Open AccessArticle

Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models

Christer Dalen

¹ and

David Di Ruscio

^2,*

Meiersvingen 38, Skien 3744, Norway

Department of Electrical Engineering, IT and Cybernetics, University of South-Eastern Norway, Porsgrunn 3918, Norway

Author to whom correspondence should be addressed.

Algorithms 2018, 11(6), 86; https://doi.org/10.3390/a11060086

Submission received: 9 April 2018 / Revised: 9 June 2018 / Accepted: 14 June 2018 / Published: 17 June 2018

(This article belongs to the Special Issue Algorithms for PID Controller)

Download

Browse Figures

Versions Notes

Abstract

A method for tuning PI controller parameters, a prescribed maximum time delay error or a relative time delay error is presented. The method is based on integrator plus time delay models. The integral time constant is linear in the relative time delay error, and the proportional constant is seen inversely proportional to the relative time delay error. The keystone in the method is the method product parameter, i.e., the product of the PI controller proportional constant, the integral time constant, and the integrator plus time delay model, velocity gain. The method product parameter is found to be constant for various PI controller tuning methods. Optimal suggestions are given for choosing the method product parameter, i.e., optimal such that the integrated absolute error or, more interestingly, the Pareto performance objective (i.e., integrated absolute error for combined step changes in output and input disturbances) is minimised. Variants of the presented tuning method are demonstrated for tuning PI controllers for motivated (possible) higher order process model examples, i.e., the presented method is combined with the model reduction step (process–reaction curve) in Ziegler–Nichols.

Keywords:

PI control; tuning; integrating system; maximum time delay error; time delay; performance optimal; process control

1. Introduction

This paper concerns tuning of PI controllers based on Integrator Plus Time Delay (IPTD) models/systems. Further details and developments regarding the

δ

-tuning algorithm are presented in the work [1,2]. IPTD processes and close-to IPTD systems are important/typical processes/systems found in the industry. Instances of IPTD processes are pulp and paper mills, oil water gas separators, communication networks, level systems and all lag-dominant processes, which may be approximated by IPTD models (see, e.g., [3,4,5]). Reported instances are high-purity distillation columns where there are relatively large time constants for minor differences in the reference, and where the time delay comes from an analyser (see, e.g., [6,7]). In Section 6.4 in [8], an example of reboiler control in connection with a distillation column was presented.

The majority of existing PI controller tuning rules for IPTD processes,

\begin{matrix} H_{p} (s) = \frac{k}{s} e^{- τ s}, \end{matrix}

(1)

may be written as the following setting

\begin{matrix} K_{p} = \frac{α}{k τ}, T_{i} = β τ, \end{matrix}

(2)

where

K_{p}

is the PI controller proportional gain,

T_{i}

is the integral time constant, k is the gain velocity (slope) and

τ \geq 0

is the time delay.

α

and

β

in Equation (2) are dimensionless parameters. For instance, using the classical Ziegler–Nichols (ZN) PI controller tuning rules, proposed in the works [9,10,11], gives

α = \frac{π}{4.4}

β = \frac{4}{1.2}

(i.e., the ZN closed loop method). Using the Internal Model Control (IMC) PI controller tuning rules in Table 1 of [7] with closed loop time constant

T_{c} = \sqrt{10} τ

, as proposed in [6], gives parameters

α = 0.42

and

β = 7.32

. Using the Simple/Skogestad IMC (SIMC) PI controller tuning rules, presented in the works of [8,12,13], with closed loop time constant

T_{c} = τ

(i.e., is the only tuning parameter in SIMC) gives

α = 0.5

and

β = 8

To find PI controller settings with good robustness properties (i.e., one could have uncertainties in the gain velocity and time delay) and simultaneously obtain reasonable fast reference and disturbance properties, for IPTD processes, the size and balanced relation between the parameters

α

and

β

are of importance.

Using the PI controller setting in Equation (2), we may define a Method Product (MP) parameter

\bar{c}

as,

\begin{matrix} \bar{c} = α β = K_{p} T_{i} k . \end{matrix}

(3)

The defined MP parameter

\bar{c}

in Equation (3) is constant for numerous PI controller tuning methods. The SIMC PI controller settings yield an MP parameter

\bar{c} = 4

. The original ZN method gives an MP parameter

\bar{c} = 2.38

(i.e., the ZN closed loop method).

In this paper, we search for optimal MP parameters, i.e., choosing

\bar{c}

which ensures the closed loop system some optimal robustness or performance setting, e.g., minimisation of the Integrated Absolute Error (IAE) or sensitivity index

M_{s}

given a prescribed robustness. Figure 1 shows that

M_{s}

is approximately minimised for

\bar{c} = 2.0

. However, it might be argued that the changes in

M_{s}

is negligible, and that

M_{s}

is optimal over the MP parameter interval

1.5 \leq \bar{c} \leq 4.0

It has been pointed out that there is usually a high degree of trial-and-error in choosing the closed loop time constant tuning parameter

T_{c}

in SIMC and IMC (e.g., [1] for SIMC and [6] for IMC). Note that one also may focus on the maximum sensitivity peak

M_{s}

of the sensitivity function as described in [14], where some inequalities relating to the Gain Margin (GM) and the Phase Margin (PM) to the robustness

M_{s}

are proposed on p. 126. Consider that the values of the minimum robustness

M_{s}

are in the interval

1.3 \leq M_{s} \leq 2

[14].

The contributions of this work are itemised in the incoming:

The PI controller tuning method in the work of [1,2] is further developed with more optimal settings for the MP parameter as well as tuning for some special instance integrating systems.
In the instance of a small or zero time delay $τ = 0$ , we propose a variant in which the Maximum Time Delay Error (MTDE) $d τ_{m a x} > 0$ is the tuning parameter (see Section 3.2).
Two optimal settings for the MP parameter are presented in Section 4. These are optimal in the sense that they minimise a Pareto performance objective (i.e., integrated absolute error for combined step changes in output and input disturbances) on two different aspects. One additional MP parameter is deduced from approximating the time delay with a (2, 1) Pade approximation in Section 3.3.
Additional MP parameter settings are suggested for minimising a variety of given indices.
The presented method (including variants of this) is demonstrated and compared to the Pareto-Optimal (PO) and SIMC (when possible) tuned PI controllers on various motivated (possible) higher order process model examples in Section 5.

The rest of this paper is organised as follows. The preliminary theory containing the definitions and some basic theory are given in Section 2. In Section 3, we present analytical results about the MTDE and present PI controller tuning rules as a function of a prescribed MTDE. Numerical simulation examples for some (possible) higher order systems/models are presented in Section 5. The conclusion and discussion remarks are given in Section 7.

2. Preliminary Theory

2.1. Definitions

Given a PI controller

\begin{matrix} H_{c} (s) = K_{p} \frac{T_{i} s + 1}{T_{i} s}, \end{matrix}

(4)

where

K_{p}

is the proportional constant and

T_{i}

is the integral time constant.

Consider the standard feedback system with disturbances as illustrated in Figure 2. To compare the different controllers, we consider indices such as defined in [12,14,15]. Performance is measured in a feedback system by

\begin{matrix} IAE = \int_{0}^{\infty} | e | d t . \end{matrix}

(5)

Furthermore, the following is defined.

${IAE}_{v u}$ evaluates the performance in case of a step input disturbance ( $H_{v} (s) = H_{p} (s)$ ), $v = 1$ (default), with the reference, $r = 0$ .
${IAE}_{v y}$ evaluates the performance in case of a step output disturbance ( $H_{v} (s) = 1$ ), $v = 1$ (default), with the reference, $r = 0$ .
${IAE}_{r}$ evaluates the performance in case of a reference unit step, $r = 1$ , with the disturbance, $v = 0$ .

