Dead time (control technology)

Dead time element in the structure diagram

As dead time (also called maturity or transport time) in the control technology the time between the signal change at the system input and the signal response referred to the system output of a controlled system. Every change in the input signal causes a change in the output signal delayed by the dead time. A system with dead time without additional time behavior is also referred to as a dead time element.

Dead-time elements cannot be described with ordinary differential equations, but only as transcendent functions (not algebraically) via the frequency behavior. This makes it more difficult to parameterize a controller in the control loop, because transcendent functions cannot be combined with fractional rational functions of a transmission system for algebraic calculation. It can therefore make sense, depending on the programming language used, to determine dead time models that can be written approximately as fractional rational functions.

The investigation of the frequency behavior of different linear transmission systems with a dead time element when using the Bode diagram or the locus of the frequency response on the cut control loop is used

• the detection of the stability of the closed control loop,
• the system analysis of linear dynamic transfer elements G (s) and dead time behavior.

The representation of the transfer behavior in the time domain of dead time elements in connection with linear and non-linear transfer elements in the control loop can only be achieved with a reasonable calculation effort using numerical mathematics . Value sequences are calculated at a discrete time interval as a function of a given input signal . In this way, a graphic can be used to show a closed time behavior for the output variable and intermediate variables of interest. ${\displaystyle \Delta t}$ ${\displaystyle y(t)}$

Basics of dead time elements

Knowledge of the application of the transfer function and control engineering is required to understand this article . ${\displaystyle G(s)}$

The dead time element is a transmission element that occurs frequently in practice and usually works in conjunction with other delay elements. It is caused by the pure transit time or transport time (conveyor belt, pipeline) or signal transit times that arise over long distances. It behaves like a P element whose output variable arrives late by the dead time without distorting the input variable during this time. Any change in the input variable is delayed by the dead time at the output.

The dead time element is only described by one parameter . The phase shift between the output variable and the input variable is proportional to the dead time and increases with the product with increasing frequency. ${\displaystyle T_{t}}$ ${\displaystyle -\omega \cdot T_{t}}$

While delay elements ( PT1 element ) can cause a maximum phase shift of -90 ° with increasing frequency of the input signal, the phase shift in dead time elements increases constantly with increasing frequency. For a closed control loop, this phase shift can lead to instability at an early stage because, depending on the loop gain, the feedback of the controlled variable can be converted from negative feedback into positive feedback.

Dead time members are non-phase minimum systems. A linear dynamic system is phase-minimal if its poles and zeros lie in the left half-plane and it has no dead time.

The transfer function of a linear dynamic system is defined as the quotient of the Laplace transform of the output variable Y (s) and the input variable U (s):

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

Linear dynamic transfer elements are described in control engineering by broken rational functions in the image area (s area) . The independent variable allows any algebraic operations in the s-domain, but is only a symbol for a completed Laplace transformation and does not contain a numerical value. Numerical values ​​arise from the coefficients a and b of the transfer function in polynomial representation, in that the polynomials of the transfer function are broken down into linear factors (products) by decomposing zeros . ${\displaystyle G(s)}$ ${\displaystyle s=\delta +j\omega }$

Example of a 3rd degree transfer function of a linear dynamic system in time constant representation:

${\displaystyle G_{1}(s)={\frac {Y(s)}{U(s)}}={\frac {K\cdot (T_{v}\cdot s+1)}{s\cdot (T_{1}\cdot s+1)(T_{2}\cdot s+1)}}}$

If there is still a dead time element in the system , this can be added as a transcendent function multiplicatively to the fractional rational function . ${\displaystyle G_{Tt}(s)={\tfrac {Y(s)}{U(s)}}=e^{-s\cdot T_{\mathrm {t} }}}$ ${\displaystyle G_{1}(s)\cdot G_{Tt}(s)}$

${\displaystyle G_{2}(s)={\frac {Y(s)}{U(s)}}={\frac {K\cdot (T_{v}\cdot s+1)}{s\cdot (T_{1}\cdot s+1)(T_{2}\cdot s+1)}}\cdot e^{-s\cdot T_{\mathrm {t} }}}$

In the time domain, we are interested in the behavior of the output variable of a system for a given input signal. Transcendent systems are unfavorable for various controller design processes. Like broken rational systems, they cannot be treated algebraically in the s-domain.

