WO1998026335A1

WO1998026335A1 - Automatic setting method for a proportional-integral-differential controller (pid) regulating a linear servo system

Info

Publication number: WO1998026335A1
Application number: PCT/DE1997/002892
Authority: WO
Inventors: Klaus Weinzierl
Original assignee: Siemens Aktiengesellschaft
Priority date: 1996-12-11
Filing date: 1997-12-11
Publication date: 1998-06-18
Also published as: EP0944865A1

Abstract

The control parameters for a PID controller regulating a linear servo system are based on the logarithmic frequency response. For that purpose, a functional is taken which includes a term representing the module curve slope with regard to the servoing open loop close by the zero crossing, a term representing the value of the phase curve close by the zero crossing and a term representing the quantity of the zero crossing. The functional is optimized in terms of control parameters, and the controller is set based on the obtained parameters.

Description

description

Method for automatically setting a PID controller that controls a linear controlled system using the Bode diagram of the control loop

There are numerous methods for obtaining the parameters of a PID controller. If a model of the controlled system is known, it can be used to construct the frequency characteristics of the Bode diagram. An experienced control engineer can determine suitable controller parameters from this. To support this, root locus curves can also be constructed and a simulation of the closed control loop can be carried out. In practice, simplifications are often carried out on the route model, e.g. B. performed an approximation of an inert system by a simpler low-order model. For these cases, suitable controller parameters can already be determined using analytical formulas, e.g. B. after the optimum amount. Simple formulas are often sufficient, e.g. B. according to Ziegler and Nichols, for controller setting. An overview of these well-known

Methods and concepts are conveyed e.g. B. the literature reference [1].

Despite this multitude of methods, there are hardly any concepts for the design of PID controllers that apply to general

Use controlled systems and have them implemented on a digital computer for fully automatic controller calculation. The simple rules of thumb are sometimes imprecise, an approximation using low-order models leads to different controllers depending on the model used and the

Deciding on a specific model is difficult to make fully automatically. The controller design based on the Bode diagrams are only imprecisely specified by some expert rules, which also applies to the controller design using root locus curves.

Approaches for controller calculation, in which a functional defined in the time domain is minimized, are suitable for a fully automatic implementation using the digital computer for general linear controlled systems. The above-mentioned functional evaluates the deviation between the reference variable and the controlled variable over time in the case of a step-shaped reference variable, see, for example, B. [2,3]. This method is rarely used in practice because it disregards practical boundary conditions such as robustness and well-damped, quiet control loop behavior. For example, it can happen that a controller of all things does

Functionally minimized, which initially leads the controlled variable quickly against the setpoint without overshoot, then drops again significantly and only then settles. A slower, better damped controller setting would be preferable here. The increased computing effort caused by the minimization of a function, on the other hand, is no longer a decisive obstacle for powerful digital computers.

It is known from [4] to determine control parameters in such a way that

Values of the Laplace-transformed transfer function of the controlled system can be determined from several measured values XJ_, OJ_ for several values Sj = dj + jDdj (D = damping). For various, randomly selected parameter sets of the controller, the values SJ are determined, for which the transfer function of the closed control loop G (Sj) R (Sj, Pj _) = - 1, until Sj with the smallest real part dj are determined.

The problem on which the invention is based is to specify a method which is based on the minimization of a

Functionality is based, but is formulated in the Bode diagram. This problem is solved according to the features of claim 1.

According to the method, an algorithm is developed which can be used for controller calculation for general linear controlled systems. A controller set in this way fulfills the requirements better than a controller which has been calculated using the known method on the basis of a function in the time domain.

As further developments of the invention, two functionalities are specified which take into account that the amplitude curve (magnitude characteristic) of the Bode diagram is as steep as possible in the area of the zero crossing of the magnitude characteristic, the phase characteristic in the vicinity of the zero crossing of the

If possible, the absolute value curve does not run below approx. -110 degrees and the zero crossing occurs even at the highest possible frequency. The result is that with the help of these functionalities the controller parameters can be determined, with which it is achieved that the control loop has only a slight overshoot, has great stability and settles quickly. Both functionals evaluate the frequency response of the Bode diagram in the vicinity of the zero crossing and contain the zero crossing itself. By minimizing one of these functions with regard to the controller parameters, the controller can be optimally adjusted.

