Amplitude a search fluctuations should be small, since these fluctuations pass into the output signal of the object and lead to an error in determining the extremum.

Quantity component y, frequency ω 0 , separated by a bandpass filter F 1 (Fig. 11). Filter task F 1 is not to miss the constant or slowly changing component and the components of the second and higher harmonics. Ideally, the filter should pass only the component with frequency ω 0.

After filter F 1 variable component of quantity y, frequency ω 0 , fed to the multiplying link MOH(synchronous detector). The reference value is also fed to the input of the multiplier link v 1 =a sin( ω 0 t + φ ). Phase φ reference voltage v 1 selected depending on the filter output phase F 1 , since filter f 1 introduces an additional phase shift.

Multiplier output voltage u=vv 1 . With a value x<x wholesale

u = vv 1 = b sin( ω 0 t+ φ ) a sin( ω 0 t+ φ ) = ab sin 2 ( ω 0 t + φ )==ab/ 2 .

When the value of the signal at the input x>X 0PT signal value at the output of the multiplier link MOH is:

u = vv 1 = b sin( ω 0 t + φ + 180°) a sin( ω 0 t + φ ) = - ab sin 2 ( ω 0 t + φ )= = - ab/ 2 .

Rice. 12The nature of the search in the CAO with auxiliary modulation:

a - object characteristics; b- change of a phase of fluctuations; in- harmonic oscillations at the input; G- total input signal; d - signal at the output of the multiplier link.

After the multiplier signal and applied to a low pass filter F 2 , which does not pass the variable component of the signal and. DC signal and=and 1 after filter F 2 is applied to the relay element RE. The relay element controls the actuator at a constant travel speed. Instead of a relay element in the circuit, there may be a phase-sensitive amplifier; then the actuator will have a variable speed of movement.

On fig. Figure 12 shows the nature of the search for an extremum in the ACS with auxiliary modulation, the block diagram of which is shown in fig. 11. Suppose that the initial state of the system is characterized by signals at the input and output of the object, respectively X 1 and y 1 (dot M 1 in fig. 12a).

Because at the point M 1 meaning x 1 <х опт then when the extreme controller is turned on, the phases of the input and output oscillations will coincide. Let us assume that in this case the constant component at the filter output F 2 is positive ( ab/2>0), which corresponds to the movement with increasing X, i.e. dx 0 /dt>0. In this case, the SAO will move towards an extremum.

If the starting point M 2 , which characterizes the position of the system at the moment of turning on the extremal controller, is such that the input signal of the object x>x opt (Fig. 12, a), then the oscillations of the input and output signals of the object are in antiphase. As a result, the constant component at the output F 2 will be negative ( ab/2<0), что вызовет движение системы в сторону уменьшения X (dx 0 /dt<0 ). In this case, the SAO will approach the extremum.

Thus, regardless of the initial state of the system, the search for an extremum will be provided.

In systems with a variable speed actuator, the speed of the system movement to the extremum will depend on the amplitude of the output oscillations of the object, and this amplitude is determined by the deviation of the input signal X from the value X wholesale

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1. Extreme control systems

Extreme control systems are such control systems in which one of the performance indicators must be kept at the limit level (min or max).

A classic example of an extreme SU is the auto-tuning system of a radio receiver.

Fig.1.1 - Frequency response:

1.1 Statement of the problem of synthesis of extremal systems

Objects are described by the equations:

The extreme characteristic drifts in time.

It is necessary to choose such a control action that would automatically find the extremum and keep the system at this point.

U: extr Y=Y o (1.2)

Fig.1.2 - Static extreme characteristic:

It is necessary to determine such a control action that ensured the implementation of the property:

1.2 Extreme condition

A necessary condition for an extremum is the equality to zero of the first partial derivatives.

A sufficient condition for an extremum is the equality to zero of the second partial derivatives. When synthesizing an extremal system, it is necessary to estimate the gradient, but the vector of second partial derivatives cannot be estimated, and in practice, instead of a sufficient condition for an extremum, the relation is used:

Stages of synthesis of an extremal system:

Gradient estimation.

