Derive the relationship for damping ratio as a function of

Derive the output responses for all parts of Figure 4.7. [Section: 4.4]

Find the output response, c(t), for each of the systems shown in Figure P4.1. Also find the time constant, rise time, and settling time for each case. [Sections: 4.2, 4.3]

Plot the step responses for Problem 2 using MATLAB.

Find the capacitor voltage in the network shown in Figure P4.2 if the switch closes at t = 0. Assume zero initial conditions. Also find the time constant, rise time, and settling time for the capacitor voltage. [Sections: 4.2, 4.3]

Plot the step response for Problem 4 using MATLAB. From your plots, find the time constant, rise time, and settling time.

For the system shown in Figure P4.3, (a) find an equation that relates settling time of the velocity of the mass to M; (b) find an equation that relates rise time of the velocity of the mass to M. [Sections: 4.2, 4.3]

Plot the step response for Problem 6 using MATLAB. From your plots, find the time constant, rise time, and settling time. Use M = 1 and M = 2.

For each of the transfer functions shown below, find the locations of the poles and zeros, plot them on the s-plane, and then write an expression for the general form of the step response without solving for the inverse Laplace transform. State the nature of each response (overdamped, underdamped, and so on). [Sections: 4.3, 4.4] (a) \(T(s)=\frac{2}{s+2}\) (b) \(T(s)=\frac{5}{(s+3)(s+6)}\) (c) \(T(s)=\frac{10(s+7)}{(s+10)(s+20)}\) (d) \(T(s)=\frac{20}{s^2+6 s+144}\) (e) \(T(s)=\frac{s+2}{s^2+9}\) f() \(T(s)=\frac{(s+5)}{(s+10)^2}\)

Use MATLAB to find the poles of [Section: 4.2] \(T(s)=\frac{s^2+2 s+2}{s^4+6 s^3+4 s^2+7 s+2}\)

Find the transfer function and poles of the system represented in state space here. [Section: 4.10] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} 3 & -4 & 2 \\ -2 & 0 & 1 \\ 4 & 7 & -5 \end{array}\right] \mathbf{x}+\left[\begin{array}{r} -1 \\ -2 \\ 3 \end{array}\right] u(t) \\ & y=\left[\begin{array}{lll} 1 & 7 & 1 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}\)

Repeat Problem 10 using MATLAB. [Section: 4.10]

Write the general form of the capacitor voltage for the electrical network shown in Figure P4.4. [Section: 4.4]

Use MATLAB to plot the capacitor voltage in Problem 12. [Section: 4.4]

Solve for x(t) in the system shown in Figure P4.5 if f(t) is a unit step. [Section: 4.4]

The system shown in Figure P4.6 has a unit step input. Find the output response as a function of time. Assume the system is underdamped. Notice that the result will be Eq. (4.28). [Section: 4.6]

Derive the relationship for damping ratio as a function of percent overshoot, Eq. (4.39). [Section: 4.6]

Calculate the exact response of each system of Problem 8 using Laplace transform techniques, and compare the results to those obtained in that problem. [Sections: 4.3, 4.4]

Find the damping ratio and natural frequency for each second-order system of Problem 8 and show that the value of the damping ratio conforms to the type of response (underdamped, overdamped, and so on) predicted in that problem. [Section: 4.5]

A system has a damping ratio of 0.15, a natural frequency of 20 rad/s, and a dc gain of 1. Use inverse Laplace transforms to find an analytic expression of the response of the system to a unit-step input. [Section: 4.6]

For each of the second-order systems that follow, find \(\zeta, \omega_n, T_s, T_p, T_r\), and %OS. [Section: 4.6] (a) \(T(s)=\frac{16}{s^2+3 s+16}\) (b) \(T(s)=\frac{0.04}{s^2+0.02 s+0.04}\) (c) \(T(s)=\frac{1.05 \times 10^7}{s^2+1.6 \times 10^3 s+1.05 \times 10^7}\)

Repeat Problem 20 using MATLAB. Have the computer program estimate the given specifications and plot the step responses. Estimate the rise time from the plots. [Section: 4.6]

Use MATLAB’s LTI Viewer and obtain settling time, peak time, rise time, and percent overshoot for each of the systems in Problem 20. [Section: 4.6]

For each pair of second-order system specifications that follow, find the location of the second-order pair of poles. [Section: 4.6] (a) \(\% O S=12 \% ; T_s=0.6\) second (b) \(\% O S=10 \% ; T_p=5\) seconds (c) \(T_s=7\) seconds; \(T_p=3\) seconds

Find the transfer function of a second-order system that yields a 15% overshoot and a settling time of 0.7 second. [Section: 4.6]

For the system shown in Figure P4.7, do the following: [Section: 4.6] (a) Find the transfer function G(s)=X(s)/F(s). (b) Find \(\zeta, \omega_n, \% O S, T_s, T_p, T_r\), and \(C_{\text {final }}\) for a unit-step input.

