a noncausal approach for pid control q...a new approach for the improvement of the set-point...

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Journal of Process Control 16 (2006) 831–843

A noncausal approach for PID control q

Aurelio Piazzi a,1, Antonio Visioli b,*

a Dipartimento di Ingegneria dell’Informazione, University of Parma, Parco Area delle Scienze 181/A, I-43100 Parma, Italyb Dipartimento di Elettronica per l’Automazione, University of Brescia, Via Branze 38, I-25123 Brescia, Italy

Received 19 July 2005; received in revised form 23 February 2006; accepted 4 March 2006

Abstract

A new approach for the improvement of the set-point following performances achieved by a PID controller is presented. Basically, itconsists of applying a suitable command signal to the closed-loop control system in order to achieve a desired transient response whenthe process output is required to assume a new value. This command signal is determined by means of a stable input–output procedurefor which a closed-form solution is presented, thus making the technique suitable to implement in the industrial context. Simulation andexperimental results show that high performances are obtained despite the presence of model uncertainties and, above all, almost inde-pendently on the PID tuning. Thus, the PID gains can be selected in order to guarantee good load rejection performances withoutimpairing the set-point transient responses.� 2006 Elsevier Ltd. All rights reserved.

Keywords: PID control; Stable input–output inversion; Feedforward control

1. Introduction

Proportional-Integral-Derivative (PID) controllers areundoubtedly the most adopted controllers in industry,due to the good cost/benefit ratio they are able to provide.To help the operator to select the controller gains toaddress given control specifications, many tuning formulashave been devised in the past [1] and autotuning function-alities are almost always available in commercial products[2,3]. However, it is well known that good performancesboth in the set-point following and in the load disturbancesrejection task are often difficult to achieve at the same time.In order to solve this problem, which is of concern in manyapplications, the typical approach is to adopt a twodegrees-of-freedom controller, namely, to adopt a feedfor-

0959-1524/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2006.03.001

q This work has been supported partially by MIUR scientific researchfunds.

* Corresponding author. Tel.: +39 030 3715460; fax: +39 030 380014.E-mail addresses: [email protected] (A. Piazzi), antonio.visioli@ing.

unibs.it (A. Visioli).1 Tel.: +39 0521 905733; fax: +39 0521 905723.

ward (linear) compensator [4]. The use of the well-knownset-point weighting strategy [5] falls in this framework.The main disadvantage of this method is that the reductionof the overshoot is paid by a slower set-point response. Toovercome this drawback, the use of a variable set-pointweight [6,7] or of a feedforward action [3,8–10] has beenproposed.

From another point of view, it is well known that whenthe desired output trajectory is known in advance, the per-formance of a control system can be significantly improvedby adopting a feedforward action determined by means ofa stable inversion technique [11]. Note that the concept ofdynamic input–output inversion [12,13] has been alreadyproven to be effective in different areas of the automaticcontrol field, such as motion control [14,15], flight control[16], robust control [17,18].

In this context we propose to use an analytic stableinput–output inversion procedure to determine a suitablecommand signal to be applied to the closed-loop (PIDbased) control system, instead of the typical step signal,in order to achieve a high performance (i.e. low rise timeand low overshoot at the same time) when the process

mailto:[email protected]

mailto:antonio.visioli@ing. unibs.it

mailto:antonio.visioli@ing. unibs.it

erd

yC Py

y

0

1

Fig. 1. The classic one degree-of-freedom PID based control scheme.

ed

yC PrFy

y

0

1

Fig. 2. The typical two degrees-of-freedom PID based control scheme.

832 A. Piazzi, A. Visioli / Journal of Process Control 16 (2006) 831–843

output is required to assume a new value. To betterunderstand the framework of the proposed method andthe differences with the usual approaches, assume that theprocess variable is required to assume a steady-state valuey1 starting from a steady-state value y0. The standardunity-feedback control scheme based on a one degree-of-freedom PID controller is shown in Fig. 1, where P is theprocess, y is the process output and C is the PID controller.The typical two degrees-of-freedom PID based controlscheme is reported in Fig. 2, where F is a causal filter; notethat the use of a set-point weight is equivalent to filteringthe step signal to be applied to the closed-loop system witha second-order system [19]. The control scheme related tothe new technique is shown in Fig. 3. Conversely to theother methods, here a step signal is not employed, butthe knowledge in advance of y1 is adopted by a commandsignal generator block to calculate, with a suitable previewtime, a command signal r to be applied to the closed-loopPID control system in order to guarantee a high perfor-mance transient response.

Basically, the method consists of choosing a desiredpolynomial function to achieve a process output transitionfrom y0 to y1 and then determining the command functionr that causes the desired transition by inverting the closed-loop dynamics by means of a stable input–output inversionprocedure.

The paper is organised as follows. The overall method-ology is explained in Section 2 and discussed in Section3. Illustrative results are shown in Section 4, while therobustness of the approach together with the role playedby the unique design parameter are analysed in Section 5.Experimental results are presented in Section 6 and conclu-sions are drawn in the last section.

ed

yC PrCommand

signalgenerator

y

y1

0

Fig. 3. The new dynamic inversion based control scheme.

2. Methodology

2.1. Modelling

As a first step of the devised method, the process to becontrolled (assumed to be self-regulating) is modelled asa first-order plus dead-time (FOPDT) transfer function, i.e.

Pðs; K;T ;LÞ ¼ KTsþ 1

e�Ls: ð1Þ

This is a typical choice in industrial practice and a varietyof methods, based on simple experiments, for the identifica-tion of the parameters K, T and L are available (see Section5). Then, in order to have a rational transfer function, thedead-time term is approximated by means of a second-order Pade approximation. In this way, the approximatedprocess transfer function results to be

~Pðs; K;T ;LÞ ¼ KTsþ 1

1� Ls=6þ L2s2=12

1þ Ls=6þ L2s2=12: ð2Þ

Note that if the process is nonself-regulating, it can bemodelled as an integrator plus dead-time (IPDT) transferfunction, i.e.

