Rule of thumb procedure (automation technology)

As a rule of thumb method in to automation or in the control technology heuristic methods for dimensioning of a regulator called that without mathematical model of the controlled system get along. Instead of modeling , experiments usually have to be carried out on the system to be controlled.

Ziegler and Nichols method

The Ziegler and Nichols method is a heuristic method for determining controller parameters. The resulting controller can be a P, PI or PID controller.

The method is therefore only suitable for existing systems that are stable or where unstable behavior cannot cause damage. It is therefore not suitable for use in the project planning phase of a system.

Scope

The controller settings according to Ziegler-Nichols are for strongly delaying processes, as they are e.g. B. occur in process engineering , provided. The turning point in the step response is characteristic of such processes. If the controller is set according to this method, a slightly oscillating control behavior (worse than when designing the controller according to the optimum amount ) but a good disturbance behavior is achieved. It is therefore particularly suitable for processes in which mainly disturbances are to be corrected.

Procedure

The procedure is available in two variants. In the first variant (also: setting to the stability margin) no assumption is made regarding the transmission behavior of the controlled system. The control loop is closed with the help of a proportional controller and the controller gain is increased until the output of the control loop with a constant input executes a continuous oscillation with the period of the controller gain . ${\displaystyle T_{krit}}$ ${\displaystyle K_{p,krit}}$

Step response of a controlled system with turning tangent to determine the value K (value of the output variable in the steady state after the step from which the steady-state gain factor K _{s is} calculated), delay time T _u and equalization time T _g . [5]

In the second variant (also: second setting rule according to Ziegler / Nichols) the controlled system is approximated as a first-order transmission element with dead time (PT1T _t element). Its stationary gain , the time constant and the dead time must be known and, if necessary, determined experimentally using the step response (see figure, step response). It applies approximately with the values ​​from the step response , and . ${\displaystyle K_{s}}$ ${\displaystyle T}$ ${\displaystyle T_{t}}$ ${\displaystyle K=K_{s}}$ ${\displaystyle T=T_{g}}$ ${\displaystyle T_{t}=T_{u}}$

The setting rules for the gain , the integral time and the derivative action time are as shown in the following table for both methods: ${\displaystyle K_{p}}$ ${\displaystyle T_{n}}$ ${\displaystyle T_{v}}$

requirement	Regulator	Controller parameters
Critical gain and period known	P	${\displaystyle K_{p}=0{,}5\cdot K_{p,\mathrm {krit} }}$
	PI	${\displaystyle K_{p}=0{,}45\cdot K_{p,\mathrm {krit} },T_{n}=0{,}85\cdot T_{\mathrm {krit} }}$
	PD	${\displaystyle K_{p}=0{,}55\cdot K_{p,\mathrm {krit} },T_{v}=0{,}15\cdot T_{\mathrm {krit} }}$
	PID	${\displaystyle K_{p}=0{,}6\cdot K_{p,\mathrm {krit} },T_{n}=0{,}5\cdot T_{\mathrm {krit} },T_{v}=0{,}125\cdot T_{\mathrm {krit} }}$
Approximation of the distance by PT1T t- term	P	${\displaystyle K_{p}=1/K_{s}\cdot T/T_{t}}$
	PI	${\displaystyle K_{p}=0{,}9/K_{s}\cdot T/T_{t},T_{n}=3{,}33\cdot T_{t}}$
	PID	${\displaystyle K_{p}=1{,}2/K_{s}\cdot T/T_{t},T_{n}=2\cdot T_{t},T_{v}=0{,}5\cdot T_{t}}$

The reset time indicates when the effect of the I component in a step response is the same as the effect of the P component: ${\displaystyle T_{n}}$

${\displaystyle T_{n}=K_{p}\cdot T_{I}}$

The derivative action time indicates when the effect of the D component in a step response is the same as the effect of the P component: ${\displaystyle T_{v}}$

${\displaystyle T_{v}={\frac {T_{D}}{K_{p}}}}$

Differential equation of the ideal PID controller in a parallel structure with control deviation e (t):

${\displaystyle u(t)=K_{p}e(t)+{\frac {1}{T_{I}}}\int _{0}^{t}{e(\tau )}d\tau +\ T_{D}{\frac {d}{dt}}e(t)}$

${\displaystyle u(t)=K_{p}\left[e(t)+{\frac {1}{T_{n}}}\int _{0}^{t}{e(\tau )}d\tau +\ T_{v}{\frac {d}{dt}}e(t)\right]}$

Restriction

Achieving an oscillation at the stability limit as described above can only be carried out, however, where the real system deviating into the unstable area does not have any harmful consequences. An unstable cruise control on the car would alternately give full throttle and no throttle, which might still be feasible in a suitable environment, with an autopilot on a passenger aircraft the consequences would certainly not be acceptable.