Organization of movement in accordance with the condition of movement to an extremum.

Stabilization of the system at the extremum point.

Fig.1.3 - Functional diagram of the extreme system:

1.3 - Types of extreme characteristics

1) Unimodal extremal characteristic of module type

Rice. 1.4 - Extreme characteristic of module type:

2) Extreme characteristic of parabola type

Rice. 1.5 - Extreme characteristic of parabola type:

3) In the general case, the extremal characteristic can be described by a parabola of the nth order:

Y = k 1 |y-y o (t)| n + k 2 |y-y o (t)| n -1 + …+k n | y-y o (t)| + k n +1 (t).(1.9)

4) Vector-matrix representation:

Y = y T By(1.10)

1.4 Gradient Estimation Methods

1.4.1 Method of dividing derivatives

Consider it on a unimodal characteristic, y is the output of the dynamic part of the system.

yR 1 , Y = Y(y,t)

Let's find the total derivative with respect to time:

When drifting slowly, in this way

Advantage: simplicity.

Disadvantage: for small 0, the gradient cannot be determined.

differentiating filter.

Rice. 1.6 - Scheme for estimating the partial derivative:

1.4.2 Discrete Gradient Estimation

Rice. 1.7 - Scheme for discrete estimation of the partial derivative:

1.4.3 Discrete estimate of the sign of the gradient

For a small sampling step, we replace:

1.4.4 Synchronous detection method

The method of synchronous detection involves adding an additional sinusoidal signal of low amplitude, high frequency to the input signal to the extreme object and extracting the corresponding component from the output signal. According to the ratio of the phases of these two signals, we can conclude about the sign of the partial derivatives.

Rice. 1.8 - Functional scheme for estimating the partial derivative:

Rice. 1.9 - Illustration of the passage of search oscillations to the system output:

y 1 - operating point, while the phase difference of the signals is 0.

y 2 - the phase difference of the signals, as the simplest PFC, you can use the multiplication block.

Rice. 1.10 - Illustration of the operation of the FCU:

As a filter, a period-averaging filter is chosen, which makes it possible to obtain an output signal proportional to the value of the partial derivative.

Rice. 1.11 - Linearization of the static characteristic at the operating point:

Therefore, the equation of the extremal curve can be replaced by the equation of a straight line:

PFC output signal:

k - coefficient of proportionality - the tangent of the angle of inclination of the straight line.

Filter output signal:

In this way:

The synchronous detection method is suitable for determining not only one partial derivative, but also the gradient as a whole, while several oscillations of different frequencies are fed to the input. Appropriate output filters highlight the response to a specific search signal.

1.4.5 Custom Gradient Estimation Filter

This method involves the introduction of a special dynamic system into the system, the intermediate signal of which is equal to the partial derivative.

Rice. 1.12 - Scheme of a special partial derivative estimation filter:

T- filter time constant:

To estimate the total derivative of Y, a DF is used - a differentiating filter, and then this estimate of the total derivative is used to estimate the gradient.

1.5 Organization of the movement to the extremum

1.5.1 First order systems

We organize the control law in proportion to the gradient:

We write the equation of a closed system:

This is an ordinary differential equation that can be investigated by TAU methods.

Consider the equation of the statics of the system:

If the stability of a closed system is ensured with the help of the gain k, then automatically in statics we will come to an extremum point.

In some cases, using the coefficient k, in addition to stability, it is possible to provide a certain duration of the transient process in a closed system, i.e. ensure the specified time to reach the extremum.

Where k is stability

Rice. 1.13 - Functional scheme of the gradient extremal system of the first order:

This method is suitable only for unimodal systems, i.e. systems with one global extremum.

1.5.2 Heavy ball method

By analogy with a ball that rolls into a ravine and overshoots the points of local extrema, the AC system with oscillatory processes also overshoots local extrema. To ensure oscillatory processes, we introduce additional inertia into the first-order system.

