Sensitivity of control systems. Examples of modeling, practical implementation and applications

UDC 330.131.7

Kotov V.I.

investment project to risks

To quantify the resistance of an investment project to the effects of risk events, you can use sensitivity functions. However, in the economic literature they often write (for example, in) that a significant drawback of this method “is its one-factor nature, i.e., its focus on changes in only one factor of the project, which leads to underestimation of the possible connection between individual factors or underestimation of their correlation.” As will be shown below, this disadvantage can be completely overcome if, when choosing a set of risk parameters (factors), we select those for which the interdependence is significant and take it into account. Most of the factors are practically independent and direct calculation of sensitivity based on them is quite justified.

One more note about the use of the term “sensitivity.” For the selected objective function, their maximum permissible values ​​are usually determined by alternately changing the risk parameters. The given algorithm for such calculation is implemented in software package For some reason Project Expert 6 and some authors call it project sensitivity analysis. The following definition is given: “Sensitivity analysis. A method showing how one factor changes depending on another...” Strictly speaking, this is not a sensitivity analysis, but simply an analysis of the dependence of the function Y on several variables that form the vector x. Note that sensitivity in systems theory refers to the corresponding differential indicators, namely: the absolute sensitivity of some target function Y(t,x) is defined as its partial derivative with respect to the risk parameter x(i, t):

The capabilities of the risk analysis method based on sensitivity functions, in our opinion,

underestimated. This article will present computer model To calculate sensitivity functions, the types and properties of these functions are considered. It is shown that the approach to sensitivity as a dynamic characteristic within the entire planning horizon gives important information on the impact of risk events on the financial performance of investment projects.

Definition and calculation model of sensitivity functions

First, let's define the sensitivity function. Let us denote the target function of the project by Y(r, x), where z is time, x(r) is a vector of variable parameters that model the impact of certain risk events. The relative sensitivity of the objective function is the ratio of the relative deviation of the function to the relative deviation of the argument (risk parameter):

^ _ dU / U _ AU / U _ A7

X dx; /X; Аx¡/X; U AH;

Here and below, time is omitted for simplicity. Due to the fact that relative sensitivities are dimensionless, they are more convenient for analysis, so in what follows we will use only them, and for the sake of brevity we will omit the adjective “relative”. The greater the sensitivity, the stronger the influence of the corresponding risk parameter on the objective function of the investment project. Numerically, the sensitivity function shows by how many percent the objective function will change when the risk parameter changes by one percent.

IN economic theory there is a concept similar to sensitivity - “elasticity” (of demand, etc.), which is calculated using a formula similar to (2). Elasticity as an indicator characterizes the external environment of a business and usually

Rice. 1. Block diagram of the sensitivity function calculation model

is not considered as a function of time, but is a static parameter. We will adhere to the term “sensitivity”, firstly, because it characterizes the internal environment of a business and is a characteristic of an investment project, and secondly, so as not to confuse the known context of using the term “elasticity” with the dynamic characteristic of sensitivity when analyzing the impact of risks.

Let us present a block diagram of a model for calculating sensitivity functions, which is based on a dynamic model of the project’s financial flows (Fig. 1). This model was implemented in an electronic environment EXCEL tables and allowed simultaneous calculations for five variants of objective functions, which will be discussed below.

Here, the main Cash-Flow model is used to calculate the selected investment project scenario, i.e., to obtain all the necessary indicators and the value of the selected objective function (one or more) in a Status Quo situation. A copy of the model is used to calculate the changed value of the objective functions under the influence of any risk parameter.

All constants are automatically transferred from the main model to the copy (using appropriate links). The copy provides for alternating changes in risk parameters and selecting the duration of exposure to each risk. Now, if you change any risk parameter in the copy, then at its output we will get the changed value of the objective function. To the block for calculating sensitivity functions from the main model

the original values ​​of the risk parameter and the objective function are received, and the corresponding modified values ​​are received from the copy. As a result, based on (2), we obtain sensitivity functions in the form of tables and corresponding graphs for the entire planning horizon.

Target functions of the project

The choice of objective function largely depends on the tastes and desires of the developers of the business plan for the investment project. Various indicators can be proposed as an objective function, for example:

NPV(T) - net present value of the project at time T;

Accumulated Discount Net Cash-Flow ADNCF(T), generated by the project by time T;

Accumulated Net Cash-Flow ANCF(T), generated by the project by time T (without discounting);

Accumulated Net Profit ANP(T), generated by the project by time T;

Accumulated balance of financial flows (state of the project current account) (Accumulated Saldo Cash-Flow) ASCF(T) by time T.

