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ACS6116 Advanced Control: Assignment
Assignment weighting
25% of the total mark for ACS6116.
Assignment released
12:00 on Wednesday 27th March 2025 (Week 7).
Assignment due
23:59 on Monday 12th May 2025 (Week 11).
Penalties for late submission
Late submissions will incur the usual penalties of a 5% reduction in the mark for every working day (or part thereof ) that the assignment is late and a mark of zero for submission more than 5 working days late. For more information see https://students.sheffield.ac.uk/assessment/late-submission.
Feedback
This will include the overall mark, individual component marks and comments on perfor mance on the assignment. The attached assessment criteria (at the back of this document) provide a guide to the areas on which the feedback will be provided. Note that marks may be subject to change as a result of unfair means.
Academic misconduct
The assignment must be completed individually. You must not work together to complete the assignment—it must be wholly your own work. References must be provided to any other work that is used as part of this assignment. Any suspicions of the use of academic misconduct will be investigated and may lead to penalties. See https://www.sheffield.ac. uk/new-students/academic-misconduct for more information.
Extenuating circumstances
If you have extenuating circumstances that cause you to be unable to submit this assignment on time or that may have affected your performance, please complete and submit an extenuating circumstances form—see https://students.sheffield.ac.uk/extenuating-circumstances for information. The form may be found here.
Assignment aim and assessed learning objectives
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 design, simulation and analysis exercise, with some open-ended elements.
This assignment assesses learning outcomes 2, 3, 6 and 7 from the following list of module learning objectives.
1. Describe and explain the principles of more than one advanced control technique.
2. Analyse practical performance specifications and convert these into functional require ments on controllers.
3. Design, implement and evaluate an advanced control system against these requirements.
4. Compare and contrast different advanced control solutions to a particular control problem or application.
5. Describe the receding-horizon principle, and hence compare and contrast LQ-optimal control and MPC.
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.
Assignment briefing
This assignment comprises a number of tasks. Complete the tasks, and produce and submit (via Turnitin) a short report, as a PDF, containing your answers.
Guidance on the technical report
Structure
For each separate task defined in this assignment briefing, describe concisely what you did, why, and what result(s) you obtained: provide enough evidence (i.e., in the form of equations, descriptions, justifications, code, plots) to show that you have engaged with, and completed, the task.
Code
Include all MATLAB code you write in your report. Please copy and paste your code from MATLAB into your report, and label it (i.e., with the number of the relevant task).
Format
Produce and submit your report in the PDF format.
Guidance on completing the assignment tasks
• This assignment briefing and the module lecture notes provide the main information that is required to complete this assignment. Although we will cover in lectures the basic theory and techniques needed to complete the assignment, you may need to read ahead of the lectures—if you wait entirely until we “cover” a particular topic, you might not leave enough time for a comprehensive attempt at the assignment.
• You may also wish to consult the literature relevant to your problem and review it in your report.
• Basic MATLAB programming is required, including the use of functions and loops; however, in tackling the assignment you may use the MPC-specific MATLAB functions (used in the computer exercises) available on the Blackboard page for ACS6116, plus any code you have developed to answer the computer exercises. You should not use the MPC toolbox in MATLAB/SIMULINK.
• 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.
• Please note that, in order to achieve the very highest marks, you will need to go beyond simply implementing the methods that you have learned in the lectures and practised in the computer exercises. Surprise me!
• If you do not manage to achieve a working controller or simulation, simplify things, get something simpler working, and build systematically towards a compre hensive 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
[email protected]).
Submit your report via Blackboard by 23:59 on 12th May 2025
A benchmark problem in control design
Figure 1: The Wie–Bernstein two-cart control benchmark system.
The assignment considers a benchmark problem in (robust) control design. Introduced by Wie and Bernstein in the early 1990s,1
the benchmark system has two carts connected by a spring, on a frictionless surface. Actuation may be applied to the first cart, but the performance output is the position of the second cart. The system is an abstraction of practical engineering systems that have a rigid body mode and a vibration mode.
System description and assumptions
The dynamics of the system are modelled as
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.
2. The full state X = [x1 x2 x3 x4] may be measured.
3. The dynamics are discretized using zero-order hold and a sampling period of 0.1 s.
2
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 assignment tasks
You are to design, implement, simulate, analyse and evaluate a model predictive controller in order to achieve the a specification; this is expressed as a series of tasks that you should complete in a sequential order.