Similarly, we define the Integrated Time-weighted Absolute Error (ITAE), Integrated Square Error (ISE) and Integrated Time-weighted Square Error (ITSE) and Total input Value (TV) as the following i.e.,

\begin{matrix} ITAE & = & \int_{0}^{\infty} t | e | d t, \end{matrix}

(6)

\begin{matrix} ISE & = & \int_{0}^{\infty} e^{2} d t, \end{matrix}

(7)

\begin{matrix} ITSE & = & \int_{0}^{\infty} t e^{2} d t, \end{matrix}

(8)

\begin{matrix} TV & = & \int_{0}^{\infty} | Δ u_{k} | d t, \end{matrix}

(9)

where

Δ u_{k} = u_{k} - u_{k - 1}

Robustness is quantified according to the maximum sensitivity peak

\begin{matrix} M_{s} & = & max_{0 \leq ω < \infty} | S (j ω) | = {| | S (j ω) | |}_{\infty}, \end{matrix}

(10)

where,

S (j ω) = \frac{1}{1 + H_{p} (j ω) H_{c} (j ω)}

, and

| | \cdot {| |}_{\infty}

is the

H_{\infty}

-norm.

2.2. Lag-Dominant Systems

Given a system approximated with a FOPTD model

\begin{matrix} H_{p} (s) = \frac{K}{1 + T s} e^{- τ s}, \end{matrix}

(11)

where K is the process gain,

τ

is the time delay and T is the time constant. The system in Equation (11) may be defined as lag-dominant when

T > τ

which is the instance for numerous systems. It is known that, when

T ≫ τ

then Equation (11) may be approximated with an IPTD model (see [6,7]). From Equation (11), we write,

\begin{matrix} H_{p} (s) = \frac{K}{T} \frac{1}{s + \frac{1}{T}} e^{- τ s} . \end{matrix}

(12)

Hence, when the system is lag-dominant and T “large”, we may approximate Equation (12) as an IPTD system (Equation (1)) where

k = \frac{K}{T}

is the gain velocity (slope) and

τ

the time delay.

2.3. SIMC Tuning Rules

Given the FOPTD process in Equation (11). The standard SIMC PI controller settings [8,12,13] are as follows,

\begin{matrix} K_{p} = \frac{T}{K (T_{c} + τ)}, T_{i} = min (T, 4 (T_{c} + τ)), \end{matrix}

(13)

where

T_{c}

is the prescribed time constant for the reference response chosen as

- τ < T_{c} < \infty

Similarly, for an IPTD process as in Equation (1), we have the following PI controller settings,

\begin{matrix} K_{p} = \frac{1}{k (T_{c} + τ)}, T_{i} = 4 (T_{c} + τ) . \end{matrix}

(14)

3. Tuning for Maximum Time Delay Error

To get some understanding of the PM of the closed loop system and the MTDE,

d τ_{m a x}

, we work out some analytic results in the following, which give a PI controller tuning method for IPTD processes.

3.1. Integrator Plus Time Delay Process

Consider an IPTD system where k is the gain velocity and

τ

is the time delay, and a PI controller. The loop transfer function,

H_{0} (s) = H_{c} (s) H_{p} (s)

, is

\begin{matrix} H_{0} (s) = K_{p} \frac{1 + T_{i} s}{T_{i} s} k \frac{e^{- τ s}}{s} . \end{matrix}

(15)

The frequency response is given by

H_{0} (j ω) = | H_{0} (j ω) | e^{j ∠ H_{0} (j ω)}

, where the magnitude is

| H_{0} (j ω) | = \frac{K_{p} k}{T_{i} ω^{2}} \sqrt{1 + {(T_{i} ω)}^{2}}

and the phase angle is

∠ H_{0} (j ω) = - τ ω - π + arctan (T_{i} ω)

. We obtain the gain crossover frequency

ω_{c}

analytically as

| H_{0} {(j ω)}_{c} | = 1

. From this, we obtain analytically that

PM = ∠ H_{0} (j ω_{c}) + π

, and the MTDE

d τ_{\max}

, such that,

0 = PM - d τ_{\max} ω_{c}

A factor f is defined as

\begin{matrix} f = \frac{1 + \sqrt{1 + \frac{4}{{(K_{p} T_{i} k)}^{2}}}}{2} = \frac{1 + \sqrt{1 + \frac{4}{{(α β)}^{2}}}}{2} . \end{matrix}

(16)

The gain crossover frequency is analytically given by

\begin{matrix} ω_{c} = \sqrt{f} K_{p} k . \end{matrix}

(17)

See previous paper [1] for proof of Equation (17).

The gain crossover frequency is then given by

ω_{c} = \sqrt{f} \frac{α}{τ}

. We obtain the PM analytically as

PM = - \sqrt{f} α + arctan (\sqrt{f} α β)

, and the MTDE as

d τ_{\max} = \frac{PM}{ω_{c}} = δ τ

, where

δ

is defined as

\begin{matrix} δ = \frac{- \sqrt{f} α + arctan (\sqrt{f} α β)}{\sqrt{f} α} = \frac{arctan (\sqrt{f} α β)}{\sqrt{f} α} - 1 . \end{matrix}

(18)

Consider the instance in which the MP parameter

\bar{c} = α β

is constant, then Equation (18) may be written as,

δ = a \frac{1}{α} - 1

, and,

δ = \frac{a}{\bar{c}} β - 1

, where

\begin{matrix} a = \frac{arctan (\sqrt{f} α β)}{\sqrt{f}}, \end{matrix}

(19)

is a function of

\bar{c} = α β

and constant. Notice that the parameter f is defined by Equation (16).

We have the following Algorithm 1.

Algorithm 1

(Max time delay error tuning).

The MP parameter is defined as

\begin{matrix} \bar{c} = α β . \end{matrix}

(20)

We express β as a linear function of a prescribed Relative Time Delay Error (RTDE)

δ > 0

, to ensure stability of the closed loop system. We have

\begin{matrix} β = \frac{\bar{c}}{a} (δ + 1), \end{matrix}

(21)

where parameter a is given by Equation (19). Note that α can be expressed by

\begin{matrix} α = \frac{\bar{c}}{β} = \frac{a}{δ + 1} . \end{matrix}

(22)

or with regard to the PI controller parameters

\begin{matrix} T_{i} = \frac{\bar{c}}{a} (δ + 1) τ, \end{matrix}

(23)

\begin{matrix} K_{p} = \frac{a}{k τ (δ + 1)} . \end{matrix}

(24)

Note that Algorithm 1 is written as a MATLAB m-file function given in Appendix C in a previous paper [1].

Before advancing, we demonstrate the above algorithm in an instance to enhance the robustness of the classical closed loop ZN PI controller tuning.

Example 1

(ZN with increased margins).

Given the classical ZN PI controller tuning (closed loop method), in which

α = \frac{π}{4.4}

β = \frac{4}{1.2}

, where the RTDE

\frac{d τ_{m a x}}{τ} = δ \approx 0.56

and the robustness

M_{s} \approx 2.86

For the original ZN method, we have the MP parameter

\bar{c} = 2.38

. Specifying an RTDE parameter,

δ = \frac{d τ_{m a x}}{τ} = 1.6

. Using Equations (21) and (22) gives the altered ZN PI controller parameters

\begin{matrix} α = 0.42, β = 5.55 . \end{matrix}

(25)

The altered ZN PI controller tuning,

K_{p} = \frac{α}{k τ}

and

T_{i} = β τ

, for an IPTD process has margins

G M = 3.35

, robustness

M_{s} = 1.66

and prescribed

\frac{d τ_{m a x}}{τ} = 1.6

. The altered ZN PI controller tuning has relatively smooth closed loop responses with a relative damping slightly less than one. The ZN method parameter

\bar{c} = 2.38

is not too far from one of the recommended optimal parameters (see below).