When choosing an objective function, you can use not accumulated indicators, but indicators of financial results in individual periods. However, we give preference to accumulated

indicators, since this makes it possible to more strictly take into account the consequences of risk events after the end of their action throughout the entire planning horizon.

A comparison of the sensitivities of accumulated net cash flow and its discounted counterpart showed that they are almost identical, since the differences were only a fraction of a percent. This is not surprising, since when calculating the sensitivity function according to (2), both the numerator (AU) and the denominator (Y) are discounted, which partially leads to compensation for the discounting procedure.

If MRU(T) is used as an objective function, then it should be borne in mind that near the payback point, when MRU = 0, the sensitivity function suffers a discontinuity of the second kind, i.e., it goes to infinity by definition (2). This makes it difficult to use the MRU as an objective function near the specified point, but there are no design problems outside of it.

If we choose the accumulated balance of financial flows as the objective function, we obtain

Y (x, T) _ £ [ (x, z) - C^ (x, z)]. (3)

Knowledge of the sensitivity functions of this objective function will be very useful for the operational management of the state of the project’s current account under the influence of risks.

Local and global sensitivity functions

When calculating sensitivity functions, one should distinguish between short-term and long-term exposure to risk events. Accordingly, we define two types of sensitivity functions.

Local sensitivity - sensitivity under local (short-term in time) influence of the risk parameter, i.e. when the deviation occurs only during one or several periods significantly shorter than the overall planning horizon, as shown in Fig. 2, a.

Global sensitivity - sensitivity under global (long-term) influence of a risk parameter, i.e. when a deviation can occur over the entire horizon

planning, starting from a certain moment (Fig. 2, b).

Which of the given sensitivity options should be chosen depends on how long certain risk events will last in a real situation.

An analogy with the analysis of the response of linear systems based on the impulse and transient characteristics of the latter is appropriate here. If the Dirac delta function 8(r - t) is used as a single impact at time t, then the reaction of the system at zero initial conditions will be numerically equal to impulse response systems g(t - t). If the Heaviside function (unit jump) 1(g - t) is used as a single impact at some point in time, then the system response at zero initial conditions will be numerically equal to the transition characteristic of the system H(g - t).

In our case, the role of the delta function can be played by a local jump in the risk parameter LS(t - t), then the response of the investment project will be proportional to the local sensitivity LS(t - t) to a given impact. The Heaviside function 1(g - t) will correspond to the global change in time of the risk parameter Ox(g - t), which will give a reaction proportional to the global sensitivity function 08(g - t). In Fig. 3 shows the corresponding functional analogies.

As is known, for linear systems the principle of superposition is valid, namely: the reaction of the system to a set of influences is equal to the sum of the reactions to each influence separately. Based on this principle, knowing the characteristics of the system g(t) or H(g), one can find the connection between them and the system’s response to any type of impact. In our case, from the superposition principle we can obtain a connection between global and corresponding local sensitivity functions. Let time change discretely:

r = 0, 1, 2, ... n, ... M,

where r = M - planning horizon; r = k - the moment of the beginning of the impact of global risk; r = k + ], (] = 0, 1, ... p - k) - moments of existence of local risks; g = n > k + ] - arbitrary (current) moment of observing the system’s reaction to a given impact.

45 40 35 30 25 20 15 10 5 0

50 45 40 35 30 25 20 15 10 5 0

t Sh and I "Ch---*----- p p p.......

6 7 8 Period

10 11 12 13 14 15

\ " ^ -1>--O--0 0 0 0 0-- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Rice. 2. Deviation of the values ​​of the objective function a - for local and b - for global impact

1 - -O; 2 - x + ah; 3 - U; 4 - U + aU

Linear system

Financial model

A bB(g - t) (local sensitivity)

Linear system

Financial model

GdX(g - t) IP

A GS(g - t) (global sensitivity)

Rice. 3. Analogies with linear systems: a - local, b - global

Then the global sensitivity, which describes the system’s response to the impact of a global risk event that began at the moment r = k and lasts up to the planning horizon, can be expressed as a superposition of local sensitivities corresponding to the totality of the impacts of local (lasting one period) risks appearing at moments from r = k and to g = k + / (/ = 0, 1, ... n - k):

