Non-overshooting control of linear system by a simple asymptotic gain scheduling method

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

Non-overshooting control asymptotic gain scheduling Posicast control dominant pole placement phase lag

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 $K_{i}$ is added to the reference input of $G (s)$ , as shown in Figure 1. The default value for $K_{i}$ is $K_{1}$ , and $K_{i}$ will remain unchanged till a schedule signal is given. $G (s)$ has a percentage overshoot of P, a settling time of $τ_{0}$ , and a steady state gain of 1 when u is a unit step signal.

Figure 1.

Block diagram of a system with an adjustable gain $K_{i}$ .

When a unit step signal is added to the reference input, that is, $R (s) = 1 / s$ , the peak output response of $G (s)$ is $K_{1} (1 + P)$ . Make this peak amplitude equals 1, we get

K_{1} = \frac{1}{(1 + P)},

(1)

After time $t_{0} >> τ_{0}$ , the transient will calm down thoroughly and output of $G (s)$ will become steady state value $K_{1}$ . Schedule $K_{1}$ into $K_{2}$ , which is equivalent to add a step signal with amplitude of $(K_{2} - K_{1})$ to $G (s)$ . The peak response after scheduling is $K_{1} + (K_{2} - K_{1}) (1 + P)$ . Again, make this peak amplitude equals 1, we get

K_{2} = \frac{1}{(1 + P)} + \frac{P}{{(1 + P)}^{2}},

(2)

After time $2 t_{0}$ , the steady state output will become $K_{2}$ .

Repeat the above process, schedule $K_{2}$ into $K_{3}$ at time $2 t_{0}$ , …, schedule $K_{i - 1}$ into $K_{i}$ , at time $(i - 1) t_{0}$ , …, make the peak output responses equal 1 each time after the gain scheduling, we have

K_{i} = \sum_{j = 1}^{j = i} \frac{P^{(j - 1)}}{{(1 + P)}^{j}},

(3)

For

K_{n} |_{n = \infty} = lim_{n \to \infty} \sum_{j = 1}^{n} \frac{P^{(j - 1)}}{{(1 + P)}^{j}} = 1, P < 1,

(4)

that is, steady state output of $G (s)$ becomes 1 after time $n t_{0}$ . Hence, non-overshooting step response is realized without steady state error.

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 $t_{0}$ can be greatly reduced by improved asymptotic gain scheduling.

Posicast control

Consider the standard second order underdamped system

\frac{Y (s)}{R (s)} = \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}, ζ < 1

(5)

where $ζ$ is the damping ratio, and $ω_{n}$ is the undamped natural frequency.

According to Posicast control, step input with amplitude of $r_{1}$ is added to the above system at time $t = 0$ , and its output response is denoted by $y_{1} (t)$ . Step input with amplitude of $r_{2}$ is added at time $t = τ$ , and its output response is denoted by $y_{2} (t)$ . The compound output response is

\begin{matrix} y (t) = y_{1} (t) + y_{2} (t) \\ = r_{1} [1 - \frac{1}{\sqrt{1 - ζ^{2}}} e^{- ζ ω_{n} t} \cos (ω_{d} t + φ)] \\ + r_{2} [1 - \frac{1}{\sqrt{1 - ζ^{2}}} e^{- ζ ω_{n} (t - τ)} \cos (ω_{d} (t - τ) + φ)] \cdot 1 (t - τ), (t \geq 0) \end{matrix}

(6)

where

$ω_{d} = ω_{n} \sqrt{1 - ζ^{2}},$ and $φ = \arctan (\sqrt{1 - ζ^{2}} / ζ)$

There exists that

y (t) = 1, (t \geq τ),

(7)

if and only if $r_{1}$ , $r_{2}$ and $τ$ satisfy that

r_{1} = {(1 + P)}^{- 1},

(8)

r_{2} = r_{1} P,

(9)

