science

Ross–Fahroo pseudospectral method

Ross–Fahroo pseudospectral method is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ross–Fahroo pseudospectral method rather than just read about it. In short: Introduced by I. Michael Ross and F.

Key takeaways

Ross–Fahroo pseudospectral method belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ross–Fahroo pseudospectral method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ross–Fahroo pseudospectral method from memory before moving on to harder problems.

Reference excerpt

Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral knotting method, the flat pseudospectral method, the Legendre-Gauss-Radau pseudospectral method and pseudospectral methods for infinite-horizon optimal control.

Overview The Ross–Fahroo methods are based on shifted Gaussian pseudospectral node points. The shifts are obtained by means of a linear or nonlinear transformation while the Gaussian pseudospectral points are chosen from a collection of Gauss-Lobatto or Gauss-Radau distribution arising from Legendre or Chebyshev polynomials. The Gauss-Lobatto pseudospectral points are used for finite-horizon optimal control problems while the Gauss-Radau pseudospectral points are used for infinite-horizon optimal control problems.

Mathematical applications The Ross–Fahroo methods are founded on the Ross–Fahroo lemma; they can be applied to optimal control problems governed by differential equations, differential-algebraic equations, differential inclusions, and differentially-flat systems. They can also be applied to infinite-horizon optimal control problems by a simple domain transformation technique. The Ross–Fahroo pseudospectral methods also form the foundations for the Bellman pseudospectral method.

Flight applications and awards The Ross–Fahroo methods have been implemented in many practical applications and laboratories around the world. In 2006, NASA used the Ross–Fahroo method to implement the "zero propellant maneuver" on board the International Space Station. In recognition of all these advances, the AIAA presented Ross and Fahroo, the 2010 Mechanics and Control of Flight Award, for "... changing the landscape of flight mechanics." Ross was also elected AAS Fellow for "his pioneering contributions to pseudospectral optimal control."

Distinctive features A remarkable feature of the Ross–Fahroo methods is that it does away with the prior notions of "direct" and "indirect" methods. That is, through a collection of theorems put forth by Ross and Fahroo,

they showed that it was possible to design pseudospectral methods for optimal control that were equivalent in both the direct and indirect forms. This implied that one could use their methods as simply as a "direct" method while automatically generating accurate duals as in "indirect" methods. This revolutionized solving optimal control problems leading to widespread use of the Ross–Fahroo techniques.

Software implementation The Ross–Fahroo methods are implemented in the MATLAB optimal control solver, DIDO.

See also Bellman pseudospectral method DIDO - MATLAB tool for optimal control named after Dido, the first queen of Carthage Ross' π lemma Ross–Fahroo lemma

References

Worked examples

Example 1 — a first encounter with Ross–Fahroo pseudospectral method

Start with the simplest possible case. Write down what Ross–Fahroo pseudospectral method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ross–Fahroo pseudospectral method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ross–Fahroo pseudospectral method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ross–Fahroo pseudospectral method

In research: Ross–Fahroo pseudospectral method appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ross–Fahroo pseudospectral method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ross–Fahroo pseudospectral method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Numerical analysis, Optimal control, so understanding it makes those chapters shorter.
In everyday life: Look for Ross–Fahroo pseudospectral method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Ross–Fahroo pseudospectral method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ross–Fahroo pseudospectral method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ross–Fahroo pseudospectral method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ross–Fahroo pseudospectral method in simple terms?

Introduced by I. Michael Ross and F.

Why does Ross–Fahroo pseudospectral method matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ross–Fahroo pseudospectral method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ross–Fahroo pseudospectral method.