OB7^ (n - k) _ (n - k - /), n > k + /. (4)

It should be noted that local functions sensitivities always decrease faster than those of the same name global functions, for all time periods. This is explained by the fact that the local effect of any risk lasts a short time, and the global risk (equal to the sum of local risks) operates all the time from the moment of its occurrence and its effect accumulates from period to period. We can say that global sensitivity functions reflect the strategic consequences of the influence of long-term deviations of parameters on an investment project. At the same time, local sensitivities reflect the tactical consequences of short-term changes in the external and internal business environment. Local sensitivity functions most often have a maximum at the moment of exposure to a particular risk and then decrease relatively quickly compared to global sensitivity for the same risk parameter.

When using the analytical apparatus for analyzing linear systems, it should be borne in mind that the financial model of an investment project may not be strictly linear, however, as experiments on many different investment projects have shown, even within a wide range of variations in risk parameters, the accuracy of sensitivity analysis remained quite acceptable. In and it is proposed, in addition to first-order sensitivities (2), to use second-order sensitivities in cases where the nonlinearity of the objective function with respect to some risk parameters is significant and cannot be neglected.

Properties of sensitivity functions

If the sales prices of manufactured goods during the implementation of an investment project are chosen as risk parameters, then in each planning period the objective function (for example, the accumulated net financial flow in the case of two goods) will have the form

Y _ a(+ p^) + b,

where p12 - prices; 612 - natural sales volumes. If we select revenue from each product p1b1 as the risk parameters, then using (2) we obtain the sensitivity functions for the period under consideration:

It is easy to see that the ratio of these sensitivity functions will be equal to the ratio of sales volumes in monetary terms of the corresponding goods in a given period. Consequently, the structure of sensitivity functions for sales volumes will exactly correspond to the structure of sales volumes in monetary terms:

This conclusion is valid for any number of products included in the assortment. If individual groups of goods available in the assortment have different VAT rates, then the above conclusion will be valid if prices without VAT are used in calculations of sensitivity and in calculations of the structure of sales volumes. The indicated property (7) of sensitivity functions allows one to significantly reduce the amount of calculations of the latter in the case of a wide range of goods, when it is necessary to know the sensitivities for all goods.

Let's consider the sign of the sensitivity function. The sensitivity function will be positive for all points in time if, with an increase (decrease) in the deviation of the risk parameter, the value of the objective function increases (decreases) provided that the objective function itself is positive. For example, sensitivity

Rice. 4. Sensitivity functions of the balance of financial flows of the project 1,2, 3 - sales volumes, respectively; 4 - semi-fixed and 5 - semi-variable costs

the accumulated balance of financial flows to prices and natural sales volumes of manufactured goods are always positive, and the sensitivity of the same objective function to deviations of any costs, as well as to bank lending rates, is always negative. An exception to this rule will be periods when there are losses instead of net profit. In Fig. Figure 4 shows examples of sensitivity functions.

As we can see, the most “dangerous” is the eighth period of the project, since in this period all sensitivity functions will be maximum. During such periods, managers' attention to the progress of the project should be greatest in order to keep performance indicators close to planned.

If MRU is chosen as the target function, then its sensitivity to prices or natural sales volumes of manufactured goods in the “dead zone” (with MRU< 0) будет отрицательной, а после срока окупаемости - положительной. Знаки чувствительности МРУ к издержкам будут обратными.

Features of sensitivity functions to price fluctuations and natural sales volumes

When determining sensitivity functions, we have so far assumed that all risk parameters are independent. Given

the assumption for most parameters is quite justified, but in some cases the mutual dependence cannot be neglected. For example, if among the many risk parameters there are prices p and natural sales volumes Q of goods produced within the framework of an investment project, then when calculating such sensitivity functions as the accumulated balance of financial flows, the accumulated net financial flow (with or without discounting) or MRU , it is necessary to take into account dependence 2(p). If it is difficult to evaluate this dependence, then when analyzing sensitivity, you can select natural sales volumes (0 or revenue from each product group (pQ) as risk parameters. For these risk parameters, the specified objective functions are linear.

Thus, sensitivity functions as dynamic characteristics of an investment project together with performance indicators provide a more complete picture for comparing projects or scenarios with each other. Using the calculated sensitivity functions, it is possible to determine those periods of the “life” of an investment project when the influence of risk parameters is greatest, that is, the most “dangerous” stages of the project. As numerous calculations have shown, the extreme values ​​of all sensitivity functions for the selected project practically coincide in time.