τ = \frac{π}{ω_{d}} = \frac{π}{ω_{n} \sqrt{1 - ζ^{2}}},

(10)

where

P = e^{- π ζ / \sqrt{1 - ζ^{2}}}

The above can be explained in Figure 2. $y_{1} (t)$ and $y_{2} (t)$ are the responses of step inputs with amplitude $r_{1}$ and $r_{2}$ , respectively. $y_{2} (t)$ starts when peak response of $y_{1} (t)$ appears. When added, the descending trend of $y_{1} (t)$ is counteracted exactly by ascending trend of $y_{2} (t)$ , non-overshooting step response can be realized in compound response $y (t)$ . The peak and trough response of $y_{1} (t)$ are marked by $A_{1}$ and $B_{1}$ . The starting point and peak response of $y_{2} (t)$ are marked by $A_{2}$ and $B_{2}$ . Dotted line $A_{1} B_{1}$ and $A_{2} B_{2}$ can be considered as an approximation of the response curve between them.

Figure 2.

Step responses of equation (5), with $ζ = 0.5$ and $ω_{n} = 1$ .

The slope of dotted line $A_{1} B_{1}$ and $A_{2} B_{2}$ are

k_{A_{1} B_{1}} = - k_{A_{2} B_{2}} = - \frac{r_{1} P (1 + P)}{τ},

(11)

Define $k_{Σ}$ as sum of the two slope values,

k_{Σ} = k_{A_{1} B_{1}} + k_{A_{2} B_{2}} = 0,

(12)

$k_{Σ}$ can used as an index for determining whether overshoot exists or not. If $k_{Σ} \neq 0$ , overshoot or undershoot is bound to exist in the compound step response.

With $K_{1} = r_{1}$ and $K_{2} = r_{1} + r_{2}$ , Posicast control can be considered as a two-step gain scheduling control, whose scheduling time τ is much shorter than $t_{0}$ in the asymptotic gain scheduling control. Though non-overshooting step response can be realized by Posicast control, it is effective only for second order system. However, the step response of higher order system having a pair of dominant complex conjugate poles resembles that of the second order system, which means that Posicast control method can still be used. To reduce the overshoot and undershoot as much as possible, dominance of the specified poles should be guaranteed, that is, the zeros and nondominant poles should be placed to the far left of the dominant poles in s plane. Therefore, designing of a good dominant pole control system is discussed in the following content.

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.

Figure 3.

Modified PID controllers in control systems.

According to Figure 3, transfer functions for these systems are as follows,

Standard PID control system

\frac{Y (s)}{R (s)} = (k_{p} + \frac{k_{i}}{s} + k_{d} s) \frac{G_{p} (s)}{1 + (k_{p} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)}

(13)

PI-D control system

\frac{Y (s)}{R (s)} = (k_{p} + \frac{k_{i}}{s}) \frac{G_{p} (s)}{1 + (k_{p} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)}

(14)

I-PD control system

\frac{Y (s)}{R (s)} = \frac{k_{i}}{s} \frac{G_{p} (s)}{1 + (k_{p} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)}

(15)

PI-PD control system

\frac{Y (s)}{R (s)} = (k_{p 1} + \frac{k_{i}}{s}) \frac{G_{p} (s)}{1 + (k_{p 1} + k_{p 2} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)}

(16)

PD-PID control system

\begin{matrix} \frac{Y (s)}{R (s)} = [(k_{p} + k_{p 1}) + \frac{k_{i}}{s} + (k_{d} + k_{d 1}) s] \\ \frac{G_{p} (s)}{1 + (k_{p} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)} \end{matrix}

(17)

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 $(k_{p 1} + k_{p 2})$ by $k_{p}$ , then (16) is changed into the following

\frac{Y (s)}{R (s)} = (k_{p 1} + \frac{k_{i}}{s}) \frac{G_{p} (s)}{1 + (k_{p} + \frac{k_{i}}{s} + k_{d} s) G_{p} (s)}

(18)

Characteristic polynomial of (18) is the same as (13), therefore, the dominant pole placement method remains effective here. Suppose real pole $- p$ , produced after dominant pole placement, is the nearest to the specified dominant poles. Place the zero $- k_{i} / k_{p 1}$ brought in by PI-PD controller to $- p$ , then $k_{p 1}$ can be determined as follows

k_{p 1} = \frac{k_{i}}{p},

(19)

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 $j ω$ axis.