Rice. 1.14 - Illustration of the "heavy" ball method:

Closed system equation;

Characteristic equation of the system:

The smaller d, the longer the transition process.

Analyzing the extreme characteristic, the necessary overshoot and the duration of the transient process are set, from which:

1.5.3 General single-channel systems

Control law:

Substituting the control law into the control of the object, we obtain the equation of a closed system:

In the general case, to analyze the stability of a closed system, it is necessary to use the second Lyapunov method, which is used to determine the controller gain. Because The 2nd Lyapunov method gives only a sufficient stability condition, then the chosen Lyapunov function may turn out to be unsuccessful and a regular procedure for calculating the controller cannot be proposed here.

1.5.4 Systems with the highest derivative in control

General case of object extremum:

The functions f, B, and g must satisfy the existence and uniqueness conditions for a solution to the differential equation. The function g - must be multiply differentiable.

С - matrix of derivatives

The synthesis problem is solvable if the matrix of products is not degenerate, i.e.

Analysis of the solvability condition for the synthesis problem allows us to determine the derivative of the output variables, which explicitly depends on the control action.

If condition (1.31) is satisfied, then such a derivative is the first derivative, and therefore the requirements for the behavior of a closed system can be formed in the form of a differential equation for y of the corresponding order.

Let us form the control law of a closed system, for which we will form the control law by substituting in the right side of the control for:

Closed-loop equation with respect to the output variable.

Consider the situation when

With an appropriate choice of the gain, we get the desired equation and an automatic exit to the extremum.

The controller parameters are selected based on the same considerations as for conventional automatic control systems, i.e. (SVK) i = (20*100), which makes it possible to provide the corresponding error.

Rice. 1.15 - Scheme of the system with the highest derivative in control:

In a system for estimating the total time derivative, a differentiating filter is introduced into the system, so it is convenient to use a gradient estimation filter to estimate gradients in such systems. Because both of these filters have small time constants, then different-tempo processes can occur in the system, which can be distinguished using the motion separation method, and slow motions will be described by equation (1.34), which corresponds to the desired at. Fast movements need to be analyzed for stability, and depending on the ratio of the time constant of the DF and the partial derivative estimation filter (PDE), the following types of movements can be distinguished:

1) The time constants of these filters are comparable.

Fast movements describe the combined processes in these two filters.

2) The time constants differ by an order of magnitude.

In addition to slow motions, fast and superfast motions corresponding to the smallest time constant are observed in the system.

Both cases need to be analyzed for stability.

2. Optimal systems

Optimal systems are systems in which the specified quality of work is achieved by maximizing the use of the capabilities of the object, in other words, these are systems in which the object operates at the limit of its capabilities. Consider an aperiodic link of the first order.

For which it is necessary to ensure the minimum transition time y from the initial state y(0) to the final state y k . The transition function of such a system for K=1 is as follows

Rice. 2.1 - Transition function of the system at U= const:

Consider the situation when we apply the maximum possible control action to the input of the object.

Rice. 2.2 - Transition function of the system at U=A= const:

t 1 is the minimum possible transition time y from the zero state to the final state for the given object.

To obtain such a transition, there are two control laws:

The second law is more preferable and makes it possible to provide control under interference.

Rice. 2.3 - Structural diagram of a system with a control law of the feedback type:

2.2 Statement of the problem of synthesis of optimal systems

2.2.1 Mathematical model of the object

The object is described by state variables

Where the function f(x,u) is continuous, differentiable with respect to all arguments, and satisfies the existence and uniqueness condition for a solution to the differential equation.

This function is non-linear, but stationary. As special cases, the object can have the form of a nonlinear system with additive control:

Or a linear system

The object must be presented in one of the three forms presented above.

2.2.2 Set of initial and final states

The problem of optimal transition from the initial state to the final state is a boundary value problem

Where the start and end points can be specified in one of the four ways shown in fig. 2.4.

a) a problem with fixed ends,

b) a problem with a fixed first end (fixed starting point and set of final values),

c) a problem with a fixed right end,

d) a problem with moving ends.

