Large signal behavior

The terms large-signal behavior and small-signal behavior are relevant in connection with non-linear transmission systems . In contrast to the small-signal behavior of a transmission system, which relates to an operating point of the characteristic curve of the input / output behavior of a transmission system, the large-signal behavior means that the input variable of the system can assume all values between zero and maximum. In the large-signal behavior of a mixed linear and non-linear dynamic system, it is of interest how the output variable of the overall system relates to the input variable of the system after a sufficiently long time. With the help of suitable linearization methods, well-approximated linear results can be achieved with non-linear individual systems.

In a linear time-invariant dynamic transmission system, the signal transmission of the output signal behaves proportionally to its input signal after a sufficiently long time. In this form of the transmission system, the large-signal behavior is identical to the small-signal behavior . ${\ displaystyle \ left.K = {\ frac {\ Delta y (t)} {\ Delta u (t)}} \ right | _ {\,} {t \ to \ infty}}$

The calculation of the input / output behavior in a transmission system as series connection, parallel connection and returned systems ( control loops ) with mixed linear and non-linear systems is carried out exclusively using numerical mathematics.

Definition of dynamic system

Block diagram of a transmission system as a single and multiple variable system.

A dynamic system is a delimited functional unit with a specific time behavior and has at least one signal input and one signal output.

Models ( modeling ) of a real dynamic transmission system are mathematically described by:

Differential Equations , , ${\ displaystyle a_ {n} y ^ {(n)} + \ dotsb + a_ {1} y ^ {(1)} + a_ {0} y = b_ {m} u ^ {(m)} + \ dotsb + b_ {1} u ^ {(1)} + b_ {0} u}$
Transfer function and frequency response , , ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}}}$
State space representation (including state equations),
Difference equations for linear dynamic systems, ${\ displaystyle {\ dot {y}} (t) \ approx {\ frac {y _ {(k)} - y _ {(k-1)}} {\ Delta t}}}$
Static non-linearity is often described by so-called logical IF - THEN - ELSE statements or table values.

The transfer function , the most common description of linear transfer systems, represents the dependence of the output signal of a linear, time-invariant system (LZI system) on its input signal in the image area (frequency area, s area). It describes the intrinsic behavior of the transfer system completely and independently of the signals . A transfer function thus makes it possible to calculate the output signal of the transmission system from the input signal and the transfer function.

The transfer function is defined as the quotient of the Laplace transformed output variable to the transformed input variable : ${\ displaystyle Y (s)}$ ${\ displaystyle U (s)}$

{\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}}}

.

When the dynamic system is in the idle state, the internal energy stores have the value zero. Under this condition that the initial conditions of the differential equation describing the system are at the point in time under consideration , the transfer function of the system corresponds to the Laplace-transformed differential equation of the system. ${\ displaystyle t = 0}$

Definition of the static system

A static linear transmission system is described by an algebraic model and has no time behavior.

Nonlinear dynamic transmission systems are mostly unique and can usually only be treated arithmetically with numerical mathematics. Such a system can be broken down into a static nonlinear system and a dynamic linear system according to the Hammerstein model . So-called logical IF - THEN - ELSE statements or tabular values are often used to describe a nonlinear model.

Behavior of the non-linear system

In a non-linear static or dynamic transmission system there is no proportionality to the input-output behavior. In practice, non-linear transmission systems very often come in various forms. ${\ displaystyle t> 0}$

Transmission systems with mixed linear and non-linear systems can only be calculated using numerical methods. Difference equations that describe the behavior of the linear terms are suitable for this. The input variables of each transfer element are calculated as subsequent elements point by point to output variables at a discrete time interval . ${\ displaystyle \ Delta t}$ ${\ displaystyle k}$

A difference equation is a numerically solvable calculation rule for a discretely defined sequence of subsequent equations which calculate variables for consecutive numbered events or numbered points in time at an interval . ${\ displaystyle k = [0, \ 1, \ 2, \ 3, ...]}$ ${\ displaystyle y _ {(k)}}$ ${\ displaystyle \ Delta t}$

The non-linear static function is defined as a mathematical model and the associated output variable is calculated point by point in the form of a table, also for a given input signal. The overall result of a chain of individual links is available in the form of a table with all input and output calculation points. The table describes the period of the calculation sequences . ${\ displaystyle n \ cdot \ Delta t}$

Examples of mathematical models of nonlinear transmission systems.

Representation of the most common non-linear functions

In practice, a technical controlled system usually consists of hardware systems which, due to the components used, rarely have ideal mathematically describable properties. These include B. motors, actuators, heating elements, springs, linear valves, switching valves, transducers and others. If the non-linear components are small compared to the overall behavior of a system, they can be neglected.

A valve can only be 100% open and an electric motor must not be operated at its maximum output. Limitations of the controller and the system must be coordinated with one another due to the dynamic control. Signal or manipulated variable limits have a dampening, time-delaying effect on the large-signal behavior of the controlled variable. The numerical calculation of the limiting function can be easily achieved using logical criteria. The gear backlash of mechanical gears can often be described by the behavior of hysteresis and dead zone with logical commands.

