PI0 CONTROL

PID controllers: recent tuning methods and design to specification

P.Cominos and N.Munro

Abstract: PID control is a control strategy that has been successfully used over many years. Simplicity, robustness, a wide range of applicability and near-optimal performance are some of the reasons that have made PID control so popular in the academic and industry sectors. Recently, it has been noticed that PID controllers are often poorly tuned and some efforts have been made to systematically resolve this matter. In the paper a brief summary of PID theory is given, then some of the most used PID tuning methods are discussed and some of the more recent promising techniques explored.

1 Introduction

A PID controller, Fig. 1, is described by the following transfer function in the continuous s-domain

(1) K.

G ~ ( s ) = P + I + D = Kp +'+ K ~ s S

or

where Kp is the proportional gain, Ki is the integration coefficient and Kd is the derivative coefficient. T, is known as the integral action time or reset time and Td is referred to as the derivative action time or rate time.

In practice, the following realisation is usually employed Ki K ~ s

GJs) = Kp + - + ~

s 1 + T,s ( 3 )

This realisation, although it has the same response at low frequencies as (l) , includes a lowpass filter with the derivative term to reduce noise amplification. Tn represents the filter's time constant.

The PID realisation given by (3) is known as the parallel form and admits complex zeros. One can also have the series PID form described by

GcG) = GPl(S)GPD(S) (4) or

It is also required that p < 1 to obtain the phase lead action. This configuration is less flexible than the parallel form

due to a stronger interaction between the design para- meters. Note, that it is possible to convert from the series

0 IEE, 2002 IEE Proceedings online no. 20020103 DOI: 10.1049/ip-cta:20020103 Paper received 7th January 2002 The authors are with the Control Systems Centre, UMIST Sackville Street, Manchester M60 lQD, UK

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form to the parallel form and vice versa under certain conditions. It is also very important to know the controller implementation before entering the PID parameters.

In cases where both good set-point tracking and good load disturbance are essential, the set-point weighting technique [ l ] can be used. This method effectively results in a two-degrees-of-freedom (2DF) system. Every input to each of the PID terms can be a weighted version of the error signal. Set-point weighting is not used with propor- tional-only control since it results in static control errors. The general 2DF PID configuration is shown in Fig. 2.

The ISA realisation is a particular case of a 2DF configuration. Here, the controller takes the form

G&) = GSP(S>YSP(S) - Gs(S)Y(S) (6)

where GsP(s) is the signal transmission from the set- point to the control variable and Gs(s) is the signal transmission from the process output to the control variable [2].

Fig. 1 I I

Block diagram representation of a PID controller

Fig. 2 2DF PID controller

IEE Proc.-Control Theory Appl.. Vol. 149, No. I, January 2002

The remainder of this paper describes different €'ID parameter tuning methods together with a discussion on some of their advantages and disadvantages.

ing critical period. In [ 11, area methods are also suggested to improve estimation of the system's parameters giving a less sensitive technique.

2 Ziegler-Nichols (ZNI method 3 Kappa-tau tuning

The PID tuning method presented by Ziegler and Nichols [3] is based on the system's open-loop step response. It uses the fact that many systems in the process industry can be approximated by a first-order lag plus a time delay as

where c( and L can be determined by simply plotting the step response of the plant [l]. The PID tuning parameters obtained by the ZN step response method are shown in Table 1.

Ziegler and Nichols later introduced a method based on .the frequency response of the closed-loop system under pure proportional control. Here, the gain is increased until the closed-loop system becomes critically stable. At this point the ultimate gain, K,, is recorded together with the corresponding period of oscillation, Tu, known as the ultimate period. Based on these values Ziegler and Nichols calculated the tuning parameters shown in Table 2.

The ZN methods were designed to give good responses to load disturbances. A quarter amplitude-damping criter- ion was used in the design giving a damping ratio close to 0.2. This is not satisfactory for many systems, since it does not give satisfactory phase and gain margins. The maxi- mum sensitivity is also large, giving systems sensitive to parameter variations. Additionally, ZN methods are not easy to apply in their original form on working plants. When critical processes are involved, sudden changes in the control signal or operation at the stability limit are not acceptable. Relay feedback and describing function analy- sis [l] are often applied for parameter identification to overcome the above problems.