Fig. 2.4 - Phase portraits of the system transition from the initial state to the final state for various tasks:

For an object, the set of initial states can generally coincide with the entire set of states or with the work area, and the set of final states is a subspace of the set of states or the work area.

Example 2.1 - Can an object described by a system of equations be transferred to any point in the state space?

Substituting the value U from the first equation u = x 2 0 - 2x 1 0 into the second equation, we get -5x 1 0 + x 2 0 = 0;

We got a set of final states described by the equation x 2 0 = 5x 1 0 ;

Thus, the set of final states specified for an object (system) must be realizable.

2.2.3 Constraints on states and control

Rice. 2.5 - General view of the workspace of the state space:

The working area of ​​the state space is allocated, which is negotiated. Typically, this area is described by its boundaries using modular conventions.

Fig.2.6 - View of the workspace of the state space, defined by modular agreements:

U is also set - the range of permissible values ​​of the control action. In practice, the region U is also specified using modular relations.

The problem of designing an optimal controller is solved subject to restrictions on control and a limited resource.

2.2.4 Optimality criterion

At this stage, the requirements for the quality of the work of a closed system are specified. The requirements are specified in a generalized form, namely, in the form of an integral functional, which is called the optimality criterion.

General view of the optimality criterion:

Particular types of optimality criterion:

1) the optimality criterion that ensures the minimum time of the transient process (the problem of optimal performance is solved):

2) the criterion of optimality, providing a minimum of energy costs:

For one of the components:

For all state variables:

For one control action:

For all control actions:

For all components (in the most general case):

2.2.5 Form of result

It is necessary to specify in what form we will seek the control action.

There are two options for optimal control: u 0 = u 0 (t), used in the absence of perturbation, u 0 = u 0 (x), optimal control in the form of feedback (closed control).

The formulation of the problem of synthesis of the optimal system in general form:

For an object described by variable states with given constraints and a set of initial and final states, it is necessary to find a control action that ensures the quality of processes in a closed system that meets the optimality criterion.

2.3 Dynamic programming method

2.3.1 Principle of optimality

Initial data:

It is necessary to find u 0:

Rice. 2.7 - Phase portrait of the transition of the system from the starting point to the final point in the state space:

The trajectory of the transition from the starting point to the final one will be optimal and unique.

Statement of the principle: The final section of the optimal trajectory is also the optimal trajectory. If the transition from the intermediate point to the final point were not carried out along the optimal trajectory, then it would be possible to find its own optimal trajectory for it. But in this case, the transition from the initial point to the final one would pass along a different trajectory, which should have been optimal, and this is impossible, since there is only one optimal trajectory.

2.3.2 Basic Bellman equation

Consider an arbitrary control object:

Consider a state-space transition:

Rice. 2.8 - Phase portrait of the transition of the system from the initial point to the final one x(t) - current (initial) point, x(t + Дt) - intermediate point.

Let's transform the expression:

Let's replace the second integral with V(x(t+Дt)):

For a small value of Дt, we introduce the following assumptions:

2) Expand the auxiliary function

Performing further transformations, we get:

Where min V(x(t)) is the optimality criterion J.

As a result, we got:

Divide both parts of the expression by Dt and eliminate Dt to zero:

We get the basic Bellman equation:

2.2.3 Calculation ratios of the dynamic programming method:

The basic Belman equation contains (m + 1) - unknown quantities, because U 0 R m , VR 1:

Differentiating m times, we get a system of (m + 1) equations.

For a limited range of objects, the solution of the resulting system of equations gives an exact optimal control. Such a problem is called the AKOR problem (analytical design of optimal controllers).

The objects for which the AKOR task is considered must meet the following requirements:

The optimality criterion must be quadratic:

Example 2.2

For an object described by the equation:

It is necessary to ensure the transition from x(0) to x(T) according to the optimality criterion:

After analyzing the object for stability, we get:

U 0 \u003d U 2 \u003d -6x.

