Solved: Using the Routh-Hurwitz criterion and the unity

Tell how many roots of the following polynomial are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis: [Section: 6.2] \(P(s)=s^5+3 s^4+5 s^3+4 s^2+s+3\)

Tell how many roots of the following polynomial are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis: [Section: 6.3] \(P(s)=s^5+6 s^3+5 s^2+8 s+20\)

Using the Routh table, tell how many poles of the following function are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. [Section: 6.3] \(T(s)=\frac{s+8}{s^5-s^4+3 s^3-3 s^2+3 s-2}\)

The closed-loop transfer function of a system is [Section: 6.3] \(T(s)=\frac{s^3+2 s^2+7 s+21}{s^5-2 s^4+3 s^3-6 s^2+2 s-4}\) Determine how many closed-loop poles lie in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis.

How many poles are in the right half-plane, in the left half-plane, and on the\(j \omega\)-axis for the open-loop system of Figure P6.1? [Section: 6.3]

How many poles are in the right half-plane, the left half-plane, and on the \(j \omega\)-axis for the open-loop system of Figure P6.2? [Section: 6.3]

Use MATLAB to find the pole locations for the system of Problem 6.

Use MATLAB and the Symbolic Math Toolbox to generate a Routh table to solve Problem 3.

Determine whether the unity feedback system of Figure P6.3 is stable if [Section: 6.2] \(G(s)=\frac{240}{(s+1)(s+2)(s+3)(s+4)}\)

Use MATLAB to find the pole locations for the system of Problem 9.

Consider the unity feedback system of Figure P6.3 with \(G(s)=\frac{1}{4 s^2\left(s^2+2\right)}\) (a) Using the Routh-Hurwitz criterion, find the region of the s-plane where the poles of the closed-loop system are located. (b) Use MATLAB to verify your answer and find the poles of T(s). Indicate whether this system is stable, unstable, or marginally stable. [Section: 6.3]

In the system of Figure P6.3, let \(G(s)=\frac{K(s+1)}{s(s-2)(s+3)}\) Find the range of K for closed-loop stability. [Section: 6.4]

Given the unity feedback system of Figure P6.3 with \(G(s)=\frac{84}{s\left(s^7+5 s^6+12 s^5+25 s^4+45 s^3+50 s^2+82 s+60\right)}\) tell how many poles of the closed-loop transfer function lie in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. [Section: 6.3]

Using the Routh-Hurwitz criterion and the unity feedback system of Figure P6.3 with \(G(s)=\frac{1}{2 s^4+5 s^3+s^2+2 s}\) tell whether or not the closed-loop system is stable. [Section: 6.2]

Given the unity feedback system of Figure P6.3 with \(G(s)=\frac{8}{s\left(s^6-2 s^5-s^4+2 s^3+4 s^2-8 s-4\right)}\) tell how many closed-loop poles are located in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. [Section: 6.3]

Repeat Problem 15 using MATLAB.

Consider the following Routh table. Notice that the \(s^5\) row was originally all zeros. Tell how many roots of the original polynomial were in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. [Section: 6.3]

For the system of Figure P6.4, tell how many closed loop poles are located in the right half-plane, in the left half-plane, and on the\(j \omega\)-axis. Notice that there is positive feedback. [Section: 6.3]

Using the Routh-Hurwitz criterion, tell how many closed-loop poles of the system shown in Figure P6.5 lie in the left half-plane, in the right half-plane, and on the \(j \omega\)-axis. [Section: 6.3]

Determine if the unity feedback system of Figure P6.3 with \(G(s)=\frac{K\left(s^2+1\right)}{(s+1)(s+2)}\) can be unstable. [Section: 6.4]

or the unity feedback system of Figure P6.3 with \(G(s)=\frac{K(s+6)}{s(s+1)(s+4)}\) determine the range of K to ensure stability. [Section: 6.4]

In the system of Figure P6.3, let \(G(s)=\frac{K(s-a)}{s(s-b)}\) Find the range of K for closed-loop stability when: [Section: 6.4] (a) a<0, b<0 (b) a<0, b>0 (c) a>0, b<0 (d) a>0, b>0

For the unity feedback system of Figure P6.3 with \(G(s)=\frac{K(s+3)(s+5)}{(s-2)(s-4)}\) determine the range of K for stability. [Section: 6.4]

Repeat Problem 23 using MATLAB.