Task 1: In the absence of disturbances (d1 = d2 = 0), the controller is able to regulate an initial state x(0) = [1 1 −0.2 0.2]
⊤
to the origin while satisfying the constraints
|u| ≤ 1, |x3| ≤ 0.5, |x4| ≤ 0.5.
For this task, it is strongly recommended that you build your controller in an incremental way; i.e., starting with no constraints present, before adding just input constraints, and then finally state constraints.
Task 2: The controller should be designed/tuned so that, when completing Task 1:
• The output y = x2 reaches a near-zero value in a reasonable amount of time.
• The value function exhibits a monotonic decrease with time.
• The horizon length, N, and hence the size of the MPC QP, is not excessive.
Task 3: Investigate and evaluate the limits of operation for your regulating controller: for which initial states is the MPC optimization problem feasible? For which initial states does the MPC optimization problem remain feasible at all times? Which initial states can be successfully controlled to the origin?4
(If necessary, refine or improve your design to create a stronger link between feasibility and stability.)
Task 4: With the system started from rest (x(0) = [0 0 0 0]
⊤
), the controller should be able to achieve offset-free tracking of a constant reference r = 1 with ‘good’ performance —
i.e., achieve y = r in steady state, with a ‘reasonable’ settling time and mimimal overshoot — in the presence of constant disturbances5 d1 = 0.25 and/or d2 = 0.7 and, of course, the constraints.
Investigate the limits of operation for your tracking controller: what are the maximal disturbances that can be rejected while ensuring y = r in steady state?
Task 5: Investigate one further direction or aspect to this assignment problem. You should engage with advanced concepts and/or techniques from the module and wider literature — this is your opportunity to explore something interesting and original. A non-comprehensive list of suggestions:
• Robustness — e.g., to modelling error, parameter uncertainty, or noise.
• Adaptation — e.g., if a parameter value drifts over time, can your controller adjust?
• Rejection of non-constant disturbances.
• Output feedback — i.e., when the whole state is not available for feedback.
• Unmeasurable disturbances — i.e., suppose that d1 and d2 are not known.
• Terminal set design.
In this case, does y = x2 contain enough information? What else needs to be measured?
Mark scheme
This mark scheme is is provided as a guide and is purely indicative: it is possible to gain a certain mark for reasons other than those given, based on academic judgement.
Mark of 0–49
• Little or no evidence of completion of Task 1 (or any other task).
• Code is not functional, or simulations do not demonstrate any regulation.
• Report is missing or severely inadequate.
Mark of 50–54
• Task 1 completed.
– Code correctly implements the MPC controller (perhaps without constraints).
– Simulation results demonstrate basic regulation to the origin.
– Report is present and contains basic simulation results.
• Constraints may not be implemented correctly.
Mark of 55–59
• Tasks 1 and 2 completed to a basic level.
– Code correctly implements the MPC controller with constraints.
– Simulation results demonstrate regulation to the origin while respecting con straints.
– Value function is present, but analysis of its monotonicity may be superficial.
– Parameter selection/design may be poor, or poorly justified.
• Task 3 or 4 are not completed, or are poorly completed.
Mark of 60–64
• Tasks 1 and 2 completed to a good standard.
– Code correctly implements the MPC controller, demonstrating competency in MPC implementation.
– Simulation results demonstrate regulation to the origin, respecting constraints, and showing a reasonable settling time.
– Value function is analysed and shown to decrease monotonically.
– Parameter selection is logical and justified.
• Task 3 is completed to an acceptable standard.
– Evidence of investigation of operating regions.
• Task 4 is not completed, or is poorly completed.
Mark of 65–69
• Tasks 1, 2, and 3 completed to a high standard.
– Code is well-structured and efficient.
– Simulations provide detailed analysis of performance.
– Feasibility and stability are analysed, and the relation between them is discussed.
• Task 4 is completed to an acceptable standard.
– Offset-free tracking is achieved, and basic disturbance rejection is achieved.
• Task 5 is not completed, or is poorly completed.
Mark of 70–100
• Tasks 1, 2, 3, and 4 completed to a high standard.
– Comprehensive analysis of limits of operation for both regulating and tracking controllers.
– Clear and well-presented report.
• Task 5 completed, demonstrating independent research and critical thinking.
– 71-80: Task 5 is completed to a good standard, demonstrating understanding of an advanced topic.
– 81-90: Task 5 is completed to a high standard, demonstrating significant indepen dent research.
– 91-100: Task 5 is completed to a very high standard, demonstrating exceptional independent research, critical thinking, and a deep understanding of the chosen topic.