2.4 Pontryagin's maximum principle

Let us introduce an extended state vector, which is expanded due to the zero component, for which we choose the optimality criterion. zR n+1

We also introduce an extended vector of right-hand sides, which is extended by the function under the integral in the optimality criterion.

Let's introduce W - the vector of conjugate coordinates:

Let us form the Hamiltonian, which is the scalar product of W and u(z, u):

H(W,z,u) = W*u(z,u),(2.33)

Equation (2.34) is called the basic equation of the Pontryagin maximum principle, based on the dynamic programming equation. The optimal control is the one that delivers the maximum of the Hamiltonian on a given time interval. If the control resource were not limited, then necessary and sufficient extremum conditions could be used to determine the optimal control. In a real situation, to find the optimal control, it is necessary to analyze the value of the Hamiltonian at the limiting level. In this case, U 0 will be a function of the extended state vector and the vector of conjugate coordinates u 0 = u 0 .

To find conjugate coordinates, it is necessary to solve the system of equations:

2.4.1 The procedure for calculating the system according to the Pontryagin maximum principle.

The equations of the object must be reduced to the standard form for the synthesis of optimal systems:

It is also necessary to specify the initial and final states and write down the optimality criterion.

Extended state vector is introduced

Extended vector of right parts:

And the vector of conjugate coordinates:

We write the Hamiltonian as a dot product:

Finding the maximum of the Hamiltonian with respect to u:

By which we determine the optimal control u 0 (Ш,z).

We write down the differential equations for the vector of conjugate coordinates:

Find conjugate coordinates as a function of time:

6. We determine the final optimal control law:

As a rule, this method allows one to obtain a program control law.

Example 2.3 - For the object shown in fig. 2. 9. it is necessary to ensure the transition from the initial point y(t) to the final point y(t) in T= 1c with the quality of the process:

Rice. 2.9 - Object model:

To determine the constants b 1 and b 2, it is necessary to solve the boundary value problem.

We write the equation of a closed system

Let's integrate:

Consider the end point t=T=1s. as x 1 (T)=1 and x 2 (T)=0:

1= 1/6 b 1 + 1/2 b 2

We got a system of equations, from which we find b 2 \u003d 6, b 1 \u003d -12.

Let's write down the control law u 0 = -12t + 6.

2.4.2 Optimal control problem

For a general object, it is necessary to ensure the transition from the initial point to the final one in the minimum time with a limited control law.

Features of the optimal speed problem

Speed ​​Hamiltonian:

Relay control:

This feature takes place for relay objects.

The theorem on the number of switchings of the control action:

This theorem is valid for linear models with real roots of the characteristic equation.

Det (pI - A) = 0 (2.51)

L(A) - vector of real eigenvalues.

Statement of the theorem:

In the problem of optimal speed with real roots of the characteristic equation, the number of switchings cannot be more than (n-1), where n is the order of the object, therefore, the number of intervals of control constancy will not be more than (n-1).

Rice. 2.10 - Type of control action for n=3:

Example 2.4 - Consider an example of solving the problem of optimal performance:

W \u003d [W 1, W 2]

H b \u003d W 1 x 2 + W 2 (-2dx 2 -x 1 + u)

At - real roots:

The sum of the two exponents is:

If, then the roots are complex conjugate and the solution will be a periodic function. In a real system, there are no more than 5 - 6 switchings.

2.4.3 Switching surface method

This method allows you to find the control of the functions of the state variable for the case when the optimal control is of a relay nature. Thus, this method can be used in solving problems of optimal performance, for an object with additive control

The essence of the method is to select points in the entire state space where the control sign is changed and combine them into a common switching surface.

Switching surface

The control law will have the following form:

To form the switching surface, it is more convenient to consider the transition from an arbitrary starting point to the origin

If the end point does not coincide with the origin, then it is necessary to choose new variables for which this condition will be true.

We have an object of the form

Consider the transition, with the optimality criterion:

This criterion allows us to find a control law of the following form:

With the unknown, the initial conditions are also unknown to us.

Considering the transition:

Reverse time method (backward movement method).

