Criterion of Hurwitz. Criteria for stability of Wald, Hurwitz, Savage

The article considers such concepts as the criteria of Hurwitz, Savage and Wald. The emphasis is mainly on the first. The criterion of Hurwitz is described in detail both from an algebraic point of view, and from the point of view of making a decision under conditions of uncertainty.

It is worth starting with the definition of the concept of stability. It characterizes the ability of the system to return to the equilibrium state at the end of the perturbation, which violated the previously formed equilibrium.

It is important to note that his opponent - an unstable system - is constantly moving away from his equilibrium state (oscillates around him) with a returning amplitude.

Sustainability criteria: definition, types

This is a set of rules that allow us to judge the existing signs of the roots of the characteristic equation without finding a solution to it. And the latter, in turn, provide an opportunity to judge the stability of a particular system.

As a rule, they can be:

• Algebraic (composition of a specific characteristic equation of algebraic expressions using special rules that characterize the stability of the automatic control system);
• Frequency (the object of study - frequency characteristics).

Criterion of Hurwitz stability from the algebraic point of view

It is an algebraic criterion, which implies the consideration of a certain characteristic equation in the form of a standard form:

A (p) = aᵥpᵛ + aᵥ₋₁pᵛ¯¹ + ... + a₁p + a₀ = 0 .

Through its coefficients, the Hurwitz matrix is formed.

The rule of compiling the Hurwitz matrix

In the direction from top to bottom all the coefficients of the corresponding characteristic equation, starting from aᵥ₋₁ to a0, are written out in order. In all columns down from the main diagonal indicate the coefficients of increasing powers of the operator p, then upwards - decreasing. Missing elements are replaced by zeros.

It is generally accepted that the system is stable when all the diagonal minors of the matrix in question are positive. If the principal determinant is zero, then we can speak of finding it at the stability boundary, where aN = 0. If other conditions are met, the system under consideration is located on the boundary of the new aperiodic stability (the penultimate minor is equated to zero). With a positive value of the remaining minors - on the boundary of the already oscillatory stability.

Decision making in a situation of uncertainty: the criteria of Wald, Hurwitz, Savage

They are the criteria for choosing the most appropriate strategy variation. The Savage criterion (Hurwitz, Wald) is applied in a situation where there are indeterminate a priori probabilities of states of nature. Their basis is analysis of the risk matrix or payment matrix. If the probability distribution of future states is unknown, all available information is reduced to a list of possible options.

So, it's worth starting with Wald's maximin test. He acts as a criterion for extreme pessimism (a cautious observer). This criterion can be formed for both pure and mixed strategies.

He got his name on the basis of the supernumerary assumption that nature can realize states in which the size of the win is equated to the smallest value.

This criterion is identical to the pessimistic criterion, which is used in solving matrix games, most often in pure strategies. So, first you need to select from each line the minimum value of the element. Then the DPR strategy is highlighted, which corresponds to the maximum element among the already selected minimum ones.

The options selected by the criterion under consideration are without risk, since the decision maker does not face a worse result than the one that acts as a benchmark.

So, the most acceptable, according to Wald's criterion, is a pure strategy, since in the worst conditions it guarantees the maximum possible winnings.

Next, consider the Savage criterion. Here, when choosing one of the available solutions in practice, they usually stop at one that will lead to minimal consequences if the choice is still erroneous.

According to this principle, any solution is characterized by a certain amount of additional losses arising in the course of its implementation, compared with the correct one with the existing state of nature. It is obvious that the correct solution can not bear additional losses, so that their value is equated to zero. Thus, in the role of the most appropriate strategy is adopted, the amount of loss in which is minimal at the worst confluence of circumstances.

Criterion of pessimism-optimism

So in another way is called the Hurwitz criterion. In the process of choosing a solution, in assessing the current situation, instead of the two extremes, they adhere to the so-called intermediate position, which takes into account the probability of both favorable and worst behavior of nature.

This compromise variant was proposed by Hurwitz. According to him, for any solution it will be necessary to establish a linear combination of min and max, then choose a strategy that corresponds to their highest value.

When is the application of the criterion justified?

Use the Hurwitz criterion is advisable in a situation characterized by the following symptoms:

1. There is a need to take into account the worst-case scenario.
2. Lack of knowledge about the probabilities of states of nature.
3. Let's assume some risk.
4. A sufficiently small number of solutions is realized.

Conclusion

Finally, it will be useful to recall that the article examined the criteria of Hurwitz, Savage and Wald. The criterion of Hurwitz is described in detail from different points of view.