Control of HIV/AIDS. The linearized model of HIV infection

Consider the following open-loop transfer functions, where G(s) = Y(s) / U(s) . Y(s) is the Laplace transform of the output, and U(s) is the Laplace transform of the input control signal: i. \G(s) = \(\frac{(s+3)}{(s+4)^2}\) ii. G(s) = \(\frac{s}{(s+5)(s+7)}\) iii. G(s) = \(\frac{20 s(s+7)}{(s+3)(s+7)(s+9)}\) iv. G(s) = \(\frac{30(s+2)(s+3)}{(s+4)(s+5)(s+6)}\) v. G(s) = \(\frac{s^2+8 s+15}{\left(s^2+4 s+10\right)\left(s^2+3 s+12\right)}\) For each of these transfer functions, do the following: [Section: 12.2] a. Draw the signal-flow graph in phase-variable form. b. Add state-variable feedback to the signal-flow graph. c. For each closed-loop signal-flow graph, write the state equations. d. Write, by inspection, the closed-loop transfer function, T(s), for your closed-loop signal-flow graphs. e. Verify your answers for T(s) by finding the closed loop transfer functions from the state equations and Eq. (3.73).

The following open-loop transfer functions can be represented by signal-flow graphs in cascade form. i. G(s) = \(\frac{30(s+2)(s+7)}{s(s+3)(s+5)}\) ii. G(s) = \(\frac{5\left(s^2+3 s+7\right)}{(s+2)\left(s^2+2 s+10\right)}\) For each, do the following: [Section: 12.4] a. Draw the signal-flow graph and show the state-variable feedback. b. Find the closed-loop transfer function with state-variable feedback.

The following open-loop transfer functions can be represented by signal-flow graphs in parallel form. i. G(s) = \(\frac{50\left(s^2+7 s+25\right)}{s(s+10)(s+20)}\) ii. G(s) = \(\frac{50(s+3)(s+4)}{(s+5)(s+6)(s+7)}\) For each, do the following: [Section: 12.4] a. Draw the signal-flow graph and show the state-variable feedback. b. Find the closed-loop transfer function with state-variable feedback.

Given the following open-loop plant, [Section: 12.2] G(s) = \(\frac{50}{(s+1)(s+3)(s+10)}\) design a controller to yield a 10% overshoot and a settling time of 0.5 second. Place the third pole 10 times as far from the imaginary axis as the dominant pole pair. Use the phase variables for state-variable feedback.

Section 12.2 showed that controller design is easier to implement if the uncompensated system is represented in phase-variable form with its typical lower companion matrix. We alluded to the fact that the design can just as easily progress using the controller canonical form with its upper companion matrix. [Section: 12.2] a. Redo the general controller design covered in Section 12.2, assuming that the plant is represented in controller canonical form rather than phase variable form. b. Apply your derivation to Example 12.1 if the uncompensated plant is represented in controller canonical form.

Given the following open-loop plant: [Section: 12.2] G(s) = \(\frac{100(s+2)(s+25)}{(s+1)(s+3)(s+5)}\) design a controller to yield 10% overshoot with a peak time of 0.5 second. Use the controller canonical form for state-variable feedback.

Given the following open-loop plant: [Section: 12.2] G(s) = \(\frac{20(s+2)}{s(s+5)(s+7)}\) design a controller to yield a 10% overshoot and a settling time of 2 seconds. Place the third pole 10 times as far from the imaginary axis as the dominant pole pair. Use the phase variables for state-variable feedback.

Repeat Problem 4 assuming that the plant is represented in the cascade form. Do not convert to phase-variable form. [Section: 12.4]

Repeat Problem 7 assuming that the plant is represented in the parallel form. Do not convert to phase-variable form. [Section: 12.4]

a. Given the plant shown in Figure P12.1, what relationship exists between \(b_1\) and \(b_2\) to make the system uncontrollable? b. What values of \(b_2\) will make the system uncontrollable if \(b_1\) = 1? [Section: 12.3]

For each of the plants represented by signal-flow graphs in Figure P12.2, determine the controllability. If the controllability can be determined by inspection, state that it can and then verify your conclusions using the controllability matrix. [Section: 12.3]

Use MATLAB to determine the controllability of the systems of Figure P12.2(d) and (f).

