Maximum Sensitivity Based Robust Tuning for Two-Degree-of

Ind. Eng. Chem. Res. 2010, 49, 5415–5423

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Maximum Sensitivity Based Robust Tuning for Two-Degree-of-Freedom Proportional-Integral Controllers V. M. Alfaro,*,† R. Vilanova,*,‡ and O. Arrieta*,‡ Departamento de Automa´tica, Escuela de Ingenierıá Elećtrica UniVersidad de Costa Rica, San Jose´ 11501-2060, Costa Rica, and Departament de Telecomunicacio´ i d’Enginyeria de Sistemes, UniVersitat Auto`noma de Barcelona, 08193 Bellaterra, Barcelona, Spain

A tuning method for two-degree-of-freedom (2-DoF) proportional-integral (PI) controllers for first-orderplus-dead-time (FOPDT) models is presented. It allows the designer to deal with the closed-loop control system performance-robustness trade-off specifying the lowest robustness level allowed by selecting a maximum sensitivity in the 1.4-2.0 range. In addition, a smooth performance is obtained by forcing the regulatory and servo-control closed-loop transfer functions to perform as close as possible to a target nonoscillatory dynamics. Controller tuning rules are provided for FOPDT models with normalized deadtimes from 0.1 to 2.0. Comparative examples show the effectiveness of the proposed tuning method. Introduction As has been widely reported, proportional-integral-derivative (PID) type controllers are with no doubt, the controllers most extensively used in the process industry, most of them actually being proportional-integral controllers (PI). Their success is mainly due to their simple structure which made it easier to understand for the control engineer than other more advanced control approaches. Since Ziegler and Nichols1 presented their PID controller tuning rules, a great number of other procedures have been developed as revealed in O’Dwyer’s review.2 Some of them consider only the system’s performance such as the classical tuning methods of Lo´pez et al.3 and Rovira et al.,4 its robustness by authors such as Åstro¨m and Ha¨gglund5 and Ho,6 or a combination of performance and robustness.7–10 Recently, tuning methods based on optimization approaches with the aim of ensuring robust stability have received attention in the literature.11,12 Also, great advances on optimal methods based on stabilizing PID solutions have been achieved.13–15 However these methods, although effective, rely on somewhat complex numerical optimization procedures and do not provide autotuning rules. Instead, the tuning of the controller is defined as the solution of the optimization problem. In industrial process control applications, the set-point normally remains constant and good load-disturbance rejection is required, usually known as regulatory-control. In addition due to process operation conditions, eventually the set-point may need to be changed and then a good transient response to this change is required, known as serVo-control operation. Attempts to get a compromise design by using one-degree-of-freedom (1-DoF) controllers have been recently reported,16 where a serVo-regulatory control trade-off tuning is proposed. However, because these two demands cannot be simultaneously satisfied with a one-degree-of-freedom controller, the use of a twodegree-of-freedom (2-DoF) controller allows one to tune the controller considering the regulatory control closed-loop performance and robustness and use the extra parameter that it provides to improve the servo-control behavior. * To whom correspondence should be addressed. E-mail: Victor. [email protected] (V.M.A.); [email protected] (R.V.); Orlando. [email protected] (O.A.). † Universidad de Costa Rica. ‡ Universitat Auto`noma de Barcelona.

This second degree of freedom is aimed at providing additional flexibility to the control system design. See for example Araki’s work17–19 and its characteristics revised and summarized,20,21 as well as different tuning methods that have been formulated over the last years.22–27 The control system design procedure is usually based on the use of low-order linear models identified at the desired closedloop control system normal operating point. Due to the nonlinear characteristics found in most of the industrial process, it is necessary to consider the expected changes in the process characteristics assuming certain relative stability margins, or robustness requirements, for the resulting control system. Therefore the design of the closed-loop control system with 2-DoF PI controllers must take into account the system performance to load-disturbances and set-point changes as well as its robustness to variation on the controlled process characteristics. There are methods such as the kappa-tau28 and AMIGO29 of Åstro¨m and Ha¨gglund that provide tuning rules to design 2-DoF PI closed-loop control systems subject to a robust constraint but for only one or two maximum sensitivity values. The kappa-tau tuning method is based on an empirical closedloop pole placement technique applied to a batch of plants to obtain closed-loop system with robustness corresponding to Ms ) 2.0 and 1.4. The controlled plants in the batch are characterized with their gain, time constant, and dead-time obtained from the processes reaction curves or by their critical parameters. The AMIGO (Approximated MIGO) method presents simple tuning rules obtained from maximization of the controller’s integral gain for minimization of the integrated error to a step load-disturbance, subject to a maximum sensitivity restriction applied to the same batch of plants. The design robustness level is set to Ms ) 1.4. An alternative tuning method for 2-DoF PI controllers is presented in this communication. This approach explicitly considers the control system performance-robustness trade-off aiming to obtain a smooth response to both disturbance and setpoint step changes and at the same time to guarantee a minimum robustness level. The distinctive feature of the resulting tuning procedure is that the designer may select one of four different robustness levels in the range 1.4 e Ms e 2.0 for first-orderplus-dead-time controlled process models with a normalized dead-time in the range 0.10 e τo e 2.0.

