Birgitta PID Control

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56 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2006

to sensor noise. Our rules are applicable to many kinds of plants,including stable plants with all poles on the negative real axis, aswell as plants with integral action. The resulting closed-loop sys-tems offer good midfrequency (MF) robustness combined withthe best possible tradeoff between performance and controlactivity. The tuning rules introduced by this study demandmoderate plant knowledge, while providing freedom for usersto make tradeoffs between such factors as stability margin, out-put performance, and control cost. In almost every situation,derivative action with a lowpass filter improves the propertiesof the controlled system compared to PI control. We show thatdesigning a PID controller with a filter is no more difficult thandesigning an ordinary PI controller.

In many design methods, not all of the available degrees of freedom are utilized. For example, with no motivation, theratio between integral and derivative time constants is oftenfixed to four [11][13], and the filter time constant is set to atenth of the derivative time constant [14][16]. Experience inoptimizing PID controllers for a variety of plants with all

design parameters free shows that the best tradeoffs amongperformance, robustness, and control activity often result incomplex controller zeros and a filter time constant that is larg-

er than that which is usually recommended [17]. A reformula-tion of the classical PID controller for design purposes is thusmotivated. This article also investigates the performance sensi-tivity related to various controller parameters, in particular,the influence of the damping factor and the time constant of the controller zeros.

When plants experience uncertain time delays or unmod-eled high-frequency (HF) resonances, or when measurementnoise is severe, the controller must have low gain in the HFrange. This requirement can be met by introducing rolloff inthe controller. A PI or, preferably, a PID controller equippedwith additional lowpass filtering (F), then becomes an attrac-tive alternative. The resulting PIDF controller with five tuningparameters can also be tuned according to the rules presentedin this article.

Aside from the Ziegler and Nichols tuning rules [11], themost commonly used tuning method is probably the approach

based on internal model control (IMC) [18]. When IMC is used,the identified or derived plant model must be reduced to first

or second order, respectively, before the PI or PID design rulescan be applied. This reduction can be accomplished in variousways [19], [20], and the choice of reduction method greatly

a Ratio T i / T d b Ratio T d / T f C i Arbitrary constants, i = 1 , 2, . . .e (t ) Control errorG (s ) Plant model transfer functionG m Gain marginGM S Generalized maximum sensitivity, stability margin

HF High frequencyIAE Integrated absolute errorJ HF High-frequency criterionJ u Control activity criterionJ u ec Economic limit for J u J v Performance criterionK Static gainK (s ) , K x (s ) Controller transfer function, x = PI, PID, PIF,

or PIDFk i , K i Controller integral gainK p Proportional controller gaink Controller high-frequency gainL(s ) Loop transfer function

Ld Delay timeLF Low frequencym Rolloff rateMF MidfrequencyMH F Mid to high frequencyM S Infinity norm of the sensitivity functionM T Infinity norm of the complementary sensitivity

function

r (t ) Reference signalS (s ) Sensitivity functionS u (s ) Control sensitivity functionS v (s ) Disturbance sensitivity functionT Time constantT (s ) Complementary sensitivity functionT d Derivative time constant

T eq Equivalent time constant ( T 63 Ld )T f Filter time constantT i Integral time constantT 63 The time for a step response to reach 63% of

its steady-state valueu (t ) Control signalv (t ) Process (load) disturbance signalw (t ) Sensor noisey (t ) Controlled outputz Plant model zero M S / M T Filter constant k /( k i ) Controller zero damping

f Controller pole damping , 150 , i 150 Parameter characterizing the dynamics of a

plant modelm Phase margin Vector of tuning parameters150 G , 180 G Frequency at which the phase lag of G (s ) is

150 or 180 , respectively

Notation


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FEBRUARY 2006 IEEE CONTROL SYSTEMS MAGAZINE 57

influences the resulting design. A comparison between themain tuning rules presented in this article and other tuningmethods, including different versions of IMC, is given in [21].

Since system properties are not independent, improvementof the control in one respect often implies deterioration inanother. A fair method for comparing two controllers must,accordingly, ensure that all aspects not immediately comparedare equally restricted during the comparison. Such an evalua-tion method, which was presented in [17], [22], and [23], isused in this article to compare tuning rules. This evaluationmethod is described in the next section.

EVALUATION OF CONTROLLERS

We present a systematic procedure for evaluating controlstructures and design parameters. Our procedure is based onfour criteria representing low-frequency (LF), MF, mid- tohigh-frequency (MHF), and HF performance and robustnessproperties. By evaluating these criteria, the tradeoff between,for example, output performance and control activity can be

investigated while passband robustness (stability margin) iskept constant. Another objective is to keep the MF, MHF, andHF criteria equal, or at least equally bounded, to evaluate theimprovement in LF performance in an objective manner. Theintroduced evaluation criteria, which are based on H -norms,also facilitate comparisons of discrete-time and multi-input,multi-output controllers. For further details, see [21] and [23].

Evaluation Criteria

Consider the single-input, single-output (SISO) system in Fig-ure 1, where the plant G(s) is controlled by the controller K (s).The closed-loop system has three inputs, the reference signalr( t) , the process disturbance v( t) , and the sensor noise w( t) . Rel-

evant outputs are the controlled output y( t) , the control signalu( t) , and the control error e( t) = r( t) y( t) . The loop transferfunction is L(s) = G(s)K (s), while the sensitivity functions are

Sensitivity function

S(s) =1

1 + L(s)= Ger(s) ,

Complementary sensitivity function

T (s) =L(s)

1 + L(s)= G yr(s) = G yw(s) ,

Disturbance sensitivity function

Sv(s) =G(s)

1 + L(s)= G yv(s) ,

Control sensitivity function

Su(s) =K (s)

1 + L(s)= Gur (s) = Guw(s) .