In Section 12.4, we discussed how to design a controller for systems not represented in phase-variable form with its typical lower companion matrix. We described how to convert the system to phase-variable form, design the controller, and convert back to the original representation. This technique can be applied just as easily if the original representation is converted to controller canonical form with its typical upper companion matrix. Redo Example 12.4 in the text by designing the controller after converting the uncompensated plant to controller canonical form. [Section: 12.4]

Consider the following transfer function: G(s) = \(\frac{(s+6)}{(s+3)(s+8)(s+10)}\) If the system is represented in cascade form, as shown in Figure P12.3, design a controller to yield a closed loop response of 10% overshoot with a settling time of 1 second. Design the controller by first transforming the plant to phase variables. [Section: 12.4]

Use MATLAB to design the controller gains for the system given in Problem 14.

Repeat Problem 14 assuming that the plant is represented in parallel form. [Section: 12.4]

The open-loop system of Problem 14 is represented as shown in Figure P12.4. If the output of each block is assigned to be a state variable, design the controller gains for feedback from these state variables. [Section: 12.4]

If an open-loop plant G(s) = \(\frac{100}{s(s+4)(s+10)}\) is represented in parallel form, design a controller to yield a closed-loop response of 20% overshoot and a peak time of 0.2 second. Design the controller by first transforming the plant to controller canonical form. [Section: 12.4]

For a specific individual, the linear time-invariant model of the hypothalamic-pituitary adrenal axis of the endocrine system with five state variables has been found to be (Kyrylov, 2005) \(\left[\begin{array}{c} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \\ \dot{x}_5 \end{array}\right]=\left[\begin{array}{ccccc} -0.014 & 0 & -1.4 & 0 & 0 \\ 0.023 & -0.023 & -0.023 & 0 & 0 \\ 0.134 & 0.67 & -0.67 & 0.38 & 0.003264 \\ 0 & 0 & 0.06 & -0.06 & 0 \\ 0 & 0 & 0.0017 & 0 & -0.001 \end{array}\right]\) \(\times\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{array}\right]+\left[\begin{array}{l} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] d_0\) The State-variable definitions were given in Problem 23, Chapter 3. a. Use MATLAB to determine if the system is controllable. b. Use MATLAB to express the matrices A and B in phase-variable form.

Consider the plant G(s) = \(\frac{1}{s(s+2)(s+5)}\) whose state variables are not available. Design an observer for the observer canonical variables to yield a transient response described by \(\zeta\) = 0.6 and \(\omega_n\) = 50. Place the third pole 10 times farther from the imaginary axis than the dominant poles. [Section: 12.5]

Design an observer for the plant G(s) = \(\frac{10}{(s+3)(s+7)(s+15)}\) operating with 10% overshoot and 2 seconds peak time. Design the observer to respond 10 times as fast as the plant. Place the observer third pole 20 times as far from the imaginary axis as the observer dominant poles. Assume the plant is represented in observer canonical form. [Section: 12.5]

Repeat Problem 20 assuming that the plant is represented in phase-variable form. Do not convert to observer canonical form. [Section: 12.7]

Consider the plant G(s) = \(\frac{(s+2)}{(s+5)(s+9)}\) whose phase variables are not available. Design an observer for the phase variables with a transient response described by \(\zeta\) = 0.6 and \(\omega_n\) = 120. Do not convert to observer canonical form. [Section: 12.7]

Determine whether or not each of the systems shown in Figure P12.2 is observable. [Section: 12.6]

Use MATLAB to determine the observability of the systems of Figure P12.2 (a) and (f).

a. Given the plant of Figure P12.5, what relationship must exist between \(c_1\) and \(c_2\) in order for the system to be unobservable? b. What values of \(c_1\) will make the system uncontrollable if \(c_2\) = 1? [Section: 12.6]

Design an observer for the plant G(s) = \(\frac{1}{(s+5)(s+13)(s+20)}\) represented in cascade form. Transform the plant to observer canonical form for the design. Then transform the design back to cascade form. The characteristic polynomial for the observer is to be \(s^3\) + \(600 s^2\) + 40,000s + 1,500,000.

Use MATLAB to design the observer gains for the system given in Problem 27.

Repeat Problem 27 assuming that the plant is represented in parallel form. [Section: 12.7]

Design an observer for G(s) = \(\frac{45}{(s+3)(s+5)(s+10)}\) represented in phase-variable form with a desired performance of 10% overshoot and a settling time of 0.5 second. The observer will be 10 times as fast as the plant, and the observer's non dominant pole will be 10 times as far from the imaginary axis as the observer's dominant poles. Design the observer by first converting to observer canonical form. [Section: 12.7]

