RU164178U1 - DEVICE FOR OPTIMUM CONTROL OF NONLINEAR DYNAMIC OBJECTS - Google Patents

Abstract

An optimal control device for nonlinear dynamic objects, comprising a serially connected master device, a coefficient calculation unit, a control function calculation unit, a control function implementation unit, a control object, a current coordinate measuring unit, which is included in feedback with the control function calculation unit, characterized in that the control function calculation unit contains a direct and dual system of nonlinear ordinary differential equations of a mathematical model of EKTA, forming a two-point boundary value problem, the results of which solutions are included in the calculation expression control function, optimal in the sense of minimizing the quadratic criterion specified.

Description

DEVICE FOR OPTIMUM CONTROL OF NONLINEAR DYNAMIC OBJECTS

The utility model relates to the field of automatic control, namely, optimal control devices with feedback by non-linear dynamic objects (NDO), subject to the influence of various kinds of disturbances (both internal, related to the properties of the object, and external).

The technical result of the proposed utility model is to increase the accuracy of the implementation of the required operating mode by non-linear dynamic objects and reduce the energy consumption for control in the presence of disturbing factors. This is achieved through the formation of a vector of control actions that is optimal in the sense of a given quadratic quality criterion.

The utility model can be applied in any industry where automatic control of the coordinates of the state vector of the object is necessary in order to ensure a given quality of tracking the required operating mode.

The objects of control can be technological processes of industrial production, economic processes, moving objects, the dynamics (functioning) of which in time is described by a system of nonlinear ordinary differential equations of arbitrary order.

The construction of a feedback operator (solving the synthesis problem) is the central problem of any control process. The synthesis task, including the construction of a mathematical model of the object and the implementation of the control process using this model.

Known devices for controlling dynamic objects, forming on the basis of measuring the difference between the current and given parameters of the state vector one of the typical laws

control (proportional using the P-controller, integral using the I-controller, differential using the D-controller and their combinations) [1 p. 80], [2 p. 152]. Basically, regulators of this type are used to control linear objects. Also for such objects, devices are used that implement the method of analytical design of optimal controllers [3, p. 177]. Appropriate analytical design methods for optimal controllers and software are well developed. However, for multidimensional nonlinear objects, control laws and devices that implement them are cumbersome.

Other well-known methods for obtaining a solution in the form of feedback control include the dynamic programming method [3, p. 150] and the Pontryagin minimum principle [4, p. 263], [5 p. fifteen]. The application of the first is difficult due to the exorbitant increase in the volume of calculations with an increase in the dimension of the problem. The use of the second leads to the necessity of solving the two-point boundary value problem (DKZ).

When conducting an information search in the prior art revealed the closest analogue of the claimed utility model, taken as a prototype, "United proportional-integral and neural network controller" (US 7117045 B2 G05B 13/02 published 03.10.2006), which disclosed the optimal device control of nonlinear dynamic objects. The signs of the prototype, which coincide with the features of the proposed utility model, are the presence of a serially connected driver, a coefficient calculation unit, a control function calculation unit, a control function implementation unit, a current coordinate measurement unit, which is included in the feedback from the control function calculation unit.

The main disadvantage of this device is that for the control of NDO, the control function calculated even after the end of the learning process

is linear. This does not allow organizing the process of managing the NDO in an optimal way.

In addition, the disadvantages of this device include:

1. High requirements for the speed of computing tools that search for the optimal parameters of the PI controller in the process of training the network, which implement many iterative cycles.

2. High requirements for the degree of adequacy of the mathematical description of the control object and the conditions for its functioning.

A distinctive feature of the proposed device is that the obtained control function for the NDO, which is optimal in the sense of the minimum of the given quadratic criterion, is non-linear. Such a property of the function is characteristic of objects described by a system of non-linear ordinary differential equations. This corresponds to the physics of the process of controlling objects of this type, allows you to fully take into account the internal properties of NDOs characterized by significant non-linearities and, as a result, increases the accuracy of the object's implementation of the required mode of operation and reduces the energy consumption for control in the presence of disturbing factors.

The functional diagram of the device for optimal control of nonlinear dynamic objects is shown in FIG. one.

The device comprises a serially connected master device 1 (memory), a coefficient calculation unit 2 (BVK), a control function calculation unit 3 (BVFU), a control function implementation unit (DUT actuator) 4 (BRFU), a control object 5 (OS), a unit measuring current coordinates 6 (BITK). The unit for measuring the current coordinates 6 (BITC) is included in the feedback from the unit for calculating the control function 3 (BVFU).

As a master device 1 (memory) can be applied to a device that generates a given signal at certain intervals

time, as a block for calculating the coefficients 2 (BVK), a block for calculating the control function 3 (BFFS), computer and software can be used, as a block for implementing the control function 4 (BRFU), a functional converter can act as a block for measuring current coordinates 6 (BITK) digital sensors can serve as control object 5 (OS) technological processes of industrial production, economic processes, moving objects, the dynamics (functioning) of which in time is described by a system of nonlinear ordinary differential equations of any order.

The master device 1 (memory) generates a master action for the control object, has two outputs. From the first output, the information enters the coefficient calculation unit 2 (BVC), where, as a result of solving the matrix Riccati equation, the transmission coefficients are calculated. From the second output, the information enters the control function calculation unit 3 (BVFU) to form the boundary conditions of the boundary value problem. BVFU contains a direct and dual system of nonlinear ordinary differential equations of a mathematical model of an object that form a two-point boundary value problem, the results of which are included in the expression for calculating a control function that is optimal in the sense of a minimum of a given quadratic criterion at each step. In the block for the implementation of the control function 4 (BRFU), the resulting control function is converted into control action signals supplied to the control object 5 (OS). From the unit for measuring the current coordinates 6 (BITC) connected to the output 5 (OU), the unit for calculating the control function 3 (BVFU) receives the values of the deviations of the current coordinates of the state vector of the control object from the set for the formation of boundary conditions for the boundary value problem.

