How to Design Pid Controller Using Root Locus

Root Locus

In addition, the root locus method allows us to determine the specific value of the system coefficient for a desired pole location and with it for a desired dynamic response.

From: Linear Feedback Controls (Second Edition) , 2020

Root locus

Yazdan Bavafa-Toosi , in Introduction to Linear Control Systems, 2019

5.6 Summary

The root locus of a system refers to the locus of the poles of the closed-loop system. In this Chapter we have dissected the method of root locus by which we could draw the root locus using the open-loop information of the system without computing the closed-loop poles. The method has some simple rules which have been fully detailed. The nitty-gritties of the subject have been girdled. Then we have extended the root locus method to the case that more than one parameter in the system varies, the so-called root contour problem. The root locus method also gives us guidelines for controller design. We have tendered and reflected on a modus operandi for controller design in the two frameworks of difficult and simple systems in details with the aid of several illuminative examples. Numerous worked-out problems which follow at the end of the chapter enhance digestion of the subject.

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The root locus method

Mark A. Haidekker , in Linear Feedback Controls (Second Edition), 2020

Abstract

The root locus method is one of the most powerful tools in the design engineer's toolbox to design a feedback control system that meets given design specifications. The significance of s-plane poles for a system dynamic response was highlighted in Chapter 6. The root locus method allows us to determine the traces of the poles in the s-plane as any one coefficient of the closed-loop system (for example the controller gain) is varied. In addition, the root locus method allows us to determine the specific value of the system coefficient for a desired pole location and with it for a desired dynamic response.

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Dynamic Response Compensation

Dennis L. Feucht , in Handbook of Analog Circuit Design, 1990

6.3 Feedback Circuit Response Representation

The feedback techniques of Chapter 3 derived closed-loop response from loop gain. The closed-loop gain A _v(s) is also determined from the loop gain GH(s). Feedback in the s-domain is the subject of control theory, found typically in control and circuits textbooks, and will not be systematically developed here.

developed here. Instead, basic aspects of amplifier stability and good dynamic response are explained, leading to methods for compensation of amplifiers that have undesirable responses.

Of the representations of A _v(s), the Bode, polar (or Nyquist), and root-locus plots are the most commonly used. Bode plots are already familiar and present the frequency and phase response. Polar plots of the imaginary (jω-axis) and real (σ-axis) components of GH with ω as the parameter are an alternative representation in polar form. For each of these representations, closed-loop performance is determined by the loop-gain characteristics.

Root-locus diagrams are s-plane plots of the loci of closed-loop poles with open-loop gain K as a parameter. As K increases from zero, the closed-loop poles begin at open-loop poles and proceed toward open-loop zeros (some of which may be at infinity). When these poles leave the left half-plane, the feedback circuit becomes unstable. The pole loci can be found by setting the denominator of A _v(s) to zero. Then,

$1+G(s)H(s)=0$

or GH = −1 = 1e ^±π. In polar form, the locus conditions are

(6.10) $\Vert GH\Vert =1,\angle (GH)=\pm {180}^{°}$

Locating the loci in the s-plane is simplified by root-locus rules. These rules are constraints imposed on the location of the closed-loop poles by (6.10). Some of the more commonly used (and easily remembered) rules are the following:

1.

The root loci start at the poles of GH (for K = 0).

2.

The root-loci terminate at the zeros of GH.

3.

There are as many separate root loci as poles of GH.

4.

The loci are symmetrical about the real axis.

5.

The root loci are on the real axis to the left of an odd number of real poles and zeros of GH.

6.

The sum of the closed-loop poles is constant. (The centroid of the loci remains constant.)

Other rules can be constructed from (6.10).

The Bode and root-locus plots for an amplifier with a frequency-independent H and a single, real pole − p are shown in Fig. 6.5. The amplifier gain is

(6.11) $G(s)=\frac{K}{s/p+1}$

The closed-loop gain for positive K and H is then

(6.12) $A_{\text{v}}(s)=\frac{G}{1+GH}=\left(\frac{K}{KH+1}\right)\frac{1}{s/(KH+1)p+1}$

The closed-loop response is also that of a single, real pole, but at the frequency of ω_bw = (KH + 1)p. The bandwidth has been extended by KH + 1. This response is unconditionally stable. [Whenever steady-state frequency response (jω-axis response) is related to pole locations in s, it is assumed that the positive value of the real component of the pole location is used in relation to the steady-state frequency. To be precise, ω_bw = (KH + 1)|−p| for real poles. Since frequency response involves only positive frequencies, and p > 0 for negative poles, no confusion should result.] The root-locus plot is shown in Fig. 6.5b. The open-loop pole at −p moves toward and terminates at the closed-loop pole −(KH + 1)p.