Such systems combined as a series connection can be calculated using various methods for the representation in the time domain for the part of the fractional rational function. The dead time function with is added graphically to the calculated time function. This does not apply to the closed control loop with a dead time element in the controlled system. ${\displaystyle T_{t}}$

Frequency behavior of the dead time element

In contrast to the linear dynamic systems, a dead time element cannot be described with an ordinary differential equation . A simpler connection between the input and output behavior results in the image area as a transfer function . ${\displaystyle G(s)}$

The functional relationship of a dead time element in the time domain is:

${\displaystyle \displaystyle y(t)=u(t-T_{\mathrm {t} })}$

The input signal appears, delayed by the dead time, unchanged at the output. This gives the transfer function in the image area :

${\displaystyle G(s)={\frac {Y(s)}{U(s)}}=e^{-s\cdot T_{\mathrm {t} }}}$

The step response results from:

${\displaystyle a(t)=\sigma (t-T)={\begin{cases}0,&t<T\\1,&t\geq T\end{cases}}}$

To calculate the phase shift, the transfer function on the imaginary axis of the image area (which corresponds to the frequency response) is considered. ${\displaystyle G(s)}$

${\displaystyle G(s=0+\mathrm {i} \omega )=e^{-\mathrm {i} \omega \cdot T_{\mathrm {t} }}}$

The phase angle can now be read off directly (see also Euler's identity ).

${\displaystyle \varphi (\omega )=-\omega \cdot T_{\mathrm {t} }}$

With results from the phase shift as a function of the frequency: ${\displaystyle \omega =2\pi f}$

${\displaystyle \varphi (f)=-2\pi f\cdot T_{\mathrm {t} }}$

A pure dead time element has the gain 1 or the attenuation D = 0 [dB]. The phase shift between the input signal and the output signal increases with increasing frequency with a lag , i.e. proportional to the frequency. ${\displaystyle u(t)}$ ${\displaystyle y(t)}$ ${\displaystyle \varphi (\omega )=-\omega \cdot T_{\mathrm {t} }}$

With increasing dead time as a parameter, a control loop becomes unstable, which forces the loop gain to be reduced. This makes the control loop sluggish with regard to changes in reference variable and disturbance variable influences.

Stability analysis of a cut control loop for systems with dead time

The classic representation of a dynamic system with dead time is the Bode diagram and the locus of the frequency response. Both graphical methods are suitable for determining the stability by means of a cut control loop for the closed control loop. The transfer function with can be transferred to the frequency response or at any time without loss of information . ${\displaystyle G(s)}$ ${\displaystyle s=\delta +j\cdot \omega }$ ${\displaystyle G(j\omega )}$ ${\displaystyle F(j\omega )}$

Example of a Bode diagram for 2 PT1 elements with the corner frequencies at and . ${\displaystyle \omega _{E}=10^{3}}$ ${\displaystyle \omega _{E}=10^{6}}$

Stability condition in the Bode diagram with the simplified stability criterion from Nyquist

In the Bode diagram , the magnitude and phase angle are plotted in two separate diagrams, as amplitude response and phase response. The Bode diagram has a logarithmic scale. In the case of the amplitude response (double logarithmic), the amount F (jω) is plotted on the ordinate, the angular frequency ω on the abscissa. In the phase response , the phase angle (linear) is plotted on the ordinate, the angular frequency ω on the abscissa (logarithmic).

The advantages of this method are the direct drawing of the asymptotes as straight lines of the amplitude response, the convenient multiplication by logarithmic addition, the direct reading of the time constants and the quick recognition of the stability of the closed control loop. In the case of phase-minimal systems, the phase response can be calculated from the amplitude response and does not necessarily need to be drawn. This does not apply to systems with a dead time element.

