Concept explainers
Figure P5.33 shows a noninverting operational amplifier.
FIGURE P5.33 a. Noninverting amplifier; b. block diagram
Assuming the operational amplifier is ideal,
a. Verify that the system can be described by the following two equations:
b. Check that these equations can be described by the block diagram of Figure P5.33(b).
c. Use Mason's rule to obtain the closed-loop system transfer function
d. Show that when
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Chapter 5 Solutions
CONTROL SYSTEMS ENGINEERING
- Consider a physical system with a three-dimensional state space. An orthonormal basis of the state space is chosen. In this basis the Hamiltonian is represented by the matrix H = [2 1 0 1 2 0 0 0 3] What are the possible results when the energy of the system is measured? A particle is in the state| psi > which is represented in this basis as, with i = Squareroot (-1), 1/Squareroot 3 [i -i i] Find , , and Delta H.arrow_forwardThe state z(t) of a dynamical system is solution of equation ż(t) + aż(t) + 25z(t) = 12, with a = 7.3. Calculate the Peak time of the response.arrow_forwardQuestion 3. Consider a mass-spring translational mechanical system in series. The spring has a nonlinear characteristics such that the relationship between the spring force and the spring displacement can be described mathematically f.(t) = 3x. Assume that the applied force is f(t) = 6+ 8f (t), where 8f(t) is a small force about the 6 Newton constant value. Assuming the output to be the displacement of the mass, obtain the state space representation of the system about the equilibrium displacement.arrow_forward
- Consider the following mechanical system: k m +f b d²y(t) +b- dy(t) + ky(t) = f (t) m %3D dt? dt Obtain the state space model of the system with input f (t) and output y(t). Calculate the system matrices for m = 1, k = 1 and b = 2. Check the stability by using the second method of Lyapunov. 3.arrow_forwardThe Gilles & Retzbach model of a distillation column, the system model includes the dynamics of a boiler, is driven by the inputs of steam flow and the flow rate of the vapour side stream, and the measurements are the temperature changes at two different locations along the column. The state space model is given by: x = 0 00 -30.3 0.00012 -6.02 0 0 0 -3.77 00 0 -2.80 0 0 Is the system?: a. unstable b. C. not unstable x+ 6.15 0 0 0 0 3.04 0 0.052 not asymptotically stable d. asymptotically stable -1 u y = 0 0 0 0 -7.3 0 0 -25.0 Xarrow_forward1. Consider the state equation d x1 = dt X2 0 -8 -6 1 ] ( 2 ) x1 x2 (1) where x1 and x2 are state variables. (a) Determine the eigenvalues and eigenvectors of the state matrix. (b) Let (t) be the state transition matrix. Determine (t) through eigenvector expansion. Hint: A useful formula is the inversion of 2 × 2 matrix as follows. a b C d = 1 d -b ad - bc -C a (c) The state equation (1) is subjected to the following initial conditions x(0) = (2) (3) Determine the solution of (1) through use of the state transition matrix. (d) Determine the stability of the system. Will the response of the system oscillate? Explain why.arrow_forward
- Represent the translational mechanical system shown in state space, where x3(t) is the output.arrow_forward1 / 1 Problem No. 1 1A. 100% + 1B. Consider the translational mechanical system shown in Figure P4.17. A 1-pound force, f(t), is applied at t = 0. If fy = 1, find K and M such that the response is characterized by a 4-second settling time and a 1-second peak time. Also, what is the resulting percent overshoot? [Section: 4.6] 70) 0000 31/1 10000 K FIGURE P4.17 Given the translational mechanical system of Figure P4.17, where K = 1 and f(1) is a unit step. find the values of M and ƒ, to yield a response with 17% overshoot and a settling time of 10 seconds. [Section: 4.6]arrow_forward2. The state-equations of the control system shown in Fig.1 is: U(s) 2 Y(s) s(s? + 3s + 12)arrow_forward
- P2. At first-order system and its response to a step input are shown in fig. P2; what are the system parameters? c(+), 2 K 0.2 Fig. P2 + K s+xarrow_forward2. Consider the state equation x1 1 20 x1 d x2 = 0 10 x2 dt x3 001 x3 where x1, x2 and 23 are state variables. Please answer the following questions. (a) The state matrix (4) 1 20 A = 0 1 0 (5) 0 0 1 has three-fold eigenvalues with \₁ = = A2 A3 1. Find all independent eigenvectors corresponding to this eigenvalue. (b) Find the modal matrix M associated with the state matrix A. Does M-1 AM lead to a Jordan form or not? Hint: The modal matrix M turns out to be a diagonal matrix. For a diagonal matrix, its inverse is given by a 00 0b0 -1 1/a 0 0 = 0 1/b 0 00 с 0 0 1/c 1 (6) (c) Find the state transition matrix (t). (d) Determine the stability of the system. Please justify your answer.arrow_forwardA certain mass is driven by base excitation through a spring (Figure P4.13). Its parameter values are m = 100 kg, c = 1000 N * s/m, and k = 10,000 N/m. Determine its peak frequency w_p, it’s peak M_p, and its bandwidth.arrow_forward
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