10.1021/ie901617y  2010 American Chemical Society Published on Web 04/30/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010

Figure 1. Closed-loop control system.

This second degree of freedom is found in the presented literature as well as in commercial PID controllers under the form of the well-known set-point weight factor (usually called β) that ranges within 0 e β e 1.0, being the main purpose of this parameter to avoid excessive proportional control action with controllers with high gain when a reference change takes place; therefore, using just a fraction of the reference. When the use of a generic reference processing is possible, other approaches may provide better results. See for example the works of refs 30–35 for 2-DoF controller design on a general setting. The rest of the paper is organized as follows: the transfer functions of the controlled process model, the controller, and the closed-loop control system are presented in the next section; then, a general robust tuning method is obtained and the proposed specific maximum sensitivity robustness level tuning rules are described. Comparative examples are provided. The paper ends with some conclusions and suggestions for future research.

Figure 2. Control system with a two-degree-of-freedom controller.

PI2: 2-DoF Proportional-Integral Controller. The process will be controlled with a two-degree-of-freedom proportionalintegral controller (PI2)28 whose output is

{

u(t) ) Kc βr(t) - y(t) +

y(s) ) Myr(s)r(s) + Myd(s)d(s)

(1)

where Myr(s) is the transfer function from the set-point to the controlled process variable, the servo-control closed-loop transfer function, and Myd(s) is the one from the load-disturbance to the controlled process variable, the regulatory-control closedloop transfer function. The regulatory-control main objective is the load-disturbance rejection, this is, to return the controlled variable to its setpoint if a disturbance enter to the control system. For the servocontrol, it is intended to follow a set-point change, this is, to bring the controlled variable to its new set-point. These two different responses will depend on the closed-loop transfer functions in eq 1, that may not be selected independently if a one-degree-of-freedom (1-DoF) controller is used but may be selected with a constrained independence in the case that a twodegree-of-freedom (2-DoF) controller is used. Controlled Process Model. It is assumed, as it is usual, that the controlled process dynamics could be adequately reproduced by a first-order-plus-dead-time (FOPDT) model given by

∫ [r(τ) - y(τ)] dτ} t

0

(3)

or

{

u(s) ) Kc βr(s) - y(s) +

}

1 [r(s) - y(s)] Tis

(4)

While the previous expressions correspond to the industrial implementation of a 2-DoF PI controller, for mathematical convenience they are rewritten as follows (note that these expressions are not suitable for direct implementation)

(

u(s) ) Kc β +

Problem Formulation Consider the closed-loop control system in Figure 1 where P(s) and C(s) are the controlled process model and the controller transfer function respectively. In this system, r(s) is the setpoint, u(s) is the controller output signal, d(s) is the loaddisturbance, and y(s) is the controlled process variable. The closed-loop control system output, y(s), to a change in its inputs, r(s) and d(s), is given by

1 Ti

)

(

)

1 1 r(s) - Kc 1 + y(s) Tis Tis

(5)

and in a compact form as u(s) ) Cr(s)r(s) - Cy(s)y(s)

(6)

where

(

Cr(s) ) Kc β +

1 Tis

)

(7)

is the set-point controller transfer function and

(

Cy(s) ) Kc 1 +

1 Ti s

)

(8)

is the feedback controller transfer function. In eqs 7 and 8, Kc is the controller gain, Ti is the integral time constant, and β is the set-point weight factor. The closed-loop control system with the 2-DoF PI controller is shown in Figure 2. The servo-control and the regulatory control closed-loop transfer functions in eq 1 are now Myr(s) )

Cr(s)P(s) 1 + Cy(s)P(s)

(9)

Myd(s) )

P(s) 1 + Cy(s)P(s)

(10)

and

which are related by Kpe-Ls , P(s) ) Ts + 1

τo ) L/T

(2)

where Kp, T, and L are the model’s gain, time constant, and dead-time, respectively, and τo is its normalized dead-time. The parameters of eq 2 may be identified from the process reaction curve or from its critical information.29,36

Myr(s) ) Cr(s)Myd(s)

(11)

Closed-Loop Target Transfer Functions. For the development of the proposed tuning method, it is important to have the lowest possible number of design parameters. On that basis, the desired control system responses to load-disturbance and set-point step changes are selected nonoscillatory; for a smooth


response; and with no steady-state error as the ones corresponding to the following closed-loop transfer functions Mtyd(s) )