Arguably, the most important characteristic of a PI controller setting is the robustness vs. model uncertainty in connection with a reasonably smooth and fast closed loop reference and disturbance responses. An MTDE

d τ_{m a x} = 1.6 τ

is reasonable. This is approximately equal to the MTDE for the SIMC setting,

d τ_{m a x} = 1.59 τ

. One idea may be to find theoretical arguments for setting the MP parameter

\bar{c}

such that the closed loop system gets some optimal settings, e.g., minimise the robustness

M_{s}

given prescribed robustness

δ

. Consider using the PI controller tuning rules deduced in [1] which gives the MP parameter

\bar{c} = 2.76

The MP parameter

\bar{c} = α β

may be seen as a tuning parameter. SIMC uses a MP parameter

\bar{c} = 4

and the corresponding

G M \approx 2.96

, which is below the recommended margin, but the MTDE is acceptable, i.e.,

d τ_{m a x} = 1.59 τ

. Based on the numerical simulations in this and previous works [1,2], we suggest a relatively broad interval for choosing the MP parameter

\bar{c}

, i.e.,

\bar{c} \in [1.5, 4.0]

Furthermore, we propose choosing the RTDE

δ > 0

to unsure stability, and choosing

δ

\bar{c} \in [1.1, 3.4]

for robustness and to make certain that

1.3 \leq M_{s} \leq 2.0

(p. 125 in [14]) is reasonable.

3.2. Pure Integrating Process

Consider the limiting case of an integrating process, i.e.,

τ = 0

(no delay), or a time constant system with a large time constant such that

\frac{1}{T} \approx 0

, i.e., we consider a process model,

H_{p} (s) = \frac{k}{s}

. Using the definition for the RTDE tuning parameter,

δ = \frac{d τ_{\max}}{τ}

, and the PI controller tuning Equations (23) and (24), we find the PI controller tuning

\begin{matrix} T_{i} = \frac{\bar{c}}{a} (\frac{d τ_{\max}}{τ} + 1) τ = \frac{\bar{c}}{a} (d τ_{\max} + τ), \end{matrix}

(26)

\begin{matrix} K_{p} = \frac{a}{k τ (\frac{d τ_{\max}}{τ} + 1)} = \frac{a}{k (d τ_{\max} + τ)} . \end{matrix}

(27)

Notice that Equations (26) and (27) are tuning variants in which the MTDE

d τ_{\max} > 0

is the tuning parameter instead of the RTDE

δ

Consider the limiting case of an integrating process, i.e.,

τ = 0

(no delay). From Equations (26) and (27), we find the PI controller tuning

\begin{matrix} T_{i} = \frac{\bar{c}}{a} d τ_{\max}, \end{matrix}

(28)

\begin{matrix} K_{p} = \frac{a}{k d τ_{\max}} . \end{matrix}

(29)

Notice that

P M = a \sqrt{f}

in this case.

3.3. Using a ( $2, 1$ ) Pade Approximation

Consider the disturbance response with PI control,

\begin{matrix} \frac{y}{v} (s) & = & \frac{H_{p}}{1 + H_{c} H_{p}} = \frac{k \frac{e^{- τ s}}{s}}{1 + K_{p} \frac{1 + T_{i} s}{T_{i} s} k \frac{e^{- τ s}}{s}} \\ = & \frac{k s e^{- τ s}}{s^{2} + \frac{K_{p} k}{T_{i}} (1 + T_{i} s) e^{- τ s}} . \end{matrix}

(30)

Consider a (

2, 1

) Pade approximation,

e^{x} = \frac{6 + 4 x + x^{2}}{6 - 2 x}

, i.e., with a second order numerator polynomial and a first order denominator polynomial, i.e., an approximation,

\begin{matrix} e^{- τ s} \approx \frac{1 - b_{1} s + b_{2} s^{2}}{1 + a_{1} s}, \end{matrix}

(31)

where

a_{1} = \frac{τ}{3}

b_{1} = \frac{2 τ}{3}

and

b_{2} = \frac{τ^{2}}{6}

Using the same procedure as in Section 5.2 in [1], and with unit relative damping, we find a third order polynomial for the closed loop response,

\begin{matrix} \frac{y}{v} (s) = \frac{T_{i} s}{K_{p}} \frac{b_{2} s^{2} - b_{1} s + 1}{(\frac{a_{1} T_{i}}{k K_{p}} + b_{2} T_{i}) s^{3} + (b_{2} - b_{1} T_{i} + \frac{T_{i}}{k K_{p}}) s^{2} + (T_{i} - b_{1}) s + 1} . \end{matrix}

(32)

We prescribe a third order polynomial

\begin{matrix} Π (s) & = & (1 - τ_{0} s) (τ_{0}^{2} s^{2} + 2 τ_{0} s + 1) = {(1 + τ_{0} s)}^{3} \\ = & τ_{0}^{3} s^{3} + 3 τ_{0}^{2} s^{2} + 3 τ_{0} s + 1 . \end{matrix}

(33)

When comparing Equations (32) and (33), we find that

\begin{matrix} τ_{0}^{3} & = & T_{i} (\frac{a_{1}}{k K_{p}} + b_{2}), \end{matrix}

(34)

\begin{matrix} 3 τ_{0}^{2} & = & b_{2} - T_{i} (b_{1} - \frac{1}{k K_{p}}), \end{matrix}

(35)

\begin{matrix} 3 τ_{0} & = & T_{i} - b_{1} . \end{matrix}

(36)

By inserting Equations (34) and (36) into Equation (35), it can be shown that

\begin{matrix} {(\frac{τ_{0}}{τ})}^{3} - {(\frac{τ_{0}}{τ})}^{2} - \frac{7}{6} (\frac{τ_{0}}{τ}) - \frac{11}{54} = 0 . \end{matrix}

(37)

We solve the third order polynomial in Equation (37) with respect to

\frac{τ_{0}}{τ}

, and find a real positive solution,

\frac{τ_{0}}{τ} \approx 1.7385

Furthermore, we find that the PI controller parameters

\begin{matrix} T_{i} = 3 τ_{0} + b_{1}, K_{p} = \frac{a_{1} T_{i}}{k (τ_{0}^{3} - b_{2} T_{i})}, \end{matrix}

(38)

where

τ_{0}

may be seen as a tuning parameter.

When assuming that the response time constant

τ_{0} = c τ

, then we may express the PI controller parameters

K_{p} = \frac{α}{k τ}

and

T_{i} = β τ

with

\begin{matrix} β & = & \frac{9 c + 2}{3} = 3 c + \frac{2}{3}, \end{matrix}

(39)

\begin{matrix} α & = & \frac{2 (9 c + 2)}{18 c^{3} - 9 c - 2} = \frac{c + \frac{2}{9}}{c^{3} - \frac{1}{2} c - \frac{1}{9}}, \end{matrix}

(40)

where the product

\bar{c} = α β

is a nonlinear function of the tuning parameter c. We find that it makes sense to choose c in the interval,

1.4 \leq c \leq 2.5

From the PI controller setting in Equation (38) with

τ_{0} = 1.7385

, we find the MP parameter

\bar{c} = α β = K_{p} k T_{i} \approx 2.6985

For reducing the complexity of the problem, the (1, 2) Pade approximation was used; e.g., a (2, 2) Pade would result in a fourth order polynomial. Notice that a (1, 1) Pade approximation was used in the earlier work of [1] in Section 5.2.

4. Optimal Performance Settings

Consider the following Pareto performance objective defined as

\begin{matrix} J (p) = s_{r} \frac{{IAE}_{x} (p)}{{IAE}_{x}^{o}} + (1 - s_{r}) \frac{{IAE}_{v u} (p)}{{IAE}_{v u}^{o}}, \end{matrix}

(41)

where

s_{r}

is the servo-regulator parameter originally introduced in [2], and is chosen in the interval

0 \leq s_{r} \leq 1.0

for the weighting between output disturbance (servo) weighting

s_{r} = 1.0

and input disturbance (regulator) weighting

s_{r} = 0

. In Equation (41), the function argument is

p = {[K_{p}, T_{i}]}^{T}

. In this work, we set

s_{r} = 0.5

([16]). Furthermore, we set

x = v y

, which was argued in [17] to be the equivalent of setting

x = r

, which was used in the original work of [16] and also [2]. The reference/weight values are calculated as following,

{IAE}_{v y}^{o} = {min}_{p} {IAE}_{v y} (p)

, and,

{IAE}_{v u}^{o} = {min}_{p} {IAE}_{v u} (p)

, for a prescribed robustness

M_{s}^{p r e}

. We set

M_{s}^{p r e} = 1.59

which is the robustness value corresponding to a SIMC-tuned PI controller with

T_{c} = τ

for a FOPTD process where

K = T = τ = 1

([16]).