Adjustment rules according to Chien, Hrones and Reswick

The setting rules according to Chien, Hrones and Reswick are a procedure developed in 1952 for the favorable setting of controllers. They are considered a further development of the second method by Ziegler and Nichols. It is advantageous that the control parameters are separate for a favorable disturbance and control behavior. They are also subdivided for aperiodic or periodic regulations.

Scope

The rules apply to systems of a higher order, of which the parameters: stationary gain , delay time and equalization time must be known. (See step response picture in previous section) ${\displaystyle K_{s}}$ ${\displaystyle T_{u}}$ ${\displaystyle T_{g}}$

Procedure

The rules for setting the gain , the integral action time and the derivative action time are as given in the following table. ${\displaystyle K_{p}}$ ${\displaystyle T_{n}}$ ${\displaystyle T_{v}}$

Regulator		Aperiodic control curve		Control curve with 20% overshoot
		Disorder	guide	Disorder	guide
P	${\displaystyle K_{p}}$	${\displaystyle 0{,}3\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}3\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}7\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}7\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$
PI	${\displaystyle K_{p}}$	${\displaystyle 0{,}6\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}35\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}7\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}6\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$
	${\displaystyle T_{n}}$	${\displaystyle 4\cdot T_{u}}$	${\displaystyle 1{,}2\cdot T_{g}}$	${\displaystyle 2{,}3\cdot T_{u}}$	${\displaystyle 1\cdot T_{g}}$
PID	${\displaystyle K_{p}}$	${\displaystyle 0{,}95\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}6\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 1{,}2\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$	${\displaystyle 0{,}95\cdot {\frac {T_{g}}{T_{u}\cdot K_{s}}}}$
	${\displaystyle T_{n}}$	${\displaystyle 2{,}4\cdot T_{u}}$	${\displaystyle 1\cdot T_{g}}$	${\displaystyle 2\cdot T_{u}}$	${\displaystyle 1{,}35\cdot T_{g}}$
	${\displaystyle T_{v}}$	${\displaystyle 0{,}42\cdot T_{u}}$	${\displaystyle 0{,}5\cdot T_{u}}$	${\displaystyle 0{,}42\cdot T_{u}}$	${\displaystyle 0{,}47\cdot T_{u}}$

Restriction

The same restrictions apply as for the Ziegler and Nichols method. In order to determine the system parameters, it must be possible to carry out experiments on the unregulated process without damaging it.

Improvements to the Chien, Hrones, and Reswick process

Further optimization of the controller parameters was carried out under the direction of Samal. They are listed in the book "Practical Control Engineering" by Wolfgang Schneider and Berthold Heinrich under "Empirical setting values ​​according to Samal".

T-sum rule

This rule applies to routes with low-pass behavior that have an S-shaped step response. You are by the transfer function

${\displaystyle G(s)=K_{S}{\frac {(1+T_{D,1}s)(1+T_{D,2}s)\ldots (1+T_{D,m}s)}{(1+T_{1}s)(1+T_{2}s)\ldots (1+T_{n}s)}}e^{-sT_{t}}}$

described. The total time constant is formed as the sum of all delaying time constants minus all differentiating time constants: ${\displaystyle T_{\Sigma }}$

${\displaystyle T_{\Sigma }=T_{1}+T_{2}+\ldots +T_{n}-T_{D,1}-T_{D,2}-\ldots -T_{D,m}+T_{t}.}$

The cumulative time constant can also be determined directly from the experimentally determined transition function. It applies

${\displaystyle T_{\Sigma }=\int _{0}^{\infty }\left(1-{\frac {y(t)}{K_{S}}}\right)dt}$

with step response function of the controlled system . ${\displaystyle y(t)\;}$

The following then applies to the controller settings:

• PI controller: ${\displaystyle K_{P}=0{,}5/K_{S},\;T_{N}=0{,}5T_{\Sigma }}$
• PID controller: ${\displaystyle K_{P}=1/K_{S},\;T_{N}=0{,}66T_{\Sigma },\;T_{V}=0{,}167T_{\Sigma }}$

or for a faster control process:

• PI controller: ${\displaystyle K_{P}=1/K_{S},\;T_{N}=0{,}7T_{\Sigma }}$
• PID controller: ${\displaystyle K_{P}=2/K_{S},\;T_{N}=0{,}8T_{\Sigma },\;T_{V}=0{,}194T_{\Sigma }}$