In addition, by comparing sensitivity functions for individual risk parameters, it is possible to rank the risks and identify among them the most significant ones, on which the main attention of product managers should be focused.

ekta. If a financial forecast model with a sensitivity analysis block is built, then it is possible to carry out simulation modeling of the influence of a set of risk parameters on the selected objective function of the investment project.

BIBLIOGRAPHY

1. Kotov V.I. Risk analysis of investment projects based on sensitivity and fuzzy set theory. St. Petersburg: Shipbuilding, 2007. 128 p.

2. Kotov V.I., Lovtsyus V.V. Development of a business plan: Textbook. allowance. St. Petersburg: Link, 2008. 136 p.

3. Risk analysis of an investment project: Textbook for universities / Ed. M.V. Gracheva. M.: Unity-Dana, 2001. 351 p.

4. Business analysis with using Microsoft Excel: Transl. from English M.: Williams, 2005. 464 p.

5. Methods of sensitivity theory in automatic control / Ed. E.N. Rosenwasser and R.M. Yusupova. L.: Energy. 1971. 344 p.

6. Tomovich R., Vukobratovich M. General theory of sensitivity. M.: Sov. radio, 1972.

7. Kuruc A. Financial Geometry // A geometric approach to hedging and risk management. Pearson Education Limited, 2003. 381 p.

8. System sensitivity and adaptability. Preprints Second IFAC Symposium, Dubrovnih, Ygaslavia, 1968.

9. Tomavic R. Sensitivity analysis of dynamic systems. Belgrade, 1963.

10. Zadeh L., Desoer Ch. Theory of linear systems. (State space method): Transl. from English / Ed. G.S. Pospelov. M.: Nauka, 1970. 704 p.

The actual values ​​of the control system parameters almost always differ from the calculated ones. This may be caused by inaccuracy in the manufacture of individual elements, changes in parameters during storage and operation, changes in external conditions, etc.

Changing parameters can lead to changes in the static and dynamic properties of the system. It is advisable to take this circumstance into account in advance during the design and configuration of the system.

parameter.

or scale derivatives of the quality criterion used / therefore parameter,

The zero index above indicates the fact that partial derivatives should be taken equal to the values ​​corresponding to the memorial (calculated) parameters.

Timing sensitivity functions. Using these sensitivity functions, the influence of small deviations of system parameters from calculated values ​​on the time characteristics of the system (transition function, weight function, etc.) is assessed.

The initial system is a system in which all parameters are equal to the calculated values ​​and have no variations. This system corresponds to the so-called basic movement.

A varied system is a system in which variations in parameters have occurred. Its movement is called varied movement.

Additional movement is the difference between the varied and main movement.

Let the original system be described by a set of first-order nonlinear equations

If parameter changes do not cause changes

order of the differential equation, then the varied motion will be described by a set of equations

the additional motion can be expanded into a Taylor series.

For small variations of parameters, it is permissible to limit ourselves to linear terms of the expansion. Then we obtain the first approximation equations for the additional motion

Partial derivatives in parentheses must be equal to their values

Thus, a first approximation for the additional motion can be found with known sensitivity functions. Note that the use of sensitivity functions is more convenient for finding additional motion compared to the direct formula (8.98), since the latter in many cases can give large errors due to the need to subtract two close quantities.

may be necessary use second approximation with retention in the Taylor series, in addition to linear, also quadratic terms.

leads to the so-called sensitivity equations

However, equations (8.100) turn out to be complex and their solution is difficult. A more appropriate way is to construct a structural model used to find sensitivity functions.

parameter.

In some cases, sensitivity functions are obtained by differentiating a known time function at the system output. So, if the transfer function of the system corresponds to a second-order aperiodic link, then (see Table 4.2)

■ 1(0 output will be

will give a sensitivity function for this parameter

Let the system under consideration be described by a set of first-order equations

then equations (8.102) correspond to zero initial conditions.

associated with the setting influence by dependence

Image of the reference influence.

Here the sensitivity function of the transfer function is introduced

These dependencies are valid in the case when the variation of parameter a. does not change the order of the characteristic equation of the system.