~Pðs; K;LÞ ¼ Ks

1� Ls=6þ L2s2=12

1þ Ls=6þ L2s2=12: ð3Þ

Then, the methodology is basically the same for theFOPDT case and details for this case will be omitted inthe following (an example will be presented in Section 4.3).

2.2. Tuning the PID controller

In order to apply the input–output inversion basedmethodology that will be presented in the following, thePID controller can be tuned according to any of the manymethods proposed in the literature or even by a trial-and-error procedure. However, since the purpose of the pro-posed procedure is the attainment of high performancesin the set-point following task, disregarding of the control-ler gains, it is sensible to select the PID parameters aimingonly at obtaining good load rejection performances.

The PID controller transfer function be denoted asfollows:

Cðs; Kp; T i; T d; T fÞ ¼ Kp 1þ 1

T isþ T ds

� �1

T fsþ 1ð4Þ

where Kp is the proportional gain, Ti is the integral timeconstant, Td is the derivative time constant and Tf is thetime constant of a first-order filter that makes the transferfunction proper.

2.3. Output function design

As a desired output function that defines the transitionfrom a set-point value y0 to another y1 (to be performedin the time interval [0,s]) we choose a ‘‘transition’’ polyno-

A. Piazzi, A. Visioli / Journal of Process Control 16 (2006) 831–843 833

mial [21], i.e. a polynomial function that satisfies boundaryconditions and that is parameterised by the transition times. In the following, without loss of generality we willassume y0 = 0, Formally, define

ydðtÞ ¼ c2kþ1t2kþ1 þ c2pkt2k þ � � � þ c1t þ c0

The polynomial coefficients can be uniquely found by solv-ing the following linear system, in which boundary condi-tions at the endpoints of interval [0,s] are imposed:

ydð0Þ ¼ 0; ydðsÞ ¼ y1

yð1Þd ð0Þ ¼ 0; yð1Þd ðsÞ ¼ 0

..

.

yðkÞd ð0Þ ¼ 0; yðkÞd ðsÞ ¼ 0

8>>>>><>>>>>:The results can be expressed in closed-form as follows(t 2 [0,s]):

ydðt; sÞ ¼ y1

ð2k þ 1Þ!k!

�X2kþ1

i¼kþ1

ð�1Þi�k�1

ði� k � 1Þ!ð2k þ 1� iÞ!its

� �i

ð5Þ

Expression (5) represents a monotonic function with nei-ther undershooting nor overshooting and its use is there-fore very appealing in the context of process control.

The order of the polynomial can be selected by imposingthe order of continuity of the command input that resultsfrom the dynamic inversion procedure [21]. Specifically, ifthe plant is modelled as a FOPTD transfer function (see(1)), its relative degree is equal to one. Taking into accountthat the relative degree of the PID controller is zero, therelative degree of the overall closed-loop system is one.Thus, a third order polynomial (k = 1) suffices if a contin-uous command input function is required, i.e.

ydðt; sÞ ¼ y1 �2

s3t3 þ 3

s2t2

� �. ð6Þ

Outside the interval [0,s] the function y(t;s) is equal to 0for t < 0 and equal to y1 for t > s.

2.4. Stable input–output inversion algorithm

At this point we address the problem of finding the com-mand signal r(t; K,T,L,Kp,Ti,Td,Tf,s) that provides thedesired output function (6). For the sake of clarity of nota-tion, in the following we will omit the dependence of thefunctions and of the resulting coefficients from the param-eters K, T, L, Kp, Ti, Td, Tf. The closed-loop transfer func-tion be denoted as

HðsÞ :¼ CðsÞ~P ðsÞ1þ CðsÞ~P ðsÞ

¼ K1

bðsÞaðsÞ ð7Þ

where b(s) and a(s) are monic polynomials. As H(s) is non-minimum phase, a stable dynamic inversion procedure isnecessary, that is a bounded input function has to be found

in order to produce the desired output [20]. Denote by Bthe behaviour set [22] of the closed-loop system:

B :¼fðrð�Þ; yð�ÞÞ 2 PCðRÞ � PCðRÞ: yð�Þ is the response output to input uð�Þfor system HðsÞg

where PC(R) denotes the set of real piecewise-continuousfunctions defined over R ¼ ð�1;þ1Þ. Now, in order toperform the stable inversion, we rewrite the numerator ofthe transfer function (7) as follows:

bðsÞ ¼ b�ðsÞbþðsÞwhere b�(s) and b+(s) denote the polynomials associated tothe zeros with negative real part (i.e. those of the PID con-troller) and positive real part (i.e. those of the Pade approx-imation) respectively. From (2) we have

bþðsÞ ¼ ðs� zþR Þ2 þ zþ

2

I ð8Þwhere ZþR ¼ 3=L; ZþI ¼

ffiffiffi3p

=L correspond to the complexzeros zþR � jzþI 2 Cþ. From (4) we distinguish three cases(depending on the selected PID parameters):

b�ðsÞ ¼ ðs� z�1 Þðs� z�2 Þ ð9Þb�ðsÞ ¼ ðs� z�Þ2 ð10Þb�ðsÞ ¼ ðs� z�R Þ

2 þ z�2

I ð11Þcorresponding to real distinct zeros (9), real coincidentzeros (10), and complex zeros (11), respectively. Now, con-sider the inverse system of (7) whose transfer function canbe written as

HðsÞ�1 ¼ c0 þ c1sþ H 0ðsÞ ð12Þwhere c0 and c1 are suitable constants and H0(s), a strictlyproper rational function, represents the zero dynamics.This can be uniquely decomposed according to

