Root locus

The root locus curve ( WOK ) is a graphic representation of the position of the poles and zeros of the complex control transfer function F ₀ ( s ) of a control loop as a function of a parameter and is used in the field of control engineering for stability studies. The method of the root locus can be used both in the complex s-plane , for continuous systems, and in the complex z-plane for time-discrete systems. The process was developed in 1948 by the American control engineer Walter Richard Evans .

General

System A _ol (controlled system) with feedback via a P element with the factor β

The stability of a system is determined by the position of its poles in the complex plane. For example, and shown on the basis of the adjacent figure, in the time-continuous system A _ol cause poles in the transfer function in the right half-plane, i.e. H. with positive real part, an instability. With the possibility of feedback , the position of these poles in the complex plane can be shifted in the feedback system; the root locus graph represents a way of illustrating this shift in pole.

In the right figure, the feedback is shown in the simplest case with a constant factor β. The system function is exemplary and given as a first-order system for the sake of simplicity:

${\displaystyle A_{OL}={\frac {b}{s-a}}}$

For this system is unstable because the pole is to the right of the imaginary axis. The transfer function of the feedback system between input and output is: ${\displaystyle a>0,b>0}$

${\displaystyle G(s)={\frac {A_{OL}}{1+\beta A_{OL}}}={\frac {b}{s-a+\beta \cdot b}}}$

and has a pole position at . The displacement of the pole as a function of the gain factor β is graphically displayed in the root locus. In this case, the feedback system becomes stable for gain values. Conversely, a stable system A _ol with the above system function and the parameters with a positive feedback can be converted into an unstable feedback system. ${\displaystyle s=a-\beta \cdot b}$ ${\displaystyle \beta \geq {\frac {a}{b}}}$ ${\displaystyle a<0,b>0}$ ${\displaystyle \beta <0}$

The root locus shows the shift of the poles as a function of the parameters of the feedback or the controller and thus enables conclusions to be drawn about the stability behavior and the dynamics of the control loop. In higher-order systems, the shifting of the poles in the complex plane takes on more complicated forms and requires the following construction specifications for determination.

definition

In the following, let G _{0 be} the transfer function of the open (not feedback) system. The root locus includes all points on the complex plane, which have the characteristic equation:

${\displaystyle 1+k\cdot G_{0}(s)=0}$

fulfill. If it applies , then it is the actual root locus , otherwise it is the complementary (or improper ) root locus . A solution to the equation for fixed k is called the root location . ${\displaystyle \quad 0\leq k<\infty }$

properties

The root locus is symmetrical to the real axis. It begins for in the poles of the open, corrected circle (L (s) = H (s) * G (s)) and ends for in its zeros . Solutions for belong to the actual (positive) WOK, the curves for belong to the complementary (negative) WOK. ${\displaystyle k=0}$ ${\displaystyle s_{i}}$ ${\displaystyle k\to \pm \infty }$ ${\displaystyle s_{0i}}$ ${\displaystyle k\geq 0}$ ${\displaystyle k<0}$

Exact construction

Example of a root locus with four poles and four zeros

For the exact construction of the root locus, the transfer function of the open chain is broken down as follows:

${\displaystyle G_{0}(s)=k{\hat {G}}_{0}(s)=k{\frac {\Pi _{i=1}^{q}(s-s_{0i})}{\Pi _{i=1}^{n}(s-s_{i})}}}$

This denotes the system order and the number of zeros in the system. The root locus includes all complex points that meet the amplitude condition and the phase condition: ${\displaystyle n}$ ${\displaystyle q}$

Amplitude condition: , ${\displaystyle {\frac {|\Pi _{i=1}^{q}(s-s_{0i})|}{|\Pi _{i=1}^{n}(s-s_{i})|}}={\frac {1}{|k|}}}$

Phase condition: , ${\displaystyle \sum _{i=1}^{q}\Phi _{0i}-\sum _{i=1}^{n}\Phi _{i}=(2l+1)\pi ,\quad l=0,1,2,...}$

where and for each zero point or each pole denote the mathematically positive counted angles between an imaginary horizontal ray pointing to the right of the zero point or pole and the point to be checked. ${\displaystyle \Phi _{0i}}$ ${\displaystyle \Phi _{i}}$

The amplitude condition can also be used to determine the associated gain for a given point on the root locus . ${\displaystyle k}$

Rules for sketching

The amplitude and phase conditions can be used by a computer for the numerical construction of the root locus. Their use for manual sketching is unwieldy, so the following design rules were derived.

