Method of harmonic balance

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The method of harmonic balance - used in control engineering - consists in approximating the continuous oscillations of a non-linear, feedback dynamic system with a harmonic oscillation . The non-linear subsystem is represented by a description function . With the method, the parameters of the vibration state and the stability limit can be calculated. NM Krylow and NN Bogoljubow used the term harmonic balance as early as 1937 for this method they developed.

A nonlinear dynamic system can be broken down into a static nonlinear system and a dynamic linear system according to the Hammerstein model . If the output of the linear system is negatively fed back to the system input of the non-linear system and thus switched to a control loop , the entire system can oscillate.

The method of harmonic linearization is based on an oscillating non-linear control loop, the output signal of which, due to the low-pass behavior of the controlled system, executes an approximate harmonic oscillation, which is fed back negatively into the system input. Only under these conditions may the non-linear static system be defined as a linear transfer element with the description function, which only depends on the sinusoidal harmonic input oscillation with the amplitude and not on the complex frequency .

With the equation of the harmonic balance , the relationships of the descriptive function of the static non-linear system and the dynamic linear system are put into a relationship. From this, the two critical system variables of the harmonically oscillating control loop - the input amplitude and the critical frequency at the stability limit - can be calculated or graphically determined using the two-locus method.

The application of harmonic balance to test non-linear control loops with the clear two-locus method to determine when permanent vibrations occur and how these can be avoided does not require any special mathematical knowledge. The required description function of the non-linear static system for the construction of the locus is shown in many variants in the technical literature on control engineering. This applies in particular to the various forms of characteristic curve controllers.

Basics of the linear and non-linear oscillating system

Strictly speaking, almost all control loops are non-linear systems. The most common static non-linearity is the saturation property of the controller's manipulated variable. In the case of a P controller , for example, despite the increased control deviation, no increase in the signal of the manipulated variable is possible because there is a signal limitation. In this simple case of signal limitation, the controlled variable is not reached as quickly as without a manipulated variable limitation in the event of a jump in the reference variable or a disturbance variable that suddenly occurs.

If the P gain of a stable control loop is increased further, then - provided the controlled system is a linear system higher than the 2nd order - the control loop becomes unstable at the stability limit f (P gain). The controlled variable will then oscillate with constant or increasing amplitude. It is not a matter of indifference whether there is a manipulated variable limitation or not.

"Continuous oscillations" (also limit oscillations and limit cycles) are periodic, sinusoidal time processes in a steady state with constant amplitude. They differ from rising and falling vibrations.

Rest positions and stability

Model of the system rest positions: State 1 is stable with respect to small disturbances and changes to state 3 in the case of large disturbances. State 2 is unstable.

A rest position of a dynamic system in the vibration-free state is asymptotically stable if the output variable returns to the rest position after a signal disturbance . A stable system tends to maintain its current state, even when there is external interference. Each stability area has a rest position and a catchment area for the rest position. If a non-linear control loop only has one rest position, one can assume that its output variable is globally asymptotically stable, provided that no continuous oscillations occur.

In contrast to linear systems, non-linear dynamic systems are dependent on initial conditions and the input signal of the static non-linearity. Depending on the function of the static non-linearity, stable and unstable behavior can occur. When using so-called characteristic curve controllers in the control loop, non-linear systems can cause desired oscillations. Such stationary continuous vibrations are also referred to as limit vibrations. Limit oscillations can be stable, unstable and semi-stable.

In contrast to linear systems, non-linear control systems can have several positions of rest. A pendulum with its masses of potential and kinetic energies is described by a differential equation of the 2nd order, in which the angle occurs  as a 2nd derivative and nonlinearly. If the angle of the fastening point is rated at 90 °, the pendulum has a rest position of 270 ° and a catchment area of ​​the rest position> 90 ° and <90 ° and an unstable rest position of 90 ° in the coordinate system .

A non-linear dynamic system with the output variable has the following behavior when deflected from an unstable rest position:

  • goes into another position of rest,
  • strives towards
  • makes a continuous oscillation.

The harmonic balance can be used to clarify whether limit oscillations can occur and whether they are stable or unstable and what frequency and (input) amplitude they have.

Stability limit of the linear dynamic system

A dynamic system behaves linearly if the effects of two linearly superimposed input signals are linearly superimposed in the same way at the output of the system. The system behaves linearly if it fulfills the superposition principle (see also: superposition principle in physics ) and the reinforcement principle.