We consider the reference example where we are given an IPTD process with

k = τ = 1

. We find the same reference values as in [18], viz.

{IAE}_{v y}^{o} = 2.17

where

K_{p} = 0.5

and

T_{i} = \infty

, and

{IAE}_{v u}^{o} = 15.10

where

T_{i} = 5.8

and

K_{p} = 0.4

The following main performance objective is defined in a mean square error sense,

\begin{matrix} V_{M} (x, y) = \frac{1}{M} \sum_{i = 1}^{M} {(J_{x} (i) - J_{y} (i))}^{2}, \end{matrix}

(42)

where x is a tuning method and,

y = PO

(default) and M = length( $M_{s}$ ).

A couple of optimal suggestions for the choice of the MP parameter are worked out in the following. The first MP parameter setting may be found by solving the following optimization problem,

\begin{matrix} \bar{c} & = & arg min_{\bar{c}} V_{M} (Alg . 1 (\bar{c}), Alg . 1^{o}) = 2.7, \end{matrix}

(43)

where Alg. 1 (

\bar{c}, δ_{i}

) and Alg. 1^o (

δ_{i}

) is pre-calculated as follows

\begin{matrix} J_{Alg . 1_{i}} = min_{\bar{c}} J_{Alg . 1} (\bar{c}, δ_{i}) \forall 1.1 \leq δ_{i} \leq 3.4 . \end{matrix}

(44)

Interestingly, the MP parameter setting in Equation (43) is approximately equal to the setting which is deduced in Section 3.3. Additional MP parameter settings are given in Table 1 based on solving Equation (43) for different servo-regulator parameters

0 \leq s_{r} \leq 1.0

in the Pareto performance objective J (Equation (41)).

The second MP parameter is found by

\begin{matrix} \bar{c} = arg min_{\bar{c}} V_{M} (Alg . 1 (\bar{c}), PO) = 2.5, \end{matrix}

(45)

where Alg. 1 (

\bar{c}, δ (M_{s}^{i})

) and PO (

M_{s}^{i}

) are pre-calculated as follows

\begin{matrix} J_{P O_{i}} = min_{p} J (p, M_{s}^{i}) \forall 1.1 \leq M_{s}^{i} \leq 3.4 . \end{matrix}

(46)

Notice that

\bar{c} = 2.5

is equal to the recommended MP parameter in [2]. However, the MP parameter in this paper results from an optimization problem, while the one proposed in [2] originated from an ad hoc approach.

Figure 3 illustrates the two MP parameters described above. In terms of the main performance objective

V_{M}

(Equation (42)), Table 2 shows that

\bar{c} = 2.5

\frac{V_{\bar{c} = 4}}{V_{\bar{c} = 2.5}} = 3 e + 4

times better than SIMC (arguably

\bar{c} = 4

), and

\frac{V_{\bar{c} = 2.7}}{V_{\bar{c} = 2.5}} = 78

times better than

\bar{c} = 2.7

Based on numerical simulations, we present the recommended settings for choosing the MP parameter

\bar{c}

as proposed in Table 3.

Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with varying gain velocity, k, and time delay

τ \geq 0

. Table 4 and Table 5 illustrate the

{\bar{c}}_{m i n} = {a r g m i n}_{\bar{c}} M_{s}

, i.e., the minimum of

M_{s}

{IAE}_{v u}

ITAE

{ITAE}_{v u}

and

{IAE}_{r}

TV

ISE

ITSE

{ITAE}_{r}

, respectively, as a function of

\bar{c}

Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, where

k = 1

and time delay

τ = 1

. Figure 4 shows the indices

M_{s}

{ITAE}_{v u}

{IAE}_{r}

{ITAE}_{r}

{IAE}_{r}

TV

ISE

{ITAE}_{v u}

and

IAE

as a function of varying the MP parameter

\bar{c} \in [1.5, 4.0]

and with prescribed RTDE

δ = 1.6

5. Simulation Examples

In the following simulations (if possible), we compare Algorithm 1, with the recommended MP parameter settings, vs. the SIMC tuning rule [12].

We continue with studying the reference example considered in Section 4. See also [1] for additional details on this example.

Example 2

(Reference Example).

The same IPTD example as in [1] is used, i.e., a process model,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with gain velocity

k = 1

and time delay

τ = 1

is considered.

The time-domain responses given a prescribed robustness,

M_{s} = 1.59

, are illustrated in Figure 5. The corresponding PI controller parameters, indices and margins are given in Table 6. The margins for the controllers are all acceptable, i.e.,

G M > 2

and

P M > 30

as in [14].

Example 3

(Lag-dominant system).

An air-heater was studied in [19] and it was found that a FOPTD model with process gain

K = 5.7

, time delay

τ = 4

and time constant

T = 60

, gives a sufficient model approximation. We approximate the FOPTD model as an IPTD process where the gain velocity (slope)

k = \frac{K}{T} = 0.095

and time delay,

τ = 4

The Pareto performance objective J vs.

M_{s}

trade-off curves are shown in Figure 6. In terms of the main performance objective

V_{M}

it can be seen in Table 7 that

\bar{c} = 2.5

\frac{V_{\bar{c} = 2.7}}{V_{\bar{c} = 2.5}} = 1.9

times better than

\bar{c} = 2.7

and

\frac{V_{SIMC}}{V_{\bar{c} = 2.5}} = 12.2

times better than SIMC.

The time-domain responses, given a prescribed robustness,

M_{s} = 1.59

, are illustrated in Figure 7. The corresponding PI controller parameters, indices and margins are given in Table 8. The margins for the controllers are all acceptable, i.e.,

G M > 2

and

P M > 30

as in [14]. Notice, that the prescribed MTDE,

d τ_{m a x} = δ τ = 7.16

is almost equal the exact

D M = 7.51

Some results regarding a couple of motivated higher order processes are presented in the following examples. Notice that SIMC offers the half-rule model reduction technique. However, for our case, we approximate the higher order systems by identifying two parameters, the unit reaction rate

R_{1}

and the lag L from a step response, i.e., the Process–Reaction Curve (PRC) as presented in the work of ZN [9,10,11]. We denote the variant as follows: PRC + Algorithm 1.

Example 4

(Higher order process).

A distillation column studied in [20] (p. 591) is partly described by the following process model,

\begin{matrix} H_{p} (s) = \frac{34}{(54 s + 1) {(0.5 s + 1)}^{2}} . \end{matrix}

(47)

By identifying the lag L and the maximum slope (unit reaction rate)

R_{1}

from the PRC method we may approximate the process model as an IPTD model with gain velocity

k = R_{1} = 0.597

and time delay

τ = L = 0.923

Using the half-rule technique in SIMC, we approximate a FOPTD model where the gain

K = 34

, time constant

T = 54 + \frac{0.5}{2} = 54.25

, and time delay

τ = 0.5 + \frac{0.5}{2} = 0.75

The Pareto Performance objective J vs. robustness

M_{s}

trade-off curves are illustrated in Figure 8. Notice, that

\bar{c} = 2.7

is the closest to optimal on the most robust part of the

M_{s}

-interval. SIMC is crossing

\bar{c} = 2.7

around

M_{s} = 1.64

and is the closest to optimal on the less robust part. In terms of the main performance objective

V_{M}

, we show in Table 9 that

\bar{c} = 2.7

\frac{V_{\bar{c} = 2.5}}{V_{\bar{c} = 2.7}} = 2.3

times better than

\bar{c} = 2.5

, and

\frac{V_{SIMC}}{V_{\bar{c} = 2.7}} = 20.7

times better than SIMC.