For the system shown in Figure P4.8, a step torque is applied at \(\theta_1(t)\). Find: (a) The transfer function, \(G(s)=\theta_2(s) / T(s)\) (b) The percent overshoot, settling time, and peak time for \(\theta_2(t)\). [Section: 4.6]

The derivation of Eq. (4.42) to calculate the settling time for a second-order system assumed an underdamped system \((\zeta<1)\). In this problem you will calculate a similar result for a critically damped \(\operatorname{system}(\zeta=1)\). (a) Show that the unit-step response for a system with transfer function \(\frac{C(s)}{R(s)}=\frac{a^2}{(s+a)^2}\) is \(c(t)=1-e^{-a t}(1+a t)\). (Note: \(\mathscr{L}\left\{\frac{1}{(s+a)^2}\right\}=t e^{-a t}\). Optional: You can derive this result similarly to Example 2.2.) (b) Show that the settling time can be found by solving for \(T_s\) in \(e^{-a T_s}\left(1+a T_s\right)=0.02\). (c) Use MATLAB to plot \(e^{-x}(1+x)= 0.02\) Vs. x. Use the plot to show that \(T_s=\frac{5.834}{a}\).

An autonomous robot to pick asparagus (Dong, 2011) capable of following planting rows has an orientation system with transfer function \(\frac{\theta}{\theta_{r e f}}=\frac{53.176}{4.6 s^2+31.281 s+53.176}\) Make a sketch of \(\theta(t)\) in response to \(\theta_{r e f}(t)=3 u(t)\). Indicate in your plot \(C_{\text {final }}, C_{\max }, T_p\), and \(T_s\). (Hint: You may use the result of Problem 27c).

Figure P4.9 shows five step responses of an automatic voltage regulation system as one of the system parameters varies (Gozde, 2011). Assume for all five responses that they are those of a second-order system with an overshoot of 20%. Make a sketch of the positions of the poles in the complex plane for each one of the responses. Label the curves A through E from left to right.

Derive the unit step response for each transfer function in Example 4.8. [Section: 4.7]

Find the percent overshoot, settling time, rise time, and peak time for \(T(s)=\frac{14.65}{\left(s^2+0.842 s+2.93\right)(s+5)}\) [Section: 4.7]

For each of the three unit step responses shown in Figure P4.10, find the transfer function of the system. [Sections: 4.3, 4.6]

For the following response functions, determine if pole zero cancellation can be approximated. If it can, find percent overshoot, settling time, rise time, and peak time. [Section: 4.8]. (a) \(C(s)=\frac{(s+4)}{s(s+2)\left(s^2+3 s+10\right)}\) (b) \(C(s)=\frac{(s+2.5)}{s(s+2)\left(s^2+4 s+20\right)}\) (c) \(C(s)=\frac{(s+2.2)}{s(s+2)\left(s^2+s+5\right)}\) (d) \(C(s)=\frac{(s+2.01)}{s(s+2)\left(s^2+5 s+20\right)}\)

Using MATLAB, plot the time response of Problem 33a and from the plot determine percent overshoot, settling time, rise time, and peak time. [Section: 4.8]

Find peak time, settling time, and percent overshoot for only those responses below that can be approximated as second-order responses. [Section: 4.8] (a) \(c(t)= & 0.003500-0.001524 e^{-4 t}\) \(\begin{aligned} & -0.001976 e^{-3 t} \cos (22.16 t) \\ & -0.0005427 e^{-3 t} \sin (22.16 t) \end{aligned}\) (b) \(c(t)=0.05100-0.007353 e^{-8 t}\) \(\begin{aligned} & -0.007647 e^{-6 t} \cos (8 t) \\ & -0.01309 e^{-6 t} \sin (8 t) \end{aligned}\) (c) \(c(t)=0.009804-0.0001857 e^{-5.1 t}\) \(\begin{aligned} & -0.009990 e^{-2 t} \cos (9.796 t) \\ & -0.001942 e^{-2 t} \sin (9.796 t) \end{aligned}\) (d) \(c(t)=0.007000-0.001667 e^{-10 t}\) \(\begin{aligned} & -0.008667 e^{-2 t} \cos (9.951 t) \\ & -0.0008040 e^{-2 t} \sin (9.951 t) \end{aligned}\)

For each of the following transfer functions with zeros, find the component parts of the unit step response: (1) the derivative of the response without a zero and (2) the response without a zero, scaled to the negative of the zero value. Also, find and plot the total response. Describe any nonminimum-phase behavior. [Section: 4.8] (a) \(G(s)=\frac{s+2}{s^2+3 s+36}\) (b) \(G(s)=\frac{s-2}{s^2+3 s+36}\)

