Abstract
In this paper, a simple yet effective method has been raised for non-overshooting control of linear higher order plant. It is based on Posicast control, asymptotic gain scheduling and dominant pole placement by modified proportional-integral-derivative (PID) controllers, including PI-D, I-PD, PI-PD and PD-PID. The control system is composed by two closed-loops, that is, the inner loop where modified PID controllers are used to stabilize the plant by dominant pole placement, and the outer loop where asymptotic gain scheduling is used to shape the non-overshooting step response. Use of the modified PID controllers is the key to secure success of asymptotic gain scheduling, for dominance of the specified poles and phase lag dominant pole control system can be designed by these controllers in the inner loop. Three numerical examples are used to validate the method; results show that a non-overshooting control with relatively short settling time and small undershoot can be realized.
Keywords
Introduction
The designing of a control system that has a non-overshooting step response has been attracting the attention of many researchers for decades. It has great significance in such practical applications as, but not limited to, chemical process, high precision manufacturing, and temperature control in dangerous environments.
By far, lots of methods have been proposed for designing controllers that can achieve non-overshooting step response for linear plant models. Among them, the linear method with linear controllers designed are the most talked about. A necessary and sufficient condition has been offered by Aaron and Segers (1958) for a bounded nondecreasing step response, which can be applied to transfer functions with real poles. Several sufficient conditions have been given by Zemanian (1960) for linear systems with complex poles to have monotone non-overshooting step response. Since the convolution of positive impulse response functions is also positive, sufficient conditions based on this simple principle have been proposed by Jayasuriya and Franchek (1991), Rachid (1995), and Jayasuriya and Song (1996) for systems having monotone nondecreasing step response. A bounding theorem derived by El-Khoury et al. (1993) can determine the number of extrema that exist in the step response. The conditions or theorems proposed in the aforementioned works are given in terms of pole zero configurations.
The necessary and sufficient conditions, which are given in terms of the transfer function’s coefficients, were derived by Lin and Fang (1997) for third order linear systems that have non-overshooting step response. Non-overshooting control of multivariable systems were studied by Schmid and Ntogramatzidis (2009, 2010), whose methods are essentially the pole placement method no matter what type of feedback is used. By adjusting two parameters, one is the time constant and the other is called “stability index” in coefficient diagram method (CDM) or “characteristic ratio” in characteristic ratio assignment (CRA), both CDM methods (Manabe, 1998) and CRA method (Kim et al., 2003) can realize monotone nondecreasing step response. However, higher order controllers are bound to be designed by the CRA method, for realizing pole placement and pole zero cancellation at the same time. Posicast control proposed by Smith (1957) can also realize non-overshooting step response by dividing the step input into two one-half cycle spaced step inputs. The problem for Posicast control is that the non-overshooting step response is limited to second order system. Despite this, input shaping based on Posicast control is a frequently used method of reducing residual vibrations for higher order systems (Pao and Singhose, 1998; Singhose and Pao, 1997; Singhose et al., 1994).
Several nonlinear control methods also have been proposed for non-overshooting step response. Tran et al. (2007) offered a cascade sliding mode PID control method for robust non-overshooting step response. Zhu and Cai (2012) used the switching control method to achieve a non-overshooting step response. Zhao and Wang (2015) used the reset controller to achieve non-overshooting step response with an arbitrarily tuned rise time under mild assumption.
A common feature for these linear and nonlinear methods is that higher order controller will be designed when the plant model is also of higher order. Though low order controllers and plant are demonstrated in switching control method by Zhu and Cai (2012), controlling of higher order plant by lower order controller is not discussed in their paper. As we know, analysis and synthesis of higher order controllers are undesirable and more difficult both from economic and computational considerations. Thus, to save cost and time, simplify implementation, low order controllers are highly desirable for engineers (Bandyopadhyay et al., 1998). Besides, the system with the lowest possible order is found to be a strong guarantee of robustness (Manabe, 1998).
In this paper, we try to design a controller that is as simple as possible for non-overshooting control of higher order plant model. PID controllers is adopted as the main controller for its simple structure and straight forward microprocessor implementation (Lu and Cheng, 2005). The designed system is composed by two closed-loops, that is, the inner loop where modified PID controllers are used to stabilize the plant by dominant pole placement, and the outer loop where asymptotic gain scheduling is used to shape the non-overshooting step response.
The rest of the paper is organized as follows. In Section 2, basic principles for asymptotic gain scheduling and Posicast control are given. In Section 3, designing of dominant pole control system by modified PID controllers are analyzed. Section 4 presents the improved asymptotic gain scheduling control. In Section 5, procedures for non-overshooting control are offered. Illustrative examples and conclusions are presented in Sections 6 and 7, respectively.
Basic principles
Two basic principles introduced are the asymptotic gain scheduling control, a multi-step control method for non-overshooting step response, and Posicast control (Smith, 1957), also called overlapping control (Han et al., 2017; Xiao, 1979), a two-step control method for non-overshooting step response.
Asymptotic gain scheduling control
Suppose that an adjustable gain