Comparison of the procedures

Comparison of the rule of thumb using an example controlled system

The following parameters for the model controlled system were determined with a Scilabscript :

==========================================================
 Parameter der Regelstrecke
 T1=2.400000 T2=1.200000 T3=0.600000 T4=0.100000
                     1
    -----------------------------------
                    2        3         4
    1 + 4.3s + 5.46s + 2.232s + 0.1728s
 Tu=1.030072 Tg=5.183502 Tg/Tu=5.032175 TSum=4.348428
 ==========================================================
 Ziegler-Nichols
 KR=6.038610 Tn=2.060144 Tv=0.515036
 ==========================================================
 Chien/Hrones/Reswick (aperiodisch)
 KR=3.019305 Tn=5.183502 Tv=0.515036
 ==========================================================
 Chien/Hrones/Reswick (überschwingen)
 KR=4.780567 Tn=6.997728 Tv=0.484134
 ==========================================================
 T-Summe
 KR=1.000000 Tn=2.900402 Tv=0.726187
 ==========================================================
 T-Summe (schnell)
 KR=2.000000 Tn=3.478742 Tv=0.847943
 ==========================================================

Empirical dimensioning

In industrial practice, control loops are often implemented by trying out controller settings without using a model. Proportional-integral-differential controllers ( PID controllers ) are mostly used. The parameters for the proportional, integral and differential components are preselected according to practical experience and then varied.

Different control variable curves (actual values) after a manipulated variable jump with different controller settings.

The control loop can be optimized using the actual value curves:

• Violet: The actual value only slowly approaches the setpoint.
  Setting rule: Increase the proportional share. If this leads to an improvement, then reduce the integration time. Repeat this until a satisfactory controller result is achieved.
• Blue: The actual value only slowly approaches the setpoint with slight fluctuations.
  Setting rule: Increase the proportional share. If this leads to an improvement, then reduce the derivative time (differentiation time). Repeat this until a satisfactory controller result is achieved.
• Light blue: The actual value approaches the setpoint without significantly overshooting.
  Optimal controller behavior for processes that do not allow overshoot.
• Green: The actual value is approaching the setpoint with a slight dampened overshoot.
  Optimal controller behavior for fast control and for correcting disturbance components.
  Setting rule: The first overshoot should not exceed 10% of the setpoint jump.
• Red: The actual value quickly approaches the setpoint, but swings far beyond it. The vibrations are dampened and therefore just about stable.
  Setting rule: Reduce the proportional component. If this leads to an improvement, then increase the integration time. Repeat this until a satisfactory controller result is achieved.

literature

• Jan Lunze: Control engineering , Springer Verlag, Vol. 1 (2005) ISBN 3-540-28326-9 , Vol. 2 (2006) ISBN 3-540-32335-X
• Heinz Unbehauen: Control engineering , Vieweg, Braunschweig / Wiesbaden, Vol. 1 (2005) ISBN 3-528-93332-1 , Vol. 2 (2000) ISBN 3-528-73348-9
• Manfred Schleicher: Control Technology for Practitioners (2006), JUMO GmbH & Co, ISBN 3-935742-00-2
• Berthold Heinrich [Hrsg.]: Measuring, controlling, regulating (2005), Vieweg Verlag, Wiesbaden, ISBN 3-8348-0006-6
• Michael Glöckler: Basics of automation: sensors, regulation, control (2015), Springer Fachmedien, Wiesbaden, ISBN 978-3-658-05961-3

swell

1. ↑ Ziegler, JG; Nichols, NB: Optimum settings for automatic controllers, Trans. ASME , 64 (1942), pp. 759-768
2. ^ Kun Li Chien, JA Hrones, JB Reswick: On the Automatic Control of Generalized Passive Systems. In: Transactions of the American Society of Mechanical Engineers. , Vol. 74, Cambridge (Mass.), USA, Feb. 1952, pp. 175-185
3. ^ Manfred Reuter, Serge Zacher: Control engineering for engineers , Vieweg Verlag, 11th edition (2004), ISBN 3-528-05004-7
4. ^ Schneider, Wolfgang and Heinrich, Berthold: Practical control engineering: Effective learning through examples , Springer-Verlag, (2017), ISBN 978-3-658-16992-3
5. ↑ Udo Kuhn: A practical setting rule for PID controllers: The T-Sum rule, Automation Technology Practice , No. 5, 1995, pp. 10-16
6. ↑ Jürgen Müller: Rules with SIMATIC , Publicis Corporate Publishing, Erlangen (2004), ISBN 3-89578-248-3