Use MATLAB's Simulink to obtain the step response of a system, \(G(s)=\frac{1}{s^2+3 s+10}\) under the following conditions: [Section: 4.9] (a) The system is linear and driven by an amplifier whose gain is 10. (b) An amplifier whose gain is 10 drives the system. The amplifier saturates at \(\pm 25\) volts. Describe the effect of the saturation on the system’s output. (c) An amplifier whose gain is 10 drives the system. The amplifier saturates at \(\pm 25\) volts. The system drives a 1:1 gear train that has backlash. The deadband width of the backlash is 0.02 rad. Describe the effect of saturation and backlash on the system’s output.

A system is represented by the state and output equations that follow. Without solving the state equation, find the poles of the system. [Section: 4.10] \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{rr} -1 & 3 \\ -4 & -2 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 3 \\ 1 \end{array}\right] u(t) \\ y & =\left[\begin{array}{ll} 5 & 1 \end{array}\right] \mathbf{x} \end{aligned}\)

A system is represented by the state and output equations that follow. Without solving the state equation, find [Section: 4.10] (a) the characteristic equation; (b) the poles of the system \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{lll} 0 & 2 & 3 \\ 0 & 6 & 5 \\ 1 & 4 & 2 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] u(t) \\ y & =\left[\begin{array}{lll} 1 & 2 & 0 \end{array}\right] \mathbf{x} \end{aligned}\)

Given the following state-space representation of a system, find Y(s): [Section: 4.10] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rr} 1 & 2 \\ -3 & -1 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \sin 3 t \\ & y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 3 \\ 1 \end{array}\right] \end{aligned}\)

Given the following system represented in state space, solve for Y(s) using the Laplace transform method for solution of the state equation: [Section: 4.10] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} 0 & 1 & 0 \\ -2 & -4 & 1 \\ 0 & 0 & -6 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] e^{-t} \\ & y=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}\)

Solve the following state equation and output equation for y(t), where u(t) is the unit step. Use the Laplace transform method. [Section: 4.10] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rr} -3 & 0 \\ -1 & -1 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 2 \\ 1 \end{array}\right] u(t) \\ & y=\left[\begin{array}{ll} 1 & 0 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 2 \\ 0 \end{array}\right] \end{aligned}\)

Solve for y(t) for the following system represented in state space, where u(t) is the unit step. Use the Laplace transform approach to solve the state equation. [Section: 4.10] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} -3 & 1 & 0 \\ 0 & -6 & 1 \\ 0 & 0 & -5 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] u(t) \\ & y=\left[\begin{array}{lll} 0 & 1 & 1 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}\)

Use MATLAB to plot the step response of Problem 43. [Section: 4.10]

Repeat Problem 43 using MATLAB’s Symbolic Math Toolbox and Eq. (4.96). In addition, run your program with an initial condition, \(\mathbf{x}(0)=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right] \cdot\) [ Section : 4.10]

Using classical (not Laplace) methods only, solve for the state-transition matrix, the state vector, and the output of the system represented here. [Section: 4.11] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rr} 0 & 1 \\ -1 & -5 \end{array}\right] \mathbf{x} ; \quad y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] \mathbf{x} \\ & \mathbf{x}(0)=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \end{aligned}\)

Using classical (not Laplace) methods only, solve for the state-transition matrix, the state vector, and the output of the system represented here, where u(t) is the unit step: [Section: 4.11] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \end{array}\right] u(t) \\ & y=\left[\begin{array}{ll} 3 & 4 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] \end{aligned}\)

Solve for y(t) for the following system represented in state space, where u(t) is the unit step. Use the classical approach to solve the state equation. [Section: 4.11] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} -2 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -6 & -1 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] u(t) \\ & y=\left[\begin{array}{lll} 1 & 0 & 0 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}\)

Repeat Problem 48 using MATLAB’s Symbolic Math Toolbox and Eq.(4.109). In addition, run your program with an initial condition, \(\mathbf{x}(0)=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right] \cdot\) [Section :4.11]

Using methods described in Appendix H.1 located at www.wiley.com/college/nise, simulate the following system and plot the step response. Verify the expected values of percent overshoot, peak time, and settling time. \(T(s)=\frac{1}{s^2+0.8 s+1}\)

Use MATLAB to simulate the following system and plot the output, y(t), for a step input. Mark on the plot the steady-state value, percent overshoot, and the rise time, peak time, and settling time. \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{rrr} 0 & 1 & 0 \\ -12 & -8 & 1 \\ 0 & 0 & -2 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] u(t) \\ y(t) & =\left[\begin{array}{lll} 1 & 1 & 0 \end{array}\right] \mathbf{x} ; \mathbf{x}(0)=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}\)

A human responds to a visual cue with a physical response, as shown in Figure P 4.11. The transfer function that relates the output physical response, P(s), to the input visual command, V(s), is (Stefani, 1973). \(G(s)=\frac{P(s)}{V(s)}=\frac{(s+0.5)}{(s+2)(s+5)}\) Do the following: (a) Evaluate the output response for a unit step input using the Laplace transform. (b) Represent the transfer function in state space. (c) Use MATLAB to simulate the system and obtain a plot of the step response.