Next, consider an amplifier with two poles:

(6.13) $G(s)=\frac{K}{(s/p_1+1)(s/p_2+1)}$

For H constant with frequency, the closed-loop response is

(6.14) $A_{\text{v}}(s)=\left(\frac{KH}{KH+1}\right)·\frac{1}{s^2/(KH+1){\omega }_{\text{n}}^2+2\zeta s/(KH+1){\omega }_{\text{n}}+1}$

A _v is also a quadratic pole response. The closed-loop parameters are

(6.15) ${\omega }_{\text{nc}}={\omega }_{\text{n}}\sqrt{KH+1}=\sqrt{p_1{p}_2(KH+1)}$

and

(6.16) ${\zeta }_{\text{c}}=\frac{\zeta }{\sqrt{KH+1}}=\frac{p_1+p_2}{2{\omega }_{\text{nc}}}$

For complex poles, both pole angle and magnitude depend on the dc loop gain, as did the single-pole response. That is why dc loop gain is the parameter of closed-loop pole movement for root-locus plots. For both first- and second-order loop gain, stability is unconditional. Response can become unacceptably underdamped for excessive loop gain in (6.14), but the poles remain in the left half-plane. The Bode magnitude and root-locus plots are shown for second-order loop gain in Fig. 6.6.

FIG. 6.6. A two-pole feedback amplifier Bode plot (a) and root locus (b). As K increases, the poles become complex.

The Bode plot of ‖G‖ and ‖1 ≠ H‖, for G of (6.11) and constant H, is shown in Fig. 6.7. Because the magnitude axis is logarithmic, the difference between the ‖G‖ and 1/H plots is the loop gain. That is,

FIG. 6.7. The 1/H curve can be used as the unity-gain "axis" for analyzing loop gain.

(6.17) $\log \Vert G\Vert -\log (1/H)=\log \Vert GH\Vert$

These Bode plots are an alternative to calculation and plotting of ‖GH‖ to determine response characteristics. We need only plot ‖G‖ and 1/H separately and then use 1/H as the unity-gain axis. This applies also for ‖H(jω)‖. In Fig. 6.7, the open- and closed-loop gains intersect at ω_bw of (6.12). ‖A _v‖ rolls off with ‖GH‖ above this closed-loop bandwidth.

The closed-loop bandwidth can be calculated from Fig. 6.7. The dc gain magnitude of G is K, and since 1/H is constant, the difference between them is KH on a Bode plot. The slope of ‖G‖ due to the pole at p is −1. Since the ω axis is also logarithmic, a logarithmic frequency difference is a ratio, and ω_bw/p = KH + 1. The bandwidth is then

(6.18) ${\omega }_{\text{bw}}=\left(KH+1\right).P$

and is the same as for the plot of ‖GH‖ in Fig. 6.5a.

Figure 6.8 shows some Bode and root-locus plots for circuits with up to three poles and two zeros. Bode plots show the gain-phase relationship with frequency directly and are most useful for compensating fixed-gain amplifiers. Root-locus plots show the closed-loop poles in the s-plane and how these poles vary with loop gain. For circuits with three or more poles, the closed-loop poles can leave the left half-plane with increasing K. The addition of zeros tends to "bend" the loci back from the jω-axis. This effect is a basis for response compensation.

FIG. 6.8. Several common pole-zero configurations: Bode and root-locus plots.

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Feedback Control Systems

Revised by William R. Perkins , in Reference Data for Engineers (Ninth Edition), 2002

Root-Locus Method

Root locus is a method of design due to Evans. It is based on the relation between the poles and zeros of the closed-loop system function and those of the open-loop transfer function. The rapidity and ease with which the loci can be constructed form the basis for the success of root-locus design methods, in much the same way that the simplicity of the gain and phase plots (Bode diagrams) make design in the frequency domain so attractive. The root-locus plots can be used to adjust system gain, guide the design of compensation networks, or study the effects of changes in system parameters.