The controller K (s) can be either strictly proper or exactlyproper. Including integral action in K (s) implies the asymptoticproperties

K (s)

k is , s 0,

k sm

, s ,(1)

where k i is the integral gain, k is the HF gain, and m is the

rolloff rate of the controller.The performance criteria, which are based on the sensitivityfunctions, are

Performance criterion

J v =1s

G yv(s)

=1s

Sv(s)

,

Passband robustness criterion

GMS = max ( S(s) , T (s) ),

Control activity criterion

J u = Gur (s) = Guw(s) = Su(s) ,

HF criterion J HF = smGur (s) = smGuw(s) = smSu(s) ,

where || .|| denotes the H -norm.The criterion J v is a measure of the systems ability to han-

dle LF load disturbances. This criterion is a frequency domainalternative to criteria based on an integral function of the errorsignal, for example, IAE =

0 |e( t) | dt [4], [24]. For servo prob-lems, a more relevant criterion is obtained by replacing G yv(s)with Ger(s) .

The passband robustness criterion given by generalizedmaximum sensitivity GMS is a combination of the constraints

S = max | S( ) | MS and T MT . The parameter = MS/ MT transforms the restriction on || T || to a similarrestriction on || S|| . The maximum S of the sensitivityfunction is often used as a robustness measure [4], [25], [26]since S is equal to the inverse of the minimum distancefrom the loop transfer function to the critical point ( 1, 0) inthe Nyquist plot.

The components of GMS correspond to two circles in thecomplex plane given by || S|| = MS and || T || = MT , from

FIGURE 1 Closed-loop SISO system. The plant G (s ) is controlled bythe controller K (s ) . The system has three input signals, namely, thereference signal r (t ) , the process disturbance V (t ) , and the sensornoise w (t ) , as well as two output signals, namely, the controlledoutput y (t ) and the control signal u (t ) .

G(s) K(s) r +

-

+ w

v

y u + +

-


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which the Nyquist plot of L( ) is excluded to enforce aspecified stability margin. In Figure 2, these circles are shown forthe default values, specifically, MS = 1.7 and MT = 1.3. Thesevalues represent an empirical compromise. A larger value of MSimplies a faster response with a larger overshoot in the referencestep response and an undershoot in the disturbance stepresponse; a smaller value implies a slower system with a largerIAE. The chosen value of MT guarantees a minimum phase mar-gin of 45 . A criterion similar to GMS is formulated in [27]. Thecombined measure GMS replaces the two classical measures,phase margin PM and gain margin GM, which are commonlyused to characterize MF robustness [28], [29].

The criterion J u is related to the MHF region, which is close toand slightly above the closed-loop bandwidth. The control sensi-tivity function most often reaches its maximum in this range.

In the HF region, two requirements are paramount, namelyrobustness toward model uncertainties and reduction of theeffects of sensor noise. According to the small gain theorem[30], plants with significant model uncertainty can be ren-

dered closed-loop stable by keeping the complementary sensi-tivity function T (s) small. Since T (s) = G(s)K (s)S(s) and,consequently, T (s)G(s) 1 = Su(s) = Gur (s) , keeping Su(s) smallalso keeps T (s) small. Since Su = Guw , this restriction alsoreduces the influence of sensor noise on the control signal.These facts motivate the HF criterion J HF = || smSu(s) || .For high frequencies, | S( ) | 1, which implies| Su( ) | | K (s) | k m according to (1). Hence J HF , as wellas J v, is almost independent of the nominal plant model. Whenm = 0, which is valid for most PI and PID controllers withoutextra filters, J HF = J u.

Evaluation Procedure

An objective evaluation of different controllers is obtainedwhen three of the four criteria are kept equal or at least

bounded from above, while the tuning parameters (collected

in a vector ) are modified to minimize the fourth criterion.For example, the LF performance is evaluated by solving theconstrained optimization problem

min

J v(), GMS() C1, J u() C2, J HF () C3, ( 2)

where C1, C2, and C3 are constants. The default value of C1 is1.7, while the values of C2 and C3 can vary. The last restrictionin (2) is relevant only for strictly proper controllers. By thisoptimization procedure, completely different controllers can

be compared under equal conditions. This type of evaluationmethod is discussed in [31]. Similar ideas with different crite-ria and constraints are presented in [32] and [33]. Alternative-ly, the loop-shaping strategy in [34] and [35] focuses on L( ) ,assuming complete knowledge of the frequency response of the plant. The expression optimal controller refers to a con-troller optimized according to (2), with all available parame-ters included in the tuning vector . We used the MATLABOptimization Toolbox for the computations.

CONTROLLER AND PLANT MODELS

Before evaluating the closed-loop properties by the abovecriteria, we discuss the parameterization of PI and PID con-trollers as well as the dynamic characteristics of the plantmodels included in this investigation.

Parameterization of Controllers

There are alternative ways to formulate and parameterize thetransfer function of a PID controller. We consider one-degree-of-freedom controllers, as shown in Figure 1. Our goal is todesign PID controllers to reject process disturbances, as inmost applications [15], [36]. In certain cases, such as in model

predictive systems, where the PID controllers serve at lowerlevels in hierarchical structures with more sophisticated con-trollers at higher levels, or when good servo properties aredemanded, the controller can be augmented by a filter in thefeedforward path.