The so-called logarithmic sensitivity function can also be used

The transfer function of a real object P(s) can change during operation by the amount DP(s), for example, due to changes in the load on the motor shaft, the number of eggs in the incubator, the level or composition of the liquid in the autoclave, due to aging and wear of the material, the appearance of backlash, changes in lubrication, etc. A well-designed system automatic regulation must maintain its quality indicators not only under ideal conditions, but also in the presence of the listed harmful factors. To assess the influence of a relative change in the transfer function of the object DP/P on the transfer function of the closed-loop system Gcl

y(s) = r(s), Gcl(s) = (8)

let's find the differential dGcl:

Dividing both sides of this equality by Gcl and substituting Gcl = PR/(1+PR) into the right side, we obtain:

Figure 17 - Estimation of gain and phase margins for a system with the hodograph shown in Figure 15

From (10) the meaning of the coefficient S is visible - it characterizes the degree of influence of a relative change in the transfer function of an object on a relative change in the transfer function of a closed loop, that is, S is the sensitivity coefficient of a closed loop to variations in the transfer function of an object. Since the coefficient S = S(jш) is frequency-dependent, it is called the sensitivity function.

As follows from (10),

Let us introduce the notation:

The value T is called complementary (additional) sensitivity function, since S + T = 1. The sensitivity function allows you to evaluate the change in the properties of the system after the closure feedback. Since the transfer function of an open-loop system is equal to G = PR, and of a closed-loop system Gcl = PR/(1+PR), then their ratio is Gcl/G = S. Similarly, for an open-loop system, the transfer function from the disturbance input d to the output of a closed-loop system is equal to (see) P(s)/(1 + P(s)R(s)), and open-loop - P(s), therefore, their ratio is also equal to S. For the transfer function from the measurement noise input n to the system output, the same ratio can be obtained S.

Thus, knowing the form of the function S(jш) (for example, Figure 18), we can say how the suppression of external influences on the system will change for different frequencies after closing the feedback circuit. Obviously, noise lying in the frequency range in which |S(jш)| > 1, after closing the feedback will increase, and noise with frequencies at which |S(jш)|< 1, после замыкания обратной связи будут ослаблены.

The worst case (the greatest increase in external influences) will be observed at the maximum frequency Ms of the module of the sensitivity function (Figure 18):

The maximum of the sensitivity function can be associated with the stability margin sm (Figure 15). To do this, let us pay attention to the fact that |1 + G(jш)| represents the distance from the point [-1, j0] to the current point on the hodograph of the function G(jш). Therefore, the minimum distance from point [-1, j0] to

function G(jш) is equal to:

Comparing (13) and (14), we can conclude that sm = 1/Ms. If the modulus G(jш) decreases with increasing frequency, then, as can be seen from Figure 15, (1-sm) ? 1/gm. Substituting here the ratio sm = 1/Ms, we obtain an estimate of the gain margin, expressed through the maximum of the sensitivity function:

Similarly, but with rougher assumptions, we can write an estimate of the phase margin in terms of the maximum of the sensitivity function:

For example, with Ms = 2 we get gm? 2 and? 29°.

Figure 18 - Sensitivity functions for a system with hodographs shown in Figure 13

Robustness is the ability of a system to maintain a given margin of stability when its parameters vary due to changes in load (for example, when the furnace load changes, its time constants change), technological dispersion of parameters and their aging, external influences, calculation errors and object model errors. Using the concept of sensitivity, we can say that robustness is the low sensitivity of the stability margin to variations in object parameters.

If the object parameters change within small limits, when the replacement of the differential by a finite increment can be used, the effect of changes in the object parameters on the transfer function of the closed-loop system can be assessed using the sensitivity function (10). In particular, we can conclude that at those frequencies where the modulus of the sensitivity function is small, the influence of changes in object parameters on the transfer function of the closed-loop system and, accordingly, on the stability margin will be small.

To assess the impact of large changes in the parameters of an object, we present the transfer function of the object in the form of two terms:

P = P0 + DP, (17)

where P0 is the calculated transfer function, DP is the magnitude of the deviation from P0, which must be a stable transfer function. Then the loop gain of the open-loop system can be represented as G = RP0 + RДP = G0 + RДP. Since the distance from the point [-1, j0] to the current point A on the hodograph of the unperturbed system (for which DP = 0) is equal to |1 + G0| (Figure 19), the stability condition of the system with deviation of the loop gain RДП can be represented as:

|RДP|< |1+G0|,

where T - additional function sensitivity (12). Finally, we can write the relationship:

which must be fulfilled in order for the system to remain stable when the process parameters change by the amount DP(jш).