Observability and controllability properties depend on the state-space representation chosen for a given system. In general, observability and controllability are affected when pole-zero cancellations are present in the transfer function. Consider the following two systems with representations: \(\dot{x}_i\) = \(\mathbf{A}_i \mathbf{x}_i=\mathbf{B}_i r\) y = \(\mathbf{C}_i \mathbf{x}_i\) \(\mathbf{A}_1=\left[\begin{array}{rr} 0 & 1 \\ -2 & -3 \end{array}\right] ; \quad \mathbf{B}_1=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] ; \quad \mathbf{C}_1=\left[\begin{array}{ll} 2 & 0 \end{array}\right]\) \(\mathbf{A}_2=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -11 & -6 \end{array}\right] ; \mathbf{B}_2=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] ; \mathbf{C}_2=\left[\begin{array}{lll} 6 & 2 & 0 \end{array}\right]\) a. Show that both systems have the same transfer function \(G_i(s)\) = \(\frac{Y(s)}{R(s)}\) after pole-zero cancellations. b. Evaluate the observability of both systems.

Given the plant \(\dot{\mathbf{x}}=\left[\begin{array}{rr} -1 & 1 \\ 0 & 2 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \mathrm{u} ; \quad y=\left[\begin{array}{ll} 1 & 1 \end{array}\right] \mathbf{x}\) design an integral controller to yield a 15% overshoot, 0.6-second settling time, and zero steady-state error for a step input. [Section: 12.8]

Repeat Problem 32 for the following plant: [Section: 12.8] \(\dot{\mathbf{x}}=\left[\begin{array}{rr} -2 & 1 \\ 0 & -5 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \mathbf{u} ; \quad y=\left[\begin{array}{ll} 1 & 1 \end{array}\right] \mathbf{x}\)

Problem 22 in Chapter 3 introduced the model for patients treated under a regimen of a single day of Glargine insulin (Tarín, 2005). The model to find the response for a specific patient to medication can be expressed in phase-variable form with \(\mathbf{A}=\left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -501.6 \times 10^{-6} & -128.8 \times 10^{-3} & -854 \times 10^{-3} \end{array}\right]\); \(\mathbf{B}=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right] ; \quad \mathbf{C}=\left[\begin{array}{lll} 0.78 \times 10^{-4} & 41.4 \times 10^{-4} & 0.01 \end{array}\right]\) \(\mathbf{D}=0\) The state variables will take on a different significance in this expression, but the input and the output remain the same. Recall that u = external insulin flow, and y = plasma insulin concentration. a. Obtain a state-feedback gain matrix so that the closed-loop system will have two of its poles placed at -1/5 and the third pole at -1/2.

Figure P12.6 shows a continuous stirred tank reactor in which an aqueous solution of sodium acetate \(\left(\mathrm{CH}_3 \mathrm{COONa}\right)\) is neutralized in the mixing tank with hydrochloric acid (HCl) to maintain a particular pH in the mixing tank. The amount of acid in the mix is controlled by varying the rotational speed of a feeding peristaltic pump. A nominal linearized transfer function from HCl flow rate to pH has been shown to be (Tadeo, 2000) G(s) = \(\frac{-0.9580 \times 10^{-4} s-0.01197 \times 10^{-4}}{s^3+0.5250 s^2+0.01265 s+0.000078}\) a. Write the system in state-space phase-variable form. b. Use state-feedback methods to design a matrix K that will yield an overdamped output pH response with a settling time of \(T_s \approx 5 \mathrm{~min}\) for a step input change in pH. c. Simulate the step response of MATLAB the resulting closed-loop ML system using MATLAB.

In the dc-dc converter of Problem 67, Chapter 4 (Van Dijk, 1995 ), with L = 6 mH, C = 1 mF, R = 100 \(\Omega\), a 50% PWM duty cycle, and assuming the system's output is the voltage across the capacitor, the model can be expressed as \(\begin{aligned} {\left[\begin{array}{c} \hat{i}_L \\ \dot{u}_C \end{array}\right] } & =\left[\begin{array}{cc} 0 & -83.33 \\ 500 & -10 \end{array}\right]\left[\begin{array}{l} i_L \\ u_C \end{array}\right]+\left[\begin{array}{c} 166.67 \\ 0 \end{array}\right] E_s \\ y & =\left[\begin{array}{ll} 0 & 1 \end{array}\right]\left[\begin{array}{l} i_L \\ u_C \end{array}\right] \end{aligned}\) a. Find the system’s transfer function. b. Express the system’s state equation in phase-variable form. c. Find a set of state-feedback gains to obtain 20% overshoot and a settling time of 0.5 second in the phase-variable system. d. Obtain the corresponding set of state-feedback gains in the original system. e. Verify that the set of gains in Part d places the closed-loop poles at the desired positions. f. Simulate the unit step response of the system using MATLAB.