This method allows you to define switching surfaces.

The essence of the method is that the initial and final points are interchanged, while instead of two sets of initial conditions, one remains for.

Each of these trajectories will be optimal. First, we find the points where the control changes sign and combine them into a surface, and then we change the direction of movement to the opposite.

Example - The transfer function of an object is:

Performance optimality criterion:

Control restriction.

Consider the transition:

Optimal control will have a relay character:

Let's go to the opposite time (i.e.). In reverse time, the problem will look like this

Consider two cases:

We obtain the equations of a closed system:

We use the method of direct integration, we obtain the dependence on and since -, then we have

Because the start and end points are swapped, then we get similarly:

Let's build the result and use the phase plane method to determine the direction

Applying the method of direct integration, we obtain:

The function will look like:

Changing direction:

Sign change point (switching point).

General analytical expression:

Surface equation:

Optimal control law:

Substituting the surface equation, we get:

2.5 Sub-optimal systems

Suboptimal systems are systems that are close in properties to optimal ones.

It is characterized by the criterion of optimality.

Absolute error.

Relative error.

A process that is close to optimal with a given accuracy is called suboptimal.

Suboptimal system - a system where there is at least one suboptimal process.

Suboptimal systems are obtained in the following cases:

when approximating the switching surface (using piecewise linear approximation, approximation using splines)

At , an optimal process will arise in a suboptimal system.

limitation of the working area of ​​the state space;

3. ADAPTIVE SYSTEMS

3.1 Basic concepts

Adaptive systems are such systems in which the controller parameters change following the change in the parameters of the object, so that the behavior of the system as a whole remains unchanged and corresponds to the desired:

There are two directions in the theory of adaptive systems:

adaptive systems with a reference model (ASEM);

adaptive systems with an identifier (ASI).

3.2 Adaptive systems with an identifier

Identifier - a device for estimating the object's parameters (parameters must be evaluated in real time).

AR - adaptive regulator

OS - control object

U - identifier

The part that is highlighted by the dotted line can be implemented digitally:

V, U, X - can be vectors. The object can be multichannel.

Consider the operation of the system.

In the case of constant object parameters, the structure and parameters of the adaptive controller do not change, the main feedback acts, the system is a stabilization system.

If the parameters of the object change, then they are evaluated by the identifier in real time and the structure and parameters of the adaptive controller are changed so that the system behavior remains unchanged. The main requirements are imposed on the identifier (performance, etc.) and on the identification algorithm itself. This class of systems is used to control objects with slow non-stationarity. If we have a non-stationary generic object:

;.The simplest responsive view would be:

Requirements for the system:

Where and are matrices of constant coefficients.

In reality we have:

If we equate, then we get a relation for determining the parameters of the controller

3.3 Adaptive systems with a reference model

In such systems, there is a reference model (EM), which is placed parallel to the object. BA - adaptation block.

Fig 2 - Functional diagram of ASEM:

Consider the operation of the system:

In the case when the object parameters do not change or the output processes correspond to the reference ones, the error is:

autotuning control programming

The adaptation block does not work and the adaptive controller is not rebuilt, the system has a smooth feedback.

If the behavior is different from the reference, this happens when the object's parameters are changed, in which case an error appears.

The adaptation block is turned on, the structure of the adaptive controller is rebuilt in such a way as to reduce it to the reference model of the object.

The adaptation block should reduce the error to zero ().

The algorithm embedded in the adaptation block is formed in various ways, for example, using the second Lyapunov method:

If this is true, then the system will be asymptotically stable and.

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The need for adaptive (adaptable) control systems arises in connection with the complication of control problems in the absence of the practical possibility of a detailed study and description of the processes occurring in control objects in the presence of changing external disturbances. The effect of adaptation is achieved due to the fact that part of the functions for receiving, processing and analyzing processes in the control object is performed during the operation of the system. This division of functions contributes to a more complete use of information about ongoing processes in the formation of control signals and can significantly reduce the impact of uncertainty on the quality of control. Thus, adaptive control is necessary in cases where the influence of uncertainty or "incompleteness" of a priori information about the operation of the system becomes significant to ensure the specified quality of control processes. Currently, there is the following classification of adaptive systems: self-adjusting systems, systems with adaptation in special phase states, and learning systems.