Typical forms of the input-output behavior of non-linear systems:

Limiting effects of signals,
Quadratic or exponential behavior,
Hysteresis behavior,
Dead zone.

Method of linearizing non-linear functions

Non-linear transmission systems can be linearized as individual systems with various measures. In control engineering in particular, linear transmission systems are desired for stability analysis. This facilitates the stability analysis using common conventional methods. The same applies to simulations of the control loop using numerical mathematics. The behavior of control loops with controller and controlled system can be completely reproduced by simulation on a computer by calculating discrete-time difference equations for the description of the linear transfer elements.

Since nonlinear systems are unique in their diverse manifestations, the mathematical description is often only possible with the help of logical commands or tables.
In the case of discontinuous controls, non-linear effects such as hysteresis and dead zone are desired for the controller with the two-point method and three-point method. They can be easily described with the logical IF - THEN - ELSE statements.
The inclusion of various components of the controlled system in the control loop through the feedback forces a proportionality.
Cascade control offers more complex solutions for many very different individual components of the controlled system.

Linearization of a non-linear function with a compensation function

Linearization measures:

The task of the linearization measures of a non-linear static mostly hardware transmission system is to achieve an acceptable linearization of the transmission system with the help of additional hardware or software. Several measures are available for this.

Linearization through feedback:

If there is a constant non-linearity of the controlled system in the forward branch of a control loop, the feedback of the controlled variable forces the output-input behavior of the control loop to be linear.

If a non-linear component can be separated from the overall system of the controlled system, it can be integrated into an auxiliary control loop in the forward branch for linearization. This achieves proportionality of the input-output behavior of the auxiliary control loop.

A control loop with a continuous nonlinear function in the controlled system, e.g. B. with a function with exponential behavior cannot be optimally controlled, because the parameterization of the controller must be set to the greatest gain of the system, otherwise the control loop can become unstable depending on the number of delay elements.

An inexact inversion function of the non-linearity of the system can be introduced into a controller. This achieves an approximately constant loop gain and a relatively optimal parameterization of the controller.

Compensation with "inverse non-linearity"

A compensation function (inverse non-linearity) as a non-linear network as hardware in the input of the non-linear system can help. This is done in such a way that a mirror-image nonlinear inverse function compensates for the behavior of the nonlinear function.

For a digital controller, this would only be a table that must be taken into account so that the non-linearity can be converted into a linear function with proportional behavior.

Step responses of a controlled variable with different manipulated variable limits

Signal limitation

A manipulated variable limitation of a controller is unavoidable if the output power level of the controller, e.g. B. the output voltage, cannot grow any further. Depending on the size of the limitation, a step response can be significantly delayed.

As with all nonlinear functions, limiting effects cannot be described with the transfer function G (s).

The numerical calculation of the limiting function can be easily achieved using so-called logical IF - THEN - ELSE instructions.

Dead zone (sensitivity)

With measuring sensors, the effect sometimes occurs that a certain value of the measured value has to be exceeded in order for the measuring sensor to emit a signal. For a control loop, this means that small setpoints and thus small controlled variables cannot be set. If the response sensitivity can be reproduced, the characteristic curve of the sensor can be offset with logical commands.

Hysteresis

By friction on valves, by magnetic effects e.g. B. with relays or through coupling to operational amplifiers, the hysteresis effect can occur. The hysteresis can occur with unsteady and constant signal processing.

The hysteresis function is desirable for unsteady controllers. Two-point controllers compare the controlled variable with a switching criterion that is usually subject to hysteresis and only know two states: "On" or "Off". These two-position controllers defined in this way theoretically have no time behavior.

The hysteresis effect is very undesirable for continuous regulation.

Remedy: The influence of the hysteresis can be compensated by adding or subtracting a signal amount to the output of the system for the signal direction to be determined.

literature

Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Verlag Europa-Lehrmittel, 2019, ISBN 978-3-8085-5869-0 .
Gerd Schulz: Control engineering 1 . 3. Edition. Oldenbourg Publishing House, 2004.
Serge Zacher, Manfred Reuter: Control technology for engineers . 14th edition. Springer Vieweg Verlag, 2014, ISBN 978-3-8348-1786-0

Individual evidence

↑ Lutz / Wendt: Pocket book of control engineering with MATLAB and Simulink: See chapter "Mathematical methods for calculating digital control loops".

↑ M. Reuter, S. Zacher: Control technology for engineers: 12th edition: See chapter "Non-linear elements in the control loop".

↑ Lutz / Wendt: Pocket book of control engineering with MATLAB and Simulink, chapter: Method of linearization .

[1] Lutz / Wendt: Pocket book of control engineering with MATLAB and Simulink: See chapter "Mathematical methods for calculating digital control loops".

[2] M. Reuter, S. Zacher: Control technology for engineers: 12th edition: See chapter "Non-linear elements in the control loop".