A transmission system is internally stable if all (partial) transmission functions only have poles in the left half-plane. A transmission system is said to be externally stable if any limited input signal to the system also produces a limited output signal. (See BIBO stability )

Block diagram of a simple standard control loop

According to Nyquist's simplified stability criterion, the stability limit of a linear control loop is reached when the locus of the frequency response (see also PT2 element ) of the open control loop precisely intersects the critical point −1 of the abscissa of the real part. For values ​​of the intersection point , increasing oscillations of the closed control loop arise.

The following applies to the stability of the limit oscillation of the critical frequency of the open control loop:

To determine the locus, the complex frequency response is broken down into real and imaginary parts and entered for different values ​​of the frequency in the Gaussian plane .

  • From a given transfer function or a frequency response in product representation, the denominator of the frequency response is multiplied as a polynomial and sorted into real part and imaginary part .
  • So that the imaginary part in the denominator of the frequency response disappears, the real and imaginary parts of the denominator must be multiplied by themselves but with the opposite sign with the frequency response equation.
Example:
[Note: ]
  • The numerator is multiplied by the real and imaginary parts of the denominator so that the equation is unchanged from the original statement.
  • The numerator is sorted according to the real part and the imaginary part.
Locus of the frequency response

Construction of a locus of the frequency response

Given: 3rd order transfer function with global I-behavior and the gain K.

If the real part of the denominator is designated with the auxiliary variable and the imaginary part of the denominator with the auxiliary variable , the frequency response for the real part and the imaginary part can be calculated as follows:

First, the numerical values ​​of the auxiliary variables and are calculated as a function of to in steps of values ​​and inserted into the frequency response equation below . The real and imaginary components of the frequency response calculated in this way are entered in the diagram of the Gaussian plane of numbers as the locus of the frequency response.

Locus of the frequency response

Step response before and at the stability limit.

Numerical example of the stability limit of a linear control loop:

Given: Open loop with a P-controller: and a controlled system 3rd order :

Wanted: Behavior of the step response for the reference variable of the closed control loop.

After closing the control loop with this unfavorable controlled system with three equal time constants, the stability limit for a constant amplitude of the controlled variable is already reached during amplification . The locus runs through the 4th, 3rd and 2nd quadrants starting at with the value and intersects the abscissa of the real part at exactly .

The shape of the locus does not change as long as the three time constants have any value but are the same among each other . The size of the time constant is important when the timing of the control loop z. B. should be displayed as a step response of the reference variable. Three equal time constants in a controlled system with three delay elements represent the most unfavorable time behavior with regard to the stability of the control loop.

If the gain is increased, then progressively larger amplitudes occur with increasing time. The locus of the frequency response intersects the abscissa at values ​​<−1. If the gain is reduced, the amplitudes of the controlled variable take an aperiodic damped course and the controlled variable assumes a constant value after the settling time. The locus intersects the abscissa at values> −1.

If the controlled system only consists of a 2nd order delay system with a P controller of any gain, theoretically continuous oscillations cannot occur because the locus of the frequency response only runs through the 4th and 3rd quadrant of the Gaussian numerical level and does not hit the critical point can.

Note: For the calculation of the oscillation state of a non-linear dynamic system according to the two-locus method, knowledge of the construction of the locus of the frequency response of the linear system is required!

Behavior of the nonlinear dynamic system

General representation of the non-linear control loop

Dynamic systems are described by differential equations (DGL). If the ODE or the function sought or its derivatives contain a power or products of the function sought or trigonometric functions, logarithms, etc. in the arguments, then it is a non-linear ODE and the system behaves non-linearly. Nonlinear differential equations can only be solved analytically in very rare exceptional cases. However, they can be calculated relatively easily using numerical discrete-time methods or commercial computer programs ( Simulink ).

In the case of a non-linear static system , the transfer characteristic can be viewed as a gain that changes within wide limits. The system usually responds to a sinusoidal input signal with a distortion of the oscillation of the input signal of the same frequency and a phase shift. Only a symmetrical two-point element ( two-point controller ) without hysteresis, without dead zone and signal limitation responds to a sinusoidal input oscillation with a square wave of the selected square wave amplitude of the same frequency without a phase shift. ( is the characteristic curve auxiliary point of the non-linear characteristic curve for a sinusoidal input oscillation )

The simple symmetrical two-point element with the rectangular amplitude height is defined as a sign function with the output variable:

The two-point element has two rest positions. It generates two manipulated variables depending on the input signal . The state for is only a stationary value if a dead zone ( is the characteristic curve auxiliary point) is set up for the two-point element . This would create a three-point element from the two-point element.