Reducing zeros and poles. Since the transfer function of an open-loop system G = RP is the product of two transfer functions, which in the general case have both a numerator and a denominator, it is possible to reduce poles that lie in the right half-plane or close to it. Since in real conditions, when there is a scatter of parameters, such a reduction is carried out inaccurately, a situation may arise when a theoretical analysis leads to the conclusion that the system is stable, although in fact, with a small deviation of the process parameters from the calculated values, it becomes unstable.

Therefore, every time the poles are shortened, it is necessary to check the stability of the system with a real scatter of object parameters.

Figure 19 - Explanation of the conclusion of relation (18)

The second effect of pole reduction is the appearance of a significant difference between the settling time of the transient process in a closed system when exposed to a setpoint signal and external disturbances. Therefore, it is necessary to check the response of the synthesized controller when exposed not only to the setpoint signal, but also to external disturbances.

Shockless switching of control modes. PID controllers may have modes when their parameters change abruptly. For example, when in a running system it is necessary to change the integration constant or when, after manual control of the system, it is necessary to switch to auto mode. In the described cases, unwanted emissions of controlled quantities may occur if special measures are not taken. Therefore, the problem arises of smooth (“shockless”) switching of operating modes or controller parameters. The main method for solving the problem is to construct a controller structure such that the parameter changes are performed before the integration stage. For example, with a changing parameter Ti = Ti (t), the integral term can be written in two forms:

I(t) = or I(t) = .

In the first case, with an abrupt change in Ti (t), the integral term will change abruptly; in the second case, it will change smoothly, since Ti (t) is under the sign of the integral, the value of which cannot change abruptly.

A similar method is implemented in the incremental form of the PID controller (see subsection “Incremental form of the digital PID controller”) and in the sequential form of the PID controller, where integration is performed at the final stage of calculating the control action.

Knowledge of the sensitivity functions of this objective function will be very useful for the operational management of the state of the company’s current account under the influence of risks.

3.3. Types and properties of sensitivity functions

When calculating sensitivity functions, one should distinguish between short-term and long-term exposure to risk events. Accordingly, we define two types of sensitivity functions:

Local sensitivity– sensitivity to local (short-term in time) influence of the risk parameter, i.e. when the deviation occurs only during one or several periods significantly shorter than the overall planning horizon (Fig. 3.2).

System response to local impact

Fig.3.2. Towards the determination of local sensitivity

Global sensitivity – sensitivity under global (long-term) influence risk parameter, those. when a deviation can occur over the entire planning horizon, starting from a certain moment (Fig. 3.3).

System response to global impact

Fig.3.3. Toward the determination of global sensitivity

Which of the given sensitivity options should be chosen depends on how long certain risk events will last in a real situation.

An analogy with the analysis of the response of linear systems based on the impulse and transient characteristics of the latter is appropriate here. If delta is used as a single effect at time τ

Dirac function - δ (t-τ), then the reaction of the system under zero initial conditions will be numerically equal to the impulse response of the system g(t-τ). If the Heaviside function (unit jump) - 1(t-τ) is used as a single impact at some point in time, then the system response at zero initial conditions will be numerically equal to the system’s transient response h(t-τ).

In our case, the role of the delta function can be played by a local jump in the risk parameter LdX(t-τ), then the response of the investment project will be proportional to the local sensitivity LS(t-τ) to a given impact. The Heaviside function 1(t-τ) will correspond to the global time change in the risk parameter GdX(t-τ), which will give

response proportional to the global sensitivity function GS(t-τ). Figure 3.2 shows the corresponding functional analogies.

Local analogy

Global analogy

Fig.3.4. Analogies with linear systems

As is known, for linear systems the principle of superposition is valid, namely: the reaction of the system to a set of influences is equal to the sum of the reactions to each influence separately. Based on this principle, knowing the characteristics of the system g(t) or h(t), you can find both the connection between them and the system’s response to any type of impact. In our case, from the superposition principle we can obtain a connection between global and corresponding local sensitivity functions. Let time change discretely:

t = 0, 1, 2, … n, … N,

where t = N – planning horizon;

t = k – moment of the beginning of the impact of global risk;

t = k+j, (j = 0, 1, … n–k) – moments of existence of local risks;

t = n ≥ k+j – arbitrary (current) moment of observing the system’s response to a given impact.