For PD-PID control system, two adjustable zeros are brought in by the PD-PID controller. Suppose complex conjugate poles $- σ \pm j ω$ are the nearest to the specified dominant poles. Place the adjustable zeros to $- σ \pm j ω$ , $k_{p 1}$ and $k_{d 1}$ can be determined by

k_{p 1} = - k_{p} + \frac{2 σ k_{i}}{σ^{2} + ω^{2}},

(20)

k_{d 1} = - k_{d} + \frac{k_{i}}{σ^{2} + ω^{2}},

(21)

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

\begin{matrix} G (s) = \frac{C}{(s + p_{0}) (s + {\bar{p}}_{0})} \frac{(s + z_{1}) (s + z_{2}) \dots (s + z_{m})}{(s + p_{1}) (s + p_{2}) \dots (s + p_{n})}, \\ m \leq (n + 1) \end{matrix}

(22)

where C is a scalar, $- p_{0}$ and $- {\bar{p}}_{0}$ are the specified dominant poles, $- z_{1}, \dots, - z_{m}$ and $- p_{1}, \dots, - p_{n}$ are zeros and nondominant poles. $G (s)$ is a strictly proper transfer function with $m \leq (n + 1)$ .

Define angle $β$ as

\begin{matrix} β = ∠ (z_{1} - p_{0}) + ∠ (z_{2} - p_{0}) + \dots + ∠ (z_{m} - p_{0}) \\ - [∠ (p_{1} - p_{0}) + ∠ (p_{2} - p_{0}) + \dots + ∠ (p_{n} - p_{0})], \end{matrix}

(23)

$G (s)$ is defined as phase lead dominant pole control system when $β > 0$ , and phase lag dominant pole control system when $β < 0$ .

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 $t_{A}$ , $t_{C}$ , $t_{E}$ and $t_{B}$ , $t_{D}$ , $t_{F}$ , and the percentage overshoot as $P_{A}$ , $P_{C}$ and $P_{E}$ .

Figure 4.

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 $t_{C} < t_{A} < t_{E}$ and $P_{C} > P_{A} > P_{E}$ . After the peak time, the curves of phase lead and phase lag systems resemble the curve of the second order system more with the elapsing of time. Hence, $(t_{B} - t_{A}) \approx (t_{D} - t_{C}) \approx (t_{F} - t_{E})$ can be found.

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 $y (t)$ are shown in Figure 5.

Figure 5.

Posicast control for dominant pole control systems.

According to analysis from Figure 4, slope $k_{C_{1} D_{1}}$ and $k_{C_{2} D_{2}}$ in subfigure (a), $k_{E_{1} F_{1}}$ and $k_{E_{2} F_{2}}$ in subfigure (b) of Figure 5, together with $k_{A_{1} B_{1}}$ and $k_{A_{2} B_{2}}$ in Figure 2 follow that

k_{C_{2} D_{2}} > | k_{C_{1} D_{1}} | \approx | k_{E_{1} F_{1}} | \approx | k_{A_{1} B_{1}} | = k_{A_{2} B_{2}} > k_{E_{2} F_{2}},

(24)

For phase lead system, $k_{Σ} = k_{C_{1} D_{1}} + k_{C_{2} D_{2}} > 0$ , and for phase lag systems $k_{Σ} = k_{E_{1} F_{1}} + k_{E_{2} F_{2}} < 0$ . Therefore, overshoot and undershoot are bound to exist in $y (t)$ . The overshoot of $y (t)$ is always larger and appears earlier in phase lead system than in phase lag system, which can be seen in subfigure (a) and (b).