The traditional PID controller K PID (s) = K p + K i/ s + K ds,with the parameters proportional gain K p, integral gain K i, andderivative gain K d, has the drawback of being improper. Tolimit the HF gain, the PID controller is often augmented by alowpass filter on the derivative part, and may then be formu-lated with a proportional gain K p, integral time T i, derivativetime T d, and filter time constant T f as

K PID (s) = K p 1 + 1sT i+ sT d1 + sT f

. (3)

This formulation has the advantage that a change in the gain mar-gin requires adjustment of only K p. With a = T i/ T d and b = T d/ T f ,a double zero in the controller corresponds to a = 4 when b = ,and a slightly higher value of a when b < (T f > 0).

In [22] and [23], it is shown that a PID controller, optimizedwith the filter included and all parameters free, often has com-plex zeros. Consequently, we reformulate the PID controllerwith a first-order, lowpass filter as

FIGURE 2 Definition of generalized maximum sensitivity GM S . Whena system satisfies a given value of GM S , the Nyquist plot of its looptransfer function L( ) does not enter into the M S circle (defaultM S = 1 .7) and the M T circle (default M T = 1 .3).

0

0

1

2

M S

M T

|L( j )|

1

24 3 2 1


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optimal PID controller, the J v / J u curve tends to flatten as J ugrows, as in Figure 3. In fact, for nonminimum phase plants,there is a theoretical lower limit for J v. Although the user hasfull freedom to choose the level of control activity J u , there isno advantage in increasing J u above a certain level J u ec, wherethe benefit of decreased J v is insignificant [17]. More complexdynamics (higher values of ) imply lower J u ec. The tuningrules are formulated to achieve control activities close to J u ec.For the PID controller, J u is typically equal to k , which meansthat the required level of J u determines the HF gain k .

Damping Factor In all cases considered here, the optimal PID controller for astable, nonoscillating plant has a pair of complex zeros. Thedamping factor has no clear dependence either on the plantdynamics, as measured by 150 , or on GM S or J u . When J u ischosen near J u ec, the optimal value of normally ranges from0.70 to 0.85. Similarly, as in Figure 4, = 0.75 is often a goodchoice. Damping that is too low, however, implies undershoot

in the response to a disturbance step as in Figure 4(b) and a

sluggish response to a reference step, while too-high dampingimplies a somewhat sluggish response to a disturbance stepand an undershoot in the response to a reference step.

Zero Time Constant The optimal zero time constant depends on the plantdynamics. In fact, after normalizing the time scale of the plant,1/ increases almost linearly along with 150 . However, isoften proportional to T 63 , the length of time that a stepresponse takes to reach 63% of its final value. Figure 5 showsthat, for plant models with 150 0.09, a reasonable choice is = T 63/ 3 . This figure also illustrates that plants with first-order dynamics behave otherwise and consequently must betreated separately.

Furthermore, the optimal value of is largely independentof J u , with the exception of small values of J u , as seen by com-paring the constant value = T 63/ 3 0.8 with the optimalsolution in Figure 6(a). Likewise, Figure 7 shows that theoptimal value of increases slightly when J u = 5. This value

also marginally increases with decreasing demand on GM S .

TABLE 1 Typical plant models and their values of 150 , or, in the case of plant integral action, values of i 150 . The parameter150 , or alternatively i 150 , which is usually between zero and one, is used to characterize the dynamics of a plant. The higherthe value, the more difficult the plant is to control. This parameter can be used to formulate simple design rules.

G(s) 150 G(s) i 1501

(1 + 0 .2 s )( 1 + s )0.038

1

s (1 + s )0.500

1

(1 + s )2 0.067

e 0:3 s

s (1 + s )0.643

e 0 .1

1 + s 0.088

1

s (1 + s )( 1 + 0 .2 s )0.657

1 0 .1 s

(1 + s )2 0.141

1

s (1 + s ) 2 0.749

e 0 .3 s

1 + s 0.228

1 0 .1 s

s (1 + s ) 2 0.774

e 0 .3 s

(1 + s )2 0.249

e 0 .3 s

s (1 + s ) 2 0.810

1

(1 + s )3 0.266

1

s (1 + s )( 1 + 0 .7 s )( 1 + 0 .7 2 s )( 1 + 0 .7 3 s )0.852

1

(1 + s )( 1 + 0 .7 s )( 1 + 0 .7 2 s )( 1 + 0 .7 3 s )0.352

1 0 .25 s

s (1 + s ) 3 0.857

1 0 .25 s

(1 + s )3 0.359

1

s (1 + s ) 4 0.870

1

(1 + s )4 0.395

e s

s (1 + s )0.877

e s

(1 + s )2 0.488

e s

s (1 + s ) 3 0.902

e s

(1 + s )3 0.527

e s

s (1 + 0 .1 s )0.997

e s

(1 + s )0.528

1

(1 + s )8 0.646

1 2 s

(1 + s )4 0.907

e s

1 + 0 .1 s 0.973


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Figure 6(b) illustrates that high values of imply slowlydecaying responses to step disturbances, while low valuesimply undershoot.

Figure 7(a) illustrates a minimum value of J v with respectto . When this minimum is not reached, a value of that ismarginally higher than the optimal value is preferable to alower value. Figure 7 also confirms that the economic limitof J u for this plant ranges between 10 and 15, while the deriva-tive action in the PID controller decreases J v by a factor of twocompared to PI control.