A further modification to the ZN methods can give a substantially improved system performance [ 11. One can start with a given point on the Nyquist diagram say, G,( jo) = rpe'(K+9(3 and then try to find a regulator to move this point to B=r,,e'("'4'). An amplitude margin corresponds to 4% = 0, r = 1 /Am and a gain margin corre- sponds to r, = 1, 4, = I$,,, , where A, , (bm are the gain and phase margins, respectively. Based on this simple modifi- cation, a better system response results. If the plant's model is well known then it is also possible to apply Routh's array technique to find the critical gain and then the correspond-

Table 1: ZN PID step response tuning parameters

Controller K T, T d

P 1 l a

PI 0.9la 3L

PID 1.21E 2 L u2

Table 2: ZN PID frequency response tuning parameters

Controller K Ti Td

P 0.5Ku

PI 0.4K, 0.8Tu

PID 0.6K, 0.5T,, 0.12T,

The dynamics of a system can be described more accu- rately if three parameters are used in the design instead of two. The kappa-tau tuning method [ 1,4] is a method in that direction and is used in automatic tuning. As in the ZN method it comes in two versions. One is based on the step response, in which the process is characterised by a static gain Kp, a gain a (the gain of the transient part of the open- loop response), and a dead time L. The controller para- meters are a function of the normalised dead time z [4, 51 given by

L L + T

z=-

with T being the dominant time constant of the process. The second method is based on the frequency response;

the process is characterised by a static gain K p , an ultimate gain Ku and an ultimate period Tu. Here, the controller parameters are a function of the gain ratio k, [4, 51, where

Maximum sensitivity is used as the design objective in both cases.

4 Pole placement

Analytical pole placement methods are mostly used when the system under consideration is of low order. A common approach is to adopt a second-order model and then specify a desired damping ratio and natural frequency for the system. These specifications can then be fulfilled by locating the two system poles at positions that give the required closed loop performance. As an example, the characteristic equation for a system approximated by a first-order model

(10) KP G ~ ( s ) ~

1 + S T ,

under PI control will take the form:

This can then be compared with the general second-order model

s2 + 2go + o2 = 0 (12)

and thus obtain

In the case where a second-order system of the following form is used:

a PID controller of the form

IEE Proc.-Control Theoiy Appl., Vol. 149, No. 1. January 2002 47

can arbitrarily place all closed-loop poles. The system’s characteristic equation then becomes

This can be compared with the following, general, third- order characteristic equation [4]:

(s + Cto)(s2 + 2q3s + 0 2 ) = 0 (17)

to get the PID parameters, as for the PI case earlier. Dominant pole design is another, simplified, pole place-

ment technique employed when one wants to obtain a PID controller for high-order systems. This method is based on the positioning o f the system’s dominant poles in the complex plane. For example, by taking the transfer func- tion of a unity feedback system

one can then readily find the poles and zeros of the resulting closed-loop system. In many cases, the dominant system dynamics can be approximated by the simple pole- zero configuration shown in Fig. 3. The pair of poles P I , P2 are known as the dominant poles. Poles and zeros which have real parts much more negative than those of the dominant poles have little influence on the overall system response. For a PI controller, as an example, the following selection of P I , P2 can be made [l]:

These give a system with the required damping c, and response speed given by coo.

For an extensive analysis of pole placement techniques with emphasis on uncertain systems, the reader is referred to [6].

5 specifications

Design based on gain and phase margin

Usually one wants to design for specific gain and phase margins [7], where the phase margin is related to the damping of the system. By definition, the gain margin of

t’.

Fig. 3 Dominunt poles

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a system is given by the solution of the following set of equations:

arg[GdJwP)G~(J~,>l = --7c (20)

Correspondingly, the phase margin is given by

where cop is the phase crossover frequency, cog is the gain crossover frequency, and Gp(s), Gc(s) are the process and the controller transfer function, respectively. It is now apparent that, depending on the plant’s model, the solution of the above sets of equations can be extremely difficult to carry out analytically. Numerical methods are usually employed.