Other developments of the invention result from the dependent claims.

The invention is further explained on the basis of exemplary embodiments which are shown in the figures.

Show it

1 is a block diagram of a control loop,

2 is an illustration of the Bode diagram, the route jump response and the leadership jump response; Fig. 3 of Fig. 2 corresponding diagrams that serve to explain the functional.

1 shows a basic circuit diagram of a control circuit which is generally known. It shows a controlled system RS that is to be controlled with the aid of a controller RG. The controller RG generates a manipulated variable Y, which is fed to the controlled system RS together with disturbance variables Z. The controlled system RS emits the controlled variable X, which is fed to the controller with a reference variable W. The manipulated variable Y is generated from the difference between the controlled variable X and the reference variable W. For this it is necessary that the controller RG is set according to the properties of the controlled system RS with controller parameters RP. These controller parameters RP are with

Determine controller design according to the controlled system RS and determine the behavior of the control loop.

The Bode diagram of the controlled system RS is used to design the controller using the frequency characteristic curve method. It is assumed that the controlled system RS is linear, time-invariant and stable. The transfer behavior of the controlled system RS can be described by a transfer function H _s (s), which represents a complex-valued function of the complex variables s. The frequency response is obtained for s = jω, ω> 0. If one plots the amount logarithmically, ie log (| H _S (jω) |), and the phase arg (H _s (jω)) over log.ω), the well-known Bode diagram is created. Here log is the tens logarithm and arg is the angle of a complex number in degrees.

The logarithmic frequency will now be used as abbreviations

x: = log (ω) (1) as well as the functions

b _s (x): = log (| Hs (lO ^X ) |) (2) and p _s (x): = arg ^ H _s (jl0 ^x ) | (3)

introduced. The function b _s (x) thus characterizes the absolute frequency response (absolute characteristic) RS__B (Fig.2a) of the controlled system RS in the Bode diagram, the function p _s (x) the associated phase response (phase characteristic) RS_P, Fig.2b.

The PID controller can also be shown in the bottom diagram. The transfer function of the PID controller is due to the relationship

Hr (jω) = κ _p (l + jωT. + L / (jωT) (4)

given. In this case, the controller parameters Kp, T _v and T _n occur in equation (4), where Kp means the proportional gain, T _n the reset time and T _v the derivative action time. This

Controller parameters are usually positive constants. The associated functions of the Bode diagram for the PID controller are b _r (x) and p _r (x) and correspond analogously to the functions of the controlled system, which are given in equation (2) and equation (3). In equations (2) and (3) only the index s has to be replaced by the index r. The corresponding magnitude characteristic RG_B and phase characteristic RG_P are also shown in FIGS. 2a, b.

For the stability of the closed circle is the

Frequency response of the open loop of importance, which one according to

H ₀ (jω): = H _r (jω) H _s (jω) (5) receives. If the functions b ₀ (x) and p ₀ (x) are introduced in the same way as the functions b _r (x) and p _r (x) as well as b _s (x) and p _s (x), we can one determines the two functions b ₀ (x) and p ₀ (x) based on equation (5) and logarithmization by a simple summation, ie

b ₀ = b _r (x) + b _s (x) (δ) and

Po = Pr ( ^x ) + Ps ( ^x ) ⁽ 7 ⁾

In the following it is assumed that b ₀ (x) has exactly one zero x ₀ with b _o (x _o ) = 0. The closed control loop is roughly stable according to the Nyquist criterion if the associated phase function at point x _{0 is} greater than -180 °, ie p _o (x _o )> - 180 °. The control loop is unstable if p _o (x _o ) <- 180 °. Details of these requirements are generally known and can, for. B. [1] can be removed.

As a rule of thumb, it is known that a well steamed

Behavior of the closed control loop occurs when p _o (x _o )> - 110 °. Furthermore, the closed control loop swings in faster, the larger x ₀ . These rules of thumb apply especially if the amount function b ₀ (x ₀ ) falls sufficiently quickly in an environment of x ₀ .