H 0ðsÞ ¼ H�0 ðsÞ þ Hþ0 ðsÞ ¼cðsÞ

b�ðsÞþ dðsÞ

bþðsÞð13Þ

where c(s) = c1s + c0 and d(s) = d1s + d0 are first-orderpolynomials with coefficients depending on K, T, L, Kp,Ti, Td and Tf. The modes associated to b�(s) and b+(s)be denoted by m�i ðtÞ; i ¼ 1; 2, and by mþi ðtÞ; i ¼ 1; 2,respectively. More specifically, the unstable zero modesare given by

mþ1 ðtÞ ¼ ezþR t cos zþI t mþ2 ðtÞ ¼ ezþR t sin zþI t ð14Þwhile the stable zero ones are given according to the cases(9)–(11) by

m�1 ðtÞ ¼ ez�1

t; m�2 ðtÞ ¼ ez�2

t ð15Þm�1 ðtÞ ¼ ez�t; m�2 ðtÞ ¼ tez�t ð16Þm�1 ðtÞ ¼ ez�R t cos z�I t; m�2 ðtÞ ¼ ez�R t sin z�I t ð17Þ

Being L the Laplace transform operator, define:

g�0 ðtÞ :¼L�1½H�0 ðsÞ�; gþ0 ðtÞ :¼L�1½Hþ0 ðsÞ�.The following propositions and the following theorem rep-resent the solution to the stable dynamic inversionproblem.


Proposition 1Z t

0

gþ0 ðt � vÞydðv; sÞdv ¼ Hþ0 ð0Þydðt; sÞ

þ 1

s3ðpþ1 ðsÞmþ1 ðtÞ þ pþ2 ðsÞmþ2 ðtÞÞ

þ 1

s3Tþ0 ðt; sÞ ð18Þ

where

Tþ0 ðt; sÞ ¼sþ0 ðtÞ þ sþ1 ðtÞs if t 2 ½0; s�qþ1 ðsÞmþ1 ðt � sÞ þ qþ2 ðsÞmþ2 ðt � sÞ if t > s

�

ð19Þand pþi ðsÞ; qþi ðsÞ; i ¼ 1; 2 are suitable s-polynomials and

sþi ðtÞ; i ¼ 0; 1 are suitable t-polynomials.

Proposition 2Z t

0

g�0 ðt � vÞydðv; sÞdv ¼ H�0 ð0Þydðt; sÞ

þ 1

s3ðp�1 ðsÞm�1 ðtÞ � p�2 ðsÞm�2 ðtÞÞ

þ 1

s3T�0 ðt; sÞ ð20Þ

where

T�0 ðt; sÞ ¼s�0 ðtÞ þ s�1 ðtÞs if t 2 ½0; s�q�1 ðsÞm�1 ðt � sÞ þ q�2 ðsÞm�2 ðt � sÞ if t > s

�

ð21Þand p�i ðsÞ; q�i ðsÞ; i ¼ 1; 2 are suitable s-polynomials and

s�i ðtÞ; i ¼ 0; 1 are suitable t-polynomials.

Theorem 1. The function r(t;s) defined as

rðt; sÞ ¼ � 1

s3ðpþ1 ðsÞmþ1 ðtÞ þ pþ2 ðsÞmþ2 ðtÞ

� qþ1 ðsÞmþ1 ðt� sÞ � qþ2 ðsÞmþ2 ðt� sÞÞ if t < 0 ð22Þrðt; sÞ ¼ c1 _ydðt; sÞ þ c0ydðt; sÞ þH 0ð0Þydðt; sÞ

þ 1

s3ðsþ0 ðtÞ þ s�0 ðtÞ þ sþ1 ðtÞsþ s�1 ðtÞs

� qþ1 ðsÞmþ1 ðt� sÞ � qþ2 ðsÞmþ2 ðt� sÞ þ p�1 ðsÞm�1 ðtÞþ p�2 ðsÞm�2 ðtÞÞ if t 2 ½0; s� ð23Þ

rðt; sÞ ¼ c0 þH 0ð0Þ þ1

s3ðp�1 ðsÞm�1 ðtÞ þ p�2 ðsÞm�2 ðtÞ

þ q�1 ðsÞm�1 ðt� sÞ þ q�2 ðsÞm�2 ðt� sÞÞ if t > s ð24Þ

is bounded over (�1,+1) and ðrðt; sÞ; ydðt; sÞÞ 2 B.

Proofs of the above propositions and of the above the-orem can be derived straightforwardly from proofsreported in [20]. Explicit closed-form expressions of thepolynomials involved in (22)–(24) are reported in theAppendix.

Summarizing, the determined function r(t; K,T,L,Kp,Ti,Td,Tf,s) exactly solves the stable inversion problemfor FOPDT plants (in which the dead-time term has been

approximated) controlled by a PID controller (4) and fora family of output functions, which depend on the freetransition time s.

Actually, from a practical point of view, since the syn-thesised function (22)–(24) is defined over the interval(�1,+1), it is necessary to adopt a truncated functionra(t;s), resulting therefore in an approximate generationof the desired output yd(t;s). In particular, a preactuationtime ts and a postactuation time tf can be selected so thatra(t;s) = 0 for t < ts and ra(t;s) = y1 for t > tp. By takinginto account that the preactuation and postactuationinputs (i.e. the input defined for t < 0 and t > s, respec-tively) converge exponentially to zero at time t!�1and to y1 at time t! +1, an arbitrarily precise approxi-mation can be accomplished [20]. Practically, the methodsuggested in [23] can be adopted. It consists of selecting

ts ¼ �10

Drhp

; tp ¼10

Dlhp

ð25Þ

where Drhp and Dlhp are the minimum distance of the rightand left half plane poles, respectively from the imaginaryaxis of the complex plane.

Hence, the approximate command signal to be actuallyused is

raðt; sÞ :¼0 for t < ts

rðt; sÞ for ts 6 t 6 tp

y1 for t > tp:

8><>:

Remark 1. It is worth highlighting that the preactuationtime depends only on the (apparent) dead time of theprocess, as this determines the unstable zeros of the closed-loop systems by means of the Pade approximation.Conversely, the postactuation time depends on the tuningof the PID parameters because the stable zeros of theclosed-loop systems are those of the controller.

