Describing function

N.M. Krylov and N.N. Bogolyubov already used in 1937 the term harmonic balance for the procedures developed. It is about to approach the duration of oscillations given by a non-linear back -coupled dynamic system by a harmonic oscillation and thus to make the parameters of the vibrational state and the stability limit computable.

A non-linear dynamic system, N = f ( j?, Non-linearity ) and a dynamic linear system G ( j? ) Are broken down according to the Hammerstein model in a static nonlinear system. The output of the linear system negatively fed back to the system input of the nonlinear system and thus connected to form a loop, the entire system can oscillate.

With the method of the harmonic linearization is assumed to be non-linear oscillating control circuit whose output carries an approximately harmonic by the low pass response of the controlled system, which is negatively fed back to the system input. Only under these conditions the non-linear static system may be defined as a linear transmission member with the describing function N (A), which depends only on the j.omega sinusoidal harmonic oscillation with the input amplitude A and not of the complex frequency.

With the equation of harmonic balance relations of the describing function N (A ) of the static nonlinear system and the dynamic linear system G0 ( j? ) Are put into a relationship. The input amplitude A and the critical frequency ωKRIT at the stability limit - - Hence the two critical system variables of the harmonically oscillating loop can be calculated or determined graphically according to the two - locus method.

The application of harmonic balance for the analysis of nonlinear control loops with the illustrative two- locus method, when sustained oscillations occur and how to avoid permanent oscillations, requires no special mathematical knowledge. The required description function of the nonlinear static system for the construction of the locus is shown in many variants in the literature of control theory. This applies particularly to the various forms of the characteristic curve control.

• 2.1 Specification of the signal variables, parameters and basic parameters of the characteristic structures
• 2.2 method of harmonic linearization 2.2.1 Simplification of non-linear describing function in a state of constant oscillation

• 3.1 Formula Standard solution of the equation of balance Hamonischen
• 3.2 Two - locus method of harmonic balance
• 3.3 Stability of the boundary oscillations

• 4.1 Compensation carrier PT1 elements by PD1- members
• 4.2 Example of a vibration-free control loop with a three-point controller

Basics of linear and nonlinear oscillating system

Strictly speaking, almost all control loops nonlinear systems. The most common static nonlinearity is the saturation property of the control value. Thus, no signal increase of the manipulated variable is at a P controller despite increased deviation possible because a signal limitation is present. For this simple case, the signal limiting the controlled variable is not as fast achieved in a reference variable jump or sudden disturbance of the nominal value of the control variable, as if no command value limitation would be present.

If the P gain of a stable control loop further increased, then - provided the controlled system is a linear system > 2nd order - the control loop at the stability limit f (P gain ) unstable and the control variable will oscillate with constant or increasing amplitude. It does not matter whether a manipulated variable limiting or not.

The term " continuous oscillation " (also limit oscillations and limit cycles) periodic sinusoidal time operations in the steady state with a constant amplitude, which are different from on and decaying vibrations.

Positions of rest and stability

A rest position ( = steady state, steady- state equilibrium ) of a dynamic system in a vibration-free state is asymptotically stable if the output variable y (t ) returns to its rest position after signal failure again. A stable system tends to retain its current state, even if interference acting from outside. For each stability region belongs to a rest position and a catchment area of the tranquility. Has a non-linear control loop only a quiet neighborhood, so one can assume that its output y (t ) is globally asymptotically stable, if not permanent oscillations occur.

Nonlinear dynamical systems are dependent, in contrast to linear systems of initial conditions and the input signal of the static nonlinearity. Depending on the function of the static non-linearity, stable and unstable behavior may occur. Nonlinear systems can cause unwanted oscillations in the control loop with the use of so-called characteristic controllers. Such permanent stationary oscillations are referred to as boundary oscillations. Boundary oscillations can be stable, unstable and semi- stable.