Use MATLAB and the Symbolic Math Toolbox to generate a Routh table in terms of K to solve Problem 23.

Find the range of K for stability for the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K(s+4)(s-4)}{\left(s^2+3\right)}\)

For the unity feedback system of Figure P6.3 with \(G(s)=\frac{K(s+1)}{s^4(s+4)}\) find the range of K for stability. [Section: 6.4]

Find the range of gain, K, to ensure stability in the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K(s-2)(s+4)(s+5)}{\left(s^2+12\right)}\)

Find the range of gain, K, to ensure stability in the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K(s+2)}{\left(s^2+1\right)(s+4)(s-1)}\)

Using the Routh-Hurwitz criterion, find the value of K that will yield oscillations for the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K}{(s+77)(s+27)(s+38)}\)

Use the Routh-Hurwitz criterion to find the range of K for which the system of Figure P6.6 is stable. [Section: 6.4]

Repeat Problem 31 for the system of Figure P6.7. [Section: 6.4]

Given the unity feedback system of Figure P6.3 with \(G(s)=\frac{K(s+4)}{s(s+1.2)(s+2)}\) find the following: [Section: 6.4] (a) The range of K that keeps the system stable (b) The value of K that makes the system oscillate (c) The frequency of oscillation when K is set to the value that makes the system oscillate

Repeat Problem 33 for [Section: 6.4] \(G(s)=\frac{K(s-1)(s-2)}{(s+2)\left(s^2+2 s+2\right)}\)

For the system shown in Figure P6.8, find the value of gain, K, that will make the system oscillate. Also, find the frequency of oscillation. [Section: 6.4]

Given the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K s(s+2)}{\left(s^2-4 s+8\right)(s+3)}\) (a) Find the range of K for stability. (b) Find the frequency of oscillation when the system is marginally stable.

Repeat Problem 36 using MATLAB.

For the unity feedback system of Figure P6.3 with \(G(s)=\frac{K(s+2)}{\left(s^2+1\right)(s+4)(s-1)}\) find the range of K for which there will be only two closed-loop, right-half-plane poles. [Section: 6.4]

For the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K}{(s+1)^3(s+4)}\) (a) Find the range of K for stability. (b) Find the frequency of oscillation when the system is marginally stable.

Given the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K}{(s+49)\left(s^2+4 s+5\right)}\) (a) Find the range of K for stability. (b) Find the frequency of oscillation when the system is marginally stable.

Using the Routh-Hurwitz criterion and the unity feedback system of Figure P6.3 with [Section: 6.4] \(G(s)=\frac{K}{s(s+1)(s+2)(s+6)}\) (a) Find the range of K for stability. (b) Find the value of K for marginal stability. (c) Find the actual location of the closed-loop poles when the system is marginally stable.

Find the range of K to keep the system shown in Figure P6.9 stable. [Section: 6.4]

Find the value of K in the system of Figure P6.10 that will place the closed-loop poles as shown. [Section: 6.4]

The closed-loop transfer function of a system is \(T(s)=\frac{s^2+K_1 s+K_2}{s^4+K_1 s^3+K_2 s^2+5 s+1}\) Determine the range of \(K_1\) in order for the system to be stable. What is the relationship between \(K_1\) and \(K_2\) for stability? [Section: 6.4]

For the transfer function below, find the constraints on \(K_1\) and \(K_2\) such that the function will have only two \(j \omega\) poles. [Section: 6.4] \(T(s)=\frac{K_1 s+K_2}{s^4+K_1 s^3+s^2+K_2 s+1}\)

The transfer function relating the output engine fan speed (rpm) to the input main burner fuel flow rate \((\mathrm{lb} / \mathrm{h})\) in a short takeoff and landing (STOL) fighter aircraft, ignoring the coupling between engine fan speed and the pitch control command, is (Schierman, 1992) [Section: 6.4] \(G(s)=\frac{1.3 s^7+90.5 s^6+1970 s^5+15,000 s^4+3120 s^3-41,300 s^2-5000 s-1840}{s^8+103 s^7+1180 s^6+4040 s^5+2150 s^4-8960 s^3-10,600 s^2-1550 s-415}\) (a) Find how many poles are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. (b) Is this open-loop system stable?