Frequency behavior of control loop elements:

• A PT1 delay element shows in the amplitude response with increasing frequency from the corner frequency (intersection of the asymptotes) an amplitude ratio that decreases by 45 °. The phase response of the sinusoidal output signal is lagging behind the sinusoidal input signal by a maximum of φ = -90 °. ${\displaystyle \omega }$
• A PD1 element shows an amplitude ratio increasing by 45 ° with increasing frequency from the corner frequency. The phase response of the sinusoidal output signal is advanced compared to the sinusoidal input signal by a maximum of φ = 90 °. ${\displaystyle \omega }$
• In the amplitude response with increasing frequency, an I-element shows a straight line falling linearly with φ = 45 ° as the amplitude ratio. In the phase response, an I element shows a phase shift that increasingly lags behind φ = -90 ° with increasing frequency. ${\displaystyle \omega }$
• In the case of the amplitude response with increasing frequency, a dead time element always shows the amplitude ratio with gain 1. The phase response of the dead time element increases proportionally with the frequency, lagging into infinity. ${\displaystyle \omega }$ ${\displaystyle \omega }$

The stability criterion is derived from the Nyquist stability criterion :

A closed control loop is stable if the lagging phase shift φ from the output to the input signal of the open loop with the loop gain K = 1 and φ> −180 °. The attenuation of the closed circuit becomes more favorable, the greater the phase distance to the −180 ° line. This distance, which is above the - 180 ° line, is called the phase edge or phase reserve and should be around 50 ° ± 10 °.

Note: If the amplitude response is plotted on the ordinate in dB (decibel), 0 dB corresponds to the amplitude ratio 1. The value 20 dB corresponds to the amplitude ratio 10.

Nyquist diagram of the PT ₁ T _t element of a cut control loop.

Stability condition with the locus of the frequency response

The frequency response is a complex variable and is separated into a real part and an imaginary part for graphic representation . ${\displaystyle G(j\omega )}$ ${\displaystyle \operatorname {Re} (G)}$ ${\displaystyle j\cdot \operatorname {Im} (G)}$

The frequency response equation ( frequency response ) of the cut control loop is resolved according to the real part and the imaginary part and entered in a coordinate system. The vertical axis shows the data of the imaginary parts, the horizontal axis the real parts. According to Nyquist, the stability condition is:

If the real parts are not wrapped around or touched by the critical point (-1; j0) on the left (negative) side of the axis when passing through the locus of the cut control loop in the direction of increasing values , the closed control loop is stable. For practical reasons, the critical point (-1; j0) should be moved to (-0.5; j0) in order to achieve a certain stability reserve. ${\displaystyle F_{0}(j\omega )}$ ${\displaystyle \omega }$

The locus of the frequency response shown in the figure of an example for the cut control loop:

${\displaystyle G_{0}(j\omega )={\frac {K_{P}}{T\cdot j\omega +1}}\cdot e^{-j\omega T_{t}}}$

shows, according to the distance from the critical point (-1; j0) of the abscissa of the real part of 0.5, a stable closed control loop. The P gain can be read directly on the abscissa and corresponds to the distance between the points . ${\displaystyle K_{p}=1}$ ${\displaystyle \omega =0(+1;j0)\to \omega =\infty (0;j0)}$

Note: The locus curve for a single dead time element makes an infinite number of revolutions on a circular path with the radius in the s diagram with increasing frequency . The series connection of a dead time element with a PT1 element (semicircle in the 4th quadrant) results in the spiral course by adding the two locus curves. ${\displaystyle F(j\omega )=1}$

Basics of the numerical calculation of dynamic transmission systems

Relatively simple transmission system structures with non-linear elements can no longer be solved in a closed manner by conventional calculation methods in the continuous time domain. With commercially available personal computers, the behavior of any meshed system structures can be determined relatively easily by means of numerical calculation.