Static non-linear systems with a quadratic or exponential characteristic are special cases. A sinusoidally excited system with a quadratic characteristic curve responds with a sinusoidal oscillation of twice the frequency and a constant component (shifted operating point); a system with an exponential curve responds with a sine-like pulse with a large proportion of harmonics.

If a non-linear dynamic system of the 2nd and higher order is excited by a sinusoidal input signal , it usually responds with an approximately sinusoidal oscillation of the same frequency but different amplitude and a phase shift. If the static non-linear system is a switching controller, an approximately sinusoidal output signal is achieved in a control loop despite steep pulse edges of the square-wave controller output signal with a sufficiently high frequency from and due to the low-pass behavior of the linear dynamic system .

The "harmonic linearization" is justified by this behavior: the square wave with its large harmonic component as an input variable is only taken into account in its fundamental wave of the corresponding amplitude. The nonlinear static system can thus be described as a function of the amplitude of the sinusoidal oscillation of the input signal .

Nonlinear systems are often unique, but behavior patterns can be defined from the nonlinear static system which allow a system description function with given parameters. This includes elements with limitation, dead zone, hysteresis, progressive characteristic curve and the so-called characteristic curve controllers such as two-point and multi-point controllers, which are presented in numerous variants as follows:

  • Two-point element (two-point controller) symmetrical and asymmetrical with and without hysteresis
  • Three-point element (three-point controller) symmetrical and asymmetrical with and without hysteresis
  • Multipoint element
  • Element with progressive characteristic symmetrical and asymmetrical
  • Element with degressive characteristic, symmetrical and asymmetrical
  • Elements with limitation symmetrical and asymmetrical
  • Elements with dead zone symmetrical and asymmetrical
  • Elements with dead zone and limitation
  • Elements with prestress
  • Rectifier element
  • Amount element

According to the specialist literature, up to 40 different description functions of nonlinear static systems can be taken, which describe the nonlinear system behavior only for a periodic sinusoidal input signal with the amplitude and the output signal only with the fundamental wave.

Harmonic linearization

Definition of the signal sizes, parameters and benchmarks of the characteristic curve structures

In the well-known technical literature on control engineering and in the current lecture manuscripts at German universities, there is hardly any identical mathematical derivation from Fourier analysis, nor identical system and signal names for harmonic linearization. The cause are the different signal designations e.g. B. with Xe, Xa, E, U, X, Y, the equation structure of the Fourier analysis and the names of the geometric corner points of the non-linear characteristics.

The equation of the Fourier analysis and its simplification differ in the different use of the harmonic sine or cosine oscillation with and without constant components, different integration limits and different calculation of the coefficients. Also, the results of the description of functions under different names (often , , , ) as a function of the amplitude of the input of the nonlinear system with numerous characteristic Controller functions inconsistent because the geometric basic parameters of the characteristic description with the auxiliary variables (usually ) in different Way to be used.

Despite the different mathematical paths that lead to the harmonic balance equation, the result is identical everywhere. Understanding the locus of the frequency response of the linear system is required, as is an understanding of the locus of the descriptive function of the nonlinear static system. The numerous description functions of the non-linear static systems with the associated sketches of the characteristic curves and the locus curves can be taken from the specialist literature. The different geometrical key values ​​with the characteristic curve sketch ( ) of the description function can easily be assigned to the various sources of the specialist literature.

The following is based on the internationally used system variables of the control loop: reference variable , controller input variable , controller output variable (non-linearity) and linear controlled system output . The following mathematical derivations are based on the technical literature.

Method of harmonic linearization

According to the Hammerstein model, the method of harmonic linearization is based on the assumption that the harmonics generated by the non-linear system will be ineffective (filtered) by the subsequent linear system with low-pass behavior.

From the point of view of control  engineering, the Hammerstein model is often used for a non-linear dynamic system - in contrast to the Wiener model - because in many cases the manipulated variable belonging to the controller is non-linear.