Upper motor neuron disorder patients can benefit and regain useful function through the use of functional neuroprostheses. The design requires a good understanding of muscle dynamics. In an experiment to determine muscle responses, the identified transfer function was (Zhou, 1995) \(M(s)=\frac{2.5 e^{-0.008 s}(1+0.172 s)(1+0.008 s)}{(1+0.07 s)^2(1+0.05 s)^2}\) Find the unit step response of this transfer function.

When electrodes are attached to the mastoid bones (right behind the ears) and current pulses are applied, a person will sway forward and backward. It has been found that the transfer function from the current to the subject's angle (in degrees) with respect to the vertical is given by (Nashner, 1974) \(\frac{\theta(s)}{I(s)}=\frac{5.8(0.3 s+1) e^{-0.1 s}}{(s+1)\left(s^2 / 1.2^2+0.6 s / 1.2+1\right)}\) (a) Determine whether a dominant pole approximation can be applied to this transfer function. (b) Find the body sway caused by a \(250 \mu \mathrm{A}\) pulse of 150 msec duration.

A MOEMS (optical MEMS) is a MEMS (Micro Electromechanical Systems) with an optical fiber channel that takes light generated from a laser diode. It also has a photodetector that measures light intensity variations and outputs voltage variations proportional to small mechanical device deflections. Additionally, a voltage input is capable of deflecting the device. The apparatus can be used as an optical switch or as a variable optical attenuator, and it does not exceed \(2000 \mu \mathrm{m}\) in any dimension. Figure P4.12 shows input-output signal pairs used to identify the parameters of the system. Assume a second-order transfer function and find the system’s transfer function (Borovic, 2005).

The response of the deflection of a fluid-filled catheter to changes in pressure can be modeled using a second-order model. Knowledge of the parameters of the model is important because in cardiovascular applications the undamped natural frequency should be close to five times the heart rate. However, due to sterility and other considerations, measurement of the parameters is difficult. A method to obtain transfer functions using measurements of the amplitude software consecutive peaks of the response and their timing has been developed (Glantz, 1979). Assume that Figure P4.13 is obtained from catheter measurements. Using the information shown and assuming a second-order model excited by a unit step input, find the corresponding transfer function.

Several factors affect the workings of the kidneys. For example, Figure P4.14 shows how a step change in arterial flow pressure affects renal blood flow in rats. In the “hot tail” part of the experiment, peripheral thermal receptor stimulation is achieved by inserting the rat’s tail in heated water. Variations between different test subjects are indicated by the vertical lines. It has been argued that the “control” and “hot tail” responses are identical except for their steady-state values (DiBona, 2005). (a) Using Figure P4.14, obtain the normalized \(\left(C_{\text {final }}=1\right)\) transfer functions for both responses. (b) Use MATLAB to prove or disprove the assertion about the “control” and “hot tail” responses.

The transfer function of a nano-positioning device capable of translating biological samples within a few ?m uses a piezoelectric actuator and a linear variable differential transformer (LDVT) as a displacement sensor. The transfer function from input to displacement has been found to be (Salapaka, 2002) \(G(s)=\frac{9.7 \times 10^4\left(s^2-14400 s+106.6 \times 10^6\right)}{\left(s^2+3800 s+23.86 \times 10^6\right)\left(s^2+240 s+2324.8 \times 10^3\right)}\) Use a dominant-pole argument to find an equivalent transfer function with the same numerator but only three poles. Use MATLAB to find the actual size and approximate system unit step responses, plotting them on the same graph. Explain the differences between both responses given that both pairs of poles are so far apart.