1. Origin / end: Each branch of the root locus begins in a pole of the open chain and ends in a zero point of the open chain, or at infinity. ${\displaystyle G_{0}}$
2. Asymptotes: For large reinforcements, the branches approach straight lines asymptotically. The number of asymptotes is . The asymptotes have angles of inclination and intersect at the common point of intersection (center of gravity of the roots) . ${\displaystyle n-q}$ ${\displaystyle k>0}$ ${\displaystyle \phi _{\mathrm {As} }={\frac {\pi +l\cdot 2\pi }{n-q}},\quad l=0,1,...,n-q-1}$ ${\displaystyle s_{\mathrm {As} }={\frac {\sum _{i=1}^{n}s_{i}-\sum _{i=1}^{q}s_{0i}}{n-q}}}$
3. Real axis: The actual root locus includes precisely those points on the real axis for which the number of real critical points (zeros and poles) on the right-hand side is odd. All other points on the real axis belong to the complementary WOK ( ). Each point on the real axis is part of a WOK: either part of the actual WOK ( ) or part of the complementary WOK ( ). ${\displaystyle s}$ ${\displaystyle k<0}$ ${\displaystyle k>0}$ ${\displaystyle k<0}$
4. Branch and union points: Branch and union points are exactly those points that satisfy both the phase condition and the equation . ${\displaystyle \sum _{i=1}^{q}{\frac {1}{s-s_{0i}}}=\sum _{i=1}^{n}{\frac {1}{s-s_{i}}}}$

application

The root locus enables the stability to be analyzed when the closed control loop is given without explicitly calculating the command transfer function . If all poles and zeros are in (open left half-plane), the closed control loop is stable. If one or more poles are in (open right half-plane), the system is unstable. If one or more poles are on the imaginary axis and all remaining poles are in the left half-plane, one speaks of a conditionally stable or borderline stable system. If all poles are on the imaginary axis ( real part equal to 0), then it is an undamped system. If there is a complex conjugated pole pair, the system is capable of oscillation. If this is in the left half-plane, the system is able to vibrate in a damped manner (decaying vibration). If the complex conjugate pole pair is in the right half-plane, the system carries out an increasing (fanned) oscillation and is therefore unstable. ${\displaystyle \left(G(s)={\frac {G_{0}(s)}{1+G_{0}(s)}}\right)}$ ${\displaystyle G(s)}$ ${\displaystyle Re(-)}$ ${\displaystyle Re(+)}$

The root locus is suitable for linear and time-invariant systems such as PID controllers . The gain is usually taken as a free parameter, which is used for a controller design using the root locus method.

Amplitude margin, left-hand distance, degree of damping, root angle

For the system design, the absolute statement about the stability is a basic requirement, but is often not sufficient. In principle, a statement about the quality of the stability is also required. Specifically, measures are to be determined that state whether the system is behaving as desired - for example, its distance from the stability limit or its calming behavior.

The stability limit of the root locations is the imaginary axis. If the dominant root crosses the imaginary axis, a distance to this border crossing can be assigned to each of its locations. This distance is primarily to be found in the value range of the parameter.

The amplitude margin is a relative measure of the distance . It relates the value of the parameter to its value for limit stability and states the factor with which the parameter must be increased or decreased in order for limit stability to occur. ${\displaystyle A_{R}}$ ${\displaystyle k}$ ${\displaystyle k_{kritisch}}$ ${\displaystyle k}$

${\displaystyle A_{R}={\frac {k}{k_{kritisch}}}}$

For the parameter appears as an absolute distance measure. ${\displaystyle k_{kritisch}=0}$ ${\displaystyle k}$

If the root locus does not exceed the imaginary axis, no amplitude margin can be determined.

For such systems, however, the left-hand distance between the dominant root and the imaginary axis can possibly be regarded as a measure of quality. However, a clearly defined reference point is missing here. For example, you can put the left-hand distance of the dominant root in relation to its distance for ; you get something like a stabilization factor . ${\displaystyle k=0}$ ${\displaystyle S}$

${\displaystyle S={\frac {\sigma }{\sigma _{0}}}={\frac {re\{s\}}{re\{s_{k=0}\}}}}$

If the parameter is the loop gain of a single loop system, then is the ratio between the decay constants of the closed loop and the open loop. ${\displaystyle k}$ ${\displaystyle S}$