Nonlinear control systems, in contrast to linear systems have multiple resting positions. A pendulum is described with its masses of potential and kinetic energy by a differential equation of 2nd order in which the angle φ as 2nd derivative and occurs not linear. An assessment of the angle of the attachment point of 90 °, then the pendulum has a rest position at 270 °, and a catchment area of ​​the resting position > 90 ° and < 90 ° and an unstable rest position of 90 ° in the coordinate system.

A nonlinear dynamical system with output y (t ) has the following behavior at deflection from an unstable rest position:

• Y ( t) goes in a different resting position on,
• Y (t ) tends towards
• Y (t) makes a continuous oscillation.

With the harmonic balance can be clarified whether boundary 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 is linear if the effects of two linearly superimposed input signals are superimposed linearly in the same way at the output of the system. The system behaves linearly when the superposition principle: satisfied ( See also superposition principle of physics ) and the reinforcement principle.

A transmission system is internally stable if all the (partial) transfer functions only have poles in the left half s- plane. A transfer system is considered as an externally stable if any input signal is limited to the system also causes a limited output signal. (See BIBO stability)

The stability limit of a linear control circuit is achieved in accordance with the Nyquist stability criterion simplified if the locus of the frequency response (see also PT2 element ) of the open loop intersects exactly the critical point of the real part -1 of the abscissa. For values ​​of the intersection point Re < -1 arise aufklingende vibrations of the closed loop.

Thus valid for the stability of the vibration limit of the critical frequency ωkrit open loop: F0 ( jωkrit ) = -1.

To determine the locus of the complex frequency response is divided into real and imaginary parts and registered for different values ​​of the frequency ω in the Gaussian plane.

• Given a transfer function G (s) or a frequency response G ( j? ) In product representation of the denominator of the frequency response is multiplied out as a polynomial and sorted into a real part and an imaginary part.
• Thus the imaginary part vanishes in the denominator of the frequency response, the real and imaginary parts of the denominator with itself but needs to be multiplied with the opposite sign with the frequency response equation.

• The counter is multiplied by the real and imaginary part of the denominator so that the equation of the original message is unchanged.
• The counter is arranged according to the real part and imaginary part.

Construction of a locus of the frequency response

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

Is the real part of the denominator of the auxiliary variable a = - (T1 T2) * ω ² and the imaginary part of the denominator with the auxiliary variable b = ω -T1 * T2 * ω ³ denotes the frequency response G ( j? ) For the real part and the imaginary part can are calculated as follows:

First, the numerical values ​​of the auxiliary variables a and b as a function of ω = 0 to ω = ∞ are computed in steps of ω - values ​​and used in the equation below the frequency response G ( j? ). The thus calculated real and imaginary parts of the frequency response are plotted in the diagram of the Gaussian plane as a locus of the frequency response.

Locus of the frequency response G ( j? )

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

Given: Open loop with a P controller: K = 8 and a controlled system of 3rd order, T = 1:

Wanted: Behavior of the step response for the reference variable w ( t) = 1 the closed loop system.

After the closing of the control loop with this unfavorable control path 3 at the same time constants, the stability limit is reached for a constant amplitude of the controlled variable at this gain K = 8. The locus passes through the f ( ω = 2 * π * f) starting at f ( ω = 0) with the value of Re = 8; ω = 0; Im = 0 the 4th, 3rd and 2nd quadrant and intersects the abscissa of the real part at exactly Re = -1; In = 0

The shape of the locus does not change as long as the three time constants have an arbitrary but mutually the same value ( T1 = T2 = T3 = T). The magnitude of the time constant T then has a meaning if the timing behavior of the control loop to be represented as step response of the control input, eg. Three equal time constants in a controlled system with three delay elements provide control engineering the worst time behavior with respect to the stability of the control loop dar.

If the gain K> 8 increases, then contact progressively increasingly larger amplitudes with increasing time. The locus of the frequency response crosses the abscissa at values ​​< -1. If the gain K < 8 is reduced, then the amplitudes of the control variable take an aperiodic damped course and the rule size increases after settling to a constant value. The locus intersects the abscissa at values ​​> -1.