An interval polynomial is of the form \(P(s)=a_0+a_1 s+a_2 s^2+a_3 s^3+a_4 s^4+a_5 s^5+\cdots\) with its coefficients belonging to intervals \(x_i \leq a_i \leq y_i\), where \(x_i, y_i\) are prescribed constants. Kharitonov's theorem says that an interval polynomial has all its roots in the left half-plane if each one of the following four polynomials has its roots in the left half-plane (Minichelli, 1989): \(\begin{aligned} & K_1(s)=x_0+x_1 s+y_2 s^2+y_3 s^3+x_4 s^4+x_5 s^5+y_6 s^6+\cdots \\ & K_2(s)=x_0+y_1 s+y_2 s^2+x_3 s^3+x_4 s^4+y_5 s^5+y_6 s^6+\cdots \\ & K_3(s)=y_0+x_1 s+x_2 s^2+y_3 s^3+y_4 s^4+x_5 s^5+x_6 s^6+\cdots \\ & K_4(s)=y_0+y_1 s+x_2 s^2+x_3 s^3+y_4 s^4+y_5 s^5+x_6 s^6+\cdots \end{aligned}\) Use Kharitonov's theorem and the Routh-Hurwitz criterion to find if the following polynomial has any zeros in the right half-plane. \(\begin{aligned} & P(s)=a_0+a_1 s+a_2 s^2+a_3 s^3 \\ & 2 \leq a_0 \leq 4 ; \quad 1 \leq a_1 \leq 2 ; \quad 4 \leq a_2 \leq 6 ; \quad a_3=1 \end{aligned}\)

A linearized model of a torque-controlled crane hoisting a load with a fixed rope length is \(P(s)=\frac{X_T(s)}{F_T(s)}=\frac{1}{m_T} \frac{s^2+\omega_0^2}{s^2\left(s^2+a \omega_0^2\right)}\) where \(\omega_0=\sqrt{\frac{g}{L}}\), L= the rope length, \(m_T=\) the mass of the car, a= the combined rope and car mass, \(f_T=\) the force input applied to the car, and \(x_T=\) the resulting rope displacement (Marttinen, 1990). If the system is controlled in a feedback configuration by placing it in a loop as shown in Figure P6.11, with K>0, where will the closed-loop poles be located?

The read/write head assembly arm of a computer hard disk drive (HDD) can be modeled as a rigid rotating body with inertia \(I_b\). Its dynamics can be described with the transfer function \(P(s)=\frac{X(s)}{F(s)}=\frac{1}{I_b s^2}\) where X(s) is the displacement of the read/write head and F(s) is the applied force (Yan, 2003). Show that if the HDD is controlled in the configuration shown in Figure P6.11, the arm will oscillate and cannot be positioned with any precision over a HDD track. Find the oscillation frequency.

A system is represented in state space as \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} 0 & 1 & 3 \\ 2 & 2 & -4 \\ 1 & -4 & 3 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right] u \\ & y=\left[\begin{array}{lll} 1 & 1 & 0 \end{array}\right] \mathbf{x} \end{aligned}\) Determine how many eigenvalues are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. [Section: 6.5]

Use MATLAB to find the eigenvalues of the following system: \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 1 & -4 \\ -1 & 1 & 8 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] u \\ & y=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right] \mathbf{x} \end{aligned}\)

The following system in state space represents the forward path of a unity feedback system. Use the Routh-Hurwitz criterion to determine if the closed-loop system is stable. [Section: 6.5] \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 1 & 2 \\ -5 & -4 & -3 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right] u \\ y & =\left[\begin{array}{lll} 1 & 0 & 1 \end{array}\right] \mathbf{x} \end{aligned}\)

Repeat Problem 52 using MATLAB.