In [8], a first-order system having the form

KP e-sL G p ( S ) = - 1 +sT

is controlled with a PI controller having the following transfer function

By trying, in this case, to solve for a specific phase and gain margin one comes across equations containing arctan functions, which cannot be solved analytically. This problem is overcome by using the following simple arctan approximations

6 D-partitioning

In the D-partitioning method [9], the system’s poles are shifted into a specified region in the left halfplane. Least absolute stability is initially achieved by placing all the system’s poles to the left of a line parallel to, and at a distance d from, the imaginary axis. Then, a region with this line as its right boundary is obtained within which the specific gain and phase margins are guaranteed. Therefore, relative stability criteria are also met.

In [lo], a detailed description of how to use the D-partition method for PID tuning is given. In addition, a method of shifting the closed-loop system zeros to specified positions in the s-plane is proposed to improve the system’s tracking behaviour. Regions of absolute and relative stability in terms of the PID tuning parameters are obtained.

7 OLDPmethod

The OLDP method [ 1 11 is an extension of the D-partition- ing method, where simultaneous achievement of minimum phase and gain requirements is possible. In addition, this method allows for specified maximum gain and phase crossover frequencies. This is possible by isolating the

IEE Proc.-Control Theory Appl., Vol. 149, No. I , Januuy 2002

gain and phase margin regions in parameter space. A plant model with real coefficients of the following form is assumed:

with c, # 0. The open-loop transfer function for the PID case is then given as

Kds2 + Kps + Ki G(s) = Gp(s) = a + jb (28)

S

s is then substituted by j w and the characteristic equation becomes

-Kdw2 + jKpo + K, - R(w) - jI(cu) = 0 (29)

where R(o), I ( o ) are the real and imaginary parts of jw(a+jb)/Gp(s). To get the required gain and phase margins in parameter space, three frequency domain boundaries have to be drawn. These boundaries are obtained by considering the following three cases:

1. o = o 2. o= f o o , and 3. O < o < c o a n d O > w > - o o .

For the PI case, the characteristic equation takes the following form:

j K p o + K, - R(w) - jZ(o) = 0 (3 0)

When w = 0 the corresponding boundary is given by cmK,=O, which implies that K,=0. In the case where o= f c o , the boundary is described by Kpco( jw)'n+' - (a + jb)( jco),,+' = 0, and finally when 0 < o < oo and 0 > o > -co the boundaries are given by K, = R(o) and Kpw =I(@).

The PID case can be similarly analysed but here one of the K,, Kd parameters has to be gridded.

Points on the boundaries specified above are related to the gain and phase crossover frequencies and thus the bandwidth of the systems under consideration. In particu- lar, a point on the non-singular phase margin boundary corresponds to a specific frequency, the gain crossover frequency of the controlled system. Similarly, a point on the non-singular gain margin boundary corresponds to the phase crossover frequency.

8 Interval polynomial techniques

An interval polynomial is a polynomial of the following form:

p(s) =po +p1s+p2s2 + . . . + p n s" (3 1) with the coefficients varying in independent intervals given by

P O E [ x o > . ~ o I > . . . > ~ n E [ ~ n ~ ~ n l ~ 0 ~ [ ~ n ~ ~ n l (32) Then, Kharitonov's theorem states that every polynomial in the interval family is Hunvitz stable if and only if the following four polynomials are Hunvitz stable

K1(~)=xo+x , s+y2s2+y3s3 +x4s4+

K2(s) = xo + y l s +y2s2 + x3s3 + x4s4 + (33)

K3(s)=y~+xls+x*s2+y3S3+,v4S4+"'

K4(s) =yo + y l s +x2s2 +x,s3 +JJ4S4 + . . . IEE Proc.-Control Theory Appl, Vol 149, No 1, Junuury 2002

This theorem can be further extended to what is known as the generalised Kharitonov theorem [12, 131, which plays an important role in the area of parametric control design.