Then the global sensitivity, which describes the system’s response to the impact of a global risk event that began at the moment t = k and lasts up to the planning horizon, can be expressed as a superposition of local sensitivities corresponding to the totality of the impacts of local (lasting one period) risks appearing at moments from t = k and up to t = k +j, (j = 0, 1, … n – k), namely:

	n− k
		(n − k − j), n ≥ k + j
GSx i	(n − k) = ∑ LSx i
	j= 0

It should be noted that local sensitivity functions always decrease faster than the global functions of the same name for all time periods. This is explained by the fact that the local effect of any risk lasts a short time, and the global risk (equal to the sum of local risks) operates all the time from the moment of its occurrence and the effect from it accumulates from period to period. We can say that global sensitivity functions reflect the strategic consequences of the influence of long-term deviations of parameters on an investment project. At the same time, local sensitivities reflect the tactical consequences of short-term changes in the external and internal business environment.

Properties of objective functions of the financial flow model

When using the analytical apparatus for analyzing linear systems, it should be borne in mind that the financial model of an investment project may not be strictly linear, however, as experiments on many different investment projects have shown, even within a wide range of variations in risk parameters, the accuracy of sensitivity analysis remained quite acceptable. However, before using this technique, it is advisable to check the objective function of a specific investment project for linearity with respect to the selected risk parameters. To do this, it is enough to check the fulfillment of the following proportionality condition:

where a is some arbitrary constant.

Let's consider situations when the objective function is nonlinear:

1. NPV depends nonlinearly on the discount rate, because the latter is raised to the power “t”.

2. The objective function may nonlinearly depend on the bank loan rate in the case where there is a deferment of interest payments, because in this case, interest will be calculated according to the compound interest scheme, which will lead to non-linearity.

3. Objective function ( NPV, accumulated balance of financial flows, accumulated net financial flow, etc.) may non-linearly depend on the price of the product sold, if the natural sales volume of this product significantly depends on its price.

4. If at the initial stage of project implementation there is no net profit (losses occur), then the objective functions will be nonlinear with respect to risk parameters during these periods of time, because The dependence of net profit on risk parameters will be piecewise linear functions. After the project was released on

positive net profit, the indicated nonlinearity becomes insignificant.

The work proposes, in addition to first-order sensitivities (3.2), to use second-order sensitivities in cases where the nonlinearity of the objective function with respect to some risk parameters is significant and cannot be neglected. This approach will be discussed in more detail below in Section 3.7.

Let's continue studying the properties of objective functions. If the sales prices of manufactured goods during the implementation of an investment project are chosen as risk parameters, then in each planning period the objective function (for example, the accumulated net financial flow in the case of two goods) will have the form:

Y = a (p1 Q 1 + p 2 Q 2 ) + b

where p 1.2 are prices, and Q 1.2 are natural sales volumes. If we can neglect the dependence Q(p), then using (3.2) we obtain the sensitivity functions for the period under consideration:

		ap 1, 2 Q 1, 2
p 1, 2

It is easy to see that the ratio of these sensitivity functions will be equal to the ratio of sales volumes in monetary terms of the corresponding goods in a given period. Consequently, the structure of price sensitivity functions will exactly correspond to the structure of sales volumes in monetary terms, i.e.

p i Q i
		S x i
∑ p i Q i		∑ S x Y i

This conclusion is valid for any number of products included in the assortment. If individual groups of goods available in the assortment have different VAT rates, then the above conclusion will be valid if prices without VAT are used in calculations of sensitivity and in calculations of the structure of sales volumes.

This property of price sensitivity functions allows one to significantly reduce the volume of calculations for the latter in the case of a wide range of goods, when it is necessary to know the sensitivity for all prices.

If the above dependence Q(p) cannot be neglected, then in this case the connection between the sensitivity functions and the sales structure will remain at a qualitative level, i.e. The greater the share of a given product compared to others in total revenue, the higher its sensitivity to price.

