The purpose of an artificial pacemaker is to regulate heart rate in

The purpose of an artificial pacemaker is to regulate heart rate in those patients in which the natural feedback system malfunctions. Assume a unity-feedback system with a forward path,

as a simplified model of a pacemaker (Neogi, 2010).

a. Convert the pacemaker model to a discrete system with a sampling rate of 0.01 second.

b. Draw the root locus of the system using a computer program.

c. Use the root locus in Part b to find the range of k for which the system is closed-loop stable.

d. Use the root locus from Part b to find the value of k that will yield a 5% overshoot for a step input.

e. Simulate the unit-step input of your discretized system to verify your design.

A linear model of the α-subsystem of a grid-connected voltage-source converter with

A linear model of the α-subsystem of a grid-connected voltage-source converter with a Y-Y transformer (Mahmood, 2012) was presented in Problem 69, Chapter 8, and Problem 51, Chapter 10. The system was represented with unity-feedback and a forward path consisting of the cascading of a compensator and a plant. The plant is given by

This system is now to be digitally controlled with the following specifications: percent overshoot, %OS = 10%; settling time, TS = 0.1 second; and sampling interval, T = 0.001 second. Design a lead compensator for that system to meet these specifications.

Data From Problem 69 Chapter 8:

A linear dynamic model of the α-subsystem of a grid connected voltage-source converter (VSC) using a Y-Y transformer is showninFigureP8.20(a) (Mahmood, 2012). Here, C = 135μF; R1 = 0.016Ω; L1 = 0.14mH; R2 = 0.014Ω; L2 = 10 μH; Rg = 1.1Ω; and Lg = 0.5mH.

a. Find the transfer function GP(s) = Vα(s)/Mα(s0.

b. If GP(s) is the plant in Figure P8.20(b) and GC(s) = K, use MATLAB to plot the root locus. On a closeup of the locus (from 300 to 0 on the real axis and from 50 to 5000 on the imaginary axis), find K and the coordinates of the dominant poles, which correspond to ζ = 0.012. Plot
the output response, c(t) = vα(t), at that value of the gain when a step input, r(t)= vr (t)= 208 u(t) volts, is applied at t = 0. Mark on the time response graph, c(t), all relevant characteristics, such as the percent overshoot, peak time, rise time, settling time, and final steady-state value.

Data From Problem 51 Chapter 10:

A linear model of the α-subsystem of a grid connected voltage-source converter (VSC) with a Y-Y transformer (Mahmood, 2012) was presented in Problem 69, Chapter 8. In Figure P8.20(b), GC(s) = K and GP(s) is given in a pole zero form (with a unity gain and slightly modified parameters) as follows:

An inverted pendulum mounted on a motor-driven cart (Prasad, 2012) was the

An inverted pendulum mounted on a motor-driven cart (Prasad, 2012) was the subject of Problem 33, Chapter 9. In that problem you were asked to develop Simulink models for two feed back systems, one of which was to control the cart position, x(t). At the recommended settings, the step response of that system was expected to satisfy the following requirements: a steady-state error, e(∞) < 2%, a peak time, TP < 1.2 seconds, and a percent overshoot, %OS < 20.5%. Having concluded that the steady-state error was unacceptable, you designed a PID controller and found its recommended settings.

Digitize the Simulink model developed for PID control of the cart position in Part d of Problem 33, Chapter 9, by adding azero-order-holdsetto0.01second.Then, run a simulation to evaluate performance.

Data from Problem 33 Chapter 9

An inverted pendulum mounted on a motor-driven cart was introduced in Problem 30 in Chapter 3. Its state-space model was linearized around a stationary point, x0 = 0 (Prasad, 2012). At the stationary point, the pendulum point-mass, m, is in the upright position at t = 0, and the force applied to the cart, u0, is 0. Its model was then modified in Problem 55 in Chapter 6 to have two output variables: the pendulum angle relative to the y-axis, θ(t), and the horizontal position of the cart, x(t). MATLAB was then used to find its eigenvalues. Noting that only one pole (out of four) is located in the left half of the s-plane, we concluded that this unit requires stabilization.