With the simulation of a mathematical model of a transmission system or a control loop, it is possible to carry out a system analysis or a system optimization with suitable test signals.
The advantage of simulating on a model is obvious. No technical systems are endangered or required. The time factor does not matter, very fast or very slow processes can be optimized. The prerequisite is the mathematical description of a well-approximated model of the mostly technical controlled system.
For the numerical calculation of the time behavior of control systems with dead time, there are no other alternative methods with regard to the analysis and optimization of systems when using commercial programs or simple programs with difference equations.

For the implementation of the calculation of transmission systems or the simulation of control loops, commercial computer programs are available. With the well-known programs such as Matlab and Simulink , extensive instruction sets are available for the theoretical modeling of dynamic systems and many special control commands.

Alternatively, you can use your own computer programs to perform very efficient control loop simulations using difference equations in conjunction with logical operators. Relatively little mathematical knowledge is required.

The numerical calculation allows a complete overview of the inner movement of dynamic transmission systems in tabular and graphical form . In connection with logical program commands and value tables, non-linear, limiting and dead time systems can be simulated.

If the differentials of the output variable y (t) of a differential equation are replaced by small difference quotients with a discretized time, a numerically solvable difference equation is created that approximates the differential equation. It is useful to convert linear elementary systems (transfer functions such as I, PT1, D, PD1 elements) into difference equations. Depending on the position of the function blocks in the signal flow diagram, these can be treated recursively with non-linear systems or systems with dead time and their numerical calculation methods. ${\displaystyle \Delta y/\Delta t}$ ${\displaystyle \Delta t}$

The numerical calculation of the difference equations of the individual control loop elements takes place in tabular form, step by step, with the discrete time interval . The equations are calculated repeatedly using calculation sequences . The control deviation is at the beginning of each calculation line . Each output variable becomes the input variable in the next calculation sequence in the same line. Every single difference equation for a specific control loop element relates to the same difference equation of a previous sequence . ${\displaystyle \Delta t}$ ${\displaystyle k=(0;1;2;3\cdots k_{\mathrm {max} })}$ ${\displaystyle e_{(k)}=w_{(k)}-y_{(k-1)}}$ ${\displaystyle y_{(k)}}$ ${\displaystyle y_{(k-1)}}$

→ For detailed details see Wikibooks , "Introduction to Systems Theory", Chapter: Numerical Computation of Dynamic Systems

→ see also control engineering # Basics of numerical calculation of dynamic transmission systems

Models of controlled systems with equivalent dead time

The transient behavior of the controlled variable of a control loop with dead time or limiting effects for a given input signal can - apart from commercially available PC programs - only be calculated numerically using difference equations in combination with logical commands.

Numerical calculations ( simulations ) are carried out in tabular form. One line contains all functions (equations) of the subsystems of a control loop. In each column there is an equation for a subsystem. All lines are identical except for the time scale . The sequence of a line determines a partial result of the overall system. ${\displaystyle k_{i}\cdot \Delta t}$ ${\displaystyle k=(0;1;2;3;\dots ;k_{\mathrm {max} })}$

Difference equations can be calculated using any programming language. When using the spreadsheet - the advantage is the avoidance of program errors and the direct graphic display of the required variable - the INDEX function can approach any column values ​​within a matrix (here one column) to calculate the dead time .

The following applies to the calculation of the INDEX function for selecting a cell in the matrix area with S = column, Z = row:

${\displaystyle {\text{Zahlenwert in Zelle}}(Z,S)=\mathrm {INDEX{\bigg [}Matrixbereich(Z_{ni},S_{ni});Zeile(Z_{ni});Spalte(S_{ni}){\bigg ]}} }$

and modified for dead time calculation on one column:

${\displaystyle {\text{Zahlenwert in Zelle}}(S_{i})=\mathrm {INDEX{\bigg [}\underbrace {(S_{\mathrm {min} }:S_{\mathrm {max} })} _{\text{Spaltenbereich}};\underbrace {(S_{\mathrm {max} })-(S_{\mathrm {min} })-{\frac {T_{t}}{\Delta t}}} _{\text{Bezug}}{\bigg ]}} }$

For the numerical calculation with the spreadsheet, the cells usually contain equations and always represent numerical values. The cell position is defined by a letter and a line number. The equations begin with an equal sign (=) and contain only the addresses of cells and are linked with mathematical operators . The variables are stored in an input field and contain direct addressing. The equations in the cells of a row are calculated from left to right and for the rows from top to bottom and relate to the cell contents on the left by specifying the address (letter and row number).

