Criterion of Hurwitz. Criteria for stability of Wald, Hurwitz, Savage
In the article such notions as criteriaHurwitz, Savage and Wald. The emphasis is mainly on the first. The criterion of Hurwitz is described in detail both from the algebraic point of view, and from the position of decision-making 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 earlier equilibrium.
It is important to note that his opponent is an unstable system - he constantly moves away from his equilibrium state (oscillates around him) with a returning amplitude.
Sustainability criteria: definition, types
This is a set of rules that allow one to judgethe existing signs of the roots of the characteristic equation without finding its solution. 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
He is an algebraic criterion, implying 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 in order to be dischargedall the coefficients of the corresponding characteristic equation, starting from aᵥ₋₁ to a0. In all columns down from the main diagonal indicate the coefficients of increasing powers of the operator p, then the upward-decreasing ones. Missing elements are replaced by zeros.
It is generally accepted that the system is stable when allThe 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 the remaining 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 mostreasonable variation of strategy. 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 not known, all available information is reduced to a list of possible options.
So, we should start with Wald's maximin criterion. He acts as a criterion for extreme pessimism (a cautious observer). This criterion can be formed for both pure and mixed strategies.
He received 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 with the pessimistic one,which is used in the solution of 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 chosen by the criterion under consideration are devoid of risk, since the decision maker does not face a worse result than the one who acts as a benchmark.
So, the most acceptable, according to Wald's criterion, is a pure strategy, since it guarantees in the worst conditions 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 in the event that the choice is still erroneous.
According to this principle, any solutionis characterized by a certain amount of additional losses arising in the course of its implementation, compared with the correct in 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. So, 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:
- There is a need to take into account the worst of the options.
- Lack of knowledge regarding the probabilities of states of nature.
- Assume some risk.
- A sufficiently small number of solutions is realized.
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.