a. Design an observer for the dc-dc converter of Problem 36. The observer should have time constants 10 times smaller than those of the original system. b. Simulate your system and observer for a unit step input using simulink. Assume that the initial conditions for the original system are \(\mathbf{x}(0)=\left[\begin{array}{l}2 \\ 1\end{array}\right]\). The observer should have initial conditions \(\hat{\mathbf{x}}(0)=\left[\begin{array}{l}0 \\ 0\end{array}\right]\).

a. Design an observer for the neutralization system using the continuous stirred tank reactor of Problem 35 . The observer should have time constants 10 times smaller than those of the original system. Assume that the original state variables are those obtained in the phase-variable representation. b. Simulate your system and observer for a unit step input using Simulink. Assume that the initial conditions for the original system are \(\mathbf{x}(0)=\left[\begin{array}{r}-1 \\ -10 \\ 3\end{array}\right]\). The observer should have initial conditions \(\hat{\mathbf{x}}(0)=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]\).

a. Redesign the dc-dc converter system of Problem 36 to include integral control. b. Simulate your system for a step input using Simulink and verify that the specifications are met. In particular, verify that the system has zero steady-state error.

The use of feedback control to vary the pitch angle in the blades of a variable speed wind turbine allows power generation optimization under variable wind conditions (Liu, 2008). At a specific operating point, it is possible to linearize turbine models. For example, the model of a three-blade turbine with a 15 m radius working in 12 m/s wind-speed and generating 220 V can be expressed as \(\dot{\mathbf{x}}=\left[\begin{array}{ccccc} -5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -10.5229 & -1066.67 & -3.38028 & 23.5107 & 0 \\ 0 & 993.804 & 3.125 & -23.5107 & 0 \\ 0 & 0 & 0 & 10 & -10 \end{array}\right] \mathbf{x}\) \(+\left[\begin{array}{l} 5 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right] u\) \(y=\left[\begin{array}{lllll} 0 & 0 & 0 & 1.223 x 10^5 & 0 \end{array}\right] x\) where the state variable vector is given by \(\mathbf{x}=\left[\begin{array}{lllll} \beta & \xi & \dot{\xi} & \omega_g & \omega_{g m} \end{array}\right]\) Here, \(\beta\) = pitch angle of the wind turbine blades, \(\xi\) = relative angle of the secondary shaft, \(\omega_g\) = generator speed, and \(\omega_{g m}\) = generator measurement speed. The system input is u, the pitch angle reference, and the output is y, the active power generated. a. Find a state feedback vector gain such that the system responds with a 10% overshoot and a settling time of 2 seconds for a step input. b. Use MATLAB to verify the operation of the system under state feedback.

The study of the flexible links, such as the one shown in Figure P12.7, is important because of their application to the control of flexible lightweight robots (Saini, 2012). The flexible link angle is deflected by a servomotor. It is assumed that the base angle, \(\theta(t)\), and the tip angular deflection relative to the undeformed link, \(\alpha(t)\), can be measured. For a specific setup, a state-space model of the system was developed. The state vector is \(\mathbf{x}=\left[\begin{array}{llll}\theta & \alpha & \omega & \dot{\alpha}\end{array}\right]^T\), where \(\omega(t)=\dot{\theta}(t)\) and input u(t) is the voltage applied to the servomotor. Thus the system is represented as \(\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B} u, y=\mathbf{C x}\) where \(\mathbf{A}=\left[\begin{array}{rccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 673.07 & -35.1667 & 0 \\ 0 & -1023.07 & 35.1667 & 0 \end{array}\right]\) \(\mathbf{B}=\left[\begin{array}{c} 0 \\ 0 \\ 61.7325 \\ -61.7325 \end{array}\right]\) \(\mathbf{C}=\left[\begin{array}{llll} 1 & 1 & 0 & 0 \end{array}\right]\) It is desired to build state-feedback compensation around this system so that the system's characteristic equation becomes D(s) = \((s+10)^4\). In order to do this: a. Find the system's controllability matrix \(\mathbf{C}_{\mathbf{M}_{\mathbf{o}}}\) and show that the system is controllable. b. Find the original system's characteristic equation and use it to find a phase-variable representation of the system. c. Find the phase-variable system's controllability matrix \(\mathbf{C}_{\mathbf{M}_{\mathbf{P}}}\) and then find the transformation matrix \(\mathbf{P}=\mathbf{C}_{\mathbf{M}_0} \mathbf{C}_{\mathbf{M}_{\mathbf{P}}}^{-1}\) d. Use the phase-variable representation to find a feedback gain matrix \(\mathbf{K}_{\mathbf{P}}=\left[\begin{array}{llll}k_{1 p} & k_{2 p} & k_{3 p} & k_{4 p}\end{array}\right]\) that will place the closed-loop poles in the desired positions. e. Find the corresponding feedback gain matrix \(\mathbf{K}_{\mathbf{O}}=\mathbf{K}_{\mathbf{P}} \mathbf{P}^{-1}\).