The class of self-adjusting (extreme) automatic control systems is widespread due to a fairly simple technical implementation. This class of systems is due to the fact that a number of control objects or technological processes have extreme dependencies (minimum or maximum) of the operating parameter on control actions. These include powerful DC electric motors, technological processes in the chemical industry, various types of furnaces, aircraft jet engines, etc. Let us consider the processes occurring in the furnace during fuel combustion. With insufficient air supply, the fuel in the furnace does not burn completely and the amount of heat generated decreases. With excess air supply, part of the heat is carried away with the air. And only with a certain ratio between the amount of air and heat, the maximum temperature in the furnace is reached. In a turbojet aircraft engine, by changing the fuel consumption, it is possible to obtain the maximum air pressure behind the compressor, and, consequently, the maximum engine thrust. At low and high fuel consumption, the air pressure behind the compressor and thrust drops. In addition, it should be noted that the extreme points of control objects are "floating" in time and space.

In the general case, we can state that there is an extremum, and at what values ​​of the control action it is achieved is a priori unknown. Under these conditions, the automatic control system during operation must form a control action that brings the object to an extreme position, and keep it in this state under conditions of disturbances and the "floating" nature of extreme points. In this case, the control device is an extremal regulator.

According to the method of obtaining information about the current state of the object, extreme systems are non-search and search systems. In searchless systems, the best control is determined by using analytical relationships between the desired value of the operating parameter and the controller parameters. In search engines that are slow, finding the extremum can be done in a variety of ways. The most widespread method is synchronous detection, which is reduced to estimating the derivative dy/du, where y is the controlled (working) parameter of the control object, u is the control action. A block diagram illustrating the method of synchronous detection is shown in fig. 6.1.

Rice. 6.1 Synchronous detection structure

At the input of the control object, which has an extreme dependence y(u), together with the control action U, an insignificant perturbation is applied in the form of a regular periodic signal f(t) = gsinwt, where g is greater than zero and sufficiently small. At the output of the control object, we get y = y(u + gsinwt). The resulting value of y is multiplied by the signal f(t). As a result, signal A will take on the value

A =yf(t) = y(u+gsinwt)gsinwt.

Assuming that the dependence y(u) is a sufficiently smooth function, it can be expanded into a power series and, with a sufficient degree of accuracy, is limited to the first terms of the expansion

Y(u+gsinwt)=y(u)+gsinwt(dy/du) + 0.5g 2 sin 2 wt(d 2 y/du 2) + ….. .

Since the value of g is small, then we can neglect the terms of higher order and as a result we get

Y(u + gsinwt) » y(u) + gsinwt(dy/du).

Then, as a result of multiplication, signal A will take on the value

A \u003d y (u) sinwt + g 2 sin 2 wt (dy / du).

At the output of the low-pass filter F, we get the signal B

.

If the filter time constant T large enough, we get

.

Therefore, the signal B at the output of the filter is proportional to the derivative dy/du

The main most common types of extreme systems in which the static mode of operation of an object is optimized are extreme systems that ensure the operation of an object at the extreme point of its static characteristic.

The static characteristic should reflect the relationship between the quality function of the object and the operating parameters of the object.

Extreme self-propelled guns are advisable to use:

1. There is a quality indicator (technical and economic, characterizing the operation of the object, and this dependence has a pronounced extremum) (most often)

2. Benefits from increased quality functionality.

3. There is a possibility of the current definition of the quality functional.

The control device in this case is called an optimizer or an extremal controller.

The quality functional for setting the operating mode is written: , where is a variable that determines the operating mode of the object.

Depending on whether the extreme static characteristic is stable or changes during the operation of the object, extreme systems are divided into two groups: - static; - dynamic.