The time-domain responses, for a prescribed robustness

M_{s} = 1.59

, are illustrated in Figure 9. The corresponding PI controller parameters, indices and margins are given in Table 10. The margins for the controllers are all acceptable, i.e.,

G M > 2

and

P M > 30

, as in [14]. Notice, that the prescribed MTDE,

d τ_{m a x} = δ τ = 1.50

is almost equal to the exact

D M = 1.54

Note that the half-rule technique in SIMC is not compatible with process models containing complex poles/underdamped dynamics, hence, in such cases, we consider arguably the same algorithm as SIMC, i.e., Algorithm 1, where the MP parameter,

\bar{c} = 4

. An example of this is given in the following.

Example 5

(Underdamped system).

An unmanned submersible vehicle studied in [21] is described partly by

\begin{matrix} H_{p} (s) & = & \frac{- 2.6158 (2.299 s + 1)}{(0.8131 s + 1) (0.5 s + 1) ({(7.692 s)}^{2} + 1.738 (7.692 s) + 1)}, \end{matrix}

(48)

i.e., from commanded elevator deflection u to the pitch angle of the vehicle y. We approximate Equation (48) by an IPTD model with gain velocity

k = R_{1} = - 0.145

and time delay

τ = L = 1.729

The Pareto performance objective J vs.

M_{s}

trade-off curves are illustrated in Figure 10. In terms of the main performance objective

V_{M}

, we show in Table 11 that

\bar{c} = 2.5

\frac{V_{\bar{c} = 4}}{V_{\bar{c} = 2.5}} = 7.3

times better than

\bar{c} = 4

and

\frac{V_{\bar{c} = 2.7}}{V_{\bar{c} = 2.5}} = 1.3

times better than

\bar{c} = 2.7

. Notice that

\bar{c} = 2.5

is closest to optimal on the most robust part of the

M_{s}

-interval. Furthermore,

\bar{c} = 4

is seen crossing both

\bar{c} = 2.5

and

\bar{c} = 2.7

around

M_{s} = 1.55

and is closest to optimal on the less robust part.

The time-domain responses, given a prescribed robustness

M_{s} = 1.59

, are illustrated in Figure 11. The corresponding PI controller parameters, indices and margins are given in Table 12. The margins for the controllers are all acceptable, i.e.,

G M > 2

and

P M > 30

as in [14].

Last, we propose a tuning variant based on the PRC and Algorithm 1, as above. However, now, the gain velocity in the IPTD model is, instead, varying proportionally, i.e.,

k = R_{1} ζ

, where

ζ

is considered as a tuning parameter. TO simplify the tuning, we propose to set the RTDE

δ = \bar{c}

equal constant (i.e., an ad hoc suggestion). This means that the only tuning parameter is

ζ

. We denote this variant as follows:

ζ

-PRC + Algorithm 1.

Example 6

ζ-PRC variant).

Consider the same process model as studied in Example 5. The model is approximated by an IPTD model, where the gain velocity is varied,

k = R_{1} ζ = - 0.145 ζ

, and time delay,

τ = L = 1.729

. In this example, we set the RTDE

δ = \bar{c} = 2.7

It can be seen in Figure 12 that the PO PI curve and the ζ-PRC curve are indistinguishable. This is quite a surprising result. In terms of the main performance objective

V_{M}

, we show in Table 13 that ζ-PRC is

\frac{V_{ζ - P R C}}{V_{P R C}} = 4 e + 3

times better than

P R C

variant.

The time-domain responses, for a prescribed robustness

M_{s} = 1.59

, are illustrated in Figure 13. As a consequence of the above, these responses are also indistinguishable. The corresponding PI controller parameters, indices and margins are given in Table 14.

6. Discussion

Remarks to Section 3

It can be shown that the PM can be given as follows

\begin{matrix} P M = δ \sqrt{f} α, \end{matrix}

(49)

for the PM in radians (see also [1,2]).

7. Concluding Remarks

The discussion and concluding remarks are itemised as follows.

The method in [1,2] is further developed with more optimal MP tuning parameters as well as tuning for some special case integrating systems.
Two optimal settings for the MP parameter are presented in Section 4. These are optimal in the sense that they minimise the main performance objective $V_{M}$ on two different aspects. Interestingly, one of the MP parameters may (arguably) be deduced from approximating the time delay with a (2, 1) Pade approximation in Section 3.3.
In the case of a small or zero time delay $τ = 0$ , we propose a variant in which the MTDE $d τ_{m a x}$ is the tuning parameter.
Note that for an IPTD model, the SIMC tuned PI controllers are seen far from optimal, i.e., PO (or (almost) equivalently, Algorithm 1 with the MP parameter setting as $\bar{c} = 2.5$ ). See Section 4.
The presented method (and variants of this) is successfully demonstrated and compared to the SIMC and PO PI controllers on numerous motivated process model examples in Section 5.
Note that, for the higher order process models in Examples 4 and 5, we use the PRC model reduction technique, which is generally easier to apply than the half-rule technique proposed in [12]. The half-rule technique is not compatible with handling complex poles.
Some surprisingly optimal results are documented for Example 6, where a tuning method based on varying the gain velocity, $k = ζ R_{1}$ , ( $R_{1}$ , is the ZN unit reaction rate), i.e., the tuning parameter is $ζ$ . Note that setting the RTDE $δ = \bar{c}$ (i.e., an ad hoc choice) equal a constant is advisable.
Note that the results in Section 5 are based on the original (possible) higher order models. The approximated IPTD models are only used for the PI controller design.

Author Contributions

David Di Ruscio contributed to the conception of the research, formulated the theory and helped revise the paper. Christer Dalen wrote the paper and did the numerical simulations.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

PI	Proportional Integrating
IPTD	Integrator Plus Time Delay
FOPTD	First Order Plus Time Delay
ZN	Ziegler–Nichols
IAE	Integrated Absolute Error
ITAE	Integrated Time-weighted Absolute Error
ISE	Integrated Square Error
ITSE	Integrated Time-weighted Square Error
TV	Total input Value
MP	Method Product
IMC	Internal Model Control
SIMC	Simple/Skogestad Internal Model Control
GM	Gain Margin
PM	Phase Margin
DM	Delay Margin
MTDE	Maximum Time Delay Error
PO	Pareto-Optimal
RTDE	Relative Time Delay Error