When copying an equation in a cell into the cells below with the "Copy" command, e.g. B. 1000 times, the addresses change automatically in alphanumeric order. This is a relative addressing.

The variables contained in the equation such as and require their values ​​from an input field with direct addressing. ${\displaystyle {T_{t}}}$ ${\displaystyle \Delta t}$

Example of direct addressing of cell B2 : Cell input field . ${\displaystyle \mathrm {B2\to \$B\$2} }$

Example of an equation for calculating dead time in column K with reference to column J with 2 direct variables:

${\displaystyle =\mathrm {INDEX{\bigg [}J100:J400;300-\$B\$16/\$F\$15{\bigg ]}} }$.

• Limit value of the dead time input: The cell K400 contains the INDEX function for the dead time calculation for the sequence k = 0, i.e. the 1st line of the system calculation. For the reference given in the INDEX equation with the variables [s] and [s], a maximum of one dead time [s] can be calculated. With longer dead times, the column area must be expanded, otherwise negative references result with an error message. ${\displaystyle T_{t}=2{,}99}$ ${\displaystyle \Delta t=0{,}01}$ ${\displaystyle T_{t}=2{,}99}$
• Empty cells above the input variables: The cells in the column below J400 contain the numerical values ​​of the input variables u (t). Above the cell in column J400, there must be no characters in the cells for the specified column range J100 to J400 ; they represent the dead time . ${\displaystyle y(t)=0}$
• Dead time range for : ${\displaystyle T_{t}=0}$ If cell J400 contains a numerical value as the input variable, cells K400 to K600 show the numerical value zero for a dead time [s]. ${\displaystyle T_{t}=2}$

If the form as a fractional rational function is required for the mathematical description of the dead time element, the following approximation models are possible:

Dead time approximation with a 3rd order all-pass in series with a PT1 element.

Approach to the behavior of a dead time element using all-pass elements as a substitute dead time

The Padé approximation of the dead time already brings good results of the dead time approximation with three identical all-pass terms (n = 3).

The all-pass with a PD element in the counter with a positive zero can be broken down into known subsystems of the 1st order as a PT1 element and D element as follows :

Example of three identical all-pass elements connected in series with the proportional factor K = 1:

${\displaystyle G_{T_{t}M1}(s)=\left({\frac {1-{\frac {T_{t}}{2n}}\cdot s}{1+{\frac {T_{t}}{2n}}\cdot s}}\right)^{n}\qquad {\text{mit }}n\to \infty }$

So that the positive zero point disappears, the numerator of the all-pass is broken down as shown below.

Example: If you choose an equivalent dead time with 3 all-pass elements and = 2 [s]: ${\displaystyle T_{t}}$

${\displaystyle G_{T_{t}M1}(s)\approx {\biggl (}{\frac {1-0{,}333\cdot s}{1+0{,}333\cdot s}}{\biggr )}^{3}=K\cdot {\biggl (}{\frac {1}{1+0{,}333\cdot s}}-{\frac {0{,}333\cdot s}{1+0{,}333\cdot s}}{\biggr )}^{3}}$

The adjacent graphic shows the step response of an all-pass element 3rd order as a dead time model in series with a PT1 element . The time behavior of the step response of the overall system was calculated numerically using the difference equation assigned to each individual system. ${\displaystyle T_{tM}=2[s]}$ ${\displaystyle G(s)={\frac {1}{4\cdot s+1}}}$

Approach to the behavior of a dead time element through PTn elements as a substitute dead time

Already from n = 5 PT1 elements with the same time constants, a good approximation of a dead time element can be achieved.