1. Given an m-tuple of fixed real or complex polynomials of the form [F , (s), F2(s), . . . , Fm(s)], the polynomial family described by

The generalised Kharitonov theorem states that:

PI @)FI (s) + P2(s)F2(s) + . . . + P,(s)F,(s) (34)

with Pi(s) an interval polynomial, is Hunvitz stable if and only if all of the one-parameter polynomial families that result from replacing [Pl(s) , P2(s), . . . , P,(s)], with m an arbitrary integer, in (34) by elements S, (the family of m4m distinct generalised Kharitonov segments), are Hunvitz stable. 2. If the polynomials Fi(s) are real and of the form Fi(s) =stfais + bj)Uj(s)Qi(s), where ti 2 0 is an arbitrary integer, ai and bi are arbitrary real numbers, Ui(s) is an anti-Hunvitz polynomial, and Qi(s) is an even or odd polynomial, then it is sufficient that (34) be Hunvitz stable with all Pi replaced by the elements of K,. K, is the set of m-tuples obtained as follows: for every fixed integer i between 1 and m set Pi@) = Kf(s) for some k = 1, 2, 3, 4. 3. If the Fi are complex and

d -arg[Fi(jw)] 5 0, Vi = 1 , 2 , . . . , m d o

then it is sufficient that (34) be Hunvitz stable with the P, replaced by the elements of K, .

The generalised Kharitonov theorem can been seen as an improvement on the edge theorem [ 141. The edge theorem considers polynomials with affine linear uncertainty struc- ture; i.e. their coefficients are dependent on each other. The edge theorem states that a family of polynomials is stable if the exposed edges of the polytopic family are stable. In the generalised Kharitonov theorem, the number of edges which have to be studied for stability depends on the number of interval polynomials and not on the number of uncertain parameters, making it a computationally more efficient method.

In [ 151, the above ideas are combined with results in the area of robust parametric control and the generalised Hermite-Biehler theorem [ 161 to obtain all stabilising P, PI and PID controllers for an interval plant family. Bulut [17] has extended some of the above results. In addition to stability, he obtained PID controllers for uncertain systems that also met system performance requirements.

In [ 181, bilinear transformation and linear programming together with the above results are used to determine the set of all PID gains that can stabilise a given discrete-time plant of arbitrary order.

9 Nyquist based design

In [19], Munro introduced some fast ways of system- atically calculating the limiting values of PID parameters based on the axis crossing form of the Nyquist's stability theorem. Namely,

Z = N + P (35)

where Z is the number of closed-loop poles located in the right halfplane, P is the number of the open-loop poles of the system located in the right halfplane and N is the number of poles crossing the imaginary axis at the point where the Nyquist plot passes through the -1 point. For

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test comp. plant K

Fig. 4 Test compensator arrangement

system stability, one wants Z= 0. The system configuration shown in Fig. 4 is used.

Values of K that make each real axis crossing equal to the critical point on the Nyquist diagram are obtained and then used to scale the test compensators. This procedure is done by gridding one of the PID parameters and recording the limiting values of the other two. For systems with no explicit time delay, the procedure can be simplified further by simply solving Im[G(s)K(s)] =0, since only the real axis crossings on the Nyquist diagram are required. The method is then further improved giving a computationally efficient way of obtaining the limiting stabilising values of the PID controller. It can be proved [20] that when the Kp tuning parameter is gridded then the others are confined within convex polygons. Linear programming techniques can then be used to obtain the vertices of these polygons. This method can be applied to stable systems, unstable systems, minimum phase and non-minimum phase systems and systems with time delay. Ideas from [15, 211 are then employed to handle systems with structured parameter uncertainty resulting in robust PID controllers.

