Assign 2 in Digital Control
ENEL 809
This assignment is part of the overall course assessment. It has 60 points.
The following general points apply to all work handed in for assessment.
. Remember to include your name and assignment # on your assignment solutions.
. Hand in your the assignment by the due date.
. When including plots and figures, incorporate them in your submission with appropriate axis labels. Always label your plot axes with titles and units. Title your plot.
. When including computer code, use an appropriate font such as courier that clearly distinguishes between the letter l and number 1 (one), the upper case letter O and the number zero, 0, etc. Do not forget meaningful comments in your code.
. You solutions must include all workings to show how you derived the answer. Simply giving the numerical solutions (as is given in the textbook in the drill exercises) is worthless.
. Read the question carefully, and make sure you answer any specific questions. You may need to write a paragraph or two. Often this is more important than your numerical answer.
. Don’t ever print out long lists of numbers. Instead, give me a plot, it saves paper. Be concise.
. Even though it is not always specifically asked for, general comments and a problem critique are always a good thing, in fact they are expected in a professional report. Some of these questions are deliberately vague where you are required to make some intelligent and professional decisions regarding issues like colour maps, axis limits and so forth. Make those decisions, and justify them.
1. (a) Convert the following continuous transfer function
to the discrete G(z − 1 ) using the bilinear transform.
(b) Check your answer with c2d. Do you get exactly the same transfer function? Hint: You may need to specify an optional argument in the c2d command.
(c) Plot the response to a step function for the (i) original continuous G(s), (ii) G(z − 1 ) obtained via a zeroth-order hold, and (iii) G(z − 1 ) obtained via a first order hold.
2. A closed loop sampled system of the form shown in Fig. 1 has the plant
Figure 1: A closed loop sampled system.
(a) Determine the range of sampling period T for which the closed loop system is stable. (b) Select a sampling period T so that the system is stable and provides a rapid response.
3. The simplified (2 × 2) model of the Wood-Berry distillation column, (without the distur- bance streams or the deadtimes), is given by
as developed in [1, §3.3.5.1].
(a) Convert the transfer function matrix in Eqn. 1 to a state-space system using tf2ss or simply ss. How many states, inputs and outputs do you have?
(b) Simulate the response for a step change in both inputs with the step command. Is this the same as for the TF version?
(c) Implement a Simulink version of a discretised state-space model where you have connected the Φ, ∆, and C matrices by hand. Use a sample time of 1 minute.
Simulate the response.
Hint: See [1, Fig 2.36].
4. Vehicle traction control, which includes antiskid braking and antispin acceleration, can enhance vehicle performance and handling. The objective of this control is to maximize tire traction by preventing the wheels from locking during braking and from spinning during acceleration.
Wheel slip, the difference between the vehicle speed and the wheel speed (normalized by the vehicle speed for braking and the wheel speed for acceleration), is chosen as the controlled variable for most of the traction-control algorithm because of its strong influence on the tractive force between the tire and the road.
A model for one wheel is shown in Fig. 2 when y is the wheel slip and the vehicle model is
The goal is to minimize the slip when a disturbance occurs due to road conditions.
Figure 2: Vehicle fraction control system.
(a) Design a controller D(z) so that the ζ of the system is 1/√2 ≈ 0.7071,and determine the resulting gain K. Assume T = 0.1 s.
(b) Plot the resulting step response, and find the overshoot and settling time (with a 2% criterion).
5. Consider the third-order svstem
(a) Using the acker function, determine a full-state feed-back gain matrix and an ob-server gain matrix to place the closed-loop system poles at s1,2 = −1.4 ± j1.4, and s3 = −2 and the observer poles at s1,2 = −18 ± j5, and s3 = −20.
Hint: See [1, §8.2 & §8.3]. You may need to use [1, Table 8.1] to be able to design both gains with the same underlying function.
(b) Construct the state variable compensator using Fig. 3 or [1, Fig 8.11] as a guide. You may find it useful to use Simulink to build this controller.
Figure 3: A State variable compensator employing full-state feedback in series with a full- state observer
(c) Simulate the closed-loop system with the state initial conditions x(0) = (1, 0, 0)T and initial state estimate of x(ˆ)(0) = (0.5, 0.1, 0.1)T .
Hint: You will need to carefully specify the initial conditions for each of the two dynamic systems.
6. In an effort to open up the far side of the Moon to exploration, studies have been con- ducted to determine the feasibility of operating a communication satellite around the translunar equilibrium point in the Earth-Sun-Moon system. The desired satellite orbit, known as a halo orbit, is shown in Fig. 4.
Figure 4: (a) A Halo orbit (b) The view from Earth of the relay communications satellite.
The objective of the controller is to keep the satellite on a halo orbit trajectory that can be seen from the Earth so that the lines of communication are accessible at all times. The communication link is from the Earth to the satellite and then to the far side of the Moon.
The linearized (and normalized) equations of motion of the satellite around the translunar equilibrium point are
The 6 element state vector x is defined as the satellite position (ξ,η,ζ) and velocity, (ξ(˙), ˙(η) , ζ(˙)), and the inputs ui, i = 1, 2, 3 are the engine thrust accelerations in the ξ,η, and ζ directions, respectively.
(a) Is the translunar equilibrium point a stable location?
Hint: Find the eigenvalues of the relevant matrix.
(b) Is the system controllable from u1 alone?
Hint: Use ctrb, but only use one column of the B matrix.
(c) Repeat part (b) for u2 and also repeat for u3. Comment on what you have established as to how difficult this control problem is.
(d) Suppose that we can only observe the position in the η direction. Determine the transfer function from u2 to η. What is the order of your obtained transfer function
and is it what you expect (i.e. nx)? Explain what has happened. (Hint: Let y = [0, 1, 0, 0, 0, 0]x and use tf. )
(e) Compute a state-space representation of the transfer function in part (d) using the ss function. Verify that the resulting 4th order system is controllable.
(f) Using state feedback
u2 = −Kx,
design a controller (i.e., find K) for the system in part (f) such that the 4 closed-loop
system poles are at s1,2 = − 1 ± j and s3,4 = − 10.
(g) Simulate the controlled closed loop from u2 to η .
References
[1] David I. Wilson. Advanced Control Using Matlab: A Pragmatic Approach. Auckland University of Technology, Auckland, New Zealand, 2018.