Filter Factor A higher filter factor usually means increased control activity J u, as shown by the optimal curve in Figure 6(a). The optimalvalue of , when J u J uec, varies according to 150 , although formany common plants the optimal value of is significantlylower than the recommended b = 10 corresponding to 20[21]. For low-order and minimum-phase plants, the optimalvalue of corresponding to J uec may be as low as = 2 (seeFigure 8). The performance J v for such plants can be improved

by using a higher value of but at the cost of high controlactivity J u. Recall that a PID controller with = 1 and = 1constitutes a PI controller, which shows the limitation on con-trol activity in the PI case.

Integral Gain k i Since k i 1/ J v , the optimal integral gain k i increases withincreasing J u (see Figures 4 and 6). However, the optimalvalue of J v is higher for morecomplex plants, which impliesthat k i decreases when 150increases. The optimal value of k i also decreases when a largerstability margin (lower GMS) isdemanded. When the con-troller parameters , , and are given by tuning rules, theintegral gain k i is also the para-meter best suited for manualtuning of the desired tradeoff

between rise time (quickness)and damping (robustness) of the step response.

Traditional Tuning, a = 4, b = 10It is desirable to view a compar-ison of the optimal J v/ J u rela-

tionship for different values of . Figure 4 shows the best possi- bl e J v/ J u combination with fixed controller time ratiosa = T i/ T d = 4 as suggested in [11] and [12] and b = T d/ T f = 10,as recommended in [14] and [16]. With two parameter valuesspecified, only two parameters remain to be optimized. Fixeda = 4 produces minor deterioration compared to optimal val-ues, which for most stable plants is approximately a = 2.5.However, the often used value b = 10 is, in most cases, too high

by a factor of two to three. As Figure 4 shows, the output per-formance J v can be significantly improved with constrainedcontrol activity J u and required MF robustness GMS by optimiz-

ing all accessible parameters, including the filter time constant.

FIGURE 3 A typical relationship between J v and J u for fixed GM S .The curve has a tendency to flatten as J u increases.

FIGURE 4 (a) The dependence of the J v / J u tradeoff on the damping of the PID controller zeros.When and k i are optimized, the J v / J u relation is not greatly influenced by , as long as remainsaround 0.8. (b) The corresponding responses to disturbance steps when J u = 10 . The curves markedwith a = 4 , b = 10 represent tuning with a and b fixed, while K p and T i in (3) are optimized. The plantmodel is G (s ) = e 0 .3s /( 1 + s )2 .

0 4 8 12 16

0

2

4

J v

J u

5 10 15 200

0.5

1

1.5

2

a =4, b =10

=0.51.0

0.75

J v

J u 0 5 10 15

0

0.1

0.1

0.3

0.5a =4, b =10

=1.51.251.0

0.75

0.5

y

t

(a) (b)

Easily understood and applied tuning rules can give close-to-optimal PI

and PID controllers.


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Tuning Rules for PID Controllers Based on the above discussion, we now present three sets of tuning rules for PID controllers. These rules are intended forstable, nonoscillating plants.

Tuning Rules Based on 150When a relay experiment involving hysteresis [12] is used toobtain the required plant information, the controlled self-oscil-lating frequency is not 180G but rather is somewhat lower, typ-ically around 150G . The first tuning method, which requiresknowledge of 150 and 150G , is suitable for plants with poles(at least two) on the negative real axis. Recommended parame-ter settings for a PID controller (4) are

Tuning rules for stable nonoscillating plants of at least thesecond order

= 0.75,1

= 150G(0.44 + 0.86150),

k =1

G(0)min 3 +

2150G

, 25 ,

k i = 150GG(0)0.45

150 + 0.07 0.1 . (6)

An alternative to the expression for k is

= 2 + 14150. (7)

The HF gain k is chosen to keep J u as close to J uec as possible. Figures 5 and 8 illustrate these rules as well as optimaloutcomes for a set of plant models with values of 150

between 0.04 and 0.91. When all four controller parametersare tuned according to (6), GMS is in the range of 1.651.85

for most of the plants tested, generally near 1.7. The result-ing value of J v differs from optimal values at correspondingvalues of J u and GMS, in most cases by less than 5%. The onlyexceptions are plants with extremely long delay times Ld rel-ative to time constants T eq (Ld/ T eq > 4) leading to 150 > 0.95,as well as plants with stable zeros close to the imaginary axis(| z| < T 163 ). Precise comparisons between the optimal solu-tion and the tuning rule (6) require that GMS be kep t atexactly the same level, which is impossible with a simpletuning rule for k i.

The parameter Ld is an equivalent delay time, defined asthe time it takes for the step response to reach 5% of its finalvalue, while T eq = T 63 Ld. A good rule of thumb is that

150 0.5Ld/ T eq (see Figure 9). For plants with 150 < 0.05, k tends to be unnecessarily high, a consequence of the tendency

FIGURE 5 The dependence of / T 63 on 150 . The correspondencebetween rules (8) and (9) (solid) and optimal outcomes ( + ) forplants of order two or higher is good. The circles (o) show optimaloutcomes from first-order plants with small time delays, which signif-

icantly differ from the outcomes for plants of higher order and thusmust be treated separately.

FIGURE 6 Optimal Jv / Ju relations and load disturbance step responses for different PID controller-zero time constants . The difference inperformance between optimal (varying with J u ) and constant = T 63 / 3 0 .8 is marginal except for small values of J u . High values of yield slowly decaying disturbance step responses, while low values imply undershoot. In (a) and (b), the plant is G (s ) = e 0 .3s /( 1 + s )2 with = 0 .75 . In (b), J u = 10 . Obviously, a value of that is too high is a better choice than a value that is too low.