The behavior of the controller as a non-linear static system can be viewed as a gain that changes within wide limits, which in the case of a sinusoidal input signal causes a signal distortion and phase shift of the output signal . The dependency on is often determined by known parameters or by the magnitude of the amplitude of the sinusoidal input signal , which is why the large-signal behavior of the system has to be examined, "stability on a large scale ". Some characteristic curve controllers only respond when the input signal reaches a minimum size.

For clear non-linear characteristics such as the two-point element, i.e. H. for each static input signal there is only one static output signal , the description function is real, i.e. H. the phase shift .

Representation of the non-linear control loop in the state of harmonic balance

Simplification of the non-linear description function in the state of continuous oscillation

The input variable of the static non-linear system should be a sinusoidal oscillation without a constant component:

The amplitude of the input sinusoidal oscillation is often referred to in the specialist literature as .

The distorted output oscillation can be written as a Fourier series, as a sum of an equivalent value and the sinusoidal signals of the fundamental frequency and the harmonic harmonics as multiples of the fundamental frequency.

.

The output variable can contain a constant component depending on the behavior of the non-linear system . But only the vibration behavior is of interest.

If the downstream linear system attenuates the distorted sine-like input oscillation as a low-pass filter strong enough, the high frequencies are suppressed very strongly compared to the lower ones. For this consideration, the components of the harmonics are assumed to be negligible.

The description function of the static non-linear system results from the ratio of the fundamental waves of the output signal to the input signal . It only takes into account the fundamental component of the output signal without the direct component .

Input signal:

Output signal:

With the complex representation of the two signals, the description function results from the Euler formula:

Descriptive function of the non-linear system

The input amplitude of the sinusoidal oscillation of the non-linear system is often also referred to as.

and are the coefficients of the first fundamental oscillations of the non-linear system.

For the numerous forms of so-called characteristic curve controllers, the Fourier coefficients and the non-linear description functions can be calculated using the geometry of the characteristic curves with these coefficients (often referred to as) .

The description functions depend on the function of the non-linear characteristic element.

  • The output of a symmetrical two-point element oscillates as a function of the input oscillation synchronously with the zero crossings without a phase shift.
  • Asymmetrical multipoint elements (amplitude ) generate a constant component.
  • For characteristic elements with hysteresis functions, the description functions are complex.

The description functions can be called an “equivalent frequency response” of a non-linear system. It depends on the amplitude of the input oscillation  and the frequency of the continuous oscillation and is also defined. In contrast to the linear frequency response with the dependence on , the description function is essentially dependent on the amplitude as an independent variable. In the case of purely static non-linearity, there is no dependence on the frequency. In the following, the description functions are considered to be frequency-independent for static non-linearities .

The list and extensive derivations of the description functions of the numerous characteristic curve elements can be found in the technical literature on control technology. The respective description functions of the non-linear static systems are all derived as algebraic functions, only the relevant geometric corner points with the letters differ, but can easily be assigned using the attached characteristic curve sketches.

List of known description functions of characteristic curve elements

The following description functions only apply to applications in the state of harmonic balance.

Naming
non-linear system
Description function
Non-linear characteristic and
inverted locus
Two-point controller

Zweipunktkennlinie.png
Two-point controller with hysteresis

Two-point characteristic with hysteresis.png
Element with limitation (saturation)


Limiting characteristic.png
Element with dead zone


Dead zone.png
Three-point controller
(= two-point controller with dead zone)

Dreipunktkennlinie.png

Harmonic balance equation

For the method of harmonic balance, the closed control loop at the stability limit is considered for the reference variable.

The output variable of the linear system is:

The non-linearity behaves like a linear transfer element in the oscillation equilibrium.

The characteristic equation of the non-linear control loop results from the 3 equations, which is referred to as the harmonic balance equation :

The most common notation of the harmonic balance equation is:

If a continuous oscillation of the non-linear control loop exists, then the equation of the harmonic balance is the characteristic equation of the non-linear control. The solution of the equation of the harmonic balance can be done numerically or graphically according to the two-locus method.

Formula-based solution of the equation of harmonic balance

In the case of complex non-linear description functions or linear systems of higher order, the formula-based solution can be very difficult. In this case, the clear graphical solution of the two-locus method is chosen.

The unknown quantities sought are the amplitude of the non-linearity and the frequency with which the non-linear dynamic system oscillates.

The equation of the harmonic balance is transformed in order to first be able to calculate:

If you write the equation in real part and imaginary part:

This results in two real equations. Are there unambiguous non-linear characteristics, i. H. for each input variable there is a unique output variable as with the two-point and three-point elements, so the description function with the imaginary part becomes zero.