Next, consider the sign of the sensitivity function. The sensitivity function will be positive for all points in time if, with an increase (decrease) in the deviation of the risk parameter, the value of the objective function increases (decreases) provided that the objective function itself is positive. For example, the sensitivity of the accumulated balance of financial flows to prices and natural sales volumes of manufactured goods is always positive, and the sensitivity of the same objective function to deviations of any costs, as well as to bank lending rates, is always negative. An exception to this rule

Radiometric and photometric units can be linked together using sensitivity functions of the human eye V(X), sometimes called the luminous efficiency function. In 1924, the International Commission on Illumination, CIE, introduced the concept of the sensitivity function of the human eye in photopic vision mode for point sources of radiation and an viewing angle of 2° (CIE, 1931). This function, called functions of the MKO 1931, is still the photometric standard in the USA 0.

Judd and Woe introduced in 1978 modified function V(\)(Vos, 1978; Wyszeckl, Stiles, 1982, 2000), which in this book will be called function of the ICE 1978 The changes were associated with an incompletely correct assessment of the sensitivity of the human eye in the blue and violet spectral ranges, adopted in 1931. The modified function F(A) in the spectral range of wavelengths less than 460 nm has higher values. The CIE endorsed the introduction of the 1978 V(A) function, stipulating that “the sensitivity function of the human eye for point sources of radiation can be represented as a modified Judd's V(A) function” (CIE, 1988). Moreover, in 1990, the CIE resolved that “in cases where luminance measurements in the short wavelength range consistent with the determination of color are made by an observer normal to the radiation source, it is preferable to use the modified Judd function” (CIE, 1990).

In Fig. 16.6 shows the functions V(X) CIE 1931 and 1978. The maximum sensitivity of the eye occurs at a wavelength of 555 nm, which is in the green region of the spectrum. At this wavelength, the sensitivity of the eye is equal to 1, i.e. Y(555 nm) = 1. It can be seen that the 1931 CIE function Y(A) underestimates the sensitivity of the human eye in the blue region of the spectrum (A< 460 нм). В приложении 16.П1 приведены численные значения функций У (А) 1931 г. и 1878 г.

‘) This standard is also valid in Russia.

In Fig. Figure 16.6 also shows the function Y"(A) of the sensitivity of the human eye for the scotopic vision mode. The peak sensitivity in the scotopic vision mode occurs at a wavelength of 507 nm. This value is much less than the wavelength of the maximum sensitivity in the photopic vision mode. Numerical values ​​of the function V"(\) The ICE of 1951 is given in Appendix 16.P2.

Note that, although in a number of cases the function of the U (L) CIE 1978 is preferable, it still does not belong to the category of standards, since changing standards often leads to uncertainties. However, despite this, in practice it is used quite often (WyszeckiandStiles, 2000). The 1978 CIE function U(L), shown in Fig. 16.7 can be considered the most accurate description of variations in the sensitivity of the human eye in the photopic vision mode.

To find the sensitivity function of the human eye, use minimal flash method, which is a classic way of comparing light sources by brightness and determining

Rice. 16.6. Comparison of human eye sensitivity functions V(\) CIE 1978 and 1931 for photopic vision. The eye sensitivity function is also shown here V"(\) in scotopic vision mode, which is used for low levels external illumination

Rice. 16.7. U(L) (left ordinate) and luminous efficiency measured in lumens per watt of optical power (right ordinate). The maximum sensitivity of the human eye occurs at a wavelength of 555 nm (data from the CIE, 1978)

functions Y(A). In accordance with this method, a small round light-emitting surface is alternately illuminated (with a frequency of 15 Hz) by sources of the reference and comparison colors. Since the color fusion frequency is below 15 Hz, the colors of alternating signals will be indistinguishable. However, the luminance fusion frequency of the input signals is always higher than 15 Hz, so if two color signals differ in luminance, a visible flash is observed. The researcher's goal is to adjust the color of the radiation source being tested until the observed flare is minimal.

By changing the distribution of spectral radiation power P(L), you can achieve any desired color shade. One of the variants of this distribution is characterized by the maximum possible light output. The maximum luminous efficiency can be achieved by mixing radiation of a certain intensity from two monochromatic light sources (MaeAdam, 1950). In Fig. Figure 16.8 shows the maximum achievable luminous efficiency values ​​obtained using one pair of monochromatic radiation sources. Maximum luminous efficiency white light depends on color temperature. At color temperature

Rice. 16.8. Relationship between the maximum possible luminous efficiency (lm/W) and chromaticity coordinates (x,y) on the 1931 CIE color chart.

6500 K it is ~ 420 lm/W, and at lower color temperatures it can exceed ~500 lm/W. The exact luminous output value is determined by the position of the hue of interest within the white range on the color chart.