The step response of a dead time model with 5 PT1 elements with the model time constant still shows considerable differences in comparison with a dead time element. If the dead time model is integrated in comparison with a dead time element in a control loop with an I controller with the same loop gain, these differences in time behavior are reduced. ${\displaystyle T_{tM}={\tfrac {T_{t}}{5}}}$

Step responses of a control loop with I controller, PTn dead time model and, alternatively, dead time element.

${\displaystyle G_{T_{t}M2}={\frac {1}{\left({\frac {T_{t}}{n}}\cdot s+1\right)^{n}}}\qquad {\text{mit }}n\to \infty }$

Example:

If you choose a substitute dead time with 5 PT1 elements and = 2 [s]: ${\displaystyle T_{t}}$

${\displaystyle G_{T_{t}M2}(s)\approx {\frac {1}{(0{,}4\cdot s+1)^{5}}}}$

The graphic image on the right shows the step response of a control loop with a controlled system for a dead time model in comparison with a dead time element.

Control loop data: dead time , I controller . ${\displaystyle T_{t}=2[s]}$ ${\displaystyle K=0{,}3}$

The cut control loop with the dead time element reads: ${\displaystyle e^{-s\cdot T_{\mathrm {t} }}}$

${\displaystyle G_{0}(s)={\frac {0{,}3\cdot e^{-s\cdot 2}}{s}}}$

The cut control loop with the equivalent dead time is:

${\displaystyle G_{0M}(s)={\frac {0{,}3}{s\cdot (0{,}4\cdot s+1)^{5}}}}$

The course of the controlled variable is shown with the dead time model and the analytical function of the dead time.

Path parameters of a step response through the tangent at the turning point

System analysis of a controlled system with dead time

The step response has the advantage that it is easier to carry out and that the expected result is better known. The time-independent system gain can be read directly from controlled systems with compensation in the static state. The time behavior of the path can be determined by a model of the dead time and a model of the S-shaped increase in the transient process. ${\displaystyle K_{S}}$

The following requirements are placed on the model controlled system for a controlled system with compensation:

• The step response of the model controlled system should be largely congruent with the analytical function of the controlled system.
• The model controlled system should have a certain form of the transfer function which is easily suitable for parameterizing the controller with a good linear standard controller - for example a PID controller.
• The method should be applicable for controlled systems from the 2nd order with and without dead time.

A PID controller in the product representation (series connection) can compensate for 2 PT1 delays. Therefore, the following form of the model controlled system, which is easy to determine, is selected, which consists of a series connection of a vibration-free PT2 element and a dead time element:

Step response of a 4th order controlled system with a dominant time constant and its 2nd order model controlled system with dead time element

Transfer function model:

${\displaystyle G_{SM}(s)={\frac {e^{-T_{u}\cdot s}}{(T_{M}\cdot s+1)^{2}}}}$

The following operations must be carried out in connection with a personal computer :

• The data of the step response of the system are entered in a diagram y (t) = f (t),
• A tangent is created at the turning point of the recorded step response and the equivalent dead time = delay time is tapped at the intersection with the abscissa. ${\displaystyle T_{u}}$
• The PC should be able to generate the step response of two PT1 elements using difference equations with any computer program,
• The model transfer function is varied with the parameter according to the heuristic method “trial and error” until the S-shaped increase in the step response matches the response of the model. ${\displaystyle G_{SM}(s)={\tfrac {e^{-T_{u}\cdot s}}{(T_{M}\cdot s+1)^{2}}}}$ ${\displaystyle T_{M}}$
• The equivalent dead time and the equivalent time constants of the PT2 element are thus given. Numerical calculations show that the time behavior of the step response of the controlled system corresponds very well to the model shown. ${\displaystyle T_{tE}=T_{u}}$ ${\displaystyle T_{M}}$

→ For detailed details see controlled system # Identification of a controlled system with compensation and dead time through the step response

Regulation of a controlled system with dead time and delay elements

In the specialist literature, the "controllability" of a controlled system with increasing dead time is often presented as difficult compared to other delay elements. In fact, the regulation of a controlled system with a large dead time component is just as easy to regulate as with a small dead time component, but the dynamics of the control loop are unfavorable with increasing dead time. This can be remedied by controllers with special structures such as B. the method of Smith's predictor .