If the controlled system only from a delay system of 2nd order with a P controller arbitrarily high gain, it can not come to sustained oscillations in theory, because the locus of the frequency response passes only the 4th and 3rd quadrants of the Gaussian number plane and the critical point Re = -1 can not hit.

Note: the knowledge of the construction of the locus of the frequency response of the nonlinear system is required for the calculation of the vibration condition of a non-linear dynamic system in accordance with the two- locus method!

Behavior of the nonlinear dynamical system

Dynamic systems are described by differential equations. Contains the ODE or the unknown function and its derivatives is a power n> 1 or products of unknown function or in the arguments of trigonometric functions, logarithms, etc., is a non-linear differential equation and the system behaves nonlinearly. Non-linear differential equations are solved analytically only in very exceptional cases. They can be relatively easily solved by means of numerical time-discrete methods, or with commercial computer programs ( Simulink ).

In a non-linear static transfer characteristic of the system can be considered as a gain changing within wide limits. The system responsive to a sinusoidal input signal with a distortion of the most vibration of the input signal of the same frequency and a phase shift. Only a symmetric two- element (two-point controller) without hysteresis, without dead zone, and signal limiter responsive to a sinusoidal input vibration with a rectangular wave of the selected square-wave amplitude u (t) = ± Vmax = ± d of the same frequency without a phase shift. (d = characteristic point of the auxiliary non-linear characteristic to a sinusoidal vibration input e (t) )

The simple symmetrical two-point element with the rectangular waveform amplitude ± d is defined as the signum function with u ( t) as output:

The two-point element has two equilibrium positions. It generates two control variables in response to the input signal e (t). The state of u (t ) = 0, e (t) = 0 is only a steady-state value, if the two-point element is a dead zone TZ = c (c = characteristic auxiliary point ) is set. This would of the two-point element is a three-point element formed.

Special cases are given when the static nonlinear system has a quadratic or an exponential curve. For a sinusoidal excitation with quadratic characteristic of the system, the system responds with a double sinusoidal frequency and DC component ( shifted operating point), the system with exponential characteristic responds with a sinusoidal pulse with a large harmonic content.

Is a non-linear dynamic system and a higher order by means of a second input signal s (t) is excited with a sinusoidal vibration, so the system responds with a mostly approximately sinusoidal oscillation of the same frequency, different amplitude and phase. Is the static non-linear system, a switching controller, it is in a control loop despite its steep pulse edges of the rectangular controller output signal ± UMAX at sufficiently high frequency of e ( t) and u ( t) by the low- pass behavior of the linear dynamic system an approximate sinusoidal output signal y (t) is reached.

This behavior is due to the " harmonic linearization " by the square wave interacts with its large harmonic content as the input variable u ( t) of the linear system, is considered only in its fundamental mode corresponding amplitude. This allows the non-linear static system N will be described in dependence on the amplitude of the sinusoidal oscillation eMAX = A of the input signal e (t).

Nonlinear systems are often unique, but may be made of the non-linear static system behaviors can be defined that allow for given parameters, a system function description. This includes elements with boundary, dead zone, hysteresis, and the so-called progressive characteristic curve regulators such as two-point and multi-point controller, which are found in numerous variants as follows:

• Two- element ( two-position) symmetrically and asymmetrically with and without hysteresis
• Three- element ( three-position controller ) symmetrically and asymmetrically with and without hysteresis
• Multi-point element
• Element with progressive characteristic symmetrical and asymmetrical
• Element with a decreasing characteristic curve, balanced and unbalanced
• Elements with limiting balanced and unbalanced
• Elements with dead zone symmetrically and asymmetrically
• Elements with dead zone and limiting
• Elements with bias
• Rectifier element
• Amount of element

According to the literature up to 40 different descriptive features nonlinear static systems can be found that the non-linear system response N (A) for a periodic sinusoidal input signal e (t) = eMAX * sin (? T ) with amplitude A = eMAX and the output signal u (t ) with only the fundamental wave describe.