Remark 2. The preactuation time corresponds to a pre-view time jtsj that is the anticipation time necessary to com-mand the desired set-point transition on the controlledoutput.

3. Discussion

The overall stable input–output inversion procedurepresented in Section 2.4 can be performed by means of asymbolic computation, i.e. a closed-form expression ofr(t; K,T,L,Kp,Ti,Td,Tf,s) results. Indeed, the actual com-mand signal to be applied for a given plant and a givencontroller is determined by substituting the actual valueof the parameters into the resulting closed-form expression.

In this context, the choice of using a second-order Padeapproximation is motivated, from one side, by keeping theexpression of r(t; K,T,L,Kp,Ti,Td,Tf,s) as simple as possi-ble and, from the other side, by providing an approxima-

0 10 20 30 40 50 60 70 80 90

0

0.2

0.4

0.6

0.8

1

1.2

Com

man

d in

put r

ZN—PIDZN—PIISTE

Time

Fig. 4. The determined command signals for the three considered PIDtuning for the FOPDT system.


tion as good as possible, since the basic rationale of thismethod is to apply model-based feedforward controlaction.

In any case, it is worth noting that the presented inver-sion procedure is based on a general one [20], where H(s)can be the rational transfer function of any (stable) system,provided that there are not purely imaginary zeros. Thus,as already mentioned, the proposed approach can bestraightforwardly applied also to integral (and unstable)processes ~P ðsÞ, as it is based on the inversion of the dynam-ics of the closed-loop system H(s) (see Sections 4.3 and 4.4).Analogously, the same method can be trivially extended toPI, P and PD control. Indeed, it appears that the devisedmethod can be extended also to high-order processes.Thus, a more accurate model of the process, if available,can be fully exploited. However, in this case, the inversionprocedure has to be performed on purpose and this mightprevent the use of the method in single-station controllers.Conversely, if a FOPDT (or a IPDT) model is employed,the determined general closed-form expression ofr(t; K,T,L,Kp,Ti,Td,Tf,s) can be used.

Once the PID controller has been tuned, the only freedesign parameter is the transition time s. This value canbe selected by solving an optimisation problem where thetransition time has to be minimised subject to actuator con-straints [21]. Alternatively, from a more practical point ofview, the choice can be made by the user either directlyor through a (possibly) more intuitive reasoning. Forexample, the user might select a ratio between the band-width of the open-loop system and that of the closed-loopone, from which the value of s can be determined easily. Inany case, parameter s represent a very desirable featurefrom a user point of view, as it allows to handle thetrade-off between performance, robustness and controlactivity [24,25]. This aspect will be analysed further in Sec-tion 5.

Finally, it is worth stressing that, differently from theset-point weighting approach (i.e. differently from filteringthe set-point with a causal filter), the devised approachallows to recover the set-point following performanceseven in the case of a sluggish tuning of the PID controller(see Section 4).

4. Illustrative results

In the following examples the process output has to per-form a transition from 0 to y1 = 1.

4.1. FOPDT process

Consider the following FOPDT process:

P 1ðsÞ ¼1

10sþ 1e�6s. ð26Þ

To prove the effectiveness of the method with different PIDtunings, three sets of PID parameters have been consid-ered, namely, the one given by the Ziegler–Nichols step re-

sponse PID formula (Kp = 2, Ti = 12, Td = 3), the onegiven by the Ziegler–Nichols step response PI formula(Kp = 1.5, Ti = 18, Td = 0), and the one that results fromthe minimisation of the ISTE integral criterion for the loaddisturbance rejection [26] (Kp = 2.41, Ti = 7.33, Td = 2.74).In the first and in the third case it has been set Tf = 0.01(for a PI controller the filter is not necessary) and the tran-sition time has been always fixed to s = 10.

The resulting value of the preaction and postactiontimes in the three cases are ts = �20 and tp = 60, tp =180, tp = 54.8, respectively (note that, for convenience,the time axis has been properly shifted in order to havets = 0). The determined command functions are reportedin Fig. 4 (solid line: Ziegler–Nichols PID; dash-dot line:Ziegler–Nichols PI; dashed line: integral criterion minimi-sation) and the corresponding process outputs are plottedin Fig. 5. For the sake of comparison, the process outputsresulting with the classic method, i.e. by applying a step set-point signal, are also reported.

It appears that the inversion based methodology is ableto provide low rise times and low overshoots at the sametime but, most of all, is able to provide basically the sameresponse despite a very different PI(D) tuning, as it is evi-denced by the very different step responses they provide.Note that this is achieved with a lower control effort withrespect to the classic case since the step signal is substitutedby a smoother signal (plots of the control signals are notreported for the sake of brevity).

4.2. High-order process

As a second example the following high-order process isconsidered:

P 2ðsÞ ¼1

ðsþ 1Þ8. ð27Þ

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Proc

ess

outp

ut y

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

Proc

ess

outp

ut y

ZN—PIDZN—PIISTE

Time

Time

ZN—PIDZN—PIISTE

Fig. 5. The resulting process outputs with the new method (top) and withthe classic method (bottom) for the three considered PID tuning for theFOPDT system.

0 50 100 1500

0.2

0.4

0.6

0.8

1

Time

Proc

ess

outp

ut y

0 50 100 15000.20.40.60.8

11.2

Time

Proc

ess

outp

ut y

ZN—PIDZN—PIISTE

ZN—PIDZN—PIISTE

Fig. 7. The resulting process outputs with the new method (top) and withthe classic method (bottom) for the three considered PID tuning for thehigh-order system.


By applying the area method [3], a FOPDT transfer func-tion has been estimated, resulting in K = 1, T = 3.04 andL = 4.97. With respect to these parameters, the same tun-ing formulae as for the FOPDT example has been adopted,resulting in Kp = 0.73, Ti = 9.93, Td = 2.48 (Tf = 0.01) forthe Ziegler–Nichols PID tuning, Kp = 0.55, Ti = 14.90,Td = 0 for the Ziegler–Nichols PI tuning, and Kp = 1.06,Ti = 4.26, Td = 2.48 (Tf = 0.01) for the minimisation ofthe ISTE integral criterion. The transition time has beenfixed to s = 20 in all the cases.