The operation of the device is illustrated by the following example of the synthesis of optimal non-linear dynamic object control by the quadratic criterion.

A nonlinear dynamic object described by a differential equation is considered

where X (t) is the n-dimensional real state vector,

U (t) is the m-dimensional real control vector,

A and B are real (n × n) - and (n × m) -matrices,

f is a vector nonlinear function defined in the space R ^{n + m} .

The initial state X (t ₀ ) is specified.

The task of controlling the object on the time interval [t ₀ , T] is to minimize the functional

where T is a fixed finite time; Q and P are positive semidefinite - (n × n) matrices; R is a positive definite (n × m) matrix; E (t) is a system error, i.e. E (t) = X (t) -Z (t) for all values of t ₀ ≤t≤T, where Z (t) = (z ₁ (t), ..., z _n (t)) ^T is n-dimensional a valid vector characterizing a given mode of operation of the object, the sign <> means a scalar product.

Thus, it is necessary to find such a control U (t) that the error E (t) is "small".

We assume that the control U (t) is not limited in magnitude; therefore, cases are possible when U (t) is very large. To avoid such extreme situations (requiring very large amplifications in the circuit

control), you can include a requirement that takes into account the fact that "large signals are expensive." In other words, it is required, on the one hand, to keep the error small enough, and on the other hand, not to use "excessively large" controls.

The term L _e = (1/2) <E (t), QE (t)>, where Q is positively semidefinite, non-negative for any E (t) and equal to zero for E (t) = 0. If the error is small for any t ₀ ≤t≤T, then the integral of L _e will be small. Obviously, the cost of big mistakes is much greater than small ones. Therefore, the device is “fined” for large control errors by much more than for small ones.

The term L _u = (1/2) <U (t), RU (t)>, where R is positive definite, estimates the cost of control and “punishes” the device for large controls much more than for small ones. Since R is positive definite, the control cost is positive for any U (t). The requirement of positive definiteness, and not semi-definiteness of R, is a condition for the existence of control of a finite quantity.

The term (1/2) <E (T), QE (T)> estimates the cost of the final state. Its purpose is to guarantee the "smallness" of the error at a finite moment in time T. In other words, this term is considered when the quantity E (T) at a finite moment in time is especially important.

The optimal control U (t) at the step is given by the feedback control law

U (t) = R ^-1 B ′ (t) [H (t) -K (t) X (t)],

in which K (t) is the solution of the Riccati matrix differential equation

with boundary condition

K (T) = P,

H (t) is a solution of a linear differential equation

with boundary condition

H (T) = PZ (T).

As an example, we consider a model of an object in which the matrices A and B are independent of time [6 p. 261],

where a = 7, b = 0.4, g = 0.68, s = 0.28, w = 1.17.

We rewrite the model in the form

where X (t) = (x (t), y (t), v (t)) ′, U (t) = (u (t), 0, 0) ′.

The required mode Z (t) = (x _ass (t), y _ass (t), v _ass (t)) ′ has the form

Z (t) = (3 + arctan (10t-5), 0, 0) ^T.

The results of the proposed utility model of the device for optimal control of nonlinear dynamic objects at the current step T = 1 s with a mismatch error value at time t ₀ equal to 0.8 are shown in FIG. 2-4.

A brief description of the drawings.

FIG. 1. Functional diagram of the proposed utility model of the device for optimal control of nonlinear dynamic objects.

FIG. 2. Graphs of functions of a given mode (2) and obtained as a result of solving the tracking problem (1).

FIG. 3. The graph of changes in the control error e (t) = x (t) -x _ass (t), which at the end of the step becomes practically equal to zero.

FIG. 4. Graph of the control function u (t) obtained as a result of solving at the step a two-point boundary value problem.

Information sources.

1. Reference on the theory of automatic control. Edited by A.A. Krasovsky. - Moscow, publishing house "Science". Ch. ed. Phys.-Math. Lit., 1987 .-- 712 p.

2. Alexandrovsky N.M., Egorov S.V., Kuzin E.V. Adaptive systems of automatic control of complex technological processes. / N.M. Alexandrovsky, S.V. Egorov, E.V. Cousin. - M. "Energy", 1973. - 272 p.

3. Summer A.M. Flight dynamics and control. / A.M. Summers. - M .: Nauka, 1969 .-- 360 p.

4. Atans, M. Optimal control / M. Atans, P.L. Falb. - M.: Mechanical Engineering, 1968 .-- 764 p.

5. Pontryagin L.S. The mathematical theory of optimal processes / Pontryagin L.S., Boltyansky V.G., Gamkrelidze R.V., Mishchenko E.F. - M .: Nauka, 1983 .-- 393 p.

6. Magnitsky N.A. New methods of chaotic dynamics / N.A. Magnitsky, S.V. Sidorov. - M.: URSS editorial, 2004 .-- 320 p.

Claims (1)

An optimal control device for nonlinear dynamic objects, comprising a serially connected master device, a coefficient calculation unit, a control function calculation unit, a control function implementation unit, a control object, a current coordinate measuring unit, which is included in feedback with the control function calculation unit, characterized in that the control function calculation unit contains a direct and dual system of nonlinear ordinary differential equations of a mathematical model of EKTA, forming a two-point boundary value problem, the results of which solutions are included in the calculation expression control function, optimal in the sense of minimizing the quadratic criterion specified.

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