What this assignment does
The report models the inverted pendulum on a cart in state space, analyses its stability and controllability, designs a linear quadratic regulator with a Luenberger observer, simulates the closed loop, and compares the result against PID control.
The inverted pendulum is the most-written-about problem in undergraduate control, which makes it a hard assignment to distinguish yourself on. This answer does it by being precise about why single-loop PID structurally cannot do the job.
How it is built
Model, analysis, controller design, observer design, simulation, critical discussion. The analysis section earns its place by establishing controllability and observability before any controller is proposed — so the later designs are licensed rather than assumed.
The observer is treated as a necessity created by the hardware, not as an extra section: the rig measures two states and the control law needs four.
Annotated extracts
Three passages, each with a note on why it earns marks and what a weaker answer does in the same place — enough to teach the move, not enough to stand in for the assignment.
PID control is by far the most widely used control strategy in industry, valued for its simplicity and its well-established tuning methods (Åström and Hägglund, 2006). For the inverted pendulum, however, a single PID loop on the pendulum angle can stabilise the pendulum but cannot regulate the cart position simultaneously, so the cart drifts. Achieving both objectives requires two interacting loops, usually an inner angle loop and an outer position loop, and tuning these loops is difficult because their interaction is strong.
Why this marks well
The criticism of PID is structural, not performance-based. A single loop has one output and this plant has two objectives — so the cart drifts. That is an argument from the architecture of the controller, and it holds regardless of how well anyone tunes it.
PID is also credited properly first, with a citation, before being found unsuitable for this plant. Scoping the criticism protects the claim: the report is not saying PID is weak, it is saying this problem is multivariable and PID is not.
The follow-through about strong loop interaction is what makes it complete — it anticipates the obvious rebuttal (use two loops) and explains why that is hard too.
What a weaker answer does here
A weaker answer states that LQR outperformed PID in simulation and shows the two response curves. The comparison is empirical and shallow: it reports that one won without explaining why the other was always going to struggle.
The LQR law assumes that all four states are available, but the rig measures only cart position and pendulum angle. The two velocities must therefore be estimated. A full-order Luenberger observer reconstructs the state estimate using the model and a correction proportional to the output error (Ogata, 2010). The estimation error then evolves according to the eigenvalues of A − LC, which can be placed arbitrarily because the system is observable.
Why this marks well
The observer is motivated by a gap between what the control law assumes and what the hardware provides. Two sentences establish the need before any theory appears, which is why this section does not read as an obligatory add-on.
The closing clause is the one that earns marks: the eigenvalues can be placed arbitrarily because the system is observable. That is a property the report established earlier in the analysis section, now being cashed in. Carrying a result forward and using it to license a later design decision is exactly the structural discipline a design report is assessed on.
It is also honest about what the observer depends on — the model. An estimator is only as good as the plant model behind it, which sets up the discussion of model error.
What a weaker answer does here
A weaker answer presents the observer equations and states that the observer poles were placed five times faster than the controller poles. The rule of thumb is correct and unexplained, so the reader cannot tell whether the student knows why it exists.
The open-loop system has a pole in the right half-plane, confirming that the upright equilibrium is unstable: any disturbance grows exponentially until the linearisation ceases to be valid. The controllability matrix has full rank, so the states can be driven to any desired values by the single input, and the observability matrix likewise has full rank for the chosen measurements.
Why this marks well
Three mathematical results are each given their physical meaning in the same sentence. A right-half-plane pole means disturbances grow; full-rank controllability means one input suffices for all four states.
The caveat — until the linearisation ceases to be valid — shows the student knows the model's domain. The linearised model predicts exponential growth, but the real pendulum falls over and the small-angle approximation dies first. Knowing where your own model stops describing reality is a consistent marker of the upper band.
This section is also doing structural work: it is what makes the LQR design legal later, rather than merely conventional.
What a weaker answer does here
A weaker answer reports the eigenvalues and states that the system is controllable and observable. The computation is right and inert, because none of it is connected to the behaviour of the physical rig or used to justify what follows.
What makes this a strong answer
On the most over-written problem in control engineering, it finds a non-obvious thing to say: the limitation of PID here is architectural, not a tuning failure.
Earlier results are carried forward and spent. Observability is established in the analysis and then used to license arbitrary observer pole placement — the report has a dependency structure, not a section list.
Every mathematical result is given its physical meaning, and the limits of the linearised model are stated rather than assumed away.
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Model answer written by a Global Projects Help mentor · © Global Projects Help
Published 8 October 2026 · Questions? mia@globalprojectshelp.com