The resulting values of the preaction and postactiontimes in the three cases are ts = �16.57 and tp = 51.23,tp = 149, tp = 49.6, respectively. The command functionsdetermined by applying the input–output inversion proce-dure are reported in Fig. 6 and the corresponding process

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Com

man

d in

put r

ZN—PIDZN—PIISTE

Time

Fig. 6. The determined command signals for the three considered PIDtuning for the high-order system.

outputs are plotted in Fig. 7, together with the process out-puts resulting with the classic method, i.e. by applying astep set-point signal. The same line types as before havebeen adopted also in this example. The same consider-ations of the previous example can be done also on thiscase. Further, the robustness of the method with respectto model uncertainties appears. This aspect will be furtheranalysed in Section 5.

4.3. IPDT process

As an example for IPDT processes, the following pro-cess has been considered [27]:

P 3ðsÞ ¼0:0506

se�6s. ð28Þ

0 10 20 30 40 50 60 70 80 90 100

0

0.2

0.4

0.6

0.8

1

1.2

Time

Com

man

d in

put r

PDPID

Fig. 8. The determined command signals for the PD and PID tuning forthe IPDT system.

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

Time

Proc

ess

outp

ut y

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

Time

Proc

ess

outp

ut y

PDPID

PDPID

Fig. 9. The resulting process outputs with the new method (top) and withthe classic method (bottom) for the PD and PID tuning for the IPDTsystem.

0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Com

man

d in

put r

PIDPI

Time

Fig. 10. The determined command signals for the PID and PI tuning forthe unstable system.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1Pr

oces

s ou

tput

y

PIDPI

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

Proc

ess

outp

ut y

PIDPI

Time

Time

Fig. 11. The resulting process outputs with the new method (top) and withthe classic method (bottom) for the PID and PI tuning for the unstablesystem.


Two sets of PID parameters have been adopted, namely,the one based on the minimisation of the integrated squareerror for set-point responses [28] (which is actually a PDcontroller with Kp = 3.394 and Td = 2.94) and the one pro-posed in [27] (Kp = 2.0123, Ti = 31.2030, Td = 1.5674). Inboth cases it has been set again Tf = 0.01 and s = 10.

The resulting value of the preaction and postactiontimes are ts = �20.01 s and tp = 29.41 and tp = 295.48 forthe PD and PID case, respectively. The determined com-mand functions are reported in Fig. 8 and the correspond-ing process outputs are plotted in Fig. 9, together with theprocess outputs resulting from the application of a step set-point signal.

It appears that the same considerations that have beendone for self-regulating processes can be done also for inte-grating processes. Indeed, almost the same performancesare obtained despite the different tuning strategies thathave been employed.

4.4. Unstable process

The devised technique can be straightforwardly appliedalso to unstable processes. Consider for example the pro-cess model by the following transfer function:

P 4ðsÞ ¼1

s� 1e�0:2s. ð29Þ

Also in this case, two sets of PID parameters have been se-lected, the one that results from the minimisation of theintegrated square error of the load disturbance response[28] (Kp = 6.85, Ti = 0.36, Td = 0.12, Tf = 0.01) and theone proposed in [29], i.e. Kp = 3.46 and Ti = 1.47, whichis actually a PI controller. The desired transition time hasbeen fixed to s = 1.

Results are shown in Figs. 10 and 11. It can be seen thatthe devised methodology retains its effectiveness also forunstable processes and therefore its generality appears.

5. Practical issues

Since the proposed approach is based on the determina-tion of a suitable model-based feed-forward action, it isnecessary to evaluate the robustness of the approach withrespect to the method adopted for the estimation of aFOPDT model influences the results. In addition, the roleplayed by the unique design parameter s has to be clarified.In this context, a remarkable theoretical result is that fors! +1 the closed-loop system output tends to theinverse-based input, independently of the closed-loop sys-tem transfer function [20]. In any case, to better understand


these practical issues, a few methods for the estimation of aFOPDT transfer function have been selected among themany methods that have been proposed and proven to beeffective in the literature. They have been chosen basedon the fact that they are suitable for industrial (PID) con-trol, as they are based on simple and cost-effective experi-ments that are typically employed in industrial practice,namely, the application of a step signal to the process input(open-loop experiment) or of a relay feedback (closed-loopexperiment). Note that, since the aim of this investigation isto evaluate the application of the different methods in thenoncausal approach rather than evaluating and comparingthe effectiveness of each method in estimate an accurateFOPDT model, measurement noise is not taken intoaccount in the following.

5.1. Methods based on step response

The following methods based on the evaluation of anopen-loop step response, where different approaches areexploited, have been considered:

(a) the well-known area method [3, p. 24];(b) the least squares method proposed in [30], which is

based on the integrated process input and outputsignals;

(c) the least squares method based on the use ofLaguerre functions described in [31, chapter 2];

(d) the least squares method proposed in [32] applied toFOPDT transfer function.

It has to be noted that whereas methods (a) and (d) yielddirectly to a FOPDT transfer function, methods (b) and (c)actually yield to a rational transfer function of arbitrarilychosen order. Thus, in these cases, a fourth-order transferfunction with relative order equal to one has been esti-mated first. Then, it has been reduced to a FOPDTmodel by adopting the model reduction method proposedin [30].