At some pointintheirlivesmost people will suffer from at least one onset of low back pain. This disorder can trigger excruciating pain and temporary disability, but its causes are hard to diagnose. It is well known that low back pain alters motor trunk patterns; thus it is of interest to study the causes for these alterations and their extent. Due to the different possible causes of this type of pain, a “control” group of people is hard to obtain for laboratory studies. However, pain can be stimulated in healthy people and muscle movement ranges can be compared. Controlled back pain can be induced by injecting saline solution directly into related muscles or ligaments. The transfer function from infusion rate to pain response was obtained experimentally by injecting a 5% saline solution at six different infusion rates over a period of 12 minutes. Subjects verbally rated their pain every 15 seconds on a scale from 0 to 10, with 0 indicating no pain and 10 unbearable pain. Several trials were averaged and the data was fitted to the following transfer function: \(G(s)=\frac{9.72 \times 10^{-8}(s+0.0001)}{(s+0.009)^2\left(s^2+0.018 s+0.0001\right)}\) For experimentation, it is desired to build an automatic dispensing system to make the pain level constant as shown in Figure P4.15. It follows that ideally the injection system transfer function has to be \(M(s)=\frac{1}{G(s)}\) to obtain an overall transfer function \(M(s) G(s) \approx 1\). However, for implementation purposes M(s) must have at least one more pole than zeros (Zedka, 1999). Find a suitable transfer function, M(s) by inverting G(s) and adding poles that are far from the imaginary axis.

An artificial heart works in closed loop by varying its pumping rate according to changes in signals from the recipient’s nervous system. For feedback compensation design itis important to know the heart’s open-loop transfer function. To identify this transfer function, an artificial heart is implanted in a calf while the main parts of the original heart are left in place. Then the atrial pumping rate in the original heart is measured while step input changes are effected on the artificial heart. It has been found that, in general, the obtained response closely resembles that of a second-order system. In one such experiment it was found that the step response has a %OS=30% and a time of first peak \(T_p=127 \mathrm{sec}\) (Nakamura, 2002). Find the corresponding transfer function.

An observed transfer function from voltage potential to force in skeletal muscles is given by (Ionescu, 2005) \(T(s)=\frac{450}{(s+5)(s+20)}\) (a) Obtain the system’s impulse response. (b) Integrate the impulse response to find the step response. (c) Verify the result in Part b by obtaining the step response using Laplace transform techniques.

In typical conventional aircraft, longitudinal flight model linearization results in transfer functions with two pairs of complex conjugate poles. Consequently, the natural response for these airplanes has two modes in their natural response. The “short period” mode is relatively well-damped and has a high-frequency oscillation. The “plugoid mode” is lightly damped and its oscillation frequency is relatively low. For example, in a specific aircraft the transfer function from wing elevator deflection to nose angle (angle of attack)is (McRuer, 1973) \(\begin{aligned} & \frac{\theta(s)}{\delta_e(s)}= \\ & -\frac{26.12(s+0.0098)(s+1.371)}{\left(s^2+8.99 \times 10^{-3} s+3.97 \times 10^{-3}\right)\left(s^2+4.21 s+18.23\right)} \end{aligned}\) a. Find which of the poles correspond to the short period mode and which to the phugoid mode. b. Perform a "phugoid approximation" (dominant-pole approximation), retaining the two poles and the zero closest to the; \(\omega\)-axis. Use MATLAB to compare the step responses of the original transfer function and the approximation.

A crosslapper is a machine that takes as an input a light fiber fabric and produces a heavier fabric by laying the original fabric in layers rotated by 90 degrees. A feedback system is required in order to maintain consistent product width and thickness by controlling its carriage velocity. The transfer function from servo motor torque, \(T_m(s)\), to carriage velocity, Y(s), was developed for such a machine (Kuo, 2008). Assume that the transfer function is: \(\begin{aligned} & G(s)=\frac{Y(s)}{T_m(s)} \\ & =\frac{33 s^4+202 s^3+10061 s^2+24332 s+170704}{s^3+8 s^6+464 s^5+2411 s^4+52899 s^3+167829 s^2+913599 s+1076555} \end{aligned}\) (a) Use MATLAB to find the partial fraction residues and poles of G(s). (b) Find an approximation to G(s) by neglecting the second-order terms found in a. (c) Use MATLAB to plot on one graph the step response of the transfer function given above and the approximation found in b. Explain the differences between the two plots.

Although the use of fractional calculus in control systems is not new, in the last decade there is increased interest in its use for several reasons. The most relevant are that fractional calculus differential equations may model certain systems with higher accuracy than integer differential equations, and that fractional calculus compensators might exhibit advantageous properties for control system design. An example of a transfer function obtained through fractional calculus is: \(G(s)=\frac{1}{s^{2.5}+4 s^{1.7}+3 s^{0.5}+5}\) This function can be approximated with an integer rational transfer function (integer powers of s) using Oustaloup’s method (Xue, 2005). We ask you now to do a little research and consult the aforementioned reference to find and run an M-file that will calculate the integer rational transfer function approximation to G(s) and plot its step response.

