Root locus method

The root locus curve method (WOK) is a method for controller design from control engineering . It is based on the root locus and aims to position the poles of the closed circuit in such a way that the control circuit meets certain quality requirements. The result of the controller design with the root locus curve method is a controller that generally contains its own dynamics and can be, for example, a P, PI, PID controller, but also a higher-order controller.

Overview of the design process

When designing the controller with the root locus method, the fact is exploited that the root locus creates a graphical relationship between the poles and zeros of the open chain and the poles of the closed circle that is easy to interpret for the design engineer . The latter should be placed in a targeted manner. The open chain is a series circuit of controller and controlled system with : ${\displaystyle K(s)=k_{P}{\hat {K}}(s)}$ ${\displaystyle G(s)}$ ${\displaystyle G(0)=k_{s}}$

${\displaystyle G_{0}(s)=K(s)G(s)}$.

As long as the controller is still unknown, it is used. The design with the root locus method takes place in detail in the following steps. The controller was broken down into a purely dynamic part with , and a proportional gain . ${\displaystyle K(s)=1}$ ${\displaystyle {\hat {K}}}$ ${\displaystyle {\hat {K}}(0)=1}$ ${\displaystyle k_{P}}$

1. Implementation of the quality requirements in a suitable form,
2. Drawing the root locus,
3. Definition of the dynamics of the controller (its poles and zeros),
4. Draw the root locus and define the controller gain.
5. Simulation or practical testing of the circular behavior
6. If the result is not satisfactory, repeat from 1st or 2nd

Suitable software tools are now available for drawing the root locus, so that the main focus is on the selection of the controller.

Quality requirements

Characteristics of the behavior of a dynamic system, represented using the step response. The overshoot describes the largest percentage deviation of the controlled variable from the static end value. The overshoot time is determined by the point in time at which the first maximum of the step response occurs. The settling time is the last point in time at which the step response dips into a band with a width of ± 5%. ${\displaystyle \Delta h}$ ${\displaystyle T_{m}}$ ${\displaystyle T_{5\,\%}}$

The quality requirements are usually given in the time domain in the form of requirements for the setpoint sequence, the maximum permissible overshoot , the overshoot time , or the settling time. They have to be translated into requirements for the position of the dominant pole pair in the complex plane. The following relationships help here. ${\displaystyle \Delta h}$ ${\displaystyle T_{m}}$ ${\displaystyle T_{5\,\%}}$

1. Stability is given when the dominant pole pair has a really negative real part.
2. Setpoint sequence is achieved for abrupt reference variables if the open chain contains an integrator (pole at zero)
3. The attenuation is related to the complementary phase angle of the dominant pole pair via the following equation:, where , denotes the phase of the dominant pole pair. ${\displaystyle d}$ ${\displaystyle \cos \phi _{d}=d}$ ${\displaystyle \phi _{d}=180-\phi }$ ${\displaystyle \phi }$
4. The overshoot time satisfies the equation , where denotes the magnitude of the imaginary parts of the dominant pole pair. ${\displaystyle T_{m}}$ ${\displaystyle T_{m}={\frac {\pi }{\omega _{0}{\sqrt {1-d^{2}}}}}={\frac {\pi }{\omega _{e}}}}$ ${\displaystyle \omega _{e}}$
5. The overshoot satisfies the equation . ${\displaystyle \Delta h}$ ${\displaystyle \Delta h=e^{-{\frac {\pi d}{\sqrt {1-d^{2}}}}}=e^{-{\frac {\delta _{e}}{\omega _{e}}}\pi }=e^{-\pi \cot \phi _{d}}}$

Using these rules, a target area for the desired position of the poles of the closed circle can be derived from the maximum permissible overshoot and the overshoot time or settling time.

Determination of the dynamic component in the controller

Appropriate positioning of poles and zeroing of the controller must ensure that there is a proportional controller gain so that all quality requirements are met. This will give you gradually ${\displaystyle k_{P}}$ ${\displaystyle {\hat {K}}}$

1. To ensure the setpoint sequence for jump-shaped reference variables, a single pole must be present at the origin. If this is not already given by the system, a pole must be added and the controller receives an I component.
2. If the root locus has a shape so that a dominant pair of poles exists in the target area of ​​the complex plane by means of suitable reinforcement, you can move on to the next step 'Determination of the reinforcement' (next section).
3. If the branches of the root locus are still unsuitable, poles and zeros must be added to the controller at suitable points in the pole / zero image so that they suitably change the shape of the root locus in the sense of the target area.

The last step is time-consuming and may require repeated trials. It is made easier by knowing the construction rules of the root locus . With computer-aided design, the modified root locus is displayed immediately after placing a pole or a zero.

When compiling the controller, it must be ensured that it is realizable, i.e. causal. The result of this design step is a dynamic controller with static gain one. In general, the dynamic order of the controller should be kept as small as possible, because every additional pole brings further delays and makes implementation more difficult. ${\displaystyle {\hat {K}}}$

Determination of the gain

Once determined, the chain is open . To determine the gain , proceed as follows. ${\displaystyle {\hat {K}}(s)}$ ${\displaystyle G_{0}(s)={\hat {K}}(s)G(s)}$ ${\displaystyle k_{p}}$

1. A pole pair that is dominant and meets the quality requirements is selected in the root locus . ${\displaystyle {\tilde {s}}}$
2. The gain is determined using the formula , with the poles of the open chain including the dynamic controller and the zeros of the open chain including the dynamic controller. ${\displaystyle {\tilde {k}}={\frac {\Pi _{i=1}^{n}|{\tilde {s}}-s_{i}|}{\Pi _{i=1}^{q}|{\tilde {s}}-s_{0i}|}}}$ ${\displaystyle {\tilde {k}}}$ ${\displaystyle s_{i}}$ ${\displaystyle s_{0i}}$
3. ${\displaystyle k_{P}={\frac {\tilde {k}}{k_{s}}}}$ is the controller gain sought.

The controller is now implemented . ${\displaystyle K(s)=k_{P}{\hat {K}}(s)}$

See also

• Control engineering
• PID controller

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 3-52873348-9

Web links

• Basics of control engineering Lecture notes