5.2. Methods based on relay feedback

Relay feedback based methods [33] are the mostadopted methods where a closed-loop experiment is usedfor automatic tuning of PID controllers. Actually, the ori-ginal idea is to employ the relay feedback to estimate theultimate gain and the ultimate period of the system. How-ever, recently, various techniques have been devised inorder to estimate a FOPDT transfer function. The follow-ing ones have been considered in this paper:

(e) the standard method with a symmetric relay withouthysteresis. The FOPDT transfer function is thendetermined by straightforward calculations (see forexample [34, chapter 2]);

(f) the technique based on the use of a biased relay (withhysteresis) [33];

(g) the method described in [35] where two points on theNyquist curve (the one where the phase is �90� inaddition to that where the phase is �180�) are esti-mated. The FOPDT transfer function parametersare then estimated by assuming the knowledge ofthe process gain (note that this can be simply esti-mated by considering steady-state values of the inputand the output);

(h) the method in which an asymmetrical relay is adopted[36];

(i) the method based on the so-called curvature factor ofthe process response described in [37].

It turns out that the addressed techniques representrather different approaches.

5.3. Illustrative results

In order to evaluate the influence of the estimationmethod in the proposed noncausal approach, consider, asan example, the following system:

P 5ðsÞ ¼1

ðsþ 1Þ4ð30Þ

A FOPDT model has been estimated with all the consid-ered techniques. Then, the PID controller has been tuned,by means of a genetic algorithm [38] in which the nominalprocess transfer function has been considered, in order tominimise the integrated absolute error defined as

IAE ¼ZjeðtÞjdt ð31Þ

where e(t) is the difference between the set-point value andthe process output, when a step load disturbance occurs onthe control system. It results Kp = 3.50, Ti = 1.77, andTd = 1.47 (Tf = 0.01). Thus, in order to analyse the effectof the identification method just on the inversion approachand not on the PID controller tuning, the best PID param-eters (in the sense of those that provide the best load distur-bance response) have been adopted. Hence, the tuning ofthe PID controller depends only on the considered processand it is independent from the adopted identification meth-od (i.e. we have the same tuning disregarding the adoptedestimation method).

Then, for each estimated process transfer function thecommand signal r(t; K,T,L,Kp,Ti,Td,Tf,s) has been calcu-lated with different values of s and two values have beencalculated with respect to the inversion-based closed-loopresponse, namely, the integrated absolute error with respectto the desired final output value y1 and the integrated abso-lute error with respect to the desired output response (6),i.e.

IAEa ¼Z þ1

t0

jy1 � yðtÞjdt ð32Þ

0 2 4 6 8 10 12 14 16 18 202

3

4

5

6

7

8

9

10

IAE a

Transition time τ

Fig. 12. IAEa for different values of s obtained by considering the ninedifferent identification methods.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

IAE b

Transition time τ

Fig. 13. IAEb for different values of s obtained by considering the ninedifferent identification methods.

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

Proc

ess

outp

ut y

τ

Time

Fig. 14. Process P5(s) outputs with different values of s.

0 5 10 15 20 25

—2

0

2

4

6

8C

ontro

l var

iabl

e

τ

Time

Fig. 15. Resulting control variable signal with different values of s.


and

IAEb ¼Z þ1

t0

jydðtÞ � yðtÞjdt ð33Þ

where t0 is the time instant in which the process output y

leaves the zero value, so that the apparent dead time ofthe process and the preaction time do not bias the result.

The values of IAEa and IAEb for different values of thespecified output transition time s that have been obtainedby employing the nine considered identification methodsare plotted in Figs. 12 and 13, respectively. For clarity,the nine plots in each figure have not been labelled, becausethey are actually very difficult to distinguish in some cases.Indeed, it is worth stressing the general result more thanproviding detailed quantitative results. Note however thatthe value of IAEa obtained by applying a step set-point sig-

nal instead of the inversion based command signal is 2.32.For a better analysis of the technique, Fig. 14 shows thedifferent outputs obtained for increasing values of s (i.e.s = 1,2, . . . , 10) when the process P5(s) has been estimatedwith method (a) (see Section 5.1). The corresponding con-trol variable signals are plotted in Fig. 15.

By taking into account that results obtained with otherprocesses [39] are similar, it can be deduced that, asexpected, the desired transition time s handles the trade-off between robustness and aggressiveness (and controlactivity), independently from the adopted estimationmethod. Indeed, by increasing the value of s the desiredresponse is obtained more accurately (IAEb decreases),although this implies that IAEa increases as the rise timeincreases. In any case good performances are obtainedfor relatively small values of s (thus, excessively high valuesof s are not worth to being employed). Actually, if thePID controller is well tuned from the point of view of

0 20 40 60 80 100 120 140 160 1801.5

1

0.5

0

0.5

1

1.5

2

2.5

3

3.5

Com

man

d si

gnal

Time [s]

Fig. 17. The determined command signal for the level control experiment.


minimising the value of IAEa for a step set-point signal,then the inversion based approach might give a highervalue of IAEa than using a step set-point signal. However,it has to be taken into account that a low value of the inte-grated absolute error is often achieved at the expense ofhigh overshoots, high control activity and poor robustnessand these latter aspects are almost always of major concernin the industrial context. Indeed, they can be handled bythe devised noncausal approach. Finally, it cannot be saidthat one of the estimation methods is better than the othersfrom the point of view of the noncausal approach. Actu-ally, they provide basically the same performances and incase a method is worse than another for a given value ofs, it can be better for another value of the transition time.

6. Experimental results

In order to prove the effectiveness of the devised tech-nique in practical applications, a laboratory experimentalsetup (made by KentRidge Instruments) has beenemployed (see Fig. 16). Specifically, the apparatus consistsof small perspex tower-type tank (whose area is 40 cm2) inwhich a level control is implemented by means of a PC-based controller. The tank is filled with water by meansof a pump whose speed is set by a DC voltage (the manip-ulated variable), in the range 0–5 V, through a PWM cir-cuit. The tank is fitted with an outlet at the base in orderfor the water to return to a reservoir. The measure of thelevel of the water is given by a capacitive-type probe thatprovides an output signal between 0 (empty tank) and5 V (full tank). Since the apparent dead-time of the system

Fig. 16. The experimental setup for the level control experiment (only onetank has been adopted).

is very small with respect to its dominant time constant, inorder to provide a significant result, a time delay of 10 s hasbeen added via software at the plant input.