In linear time-invariant "lumped" systems, G(s) is a rational algebraic function, the ratio of two polynomials in s:

$G\left(s\right)=m\left(s\right)\text{/}n\left(s\right)$

From Fig. 7

$\begin{array}{c}\left(C\text{/}R\right)\left(s\right)=G\left(s\right)\text{/}\left[1+G\left(s\right)\right]\\ =\frac{m\left(s\right)\text{/}n\left(s\right)}{1+\left[m\left(s\right)\text{/}n\left(s\right)\right]}\\ =m\left(s\right)\text{/}\left[m\left(s\right)+n\left(s\right)\right]\end{array}$

The zeros of the closed-loop system are identical with those of the open-loop system function. The closed-loop poles are the values of s at which m(s)/n(s) = −1. The root-locus method is a graphical technique for determination of the zeros of m(s) + n(s) from the zeros of m(s) and n(s). Root loci are plots in the complex s plane of the variations of the poles of the closed-loop system function with changes in the open-loop gain. For the single-loop system of Fig. 7, the root loci constitute all s-plane points at which

$\angle G\left(s\right)={180}^{\circ }+n{360}^{\circ }$

where n is any integer including zero. A graphical interpretation for

$G(s)=\frac{K(s+z_1)(s+z_2)}{s(s+p_1)(s+p_2)(q+p_3)}$

is given in Fig. 12. Examples of root loci are given in Fig. 13 and 14.

Fig. 12. Graphical interpretation of G(s).

Fig. 13. Root locus for G(s) = K/[s(s + 1)]. Values of K as indicated by fractions.

Fig. 14. Root locus for G(s) = K/[s(T ₁ s + 1)(T ₂ s + 2)].

For the example of Fig. 14, K = K ₁ produces the case of critical damping. An increase in gain somewhat beyond this value causes a damped oscillation to appear. The latter increases in frequency (and decreases in damping) with further increase in gain. At gain K ₃, a sustained oscillation will result. Instability exists for gain greater than K ₃, as at K ₄. This corresponds to poles in the right half of the s plane for the closed-loop transfer function.

Various rules are available as aids in sketching root locus plots by hand. (Computer-aided packages can also be used; see section "Computer-Aided Analysis and Design.")

Intervals Along the Real Axis: The simplest portions of the plot to establish are the intervals along the negative real (−σ) axis, because then all angles are either 0° or 180°. Complex pairs of zeros or poles contribute no net angle for points along the real axis. Along the real axis, the locus will exist for intervals that have an odd number of zeros and poles to the right of the interval (Fig. 15).

Fig. 15. Root-locus intervals along the real axis.

Asymptotes: For very large values of s, G(s) ∼ K/s^n–m . The locus will thus finally approach (n − m) asymptotes at the angles (Fig. 16) given by the expression

Fig. 16. Final asymptotes for root loci. Top, 60° asymptotes for system having three poles. Bottom, 45° asymptotes for system having an excess of four poles over zeros.

$({180}^{\circ }+k{360}^{\circ }\text{/}(n-m)$

These asymptotes meet at a point s ₁ (on the negative real axis) given by

$s_1=\frac{\sum \left(\text{poles}\right)-\sum \left(\text{zeros}\right)}{\left(\text{finite poles}\right)-\left(\text{finite}\text{zeros}\right)}$

The other m branches of the locus will approach the zeros of G(s), which are the zeros of n(s).

Breakaway Points: Breakaway points from the real axis occur where the net change in angle caused by a small vertical displacement is zero. In Fig. 17, point p satisfies this condition at 1/x ₀ = (1/x ₁) + (1/x ₂).

Fig. 17. Breakaway point.

Intersections With jω Axis: The Routh-Hurwitz test applied to the polynomial m(s) + n(s) frequently permits rapid determination of the points at which the loci cross the jω axis and the value of gain at these intersections.

