One of the main tasks of the analysis of dynamic control systems is the solution to the problem of their stability. Their stability is one of the most important characteristics of the management concept. A system is considered unstable if it does not return to its original position, but continues to oscillate after it has undergone any changes at the input, or is under the influence of undesirable disturbance.
Definition of the basic concept
According to the concept of stability of systems, the state of its equilibrium is due to the absence of influence of disturbing factors on it. In this situation, the difference between the given and the actual state tends to zero. Stability is its ability to return to its original equilibrium state after the disturbance that led to its disturbance has ended. An unstable system due to the influence of perturbation moves away from the equilibrium state or makes oscillations, the amplitude of which gradually increases.
Sustainability conditions
For the stability of a system with constant time, the following two conditions must be fulfilled:
- She herself will create a limited output for each input; if there is no input, the output must be zero, regardless of any initial conditions.
- The stability of the system can be called absolute or relative stability. The presented term is used in relation to research, during which certain quantities are compared, their operating conditions. Stability is the end result created by the result.
If the output of the system is infinite, even when the final input is applied to it, then it will be called unstable, that is, stable in its essence, it has a limited completion when the limited beginning is applied to itself.
Moreover, the entry refers to various points of application of the influence of the external environment on the system. The output is the final product of its activity, which has the form of converted input data.
In a continuous linear time system, the stability condition can be written for a specific impulse response.
In the case where it is discrete, a stability index can also be recorded for a specific impulse response.
For an unstable condition, both in a continuous and in a bounded system, these expressions will be infinite.
Types of stability and disturbance
Under the static stability of the system understand its ability to restore the original (or close to the original) mode after a small disturbance. Under the presented concept in this context we consider the oscillation that affects its behavior, regardless of where the surge or drop occurs, and what is their magnitude. Based on this, these regimes close to the initial one allow us to consider it as linear.
The dynamic stability of systems is the ability of the latter to restore the initial state after a large disturbance.
By large fluctuation we understand such a movement, the nature of the influence of which and its corresponding behavior determine the time of existence, the magnitude and place of its appearance.
Based on this, the system in this range is defined as non-linear.
Sustainability criteria
The main condition for the stability of a linear system is not the nature of the perturbation, but its structure. It is believed that this stability “in the small” is determined if its boundaries are not established. Stability “in large” is determined by the limits and the correspondence of real deviations to these established frames.
The following criteria are used to determine system stability:
- root criterion;
- Stodola criterion;
- Hurwitz criterion;
- Nyquist criterion;
- Mikhailov et al. criterion
The root criterion and evaluation method Stodoly used in determining the stability of individual links and open systems. The Hurwitz criterion is algebraic; it allows one to determine the stability of closed systems without delay. The Nyquist and Mikhailov criteria are frequency. They are used to determine the stability of closed systems based on their frequency characteristics.
Root criterion
It allows you to determine the stability of the system, based on the type of transfer function. Its behavior properties are described by a characteristic polynomial (denominator of the transfer function). If we equate the denominator to zero, the roots of the resulting equation will determine the degree of stability.
According to this criterion, a linear system will be stable if all the roots of the equation are in the left half-plane. If at least one of them is located on the stability boundary, it will also be at the limit. If at least one of them is in the right half-plane, the system can be considered unstable.
Stodola criterion
It follows from the root definition. In accordance with the Stodola criterion, a linear system can be considered stable in the case when all coefficients of the polynomial are positive.
Hurwitz criterion
This criterion is used for the characteristic polynomial of a closed system. According to this technique, a sufficient condition for stability is the fact that the value of the determinant and all the main diagonal minors of the matrix is greater than zero. If at least one of them is equal to zero, it is considered at the stability boundary. In the presence of at least one negative determinant, it should be considered unstable.
Nyquist criterion
The basis of this technique is the construction of a curve connecting the ends of a variable vector that displays the transfer function. The formulation of the criterion is as follows: a closed system is considered stable if the function curve does not cover a point with coordinates (-1, j0) on the complex plane.
Financial stability system
Financial stability is a condition in which a system, that is, key markets and an institutional structure, is resistant to economic shocks and is ready to smoothly perform its main functions: cash flow intermediation, risk management and organization of payments.
Due to the mutual relationship of the interpretation provision (both at the vertical and horizontal levels), the analysis should cover the entire financial intermediation system. In other words, in addition to the banking sector, it is also necessary to analyze non-banking institutions that in one form or another participate in mediation. These include numerous types of institutions, including brokerage firms, investment funds, insurers and other (various) entities. When analyzing the financial stability system, the degree to which the entire structure is able to withstand external and internal shocks is studied. Of course, shocks do not always lead to crises, but the unstable financial environment in itself can impede the healthy development of the economy.
Various theories determine the causes of financial instability. Their relevance may vary depending on the period and countries involved in the scope of analysis. Among the problematic factors affecting the entire financial system, literature usually defines the following:
- rapid liberalization of the financial sector;
- inadequate economic policies;
- inappropriate exchange rates mechanism;
- inefficient allocation of resources;
- weak supervision;
- insufficient regulation of accounting and auditing.
Possible reasons are manifested not only collectively, but also individually or in a random combination, so the analysis of financial stability is an extremely difficult task. The focus on individual industries distorts the overall picture, so questions should be considered in their complexity in the study of financial stability.
The process of analyzing the stability of an enterprise system takes place in several stages.
Initially, absolute and relative indicators of financial stability are estimated and analyzed. At the second stage, factors are distributed in accordance with their significance, their influence is qualitatively and quantitatively evaluated.
Ratios of financial stability of enterprises
The financial condition of the company, its stability largely depends on the optimal structure of capital sources, that is, the ratio of debt to own resources, on the optimal structure of the company’s assets and, first of all, on the ratio of fixed and current units of property, as well as the balance of funds and obligations of the company.
Therefore, it is important to study the structure of sources of venture capital and assess the degree of financial stability and risk. For this purpose, system stability factors are used:
- coefficient of autonomy (independence) - the share of capital in the balance sheet;
- dependency ratio - the share of borrowed capital in the balance sheet;
- current debt ratio - the ratio of short-term financial liabilities to the balance sheet;
- financial stability ratio (long-term financial independence) - the ratio of capital and long-term debt to the balance sheet;
- debt coverage ratio (solvency ratio) - the ratio of capital to debt;
- financial leverage ratio (financial risk ratio) - the ratio of debt to capital.
The higher the level of indicators such as autonomy, financial stability, debt capital coverage, the lower the level of another group of ratios (dependence, current debt, long-term liabilities to investors) and, accordingly, the stability of the financial condition of the company. Leverage is also called leverage.