Despite the model is nonlinear (as the flow rate out ofthe tank depends on the square root of the level), a FOPDTmodel has been estimated by applying the area method tothe open-loop response with a step from 2 V to 2.5 V atthe process input. The FOPDT model obtained is

P 6ðsÞ ¼1:98

29sþ 1e�11s

Based on this model, it has been set Kp = 1.24, Ti = 31 andTd = 0. The derivative action has not been employed due tothe excessive measurement noise. In order to obtain a pro-cess output transition from 2 V to 3 V (starting when theprocess is at the steady-state), which corresponds approxi-mately to a level transition from 6 cm to 12 cm, it has been

0 20 40 60 80 100 120 140 160 1801.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Proc

ess

outp

ut [V

]

step

dynamic inversion

Time [s]

Fig. 18. The resulting process outputs with the new and the classic methodfor the level control experiment.


fixed s = 10 s and the inversion-based command signal hasbeen determined. It is shown in Fig. 17. The correspondingprocess output, together with the step response is plotted inFig. 18. A significant reduction of the overshoot and of thesettling time appears. It turns out that these experimentalresults confirms the effectiveness of the methodology.

7. Conclusions

A noncausal approach for the improvement of the set-point following performances of PID controllers has beenproposed. It is based on an input–output inversion proce-dure that allows to determine a closed-form expression ofthe command signal to be applied to the closed-loop sys-tem. The main merit of the methodology is that it providespredefined (high) performances almost independently fromthe PID parameters and from the adopted identificationmethod. The unique design parameter (i.e. the desired tran-

pþ1 ðsÞ ¼�48d0zþR zþ

3

I � 12d1zþ5

I þ 36d1zþI zþ4

R þ 24d1zþ3

I zþ2

R þ 48d0

ðzþ2

R þ zþ2

I Þ4zþI

þ 6d0zþ5

I � 24d1zþ3

I zþ3

R � 12d1zþ5

I z�R 12d0zþ3

I zþ2

R � 18d0zþI z

ðzþ2

R þ zþ2

I Þ4zþI

2þðsÞ ¼ �12d1zþ5

R � 12d0zþ4

R þ 72d0zþ2

I zþ2

R � 12d0zþ4

I þ 24d1zþ2

I zþRðzþ2

R þ zþ2

I Þ4zþI

þ�12d0zþ3

R zþ2

I þ 6d0zþ5

R þ 6d1zþ6

R þ 6d1zþ2

I zþ4

R � 6d1zþ4

I z

ðzþ2

R þ zþ2

I Þ4zþI

sþ0 ðtÞ ¼�6ð6d1zþ

4

R � 2d1zþ4

I þ 8d0zþ3

R þ 4d1zþ2

I zþ2

R � 8d0zþ2

I zþR Þzþ

8

R þ 4zþ6

R zþ2

I þ 6zþ4

R zþ4

I þ 4zþ2

R zþ6

I zþ8

I

þ�6ð4d0zþ2

I zþ2

R þ 4d1zþ5

R þ 6d0zþ4

R � 2d0zþ4

I þ 8d1zþ3

R zþ2

I

zþ8

R þ 4zþ6

R zþ2

I þ 6zþ4

R zþ4

I þ 4zþ2

R zþ6

I þ zþ8

I

þ 4d0zþ3

R zþ2

I þ 2d0zþR zþ4

I þ d1zþ6

R þ 3d1zþ4

R zþ2

I þ 3d1zþ2

R zþ4

I

zþ8

R þ 4zþ6

R zþ2

I þ 6zþ4

R zþ4

I þ 4zþ2

R zþ6

I þ zþI

sþ1 ðtÞ ¼�6ð�2d1zþ

5

R � 2d0zþ2

I zþ2

R � 4d1zþ2

I zþ3

R � 2d1zþ4

I zþR � 3d0zþRzþ

8

R þ 4zþ6

R zþ2

I þ 6zþ4

R zþ4

I þ 4zþ2

R zþ6

I þ zþ8

I

þ�6ð�2d0zþ5

R � 4d0zþ3

R zþ2

I � 2d0zþR zþ4

I � d1zþ6

R � 3d1zþ4

R z

zþ8

R þ 4zþ6

R zþ2

I þ 6zþ4

R zþ4

I þ 4zþ2

R zþ6

I þ

qþ1 ðsÞ ¼12d1zþ

5

I � 36d1zþI zþ4

R þ 48d0zþR zþ3

I � 48d0zþ3

R zþI � 24d1zþIzþI zþ

8

R þ 4zþ3

I zþ6

R þ 6zþ5

I zþ4

R þ 4zþ7

I zþ2

R þ zþ9

I

þ�12d1zþ5

I zþR � 24d1zþ3

I zþ3

R þ 6d0zþ5

I � 12d0zþ3

I zþ2

R � 12

zþI zþ8

R þ 4zþ3

I zþ6

R þ 6zþ5

I zþ4

R þ 4zþ7

I zþ2

R þ

qþ2 ðsÞ ¼�72d0zþ

2

I zþ2

R � 36d1zþ4

I zþR � 24d1zþ2

I zþ3

R þ 12d0zþ4

R þ 12d

zþI zþ8

R þ 4zþ3

I zþ6

R þ 6zþ5

I zþ4

R þ 4zþ7

I zþ2

R þ zþ9

I

þ 6d1zþ2

I zþ4

R � 12d0zRþ3zþ2

I þ 6d0zþ5

R þ 6d1zþ6

R � 6d1zþ6

I �zþI zþ

8

R þ 4zþ3

I zþ6

R þ 6zþ5

I zþ4

R þ 4zþ7

I zþ2

R

sition time s) has a clear physical meaning and it allows theuser to handle the trade-off between aggressiveness, robust-ness and control activity. Thus, the technique integratesthose features that are desirable in the industrial contextand it appears to be suitable to implement in DistributedControl Systems as well as in single-station controllers.

