Specialized course Process Control
at Norwegian University of Life Sciences (NMBU), Spring 2018
Compulsory exercise to lessons in Lecture 4
Systems theory
1.
Learning Matlab Control System Toolbox (“CST”): Work through the following
sections of this
tutorial to the Matlab Control System Toolbox: 1, 2, 3 - intro, 3.1, 3.2,
5. (You are not expected to report anything from this task.)
2.
Using CST: Stability analysis and simulation of transfer functions: See
Exercise 4.2 in the exercise
book. For each of the transfer functions: Calculate the poles, plot the
poles, determine the stability property (“manually”), and simulate both the
impulse response and the step response. [Abhilash]
3.
Using CST: Stability analysis and simulation of a state space model: See
Exercise 4.3 in the exercise
book. [Alexander]
a.
Determine the stability property of the
system from its eigenvalues. Simulate an initial state response of the state
space model, i.e. assume some non-zero initial state, and simulate the
responses in the states.
b.
Find the transfer function from u to y, and
calculate its poles. Are these poles the same as the eigenvalues of the state
space model?
4.
Using CST: Tuning and simulation
of a feedback control system: See Example 5.1 in the
text book. Assume the controller is a PI controller. Tune the controller
using the Ziegler-Nichols’ method. Plot the closed loop poles (i.e. the poles
of the closed loop, i.e. the feedback control system), and simulate the
setpoint-step response after the tuning. (Tip: Create a transfer function
model of the control system using the feedback
function of CST.) [Duo]
5.
Using CST: Discretization of a
continuous-time model: See Exercise 4.3 in the exercise
book. Discretize the given continuous-time state space model using the
ZOH method with time-step 0.1 s. Calculate the eigenvalues of the resulting
discrete-time model, and conclude about its stability property. Is the
stability property the same as for the original continuous-time model ? (Cf.
Problem 3 above.) [Karla]
6.
Symbolic linearization with Matlab
Symbolic Toolbox:
Linearize the nonlinear model (1.2) - (1.3) shown in the exercise
book. (Tip: Use the jacobian function.) Compare with (your)
manual result. [Xiaodong]
Optimization
The
Rosenbrock optimization problem (“ROP”) is a “standard” optimization problem,
cf. https://se.mathworks.com/help/optim/examples/banana-function-minimization.html.
1.
Grid search: Solve the ROP using the grid
search method. The initial guess can be set to [-1.9,2] as in the above
reference, and you may allow for a search of the optimal solution within -5
and 5 for both x1 and x2. (Template
for solution, similar to script presented in lecture 22 Feb.) [Abhilash]
2.
fmincon: The same as the above problem,
but now by using fmincon. (Template
for solution, similar to script presented in lecture 22 Feb.) [Alexander]
3.
Newton search: The same as the above problem,
but now by implementing the Newton search method, from scratch. You may derive
the gradient function and the Hessian function symbolically with the
gradient() and the hessian() functions in the Symbolic Toolbox. (Template for solution, similar to
script presented in lecture 22 Feb.)
Newton search: x_kp1 = x_k - inv[Hessian_f(x_k)]*Gradient_f(x_k), where “kp1”
means “k plus 1”. [Duo]
4.
Grid search, with constraints: Same as the grid problem above,
but now include the constraint x2 >= x1+1. [Karla]
5.
fmincon, with constraints: Same as the fmincon problem
above, but now include the constraint x2 >= x1+1. [Xiaodong]
Updated 11 March 2018 by Finn Aakre Haugen, course teacher.
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