The solution of the system of equations is simplified to the solution of two separate equations. First the poles are determined from the linear system and inserted into the right hand side of the equation below. So can be calculated.

Two-locus method of harmonic balance

The graphical method is very clear. The amplitude and the frequency of the continuous oscillation of the non-linear control loop are obtained when there is an intersection of the two locus curves of the frequency response and the negative inverted description function. It can be determined whether there are continuous oscillations in a control loop and whether several continuous oscillations are possible.

Representation of the two-locus method of harmonic balance

The negative, inverted locus of the description function shown in the equation of the harmonic balance can be entered in the same diagram of the locus of the frequency response as , with the locus parameters of . If the system oscillates, there is an intersection of the two loci. The amplitude and frequency can be read from this.

Depending on the type of non-linear system, the position of the negative inverse locus on the real axis of the coordinate system is a straight line and has no imaginary components. In the case of non-linear systems with a hysteresis, it is a straight line offset in the negative imaginary area. In the case of non-linear systems with three-point controllers and hysteresis, it is a characteristic field.

Depending on the type of locus of the frequency response and the locus of the description function, there can be several continuous oscillations because there can be several intersections. With the three-point controller, the locus of the description function is folded on the real axis (the two branches of the locus overlap), i.e. H. For a given frequency , oscillations for two different amplitudes can be possible at the point of intersection .

On the other hand, the locus of the frequency response for linear systems of larger order can wrap around the point Re = 0 and Im = 0. If the linear system contains a dead time, the locus of the frequency response runs in a spiral around the point Re = 0 and Im = 0, so that there can be many intersections and thus several continuous oscillations of different amplitudes and different frequencies .

If a control loop is, for example, a two-position controller and a linear system of the first or second order, according to the two-locus method, continuous oscillations cannot occur because the locus of the frequency response cannot come into the second quadrant. Unfortunately, from numerical calculations or from experience with two-position controllers, we know that the control loop responds with a continuous oscillation. The reason for this behavior is explained by the fact that the condition of the harmonic oscillation of the output variable of the control loop with low-pass behavior is not met. In this case, the output variable is not a harmonic oscillation.

In contrast, the locus of a two-point controller with hysteresis can very well meet the locus of the linear system of the 2nd order due to the position of the [-1 / N (A)] locus in the negative imaginary area of ​​the 3rd quadrant .

Step response of the non-linear system with three-point controller as continuous oscillation

Practical example of harmonic balance

Controlled system with global I behavior:

Symmetrical three-point controller with the description function: ± U max = ± 2; ± dead zone = ± 0.5; Reference variable w (t) = 1
Geometric corner points: U max = d; Dead zone = c

Result:

The values ​​of the locus of the negative inverted description function are calculated from the associated equation of the three-point controller and entered on the real axis Re of the locus of the frequency response. The two branches of the locus start at , end at and reverse direction at . They intersect with with and with the locus of the frequency response in real part of -0.8; .

The continuous oscillation with is unstable. The amplitude is in the border area of ​​the dead zone , in which no continuous oscillation is possible. The description function with the amplitude results in a stable continuous oscillation. In general: the stable continuous oscillation adjusts to the larger amplitude.

The diagram with the time behavior of the system variables calculated by means of numerical time-discrete methods agrees completely with the results of the harmonic balance in terms of amplitude and frequency after settling after approx. 16 s.

Stability of the limit oscillations

The locus of the frequency response of a linear system as an open control loop shows continuous oscillations if, according to the Nyquist method, the locus hits the point and in the complex plane . This is not a stable continuous oscillation because even the slightest change in the poles does not affect the critical point . The result is with a value of , a damped oscillation occurs or with a value of an increasingly increasing oscillation arises.

With the two-locus method, continuous oscillations of a non-linear dynamic feedback system can be determined if the locus of the description function of the nonlinear system and the locus of the frequency response of the linear system meet in the complex plane. If the two loci do not meet, it is a vibration-free control loop.

It must be taken into account that the linear decelerating system must be of at least the second order, otherwise the locus of the frequency response does not pass through the second quadrant and can not hit a locus lying on the real axis .

In the case of continuous oscillation, limit conditions for the description function occur. The boundary conditions of the intersection points of the locus of the description function are considered for three cases in which the amplitude is increased or decreased by a small value due to transient disturbances and then returns to the starting position. The question arises as to what happens when there is continuous oscillation and a small, temporary change in amplitude has a positive or negative effect on the equilibrium state of the continuous oscillation. The harmonic balance equation becomes an inequality for these cases.