Appendix A. Closed-form expressions

The closed-form expressions of the polynomials appear-ing in Eqs. (18)–(21) are given hereafter. First, consider thepart of the inverse transfer function (13) related to theunstable zeros given by the Pade approximation, whichcan be written as

Hþ0 ðsÞ ¼d1sþ d0

ðs� zþR Þ2 þ zþ

2

I

. ð34Þ

Then, we obtain (see Proposition 1):

zþ3

R zþI

þ4

R � 12d1zþI zþ5

R s ð35Þ

3 þ 36d1zþ4

I zþR

þ2

R � 18d0 � zþR zþ4

I � 6d1zþ6

I s ð36Þ

þ 4d1zþR zþ4

I Þ t

þ d1zþ6

I þ 2d0zþ5

R8 t2 ð37Þ

4 þ d0zþ4I Þ

þ2

I � d1zþ6

I � 3d1zþ4

I zþ2

R Þzþ

8

I

t ð38Þ

3zþ

2

R

d1zþI zþ5

R � 18d0zþI zþ4

R

zþ9

I

s ð39Þ

0zþ4

I þ 12d1zþ5

R

6d1zþ4

I zþ2

R � 18d0zþR zþ4

I

þ zþ9

I

s ð40Þ


With respect to the stable zeros, three cases have to be con-sidered, related to the presence of two different real zeros,two coincident real zeros and two complex conjugate zeros,depending of the selected tuning of the controller. In thefirst case, the part of the inverse transfer function (13) re-lated to the stable zeros can be written as

H�0 ðsÞ ¼c1sþ c0

ðs� z�1 Þðs� z�2 Þ. ð41Þ

Then, we obtain (see Proposition 1):

p�1 ðsÞ ¼�12c0z�

4

2 � 12c1z�1 z�4

2

z�4

1 ðz�1 � z�2 Þz�4

2

þ 6c1z�2

1 z�4

2 þ 6c0z�1 z�4

2

z�4

1 ðz�1 � z�2 Þz�4

2

s ð42Þ

p�2 ðsÞ ¼12c1z�

4

1 z�2 þ 12c0z�4

1

z�4

1 ðz�1 � z�2 Þz�4

2

þ�6c0z�4

1 z�2 � 6c1z�4

1 z�2

2

z�4

1 ðz�1 � z�2 Þz�4

2

s ð43Þ

sþ0 ðtÞ ¼�6ð2z�

3

1 c0 þ 2z�1 z�2

2 c0 þ 2z�3

1 z�2 c1 þ 2z�1 z�3

2 c1 þ 2z�2

1 z�2 c0 þ 2z�2

1 z�2

2 c1 þ 2z�3

2 c0Þz�4

1 z�4

2

þ�6ð2z�3

1 c1z�2

2 þ 2z�2

1 c1z�3

2 þ 2z�3

1 c0z�2 þ 2z�1 c0z�3

2 þ 2z�2

1 c0z�2

2 Þz�4

1 z�4

2

t þ�6ðz�3

1 c1z�3

2 þ z�3

1 c0z�2

2 þ z�2

1 c0z�3

2 Þz�4

1 z�4

2

t2 ð44Þ

sþ1 ðtÞ¼�6ð�z�

3

1 z�2 c0� z�2

1 z�2

2 c0� z�1 z�3

2 c0� z�2

1 z�3

2 c1� z�3

1 z�2

2 c1Þz�4

1 z�4

2

þ�6ð�z�3

1 c1z�3

2 � z�2

1 c0z�3

2 � z�3

1 c0z�2

2 Þz�4

1 z�4

2

t ð45Þ

qþ1 ðsÞ ¼12c0z�

4

2 þ 12c1z�1 z�4

2

z�4

1 ðz�1 � z�2 Þz�4

2

þ 6c1z�2

1 z�4

2 þ 6c0z�1 z42

z�4

1 ðz�1 � z�2 Þz�4

2

s ð46Þ

qþ2 ðsÞ ¼�12c0z�

4

1 � 12c1z�4

1 z�2z�4

1 ðz�1 � z�2 Þz�4

2

þ�6c1z�4

1 z�2

2 � 6c0z�4

1 z�2z�4

1 ðz�1 � z�2 Þz�4

2

s ð47Þ

In case the tuning of the controller is such that two coinci-dent (stable) real zeros result, we can write


ðs� z�Þ2. ð48Þ

Then, it results:

p�1 ðsÞ ¼48c0 þ 36z�c1

z�5 þ�18c0z� � 12c1z�2

z�5 s ð49Þ

p�2 ðsÞ ¼�12c0z� � 12c1z�

2

�5 þ 6c1z�3 þ 6c0z�

2

�5 s ð50Þ
z z
sþ0 ðtÞ ¼�6ð8c0 þ 6z�c1Þ

z�5 þ�6ð6c0z� þ 4c1z�2Þ

z�5 t

þ�6ð2c0z�2 þ c1z�

3Þz�5 t2 ð51Þ

sþ1 ðtÞ ¼�6ð�3c0z� � 2c1z�

2Þz�5

þ�6ð�2c0z�2 � c1z�

3Þz�5 t ð52Þ

qþ1 ðsÞ ¼6ð�8c0 � 6z�c1Þ

z�5 þ 6ð�5c0z� � 4c1z�2Þ

z�5 s

þ 6ð�c0z�2 � z�

3c1Þ

z�5 s2 ð53Þ

qþ2 ðsÞ ¼6ð2c0z� þ 2c1z�

2Þz�5 þ 6ðc0z�

2 þ c1z�3Þ

z�5 s ð54Þ

Finally, if the PID controller is tuned such as two complexconjugate zeros are posed, it can be written


ðs� z�R Þ2 þ z�2

I

. ð55Þ

Then, the resulting polynomials are the same of expressions(35)–(40) where d1, d0 and zþR have to be substituted with c1,c0 and z�R , respectively.

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