Angles of Departure and Arrival: The angles at which the loci leave the poles and arrive at the zeros are readily evaluated from

$\sum \angle \text{vectors from zeros to}s-\sum \angle \text{vectors from poles to}s={180}^{\circ }+n{360}^{\circ }$

For example, consider Fig. 18. The angle of departure of the locus from the pole at (−1 + j1) is desired. If a test point is assumed only slightly displaced from the pole, the angles contributed by all critical frequencies (except the pole in question) are determined approximately by the vectors from these poles and zeros to (−1 + j1). The angle contributed by the pole at (−1 + j1) is then just sufficient to make the total angle 180°. In the example shown in the figure, the departure angle is found from the relation

Fig. 18. Loci for G(s) = K(s + 2)/[s(s + 3)(s ² + 2s + 2)].

Hence, θ = −26.6°, the angle at which the locus leaves (−1 + j1).

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Vibrations Induced by Internal Fluid Flow

In Flow-induced Vibrations (Second Edition), 2014

4.1.1.3 Stability of straight pipes conveying fluid

Root loci of the complex frequencies in the case of pinned-pinned pipes with α=κ=γ=0 are shown in Fig. 4.3 [8], for (a) β=0.1 and (b) β=0.5. In Fig. 4.3(a), with increasing flow velocity, the frequency of the first mode decreases and eventually vanishes at the dimensionless velocity u=π. This is the first critical flow velocity for divergence. Similarly, the frequency of the second mode vanishes at u=2π. However, at a slightly higher flow velocity u>2π, the loci of the first and second modes coalesce on the imaginary axis and leave the axis at symmetrical points, indicating the onset of coupled-mode flutter.

Figure 4.3. Root loci of pinned-pinned pipes with increasing flow velocity for α=κ=γ=0, and (a) β=0.1 or (b) β=0.5 [8].

In Fig. 4.3(b), the first-mode frequency vanishes at u=π. However, u=2π does not correspond to buckling in the second mode, but it is the point where the system regains stability in its first mode. At a slightly higher flow velocity u>2π the first- and second-mode loci coalesce on the real axis, once again indicating the onset of coupled-mode flutter. With increasing flow velocity u, the real part of the frequency eventually vanishes. Then, by a similar process, coupled-mode flutter occurs involving the third-mode locus.

In the case of fixed-free pipes, typical root loci of the complex frequencies with β=0.5 and α=κ=γ=0 are shown in Fig. 4.4 [5]. In this figure, small flow velocities damp the system in all its modes and result in a reduction of the vibration frequencies. At higher flow velocities, the locus of at least one of the modes crosses into the unstable region, signifying that a flutter type instability occurs in this mode just above the flow velocity corresponding to a point of neutral stability. The critical flow velocity u _cr is the lowest flow velocity at which this instability occurs.

Figure 4.4. Root loci of fixed-free pipes with increasing flow velocity for α=κ=γ=0 and β=0.5 [5].

For the non-conservative fixed-free system, it has been found that low damping destabilizes the system. Figure 4.5 [13] shows the variation of the critical flow velocity u _cr versus β for progressively higher values of the viscoelastic dissipation constant α. It can be seen that, in the case of large β, the critical flow velocity decreases with the viscoelastic dissipation constant.

Figure 4.5. Variation of critical velocity of fixed-free pipes versus β for progressively higher values of the visco-elastic dissipation constant α [8].

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Analog control system design

M. Sami Fadali , Antonio Visioli , in Digital Control Engineering (Third Edition), 2020

5.1 Root locus

The root locus method provides a quick means of predicting the closed-loop behavior of a system based on its open-loop poles and zeros. The method is based on the properties of the closed-loop characteristic equation

(5.1) $1+KL\left(s\right)=0$

where the gain K is a design parameter and L(s) is the loop gain of the system. We assume a loop gain of the form

(5.2) $L\left(s\right)=\frac{\prod _{i=1}^{n_z}\left(s-z_i\right)}{\prod _{j=1}^{n_p}\left(s-p_j\right)}$

where z _i , i  =   1, 2, …, n _z , are the open-loop system zeros and p _j, j  =   1, 2, …, n _p are the open-loop system poles. It is required to determine the loci of the closed-loop poles of the system (root loci) as K varies between zero and infinity. ¹ Because of the relationship between pole locations and the time response, this gives a preview of the closed-loop system behavior for different K.