The following rule applies to most practical applications and in particular to intersections with several ω values ​​for the smallest value of ω.

The point of intersection of the two locus curves represents a stable limit oscillation if the amount of the descriptive function decreases with increasing amplitude . An unstable limit oscillation results if the magnitude of increases with the description function.

Example of a control loop with a locus curve of a three-point controller and a PT3 controlled system:

The two branches of the locus of the real description function are negatively inverted one above the other on the real axis.

Analysis of the boundary vibrations during transient deviations of the amplitude in order for three cases: taken touched, not taken.

A distinction is made between the following three cases in which the two loci meet, touch or do not meet:

  • Case 1): The two loci meet
The negatively inverted locus of the description function is hit by the locus of the frequency response . Theoretically there are two points of intersection and thus two continuous oscillations of the same frequency with different amplitudes. This applies in the event that the short arm of the description function up to (reversal point) is hit.
In the following, the branch of the descriptive function from the reversal point at a distance from the ordinate with increasing size of the amplitude to is considered.
If the amplitude is increased by a small amount as a result of a transient disturbance , the new value lies to the left of the locus of the frequency response and lies outside the locus (not looped around). According to the Nyquist criterion , the closed control loop is stable. The oscillation dies away. After the end of the disturbance, the state of stable continuous oscillation is reached again.
If the amplitude is reduced by a small amount as a result of a transient disturbance , the new value is to the right of the locus of the frequency response . The new value is wrapped around the locus and lies within the range of the locus. According to the Nyquist criterion, the closed control loop is unstable. The oscillation stops, the amplitude increases to its original value. After the disturbances have disappeared, the limit oscillation is stable.
  • Case 2): The two loci touch at the turning point of the two branches
If the points of intersection are only points of contact, then the limit oscillation is referred to as semi-stable.
If the amplitude is increased by a small amount as a result of a transient disturbance, the new value lies to the left of the locus of the frequency response and outside the locus (not looped around ). According to the Nyquist criterion, the closed control loop is stable. The oscillation dies away. After the end of the disturbance, the state of stable continuous oscillation is reached again.
If the amplitude is reduced by a small amount as a result of a transient disturbance , the new value lies to the right of the locus of the frequency response and is wrapped around by the locus . The semi-stable limit oscillation sounds. The control system is unstable.
  • Case 3): The two loci do not meet and do not touch
The closed loop is stable. There are no permanent vibrations.

Avoidance of permanent vibrations

Nonlinear dynamic systems can e.g. B. with the use of a simple two-position controller intentionally cause stable continuous oscillations. The position of the locus of the description function lies on the real axis of the locus of the frequency response at the point and starts at and ends at . Therefore, there is no way of avoiding continuous oscillations with the two-position controller.

Note: A two-position controller already oscillates in connection with a linear first-order delay element, although the two locus curves cannot meet. The harmonic balance is not valid in this case because the prerequisites for harmonic vibrations (sawtooth-like vibrations) are not given. This also applies to a second-order delay element (PT2 element).

With the use of other static non-linear systems such as For example, the three-point controller can be used to implement vibration-free control loops with asymptotic behavior. A controlled system with global I behavior is required.

The question arises as to which measures can be implemented with the help of the two-locus method.

The following measures are in place so that the equation of the harmonic balance becomes an inequality in which the two locus curves cannot meet.

The inequality of harmonic balance could be:

  • Reduce the gain of the linear system
As the simplest method, the locus of the frequency response could be designed by changing the linear system (e.g. reducing the gain) so that it does not intersect the locus of the description function. This would worsen the dynamic properties of the control loop.
  • Enlargement of the distance between the locus of the description function and the ordinate .
The locus of the description function could be changed by increasing the distance between the reversal point and the ordinate of the two arms of the description function . It is possible to reduce the amplitude or to increase the dead zone .
For technical control reasons with regard to the accuracy of the controlled variable , the dead zone must be as small as possible, because the controller does not react to control deviations within the dead zone. The reduction of the amplitude (= controller output variable ) requires the reference variable to be reduced . Therefore, this approach does not lead to any significant result.
  • Partial compensation of the delay elements of the linear system
Change (partial compensation) of the linear controlled system by using a real PD1 element. This method of improving the system speed by compensating sluggish PT1 elements with real PD1 elements is also common in linear control technology. It also leads to an acceptable result with non-linear control loops.