10 Genetic algorithms for PID tuning

Genetic algorithms are a rapidly expanding area in control systems design. A genetic tuning algorithm usually starts with no knowledge of the correct solution and depends on the responses from its environment to give an acceptable result. It has been shown that genetic algorithms are capable of locating optimal regions in complex domains avoiding the difficulties, or even erroneous results in some cases, associated with the gradient descent methods and with high-order systems. To obtain the PID tuning para- meters one usually has to minimise a performance index. This, in the majority of the cases, is one of the following:

ISE = I r(t) - y(t) I2dt I" IAE = I ~ ( t ) - ~ ( t ) I dt (36) fi

$I ITAE = tl r(t) - ~ ( t ) I dt

with r(t) being the reference input and y(t) the output of the system.

In [22], a genetic algorithm based on Gray coding is used. Each PID parameter (Kp, K,, Kd) is represented by 16 bits and a single individual is generated by concatenating the coded parameter strings. The genetic algorithm requires a population of initial approximations, which may be random, to start the search. The algorithm then checks the fitness of each individual (or chromosome), and then grades them. A selection process follows where five of the fittest individuals are chosen. The remaining individuals are selected probabilistically. The selected individuals are

50

used to produce the next population, and the process is then repeated until the design requirements are met. This method is applicable to a wide range of system models due to its adaptability. High-order systems do not present a problem with this tuning procedure.

11 PID tuning using the theory of adaptive interaction

In [23], an adaptive method based on the theory of adaptive interaction [24] is used for tuning PID controllers. Tuning is obtained through minimisation of a performance index (e.g. the error squared). The controlled system is broken down into four subsystems; namely, the plant, the propor- tional control, the integral control and the derivative control, Kp, K, and K d are viewed as the interaction parameters between the four subsystems. An adaptive interaction algorithm is then used to tune these parameters. For this purpose, the system's Frechet derivative [25] is needed. In many cases, the Frechet derivative can be replaced by a constant that will be absorbed in an adapta- tion coefficient. This gives flexibility in the design, since it reduces the dependency on the plant's model. The Kp, K,, Kd parameters are treated as variables and they are not required to converge to a constant value. In fact, they change as the set-point and the disturbances entering the system change.

The theory of adaptive interaction is based on the assumption that a complex system is composed of other subsystems called devices. The dynamics of each device are given by a functional (a transformation from a vector space X into the space of real (or complex) scalars is said to be a functional on X) F,: xn + y n , n E K, where x,, yn denote the input and output spaces, respectively. In other words, the output of each device is a function of each input, which in turn is related to the outputs of other devices. Additionally, it is assumed that the input to a device is a linear combination of the output of the other devices via connections in I,, the set of input interactions for the nth device, and possibly an external signal u,(t) described by x,(t) = u,(t) + ~ c c , , ~ ~ y ~ , . ~ ~ ( t ) , n E K, where a, is a connection weight, and ypre, is the output of another device whose output is conveyed by connection c. The dynamics of the system can now be described by y,(t) = Fn[un(t) + C, a,ypre,(t)], n E K. The target is to find the correct a, so as to minimise some performance index E(.YI,. . . ,yn, u i , . . .u,>.

For the PID case, the system is decomposed into four subsystems as explained earlier. Then, the adaptive algo- rithm requires that

where the o symbol denotes composition, i.e.

In [23], the error squared is minimised as the performance index giving the following auto-tuning parameters:

Kp = -2yeF:[x4] o y l

Ki = -2ye~:[x,] o y2 (39)

Kd = -2yeFi[x4] o y 3

IEE Proc.-Control Theory Appl., Vol. 149, No. 1, January 2002

Then, the Frkchet derivative is approximated by F‘[x] 0 h = ph, with p being a constant. This can then be substituted into (39) to get

K,] = -YeYl

Ki = -yey, (40)

Kd = -WY, The advantage of such a method is that it can be applied to both open-loop stable and unstable plants, including systems with time delay, and it is robust to changes in the system’s parameters and/or to noise entering the system.