Mathematical modeling and control of \(\mathrm{pH}\) processes are quite challenging since the processes are highly nonlinear, due to the logarithmic relationship between the concentration of hydrogen ions \([\mathrm{H}+]\) and \(\mathrm{pH}\) level. The transfer function from input \(\mathrm{pH}\) to output \(\mathrm{pH}\) is \(G_a(s)=\frac{Y_a(s)}{Y_a(s)}=\frac{14.49 e^{-3.3 s}}{1478.26 s+1}\), where we assume a delay of 3.3 seconds. \(G_a(s)\) is a model for the anaerobic process in a wastewater treatment system in which methane bacteria need the \(\mathrm{pH}\) to be maintained in its optimal range from 6.8 to 7.2 (Jiayu, 2009). Similarly, (Elarafi, 2008) used empirical techniques to model a \(\mathrm{pH}\) neutralization plant as a second-order system with a pure delay, yielding the following transfer function relating output \(\mathrm{pH}\) to input \(\mathrm{pH}\): \(G_p(s)=\frac{Y_p(s)}{X_p(s)}=\frac{1.716 \times 10^{-5} e^{-25 s}}{s^2+6.989 \times 10^{-3} s+1.185 \times 10^{-6}}\) where we assume a delay of 25 seconds. (a) Find analytical expressions for the unit-step responses \(y_a(t)\) and \(y_p(t)\) for the two processes, \(G_a(s)\) and \(G_p(s)\). (Hint: Use the time shift theorem in Table 2.2). (b) Use Simulink to plot \(y_a(t)\) and \(y_p(t)\) on a single graph.

Using wind tunnel tests, insect flight dynamics can be studied in a very similar fashion to that of man-made aircraft. Linearized longitudinal flight equations for a bumblebee have been found in the unforced case to be \(\left[\begin{array}{c} \dot{u} \\ \dot{w} \\ \dot{q} \\ \dot{\theta} \end{array}\right]=\left[\begin{array}{cccc} -8.792 \times 10^{-3} & 0.56 \times 10^{-3} & -1.0 \times 10^{-3} & -13.79 \times 10^{-3} \\ -0.347 \times 10^{-3} & -11.7 \times 10^{-3} & -0.347 \times 10^{-3} & 0 \\ 0.261 & -20.8 \times 10^{-3} & -96.6 \times 10^{-3} & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]\left[\begin{array}{l} u \\ w \\ q \\ \theta \end{array}\right]\) where u= forward velocity; w= vertical velocity, q= angular pitch rate at center of mass, and \(\theta\)= pitch angle between the flight direction and the horizontal (Sun, 2005). (a) Use MATLAB to obtain the system’s eigenvalues. (b) Write the general form of the state-transition matrix. How many constants would have to be found?

A dc-dc converter is a device that takes as an input an unregulated dc voltage and provides a regulated dc voltage as its output. The output voltage may be lower (buck converter), higher (boost converter), or the same as the input voltage. Switching dc-dc converters have a semiconductor active switch (BJT or FET) that is closed periodically with a duty cycle d in a pulse width modulated (PWM) manner. For a boost converter, averaging techniques can be used to arrive at the following state equations (Van Dijk, 1995): \(\begin{aligned} & L \frac{d i_L}{d t}=-(1-d) u_c+E_s \\ & C \frac{d u_C}{d t}=(1-d) i_L-\frac{u_C}{R} \end{aligned}\) where L and C are, respectively, the values of internal inductance and capacitance; \(i_L\) is the current through the internal inductor; R is the resistive load connected to the converter; \(E_s\) is the dc input voltage; and the capacitor voltage, \(u_C\), is the converter's output. (a) Write the converter's equations in the form \(\begin{aligned} \dot{\mathbf{x}} & =\mathbf{A x}+\mathbf{B u} \\ \mathbf{y} & =\mathbf{C x} \end{aligned}\) assuming d is a constant. (b) Using the A, B, and C matrices of Part a, obtain the converter's transfer function \(\frac{U_C(s)}{E_s(s)}\).

An IPMC (ionic polymer-metal composite) is a Nafion sheet plated with gold on both sides. An IPMC bends when an electric field is applied across its thickness. IPMCs have been used as robotic actuators in several applications and as active catheters in biomedical applications. With the aim of improving actuator settling times, a state-space model has been developed for a \(20 \mathrm{~mm} \times 10 \mathrm{~mm} \times 0.2 \mathrm{~mm}\) polymer sample (Mallavarapu, 2001): \(\begin{aligned} {\left[\begin{array}{c} \dot{x}_1 \\ \dot{x}_2 \end{array}\right] } & =\left[\begin{array}{cc} -8.34 & -2.26 \\ 1 & 0 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u \\ \mathrm{y} & =\left[\begin{array}{ll} 12.54 & 2.26 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right] \end{aligned}\) where u is the applied input voltage and y is the deflection at one of the material’s tips when the sample is tested in a cantilever arrangement. (a) Find the state-transition matrix for the system. (b) From Eq. (4.109) in the text, it follows that if a system has zero initial conditions, the system output for any input can be directly calculated from the state-space representation and the state-transition matrix using \(y(t)=\mathbf{C x}(t)=\int \mathbf{C} \Phi(t-\tau) \mathbf{B} u(\tau) d \tau\) Use this equation to find the zero initial condition unit step response of the IPMC material sample. (c) Use MATLAB to verify that your step response calculation in Part b is correct.

