Popov criterion

Key takeaways

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

Reference excerpt

In nonlinear control and stability theory, the Popov criterion is a stability criterion discovered by Vasile M. Popov for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an open-sector condition. While the circle criterion can be applied to nonlinear time-varying systems, the Popov criterion is applicable only to autonomous (that is, time invariant) systems.

System description The sub-class of Lur'e systems studied by Popov is described by:

x ˙ = A x + b u ξ ˙ = u y = c x + d ξ {\displaystyle {\begin{aligned}{\dot {x}}&=Ax+bu\\{\dot {\xi }}&=u\\y&=cx+d\xi \end{aligned}}}

u = − φ ( y ) {\displaystyle {\begin{matrix}u=-\varphi (y)\end{matrix}}}

where x ∈ Rn, ξ,u,y are scalars, and A,b,c and d have commensurate dimensions. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to open sector (0, ∞), that is, Φ(0) = 0 and yΦ(y) > 0 for all y not equal to 0. Note that the system studied by Popov has a pole at the origin and there is no direct pass-through from input to output, and the transfer function from u to y is given by

H ( s ) = d s + c ( s I − A ) − 1 b {\displaystyle H(s)={\frac {d}{s}}+c(sI-A)^{-1}b}

Criterion Consider the system described above and suppose

A is Hurwitz (A,b) is controllable (A,c) is observable d > 0 and Φ ∈ (0,∞) then the system is globally asymptotically stable if there exists a number r > 0 such that inf ω ∈ R Re ⁡ [ ( 1 + j ω r ) H ( j ω ) ] > 0. {\textstyle \inf _{\omega \,\in \,\mathbb {R} }\operatorname {Re} \left[(1+j\omega r)H(j\omega )\right]>0.}

Worked examples

Example 1 — a first encounter with Popov criterion

Start with the simplest possible case. Write down what Popov criterion 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 Popov criterion 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 Popov criterion 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 Popov criterion

In research: Popov criterion 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 Popov criterion 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: Popov criterion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Nonlinear control, Stability theory, so understanding it makes those chapters shorter.
In everyday life: Look for Popov criterion 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 Popov criterion in 20 minutes

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

Frequently asked questions

What is Popov criterion in simple terms?

In nonlinear control and stability theory, the Popov criterion is a stability criterion discovered by Vasile M. Popov for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an open-sector condition.

Why does Popov criterion 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 Popov criterion?

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 Popov criterion.