is: The missile will tumble end over end.
It is meaningless to conduct a steady-state error analysis. Thus the above results are valid. Unit-step Input: System Transfer Function with N s as Input: Forward-path Transfer function: N s Steady -State Output due to n t: Characteristic Equation: Thus, 0. Settling time: This is less s than the calculated value of 0. Y s R s Unit-Step Response: Rise Time: Routh Tabulation: This is due to the zero of G s that lies in the right-half s-plane.
D b Characteristic Equation: So we can set K to just less than 0. With this value of K, the roots of the characteristic equation are: Forward-Path Transfer Function: As the shaft increases, and the imaginary parts of the open-loop poles also increase. When the shaft is rigid, the poles of the forward-path transfer function are all real. Similar effects are observed for the roots of the characteristic equation with respect to the value of K. Unit-Step Responses: Unit-Step Response: When T z is small, the effect is lower overshoot due to improved damping.
When T overshoot becomes very large due to the derivative effect.
T z z is very large, the improves the rise time, since 1 derivative control or a high-pass filter. The system is p , the closed-loop system is stable.
Closed-loop Transfer Function: As the value of N increases, the gain of the system is increased, and the roots are more preturbed. Intersect of Asymptotes: Finite zeros of Q s: Finite zeros of P s: Root Locus Diagram: Breakaway Points: Breakaway-point Equation: Breakaway Point: P s Asymptotes: None for K c Breakaway-point Equation: The characteristic equation has two roots in the right-half s-plane.
The closed-loop system is stable. The system is unstable for all values of K. We have the maximum time delay for stability is G j For stability, the maximum value of K is K can be increased by This means that the loop gain can be increased by 10 dB from the nominal value.
The gain crossover frequency is 2. The phase margin is The gain is increased to four times its nominal value.
The magnitude curve is raised by To change the GM to 20 dB we need to increase the gain by 1. The forward-path gain for stability is Thus, the gain must be increased by 10 dB, or by a factor of 3.
With the gain at twice its nominal value, the system is stable. Since the system is type 1, the steady-state error due to a step input is 0.
Thus the steady-state error would be infinite. With a pure time delay of 0. Thus, the gain margin is 10 dB. The critical value of K for stability is Thus, the steady-state error due to a unit-step input is zero. From the expression for the ramp-error constant, we see that as a or K goes to infinity, K Thus the maximum value of K v v approaches The difficulties with very large values of K and a are that a high-gain amplifier is needed and unrealistic circuit parameters are needed for the controller.
The forward-path Transfer Function: Since the system is type 1, reducing the gain does not affect the steady-state liquid level to a step input.
Using long division and solve for zero remainder condition. From Eq. The corresponding roots of the characteristic equation roots are: In the present case, the system with the PI controller has a higher maximum value for the sensitivity function.
KP ymax 0. Although the relative damping ratio of the complex roots is 0. Then 0. For 0. Dividing the characteristic equation by the seond-order term. The following frequency-domain results substantiate the design.
Kp PM deg Mr 0. This can be chosen to be 0. Using Eq.
Let us first attempt to compensate the system with a PI controller. The following frequency-domain and various value of K PM deg P ranging from 10 to Next, we try a PID controller. The maximum overshoot is zero, and the rise time is 0. Select a value for a.
The attributes of the response are: The diagram shows that the uncompensated system is marginally stable. To realize a phase margin of 75 deg, we need more than 57 deg of additional phase. Let us add an additional 10 deg for safety. Thus, The following time-domain attributes are obtained by varying the value of T.
The maximum overshoot is Unit-step Response. The attributes of the frequency-domain characteristics are given below. The performance worsens if the value of a is less than The attributes of the frequency response are: The following attributes of the frequency-domain characteristics are obtained.
We can show that the phase margin is not very sensitive to the variation of a when a is near The optimal value of a is around , and the corresponding phase margin is The following results are obtained in the frequency domain.
Thus, the maximum phase margin of The unit-step response is plotted at the end together with those of parts b and c. Thus Root Contour Plot a varies. Process Transfer Function: The following results are obtained for various values of T ranging from 0. A higher or lower value for a will give larger overshoot. As it turns out varying the value of a from does not improve the phase margin.
Then, the value of a is chosen to be The value of T is preferrably to be large. However, if T is too large, rise and settling times will suffer. The following performance attributes of the unit-step response are obtained for various values of a and T. The corresponding frequency-domain characteristics are: Let us add a safety factor by requiring that the desired phase margin is 75 degrees.
We see that a phase margin of 75 degrees can be realized if the gain crossover is moved to 0. Thus the phase-lag controller must provide an attenuation of the new gain crossover frequency. The unit-step response is shown below.