Data from Problem 30 Chapter 3

Figure P3.17 shows a free-body diagram of an inverted pendulum, mounted on a cart with a mass, M. The pendulum has a point mass, m, concentrated at the upper end of a rod with zero mass, a length, l, and a frictionless hinge. A motor drives the cart, applying a horizontal force, u(t). A gravity force, mg, acts on m at all times. The pendulum angle relative to the y-axis, θ, its angular speed, θ´ , the horizontal position of the cart, x, and its speed, x´, were selected to be the state variables.

In Problem 24, Chapter 11, we discussed an EVAD, a device that

In Problem 24, Chapter 11, we discussed an EVAD, a device that works in parallel with the human heart to help pump blood in patients with cardiac conditions. The device has a transfer function

where Em(s) is the motor’s armature voltage, and Pao(s) is the aortic blood pressure (Tasch, 1990). Using continuous techniques, a cascaded compensator is designed in a unity feedback configuration with a transfer function

Selecting to control the device using a microcontroller, a discrete equivalent has to be found for Gc(s). Do the following:

a. Find an appropriate sampling frequency for the discretization.

b. Translate the continuous compensator into a discrete compensator using the sampling frequency found in Part a.

c. Use Simulink to simulate the continuous and discrete systems on the same graph for a unit step input. There should be little difference between the compensated continuous and discrete systems.

Data from Problem 24 Chapter 11:

An electric ventricular assist device (EVAD) that helps pump blood concurrently to a defective natural heart in sick patients can be shown to have a transfer function

The input, Em(s), is the motor’s armature voltage, and the output is Pao(s), the aortic blood pressure (Tasch, 1990). The EVAD will be controlled in the closed-loop configuration shown in Figure P11.1.

In Problem 49, Chapter 9, and Problem 39, Chapter 10, we considered

In Problem 49, Chapter 9, and Problem 39, Chapter 10, we considered the radial pickup position control of a DVD player. A controller was designed and placed in cascade with the plant in a unit feedback configuration to stabilize the system. The controller was given by

and the plant by (Bittanti, 2002)

It is desired to replace the continuous system by an equivalent discrete system without appreciably affecting the system performance.

a. Find an appropriate sampling frequency for the discretization.

b. Using the chosen sampling frequency, translate the continuous compensator into a discrete compensator.

c. Use Simulink to simulate the continuous and discrete systems on the same graph. Assume a unit step input. Are there significant differences in the system’s performance?

Data from Problem 49 Chapter 9:

Digital versatile disc (DVD) players incorporate several control systems for their operations. The control tasks include (1) keeping the laser beam focused on the disc surface, (2) fast track selection, (3) disc rotation speed control, and (4) following a track accurately. In order to follow a track, the pickup-head radial position is controlled via a voltage that operates a voice coil embedded in a magnet configuration. For a specific DVD player, the transfer function is given by

In Problem 47, Chapter 9, a steam-driven turbine governor system was implemented

In Problem 47, Chapter 9, a steam-driven turbine governor system was implemented by a unity feedback system with a forward-path transfer function (Khodabakhshian, 2005)

a. Use a sampling period of T = 0.5 s and find a discrete equivalent for this system.

b. Use MATLAB to draw the root locus.

c. Find the value of K that will result in a stable system with a damping factor of ζ = 0.7.

d. Use the root locus found in Part a to predict the step-response settling time, Ts, and peak time, Tp.

e. Calculate the final value of the closed-loop system unit step response.

f. Obtain the step response of the system using Simulink. Verify the predictions you made in Parts c and d.

Data from Problem 47 Chapter 9:

Steam-driven power generators rotate at a constant speed via a governor that maintains constant steam pressure in the turbine. In addition, automatic generation control (AGC) or load frequency control (LFC) is added to ensure reliability and consistency despite load variations or other disturbances that can affect the distribution line frequency output. A specific turbine-governor system can be described only using the block diagram of Figure P9.1 in which G(s) = Gc(s)Gg(s)Gt(s)Gm(s), where (Khodabakhshian, 2005)

Discrete time controlled systems can exhibit unique characteristics not available in continuous

Discrete time controlled systems can exhibit unique characteristics not available in continuous controllers. For example, assuming a specific input and some conditions, it is possible to design a system to achieve steady state within one single time sample without overshoot. This scheme is well known and referred to as deadbeat control. We illustrate deadbeat control design with a simple example. For a more comprehensive treatment see (Ogata, 1987).