The complex equality (5.1) is equivalent to the two real equalities:

•

Magnitude condition K|L(s)|   =   1

•

Angle condition ∠L(s)   =   ±(2m   +  1)180°, m   = 0, 1, 2, …

Using (5.1) or the preceding conditions, the following rules for sketching root loci can be derived:

1.

The number of root locus branches is equal to the number of open-loop poles of L(s).

2.

The root locus branches start at the open-loop poles and end at the open-loop zeros or at infinity.

3.

The real axis root loci have an odd number of poles plus zeros to their right.

4.

The branches going to infinity asymptotically approach the straight lines defined by the angle

(5.3) ${\theta }_a=\pm \frac{\left(2m+1\right){180}^{\circ }}{n_p-n_z},m=\mathrm{0,1,2},\dots$

and the intercept

(5.4) ${\sigma }_a=\frac{\sum _{i=1}^{n_p}{p}_i-\sum _{j=1}^{n_z}{z}_j}{n_p-n_z}$

5.

Breakaway points (points of departure from the real axis) correspond to local maxima of K, whereas break-in points (points of arrival at the real axis) correspond to local minima of K.

6.

The angle of departure from a complex pole p _n is given by

(5.5) ${180}^{\circ }-\sum _{i=1}^{n_p-1}\angle \left(p_n-p_i\right)+\sum _{j=1}^{n_z}\angle \left(p_n-z_j\right)$

The angle of arrival at a complex zero is similarly defined.

Example 5.1

Sketch the root locus plots for the loop gains

1.

$L\left(s\right)=\frac{1}{\left(s+1\right)\left(s+3\right)}$

2.

$L\left(s\right)=\frac{1}{\left(s+1\right)\left(s+3\right)\left(s+5\right)}$

3.

$L\left(s\right)=\frac{s+5}{\left(s+1\right)\left(s+3\right)}$

Comment on the effect of adding a pole or a zero to the loop gain.

Solution

The root loci for the three loop gains as obtained using MATLAB are shown in Fig. 5.1 . We now discuss how these plots can be sketched using root locus sketching rules.

1.

Using rule 1, the function has two root locus branches. By rule 2, the branches start at −1 and −3 and go to infinity. By rule 3, the real axis locus is between (−1) and (−3). Rule 4 gives the asymptote angles

$\begin{array}{c}{\theta }_a=\pm \frac{\left(2m+1\right){180}^{\circ }}{2},m=\mathrm{0,1,2},\dots \\ =\pm {90}^{\circ },\pm {270}^{\circ },\dots \end{array}$

and the intercept

${\sigma }_a=\frac{-1-3}{2}=-2$

Figure 5.1. Root loci of second- and third-order systems. (A) Root locus of a second-order system. (B) Root locus of a third-order system. (C) Root locus of a second-order system with zero.

To find the breakaway point using Rule 5, we express real K using the characteristic equation as

$K=-\left(\sigma +1\right)\left(\sigma +3\right)=-\left({\sigma }^2+4\sigma +3\right)$

We then differentiate with respect to σ and equate to zero for a maximum to obtain

$-\frac{dK}{d\sigma }=2\sigma +4=0$

Hence, the breakaway point is at σ _b   =   −2. This corresponds to a maximum of K because the second derivative is equal to −2 (negative). It can be easily shown that for any system with only two real axis poles, the breakaway point is midway between the two poles.

2.

The root locus has three branches, with each branch starting at one of the open-loop poles (−1, −3, −5). The real axis loci are between −1 and −3 and to the left of −5. The branches all go to infinity, with one branch remaining on the negative real axis and the other two breaking away. The breakaway point is given by the maximum of the real gain K

$K=-\left(\sigma +1\right)\left(\sigma +3\right)\left(\sigma +5\right)$

Differentiating gives

$\begin{array}{c}-\frac{dK}{d\sigma }=\left(\sigma +1\right)\left(\sigma +3\right)+\left(\sigma +3\right)\left(\sigma +5\right)+\left(\sigma +1\right)\left(\sigma +5\right)=3{\sigma }^2+18\sigma +23\\ =0\end{array}$

which yields σ _b   =   −1.845 or −4.155. The first value is the desired breakaway point because it lies on the real axis locus between the poles and −1 and −3. The second value corresponds to a negative gain value and is therefore inadmissible. The gain at the breakaway point can be evaluated from the magnitude condition and is given by