Figure P4.16 shows the step response of an electric vehicle’s mechanical brakes when the input is the drive torque (N-m) and the output is the hydraulic brake pressure (bar) (Ringdorfer, 2011). (a) Find the transfer function of the system. (b) Use the values of the parameters for the transfer function obtained in Part a to find an expression for the brake pressure as a function of time. (c) Find the output in bars of the system 0.2 sec after the input is applied.Check your result against Figure P4.16.

Figure P4.17 shows the free-body diagrams for planetary gear components used in the variable valve timing (VVT) system of an internal combustion engine (Ren, 2011). Here an electric motor is used to drive the carrier. Analysis showed that the electric motor with planetary gear load (Figure P4.17) may be represented by the following equation: \(\Omega_c(s)=G_e(s) E_a(s)+G_m(s) T_{c a m}(s)\) where \(\Omega_c(s)\) is the output carrier angular speed in rad/s, \(E_a(s)\) is the input voltage applied to the armature, and \(T_{\text {cam }}(s)\) is the input load torque. The voltage input transfer function, \(G_e(s)\), is \(G_e(s) \cong \frac{K_\tau}{R_m(J s+D)+K_\tau K_m}=\frac{45}{0.2 s+1}\) and the load torque input transfer function, \(G_m(s)\), is \(G_m(s) \cong \frac{-R_m k}{R_m(J s+D)+K_\tau K_m}=\frac{-5}{0.2 s+1}\) Find an analytical expression for the output carrier angular speed, \(\omega_c(t)\), if a step voltage of 100 volts is applied at t=0 followed by an equivalent load torque of \(10 \mathrm{~N}-{m}\), applied at \(t=0.4 \mathrm{sec}\).

A drive system with elastically coupled load (Figure P4.18) has a motor that is connected to the load via a gearbox and a long shaft. The system parameters are: \(J_{\mathrm{M}}=\) drive-side inertia = \(0.0338 \mathrm{~kg}-\mathrm{m}^2\), \(J_L=\) load-side inertia \(=0.1287 \mathrm{~kg}-\mathrm{m}^2\), \(K=C_T=\) torsional spring constant \(=1700 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\), and D= damping coefficient \(=0.15 \mathrm{~N}-\mathrm{m}-\mathrm{s} / \mathrm{rad}\). This system can be reduced to a simple two inertia model that may be represented by the following transfer function, relating motor shaft speed output, \(\Omega(s)\), to electromagnetic torque input (Thomsen, 2011): \(G(s)=\frac{\Omega(s)}{T_{e m}(s)}=\frac{1}{s\left(J_M+J_L\right)} \cdot \frac{\frac{J_L}{C_T} s^2+\frac{D}{C_T} s+1}{\frac{J_M J_L}{C_T\left(J_M+J_L\right)} s^2+\frac{D}{C_T} s+1}\) Assume the system is at standstill at t=0, when the electromagnetic torque, \(T_{e m}\), developed by the motor changes from zero to \(50 \mathrm{~N}-\mathrm{m}\). Find and plot on one graph, using MATLAB or any other program, the motor shaft speed, \(\omega(t)\), \(\mathrm{rad} / \mathrm{sec}\), for the following two cases: (a) No load torque is applied and, thus, \(\omega=\omega_{n l}\). (b) A load torque, \(T_L=0.2 \omega(t) \quad \mathrm{N}-\mathrm{m}\) is applied and \(\omega=\omega_L\).

An inverted pendulum mounted on a motor-driven cart was presented in Problem 30 of Chapter 3. The nonlinear state-space equations representing that system were linearized (Prasad, 2012) around a stationary point corresponding to the pendulum point-mass, m, being in the upright position (\(x_0 = 0\) at t = 0), when the force applied to the cart was zero \(\left(u_0=0\right)\). The state-space model developed in that problem is \(\dot{\mathbf{x}}=\mathbf{A} \mathbf{x}+\mathbf{B} u\) The state variables are the pendulum angle relative to the y-axis, \(\theta\), its angular speed, \(\theta^{\prime}\), the horizontal position of the cart, x, and its speed, \(x^{\prime}\). The horizontal position of m (for a small angle, \(\theta\) ), \(x_m=x+l \sin \theta=x+l \theta\), was selected to be the output variable. Given the state-space model developed in that problem along with the output equation you developed in that problem, use MATLAB (or any other computer program) to find and plot the output, \(x_m(t)\), in meters, for an input force, u(t), equal to a unit impulse, \(\delta(t)\), in Newtons.