Harmonic linearization

Determination of the signal variables, parameters and basic parameters of the characteristic structures

In the known literature of control theory as well as in the current lecture notes German universities can be found for the harmonic linearization hardly an identical mathematical derivation of the Fourier analysis, yet identical system and signal names. Is caused by the different signal designations, for example, XA, XE, E, U, X, Y, the structure of the Fourier analysis equation and the terms of the geometric corners of the non-linear characteristics.

The equation of Fourier analysis and its simplification differs by the different use of the harmonic sine or cosine wave with and without DC components, different limits of integration and different calculation of the coefficients. The results of the description of functions under different names ( often N (A), N ( U0), , ) is not consistent as a function of the amplitude of the input variable of the nonlinear system with numerous characteristic controller functions, because the geometric basic parameters of the characteristic description the auxiliary variables (generally a, b ​​, c, d ) can be used in different ways.

Despite the different mathematical paths that lead to the equation of harmonic balance, the result is the same everywhere. Understanding of the locus of the frequency response of the linear system is required and also the understanding of the locus of the describing function of the nonlinear static system. From the literature the numerous functions description of nonlinear static systems with the accompanying sketches of the characteristics and the loci can be removed. The different geometric benchmarks with the characteristic diagram ( a, b, c, d) of the describing function can be easily assigned to the various sources of literature.

Below it is assumed that the system parameters of the control loop used internationally: reference variable w ( t) = 0, the controller input e ( t), controller output variable (non-linearity ) u ( t) and linear controlled system output y (t). The following mathematical derivations are shown modeled the literature.

Method of harmonic linearization

The method of harmonic linearization is based according to the Hammerstein model from the assumption that the harmonics generated by the nonlinear system by the following linear system with low-pass behavior as ineffective (filtered).

From the viewpoint of control technology is a non-linear dynamic system often the Hammerstein model - in contrast to the Wiener model - used because in many cases the controller associated with the manipulated variable is not linear.

The behavior of the regulator as a nonlinear static system N may be viewed by a within a wide range varying gain, which with a sinusoidal input signal e ( t) = eMAX * sin (? T ) is a signal distortion, and phase shift φ of the output signal u (t) = f [ s (t ), N ( eMAX ) ] is produced. Often, the dependence of N ( eMAX ) by known parameters or by the magnitude of the amplitude of the sinusoidal input signal eMAX e ( t) is determined, therefore, the large-signal behavior of the system is to investigate " stability in the large". Some characteristic regulator to speak only when the input signal reaches a minimum size.

For unique non-linear characteristics as the two- element, i.e., for each static input signal e (t), there is only a static output signal u ( t), the describing function is real, i.e., the phase displacement φ = 0

Simplification of the nonlinear function description in a state of constant oscillation

The input e (t ) of the static non-linear system to be a sine wave without a dc component:

The amplitude eMAX the input sine wave is often referred to in technical literature with eMAX = A.

The distorted output waveform u ( t) can be written as a Fourier series as a sum of a direct value a0 and from the sinusoidal signals of the fundamental frequency a1 * sin ( ω0 * t φ1 ) and the harmonics as multiples of the fundamental frequency.

The output u ( t) may contain a DC component a0 depending on the behavior of the nonlinear system. It only interested in the oscillation behavior.

If the downstream linear system G ( j? ) The distorted sinusoidal vibration input u ( t) as a low-pass filter attenuates strong enough, the high frequencies are strongly suppressed compared to the lower. For this consideration, the shares of the harmonics are assumed to be negligible.

The description function of the static non-linear system is given by the ratio of the fundamental waves of the output signal u1 ( t) to the input signal e1 (t). It takes into account only the fundamental component of the output signal without the DC component a0.

