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(a) The control input is the elevator deflection angle u(t) = 8(t). Define the output as y(t) = 0(t), the pitch angle. The following linear time-invariant ODE expresses the relationship between u(t) and y(t) under certain conditions (it is derived from linearized equations of motion of the aircraft): ÿ +0.74 ÿ + 0.92 y R(s): C(s) State the order of this ODE. Use the ODE and a Laplace transform table to derive the open-loop (or forward-path) transfer function, H(s) = Y(s)/U(s) = (s)/A(s). Submit your calculations. (b) Compute the finite zeros and finite poles of H(s) and state the number of infinite zeros of H(s). Is the transfer function H(s) stable, marginally stable, or unstable? Justify your answer. = = 1.2 Parts (c) (e) pertain to the unity negative feedback controller shown below, in which Odes (s) is the desired pitch angle, H(s) is the transfer function computed in part (a), and K(s+1) -, a proportional-integral (PI) controller with a single control gain K. S R(s) 0- 0.18 u C(s) H(s) - Y(s)

(a) The control input is the elevator deflection angle u(t) = 8(t). Define the output as y(t) = 0(t), the pitch angle. The following linear time-invariant ODE expresses the relationship between u(t) and y(t) under certain conditions (it is derived from linearized equations of motion of the aircraft): ÿ +0.74 ÿ + 0.92 y R(s): C(s) State the order of this ODE. Use the ODE and a Laplace transform table to derive the open-loop (or forward-path) transfer function, H(s) = Y(s)/U(s) = (s)/A(s). Submit your calculations. (b) Compute the finite zeros and finite poles of H(s) and state the number of infinite zeros of H(s). Is the transfer function H(s) stable, marginally stable, or unstable? Justify your answer. = = 1.2 Parts (c) (e) pertain to the unity negative feedback controller shown below, in which Odes (s) is the desired pitch angle, H(s) is the transfer function computed in part (a), and K(s+1) -, a proportional-integral (PI) controller with a single control gain K. S R(s) 0- 0.18 u C(s) H(s) - Y(s)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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g) Suppose you want to design the controller C(s) to produce a closed-loop response to a
reference step input that has no more than 30% maximum overshoot (Mp ≤ 0.30) and a peak time
of no more than 8 sec (tp ≤ 8). Given the specifications on Mp and tp , compute the constraints
on the damping ratio ζ and the damped natural frequency ω, assuming that the closed-loop system
is approximated as a 2nd-order underdamped system. Sketch the allowable regions in the
complex plane (the s-plane) that these constraints define, and shade in the regions where the poles
should not be placed. Submit your calculations and your sketch.

(a) The control input is the elevator deflection angle u(t) = 8(t). Define the output as y(t) = 0(t), the pitch angle. The following linear time-invariant ODE expresses the relationship between u(t) and y(t) under certain conditions (it is derived from linearized equations of motion of the aircraft): ÿ +0.74 ÿ + 0.92 y State the order of this ODE. Use the ODE and a Laplace transform table to derive the open-loop (or forward-path) transfer function, H(s) = Y(s)/U(s) = (s)/A(s). Submit your calculations. R(s): C(s): (b) Compute the finite zeros and finite poles of H(s) and state the number of infinite zeros of H(s). Is the transfer function H(s) stable, marginally stable, or unstable? Justify your answer. = = 1.2 0.18 u Parts (c) (e) pertain to the unity negative feedback controller shown below, in which Odes (s) is the desired pitch angle, H(s) is the transfer function computed in part (a), and K(s+1) -, a proportional-integral (PI) controller with a single control gain K. S R(s) 0 C(s) H(s) - Y(s)
Transcribed Image Text:(a) The control input is the elevator deflection angle u(t) = 8(t). Define the output as y(t) = 0(t), the pitch angle. The following linear time-invariant ODE expresses the relationship between u(t) and y(t) under certain conditions (it is derived from linearized equations of motion of the aircraft): ÿ +0.74 ÿ + 0.92 y State the order of this ODE. Use the ODE and a Laplace transform table to derive the open-loop (or forward-path) transfer function, H(s) = Y(s)/U(s) = (s)/A(s). Submit your calculations. R(s): C(s): (b) Compute the finite zeros and finite poles of H(s) and state the number of infinite zeros of H(s). Is the transfer function H(s) stable, marginally stable, or unstable? Justify your answer. = = 1.2 0.18 u Parts (c) (e) pertain to the unity negative feedback controller shown below, in which Odes (s) is the desired pitch angle, H(s) is the transfer function computed in part (a), and K(s+1) -, a proportional-integral (PI) controller with a single control gain K. S R(s) 0 C(s) H(s) - Y(s)
(c) Suppose that you want the pitch angle to increase at a constant acceleration of 1 deg/s². To achieve this, you set the desired pitch angle to a parabolic function, r(t) = ½t² · 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 K. Submit your calculations. (d) Compute the closed-loop transfer function, Hcz (s) = Y(s)/R(s). Write the characteristic equation of the closed-loop system in the form 1 + KL(s) = 0. Submit your calculations. (e) Plot the positive root locus for the closed-loop system in MATLAB using the command rlocus (L). Use this plot to find the range of K for which the closed-loop system is stable. Submit your MATLAB plot and explain how you used it to find the range of K. W (1) Now we introduce a disturbance input W(s) = where W is a constant, into the feedback sk controller as shown below. The transfer functions H(s) and C(s) are the same as in parts (c) (e), and we set R (s) = 0. What type of disturbance input W(s) (step, ramp, or parabola) can the system reject (i.e., the steady-state error ess is a nonzero constant)? Submit your calculations. R(s) 0- C(s) + W(s) H(s) - Y(s)
Transcribed Image Text:(c) Suppose that you want the pitch angle to increase at a constant acceleration of 1 deg/s². To achieve this, you set the desired pitch angle to a parabolic function, r(t) = ½t² · 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 K. Submit your calculations. (d) Compute the closed-loop transfer function, Hcz (s) = Y(s)/R(s). Write the characteristic equation of the closed-loop system in the form 1 + KL(s) = 0. Submit your calculations. (e) Plot the positive root locus for the closed-loop system in MATLAB using the command rlocus (L). Use this plot to find the range of K for which the closed-loop system is stable. Submit your MATLAB plot and explain how you used it to find the range of K. W (1) Now we introduce a disturbance input W(s) = where W is a constant, into the feedback sk controller as shown below. The transfer functions H(s) and C(s) are the same as in parts (c) (e), and we set R (s) = 0. What type of disturbance input W(s) (step, ramp, or parabola) can the system reject (i.e., the steady-state error ess is a nonzero constant)? Submit your calculations. R(s) 0- C(s) + W(s) H(s) - Y(s)
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