$K=-\left(-1.845+1\right)\left(-1.845+3\right)\left(-1.845+5\right)=3.079$

The asymptotes are defined by the angles

$\begin{array}{c}{\theta }_a=\pm \frac{\left(2m+1\right){180}^{\circ }}{3},m=\mathrm{0,1,2},\dots \\ =\pm {60}^{\circ },\pm {180}^{\circ },\dots \end{array}$

and the intercept by

${\sigma }_a=\frac{-1-3-5}{3}=-3$

The closed-loop characteristic equation

$s^3+9s^2+23s+15+K=0$

corresponds to the Routh table

Thus, at K  =   192, a zero row results. This value defines the auxiliary equation

$9s^2+207=0$

Thus, the intersection with the j ω-axis is ±j4.796   rad/s. The intersection can also be obtained by factorizing the characteristic polynomial at the critical gain

$s^2+9s^2+25s+15+192=\left(s+9\right)\left(s+j4.796\right)\left(s-j4.796\right)$

3.

The root locus has two branches as in (1), but now one of the branches ends at the zero. From the characteristic equation, the gain is given by

$K=-\frac{\left(\sigma +1\right)\left(\sigma +3\right)}{\sigma +5}$

Differentiating gives

$\begin{array}{c}\frac{dK}{d\sigma }=-\frac{\left(\sigma +1+\sigma +3\right)\left(\sigma +5\right)-\left(\sigma +1\right)\left(\sigma +3\right)}{{\left(\sigma +5\right)}^2}\\ =-\frac{{\sigma }^2+10\sigma +17}{{\left(\sigma +5\right)}^2}\\ =0\end{array}$

which yields σ _b   =   −2.172 or −7.828. The first value is the breakaway point because it lies between the poles, whereas the second value is to the left of the zero and corresponds to the break-in point. The second derivative

$\begin{array}{c}\frac{d^{2}K}{d{\sigma }^2}=-\frac{\left(2\sigma +10\right)\left(\sigma +5\right)-2\left({\sigma }^2+10\sigma +17\right)}{{\left(\sigma +5\right)}^3}\\ =-16/{\left(\sigma +5\right)}^3\end{array}$

is negative for the first value and positive for the second value. Hence, K has a maximum at the first value and a minimum at the second. It can be shown that the root locus is a circle centered at the zero with radius given by the geometric mean of the distances between the zero and the two real poles.

Clearly, adding a pole pushes the root locus branches toward the right hand plane (RHP), whereas adding a zero pulls them back into the left hand plane (LHP). Thus, adding a zero allows the use of higher gain values without destabilizing the system. In practice, the allowable increase in gain is limited by the cost of the associated increase in control effort and by the possibility of driving the system outside the linear range of operation.

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The PID controller

Mark A. Haidekker , in Linear Feedback Controls (Second Edition), 2020

15.2.2 PID control of a second-order process

We now turn to a second-order process, again with unit DC gain, with the transfer function

(15.11) $G(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2},$

which describes, for example, a spring-mass-damper system, an electromagnetic actuator, or an electronic circuit with two different kinds of energy storage. Note that here the damping factor ζ and the natural resonance frequency ${\omega }_n$ are properties of the process and not the auxiliary variables used in the previous section in Eq. (15.10). The closed-loop transfer function for the second-order process is

(15.12) $\frac{Y(s)}{X(s)}=\frac{{\omega }_n^2(k_D{s}^2+k_{p}s+k_I)}{s^3+(2\zeta {\omega }_n+k_D)s^2+({\omega }_n^2+k_p)s+k_I}.$

Unlike the first-order system, for which the PID controller provided redundant controller coefficients, the second-order system and its corresponding third-order closed-loop system has no redundant controller coefficients with $k_D$ associated with $s^2$ , $k_p$ associated with $s^1$ , and $k_I$ associated with $s^0$ . Closed-term solutions for the roots of the characteristic polynomial are relatively complex. However, we can use the root locus method to get an idea how the individual parameters of the PID controller influence the poles of the closed-loop system. The root locus form of the characteristic polynomial for $k_I$ is

(15.13) $1+k_I\cdot \frac{1}{s^3+(2\zeta {\omega }_n+k_D)s^2+({\omega }_n^2+k_p)s}=0.$

The root locus diagram has one open-loop pole in the origin and either two real-valued poles or a complex conjugate pole pair, depending on $k_p$ and $k_D$ . Furthermore, the root locus diagram has three asymptotes: one along the negative real axis and two at ±60° with the positive real axis. We can therefore qualitatively describe the influence of $k_I$ :

•

For a large enough $k_I$ , a pole pair will move into the right half-plane, making the closed-loop system unstable.