Input signal:

Output signal:

The complex representation of the two signals, the description of function results according to the Euler formula:

Description function N ( A) of the non-linear system

The input amplitude of the sinusoidal oscillation eMAX the nonlinear system is often referred to as A.

A1 and b1 are the coefficients of the first fundamental waves of the non- linear system.

For many forms of so-called characteristic curve control can be based on the geometry of the characteristics by these coefficients ( commonly labeled A, B, C, D, hereinafter), the Fourier coefficients A1 and B1 of the nonlinear functions Description N (A) can be calculated.

The specific functions dependent on the function of the non-linear characteristic element.

• The output of a balanced two- element oscillates as a function of input vibration in synchronism with the zero crossings without the phase shift.
• Unbalanced multiple-point elements (amplitude d ≠ | d | ) produce a DC component.
• With characteristic elements with hysteresis functions describing functions are complex.

The description of functions can be called a "substitute frequency response " of a nonlinear system. It depends on the amplitude A of the vibration input e (t) and the frequency of the continuous oscillation ω = 2 * π * f, and is defined as N ( A ω ). Unlike the linear frequency response with the dependence of ω, the description of the main function of the amplitude A as the independent variable is dependent. If only static non-linearity of the dependency is omitted from the frequency. Hereinafter, the description will be regarded as frequency functions independently of static nonlinearities than N (A).

The preparation and extensive derivations describing functions N (A) of the many characteristic elements can be found in the literature of control theory. The respective description of functions N (A) of the nonlinear static systems are all derived as algebraic functions, only the relevant geometric vertices with the letters a, b ​​, c, d are different, but can be easily associated with the accompanying characteristic sketches.

Description list of known functions of characteristic elements

The following description of functions N (A) apply only to cases in the state of harmonic balance.

Equation of harmonic balance

For the method of harmonic balance of the closed loop system at the stability limit for the reference variable w ( t) = 0 is considered.

The output of the linear system applies:

The nonlinearity behaves in the vibrational equilibrium as a linear transmission member.

E ( j? ) = Y ( j? ) Results from the equation 3, the characteristic equation of the non-linear control circuit, referred to with the harmonic balance equation:

The most common spelling of the equation of harmonic balance is:

If a continuous oscillation of the non-linear control loop exists, then the equation of Harmon 's balance as the characteristic equation of the non- linear control represents the solution of the equation, the harmonic balance can be a formula - numerically or graphically by the two - locus method.

Standard formula solution of the equation of balance Hamonischen

In complicated nonlinear description functions or linear higher-order systems, the formulaic solution can be very difficult. In this case, one chooses the descriptive graphical solution of the two- locus method.

The desired unknown values ​​are the amplitude A of the non-linearity and ω the frequency of oscillation, the non-linear dynamic system.

The equation for the harmonic balance is converted to first calculate A can:

To write the equation in real and imaginary parts:

Thus, two real equations arise. If it is a clear non-linear characteristics, that is, for each input variable there exists a unique output as in the two-point and three-point elements, so the description will function with the imaginary part to zero.

The solution of the system of equations reduces to the solution of two separate equations. First, the poles are determined by ω of the linear system of In 1 / G ( j? ) = 0 and inserted into the right side of the following equation. Then A can be calculated.

Two - locus method of harmonic balance

The graphical method is very descriptive. This gives the amplitude A and the frequency ω of the steady oscillation of the nonlinear control loop, if there is an intersection of the two loci of the frequency response G ( j? ) And the negative inverse describing function -1 / N ( A). It can be stated, whether there will be permanent oscillations in a control loop and whether more sustained oscillations are possible.

Negative, inverted locus shown in the equation of the harmonic balance of the describing function N ( A) can be registered in the same graph of the locus of the frequency response G ( j? ) And -1 / N (A), with the locus parameters of A. the system oscillates, there is an intersection of the two loci. From this, the amplitude A and the frequency ω = 2 * π * f can be read.

