Design and analysis of root locus based controllers

The root locus method is the classical method for analyzing the variation of the position of the poles of a closed-loop control system transfer function in the complex plane. It is also the basis for guiding the design of various controllers. Starting from the nature of the root locus, this paper takes a standard form of closed-loop feedback control system as an example, and analyzes in detail the internal logic of designing a controller and compensator by the root locus method. At the same time, software such as MATLAB/Simulink, and Octave are used. The working principles and properties of the PD controller; the lead and lag compensator, their interconnections, advantages, and characteristics, as well as the adjustment of parameters to better suit practical needs, are sorted out. The PD controller improves the system response, the lead compensator eliminates the PD controller’s defects, and the lag compensator reduces the steady-state error of the system. It provides help for understanding the essence of controller design and provides design ideas for control problems in actual engineering.


Introduction
Automatic control is an indispensable and important part of modern society, and everyone's living environment is surrounded by automatic control systems [1,2].From the fresh air system that automatically adjusts the indoor temperature and humidity to the robot control servo system that completes the assembly of components, the application of control theory makes human life more convenient and beautiful [3].The controller is the core of all-scale automatic control systems.Through comparison with each other, it automatically adjusts and reduces the difference between controlled variables and participating variables.However, the design of the controller and the adjustment and selection of parameters are difficult problems for many system designers.
In order to design a controller that is completely suitable for the needs of the control system and has the best performance at the same time, many methods of designing controllers have been found, and they can work well in various fields.Examples include total least squares and data conversion methods [4,5].However, with the continuous development of modern control theory, the classical root locus method seems to have faded from people's sight [6].The root locus is a new method to solve the root of the characteristic equation of the closed-loop system using the graph method.From the perspective of the time domain, this paper analyzes its application in the standard unit feedback control system [7] in detail by using the root locus method [8] and compares the advantages and characteristics of different controllers designed.

Literature Review
A fundamental concept in automatic control is feedback, and the application of feedback control by mankind since very early times.During the industrial revolution, the British invented the centrifugal flying hammer governor for steam engines, and since then automatic control has attracted attention, and automatic control theory and technology have been continuously developed and improved.Nyquist proposed a frequency judgment basis for stability in 1932 and Bode introduced the logarithmic coordinate system to the frequency method in 1940.Then, Harris introduced the concept of transfer functions in 1942, which provided the basis for the root-trajectory and frequency methods widely used in the field of classical control.By 1948, W. R.Evans proposed the root locus method [9,10], which is one of the basic methods of classical control theory.Their joint efforts led to the maturation of classical control theory.

Basic properties of root locus
For a standard unit feedback closed-loop control system, the root locus studies the variation of the roots of the characteristic equations of the closed-loop control system [11], which is also the location of the poles of the closed-loop transfer function in the complex plane, as the proportional gain varies from 0 to positive infinity.The standard form of the root locus is: This paper takes a basic control system as an example.As shown in Figure 1, adding negative feedback to an open-loop control system gives the standard form of a unit feedback control system, as well as a proportional control system.
This system has an open loop transfer function G(s) = 1 (s+1)(s+3) and gain K, and has a unit step function as its reference input.
Nowadays, manually drawing the root trajectory is an unnecessary task, and a few lines of code can be used instead in many software [12].Therefore, the focus needs to be on understanding the logic behind the root change pattern as a basis for guiding controller design.For a closed-loop transfer function G(s) = N(s) D(s) , the modulus and complex angles can be obtained from the zeros and poles.

M =
∏ Distance from zeros to s ∏ Distance from poles to s s=σ+jω (2) φ = ∑ Angle from zeros to s − ∑ Angle from poles to s (3) The conditions for determining whether a root is on the root locus can be obtained from equations ( 1), (2), and (3), and can also be adjusted to place an appropriate value on the root locus.

Controller analysis using root locus
Using the geometric properties of the root locus, the root locus of the closed-loop transfer function G cl (s) of the system is changed by adding zeros or poles to the open-loop transfer function, thereby changing the dynamic response of the system so as to design the controller.Different values of K will have different system performance.
(1).When K is less than 10, there are two real roots, p 1 and p 2 (p 1 < −2 < p 2 < −1).The final time response of the system at this point is (5) Where the constant 1 comes from the reference input, the other two converge to zero as time increases, the latter converges more slowly, so p 2 is the dominant pole.
(2).When K is greater than 10 and the roots are in the complex plane, the output of the system at this time is x(t) = 1 + C 3 e −2t sin(ωt + φ) (6) The system shows attenuation when oscillating, and the K value change at this time will not change the convergence rate.