0 5 10 15 20

0

0.2

0.4

0.6

=0.40.6

0.8

1.0

1.2

y

t5 10 15 20

0

0.5

1

1.5

2

= 0.6

0.7

1.0

Optimal

T 63 /3

=35

8 10

J v

J u

(a) (b)

0 0.5 10

0.2

0.4

150

/T 63


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of the optimal J v 1/ k i to be extremely small. Therefore, themaximum value k = 25 is used in (6).

The rules (6) are recommended for autotuning [4], as wellas for finding appropriate initial values for optimization.

The placement of the controller zeros is not as crucial as thetradeoff between k i and k in attending the specified value of GM S . Thus, one option is to use (6) to select only and . There-after, a suitable control activity k is chosen ( J u k ), with k according to (6) as a first choice. Finally, the integral gain k i ismanually tuned until the desired tradeoff between rise time (out-put performance) and damping of the step response (robustness)is achieved. Both good performance and low control activity(small k ) can be obtained. The low control activity is a conse-quence of including the lowpass filter in the controller design.

Tuning Rules Based on T 63An alternative tuning procedure is based on the step response.Figure 5 shows that, except for plants with 150 < 0.09,approximately one-third of T 63 serves as an appropriate value

for . For plant models with 150 < 0.09, the optimal value of decreases as

= T 63(2.65150 + 0.11). ( 8)

Given a step response, the controller zeros of a PID controllercan be chosen as

Extremely simple tuning rules for PID controller zeros

= 0.35T 63 T 63/ 3, = 0.75. (9)

The remaining parameters k and k i can be used to tune thetradeoff between output performance and control activity.

Start with between five and ten, noting that the con-trol activity is increased by using a larger value of anddecreased by using a smaller one. A higher value of isrecommended, according to Figure 8, for plants withmore complicated dynamics (higher 150) (see Table 1).Optimal values of as well as of are almost indepen-dent of .

Finally, increase k i until acceptable overshoot (damp-ing) is achieved in the closed-loop step response.

FIGURE 7 Optimal performance J v = f ( ) and load disturbance step responses for different control activities J u . The value of that mini-mizes J v is largely independent of J u except for small values. The economic limit of J u for the plant G (s ) = e 0 .3s /( 1 + s )2 ranges from 10to 15. The derivative action in the PID controller decreases J v (J u J u ec ) by a factor of two compared to optimal PI control ( = 1 , = 1).

0.7 0.9 1.1 1.3 1.50.4

0.8

1.2

J v

PI

J u = 5

10

15

0 5 10 15

0.1

0.3

0.5

PI

J = 5

10

15

y

t(a) (b)

FIGURE 8 Normalized PID controller parameters k i , k , and . The suggested tuning rules (6) and (7) for PID controllers based on 150(solid) produce parameters with close-to-optimal outcomes ( + ) for most plants with all poles on the negative real axis.

0 0.5 10

2.5

5

150 0 0.5 1

0

10

20

30

k G

( 0 )

0 0.5 10

5

10

15

(a) (b) (c)

k i G ( 0 ) / 1 5 0

G

150

150


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For plants with 150 > 0.09 (Ld > 0.18T eq), including plantswith highly nonminimum phase behavior, the results of thistuning procedure are remarkably close to optimal. The onlyproblems arise from 1) plants with extremely long time delaysand correspondingly high values of ( > 0.95), 2) plantswith stable zeros close to the imaginary axis, and 3) plantswith first-order dynamics and moderate time delay (see

below). In Figure 10, responses are given to reference and dis-turbance steps, when a PID controller for G(s) = e 0.3s/( 1 + s)2

is tuned using (9) and = 6.4. The integral gain k i should bemanually tuned to GMS = 1.7. These step responses turn out

to be nearly optimal. Figure 10 confirms that nearly optimal behavior is obtained by the tuning rule (6).

First-Order Plants

A special group of plant models constitutes those with onetime constant and moderate time delay, represented by thetransfer function

G(s) = K e sLd

1 + sT . (10)

First-order plants with extremely short time delays havesimple dynamics and thus are not considered.

PID tuning rules for first-order plants with time delay

=1

0.68 + 0.84Ld/ T 0.25(Ld/ T )2,

= T 0.06 + 0.68LdT

0.12LdT

2

, Ld/ T [0.05, 2],

k =1K

min 5 + 2 T Ld

, 25 , k i =1

KT 1.1

T Ld

0.1 ,

Ld/ T [0.25, 2]. (11)

In Figure 11, step responses show that results based on (11)almost coincide with the optimal solution given by (2).

PI Controllers

In a PI controller, = = 1. Only two parameters need to betuned. Unlike the PID case, an optimal PI controller yields a

FIGURE 10 Reference (a) and disturbance (b) step responses. The closed-loop behavior of a controller tuned in accordance with either thetuning rules (6) (KL 150) or (9) (KLT63) is remarkably similar to the optimal response. The plant considered here is G (s ) = e 0.3s /( 1 + s )2 .

0 5 10

0.1

0.1

0.3

y

t

KLT63

Optimal

KL 150

0 5 100

0.5

1

y

t

Optimal

KLT63

KL 150

(a) (b)

FIGURE 9 Approximating the tuning paramter 150 . A good rule ofthumb for computing 150 is 150 0.5 Ld / T eq .