Find an equation that relates 2% settling time to the value of \(f_v\) for the translational mechanical system shown in Figure P4.19. Neglect the mass of all components. [Section: 4.6]

Consider the translational mechanical system shown in Figure P4.20. A 1-pound force, f(t), is applied at t=0. If \(f_v=1\), 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]

Given the translational mechanical system of Figure P4.20, where K=1 and f(t) is a unit step, find the values of M and \(f_v\) to yield a response with 17% overshoot and a settling time of 10 seconds. [Section: 4.6]

Find J and K in the rotational system shown in Figure P4.21 to yield a 30% overshoot and a settling time of 3 seconds for a step input in torque. [Section: 4.6]

Given the system shown in Figure P4.22, find the damping, D, to yield a 30% overshoot in output angular displacement for a step input in torque. [Section: 4.6]

For the system shown in Figure P4.23, find \(N_1 / N_2\) so that the settling time for a step torque input is 16 seconds. [Section: 4.6]

Find M and K, shown in the system of Figure P4.24, to yield x(t) with 16% overshoot and 20 seconds settling time for a step input in motor torque, \(T_m(t)\). [Section: 4.6]

If \(v_i(t)\) is a step voltage in the network shown in Figure P4.25, find the value of the resistor such that a 20% overshoot in voltage will be seen across the capacitor if \(C=10^{-6} \mathrm{~F}\) and L=1 H. [Section: 4.6]

If \(v_i(t)\) is a step voltage in the network shown in Figure P4.25, find the values of R and C to yield a 20% overshoot and a 1 ms settling time for \(v_c(t)\) if L=1 H. [Section: 4.6]

Given the circuit of Figure P 4.25, where \(C=10 \mu \mathrm{F}\), find R and L to yield 15% overshoot with a settling time of 7 ms for the capacitor voltage. The input, v(t), is a unit step. [Section: 4.6]

For the circuit shown in Figure P4.26, find the values of \(R_2\) and C to yield 8% overshoot with a settling time of 1 ms for the voltage across the capacitor, with \(v_i(t)\) as a step input. [Section: 4.6]

Control of HIV/AIDS. In Chapter 3, Problem 31, we developed a linearized state-space model of HIV infection. The model assumed that two different drugs were used to combat the spread of the HIV virus. Since this book focuses on single-input, single-output systems, only one of the two drugs will be considered. We will assume that only RTIs are used as an input. Thus, in the equations of Chapter 3, Problem 31, \(u_2 = 0\) (Craig, 2004). (a) Show that when using only RTIs in the linearized system of Problem 31, Chapter 3, and substituting the typical parameter values given in the table of Problem 31c, Chapter 3, the resulting state-space representation for the system is given by \(\begin{aligned} {\left[\begin{array}{c} \dot{T} \\ \dot{T}^* \\ \dot{v} \end{array}\right]=} & {\left[\begin{array}{ccc} -0.04167 & 0 & -0.0058 \\ 0.0217 & -0.24 & 0.0058 \\ 0 & 100 & -2.4 \end{array}\right] } \\ & \times\left[\begin{array}{c} T \\ T^* \\ v \end{array}\right]+\left[\begin{array}{c} 5.2 \\ -5.2 \\ 0 \end{array}\right] u_1 \\ y= & {\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} T \\ T^* \\ v \end{array}\right] } \end{aligned}\) b. Obtain the transfer function from RTI efficiency to virus count; namely, find \(\frac{Y(s)}{U_1(s)}\). (c) Assuming RTIs are 100% effective, what will be the steady-state change of virus count in a given infected patient? Express your answer in virus copies per ml of plasma. Approximately how much time will the medicine take to reach its maximum possible effectiveness?

Hybrid vehicle. Assume that the car motive dynamics for a hybrid electric vehicle (HEV) can be described by the transfer function \(\frac{\Delta V(s)}{\Delta F_e(s)}=\frac{1}{1908 s+10}\) where AV is the change of velocity in m/sec and \(\Delta F_e\) is the change in excess motive force in N necessary to propel the vehicle. (a) Find an analytical expression for \(\Delta v_{(t)}\) for a step change in excess motive force \(\Delta F_e=2650 \mathrm{~N}\). (b) Simulate the system using MATLAB. Plot the expression found in Part a together with your simulated plot.