Depending on the nature of the 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. For nonlinear systems with hysteresis is an imaginary straight line offset in the negative range. For nonlinear systems with three-position controller and hysteresis is a characteristic field.

Depending on the nature of the locus of the frequency response and the locus of the describing function may lead to more permanent oscillations, because there may be several intersections. In the three-point controller, the locus of the describing function is folded on the real axis (the two branches of the locus overlap ), that is, for a given frequency ω can in the intersection of oscillations for two different large amplitudes A be possible.

On the other hand, the locus of the frequency response for linear systems greater order to the point Re = 0 and Im = 0 entwine. The linear system containing a dead time, then the locus of the frequency response extends helically around the point Re = 0, and in = 0, so that A and different frequencies may ω many intersections, and thus a plurality of continuous oscillations of different amplitudes.

If it is in a loop for example, a two -position controller and a linear system G0 ( s ) 1st or 2nd order, it can not come to sustained oscillations, according to the two- locus method, because the locus of the frequency response does not occur in the second quadrant can. Unfortunately, we know by numerical calculation or from experience with two-position controllers, that the control loop responds with a sustained oscillation. The reason of this behavior is explained that the condition of the harmonic oscillation of the output variable y ( t) of the control circuit with low-pass behavior is not met. The output y (t ) is not a harmonic oscillation in this case.

In contrast, the loci of a two-point controller with hysteresis can by the position of [-1 / N ( A)] locus in the negative imaginary area of ​​the third quadrant very well with the locus of the linear system G ( j? ) Meet the 2nd order.

Practical example of harmonic balance

Controlled system with global I- behavior:

Symmetrical three-point controller with the function description: UMAX ± = ± 2; ± deadband = ± 0.5; Reference variable w ( t) = 1 Geometric key points: UMAX = d; Deadband = c

Result:

The values ​​of the locus of the negative inverse describing function -1 / N (A) as f ( A) from the corresponding N Equation ( A) of the three-position controller is calculated and entered on the real axis Re of the locus of the frequency response. The two branches of the locus start at A = c, ends at A = ∞ and reverse the direction at A = 1.41 * c. They intersect at -1 / N ( A) = 0.8 with A = 1.97 and A = 0.517 with the locus of the frequency response at Re -0.8; ω = 1

The continuous oscillation with A = 0.517 is unstable. The amplitude is A = 0.517 c = 0.5, in which no continuous oscillation is possible even at the limit of the dead zone. The description of function with amplitude A = 1.97 results in a stable continuous oscillation. In general, the stable continuous oscillation adjusts to the larger amplitude A.

The calculated by means of numerical methods of discrete time diagram showing the time response of the system parameters determined with respect to the amplitude A and the frequency ω = 2 * π * f after settling after about 16 s fully consistent with the results of the harmonic balance match.

Stability of the boundary oscillations

The locus of the frequency response of a linear system as an open loop indicates continuous oscillation when, after the Nyquist method in the complex plane the locus of the point Re = -1 and Im = 0 is true. This is not a stable sustained oscillations because at slightest changes to the poles of G ( j? ) Is the critical point Re = -1 ± ΔRe not taken. The result is a value of Re = -1 ΔRe, a damped oscillation occurs or at a value of Re = -1 ΔRe arises increasingly aufklingende oscillation.

With the two- locus method, sustained oscillations of a nonlinear dynamic system can be fed back to determine when in the complex plane meet the loci of the describing function of the nonlinear system and the locus of the frequency response of the linear system. Meeting, the two loci not, there is a vibration-free control loop.

It should be noted that the linear retarding system G ( j? ) Must be at least > 2nd order otherwise passes through the locus of the frequency response is not the second quadrant, and can not make a located on the real axis locus N ( A).