PD controller
To change the response speed of the system, the root locus needs to be changed.This can be achieved by connecting a controller or compensator in series.The case of a pole at s = −3 + 2√3j is chosen here for analysis, and if this can be achieved, the system can respond faster.
According to the geometric properties of the root locus in equation ( 4), the angle to be adjusted can be calculated.
In order to satisfy the condition of the root locus so that the point is located on the root locus, the simplest way is to add a zero point of G(s) and satisfy the angle from the point to s is + 1 6 π.The geometric relationship indicates that the zero point is at position -9 of the real axis, as shown in Figure 3.The added controller corresponds to the transfer function of C(s) = s + 9, which is connected in series with the original control system, as shown in Figure 5.This is also a proportional-differential controller.

Lead compensator
Although the use of proportional-differential control can improve the response speed of the system, it also has some obvious disadvantages.Firstly, the PD controller cannot be implemented through passive components, which means that an additional source of energy is required.Secondly, it is too sensitive to high frequency noise.
To avoid the pitfalls of PD control, it is possible to add both a pole and a zero point to the system and to position the zero point closer to the imaginary axis than the pole.This is done by adding a lead compensator, which is expressed as: In the 3.2 example, additional 1 6 π needs to be added, where the difference between the zero-to-s angle and the pole-to-s angle needs to be equal to 1 6 π, the new zero-to-pole pair to satisfy the condition of the root locus.The design idea of using a lead compensator, the root locus of the system after using the lead compensator and bode diagram of C(s) are shown in Figures 6, 7 and 8.   Comparing Figure 7 with Figure 4, it can be seen that the intersection of the asymptote with the real axis is shifted away from the imaginary axis by the lead compensator, so not only does it improve the stability of the system, but it also speeds up the response time.The Bode diagram in Figure 8 proves that the lead compensator does not amplify the high frequency term indefinitely, thereby solving the problem of the PD controller.Besides, it advances the phase somewhat, and the phase is positive, which is why it is called lead.

Lag compensator
For standard feedback closed loop control system containing a compensator, its error can be calculated.
D(s) into equation ( 9), this lead to Use the final value theorem for equation (10) to calculate the steady state error of the system e ss = lim s→∞ sE(s) = D(0) D(0) + KN(0) s zc s pc (11) This shows that the larger the , the smaller the e ss .Therefore, if the designed compensator is made s zc < s pc < 0, the steady-state error of the system can be reduced.In particular, as , tends to infinity, e ss equals 0. At this point, C(s) = 1 − s zc s , this is a proportional-integral controller that eliminates the steady-state error completely.In contrast to the lead compensator, the pole of this compensator is closer to the imaginary axis than the zero in the complex plane, and is called a lag compensator.
Continuing the analysis of the example in 3.3, change the compensator and let K = 2, s zc s pc = 10.The steady state error of the system will be reduced from 0.6 to 0.13.However, there are different options for controllers with a ratio of 10, for example: Put the compensator in both cases together with the original system and compare its output, as shown in Figures 9 and 10.Both lag compensators achieve a reduction in steady-state error, but there are significant differences between the two.C 2 (s) system output at the beginning behaves in the same way as the original system, after which the compensator corrects the steady-state error.The system output of C 1 (s), on the other hand, has a completely different behavior.It is clear that the second compensator will be chosen for practical applications.Draw a graph of their root locus to analyze the reasons for this, as shown in Figure 11.Although the newly added poles and zeros are 10 times different, the pole zero in C 2 (s) is closer to the imaginary axis.Its root locus diagram is very similar to the original system and maintains the original transient response, whereas C 1 (s), which changes greatly with the shape of the root locus.

Conclusion
This paper studies and analyzes the design of three different controllers and compares their effects on the response of the closed-loop feedback control system using the root locus method.These three controllers have different advantages and characteristics for different applications.The proportional differential controller can significantly improve the system response, the lead compensator can avoid the shortcomings of PD control, and the lag compensator can reduce the steady-state error of the system.In addition, this paper also analyses the adjustment of relevant parameters in the design of the controllers and explains the reasons for these changes in terms of root locus, which provides some ideas for their application in practical engineering problems.However, there are limitations to these types of controllers.For complex systems, using root trajectories alone may not yield the desired results.The performance can be further improved using a state space equation approach in combination with an observer.On this basis, more advanced controllers can be designed and put into use.

Figure 1 .
Figure 1.Block diagram of proportional control.

4. 1 .
Original system analysis G(s) has two poles, -1 and -3, with 0 zeros.The root locus of the closed-loop transfer function G cl (s) is plotted as shown in Figure2.

Figure 3 .
Figure 3. Location of the new zero point.The changed root locus is shown in Figure4.The root locus crosses the point s, while the convergence rate still has the potential to increase as the value K increases, and the pole can continue to move to the left.

Figure 5 .
Figure 5. Block diagram after adding the controller.

Figure 6 .
Figure 6.Location of new zero and pole.

Figure 9 .
Figure 9. Block diagram of three different cases.