•

When the closed-loop system has three real-valued poles as $k_I\to 0$ , a branchoff point between the pole in the origin and the slower pole exists. Further increasing $k_I$ leads to a dominant complex pole pair with relatively slow transient response (Fig. 15.2A).

Figure 15.2. Root locus plots for k _I for two sample second-order systems. In (A) the process under PID control has three real-valued poles, and increasing k _I first moves the slowest pole out of the origin but rapidly leads to overshoot and then to instability. In (B) the process under PID control has one complex conjugate pole pair. Low values for k _I actually move the poles closer to the real axis, thus improving dynamic response. Further increasing k _I , however, worsens the overshoot and eventually leads to instability.

•

When the closed-loop system has a complex pole pair as $k_I\to 0$ , low values of $k_I$ can improve the dynamic response (Fig. 15.2B).

The root locus form of the characteristic polynomial for $k_p$ is

(15.14) $1+k_p\cdot \frac{s}{s^3+(2\zeta {\omega }_n+k_D)s^2+{\omega }_n^{2}s+k_I}=0.$

With one zero at the origin and three poles that are configured similar to Eq. (15.13), the root locus for $k_p$ has two asymptotes. Unlike $k_I$ , a large value for $k_p$ will not make a stable closed-loop system unstable. However, large overshoots and oscillations may occur. The three roots of the denominator polynomial in Eq. (15.14) can either be all real-valued or form a complex conjugate pair with one real-valued pole. None of the poles is at the origin. The root locus curve for two example configurations is shown in Fig. 15.3. We can qualitatively describe the influence of $k_p$ :

Figure 15.3. Root locus plots for k _p for two sample second-order systems, analogous to Fig. 15.2. In (A) the process has three real-valued open-loop poles (when k _p  = 0), whereas the process in (B) has a complex conjugate open-loop pole pair. In both cases, one closed-loop pole moves toward the zero at the origin with increasing k _p . The other two poles both emit branches of the root locus; if the pole pair is complex conjugate, then the resulting poles remain complex conjugate. If the open-loop system has only real-valued poles, then a branchoff point exists.

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For a large enough $k_p$ , one pole will follow the real-valued branch to the zero at the origin. A large value for $k_p$ paradoxically leads to a slow step response and therefore to undesirable dynamic behavior.

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When the closed-loop system has three real-valued poles as $k_p\to 0$ , a branchoff point between the two faster poles exists, and a critically damped case can be achieved. Further increasing $k_p$ leads to a complex conjugate pole pair with an oscillatory response (Fig. 15.3A). Increasing $k_p$ reduces the damping and increases the oscillatory frequency.

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When the closed-loop system has a complex conjugate pole pair as $k_p\to 0$ , the three poles can be brought in close proximity, with an overall optimal dynamic response (Fig. 15.3B).

Lastly, we examine the influence of $k_D$ . The root locus form of the characteristic polynomial for $k_D$ is

(15.15) $1+k_D\cdot \frac{s^2}{s^3+2\zeta {\omega }_n{s}^2+({\omega }_n^2+k_p)s+k_I}=0$

with an interesting double zero in the origin that "attracts" the root locus for large $k_D$ . The root locus for $k_D$ has only one asymptote, and one pole rapidly becomes nondominant for larger $k_D$ . Two branches end at the origin, either from the complex conjugate pole location or through a branchoff point on the real axis (Fig. 15.4). Although a large $k_D$ may lead to an undesirably slow system response, no branches exist in the right half-plane, and the system is always stable for arbitrarily large $k_D$ .