If, in addition to PT1 elements, the controlled system contains a noticeable dead time in relation to a dominant time constant , an I element is required within the control loop. A controlled system consisting of pure dead time can - apart from special controllers - only be controlled by an I controller. ${\displaystyle T}$ ${\displaystyle T_{t}}$

The regulation of a controlled system with global dead time (no further transfer elements) with an I controller has a special feature that the loop gain

${\displaystyle K={\frac {k_{1}}{T_{t}}}}$; with = freely selectable factor that determines the settling behavior of the controlled variable. ${\displaystyle k_{1}}$

The limit stability (constant continuous oscillations) of such a control loop also has rules. For example, according to numerical calculation, a loop gain of K = 1.566 results for [s], at which the controlled variable oscillates with a constant amplitude. If the dead time is [s], then half the value of the loop gain K = 0.783 applies to doubling the dead time. ${\displaystyle T_{t}=1}$ ${\displaystyle T_{t}=2}$

With a fixed value of , the same amount of overshoot ü results for any values . It is also clear, depending on the loop gain and the amount of overshoot above a simple relationship of the first zero crossing of the target control size . This period from the time until the first zero crossing is reached is called the rise time: ${\displaystyle k_{1}}$ ${\displaystyle T_{t}}$ ${\displaystyle K}$ ${\displaystyle y(t)}$ ${\displaystyle t_{0}}$

${\displaystyle T_{A}}$= Delay time + rise time . ${\displaystyle T_{u}}$ ${\displaystyle T_{g}}$

Regulation of a controlled system as a pure dead time element

Step responses of a control loop with 2 different dead time control systems

Using the numerical calculation, the following relationships were found when calculating a control loop with a controlled system and a controller . This makes it possible to find important parameters for a stable control loop and the settling process for the parameterization of a controller for a given controlled system with global dead time. ${\displaystyle G_{s}(s)=e^{-T_{t}\cdot s}}$ ${\displaystyle G_{R}(s)={\frac {K}{s}}={\frac {k_{1}}{T_{t}\cdot s}}}$

If the loop gain is selected for a fixed numerical value , the overshoot is ü of the transient process of the controlled variable and the rise time for ${\displaystyle K}$ ${\displaystyle k_{1}}$ ${\displaystyle T_{A}}$

• ${\displaystyle k_{1}=0{,}7\to {\text{ü}}\approx 20\,\%\quad \to T_{A}\approx 2{,}5\cdot T_{t}}$
• ${\displaystyle k_{1}=0{,}6\to {\text{ü}}\approx 10\,\%\quad \to T_{A}\approx 3\cdot T_{t}}$
• ${\displaystyle k_{1}=0{,}5\to {\text{ü}}\approx 4\,\%\quad \ \to T_{A}\approx 3{,}8\cdot T_{t}}$

Example of a given controlled system with global dead time:

Entry jump ${\displaystyle e(t)=1(t)}$

Control path given: ${\displaystyle G_{s}(s)=e^{-T_{t}\cdot s}}$

for [s] and alternatively [s]. ${\displaystyle T_{t}=0{,}5}$ ${\displaystyle T_{t}=2}$

Suitable controller: → for ü ≈ 4% ${\displaystyle G_{R}(s)={\frac {K}{s}}={\frac {0{,}5}{T_{t}\cdot s}}}$

With the election of u and the value of the loop gain can be and the value of the rise time without elaborate simulation of the control loop can be calculated. See graphical representation of the step responses . ${\displaystyle k_{1}}$ ${\displaystyle K}$ ${\displaystyle T_{A}}$

Control path with dead time and further PT1 elements

In control engineering, it is usual to parameterize a controller by compensating the PT1 delay elements with differentiating PD1 elements. This simplifies the calculation of the cut control loop.