The validity of these rules of thumb can e.g. B. Fig. 2 are removed. 2 shows the magnitude characteristics of the Bode diagram as FIG. 2a, FIG. 2b the phase characteristics, FIG. 2c the step response of the controlled system and FIG. 2d the lead step response. The result from Fig. 2a

Amount characteristic RG_B for the controller RG, the amount characteristic RS_B for the controlled system RS and the amount characteristic RK_B for the open control loop. Correspondingly, the phase characteristic RG_P for the controller RG, the phase characteristic RS_P for the controlled system RS and the result from FIG. 2b

Phase characteristic curve RK_P for the open control loop. In Figs. 2c and 2d the step response is SA and the Step response SP of the control loop with the transfer function

H (jω) = ^H ° ^(j _, (8) l + H ₀ ( _D ω)

shown. Those drawn in the Bode diagram

Frequency characteristics Fig. 2a, 2b correspond to the introduced

Amount and phase functions. One can see that the curves

RK_P, RK_B of the open control loop RK in the amount of FIG. 2a or phase FIG. 2b each form the sum of the characteristic curves RG_B or RG_P of the controller and RS_B or RS_P of the controlled system. The zero crossing x _{0 of} the magnitude characteristic RK_B of the open controlled system is characterized by its intersection with the horizontal line 10 ^ ^(χ) = 10 ° = 1. At this point, the distance of the curve RK_P from the -180 ° mark, referred to below as the phase reserve PRE, is significant. If this is sufficiently large, a well damped settling behavior of the closed circuit can be expected. In Fig. 2, the intersection is relatively far to the right, so that a rapid settling of the closed control loop can be expected. This is also the case, as shown in FIG. 2d, but only up to a value of approximately 0.8. Setpoint 1 is only creeping. In the Bode diagram, this is due to the small distance between the curve RK_B and the value 10 ⁽ - ^> = 1 to the left of the intersection x ₀ .

The example shows how the controller setting works in principle with the help of the Bode diagram. It is noteworthy that the rules of thumb mentioned on the course of the frequency characteristics only in an environment of the

Point x ₀ based. This is a known fact when designing a controller using the frequency characteristic method and simplifies the design considerably. When optimizing the controller, the following goals are now pursued: by adding a suitable magnitude characteristic RG_B of the controller to the magnitude characteristic RS_B of the route, the point x _{0 is} tried so far to shift to the right as possible. Variations are available for this by selecting the parameters of the PID controller. It is required as a secondary condition that this sum amount characteristic RK_B falls sufficiently steep in an environment of the point x ₀ and the

Summer phase characteristic RK_P has a value> -110 ° in an environment of point x ₀ . The invention now consists in formulating these known facts in the sense of an optimization task and performing the numerical optimization fully automatically on a digital computer. For this purpose, a corresponding functional is introduced with which the value x ₀ and the course of the frequency response in its environment are evaluated.

The effect of the function is illustrated in FIG. 3. There are also the magnitude characteristics Fig. 3a, the phase characteristics Fig. 3b, the

Range jump response Fig. 3c and the leadership jump response Fig. 3d shown. The functional is formulated in such a way that any course of the magnitude curve b ₀ (x) which lies within the hatched area STRAF in FIG. 3a is punished. The same also applies to the phase characteristic curve Fig. 3b. The punishment is all the greater, the deeper the curves mentioned penetrate into the penalty area STRAF and the closer the injured part of the penalty area is to the point of passage x ₀ = log (ω ₀ ). On the other hand, a high value of x ₀ , ie a high crossover frequency ω _{0, is} rewarded in the Bode diagram.

The ramp function is used for the formal description of the function

and the difference Δx: = xx _{0 (10} )

introduced. With gτ_ (Δx) and g2 (Δx), functions are designated that disappear sufficiently quickly for Δx— »± ∞, rise or fall strictly monotonously for Δx <0 and Δx> 0, and are symmetrical with respect to their origin. Such a function is e.g. B. the Gauss function. It serves as an evaluation function for the area around point x ₀ .

Taking all of these points into account leads e.g. B. the following functional

dx - x.

(11)

The term squared on the left in the integral characterizes the contribution to the penalty if the amplitude curve b ₀ penetrates into the penalty area STRAF, which in this special case is bordered by the straight line -Δx.

It should be emphasized that the ramp function r explicitly suppresses a reward or punishment if the amplitude specification is exceeded. With the function gη_ (Δx) the mentioned penalty contribution depends on the distance | Δx | weighted and the weaker the greater the distance of x ₀ .