Block diagram of a system with an adjustable gain
When a unit step signal is added to the reference input, that is,
After time
After time
Repeat the above process, schedule
For
that is, steady state output of
It takes such a long time to finish the whole control process that implementation of this method is impossible. However, with Posicast control introduced in the following part, the time interval
Posicast control
Consider the standard second order underdamped system
where
According to Posicast control, step input with amplitude of
where
There exists that
if and only if
where
The above can be explained in Figure 2.

Step responses of equation (5), with
The slope of dotted line
Define
With
Design of dominant pole control system
Dominant pole placement by modified PID controllers
Dominant pole placement by standard PID controller (parallel form) has been well studied. Effective methods with low overshoot can be found in Hagglund and Astrom (1985), Persson and Astrom (1992), Tang et al. (2006), Atherton (2009), Wang et al. (2009), Liu et al. (2015), and Dincel and Soylemez (2016). Due to its simplicity and ease of use, the method raised by Wang et al. (2009) is adopted in this paper. But a serious problem exists in Wang’s method, that is, the dominance of the specified poles is compromised by the nearby zeros introduced by standard PID controller. This has been pointed out by themselves in the conclusion part and in Li et al. (2011). We find that the modified PID controllers can avoid this problem and improve the dominance of the specified poles, above all, the dominant pole placement method remains effective in the modified PID control system. To save space, the dominant pole placement method is not repeated here, readers can refer to Wang et al. (2009).
The modified PID controllers introduced are PI-D, I-PD, PI-PD and PD-PID, as shown in Figure 3. Detailed discuss of these controllers can be found in Ogata (2010). This paper will mainly focus on the dominant pole placement by these controllers.

Modified PID controllers in control systems.
According to Figure 3, transfer functions for these systems are as follows,
Standard PID control system
PI-D control system
I-PD control system
PI-PD control system
PD-PID control system
Note that PI-D, I-PD, PD-PID and standard PID control system have the same characteristic polynomial, which means the dominant pole placement methods are the same for them.
Compared with standard PID controller bringing in two zeros nearby the dominant poles, the dominance of the specified poles can be improved by PI-D and I-PD controllers for one real zero is brought in by the PI-D controller in (14), and no zero is brought in by the I-PD controller in (15).
For PI-PD control system, substitute
Characteristic polynomial of (18) is the same as (13), therefore, the dominant pole placement method remains effective here. Suppose real pole
Dominance of the specified poles can be enhanced by pole-zero cancellation in PI-PD control system. In addition, response time of this system can be improved, for specified dominant poles can be placed further to the left of
For PD-PID control system, two adjustable zeros are brought in by the PD-PID controller. Suppose complex conjugate poles
With two poles having been cancelled, both dominance of the specified poles and system response time can be improved by the PD-PID controller.
In view of the performance in dominant pole placement, PD-PID controller is the most preferred, followed by PI-PD controller, I-PD controller, and PI-D controller rank the last for it brings in one fixed real zero. The existence of adjustable zeros in PI-PD and PD-PID controller is due to the increased number of elements in these controllers.
Phase lag dominant pole control system
Suppose transfer function of the dominant pole control system is
where C is a scalar,
Define angle
To show the difference between these two types of systems, unit step responses are drawn in Figure 4 for phase lead, phase lag and standard second systems having the same specified dominant poles. The peak and trough response are marked by A, C, E and B, D, F, respectively. Denote the peak and trough time as

Unit step response for phase lead, phase lag and standard 2nd order systems.
As we know, step response curve of the dominant pole control system is composed by a number of exponential terms and damped sinusoidal terms. Also, terms caused by the nondominant poles decay rapidly to zero with the elapsing of time. Influence of the nondominant terms is mainly reflected in the initial response period with
When Posicast control is applied to the phase lead and phase lag systems, that is, two step inputs are added to the systems, the second step input is added exactly when the peak response of the first step input appears with peak amplitude of 1, and summation of the two step inputs’ amplitude equals 1, the compound step responses