For complex dominant roots, their degree of attenuation D is often mentioned as a measure of stability, corresponding to the angle that the dominant roots enclose. Reference is made here to the behavior of the PT ₂ system with the characteristic equation ${\displaystyle 2\alpha }$

${\displaystyle 0=s^{2}+2D\omega _{0}s+\omega _{0}^{2}}$

According to the solutions s of this equation, for the degree of damping D and the angle of conjugate complex roots result: ${\displaystyle \alpha }$

${\displaystyle \tan {\alpha }=\left\vert {\frac {im\{s\}}{re\{s\}}}\right\vert }$ such as

${\displaystyle \cos {\alpha }=D={\frac {\left\vert re\{s\}\right\vert }{\left\vert s\right\vert }}}$.

The degree of damping D of the dominant pair of roots, defined by the angle, provides information about the quality of the step response; it largely determines the strength of the overshoot and thus the calming behavior. In the 2nd order system, the degree of damping D is proportional to the left-hand distance of the complex roots from the imaginary axis, whereby D also represents a direct measure of stability for this system. The following were defined as reference points:

- the limit stable case with D = 0 (here = 90 °), ${\displaystyle \alpha }$

- the aperiodic borderline case with D = 1 (here = 0 °), ${\displaystyle \alpha }$

whereby the aperiodic limit case also represents the greatest possible distance between the roots and the imaginary axis. For systems of a higher order, the situation can (but need not) be different. A smaller angle does not automatically mean a higher degree of stability here. The opening direction and position of the complex figure can be completely different from the 2nd order system. ${\displaystyle \alpha }$

For this reason, the damping factor alone can be unsuitable for characterizing the stability of an unknown system. For this reason, only the distance re {s} to the imaginary axis is usually used as a measure of stability and the maximum opening angle of the root locus is required as a further criterion.

For systems that are already known to be sufficiently stable, it is sufficient to consider the opening angle.

Systems with dead time

For systems with dead time , it is often believed that the root location method is not applicable to these systems. However, such systems also have poles whose locus can be calculated exactly. The availability of mathematical software with symbolic treatment makes this possible. Potential theory can be used as the basis for these calculations.

Remarks

The Nyquist method continues to be the gold standard for stability analysis - and not without reason.

On the one hand, the simplified Nyquist criterion (left-hand rule) provides a standardized method that provides a reliable measure of stability even without clairvoyant abilities , namely primarily the Euclidean distance of the complex frequency response of the open loop from the Nyquist point. On the other hand, you often save yourself having to test the necessary preconditions because this can be very time-consuming.

However, the left-hand rule may only be applied if the open loop is stable! The open loop is not stable if the controlled system or the controller are not stable. Examples are:

- the unstable pendulum

- the floating magnet

- the quadrocopter

A stability measure has to be found (better said: guess) for such systems with another method. However, there is no algebraically based theory for this.

It is known for control-technical function blocks whether they are inherently stable. The situation is different when designing complex integrated circuits, such as operational amplifiers. These can be extensive active systems, the loop structure of which is not apparent, so that an extensive test must always be carried out before the left-hand rule is applied. Gräfe developed the knot potential method for this.

[Martin Gräfe: "Development of an integrated infrared transmission system with the help of computer-aided analysis methods for analog circuit design", VDI-Verlag Düsseldorf, series 9 (electronics), no. 297, ISBN 3-18-329709-4 ]

The root locations are determined for the closed loop or the system as a whole, without knowledge of the function of the individual parts. The root locus does not automatically provide information about which optimization measures are to be taken. Optimization is often based on the "trial and error" principle; H. exploratory and not algebraic.

Polynomials of order greater than 4 are usually solved numerically. For closed-loop control systems, it is advisable to create the characteristic polynomial and then attempt the search for the roots (instruction: roots). If the description is available as a system of equations, the eigenvalue search is used (instruction: eig).

Example: the floating magnet

The floating magnet
Link to the picture
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Magnetic position tracking is discussed below as an example to illustrate the system behavior using the root locations. A magnet should be held at a certain height without contact, as shown in the picture on the right.

The position tracking should be done by a PID controller. On the one hand, this enables a purely intuitive design process that is easy to understand. On the other hand, all the settings suggested here can also be easily recalculated.

The floating magnet is an unstable controlled system without a steady state; A PD or PID controller is recommended. Why the other combinations don't work is explained below.

The object is held in its position by the controllable magnetic field of an electromagnet. The current position x of the object is measured by a position transmitter; the desired rest position is defined with x = 0, corresponding to an encoder voltage of 0V. The path x is measured positively upwards from this position .

