Optimal projection equations

Key takeaways

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

Reference excerpt

In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs. Moreover, the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal perturbation control of non-linear systems. The LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. Therefore, implementing the LQG controller may be problematic if the dimension of the system state is large. The reduced-order LQG problem (fixed-order LQG problem) overcomes this by fixing a-priori the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated optimal projection equations.

Mathematical problem formulation and solution

Continuous-time The reduced-order LQG control problem is almost identical to the conventional full-order LQG control problem. Let x ^ r ( t ) {\displaystyle {\hat {\mathbf {x} }}_{r}(t)} represent the state of the reduced-order LQG controller. Then the only difference is that the state dimension n r = d i m ( x ^ r ( t ) ) {\displaystyle n_{r}=dim({\hat {\mathbf {x} }}_{r}(t))} of the LQG controller is a-priori fixed to be smaller than n = d i m ( x ( t ) ) {\displaystyle n=dim({\mathbf {x} }(t))} , the state dimension of the controlled system. The reduced-order LQG controller is represented by the following equations:

x ^ ˙ r ( t ) = A r ( t ) x ^ r ( t ) + B r ( t ) u ( t ) + K r ( t ) ( y ( t ) − C r ( t ) x ^ r ( t ) ) , x ^ r ( 0 ) = x r ( 0 ) , {\displaystyle {\dot {\hat {\mathbf {x} }}}_{r}(t)=A_{r}(t){\hat {\mathbf {x} }}_{r}(t)+B_{r}(t){\mathbf {u} }(t)+K_{r}(t)\left({\mathbf {y} }(t)-C_{r}(t){\hat {\mathbf {x} }}_{r}(t)\right),{\hat {\mathbf {x} }}_{r}(0)={\mathbf {x} }_{r}(0),}

u ( t ) = − L r ( t ) x ^ r ( t ) . {\displaystyle {\mathbf {u} }(t)=-L_{r}(t){\hat {\mathbf {x} }}_{r}(t).}

These equations are deliberately stated in a format that equals that of the conventional full-order LQG controller. For the reduced-order LQG control problem it is convenient to rewrite them as

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Optimal projection equations

Start with the simplest possible case. Write down what Optimal projection equations claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Optimal projection equations 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 Optimal projection equations 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 Optimal projection equations

In research: Optimal projection equations appears in mathematics 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 Optimal projection equations 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: Optimal projection equations is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Optimal control, Stochastic control, so understanding it makes those chapters shorter.
In everyday life: Look for Optimal projection equations 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.

How to study Optimal projection equations in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Optimal projection equations 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 Optimal projection equations out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Optimal projection equations in simple terms?

In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems.

Why does Optimal projection equations matter?

Because it connects several mathematics 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 Optimal projection equations?

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 Optimal projection equations.