Static: Here, extreme control is provided, corresponding to the extremum of the static characteristic of the object with unchanged parameters set for a given extremum point, and the system is similar to a conventional mode stabilization system.

Dynamic: Here, the characteristic can shift independently and the extremum point too. In this case, two cases are possible:

It is known how the characteristic shifts, and you can get by with program control;

The displacement of the most extreme characteristic and the extremum point is random (you must first find the optimal point, then move towards it).

In extreme systems, when the extreme characteristic is shifted, there may be an automatic search for the extremum and a shift to it.

In such cases, two operations are carried out:

1. Trial search engine(determining the relationship between the current quality indicator Q and Q extr and determining the direction of movement. It boils down to determining the steepness of the characteristic: ).

2. Working(works out the found values ​​of changing the controller settings to ensure the extremum of the function)

You can determine the value and sign of the derivative or use a special step method for finding the extremum.

Depending on whether an additional signal is used to search for an extremum, the systems are divided into:

systems without an additional search signal (depending on whether the slope values ​​S 0 or the sign of the derivative of the system are divided by proportional(determined by the steepness dx slave / dt = h 0 S, i.e. performing a dependent search and the speed of moving the working body depends on the slope, which determined the “setpoint” of the regulator) and relay(the direction of movement is determined by the sign dx slave / dt=h 0 SignS= h 0 Sign, i.e. performing an “independent search” and RO moving from one state to another and back, leading the object to an extremum of the static character Here, the logic device switches when the sign of the derivative changes - this leads to a change in the controller setpoint and the corresponding movement of the controller. They are used for fast-response objects.). For inertial systems, the system is used. step type(here, at the command of the command generator, through the step Dt, the measured value of the quality index. And comparing it with the given Q, as a result, the input signal reverses or does not occur)

system with add. Search. signal (a harmonic signal and a signal from a logic device are fed to the input. The search for an extremum is carried out on the basis of a study of the phase shift of the signal X n at the output of the system. The search signal with respect to the main one is a modulating signal.

On the basis signal X superimposed harmonic. search signal and if start signal. X resp. position to the left of the extremum point (X 1), then on the output. extra link additional search signal will create a harmonic. component Q * with the same f as the search signal and there will be no phase shift. Main signal X 3 - harmonic. status on output extras links shifted rel. Search. signal per angle –pi. Main signal X 2 - harmonic. status on output extras link will have f 2 times greater than f of the original. signal. That. by phase shift m.o. def. direction movement.

Multidimensional extremal systems. are built for multi-parameter objects that have several inputs and outputs, and one of the outputs has an extremal characteristic, and restrictions are imposed on the other m/t outputs.

To construct such extreme systems. use special. methods of mathematics. programming and algorithmic optimization methods.

The condition of an extremal function of many variables is the equality to zero of all its parts. derivatives with respect to parameters

In a particular case, if the generalized quality function Q is represented. extreme. static har-coy, then for design it is multidimensional. syst. m / b used the method of simplex planning and in this case in the system. centuries device for computing. deg. extrem. specifications and device for forming. control signal.

The principle of constructing a device for calculating. deg. in the extremum search operation depends on the method of determining. private derivatives and the type of algorithm used.

The most widely used methods are:

1. of course increments

2. time derivative

3. Synchronous detection

4. application of the adaptive model

1. The finite increment method is based on the replacement of partial derivatives by the ratio of finite. increments and defining it. In this case, the cord is changed in turn. control and computing. resp. im increments. yavl. components of the function gradient.

2. The control actions are also alternately changed and the quotients are calculated. derivatives and gradient functions.

Disadvantages 1 and 2: the need to alternately change ex. impacts and calculate the gradient for each change in ex. signal. This requires additional time for calculation.

3. Control coordinates are modulated additionally. harmonic signals with different amplitudes a ni and frequencies w ni . Number of detectors def. the number of independent coordinates defining the extremum of the function Q xi . Output signal sync. detective proportional to private derivative . Because modulating signals are separated by frequency. spectrum, then comp. gradient def. parallel. Using a computer this time will be MIN.