12 Methods based on cancellation

It has been pointed out in [26, 271, that PID control is often based on a second-order model. This implies that when PID action is used for control, then a model order reduc- tion has to be employed. Internal model control (IMC) [28, 291 is useful in this case, since a plant model order reduction automatically results in controller order reduc- tion. All stabilising controllers of a system under IMC are parameterised as

where ,G(s) is the system’s transfer function and G(s) = G-(s)G+(s). Here, G+(s) includes all non-minimum phase dynamics and Q(s) is usually chosen to be Q(s)=F(s)GI’(s). F(s) is chosen to be a lowpass filter with F(O)= 1 and GI’(s) a stable inverse of G(s). The resulting controller takes the. following form

F(s)GI1 (s) C ( S ) =

1 - F ( S ) & + ( S )

Therefore, a PI controller will result when a first-order system is to be controlled, and a PID controller when a second-order system is to be controlled.

A special case of IMC is the 2-tuning method [30, 311, which was developed for processes with long dead time and is mainly described by first-order dynamics. Here the controller transfer function is simply expressed as

1 1 G,(s) = ~ -

Gp(s) 3,s (43)

where Gc(s), Gp(s) and /z represent the controller’s transfer function, the process transfer function and the desired closed-loop time constant.

The overall closed-loop transfer function will take the following form

(44) 1

1 + As = -

Now using (43) and a P, PI or PID controller, the tuning parameters can be obtained as a function of 1. Examples related to ,?-tuning are presented in [30].

It can also be shown that if the system’s model is accurate then by using 2-tuning the sensitivity is always less than two. Small 1 gives a small IAE but increases the system sensitivity. For further details of the ),-method the reader is referred to [4, 30, 311.

In a similar approach, the zeros of the PID controller are used to cancel out the system’s dominant poles. Although this may work, especially when large time delays are

IEE Proc -Control Theory Appl.. Vol 149, No 1, January 2002

present [l], it is not recommended for some cases [ 3 2 ] . Good set-point responses may be achieved, but load disturbance response can be poor. In the case where unstable poles are involved, then cancellations must be avoided because they will give an unstable response.

In [26], a simple but effective order reduction technique for use under PID control action is introduced. The plant’s model is initially reduced by retaining the slowest poles, and then by retaining the low-order coefficients. It is then proven that the former is an overestimate of the plant’s magnitude, whereas the later is an underestimate. By taking the average of these two, a reasonably good approx- imation of the plant’s model is achieved up to its bandwidth point. In [33], the results of [26] are extended to deal with discrete-time systems. The main advantage of such methods is their simplicity. For an extensive reference on model order reduction techniques, the reader is referred to [34, 351.

13 K-B parametrisation

K-B, Keviczky-Banyaz, parametrisation is a generic two- degree-of-freedom (G2DF) implementation. It provides the desired transient responses considering both tracking and regulatory performances, and it effectively opens the system’s closed loop. Figs. 5 and 6 illustrate the principle, where SE I, Ho E 1 and Ro are the process model, the noise model and the controller, and 1 is the set of all stable proper systems.

In [36], classical PID tuning techniques are combined with the K-B parametrisation method to give improved control performance.

14 Magnitude optimum multiple integration method

The magnitude optimum multiple integration method (MOMI) is based on the magnitude optimum (MO or BO) method [37] in which the frequency response from

r -

I I Ho I I

RO

Fig. 5 K-B parametvised 2DF system

A r

S - Fig. 6 K-B parametvised opened system with augmented noise model

5 1

set-point to output should be as close to one as possible. If the closed-loop transfer function is GCL(s), then the objec- tive is to find a controller such that

GCL(0) = 1

~ l G ; $ ~ ) l ~ u y o = 0 (45)

for as many n as possible [38]. This will give a fast non- oscillatory response for many cases. The MOM1 method can be used to tune a PID controller with transfer fbnction given by

in the following way. First, the parameters Kp, b l , . . . , b,, a1 , . . . , a,, Tdel in the plant's transfer function model

are obtained in the following way, using the concept of multiple integrations [39]. The process's open-loop step response is recorded for a change AU and the following areas are calculated:

' 4 1 =v1(w) (48)

with

(49)

Yk( t ) = iAk-l -Yk-1(Z)1d7 JI Now, to meet (45) the following PID parameters are used [401

(52)

This method is applicable to high-order systems, non- minimum phase systems, and systems with reasonably large time delays.