Compensation of sluggish PT1 elements with PD1 elements

By replacing slow delay elements in the linear dynamic system, the system has a faster time response.

Example of a linear controlled system:
Linear real PD1 element (compensating element):
The time constant of the route should be greater than the route. The PD element with time constant of the controller is intended to compensate the delay element with the time constant . The time constant must be smaller . This results in the new transfer function of the linear controlled system in frequency response representation as:
Step response of the non-linear control loop with three-point controller without continuous oscillations, representation of the transient process
With this measure, the course of the frequency response hugs closer to the ordinate when passing from the 3rd to the 2nd quadrant. If the parameters of the description function are chosen so that the two loci do not touch, the control loop is asymptotically stable. The minimization of the dead zone and thus the optimization of the vibration-free control loop with a three-point controller can only be determined with numerical simulation.

Example of a vibration-free control loop with a three-point controller

For a vibration-free control loop with a three-point controller, it is assumed that the linear dynamic system, the controlled system, shows a global I behavior. Typical for such controlled systems are actuators that are set to forward, reverse or idle movements by the three-position controller. The standstill corresponds to the target positioning.

If the parameters of the calculation example of the control loop with continuous vibrations are modified with the three-position controller as follows, a vibration-free control loop is created as shown in the graphic.

  • The two delays of the PT1 elements are reduced by a factor of 10 using PD1 elements,
  • The related size (1 = 100%) of the rectangular vibration amplitude is set to,
  • The related size of the dead zone is set to
  • the maximum related reference variable is set.

The size of the dead zone determines the control deviation at which the controller should respond. In this case, the time behavior (integration constant ) of the controlled system to be designed is based on the dead zone. Only a slow actuator allows a small dead zone and thus precise positioning for vibration-free operation. The smaller the dead zone, the more frequent and shorter the square wave oscillations are during the settling process of the controlled variable .

Conclusion on the application of harmonic balance

  • The harmonic balance shows interesting aspects of the occurrence of permanent oscillations in non-linear control loops and also possibilities of preventing permanent oscillations.
  • There is no calculation method for analyzing the system behavior of non-linear control loops with dead time and hysteresis elements that is nearly as powerful and at the same time as simple as the numerical recursive calculation of a non-linear system according to the Euler system with discrete time and the Calculation sequence .
Whether the static non-linearity is calculated with logical program commands and the linear factors of the linear subsystems with difference equations or the commercial numerical computer programs such as Matlab or Simulink are used, the findings from the harmonic balance can be simulated and confirmed immediately in the time domain.

literature

  • Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Europa-Lehrmittel, Haan-Gruiten 2019, ISBN 978-3-8085-5869-0 .
  • Gerd Schulz: Control engineering 1: linear and non-linear control, computer-aided controller design . 3. Edition. Oldenbourg, 2007, ISBN 3-486-58317-4 .
  • Otto Föllinger: Nonlinear Regulations II: With harmonic balance, Popow and circle criterion, hyperstability, synthesis in the state space . 8th edition. Oldenbourg, Munich / Vienna 1993, ISBN 3-486-22503-0 .
  • Heinz Unbehauen: Control technology II: With state controls, digital and non-linear control systems . 9th edition. Vieweg + Teubner, Wiesbaden 2007, ISBN 978-3-528-83348-0 .

Individual evidence

  1. Holger Lutz, Wolfgang Wendt: Pocket book of control engineering . Chapter: Harmonic linearization with the description function
  2. Holger Lutz, Wolfgang Wendt: Pocket book of control engineering . Chapter: Nonlinear Controls, Table: Nonlinear Static Elements.
  3. Jürgen Adamy: Non-linear systems and regulations . 2018, p. 68 , doi : 10.1007 / 978-3-662-55685-6 ( springer.com [accessed June 21, 2020]).
  4. Otto Föllinger: Nonlinear Regulations II, Harmonic Balance . Chapter: Solving the equation of harmonic balance
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  8. Otto Föllinger: Nonlinear Regulations II . Chapter: Nonlinear Controls, Subchapter: Stability Behavior of Continuous Vibrations.
  9. Otto Föllinger: Nonlinear Regulations II, Harmonic Balance . Chapter: Stabilization of non-linear regulations