mathematics

State-transition equation

State-transition equation is a mathematics 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 State-transition equation rather than just read about it. In short: The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by d x ( t ) d t = A x ( t ) + B u ( t ) + E w ( t ) , {\displaystyle {\frac {d\mathbf {x} (t)}{dt}}=\mathbf {Ax} (t)+\mathbf {Bu} (t)+\mathbf {Ew} (t),} with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by u…

Key takeaways

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

Reference excerpt

The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by

d x ( t ) d t = A x ( t ) + B u ( t ) + E w ( t ) , {\displaystyle {\frac {d\mathbf {x} (t)}{dt}}=\mathbf {Ax} (t)+\mathbf {Bu} (t)+\mathbf {Ew} (t),}

with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by using either the classical method of solving linear differential equations or the Laplace transform method. The Laplace transform solution is presented in the following equations. The Laplace transform of the above equation yields

s X ( s ) − x ( 0 ) = A X ( s ) + B U ( s ) + E W ( s ) {\displaystyle s\mathbf {X} (s)-\mathbf {x} (0)=\mathbf {AX} (s)+\mathbf {BU} (s)+\mathbf {EW} (s)}

where x(0) denotes initial-state vector evaluated at t = 0. Solving for X(s) gives

X ( s ) = ( s I − A ) − 1 x ( 0 ) + ( s I − A ) − 1 [ B U ( s ) + E W ( s ) ] . {\displaystyle \mathbf {X} (s)=(s\mathbf {I} -\mathbf {A} )^{-1}\mathbf {x} (0)+(s\mathbf {I} -\mathbf {A} )^{-1}[\mathbf {BU} (s)+\mathbf {EW} (s)].}

So, the state-transition equation can be obtained by taking inverse Laplace transform as

x ( t ) = L − 1 { ( s I − A ) − 1 } x ( 0 ) + L − 1 { ( s I − A ) − 1 [ B U ( s ) + E W ( s ) ] } = Φ ( t ) x ( 0 ) + ∫ 0 t Φ ( t − τ ) [ B u ( τ ) + E w ( τ ) ] d t {\displaystyle {\begin{aligned}x(t)&={\mathcal {L}}^{-1}{\Bigl \{}(s\mathbf {I} -\mathbf {A} )^{-1}{\Bigr \}}\mathbf {x} (0)+{\mathcal {L}}^{-1}{\Bigl \{}(s\mathbf {I} -\mathbf {A} )^{-1}[\mathbf {BU} (s)+\mathbf {EW} (s)]{\Bigr \}}\\&=\mathbf {\Phi } (t)\mathbf {x} (0)+\int _{0}^{t}\mathbf {\Phi } (t-\tau )[\mathbf {Bu} (\tau )+\mathbf {Ew} (\tau )]dt\end{aligned}}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with State-transition equation

Start with the simplest possible case. Write down what State-transition equation 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 State-transition equation 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 State-transition equation 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 State-transition equation

In research: State-transition equation 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 State-transition equation 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: State-transition equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, so understanding it makes those chapters shorter.
In everyday life: Look for State-transition equation 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 State-transition equation in 20 minutes

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

Frequently asked questions

What is State-transition equation in simple terms?

The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by d x ( t ) d t = A x ( t ) + B u ( t ) + E w ( t ) , {\displaystyle {\frac {d\mathbf {x} (t)}{dt}}=\mathbf {Ax} (t)+\mathbf {Bu} (t)+\mathbf {Ew} (t),}…

Why does State-transition equation 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 State-transition equation?

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 State-transition equation.