Assume in Figure 13.25(a) that Gp(s) = 1/s + 1. The purpose of the design will be to find a compensator, Gc(z), such that for a step input the system achieves steady state within one sample. We start by translating the system into the discrete domain to obtain the equivalent of Figure 13.25(c). The pulse transfer function, Gp(z)=(1-e-T)z-1/1 – e-T z-1 , is found using Eq. (13.40), since it is assumed that the compensator will be followed by a zero-order hold. In Figure 13.25(c), the closed-loop transfer function is given by C(z)/R(z) = T(z) = Gc(z)Gp(z)/1 + Gc(z)Gp(z), or, solving for the compensator, we get Gc(z) = 1/Gp(z) T(z)/1 – T(z). The desired system output is a unit step delayed by one unit sample.

Thus, C(z) = z/z – 1 z-1 = 1/z – 1. Since the input is a unit step, R(z) = z/z – 1; the desired closed-loop transfer function is C(z)/R(z) = T(z) = z-1, and the resulting compensator, found by direct substitution, is given by Gc(z) = 1/1 – e-T (z – e-T)/z – 1.

Assume now that the plant is given by Gp(s) = 1/s, and a sampling period of T = 0.05 second is used.

a. Design a deadbeat compensator to reach steady state within one time sample for a step input.

b. Calculate the resulting steady-state error for a unit-slope ramp input.

Obtaining an exact shape in metal forming can be tricky because of

Obtaining an exact shape in metal forming can be tricky because of material spring back. A feedback system has been devised in which critical deviations from specifications are measured as soon as a part is formed and automatic incremental corrections to the forming tools are made before the next part is formed. Eventually the system compensates for material spring back and results in parts compliant with specifications. A unity-feedback digital system with a forward path

can be used as a simplified representation of the system (Fu, 2013).

a. Make a sketch of the system’s root locus.

b. Find the range of k for which the system is closed loop stable.

c. Find the system’s steady-state error for a step input.

d. Find the value of k that will result in the fastest possible response.

Write a MATLAB program that can be used to find the range

Write a MATLAB program that can be used to find the range of sampling time, T, for stability. The program will be used for systems of the type represented in Figure P13.5 and should meet the following requirements:

a. MATLAB will convert G1(s) cascaded with a sample-and-hold to G(z).

b. The program will calculate the z-plane roots of the closed-loop system for a range of T and determine the value of T, if any, below which the system will be stable. MATLAB will display this value of T along with the z-plane poles of the closed-loop transfer function. Test the program on

In Problem 32, Chapter 3, we introduced the idea that when an

In Problem 32, Chapter 3, we introduced the idea that when an electric motor is the sole motive force provider for a hybrid electric vehicle (HEV), the forward paths of all HEV topologies are similar. It was noted that, in general, the forward path of an HEV cruise control system can be represented by a block diagram similar to that of Figure P3.18 (Preitl, 2007). The diagram is shown in Figure P12.8, with the parameters substituted by their numerical values from Problem 69, Chapter 6; the motor armature represented as a first-order system with a unity steady-state gain and a time constant of 50 ms; and the power amplifier gain set to 50. Whereas the state variables remain as the motor angular speed, ω(t), and armature current, Ia(t), we assume now that we have only one input variable, uc(t), the command voltage from the electronic control unit, and one output variable, car speed, v = rω/itot = 0.06154ω. The change in the load torque, Tc(t), is represented as an internal feedback proportional to ω(t).
Looking at the diagram, the state equations may be written as

a. Design an integral controller for %OS ≤ 4.32%, a settling time, Ts ≤ 4.4 sec, and a zero steady-state error for a step input To account for the effect of the integral controller on the transient response, use Ts = 4 seconds in your calculation of the value of the naturalfrequency,ωn, of the required dominant poles.

b. Use MATLAB to verify that the design requirements are met.