A Butterworth polynomial is of the form \(B_n(s)=1+(-1)^n\left(\frac{s}{\omega_c}\right)^{2 n}, n>0\) Use the Routh-Hurwitz criteria to find the zeros of a Butterworth polynomial for: (a) n = 1; (b) n = 2

An inverted pendulum, mounted on a motor-driven cart was presented in Chapter 3, Problem 30. The system's state-space model was linearized around a stationary point, \(\mathbf{x}_{\mathbf{0}}=\mathbf{0}\), corresponding to the pendulum point mass, m, being in the upright position at t=0, when the force applied to the cart \(u_0=0\) (Prasad, 2012). We'll modify that model here to have two output variables: the pendulum angle relative to the y-axis, \(\theta\), and the horizontal position of the cart, x. The output equation becomes: \(\mathbf{y}=\left[\begin{array}{c} \theta \\ x \end{array}\right]=\mathbf{C x}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right]\left[\begin{array}{c} \theta \\ \dot{\theta} \\ x \\ \dot{x} \end{array}\right]\) Using MATLAB, find out how many eigenvalues are in the right half-plane, in the left half-plane, and on the \(j \omega\)-axis. What does that tell us about the stability of that unit? [Section: 6.5]

A model for an airplane’s pitch loop is shown in Figure P6.12. Find the range of gain, K, that will keep the system stable. Can the system ever be unstable for positive values of K?

A common application of control systems is in regulating the temperature of a chemical process (Figure P6.13). The flow of a chemical reactant to a process is controlled by an actuator and valve. The reactant causes the temperature in the vat to change. This temperature is sensed and compared to a desired set-point temperature in a closed loop, where the flow of reactant is adjusted to yield the desired temperature. In Chapter 9, we will learn how a PID controller is used to improve the performance of such process control systems. Figure P6.13 shows the control system prior to the addition of the PID controller. The PID controller is replaced by the shaded box with a gain of unity. For this system, prior to the design of the PID controller, find the range of amplifier gain, K, to keep the system stable.

A transfer function from indoor radiator power, \(\dot{Q}(s)\), to room temperature, T(s), in an \(11 \mathrm{~m}^2\) room is \(P(s)=\frac{T(s)}{\dot{Q}(s)}=\frac{1 \times 10^{-6} s^2+1.314 \times 10^{-9} s+2.66 \times 10^{-13}}{s^3+0.00163 s^2+5.272 \times 10^{-7} s+3.538 \times 10^{-11}}\) where \(\dot{Q}\) is in watts and T is in \({ }^{\circ} \mathrm{C}\) (Thomas, 2005). The room's temperature will be controlled by embedding it in a closed loop, such as that of Figure P6.11. Find the range of K for closed-loop stability.

During vertical spindle surface grinding, adjustments are made on a multi-axis computer numerical control (CNC) machine by measuring the applied force with a dynamometer and applying appropriate corrections. This feedback force control results in higher homogeneity and better tolerances in the resulting finished product. In a specific experiment with an extremely high feed rate, the transfer function from the desired depth of cut (DOC) to applied force was \(\frac{F(s)}{D O C(s)}=\frac{K_C}{1+\frac{K_C}{m s^2+b s+k}-\frac{K_C}{K_f}\left(\frac{1}{T s+1}\right)}\) where \(k=2.1 \times 10^4 \mathrm{~N} / \mathrm{m}, b=0.78 \mathrm{Ns} / \mathrm{m}, m=1.2 \times 10^{-4} \mathrm{~kg}, K_C=1.5 \times 10^4 \mathrm{~N} / \mathrm{mm}\), and \(T=0.004 \mathrm{~s}\). The parameter \(K_f\) is varied to adjust the system. Find the range of \(K_f\) under which the system is stable (Hekman, 1999).

In order to obtain a low-cost lithium-ion battery charger, the feedback loop of Figure P6.3 is used, where \(G(s)=G_c(s) P(s)\). The following transfer functions have been derived for G(s) (Tsang, 2009): \(\begin{gathered} P(s)=\frac{R_1 R_2 C_1 C_2 s^2+\left(R_1 C_1+R_2 C_1+R_2 C_2\right) s+1}{C_1\left(1+R_2 C_2\right) s} \\ G_c(s)=K_p+\frac{K_I}{s} \end{gathered}\) If \(R_1=0.15 \Omega ; R_2=0.44 \Omega ; C_1=7200 \mathrm{~F}\); and \(C_2=170 \mathrm{~F}\), use the Routh-Hurwitz criteria to find the range of positive \(K_P\) and \(K_I\) for which the system is closed-loop stable.