We want to use an observer in a textile machine to estimate the state variables. The 2-input, 1-output system's model is \(\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B u} ; y=\mathbf{C x}\), where (Cardona, 2010) \(\mathbf{A}=\left[\begin{array}{ccc} 0 & 1 & 0 \\ -52.6532 & -4.9353 & -2768.1557 \\ -0.001213 & 0 & -0.06106 \end{array}\right]\) \(\mathbf{B}=\left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \\ 0.001613 & -0.001812 \end{array}\right]\) \(\mathbf{C}=\left[\begin{array}{lll} 1 & 0 & 0 \end{array}\right]\) a. Find the system's observability matrix \(\mathbf{O}_{\mathbf{M z}}\) and show that the system is observable. b. Find the original system's characteristic equation and use it to find an observable canonical representation of the system. c. Find the observable canonical system's observability matrix \(\mathbf{O}_{\mathrm{Mx}}\) and then find the transformation matrix \(\mathbf{P}=\mathbf{O}_{\mathbf{M z}}^{-1} \mathbf{O}_{\mathbf{M x}}\). d. Use the observable canonical representation to find an observer gain matrix \(\mathbf{L}_{\mathbf{x}}=\left[\begin{array}{llll}l_{1 x} & l_{2 x} & l_{3 x} & l_{4 x}\end{array}\right]^T\) so that the observer characteristic polynomial is \(D(s)=s^3+30 s^2+316 s+1160\). e. Find the corresponding observer gain matrix \(\mathbf{L}_{\mathbf{z}}=\mathbf{P L}_{\mathbf{x}}\).

An inverted pendulum mounted on a motor-driven cart was presented in Problem 30, Chapter 3. Its state space model was linearized (Prasad, 2012) around a stationary point, \(\left(x_0=0\right)\), corresponding to the pendulum point-mass, m, being in the upright position at t = 0. Its model was then modified in Problem 55, Chapter 6, to have two output variables: the pendulum angle relative to the y-axis, \(\theta(t)\), and the horizontal position of the cart, x(t). Noting that the unit requires stabilization, you were asked in Problem 33, Chapter 9, to develop Simulink models for two feedback systems: one to control the cart position, x(t), and the second to control the pendulum angle, \(\theta(t)\). Modify the second model, using state-feedback amplifiers with appropriate gains (in addition to the rate feedback amplifier and the PD controller), to improve the unit-impulse response of the angle control loop. Compare the response you get here with that obtained for Problem 33, Chapter 9.

Let the plant in the drive system with an elastically coupled load (Thomsen, 2011) shown in Figure P8.15 be \(G_p(s)\) = \(\frac{Y(s)}{M(s)}=\frac{250\left(s^2+1.2 s+12500\right)}{s^3+8.1 s^2+62003 s+31250}\) where Y(s) = \(\Omega_L(s)\), the load speed. Represent \(G_p(s)\) in observer canonical form. Then design an observer for it, such that it responds 10 times faster than the output, y(t), if \(G_C(s)\) = \(K_P\). [Section: 12.5]

Control of HIV/AIDS. The linearized model of HIV infection when RTIs are used for treatment was introduced in Chapter 4 and repeated here for convenience (Craig, 2004): \(\left[\begin{array}{c} \dot{T} \\ \dot{T}^* \\ \dot{v} \end{array}\right]=\left[\begin{array}{ccc} -0.04167 & 0 & -0.0058 \\ 0.0217 & -0.24 & 0.0058 \\ 0 & 100 & -2.4 \end{array}\right]\left[\begin{array}{c} T \\ T^* \\ v \end{array}\right]\) \(+\left[\begin{array}{c} 5.2 \\ -5.2 \\ 0 \end{array}\right] u_1\) \(y=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} T \\ T^* \\ v \end{array}\right]\) T represents the number of healthy T-cells, \(T^*\) the number of infected cells, and v the number of free viruses. a. Design a state-feedback scheme to obtain (1) zero steady-state error for step inputs (2) 10% overshoot (3) a settling time of approximately 100 days (Hint: the system's transfer function has an open-loop zero at approximately -0.02. Use one of the poles in the desired closed-loop-pole polynomial to eliminate this zero. Place the higher-order pole 6.25 times farther than the dominant pair.) b. Simulate the unit step response of your design using Simulink.