It makes sense to use the above relationship - selection of loop gain for a certain size of overshoot - for controlled systems with PT1 and dead time systems, in that the PT1 delay times are compensated by PD1 elements of the controller.

It should be noted that these are ideal PD1 links that cannot be manufactured technically. Real PD1 elements always contain so-called parasitic delays, the time constants of which in practice are about a tenth of the time constants of the PD1 elements.

Calculation example:

Control path given:

${\displaystyle G_{S}(s)={\frac {e^{-T_{t}\cdot s}}{(T_{1}\cdot s+1)(T_{2}\cdot s+1)}}}$

This controlled system is suitable for a PID controller in that the two PD elements of the controller compensate the two PT1 elements of the controlled system model.

The matching, ideal PID controller consists of two PD1 elements and an I element:

${\displaystyle G_{R}(s)={\frac {K\cdot (T_{v1}\cdot s+1)(T_{v2}\cdot s+1)}{s}}={\frac {0{,}6\cdot (T_{v1}\cdot s+1)(T_{v2}\cdot s+1)}{T_{t}\cdot s}}\quad {\text{für}}\ {\text{ü}}=10\,\%}$ Overshoot.

If you insert the numerical values ​​for a selected ü ( and ) and for the time constants , the transfer function of the cut control loop results as: ${\displaystyle k1}$ ${\displaystyle T_{t}}$ ${\displaystyle T_{v}=T}$

${\displaystyle G_{0}(s)={\frac {0{,}6\cdot e^{-T_{t}\cdot s}}{T_{t}\cdot s}}\quad {\text{für}}\ {\text{ü}}=10\,\%}$ Overshoot.

The parameters of the controller for the real controlled system are thus given as follows:

Loop gain for an overshoot of approx. 10%, time constants of the controller . The rise time of the closed control loop is . ${\displaystyle K={\tfrac {0{,}6}{T_{t}}}}$ ${\displaystyle T_{v}=T}$ ${\displaystyle T_{A}\approx 3\cdot T_{t}}$

If you want to calculate the continuous course of the step response of the controlled variable of the closed control loop, the above transfer function of the cut control loop can be used by means of numerical calculation, if the closing condition for the control deviation is additionally introduced and the corresponding difference equation is used for the integration. The INDEX function or an exact dead time model is best suited for calculating the dead time element. ${\displaystyle G_{0}(s)}$ ${\displaystyle e_{(k)}=w_{(k)}-y_{(k-1)}}$

Individual evidence

1. ↑ See reference book: Lutz / Wendt: “Pocket book of control engineering with MATLAB and Simulink; Chapter: Dead time element (PT _t element). "
2. ↑ The real part of the Laplace variable s is called differently in the specialist literature: or , specialist book author Prof. Dr.-Ing. Jan Lunze, University of Bochum, preferred . ${\displaystyle \delta }$ ${\displaystyle \sigma }$ ${\displaystyle \delta }$
3. ↑ Lutz / Wendt: Pocket book of control engineering, chapter: "Mathematical methods for calculating digital control loops in the time domain, sub-chapter: Difference equations".
4. ↑ See lecture notes University of Siegen, Prof. Dr.-Ing Oliver Nelles: Measurement and control technology I: "All-pass and non-phase-minimal systems"

literature

• G. Schulz: Control engineering 1: linear and non-linear control, computer-aided controller design . 3. Edition. Oldenbourg, Munich 2007, ISBN 978-3-486-58317-5 .
• M. Reuter, S. Zacher: Control technology for engineers: Analysis, simulation and design of control loops . 12th edition. Vieweg + Teubner, 2008, ISBN 978-3-8348-0018-3 .
• J. Lunze : Control engineering 1 - system theory basics, analysis and design of single-loop controls. Springer-Verlag, 2006, ISBN 3-540-28326-9 .
• H. Unbehauen : Control engineering 1. Vieweg Verlag, 2007, ISBN 978-3-528-21332-9 .
• H. Lutz, W. Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Europa-Verlag, 2019, ISBN 978-3-8085-5869-0 .