Figure P6.14 is a simplified and linearized block diagram of a cascade control system, which is used to control water level in a steam generator of a nuclear power plant (Wang, 2009). In this system, the level controller, \(G_{L C}(s)\), is the master controller and the feed-water flow controller, \(G_{F C}(s)\), is the slave controller. Using mass balance equations, the water level would ordinarily be regarded as a simple integration process of water flow. In a steam generator, however, steam flow rate and the cooling effect of feed-water change the dynamics of that process. Taking the latter into account and ignoring the much-less pronounced impact of changes in steam flow rate, a firstorder lag plus time delay is introduced into the transfer function, \(G_{f w}(s)\), relating the controlled level, C(s), to feed-water flow rate, \(Q_w(s)\) as follows: \(\begin{aligned} G_{f w}(s) & =\frac{C(s)}{Q_w(s)}=\frac{K_1 e^{-\tau_1 s}}{s\left(T_1 s+1\right)}=\frac{2 e^{-2 s}}{s(25 s+1)} \\ & \approx \frac{2}{s(25 s+1)\left(2 s^2+2 s+1\right)} \end{aligned}\) where \(K_1=2\) is the process gain, \(\tau_1=2\) is the pure time delay, and \(T_1=25\) is the steam generator's time constant. (The expression \(e^{-\tau_1 s}\) represents a time delay. This function can be represented by what is known as a Pade approximation. This approximation can take on many increasingly complicated forms, depending upon the degree of accuracy required. Here we use the Pade approximation, \(ee^{-x} \approx \frac{1}{1+x+\frac{x^2}{2 !}}\), and specific numerical values for the considered steam generator.) The dynamic characteristics of the control valve are approximated by the transfer function \(eG_v(s)=\frac{Q_w(s)}{Y(s)}=\frac{K_v}{T_v s+1}=\frac{1}{3 s+1}\) where \(eK_v\) is the valve gain and \(eT_v\) is its time constant. Given that: \(eG_{F C}(s)=K_{P_{F C}}+K_{D_{F C}} s=0.5+2 s\) and \(eG_{L C}(s)=K_{P_{L C}}+K_{D_{L C}} s=0.5+K s\), use the Routh-Hurwitz criterion to find the range of the level controller's derivative gain, \(eK_{D_{L C}}=K>0\), that will keep the system stable.

Look-ahead information can be used to automatically steer a bicycle in a closed-loop configuration. A line is drawn in the middle of the lane to be followed, and an arbitrary point is chosen in the vehicle's longitudinal axis. A look-ahead offset is calculated by measuring the distance between the look-ahead point and the reference line and is used by the system to correct the vehicle's trajectory. A linearized model of a particular bicycle traveling on a straight-line path at a fixed longitudinal speed is \(\left[\begin{array}{l} \dot{V} \\ \dot{r} \\ \dot{\psi} \\ \dot{Y}_g \end{array}\right]=\left[\begin{array}{cccl} -11.7 & 6.8 & 61.6 K & 7.7 K \\ -3.5 & -24 & -66.9 K & 8.4 K \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -10 & 0 \end{array}\right]\left[\begin{array}{l} V \\ r \\ \psi \\ Y_g \end{array}\right]\) In this model, V= bicycle's lateral velocity, r= bicycle's yaw velocity, \(\psi=\) bicycle's yaw acceleration, and \(Y_g=\) bicycle's center of gravity coordinate on the y-axis. K is a controller parameter to be chosen by the designer (Ö?zgüner, 1995). Use the Routh-Hurwitz criterion to find the range of K for which the system is closed-loop stable.

Figure P5.42 Shows the block diagram of an Automatic Voltage Regulator (Gozde, 2011). Assume in this diagram the following parameter values: \(K_a=10, T_a=0.1, K_e=1\), \(T_e=0.4, K_g=1, T_g=1, K_s=1\), and \(T_s=0.001\). Also assume that the PID transfer function is substituted by a simple integrator, namely \(G_{P I D}(s)=\frac{K}{s}\). Find the range of K for which the system is closed-loop stable.