Two forces act on the controlled object. On the one hand, the gravitational field causes the weight force that permanently pulls the body vertically downwards. On the other hand, the magnetically generated force also acts in the vertical direction, but the magnitude and sense of direction are variable. ${\displaystyle F_{Gew}}$ ${\displaystyle F_{Mag}}$

The resulting force accelerates the body in the x-direction and is therefore called the acceleration force according to the law of acceleration: ${\displaystyle F_{Beschl}}$

${\displaystyle F_{Beschl}=m\cdot a}$ With ${\displaystyle a={\dot {v}}={\ddot {x}}}$

If the magnetic force is greater than the weight, the body is accelerated upwards; the upward acceleration force , like the way, is counted positively. Hence:

${\displaystyle F_{Beschl}=F_{Mag}-F_{Gew}}$

The (very general) basic equation of the system follows with the law of acceleration.

${\displaystyle m\cdot {\ddot {x}}=F_{Mag}(x)-F_{Gew}}$

In this, all quantities derived from position x are functions of time t, only the weight force is invariant. We still have to determine how the magnetic force is determined from the position x. The function is the controller function . ${\displaystyle F_{Mag}}$ ${\displaystyle F_{Mag}(x)}$

For the rest position to be approached , we test a body that is at rest in the differential equation - all derivatives of the path with respect to time are zero.

${\displaystyle x'=x''=x'''...=0\quad }$

If we then receive a specific position , this is the inhomogeneous (stationary) solution - the final control value. In this position, the body is in equilibrium, because it applies ${\displaystyle x}$

${\displaystyle {\ddot {x}}=0\quad \leftrightarrow \quad F_{Mag}=F_{Gew}}$

1st attempted solution: proportional controller

The consideration of the proportional controller is simple: the further the object hangs down (x <0), the higher the magnetic force. So the controller function applies:

${\displaystyle F_{Mag}=-k_{p}\cdot x}$

We plug this into the basic equation and get the differential equation for this system.

${\displaystyle m\cdot {\ddot {x}}=-k_{p}\cdot x-F_{Gew}}$

We move the inhomogeneous part to the left, and we would also like to have positive coefficients for the homogeneous part.

${\displaystyle {\bf {-F_{Gew}=m\cdot {\ddot {x}}+k_{p}\cdot x}}}$

The _final control _value x _inh is the stationary (inhomogeneous) solution. The body is not accelerated there.

${\displaystyle -F_{Gew}=k_{d}\cdot x_{inh}}$

${\displaystyle x_{inh}=-{\frac {F_{Gew}}{k_{p}}}}$

The floating magnet will (as expected) hang down a bit in its rest position, depending on its mass and the proportional gain. But we also have to clarify whether the system actually moves to this rest position. A dynamic solution is superimposed on the stationary solution - the intrinsic behavior of the system. This dynamic solution must subside; otherwise the control end value is not approached.

The dynamic behavior is determined solely by the homogeneous part of the differential equation. The eigenvalues satisfy the characteristic equation. ${\displaystyle \lambda }$

${\displaystyle 0=m\cdot \lambda ^{2}+k_{p}}$

This equation has exactly two solutions, which are complex conjugate for all k _p > 0.

${\displaystyle \lambda _{1/2}=\pm j\omega }$

The system carries out an undamped oscillation around the rest position with the frequency . The magnet only remains in the rest position if it was already at rest there at the start of control. Every shock and every position deviation from the rest position is answered with a never-ending oscillation, which means that the proportional controller is not yet sufficient for the task at hand. ${\displaystyle \omega ={\sqrt {k_{p}/m}}}$

2nd attempted solution: adding a damping: PD controller

In addition to the forces already introduced, there is another force which counteracts the movement of the object to be held. This can, for example, be a frictional force such as would arise from a movement in a viscous liquid. ${\displaystyle v={\dot {x}}}$

Such a force can also be generated purely magnetically by the controller. This additional magnetic force increases the faster the object moves downwards. The controller now has the equation:

${\displaystyle F_{Mag}=-k_{p}\cdot x-k_{d}\cdot {\dot {x}}}$

The differential equation is now:

${\displaystyle {\bf {-F_{Gew}=m\cdot {\ddot {x}}+k_{d}\cdot {\dot {x}}+k_{p}\cdot x}}}$

The control end value is to be found where the speed and acceleration are zero. As expected, this is the rest position x _inh <0 of the system with proportional controller.