References

Di Ruscio, D. On Tuning PI Controllers for Integrating Plus Time Delay Systems. Model. Identif. Control 2010, 31, 145–164. [Google Scholar] [CrossRef] [Green Version]
Di Ruscio, D. PI Controller Tuning Based on Integrating Plus Time Delay Models: Performance Optimal Tuning. In Proceedings of the IASTED Control and Applications Conference (CA2012), Crete, Greece, 18–21 June 2012. [Google Scholar]
Arbogast, J.E.; Cooper, D.J. Extension of IMC tuning correlations for non-self regulating (integrating) processes. ISA Trans. 2007, 46, 303–311. [Google Scholar] [CrossRef] [PubMed]
Alfaro, V.; Vilanova, R. Model Reference Robust Tuning of 2Dof PI Controllers for Integrating Controlled Processes. In Proceedings of the 2012 20th Mediterranean Conference on Control & Automation (MED), Barcelona, Spain, 3–6 July 2012. [Google Scholar]
Antonio Visioli, Q.Z. Control of Integral Processes with Dead Time; Springer-Verlag: London, UK, 2011. [Google Scholar]
Tyreus, B.D.; Luyben, W.L. Tuning PI Controllers for Integrator/Dead Time Processes. Ind. Eng. Chem. Res. 1992, 31, 2625–2628. [Google Scholar] [CrossRef]
Chien, I.L.; Fruehauf, P.S. Consider IMC Tuning to Improve Controller Performance. Chem. Eng. Prog. 1990, 86, 33–41. [Google Scholar]
Skogestad, S. Probably the best simple PID tuning rules in the world. In Proceedings of the AIChE Annual Meeting, Reno, Nevada, 6 November 2001. [Google Scholar]
Ziegler, J. “On-the-job” adjustments of air operated recorder-controllers. Instruments 1941, 16, 394–397. [Google Scholar]
Ziegler, J.; Nichols, N.B. Optimum Settings for Automatic Controllers. Trans. Am. Soc. Mech. Eng. 1942, 64, 759–768. [Google Scholar] [CrossRef]
Ziegler, J.; Nichols, N.B. Process lags in automatic control circuits. Trans. Am. Soc. Mech. Eng. 1943, 65, 433–444. [Google Scholar]
Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291–309. [Google Scholar] [CrossRef] [Green Version]
Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. Model. Identif. Control 2004, 25, 85–120. [Google Scholar] [CrossRef] [Green Version]
Åström, K.; Hägglund, T. PID Controllers: Theory, Design, and Tuning; Instrument Society of America: Research Triangle Park, NC, USA, 1995. [Google Scholar]
Seborg, D.; Edgar, T.F.; Mellichamp, D.A. Process Dynamics and Ciontrol; John Wiley and Sons: Hoboken, NJ, USA, 1989. [Google Scholar]
Grimholt, C.; Skogestad, S. Optimal PI-Control and Verification of the SIMC Tuning Rule. IFAC Proc. Vol. 2012, 45, 11–22. [Google Scholar] [CrossRef]
Grimholt, C.; Skogestad, S. Optimal PID-Control on First Order Plus Time Delay Systems & Verification of the SIMC Rules. IFAC Proc. Vol. 2013, 46, 265–270. [Google Scholar] [CrossRef]
Skogestad, S.; Grimholt, C. The SIMC Method for Smooth PID Controller Tuning; Springer: London, UK, 2012. [Google Scholar]
Haugen, F. Comparing PI Tuning Methods in a Real Benchmark Temperature Control System. Model. Identif. Control 2010, 31, 79–91. [Google Scholar] [CrossRef] [Green Version]
Luyben, W. Process Modeling, Simulation, and Control for Chemical Engineers; Chemical engineering series; McGraw-Hill: New York, NY, USA, 1990. [Google Scholar]
Abbasi, I.; Ali, S.; Ovinis, M.; Naeem, W. U-Model Based Controller Design for an Unmanned Free Swimming Submersible (UFSS) Vehicle Under Hydrodynamic Disturbances; NISCAIR-CSIR: New Delhi, Delhi, India, 2017; Volume 46, pp. 742–748. [Google Scholar]

Figure 1. Consider PI control of the FOPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure shows the robustness

M_{s}

as a function of the MP parameter

\bar{c}

, given constant robustness, for the interval

1.5 \leq \bar{c} \leq 4.0

Figure 1. Consider PI control of the FOPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure shows the robustness

M_{s}

as a function of the MP parameter

\bar{c}

, given constant robustness, for the interval

1.5 \leq \bar{c} \leq 4.0

Figure 2. Consider a control feedback system where the plant model is described by the process model,

H_{p} (s)

, PI controller,

H_{c} (s) = K_{p} \frac{1 + T_{i} s}{T_{i} s}

, and the disturbance model,

H_{v} (s)

, where disturbance v at the input when,

H_{v} (s) = H_{p} (s)

, and at the output when,

H_{v} (s) = 1

. Input u, output y and reference r.

Figure 2. Consider a control feedback system where the plant model is described by the process model,

H_{p} (s)

, PI controller,

H_{c} (s) = K_{p} \frac{1 + T_{i} s}{T_{i} s}

, and the disturbance model,

H_{v} (s)

, where disturbance v at the input when,

H_{v} (s) = H_{p} (s)

, and at the output when,

H_{v} (s) = 1

. Input u, output y and reference r.

Figure 3. Reference example (Example 2). Consider PI control of the IPTD model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure illustrates the trade-off between the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It illustrates the MP parameters

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

Figure 3. Reference example (Example 2). Consider PI control of the IPTD model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure illustrates the trade-off between the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It illustrates the MP parameters

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

Figure 4. Consider PI control of an IPTD process,

H_{p} (s) = k \frac{e^{- τ s}}{s}

with process parameters

k = 1

and

τ = 1

. PI controller

H_{c} (s) = K_{p} \frac{1 + T_{i} s}{T_{i} s}

with settings as in Algorithm 1. The figure shows the indices

M_{s}

{ITAE}_{v u}

{IAE}_{r}

{ITAE}_{r}

{IAE}_{r}

TV

ISE

{ITAE}_{v u}

and

IAE

as a function of varying the MP parameter

\bar{c} \in [1.5, 4.0]

and with prescribed RTDE

δ = 1.6

Figure 4. Consider PI control of an IPTD process,

H_{p} (s) = k \frac{e^{- τ s}}{s}

with process parameters

k = 1

and

τ = 1

. PI controller

H_{c} (s) = K_{p} \frac{1 + T_{i} s}{T_{i} s}

with settings as in Algorithm 1. The figure shows the indices

M_{s}

{ITAE}_{v u}

{IAE}_{r}

{ITAE}_{r}

{IAE}_{r}

TV

ISE

{ITAE}_{v u}

and

IAE

as a function of varying the MP parameter

\bar{c} \in [1.5, 4.0]

and with prescribed RTDE

δ = 1.6

Figure 5. Example 2 (Reference example). Consider PI control of an IPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI, SIMC with prescribed closed loop time constant

T_{c} = 1.24 τ

and Algorithm 1 where the MP parameter

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 1.79

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 50

Figure 5. Example 2 (Reference example). Consider PI control of an IPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI, SIMC with prescribed closed loop time constant

T_{c} = 1.24 τ

and Algorithm 1 where the MP parameter

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 1.79

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 50

Figure 6. Example 3. Consider PI control of the FOPTD process model,

H_{p} (s) = K \frac{e^{- τ s}}{T s + 1}

, where

K = 5.7

τ = 4

and

T = 60

. The figure shows the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It illustrates the MP parameters

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 (proposed in Section 4). SIMC with set-point time constant

T_{c}

is added for comparison.

Figure 6. Example 3. Consider PI control of the FOPTD process model,

H_{p} (s) = K \frac{e^{- τ s}}{T s + 1}

, where

K = 5.7

τ = 4

and

T = 60

. The figure shows the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It illustrates the MP parameters

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 (proposed in Section 4). SIMC with set-point time constant

T_{c}

is added for comparison.

Figure 7. Example 3. Consider PI control of the FOPTD process model,

H_{p} (s) = K \frac{e^{- τ s}}{T s + 1}

, where

K = 5.7

τ = 4

and

T = 60

. The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI, SIMC with prescribed closed loop time constant

T_{c} = 1.10 τ

, and Algorithm 1 where the MP parameter

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 1.56

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 140

Figure 7. Example 3. Consider PI control of the FOPTD process model,

H_{p} (s) = K \frac{e^{- τ s}}{T s + 1}

, where

K = 5.7

τ = 4

and

T = 60

. The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI, SIMC with prescribed closed loop time constant

T_{c} = 1.10 τ

, and Algorithm 1 where the MP parameter

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 1.56

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 140

Figure 8. Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It shows the MP parameter settings

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

M_{s}

(Equation (10)). It shows the MP parameter settings

\bar{c} = 2.5

and

\bar{c} = 2.7

for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

Figure 9. Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI controller vs. SIMC with closed loop time constant

T_{c} = 1.33 τ

, and PRC + Algorithm 1 where the MP parameter setting

\bar{c} = 2.7

(proposed in Section 4) and RTDE

δ = 1.63

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 35

Figure 9. Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of the following methods: the PO PI controller vs. SIMC with closed loop time constant

T_{c} = 1.33 τ

, and PRC + Algorithm 1 where the MP parameter setting

\bar{c} = 2.7

(proposed in Section 4) and RTDE

δ = 1.63

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 35

Figure 10. Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure shows the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It illustrates the PO PI controllers and PRC + Algorithm 1 variants with MP parameter settings

\bar{c} = 4.0

\bar{c} = 2.5

and

\bar{c} = 2.7

M_{s}

(Equation (10)). It illustrates the PO PI controllers and PRC + Algorithm 1 variants with MP parameter settings

\bar{c} = 4.0

\bar{c} = 2.5

and

\bar{c} = 2.7

Figure 11. Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of following methods: the PO PI and the PRC + Algorithm 1 where the MP parameter setting

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 2.20

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 80

Figure 11. Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, of following methods: the PO PI and the PRC + Algorithm 1 where the MP parameter setting

\bar{c} = 2.5

(proposed in Section 4) and RTDE

δ = 2.20

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 80

Figure 12. Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness

M_{s}

(Equation (10)). It shows the PO PI controllers with robustness

M_{s}

and the ζ-PRC + Algorithm 1 variant where the RTDE

δ = \bar{c} = 2.7

is fixed and the main tuning parameter is ζ.