0 1 20

0.5

1

150

Ld / T eq

A PID controller can reject process disturbances

without too much control action and sensitivity to sensor noise.


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minimum in the J v/ J u plot for most plant models (see [21],[22], and Figure 12). The goal of the PI tuning rules (12) and(13) is to reach that minimum, provided that the demand onMF robustness is fulfilled. Suitable rules for tuning PI con-trollers are

PI tuning rules for plants of at least the second order

1

= 150G 0.06 + 1.6150 0.06 2150 , or = 0.8(T 63 Ld),

150 [0.05, 0.9],

k i =150GG(0)

0.2 +0.075

150 + 0.05, 150 [0.1, 0.9]. (12)

PI tuning rules for first-order plants with time delay

= 0.5T 63 = 0.5(T + Ld),

k i =1

TK 0.57

Ld/ T 0.055, Ld/ T [0.2, 1]. (13)

PI Versus PID Controllers

Rules (6)(13), especially (9), which is based on a stepresponse and manual final tuning of the integral gain k i, showthat PID controllers are no more difficult to design than PIcontrollers. The difference is that PID solutions provide addi-tional freedom in selecting appropriate control activities J u k (implicitly provided by ).

PI control, where = 1 and = 1, means that the optimalcontrol activity J u is determined by the minimum in the J v/ J ucurve [see Figure 12(a)]. Often, the value of J u can be some-

what increased by introducing derivative action with a valueof typically in the interval [3 , 10]. Performance is thenimproved significantly, without achieving excessive sensitivityto sensor noise and HF model uncertainties.

Because of the concern for excessive sensitivity to sensornoise in the control signal, derivative action is often omitted inindustrial applications. We thus optimize the tradeoff betweennoise sensitivity, the ability to compensate load disturbances,and MF robustness by fixing the controller zeros according torules (6), (9), or (11) and then sequentially tuning or k andfinally k i in the PID controller. As far as we know, this free-dom for users is not available in any other tuning method.

The IMC strategy [18], widely used in chemical processcontrol, leaves only one parameter to tradeoff robustness,control costs, and performance qualities. To yield PI or PIDcontrollers, this method demands first- or second-order plantmodels, respectively. The fundamental characteristics of a PIDcontroller designed by IMC are two real zeros, chosen to can-cel the plant poles, and a resulting closed-loop system withthe characteristic polynomial ( s + 1/) 2 or ( s + 1/)( s + z) ,where s = z is a plant nonminimum phase zero, if present. Theparameter is used for setting the desired response time.Comparisons with IMC are presented in [21].

FIGURE 11 Reference and disturbance step responses for a first-order plant with moderate time delay. These plots show that (11)yields closed-loop behavior close to the optimal solution. The plantconsidered here is G (s ) = e 0 .6s /( 1 + s ) .

FIGURE 12 Comparisons of PI and PID controllers. For both types ofcontrollers, tuning rules (6), (9), and (12) yield nearly optimal results.The price for performance J v , compared to an optimal PI controller,is modest in terms of control activity J u , when a PID controller istuned by (6) or (9); it is high when a = 4 and b = 10 are fixed.Results of rules (6) (PID KL 150) and (12) (PI KL) are marked (o) in(a). In (b), J u J u ec for the PID controllers. The plant here isG (s ) = e 0 .3s /( 1 + s )2 .

0 5 10

00.1

0.1

0.20.30.40.5y

t

Optimal

Tuning Rules

0 5 100

0.20.40.60.8

11.2y

Optimal

Tuning Rules

t

0 5 10 150

1

2

3

PID: a = 4, b = 10

Opt PI

Opt PID

J v

J u

0 5 100.1

0.1

0.3

0.5

y

t

PI KL

PI Optimal

PID: a = 4, b = 10

PID KLT63

PID Optimal

PID KL 150

= f(T 63 )

= f( 150 )

(a)

(b)



12/15

Figure 12 shows results from applying tuning rules (6), (9),and (12). It can be seen that the rules for the zeros in the PIDcase can be used over a wide range of values of J u. Note thatthe two rules for based on 150 or, alternatively, on T 63, pro-duce almost identical results in the PID case and the PI case.Also observe that tuning all accessible parameters by the 150rules (6) provides results that are close to the optimal results.Figure 12 also shows the high cost in terms of control activity J u for the improved performance ( J v) achieved by a PID con-troller compared to the performance offered by an optimal PIcontroller when a and b are held constant to the traditionalvalues 4 and 10. In contrast, the cost is moderate when the PIDcontroller is optimized with the filter included. In many com-mercial PID controllers, the filter time constant is fixed (oftenb = 10) and cannot be modified. This situation often motivatesthe exclusion of derivative action in practice.

TUNING PID CONTROLLERS

FOR PLANTS WITH INTEGRAL ACTION

When the plant has integral action, alternative tuning rules forcontrollers are needed. However, formulating tuning rules inthis case is more challenging than for stable plants. In particu-lar, the necessary normalizations are more complicated, whilea starting point near the optimal solution, which makes theoptimization routine (2) converge, is more critical.

Controller Zeros For plants with integral action, the optimal value of is oftenclose to one (corresponding to a double zero), but tends togrow with increasing complexity in terms of i150 . A value of slightly above the optimal point is a better choice than alower value. Figure 13 illustrates the relationship J v = f ( J u)

with various values of as well as with fixed a = 4 andb = 10. For values of J u larger than 6.5, the required GMS can-not be reached.

