Excercise 4: PI control for a first-order plant. Suppose you are to design a feedback controller for afirst-order plant depicted in the figure below:ControllerPlantКyиКрTS1kiThis configuration is referred to as a proportional-integral (PI) controller. You are to design the controllerto satisfy some given time-domain specifications.(a) Find the (closed-loop) transfer function Gyr from r to y (see hw02)(b) Determine the steady-state error for a unit step input (Hint: e r - y).(c) Find the transfer function Gum from n to u(d) Determine kp and ki such that the feedback controlled system has damping ratio Ç = 0.5 and fre-quency wo. (Hint: the desired denominator polynomial for a closed-loop transfer function is of theform: d(s) s22wos +w2.)

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Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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. Question 04: A process has a liquid storage tank has an open loop transfer function given by G(s). a) Since it is desired that the closed loop system presents Mp = 10% and Ts = 4.5 sec, design a PID controller using the LGR method. Explain why this controller should be used. b) With the controller designed, obtain the closed loop transfer function of the system. Note: Consider the step-type reference and unitary and negative feedback and response.
below is the representation of a linear control system in the state space.a) examine the controllability and observability of this systemb) estimate the unit digit response by finding the current eigenvalues of this systemC) find the feedback Matrix K= (k1 k2) that must be applied so that the system can be a two - layer root in-2
Consider the two feedback control strategies shown in Fig. (withG=GvGpGm) and the following transfer func-tions:Gp(s)=Gd(s)=2(1−s)/s,Gv(s)=2,Gm(s)=1(a)Design an IMC controller,G∗c,usingafilter,f=1/(τ꜀s+1).(b)Suppose that the IMC controller will be implemented as a controller Gc in the classical feedback control configurationof (a). Derive an expression forGcandreportitinastandard form (e.g., P, PI, or PID).
the unity negative feedback controller shown below, in whichR(s) = thetades(s) is the desired pitch angle, H(s) is the transfer function computed in part (a), and C(s) =K(s+1)/s, a proportional-integral (PI) controller with a single control gain K. (a) Suppose that you want the pitch angle to increase at a constant acceleration of 1 deg/s2. Toachieve this, you set the desired pitch angle to a parabolic function, r(t) = 0.5t2 * us(t), where us(t) is the unit step function. What is the system type n? Calculate the steady-state error ess of the closed-loop system as a function of ?(b) Compute the closed-loop transfer function, Hcl(s) = Y(s)/R(s). Write the charateristic equation of the closed-loop system in the form 1 + KL(s) =0
When a unit step input is applied to the closed-loop control of a system with G(s) function and controlled by proportional control, the system operates at the point A= -2.5+j3.57. A) For the relevant operating point, Calculate the proportional gain K. Find the value of ζ. Find the percent (%OS) overshoot Find the natural frequency of the point of interest. B) For the PD controller to be added when the same system is wanted to be operated at the point B= -7.8+j8.77, Find the controller zero value. Find the new K value. Find the ζ value of the relevant point. Find the percent (%OS) overshoot Find the natural frequency of the point of interest.
Given a PI system with a proportional gain constant of 2 and an integral gain constant of 3. Initiallythe controller is running at a steady state output of 50%. For time = 0 to 1s there is no error. Fortime = 1s to 2s there is a constant error of +6%. For time = 2s to 3s there is an error of -4%. Findthe output of the controller at the end of each of the three time spans
Given the error of Figure, plot a graph of a proportional-integral controller output as a function of time. Let KP=3.8, KI = 0.4 s-1, and p(0) = 19.5%. given ep(%) = 2.5 %.
Question 1) In a dynamic system with expression [img1] in the time domain, all initial conditions are assumed to be zero. This system will be used in the standard unit feedback control system, and the reference input is unit stepfunction, the controller is a PD controller given with [img2]. Accordingly, fill in the table below. The numerical solver (Configuration Parameters under Simulation in the menu) parameters for this stage are shown above. Table 1: Design Results Exceed (%) ?? Rise time (s) ?? Sitting time (s) ?? Peak time (s) ??
Explain the concept of linear time-invariant (LTI) systems and their applications in control theory and signal processing.
3. Consider the system a) Use state feedback ? = ?? to assign the eigenvalues of ? + ?? at −0.5 ± ?0.5. Plot TT x(t) = [x1(t), x2(t)] for the open- and closed-loop system with x(0) = [−0.6, 0.4] . b) Design an identity observer with eigenvalues at −? ± ?, where ? > 0. What is the observer gain ? in this case? c) Use the state estimate ?̂ from (b) in the linear feedback control law ? = ? ?̂, where ? was found in (a). Derive the state-space description of the closed-loop system. If ? = ? ?̂ + ?, what is the transfer function between ? and ?? TT d) For ?(0) = [−0.6, 0.4] and ?̂(0)= [0, 0] , plot ?(?), ?̂ (?), ?(?), and ?(?) of the closed- loop system obtained in (c) and comment on your results. Use ? = 1, 2, 5, and 10, and comment on the effects on the system response. Remark: This exercise illustrates the deterioration of system response when state observers are used to generate the state estimate that is used in the feedback control law.
Discuss the concept of transfer functions in system modeling and their significance in control theory.
below is the representation of a linear control system in the state space.a) estimate the unit digit response by finding the current eigenvalues of this systemb) find the feedback Matrix K= (k1 k2) that must be applied so that the system can be a two - layer root in-2

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