It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by \(\begin{aligned} & s^3+\left(\frac{m g h}{I}\left(q_2+k k_d\right)+q_1 g\right) s^2 \\ & +k \frac{m g h}{I} s+\frac{m g h}{I}\left(k k_i+q_1\right)=0 \end{aligned}\) where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, \(q_1\) and \(q_2\) are stabilizer flapping parameters, \(k, k_i\), and \(k_d\) are controller parameters; all constants >0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to \(m^{\prime}, h^{\prime}\), and \(I^{\prime}\), respectively, all still >0. Show that the helicopter will remain stable as long as \(\frac{m^{\prime} g h^{\prime}}{I^{\prime}}>\frac{q_1+k k_i-q_1 g k}{k\left(q_2+k k_d\right)}\)

Figure P6.15 shows the model of the dynamics of an economic system (Wingrove, 2012). In this diagram x represents the rate of growth in real Gross National Product (GNP), \(x_0\) the long-term trend (dc value) of the GNP, \(\Delta x\) the change over the long-term trend of the GNP, \(r_x\) the real and psychological disturbance inputs that affect the economy, \(r_m\) the random monetary inputs, and \(\Delta u\) fluctuations in unemployment rate. The diagram has two feedback loops: one through Friedman's model in which the economy dynamics are approximated by \(F(s)=\frac{K_x s}{\left(\frac{s}{\omega_n}\right)^2+2 \xi\left(\frac{s}{\omega_n}\right)+1}\) and a second loop through Okun's law that relates the GNP to unemployment changes. Assuming the following parameter values: \(K_x=2\) years, \(\omega_n=1.5 \mathrm{rad} /\) year, \(\zeta=0.8\), \(K_u=0.4\) and \(G_x=-0.4\). Find the range of \(G_u\) for closed loop stability.

The system shown in Figure P6.16 has \(G_1(s)= 1 / s(s+2)(s+4)\). Find the following: (a) The value of \(K_2\) for which the inner loop will have two equal negative real poles and the associated range of \(K_1\) for system stability. (b) The value of \(K_1\) at which the system oscillates and the associated frequency of oscillation. (c) The gain \(K_1\) at which a real closed-loop pole is at s=-5. Can the step response, c(t), be approximated by a second-order, underdamped response in this case? Why or why not? (d) If the response in Part d can be approximated as a second-order response, find the %OS and settling time, \(T_s\), when the input is a unit step, r(t)=u(t).

A drive system with an elastically coupled load was presented in Problems 71 and 67 in Chapters 4 and 5, respectively (Thomsen, 2011). That drive was shown in Figure P5.43 as the controlled unit in a feedback control system, where \(\Omega_L(s)=\) the load speed and \(\Omega_r(s)=\) the desired (reference) speed. If the controller transfer function is \(G_C(s)=K_p+\frac{K_I}{s}\), while all of the other parameters and transfer functions are the same as in Problem 67 in Chapter 5, find the range of \(K_P\) for stability of the system if \(K_I=0.1\).

Control of HIV/AIDS. The HIV infection linearized model developed in Problem 84, Chapter 4, can be shown to have the transfer function \(P(s)=\frac{Y(s)}{U_1(s)}=\frac{-520 s-10.3844}{s^3+2.6817 s^2+0.11 s+0.0126}\) It is desired to develop a policy for drug delivery to maintain the virus count at prescribed levels. For the purpose of obtaining an appropriate \(u_1(t)\), feedback will be used as shown in Figure P6.17 (Craig, 2004). As a first approach, consider G(s)=K, a constant to be selected. Use the Routh-Hurwitz criteria to find the range of K for which the system is closed-loop stable.

Hybrid vehicle. Figure P6.18 shows the HEV system presented in Chapter 5, where parameter values have been substituted. It is assumed here that the speed controller has a proportional gain, \(K_p\), to be adjusted. Use the Routh-Hurwitz stability method to find the range of positive \(K_p\) for which the system is closed-loop stable (Graebe, 1995).

Parabolic trough collector. The fluid temperature of a parabolic trough collector (Camacho, 2012) will be controlled by using a unity feedback structure as shown in Figure P6.11. Assume the open-loop plant transfer function is given by \(P(s)=\frac{137.2 \times 10^{-6}}{s^2+0.0224 s+196 \times 10^{-6}} e^{-39 s}\) Use the Routh-Hurwitz criteria to find the range of gain K that will result in a closed-loop stable system. Note: Pure time-delay dynamics, such as the one in the transfer function of the parabolic trough collector, cannot be treated directly using the Routh-Hurwitz criterion because it is represented by a nonrational factor. However, a Padé approximation can be used for the nonrational component. The Padé approximation was introduced in Problem 6.61, but it can appear in different forms. Here, it is suggested you use a first-order approximation of the form \(e^{-s T} \approx \frac{1-\frac{T}{2} s}{1+\frac{T}{2} s}\)