M_{s}

(Equation (10)). It shows the PO PI controllers with robustness

M_{s}

and the ζ-PRC + Algorithm 1 variant where the RTDE

δ = \bar{c} = 2.7

is fixed and the main tuning parameter is ζ.

Figure 13. Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, for the following methods: the PO PI and the ζ-PRC + Algorithm 1 variant with MP parameter and MTDE settings

\bar{c} = δ = 2.7

, and tuning parameter

ζ = 0.74

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 80

Figure 13. Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness

M_{s} = 1.59

, for the following methods: the PO PI and the ζ-PRC + Algorithm 1 variant with MP parameter and MTDE settings

\bar{c} = δ = 2.7

, and tuning parameter

ζ = 0.74

. An output disturbance unit step is presented at time

t = 0

and an input disturbance unit step at time

t = 80

Table 1. The table shows the recommended MP parameters

\bar{c}

if one wants to minimise the main performance objective

V_{M}

(Equation (42)) for different servo-regulator parameters

s_{r}

in Equation (41). The optimal

\bar{c}

values as indicated are almost constant in the interval

δ \in [1.1, 3.4]

([2]).

Table 1. The table shows the recommended MP parameters

\bar{c}

if one wants to minimise the main performance objective

V_{M}

(Equation (42)) for different servo-regulator parameters

s_{r}

in Equation (41). The optimal

\bar{c}

values as indicated are almost constant in the interval

δ \in [1.1, 3.4]

([2]).

$s_{r}$	0	0.1	0.25	0.5	0.75	1
$\bar{c}$	2.4	2.5	2.6	2.7	3.7	∞

Table 2. Reference example (Example 2), i.e., an IPTD model,

H_{p} (s) = \frac{e^{- s}}{s}

. Comparing the different settings for the MP parameters for Algorithm 1 and SIMC using the main performance objective

V_{M}

(Equation (42)).

Table 2. Reference example (Example 2), i.e., an IPTD model,

H_{p} (s) = \frac{e^{- s}}{s}

. Comparing the different settings for the MP parameters for Algorithm 1 and SIMC using the main performance objective

V_{M}

(Equation (42)).

Method	$\bar{c}$ = 2.5	$\bar{c}$ = 2.7	SIMC
$V_{M}$ /e-4	0.02	1.56	592.75

Table 3. Summary: The table shows the recommended settings for the MP parameter

\bar{c}

for minimizing the objectives in the first row.

Table 3. Summary: The table shows the recommended settings for the MP parameter

\bar{c}

for minimizing the objectives in the first row.

	$M_{s}$	${IAE}_{vu}$	${ITAE}_{vu}$	${ITAE}_{r}$	${IAE}_{r}$	$V_{M} (δ O)$	$V_{M} (t)$
$\bar{c}$	2.0	2.4	2.4	2.6	4.0	2.7	2.5

Table 4. Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with varying gain velocity, k, and time delay

τ \geq 0

. The table illustrates the

{\bar{c}}_{\min} = {\arg \min}_{\bar{c}} M_{s}

, i.e., the minimum of the

M_{s}

{IAE}_{v u}

ITAE

and

{ITAE}_{v u}

indices as a function of

\bar{c}

, with PI controller settings from Algorithm 1.

Table 4. Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with varying gain velocity, k, and time delay

τ \geq 0

. The table illustrates the

{\bar{c}}_{\min} = {\arg \min}_{\bar{c}} M_{s}

, i.e., the minimum of the

M_{s}

{IAE}_{v u}

ITAE

and

{ITAE}_{v u}

indices as a function of

\bar{c}

, with PI controller settings from Algorithm 1.

k	$τ$	$M_{s}$	${IAE}_{vu}$	$ITAE$	${ITAE}_{vu}$
1	0.1	2.0	2.45	2.45	2.45
1	0.3	2.0	2.4	2.45	2.4
1	0.5	2.0	2.4	2.45	2.4
1	1	2.0	2.4	2.4	2.4
1	2	2.0	2.4	2.4	2.4
1	4	2.0	2.4	2.4	2.4
0.1	1	2.0	2.4	2.5	2.4
0.1	2	2.0	2.4	2.45	2.4
0.1	4	2.0	2.4	2.45	2.4

Table 5. Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with varying gain velocity, k, and time delay

τ \geq 0

. The table illustrates

{\bar{c}}_{\min} = {\arg \min i n}_{\bar{c}} M_{s}

, i.e., the minimum of

{IAE}_{r}

Tv

ISE

{ITAE}_{r}

and

ITSE

indices as a function of

\bar{c}

, with PI controller settings from Algorithm 1.

Table 5. Consider PI controller settings for an IPTD system,

H_{p} (s) = k \frac{e^{- τ s}}{s}

, with varying gain velocity, k, and time delay

τ \geq 0

. The table illustrates

{\bar{c}}_{\min} = {\arg \min i n}_{\bar{c}} M_{s}

, i.e., the minimum of

{IAE}_{r}

Tv

ISE

{ITAE}_{r}

and

ITSE

indices as a function of

\bar{c}

, with PI controller settings from Algorithm 1.

k	$τ$	${ITAE}_{r}$	$ITSE$	$ISE$	$TV$	${IAE}_{r}$
1.0	0.1	2.7	3.4	4.0	4.0	4.0
1.0	0.3	2.7	3.2	4.0	4.0	4.0
1.0	0.5	2.7	3.1	3.5	4.0	4.0
1.0	1.0	2.6	3.1	3.2	4.0	4.0
1.0	2.0	2.6	3.0	3.1	4.0	4.0
1.0	4.0	2.6	3.0	3.1	4.0	4.0
0.1	1.0	2.6	3.9	4.0	4.0	4.0
0.1	2.0	2.6	3.2	4.0	4.0	4.0
0.1	4.0	2.6	3.1	3.6	4.0	4.0

Table 6. Example 2. Consider PI control of the IPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The table shows the controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

for the following methods: Alg. 1 (

\bar{c} = 2.5, δ = 1.79

), SIMC (

T_{c} = 1.24 τ

) and PO PI (

M_{s} = 1.59

Table 6. Example 2. Consider PI control of the IPTD process model,

H_{p} (s) = \frac{e^{- s}}{s}

. The table shows the controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

for the following methods: Alg. 1 (

\bar{c} = 2.5, δ = 1.79

), SIMC (

T_{c} = 1.24 τ

) and PO PI (

M_{s} = 1.59

	Alg. 1	SIMC	PO PI
$K_{p}$	0.41	0.45	0.41
$T_{i}$	6.14	8.96	6.28
${IAE}_{v y}$	4.39	4.24	4.37
${IAE}_{v u}$	15.26	20.06	15.39
J	1.52	1.64	1.52
TV	3.33	3.12	3.31
GM	3.56	3.34	3.54
PM	44.57	50.02	44.94
DM	1.79	1.90	1.80
$M_{s}$	1.59	1.59	1.59

Table 7. Example 3. The table shows the comparison of the settings for Algorithm 1 and SIMC using the main performance objective

V_{M}

(Equation (42)).

Table 7. Example 3. The table shows the comparison of the settings for Algorithm 1 and SIMC using the main performance objective

V_{M}

(Equation (42)).