Suppose peak response $D_{2}$ , $F_{2}$ overlap with the undershoot region around $D_{1}$ , $F_{1}$ in time respectively, peak overshoot of $y (t)$ can be reduced by counteract effect of $y_{1} (t)$ and $y_{2} (t)$ when added. Percentage overshoot of $y (t)$ , denoted as $P_{y}$ , can be estimated as $P_{y} \in (0, P_{c})$ for phase lead system and $P_{y} \in (0, \frac{P_{E}^{2}}{1 + P_{E}})$ for phase lag system. When $P_{E} < P_{c} \leq 10 %$ , the upper bound for estimated overshoot value is at least one order of magnitude smaller in phase lag system than that in phase lead system. Obviously, the overshoot reducing effect is better in phase lag system. Hence, phase lag dominant pole control system is adopted for realizing non-overshooting step 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 $K_{2}$ is scheduled at the peak response of the second order dominant term, and the scheduling time can be determined by (10). The advantage of scheduling $K_{2}$ at this time is that a good overshoot and undershoot reducing effect can be reached according to Posicast control. Besides, the settling time can also be minimized by the reduced undershoot, for the system response is directly driven into the allowable error band.

Other gains $K_{i}, i = 3, \dots$ , are scheduled when the peak response appears after previous scheduling with two reasons. One reason is that overshoot can be avoided, for the peak response caused by the equivalent step input $(K_{i} - K_{i - 1})$ will overlap with the undershoot region of previous compound input response. The most important thing is the overshoot refrain mechanism can be maintained each time when new gains are scheduled. This is because the nondominant term in the response can be sufficiently damped in time, and the system response resembles that of the second order underdamped system. While nondominant term could not be sufficiently damped for phase lead dominant pole control system at the peak time, the compound response will not so resemble that of the second order underdamped system after the peak response. Therefore, overshoot refrain mechanism cannot be guaranteed each time when new gains are scheduled at the peak response. This is why the dominant pole control system is chosen to be of phase lag type rather than phase lead type. Another reason is that the scheduling time can be easily determined by adding a logic element in the output part.

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 $K_{1}$ is given as

K_{1} = \frac{α}{(1 + P)},

(25)

where $α$ is a constant, with $α \in [0.95, 1)$ . By introducing coefficient $α$ , overshoot can be avoided in the compound response.

When scheduling $K_{i}$ , the output response of equivalent step input $(K_{i} - K_{i - 1})$ has a peak amplitude of $(K_{i} - K_{i - 1}) (1 + P)$ . Due to the counteract effect, the peak amplitude of compound response, denoted by $y_{p}$ , will satisfy that

y_{p} < K_{i - 1} + (K_{i} - K_{i - 1}) (1 + P),

(26)

Let the right side of (26) equals 1, $K_{i}$ can be determined as

K_{i} = \frac{1}{(1 + P)} + \frac{P K_{i - 1}}{(1 + P)},

(27)

By iteration, $K_{i}$ can be further written as

K_{i} = \frac{α}{(1 + P)} + (1 - α + P) \sum_{i = 2}^{n} \frac{P^{i - 2}}{{(1 + P)}^{i}}, (i = 2, \dots, n)

(28)

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.

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:

Step 1: Determine dominant poles for the inner modified PID control loop.

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.

Step 2: Design of a phase lag dominant pole control system by modified PID controller, and determine the value of its parameters.

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.

Step 3: Specify the value of $α$ , determine the system percentage overshoot, and calculate the gain table.

The value of $α$ can be chosen from $[0.95, 1)$ . Though system robustness can be increased by even smaller $α$ , settling time will be increased. The inner loop system percentage overshoot can be obtained by experiment or simulation. Then, gain table can be calculated by (28).

Step 4: Design of the scheduling event.

Gain $K_{1}$ is the default gain, it is scheduled each time the reference input signal is changed. Gain $K_{2}$ is scheduled by a time driven event, whose scheduling time can be calculated by (10). Other gains $K_{i}, i = 3, \dots$ are scheduled by the occurrence of the peak output response. The scheduling events for these gains can be easily designed by adding a judgement unit, as shown in Figure 6.