Posicast control for dominant pole control systems.
According to analysis from Figure 4, slope
For phase lead system,
Suppose peak response
Improved asymptotic gain scheduling control
For realizing non-overshooting step response, the scheduling time and gain amplitude of the asymptotic gain scheduling control in section 2.1 should be further modified according to Posicast control introduced above.
Gain scheduling time
The second gain
Other gains
Modified gain amplitude
To ensure non-overshooting step response, the gain amplitudes of the asymptotic gain scheduling control are modified as follows
The default gain
where
When scheduling
Let the right side of (26) equals 1,
By iteration,
The asymptotic gain scheduling control can be applied to a general class of plants that can be designed into dominant pole control systems by standard PID controllers, for it is certain that phase lag dominant pole control systems will be designed by trying different modified PID controllers. The output error is decaying geometrically, which means a relatively short settling time can be realized by asymptotic gain scheduling control.
Procedure for non-overshooting control
The block diagram for the asymptotic gain scheduling control system is shown in Figure 6.

Control system diagram.
This system can be considered as a two-degrees-of-freedom (2-DOF) system composed by two closed-loops. In the inner loop, modified PID controller is used to stabilize the plant by dominant pole placement. In the outer loop, improved asymptotic gain scheduling is used to shape the output response.
Moreover, the inner loop is also a 2-DOF control system, where dominant pole placement and pole-zero cancellation can be carried out independently.
The general procedure for non-overshooting control is summarized as follows:
The dominant poles can be determined by the specified percentage overshoot and rise time of the inner modified PID control loop. Commonly, the specified percentage overshoot is 8%–10% for industrial control applications (Astrom and Hagglund, 1995). Small overshoot is preferred for good overshoot and undershoot reducing effect can be realized.
PD-PID controller is always preferred for designing of a phase lag dominant pole control system, followed by PI-PD, I-PD, and PI-D controller. When PI-PD or PD-PID controller is used, (19) or (20) and (21) can be used to determine parameters for the controller. Whether the system is of phase lag type or not should be checked when all the parameters are known. If phase lag cannot be satisfied, other modified PID controllers should be tried.
The value of
Gain
Numerical examples
Example 1
Consider the third-order plant model given in the CDM method by Manabe (1998) as follows
Since the denominator of
The inner PI-PD control system is checked to be of phase lag type, therefore, asymptotic gain scheduling control can be used. Calculated by (10) with

Control input and unit step response of
It is easy to notice that non-overshooting step response can be realized in subfigure (b) of Figure 7. The rising times (90%) are 1.589s and 3.754s for asymptotic gain scheduling and CDM method. If error criterion is set to be 5%, then settling times will be 1.787s and 4.315s for asymptotic gain scheduling and CDM method, respectively. While, if error criterion is set to be 2%, then settling times will be 2.046s and 5.028s for asymptotic gain scheduling and CDM method. Obviously, the system response is much faster in the asymptotic gain scheduling control. Besides, the controller is simpler. Note that the CDM method is an approach for all pole transfer functions, it works in this example for the zero of
Example 2
Consider the eighth-order plant model given by Basilio and Matos (2002) as follows

Control input and unit step response of
Then, asymptotic gain scheduling control can be used after checking that the inner PD-PID control system is of phase lag type. The scheduling time for the second is
The rising times (90%) are 8.871s and 30.590s, while the settling times are 9.763s and 38.929s (95%), 10.888s and 49.845s (98%) for the asymptotic gain scheduling control and PI control method. Though non-overshooting step response has been realized by the PI controller, the prolonged settling time is unbearable. Compared with the direct step input response of
Conclusions
A simple yet effective non-overshooting control method has been presented for the higher order plant model. The controller is simple and easy to implement in the industrial application, for it is mainly composed by simple PID controller and logic unit. By using modified PID controllers, a good dominant effect can be ensured for the specified dominant poles in the inner loop. The asymptotic gain scheduling is used to shape the non-overshooting step response in the outer loop. The undershoot of this method is small, which can almost be neglected. And its settling time is relatively short and can be reached in about one period of peak time of the inner dominant pole control loop.
However, there are also limits for this method. The rising time could not be tuned arbitrarily short subjecting to the existence of the dominant poles. The undershoot in this method is unavoidable, so applications that are sensitive to undershoot are not suitable for this method.
Footnotes
Acknowledgements
The authors would like to thank the reviewers for a meticulous review of this manuscript.
Declaration of conflicting interests
The author(s) declared no potential conflict of interests with respect to the research, authorship and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