The zero time constant is crucial. If is more than marginal-ly lower than the optimal value, J v increases drastically and therequirement on GMS cannot be met, as shown in Figure 14.Except for the lowest values of J u, the optimal value of is almostindependent of J u. Figure 14(b) also illustrates that the improve-ment in output performance J v is significant when derivativeaction is included, compared to the case in which a PI controlleris used. The control activity in the latter case is J u = 0.45.

HF Gain k , Filter Factor , and Integral Gain k i As in the stable case, k is equal to J u for the closed-loop system.In general, to meet the desired stability margins, more derivativeaction is required for plants with integral action than for stableplants. The filter factor increases rapidly with increasing J u[see Figure 14(a)]. Without adverse consequences, may formost plants (except when i150 < 0.7) be fixed to 20, a value cor-responding to the well-known value b = 10. In tuning formulas,we also observe that the integral gain must be normalized twicewith respect to a time or frequency parameter, since both k i andthe plant integral gain have units of inverse time.


FIGURE 13 Optimal J v / J u relationship for various damping values of the controller zeros when the plant has integral action. The opti-mal value of for a plant with a pole at the origin is about 1.0. Ahigher value of is a better choice than a lower value. The plantis G (s ) = e 0 .3s /( s (1 + s )2 ) .

FIGURE 14 (a) Optimal J v / J u relationship for various PID zero timeconstants for a plant with integral action. The value of is crucial.(b) When this value is more than marginally lower than the optimalvalue, J v increases drastically and the requirements on GM S cannotbe met. With a suitable value of , the PID controller offers signifi-cant improvement of the output performance compared to a PI con-troller. The plant is G (s ) = e 0 .3s /( s (1 + s )2 ) .

2 7 120

20

40

1.5

1.0

1.25

0.75

a = 4, b = 10J v

J u

= 0.6

5 10 15 200

10

20

30

40J v

J v

J u = 5

J u

5.0

4.0 3.5

Optimal

30 40

0 5 10 15 200

20

40

60

80

PI

10

15

(a)

(b)

= 20

= 3.0


13/15

When the plant has integral action, suitable tuning rulesfor a PID controller are

PID tuning rules for plants with integral action and at least two time constants

i150 0.7, = 20, =1

3.5 3 i150,

1

= 0.6150G, k i =2150G

K e6.4 9 i150 ,

i150 < 0.7, k = 20150G

K , =

13.5 3 i150

,

1

= 0.68150G, k i =2150G

K e4.4 6.5 i150 . (14)

The normalization factor K = lim 0 |G( ) | .For plants with integral action, only one time constant, and

time delay of the form

G(s) =Ke sLd

s(1 + sT ),

more appropriate results are obtained by the rules

PID tuning rules for plants with integral action, one timeconstant and time delay

50 > Ld/ T 0.25, = 20, = 1.7 e Ld/( 1.3T ) , = 1.7Ld + 1.3T ,

Ld < T , k i =1

KT 22e 4.5Ld/ T

Ld T , k i =1

KL2d(0.06 + 0.004Ld/ T ),

0 Ld/ T < 0.25, k = 201

KT

, = 1 0.35e 4Ld/ T ,

= 2.4Ld + 0.9T , k i =1

KT 25e 8Ld/ T . (15)

When the plant has integral action, a PI controller, whichintroduces additional integral action, is usually inappropriate.

Figure 14(b) shows that the optimal performance offered by aPI controller corresponds to J v = 74.0, while a PID controllerwith J u = 15 can offer J v = 8.9, almost a tenth of that offered

by the PI controller. Figure 15 illustrates a typical difference between PI and PID controllers in terms of process disturbance step responses, which implies that a controller withderivative action is recommended for plants with integralaction, except for plants with large time delays ( Ld/ T > 10).

ADDITIONAL FILTER ACTION

To achieve a strictly proper controller, the PI controller can beaugmented by a lowpass filter, or the first-order filter in the PIDcontroller can be replaced by a higher order filter. This modifica-tion is advantageous in cases of significant sensor noise orunmodeled HF resonances [40]. The interesting observation isthat a lowpass filter can be added to both PI and PID controllerswith only marginal effect on LF performance [41].

When a first-order lowpass filter is included, the PI con-troller becomes a PIF controller with transfer function

K PIF = k i1 + s

s(1 + s/).

When the first-order filter in the PID controller (4) is replaced bya second-order filter, the resulting PIDF controller has the form

K PIDF (s) = k i1 + 2 s + ( s)2

s 1 + 2 f s + s

2. (16)

This formulation enables complex poles, with damping ratio f and undamped frequency / , as well as complex zeros.

Figure 16 shows an example of a plant controlled by threeexactly proper and four strictly proper controllers. All con-trollers are optimized with GMS = 1.7. The PI/PIF controllershave J u values corresponding to an optimal PI controller withouta filter. For one pair of PID/PIDF controllers, J u = 10.8, close to


FIGURE 15 Load disturbance step responses for three plants with integral action controlled by optimal PI and PID controllers and by PIDcontrollers tuned by the rules (14) and (15). A controller with derivative action is recommended for all plants with a pole at the origin.Plant G (s ) = e 0 .3s /( s (1 + s )2 ) with rule (14) is used in (a), plant G = e 0 .1s /( s (1 + s )) with (15) in (b), and G = e s /( s (1 + 0 .1s )) withrule (15) in (c).