Numerical examples

Example 1

Consider the third-order plant model given in the CDM method by Manabe (1998) as follows

G_{1} (s) = \frac{0.1 s + 1}{0.25 s^{3} + 1.25 s^{2} + s}

Since the denominator of $G_{1} (s)$ is two degree higher than its numerator, PI-PD controller is used to design the phase lag dominant pole control system. Suppose the inner PI-PD control loop has a percentage overshoot of around 10% and a rise time of about 1.4s, the desired dominant poles can be determined as $- 1.1 \pm 1.668 j$ . According to the dominant pole placement (Wang et al., 2009), $k_{p} \in (3.41, 3.72)$ can be found. If $k_{p} = 3.64$ is set, then $k_{i} = 0.422$ and $k_{d} = 1.641$ can be determined. The poles for the inner PI-PD control system will be $p_{1} = - 0.127$ , $p_{2, 3} = - 1.1 \pm 1.668 j$ and $p_{4} = - 3.329$ . Though $p_{1}$ is on the right of dominant pole $p_{2, 3}$ in s plane, with $k_{p 1} = 3.323$ and $k_{p 2} = 0.137$ obtained by (18) and (19), its influence on the dominance of $p_{2, 3}$ can be eliminated by pole-zero cancellation.

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 $ω_{d} = 1.668$ , the scheduling time for the second gain is $t = 1.884$ s. The percentage overshoot for the PI-PD control system is $P = 9.797 %$ by simulation. Suppose $α$ is set to be 0.98, then the gain table can be obtained by (25) and (28). The control input and unit step response of $G_{1} (s)$ by CDM method and asymptotic gain scheduling control are shown in subfigures (a) and (b) of Figure 7.

Figure 7.

Control input and unit step response of $G_{1} (s)$ .

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 $G_{1} (s)$ is far away from the $j ω$ axis.

Example 2

Consider the eighth-order plant model given by Basilio and Matos (2002) as follows

G_{2} (s) = \frac{1}{{(s + 1)}^{8}}

$G_{2} (s)$ is an all-pole transfer function whose poles are negative real. A direct step input to $G_{2} (s)$ will generate a non-overshooting step response, as shown in subfigure (b) of Figure 8 with dashed lines. Since there is no zero in $G_{2} (s)$ , PD-PID controller is used to improve the step response of $G_{2} (s)$ . Suppose the inner PD-PID control loop has a percentage overshoot of around 9% and a rise time of about 7s, the desired dominant poles can be determined as $- 0.28 \pm 0.358 j$ . Then, $k_{p} \in (0.24, 0.85)$ can be found by dominant pole placement method. If $k_{p}$ is set to be 0.51, then $k_{i} = 0.107$ and $k_{d} = 0.771$ can be obtained. The poles of designed inner PD-PID control system will be $p_{1, 2} = - 0.215 \pm 0.047 j$ , $p_{3, 4} = - 0.28 \pm 0.358 j$ , $p_{5, 6} = - 0.848 \pm 0.943 j$ , $p_{7, 8} = - 1.651 \pm 0.757 j$ and $p_{9} = - 2.011$ . For $p_{1, 2}$ are on the right of dominant poles $p_{3, 4}$ in s plane, $k_{p 1} = 0.437$ and $k_{d 1} = 1.431$ can be acquired by (20) and (21) to eliminate $p_{1, 2}$ ‘s influence on the dominance of $p_{3, 4}$ .

Figure 8.

Control input and unit step response of $G_{2} (s)$ .

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 $t = 8.27$ s by (10). The percentage overshoot for the inner PD-PID control system is $P = 8.409 %$ by simulation. The gain table can be obtained by (25) and (28) with $α = 0.98$ . The control input and unit step response of $G_{2} (s)$ by PI control with $k_{p} = 0.25$ and $k_{i} = 0.0625$ (Basilio and Matos, 2002) and asymptotic gain scheduling control are shown in subfigures (a) and (b) of Figure 8.

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 $G_{2} (s)$ with dashed lines in subfigure (b), practical improvement of the system step response can be offered by our asymptotic gain scheduling control method.

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.

ORCID iD

Huanchao Du

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