Figure 15.4. Root locus plots for k _D for the same second-order systems in Fig. 15.3. Here a double-zero at the origin exists. In both cases, one branch moves to the left along the only asymptote and creates a fast pole that becomes less and less dominant for larger k _D . However, two branches move toward the origin with increasing k _D , either from the complex conjugate pole location (B) or from the real axis with an excursion into the complex plane (A). No branches extend into the right half-plane.

It becomes obvious that higher-order systems under PID control gain rapidly in complexity. PID control is not necessarily the optimum control for high-order processes. However, it is a workable general-purpose solution, which rapidly leads to acceptable control behavior in many, especially lower-order, applications.

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Linear analysis of fractional point reactor kinetics

Gilberto Espinosa-Paredes , in Fractional-Order Models for Nuclear Reactor Analysis, 2021

5.6.1 Stability analysis using root locus method

The root locus (RL) method is a powerful and easy-to-use technique for investigating the closed-loop stability of a system when the proportional gain is varied from 0 to ∞. Basically, it is a locus of the closed-loop poles (which are the roots of the characteristic equation) of a system for 0  ≤ K  ≤   ∞. For integer-order systems, there are standard heuristic rules available to plot the root locus. The open-loop of fractional-order transfer functions given by Eqs. (5.15), (5.17) can be simplified as

(5.19) $G\left(s\right)=\frac{K\left(n⁎{\mathrm{\Lambda }}^{-1}\right)\left(s+\lambda \right)}{{\tau }^{\alpha }\mathrm{\Lambda }{s}^{2+\alpha }+\mathrm{\Lambda }{s}^2+{\tau }^{\alpha }{M}_1{s}^{1+\alpha }+M_{2}s+{\tau }^{\alpha }{M}_3{s}^{\alpha }+M_4}$

and

(5.20) $G\left(W\right)=\frac{K\left(n⁎{\mathrm{\Lambda }}^{-1}\right)\left(W^m+\lambda \right)}{{\tau }^{k/m}\mathrm{\Lambda }{W}^{2m+k}+\mathrm{\Lambda }{W}^{2m}+{\tau }^{k/m}{M}_1{W}^{m+k}+M_2{W}^m+{\tau }^{k/m}{M}_3{W}^k+M_4}$

where M ₁  =   Λλ  + A ₁Λ, M ₂  = λΛ   + A ₂Λ, M ₃  = A ₁Λλ  − λβ, and M ₄  = A ₂Λλ  − λβ, with A ₁  =   1/l  −   (1   − β)/Λ and A ₂  = β/Λ.

It is seen that the closed-loop system is stable for all values of K  >   0. Now using the method described above, the root loci were plotted for the loop of fractional-order transfer functions for three values of α and two values of relaxation time constants. Fig. 5.7 depicts a representative result.

Fig. 5.7. Root locus with α  =   0.8 and τ  = τ ₁.

It is seen from these plots that all the fractional-order closed-loop models are stable above some marginal value of K (K _marginal) as at this value, the RL branches cross the imaginary axis in the principal Riemann sheet and enter into the stable region. The values of K _marginal for various models are given in Table 5.5. The fractional-order models with short relaxation times are more robust as compared to those with long relaxation times.

Table 5.5. Critical values of K.

α	K marginal
	τ  = τ 1	τ  = τ 2
0.8	0.00372	7.27   ×   10−   8
0.5	0.000144	4.13   ×   10−   6
0.25	0.000335	0.000101

As 0   < α  <   1, it is convenient to think that α is the anomalous diffusion coefficient because it is a subdiffusion process of the neutron movement. As the relaxation time is directly proportional to the diffusion coefficient of neutrons, the physical meaning of τ ₁  > τ ₂ is that the neutron diffusion is higher for τ ₁, that is, it has less resistance to neutron movement than τ ₂. A control rod drop could describe the behavior of the value of τ ₁ while changes in reactor power to push or pull bar control may explain the value of τ ₂ where the reactor is more robust. However, these are not the only scenarios in nuclear reactors, for example, a reduction or increase in mass flow rate in the core of a BWR reactor, where proportional gain K represents the feedback effects due to Doppler that are physically related with fuel temperature and by void fraction.

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How to Design Pid Controller Using Root Locus

Source: https://www.sciencedirect.com/topics/engineering/root-locus

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