Method	$\bar{c}$ = 2.5	$\bar{c}$ = 2.7	SIMC
$V_{M}$ /e-2	0.57	1.08	6.96

Table 8. Example 3. The table shows the PI controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

Table 8. Example 3. The table shows the PI controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

	Alg. 1	SIMC	PO PI
$K_{p}$	1.17	1.25	1.12
$T_{i}$	22.55	33.60	19.47
${IAE}_{v y}$	15.13	13.53	15.70
${IAE}_{v u}$	17.73	25.16	15.89
J	1.39	1.52	1.37
TV	3.94	3.70	4.05
GM	3.36	3.22	3.46
PM	50.49	56.21	47.83
DM	7.51	8.08	7.26
$M_{s}$	1.59	1.59	1.59

Table 9. Example 4. The table shows the comparison of the different settings for Algorithm 1 and SIMC using the main performance

V_{M}

(Equation (42)).

Table 9. Example 4. The table shows the comparison of the different settings for Algorithm 1 and SIMC using the main performance

V_{M}

(Equation (42)).

Method	$\bar{c}$ = 2.5	$\bar{c}$ = 2.7	SIMC
$V_{M}$ /e-3	0.7	0.3	6.2

Table 10. Example 4. The table shows the PI controller parameters, indices and margins are given for prescribed robustness

M_{s} = 1.59

Table 10. Example 4. The table shows the PI controller parameters, indices and margins are given for prescribed robustness

M_{s} = 1.59

	Alg. 1	SIMC	PO PI
$K_{p}$	0.78	0.91	0.85
$T_{i}$	5.35	7.04	6.06
${IAE}_{v y}$	3.62	3.35	3.48
${IAE}_{v u}$	6.83	7.74	7.14
J	1.41	1.41	1.39
TV	3.77	3.77	3.77
GM	6.74	6.13	6.39
PM	43.63	46.74	45.19
DM	1.54	1.49	1.51
$M_{s}$	1.59	1.59	1.59

Table 11. Example 5. The table shows the different MP parameter settings for the PRC + Algorithm 1 variant with corresponding main performance

V_{M}

(Equation (42)).

Table 11. Example 5. The table shows the different MP parameter settings for the PRC + Algorithm 1 variant with corresponding main performance

V_{M}

(Equation (42)).

$\bar{c}$	2.5	2.7	4
$V_{M}$ /e-2	0.84	1.05	6.13

Table 12. Example 5. The corresponding controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

Table 12. Example 5. The corresponding controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

	PRC + Alg. 1	PO PI
$K_{p}$	−1.42	−1.70
$T_{i}$	12.18	14.90
${IAE}_{v y}$	6.41	5.88
${IAE}_{v u}$	8.59	8.72
J	1.04	1.00
TV	4.62	5.06
GM	13.80	11.84
PM	43.90	44.20
DM	3.03	2.74
$M_{s}$	1.59	1.59

Table 13. Example 6. Comparing the following variants, ζ-PRC + Algorithm 1 with MP parameter and MTDE settings

\bar{c} = δ = 2.7

and

ζ = 0.74

, and the PRC + Algorithm 1 with MP parameter setting

\bar{c} = 2.5

, using the main performance

V_{M}

defined in Equation (42).

Table 13. Example 6. Comparing the following variants, ζ-PRC + Algorithm 1 with MP parameter and MTDE settings

\bar{c} = δ = 2.7

and

ζ = 0.74

, and the PRC + Algorithm 1 with MP parameter setting

\bar{c} = 2.5

, using the main performance

V_{M}

defined in Equation (42).

Variant	PRC	$ζ$ -PRC
$V_{M}$ /e-4	83.6	0.02

Table 14. Example 6. The corresponding controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

. ζ-PRC + Algorithm 1.

Table 14. Example 6. The corresponding controller parameter, indices and margins are given for prescribed robustness

M_{s} = 1.59

. ζ-PRC + Algorithm 1.

	$ζ$ -Alg. 1	PO PI
$K_{p}$	−1.70	−1.70
$T_{i}$	14.82	14.90
${IAE}_{v y}$	5.88	5.88
${IAE}_{v u}$	8.69	8.72
J	1.00	1.00
TV	5.06	5.06
GM	11.85	11.84
PM	44.15	44.20
DM	2.74	2.74
$M_{s}$	1.59	1.59

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Dalen, C.; Di Ruscio, D. Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models. Algorithms 2018, 11, 86. https://doi.org/10.3390/a11060086

AMA Style

Dalen C, Di Ruscio D. Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models. Algorithms. 2018; 11(6):86. https://doi.org/10.3390/a11060086

Chicago/Turabian Style

Dalen, Christer, and David Di Ruscio. 2018. "Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models" Algorithms 11, no. 6: 86. https://doi.org/10.3390/a11060086

APA Style

Dalen, C., & Di Ruscio, D. (2018). Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models. Algorithms, 11(6), 86. https://doi.org/10.3390/a11060086

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models

Abstract

1. Introduction

2. Preliminary Theory

2.1. Definitions

2.2. Lag-Dominant Systems

2.3. SIMC Tuning Rules

3. Tuning for Maximum Time Delay Error

3.1. Integrator Plus Time Delay Process

3.2. Pure Integrating Process

3.3. Using a ( $2, 1$ ) Pade Approximation

4. Optimal Performance Settings

5. Simulation Examples

6. Discussion

Remarks to Section 3

7. Concluding Remarks

Author Contributions

Conflicts of Interest

Abbreviations

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

k	$τ$	${ITAE}_{r}$	$ITSE$	$ISE$	$TV$	${IAE}_{r}$
1.0	0.1	2.7	3.4	4.0	4.0	4.0
1.0	0.3	2.7	3.2	4.0	4.0	4.0
1.0	0.5	2.7	3.1	3.5	4.0	4.0
1.0	1.0	2.6	3.1	3.2	4.0	4.0
1.0	2.0	2.6	3.0	3.1	4.0	4.0
1.0	4.0	2.6	3.0	3.1	4.0	4.0
0.1	1.0	2.6	3.9	4.0	4.0	4.0
0.1	2.0	2.6	3.2	4.0	4.0	4.0
0.1	4.0	2.6	3.1	3.6	4.0	4.0

k	$τ$	${ITAE}_{r}$	$ITSE$	$ISE$	$TV$	${IAE}_{r}$
1.0	0.1	2.7	3.4	4.0	4.0	4.0
1.0	0.3	2.7	3.2	4.0	4.0	4.0
1.0	0.5	2.7	3.1	3.5	4.0	4.0
1.0	1.0	2.6	3.1	3.2	4.0	4.0
1.0	2.0	2.6	3.0	3.1	4.0	4.0
1.0	4.0	2.6	3.0	3.1	4.0	4.0
0.1	1.0	2.6	3.9	4.0	4.0	4.0
0.1	2.0	2.6	3.2	4.0	4.0	4.0
0.1	4.0	2.6	3.1	3.6	4.0	4.0

Article Menu

Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models

Abstract

1. Introduction

2. Preliminary Theory

2.1. Definitions

2.2. Lag-Dominant Systems

2.3. SIMC Tuning Rules

3. Tuning for Maximum Time Delay Error

3.1. Integrator Plus Time Delay Process

3.2. Pure Integrating Process

3.3. Using a ( 2 , 1 ) Pade Approximation

4. Optimal Performance Settings

5. Simulation Examples

6. Discussion

Remarks to Section 3

7. Concluding Remarks

Author Contributions

Conflicts of Interest

Abbreviations

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

3.3. Using a ( $2, 1$ ) Pade Approximation

k	$τ$	${ITAE}_{r}$	$ITSE$	$ISE$	$TV$	${IAE}_{r}$
1.0	0.1	2.7	3.4	4.0	4.0	4.0
1.0	0.3	2.7	3.2	4.0	4.0	4.0
1.0	0.5	2.7	3.1	3.5	4.0	4.0
1.0	1.0	2.6	3.1	3.2	4.0	4.0
1.0	2.0	2.6	3.0	3.1	4.0	4.0
1.0	4.0	2.6	3.0	3.1	4.0	4.0
0.1	1.0	2.6	3.9	4.0	4.0	4.0
0.1	2.0	2.6	3.2	4.0	4.0	4.0
0.1	4.0	2.6	3.1	3.6	4.0	4.0