It may be assumed that the term " boundary oscillations " = " permanent oscillations " comes from the boundary conditions of the locus of the describing function N ( A). The boundary conditions of the points of intersection of the locus of the describing function can be considered for three cases in which the amplitude A increases or decreases due to transient disturbances by a small value? A, and then returns into the starting position again in each case. This raises the question of what happens when a continuous oscillation is and acts positively or negatively by a small transient amplitude change on the equilibrium state of the steady oscillation. The equation of harmonic balance is for these cases to an inequality.

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

The intersection of the two loci is a stable limit oscillation when increasing amplitude with A, the amount of the describing function | N (A) | decreases. An unstable limit vibration occurs when the amount of | N (A) | A increases with the describing function.

Example of a control loop with a locus N ( A) of a three-point controller and a PT3 system:

The two branches of the locus of real describing function N (A) are negative inverted -1 / N ( A) above the other on the real axis Re.

The following three cases are distinguished, which meet, touch or do not meet the two loci:

• Case 1 ): The two loci meet

• Case 2 ): The two loci touching the turning point A = c * √ 2 of the two branches

• Case 3 ): The two loci do not meet and do not touch

Avoiding sustained oscillations

Nonlinear dynamical systems can cause willed stable sustained oscillations, for example when using a simple two -step controller. The position of the locus of the describing function - 1 / N ( A) on the real axis of the locus of the frequency response at the point Re = 0 and starts at A = 0 and ends at A = ∞. Therefore, there is no way to avoid the two-position controller continuous oscillation.

Note: A two-position controller vibrates already in conjunction with a linear 1st order time lag, although the two loci can not meet. The harmonic balance is not valid for this case, because the condition of harmonic oscillations are (sawtooth -like oscillation) to y ( t) is not given. This applies even for a second order delay element ( PT2 element ).

With the use of other static nonlinear systems, such as by the three- point controller can be vibration-free control circuits with asymptotic realize. This requires a control system with global I behavior.

This raises the question of what measures using the two- locus method can be realized.

The following measures are given so that the equation of harmonic balance to an inequality, in which the two loci can not meet.

The inequality of harmonic balance could be:

• Reducing the gain of the linear system G ( j? )

• Increasing the distance of the locus of the describing function of the ordinate at Re = 0

• Partial compensation of the delay elements of the linear system G ( j? )

Compensation carrier PT1 elements by PD1- members

By exchanging carrier delay elements of the linear dynamic system, the system obtains a faster time response.

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

A vibration-free control circuit with a three-point controller is provided that the linear dynamic system, the controlled system, a global I- behavior. Typical of such control systems are actuators that are placed on the movement forward, rewind or hibernation by the three-position controller. The rest corresponds to the desired positioning.

If the parameters of the calculation example of the afflicted with continuous oscillation control circuit modified with the three-position controller as follows, created as shown in the diagram, a vibration-free control loop.

• The two delays are reduced by PT1 elements PD1- members by a factor of 10

• The related variable ( 1 = 100 %) of the rectangular oscillation amplitude ± UMAX is set to ± 1.2,
• The size of the dead zone is related to ± c = ± 0.1 set,
• The maximum related reference variable w ( t) = 1.

The size of the dead zone determined to operate at which deviation of the regulator. In this case, the time response ( integration constant TN) addresses the controlled system to be interpreted according to the dead zone. Only a slow actuator allows for a vibration- free operation of a small dead zone and thus accurate positioning. The smaller the dead zone, the more frequent shorter and the square wave u (t) during the transient of the controlled variable y (t).

Conclusion on the application of harmonic balance

• The harmonic balance shows interesting aspects of the conclusion of sustained oscillations in nonlinear control loops and also ways of preventing permanent oscillations.
• There is no calculation method of analysis of the system behavior of nonlinear control loops with dead time and hysteresis elements, which is almost as powerful and yet so simple as the numerical recursive calculation of a nonlinear system by the Euler polygonal line method with the discrete time? T and the calculation sequence k = (0, 1, 2, 3, ... kmax ).