The squared term on the right in equation 11 describes the contribution to the penalty that occurs when the phase curve penetrates the penalty area, which is given by -110

Degree is edged up. Here too, over-performance of the phase specification is not rewarded or punished. The contribution is weighted with the function g2 (Δx) and therefore becomes weaker the greater the distance from x ₀ . Through the integral, the individual contributions are combined to form a sentence. The value x _{0 is} subtracted from this penalty term as a reward for a quick controller setting. By minimizing the mentioned function with regard to the control parameters Kp, T _n and T _v , optimum setting values for the PID controller are created with regard to the function. The result is a generally well damped, robust, yet fast controller setting with very little overshoot.

The properties of the optimal controller setting with regard to the function can still be influenced by varying the penalty areas. For example, if you allow a smaller phase reserve, i.e. H. If you move the penalty area for the phase curve down a bit, you will generally get a faster controller setting with greater overshoot. The result can also be influenced by selecting other weight functions gη_ and g2.

Another function results from equation 12, which can be used to determine the controller parameters. Here the derivative b ' ₀ (x): = db ₀ (x) / dx is used instead of the amplitude curve b ₀ (x). This avoids excessive curvature of the amplitude curve and improves the evaluation in the immediate vicinity of x ₀ . In addition, the optimization properties for critical controlled systems are improved by introducing a separate weighting of the controller parameters by means of a function. The functional used is therefore

G: = (r (bό (x) + α)) gi (Δx) + (r (-90ßα ° - _Po (x))) g ₂ (Δx) fixed

(12) Here, α and ß denote real numbers that should be selected between 1 and 1.2 for low overshoot. With increasing ßα the control speed increases, but so does the tendency to oscillate in the closed circuit. Values with ßα> 2 generally lead to an unstable control loop.

The correction function f (Kp, T _n , T _v ) includes a small reward for larger values of Kp, 1 / T _n and T _v and the quotient T _n / T _v . The functional is again minimized in a known manner with regard to the control parameters Kp, T _n and T _v . This results in the controller parameters, which are used as optimal setting values for the PID controller.

bibliography

[1] O. Föllinger, Regelstechnik, 5th edition, Hüthig Verlag Heidelberg, 1985, sections 7.4 and 7.5

A. Weinmann, regulations Volume 1, 2nd edition, Springer Verlag, 1988

[3] Dr. G. Schneider, control engineering, 32nd year, 1984, volume 9, a simple method for the design of control systems with limits, TU Graz

[4] DE 37 26 018 AI

Claims

claims

1. A method for automatically setting a PID controller that controls a linear controlled system (RS) using the Bode diagram of the control loop with the following steps:

The controller parameters (Kp, T _n , T _v ) of the controller (RG) are obtained using a functional developed from the Bode diagram for the controlled system (RS) and controller (RG).

• The functional points to a) a term that indicates the steepness of the amount characteristic

(RK_B) of the Bode diagram of the open control loop in a zero crossing environment, b) a term that evaluates the value of the phase characteristic (RK_P) of the open control loop in a zero crossing environment, c) a term that evaluates the position of the zero crossing self-rated,

• The functional is minimized with regard to the controller parameters, • The controller is set with the controller parameters obtained.

2. The method according to claim 1, in which as a functional

+ 00

= J 7Ax (b ° ^t _Λ *) ₊ Λx) ff _g _ (Ax) + (r (-110- - P ₀ (x))) ² _g _ (Λ _X ) dx - x ,.

-00 Δx

is used, where: Δx: = xx ₀ . r = ramp function; b ₀ = characteristic curve of the open control loop; gι_ and g2 an evaluation function; p ₀ = phase characteristic of the open control loop.

3. The method according to claim 1, in which as a functional (r (bό (x) + α)) _gι (Δx) + (r (-90ßα ° - _Po (x))) g ₂ (Δx) fcb

x ₀ + f (κ _p , T ^, ¹ ^)

is used, where Δx: = X-Xo; r = ramp function; b ' ₀ = derivation of the amount characteristic of the open control loop; gτ_ and g2 an evaluation function; p ₀ = phase characteristic of the open control loop α and ß = real number between 1 and 1.2.