15 Frequency loop shaping FLS)

Frequency loop shaping methods are based on the use of the sensitivity function S given by

1 1 + C(s)G(s)

S(S) = (53)

and the complementary sensitivity function T given by

T(s) = 1 - S(S) (54) 52

where G(s) is the transfer function of the plant and C(s) is the transfer function of the controller used to shape the plant's response.

The performance characteristics can be described as bounds of the above functions in the frequency domain. For good disturbance rejection and set-point following, the sensitivity function S must be small for low frequencies. For good noise attenuation and for robustness, related to high-frequency unmodelled dynamics, one wants the complementary sensitivity T to be small.

Resent research has been concentrated on the develop- ment of techniques that combine identification and control within the loop shaping framework for the design of more reliable systems.

The frequency response information of the plant can be employed for identification purposes to give its nominal model, together with estimated confidence limits. Convex optimisation techniques, like I , or 12, can then be used to tune the controller. The specifications are usually given in the form of a desired loop transfer function (LTF).

In [41], a method that integrates identification and tuning for PID control is presented. The objective of the design is to maximise the disturbance attenuation. The system to be identified is excited by a random binary signal, which is assumed to have sufficient energy content across the desired closed-loop bandwidth. The system's uncer- tainty is estimated by considering a multiplicative uncer- tainty structure. The selection of the target loop can then be done by optimising the disturbance rejection properties subject to the uncertainty-related bandwidth constraints and the fundamental limitations on the plant's model related to the locations of the plant's poles and zeros. It is also pointed out that the target LTF must contain all non- invertible dynamics present in the plant and compensator. In the case where a more systematic approach is necessary, one has to move from simple PID control to a full-order compensator design based on H, or other methods.

In the FLS PID tuning method, the PID coefficients K p , K , , Kd are tuned so that the compensated open-loop transfer function is close to the target transfer function L(s) in a I , sense.

The PID controller transfer function is rearranged in the following form:

Kls2 + K2s + K3 C(s) =

s(7s + 1) (55)

with K,, = K2 - K3s, K, = K 3 , Kd = K1 - K27 + K3z2. The advantage of such a parameterisation, as stated in

[41], is that any functional of the form 11 W(GC - L)ll,, with W a fixed weighting function, is convex in the new design parameters.

The FLS tuning of the PID controller can then be formulated as the following optimisation problem:

where So = (1 +L)-' is the target sensitivity and k is a convex set of constraints for the PID parameters.

A more detailed description of this PID tuning method and some applications of it can be found in [41, 421.

16 Concluding remarks

This paper has presented an overview of PID control, its advantages, disadvantages and different tuning methods. Only a flavour of the available PID tuning methods has been given, since it is not possible to include all available

IEE Proc.-Control Theory Appl., Vol. 149, No. 1, January 2002

methods here. Novel approaches to intelligent PID control, using concepts of normalised dead time and normalised process gain, are described in [ 5 ] . The symmetrical opti- mum (SO) method [43] is also an optimisation method of great interest. However, it must also be pointed out that PID control may not be sufficient for some cases, for example, processes with more than one oscillatory mode or processes with large time delays or with complex disturbance behaviour.

It is concluded here that PID control is still of great interest, and is a promising control strategy that deserves further research and investigation. Both industry and academia have a lot to gain from this.

17 Acknowledgments

The authors would like to thank EPSRC and TRW for supporting this research.

18 References

1 ASTROM, K.J., and HAGGLUND, T.: ‘Automatic tuning of PID controllers’ (ISA, 1988)

2 ASTROM, K.J., and HAGGLUND, T.: ‘PID control’ in ‘The control

3 ZIEGLER, J.G., and NICHOLS, N.B.: ‘Optimum settings for automatic controllers’, Trans. ASME 1942, 64, pp. 759-768

4 HAGGLUND, T., and ASTROM, K.J.: ‘Automatic tuning of PID controllers’ in ‘The control handbook’, 1996, pp. 817-826

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