0 20 400

1

2

3

y

t

PI Optimal

Tuning Rules

PID Optimal

0 20 400

0.5

1

1.5

y

t

PI Optimal

Tuning RulesPID Optimal

0 20 400

1

2

y

t

PI Optimal

PID Optimal

Tuning Rules

(a) (b) (c)


14/15

J uec; for the other pair, J u = 5.6. In both cases, f = 0.4. When theHF gaink for the PIDF as well as the PIF controller is reducedto 20, which corresponds to very low bandwidth for the PIDFcontroller, then the performance J v for the PIDF controller isdeteriorated by 48% compared to the case of a first-order filter.Nevertheless, J v is still only 53% of the value of J v for the corre-sponding PIF controller. When the HF gain isk = 80, J v forPIDF is increased by only 5.7% due to the filter, which is 40% of J v for the corresponding PIF controller (with the samek = 80).These numbers and corresponding responses in Figure 16 showthat derivative action is valuable for the output performancewhen significant rolloff is required in the controller. For moredetails about strictly proper PID controllers, see [23].

A set of simple tuning rules for PIDF controllers for stableplants can be formulated as

Tuning rules for PIDF controllers

= 10, = 0.8, = T 63/ 3, f = 0.4. (17)

If more or less control activity is preferable, adjust .

Finally, adjustk i to the desired damping of a closed-loop step response.

The simple design method for the PIDF controller can becompared to the H loop-shaping strategy described in [42].The main idea in H loop-shaping is to augment the plantwith a weight functionW and modifyW until a desired openloop shape is obtained for the augmented plantG = WG. Theresulting controller is then combined with the weight functionto obtain the final controller. When a PID filter is used for theweight function, the results of the PIDF controller and the H controller are almost equal. For details, see [41].

CONCLUSIONS

This article presents some easily understood and applied meth-ods for close-to-optimal tuning PI and PID controllers. Byopti-mal we mean good MF robustness and the best possible tradeoff between output performance and control activity. For plantswith all poles on the negative real axis, a simple step response

can provide adequate plant knowledge. For plants with integralaction, an impulse response or a relay experiment can be used.In all PID cases, the controller zeros can be fixed, the controlactivity can be varied by the filter factor, and, finally, the inte-gral gain can be adjusted to the required damping of a stepresponse for the closed-loop system. We maintain that, withthis strategy, tuning a PID controller is as easy as tuning a PIcontroller, the difference being that the PID solution gives addi-tional freedom for selecting slightly higher control activity,which significantly improves the output performance.

When the situation demands HF rolloff of the controller,the PI or PID controller can be augmented by an additionallowpass filter. In these cases, the inclusion of derivative action

is recommended. Four of the five controller parameters arethen easily found, and the remaining gain can be manuallytuned to obtain a desired tradeoff between output perfor-mance and damping (MF robustness).

The major advantage of these tuning rules, compared tothe previous rules, is that derivative action can be used with-out introducing high sensitivity to sensor noise in the controlsignal, a feature due to the inclusion of the low-pass filter inthe design procedure. This approach can also be extended toinclude parameterized (larger) uncertainties using Horowitz bounds or -analysis [43].

ACKNOWLEDGMENTS

Prof. Karl-Johan strm and Prof. Sigurd Skogestad have both provided valuable comments on this manuscript, forwhich the authors are most appreciative.

AUTHOR INFORMATION

Birgitta Kristiansson ([email protected])received the M.Sc. degree in engineering physics in 1965 andthe Ph.D. in automatic control in 2003 from Chalmers Universi-ty of Technology, Gteborg, Sweden. Her first major assign-ment was with the Swedish Council for Research Financing.


FIGURE 16 PI and PID controllers with additional filter action. Anextra lowpass filter in a PI or a PID controller can reduce the HFgain, and hence the sensitivity to sensor noise, with little deteriora-tion of the LF performance. This property is illustrated by the con-troller gain |K | and the disturbance step responses for three exactlyproper controllers and four strictly proper controllers, where k islimited to 20 and 80, respectively. The plant here isG (s ) = e 0 .3s /( 1 + s )2 .

10 1 10 1 10 30.2

1

10

|K |

PID Opt

PIDF

PI Opt

PIF

0 2 4 6 8 10 120.1

0

0.1

0.2

0.3

0.4

0.5

t

PIDF

PID Opt

PI Opt

PIFy

(a)

(b)


15/15

Later, she was responsible for technical audiology at theSahlgrenska University Hospital in Gteborg, after which shetaught control engineering and related subjects at the under-graduate and graduate levels. She is currently a senior lecturerin control engineering at Chalmers, and her research interestsrange from process and robust control to the design of PID con-trollers. She can be contacted at the Department of Signals andSystems, Campus Lindholmen, P.O. Box 8873, SE-402 72, Gte- borg, Sweden.

Bengt Lennartson received the M.Sc. degree in engineeringphysics in 1979 and the Ph.D. in automatic control in 1986 fromChalmers University of Technology, Gteborg, Sweden. In 1989,he became an associate professor at the Control Engineering Labo-ratory at Chalmers. Since 1999, he has been professor and chair of Automation in the Department of Signals and Systems. From19982004, he was vice dean of the School of Mechanical Engineer-ing; he is now dean of Education at Chalmers. His principle areasof research include robust, sampled-data, and PID control, in addi-tion to discrete event and hybrid systems for manufacturing appli-

cations. He has been a member of IFACs Technical Committee onManufacturing and an associate editor of Automatica. He is thesupervisor for six Ph.D. candidates, and he is the coauthor of two books and over 100 peer-reviewed international papers.

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