The dynamic behavior looks different now. The characteristic equation is:

${\displaystyle 0=m\cdot \lambda ^{2}+k_{d}\cdot \lambda +k_{p}}$

${\displaystyle {\bf {0=\lambda ^{2}+K_{D}\cdot \lambda +K_{P}}}}$

For the 2nd order system, the degree of damping D is defined as the stability measure:

${\displaystyle 0=\lambda ^{2}+2D\omega _{0}\cdot \lambda +\omega _{0}^{2}}$

The degree of attenuation D can be freely selected for each specified angular frequency . It is proportional to the differential gain K _D. ${\displaystyle \omega _{0}={\sqrt {K_{P}}}}$

${\displaystyle D={\frac {K_{D}}{2{\sqrt {K_{P}}}}}}$

This can go beyond the aperiodic limit case (D = 1). We can leave the determination of the roots to the computer (instruction: roots). We can also do this by hand:

${\displaystyle \lambda _{1/2}=\omega _{0}\left(-D\pm {\sqrt {D^{2}-1}}\right)}$

Complete display
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A change in does not change the relative position of the roots . For this form of representation, only the degree of attenuation D generates a meaningful root locus as a parameter; all possible operating states of the system are thereby recorded. Such a complete representation is shown in the following picture. ${\displaystyle \omega _{0}}$ ${\displaystyle \lambda _{1/2}}$

The vertical lines mark the root centers of gravity for the degrees of damping indicated above. Furthermore, the qualitatively expected time courses of the homogeneous solution are given. Only the states to the left of the imaginary axis are of interest for the design. For D = 0, the root locations start imaginarily conjugated at and move away from there on a circular path for increasing degrees of damping. For D = 1 you reach the aperiodic limit case. From this point, for D> 1, one root pulls towards negative infinity, the other asymptotically towards the origin. ${\displaystyle {\displaystyle \lambda _{1/2}}\pm j\omega _{0}}$

The degree of damping D represents a proportional measure of stability for the 2nd order system, which is associated with the root centers of gravity. For the periodic system this also corresponds to the left-hand distance of the conjugate complex roots.

For the aperiodic system, however, the left-hand distance is misleading, because for D> 1 this distance becomes smaller again. However, the dominant real root remains on the left half-plane; it is not heading for another border crossing. The distance to the only existing crossing D = 0 is also for increasing values of the parameter D is greater . It cannot be otherwise, because an increasing degree of damping also results in a higher degree of stability for the 2nd order system. All assumptions to the contrary must obviously be wrong. For this reason it was suggested above to search for the distance to a limit crossing primarily in the space of the parameter.

In the left-hand aperiodic limit case D = 1 , the root locations have the greatest possible distance from the imaginary axis, which means the smallest response time , based on a fixed natural angular frequency . K _D and K _P are then in a defined relationship to one another. ${\displaystyle \omega _{0}}$

${\displaystyle AGF:\quad K_{D,agf}=2\cdot {\sqrt {K_{P,agf}}}\quad \leftrightarrow \quad \lambda _{agf}=-D\omega _{0}=-{\frac {K_{D}}{2}}\quad \leftrightarrow \quad T_{99}\approx \left|{\frac {5}{\lambda _{agf}}}\right|={\frac {10}{K_{D}}}=10{\frac {m}{k_{d}}}\quad \quad \quad \left[T_{99}\right]={\frac {kg}{\frac {N}{\frac {m}{s}}}}={\frac {kg}{N}}{\frac {m}{s}}={\frac {s^{2}}{m}}{\frac {m}{s}}=s}$

A single overshoot can occur for D = 1, which is not insignificant due to the short settling time. The overshoot can only be reduced by increasing the degree of damping. If the system is now too sluggish, the proportional gain KP (and thus ) can be increased; then the differentiation gain KD is brought back to the desired ratio to KP. It is therefore possible to fit the system behavior into a given tolerance scheme if correspondingly high forces can be generated. ${\displaystyle \omega _{0}}$

Individual evidence

1. ↑ Manfred Reuter, Serge Zacher: Control technology for engineers . 12th edition. Vieweg + Teubner, 2008, ISBN 978-3-8348-0018-3 , p. 201 ff .
2. ↑ S. Bernhard: Numerical calculation of root locus curves for parallel and return structures with dead time. 1998.
3. ↑ D. Krauße: Synthesis of frequency response compensation networks for integrated broadband signal amplifiers. 2011.

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 .

Web links

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