science

Kalman's conjecture

Kalman's conjecture 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 Kalman's conjecture rather than just read about it. In short: Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of Aizerman's conjecture and is a special case of Markus–Yamabe conjecture.

Key takeaways

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

Reference excerpt

Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of Aizerman's conjecture and is a special case of Markus–Yamabe conjecture. This conjecture was proven false but led to the (valid) sufficient criteria on absolute stability.

Mathematical statement of Kalman's conjecture (Kalman problem) In 1957 R. E. Kalman in his paper stated the following:

If f(e) in Fig. 1 is replaced by constants K corresponding to all possible values of f'(e), and it is found that the closed-loop system is stable for all such K, then it intuitively clear that the system must be monostable; i.e., all transient solutions will converge to a unique, stable critical point.

Kalman's statement can be reformulated in the following conjecture:

Consider a system with one scalar nonlinearity

d x d t = P x + q f ( e ) , e = r ∗ x x ∈ R n , {\displaystyle {\frac {dx}{dt}}=Px+qf(e),\quad e=r^{*}x\quad x\in R^{n},}

where P is a constant n×n matrix, q, r are constant n-dimensional vectors, ∗ is an operation of transposition, f(e) is scalar function, and f(0) = 0. Suppose, f(e) is a differentiable function and the following condition

k 1 < f ′ ( e ) < k 2 . {\displaystyle k_{1}<f'(e)<k_{2}.\,}

is valid. Then Kalman's conjecture is that the system is stable in the large (i.e. a unique stationary point is a global attractor) if all linear systems with f(e) = ke, k ∈ (k1, k2) are asymptotically stable.

In Aizerman's conjecture in place of the condition on the derivative of nonlinearity it is required that the nonlinearity itself belongs to the linear sector. Kalman's conjecture is true for n ≤ 3 and for n > 3 there are effective methods for construction of counterexamples: the nonlinearity derivative belongs to the sector of linear stability, and a unique stable equilibrium coexists with a stable periodic solution (hidden oscillation). In discrete-time, the Kalman conjecture is only true for n=1, counterexamples for n ≥ 2 can be constructed. The development of Kalman's ideas on global stability based on the stability of linear approximation for a cylindrical phase space gave rise to the Viterbi problem on the coincidence of phase-locked loop ranges.

References

Further reading Leonov G.A.; Kuznetsov N.V. (2011). "Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems" (PDF). IFAC Proceedings Volumes (IFAC-PapersOnline). 18 (1): 2494–2505. doi:10.3182/20110828-6-IT-1002.03315.

External links Analytical-numerical localization of hidden oscillation in counterexamples to Aizerman's and Kalman's conjectures Discrete-time counterexample in Maplecloud Archived 2016-05-28 at the Wayback Machine

Worked examples

Example 1 — a first encounter with Kalman's conjecture

Start with the simplest possible case. Write down what Kalman's conjecture 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 Kalman's conjecture 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 Kalman's conjecture 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 Kalman's conjecture

In research: Kalman's conjecture 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 Kalman's conjecture 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: Kalman's conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Disproved conjectures, Hidden oscillation, Nonlinear control, so understanding it makes those chapters shorter.
In everyday life: Look for Kalman's conjecture 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 Kalman's conjecture in 20 minutes

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

Frequently asked questions

What is Kalman's conjecture in simple terms?

Why does Kalman's conjecture 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 Kalman's conjecture?

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 Kalman's conjecture.