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Assignment briefing
This assignment will assess your fundamental understanding of model predictive control and your ability to design MPC controllers and simulate and analyse MPC-controlled systems. The assignment comprises an open-ended design, simulation and analysis exercise:
Produce a report (limit: 4 pages in the provided template) containing your answer.
It is up to you how you tackle the problem and stucture your answer. However, it is suggested that you look at (i) the help below and (ii) the attached assessment criteria (at the back of this document) for guidance on what to include.
Assessment criteria
The assessment criteria for this exercise are derived from the module learning outcomes, which are:
1. Assignment Project Exam HelpDescribe and explain the principles of more than one advanced control technique.
2. Analyse practical performance specifications and convert these into functional requirements on controllers.
problem or application.
6. Construct a constrained finite-horizon optimal control problem—including constraint, model and cost definition—re-formulate it as an optimization problem, and recall and evaluate the analytical expression for the control law in the unconstrained case.
7. Analyse, design, implement and simulate MPC controllers with guaranteed properties, including feasibility, stability and offset-free tracking.
In particular, learning outcomes 2, 3, 6 and 7 are relevant to this assessment, and the attached assessment criteria—the marksheet that will be used to assess the assignment— are derived from these. The marksheet indicates the criteria that will be used in assessing your answer, and also the expectation for each criterion in order to achieve a mark within the specified ranges. It is suggested that you study this marksheet before completing the assignment.
These assessment criteria are deliberately broad in order to accommodate the different approaches that might be taken to solving the assignment problem.
Guidance
• The non-assessed computer exercises are good preparation for this assignment. However, the computer exercises were structured, whereas this assignment is open-ended: you need to decide what is the most appropriate approach to solve this assignment, and also how to present your results.
• For the report, you are recommended to consult the attached assessment criteria (at the back of this briefing) for guidance on what to include and to what level of detail.
Assignment Project Exam HelpIn particular, the assessment criteria suggest that your report might need to include, among other things,
– Clear reporting and discussion of results (including clear, labelled plots), and critical evaluation of the controller. (Think about more than just, for example, “Did my controller meet the spec.?” — what are the strengths and weaknesses of your solution? What could be improved?)
– Some analysis and evaluation of your design, including its theoretical properties (if any) and its performance under different scenarios.
This is not an exhaustive list, and what you should include will vary depending on the topic you choose. However, a suggested outline for any report is
1. Abstract
2. Introduction
3. Problem statement
4. Design
5. Results
6. Analysis and discussion
7. Conclusion
• If you do not manage to achieve a working controller or simulation, simplify things, get something simpler working, and build systematically towards a comprehensive solution. In fact, I advise in all cases to start tackling the problem without constraints, in order that you can get something working without having to wrestle
with infeasibility and instability issues (which often arise from incorrect implementation of constraints). Before you consider tracking and disturbance rejection problems, solve the regulation problem, from a non-zero initial state, first.
• Should you need clarification or have questions on any part of the assignment then please just ask! (Talk to me in classes, post to the discussion board, or email me at Assignment Project Exam [email protected]).
A benchmark problem in control design
Figure 1: The Wie–Bernstein two-cart control benchmark system.
System description and assumptions
The dynamics of the system are modelled asAssignment Project Exam Help
⎡𝑥1⎤ ⎡ 0 0 1 0⎤ ⎡𝑥1⎤ ⎡ 0 ⎤ ⎡ 0 0 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢
d𝑡 ⎢𝑥3⎥ ⎢−𝑘s/𝑚1 𝑘s/𝑚1 0 0⎥ ⎢𝑥3⎥ ⎢1/𝑚1⎥ ⎢1/𝑚1 0 ⎥ [𝑑2]
⎢⎣𝑥4⎥⎦ ⎢⎣ 𝑘s/𝑚2 −𝑘s/𝑚2 0 0⎥⎦ ⎣⎢𝑥4⎦⎥ ⎢⎣ 0 ⎦⎥ ⎢⎣ 0 1/𝑚2⎥⎦
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
where 𝑥1 and 𝑥2 are the respective displacements (from datums) of carts 1 and 2, 𝑥3 and 𝑥4 are the respective velocities, 𝑢 is the control input, 𝑦 is the performance output, and 𝑑1 and 𝑑2 are disturbances acting on carts 1 and 2 respectively.
Assume that:
1. The masses 𝑚1 = 𝑚2 = 1 and the (nominal) spring constant 𝑘s = 1.
3. The dynamics are discretized using zero-order hold and a sampling period of 0.1 s.
4. Because zero-order hold is an exact discretization method, the obtained discretized model is the plant for simulation purposes — you do not need to simulate the true continuous-time plant in your solution.
The control task
Design, implement, simulate, analyse and evaluate a model predictive controller in order to achieve the following specifications:
1. In the absence of disturbances (𝑑1 = 𝑑2 = 0), the controller is able to to regulate an initial state 𝑥 to the origin, in a reasonable amount of time,
while satisfying the constraints
|𝑢| ≤ 1, |𝑥3| ≤ 0.5, |𝑥4| ≤ 0.5.
2. The same controller (i.e., a controller with the same values of design parameters) should be able to perform this regulation task for values of the spring constant in the range
0.5 ≤ 𝑘s ≤ 2.
3. With the system started from rest (𝑥 ), the controller should be able to achieve offset-free tracking of a constant reference 𝑟 with ‘good’ performance — i.e., achieve 𝑦 = 𝑟 in steady state, with a ‘reasonable’ settling time and mimimal overshoot — in the presence of constant disturbances 𝑑1 and/or 𝑑2 and, of course, the constraints. It should be able to achieve this for a range of different 𝑟, 𝑑1 and 𝑑2. A particularly interesting scenario for task 3 is whereAssignment Project Exam Help
𝑟 = 1, 𝑑1 = 0.25, 𝑑2 = 0.7.
Your design should consider all, or as many, parts of this specification as possible. You should try to balance performance with computational complexity, and theoretical properties with practicality. For example, designs with guaranteed properties (e.g., recursive feasibility) are
It is strongly recommended that you build your controller in an incremental way; i.e., starting with task 1 and initially in the absence of any constraints, before adding just input constraints, etc. (You do not, however, have to describe the entirety of this incremental design process in your report.)
Finally, you should to try to provide some quantitative assessment of the performance, complexity and the limits of operation of your controller. Performance wise, consider calculating (particularly for task 3) the integral squared error (ISE) and maximum overshoot during a simulation that comprises 𝑇 steps.
𝑇
ISE = ∑ (𝑦(𝑘) − 𝑟)2 Overshoot = max𝑘∈[0,𝑇] 𝑦(𝑘) − 𝑟
𝑟
𝑘=0
A ‘good’ design should minimize both, where possible.
In terms of computational complexity, you might like to consider the size of the QP problem your controller has to solve, and how long this takes.