Kse-Ls (Tcs + 1)2

t Myr (s) )

e Tcs + 1

β ) min

(13)

where K and Tc are the regulatory control closed-loop transfer function static gain and time constant, respectively. If Tc is expressed as a function of the controlled process model time constant (Tc ) τcT), then τc ) Tc/T may be used as the nondimensional design parameter. The tuning method will require only one design parameter, τc, that is an indication of the relative closed-loop system response speed with respect to the controlled process speed. Using (12) and (13) in (1), the global target control system output yt(s) is computed as yt(s) )

Kse-Ls e-Ls r(s) + d(s) τcTs + 1 (τcTs + 1)2

(14)

Controller Design. The controller design is approached in two stages. First, the feedback controller (8) parameters (Kc, Ti) required to obtain the target regulatory control closed-loop transfer function (12) are determined for a FOPDT model with several normalized dead-times τo and design parameters τc. On a second step, the free parameter, β, in the set-point controller (7) is used to adjust the servo-control closed-loop transfer function to the target (13). Regulatory Control. It can be shown that the regulatory control closed-loop transfer function gain K in (12) is given by K ) Ti/Kc. Then, the target regulatory control closed-loop transfer function (12) can be expressed as Mtyd(s) )

Tise-Ls Kc(τcTs + 1)2

∫

∞

0

[ytd(pj, t) - yd(pj, t)]2 dt

(βTis + 1)e-Ls (τcTs + 1)2

(16)

(17)

which can be made equal to the servo-control target transfer function (13) if the set-point weight factor is selected as β)

τcT Ti

τcT ,1 Ti

(18)

(19)

In most of the published tuning rules for 2-DoF PI/PID controllers, the above constraint on the proportional set-point weight is not explicitly included, but as per authors knowledge up to this date all the manufacturers of available commercial 2-DoF PID controllers allow selection of this weight only in the 0-1 range. This will reduce the servo-control performance obtained when a highly robust regulatory control is required as pointed out by the authors.37 Optimization Results. The optimization problem setup has considered model’s normalized dead-times, τo, and the design parameters, τc, in the range 0.1 e (τo, τc) e 2.0, to obtain the controller optimum parameters pjo )[Kco, Tio] such that Jdo ) Jd(pjo) ) min Jd(pj) p¯

(20)

Note that pjo ) pjo(τo, τc). Moreover, for each obtained pjo, the control-loop robustness is measured using the maximum sensitivity Ms. From the results, it is observed that if very slow responses are specified (τc high) for models with very low normalized dead-times, negative controller parameters were obtained. Additionally, it was also found that if very fast responses are specified (τc low) the robustness of the resulting closed-loop was very poor (Ms > 2.0). Considering the above, all cases corresponding to negative controller parameters and maximum sensitivities higher than 2.0, were rejected. Using the controlled process model’s gain Kp and time constant T, the controller’s parameters were normalized as κc ) KcKp, τi )

is to be used, where ytd is the output to a step disturbance of the target regulatory control (15) and yd is the output of the regulatory control (10) for the controlled FOPDT model with a 2-DoF PI controller. As it can be seen, functional (16) corresponds to the specification of a model reference target behavior (ytd) for the closed-loop load-disturbance response. Servo-Control. As the closed-loop transfer functions (9) and (10) are related as indicated in (11), the servo-control closedloop transfer function obtained from the regulatory control target transfer function (15) is Myr(s) )

{ }

(15)

To obtain the controller parameters pj ) [Kc, Ti], the cost functional Jd(pj) )

As in commercial controllers, the set-point weight factor adjustment is restricted to have values lower than or equal to 1, its selection criteria is finally stated as

(12)

-Ls

5417

Ti T

(21)

Figure 3 shows the optimized controller’s parameters κc, τi, and β versus the desired closed-loop speed τc for 12 of the 20 normalized dead-times considered. As it can be seen, high normalized dead-times have practically no effect over the controller gain, but produce big changes on the integral time constant. Note that this situation is also observed for the robustness, Ms, of the closed-loop system in all the τc range as it is shown in Figure 3 too. If the controlled process normalized dead-time increases, a slow closed-loop response needs to be specified to obtain the same robustness. It can also be seen in Figure 3 that if very slow closed-loop systems are specified, a set-point weigh factor β > 1 may be required to achieve the target servo-control closed-loop transfer function.37 From the above information, it is possible to obtain the controller’s parameters and the control system robustness as a function of the model (τo) and design (τc) parameters. However, to simplify the design procedure, the controller’s parameters were obtained directly as a function of only the maximum sensitivity Ms. Effectively, the controller’s parameters in the optimized set for each τok ∈ [0.1, 2.0] analyzed, can be expressed as a function of τok and τc as a functional of the form (from Figure 3)

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Figure 3. PI controller parameters and robustness vs τc.

(κck, τik, βk) ) f(τok, τc)

(22)

as well as the obtained control system robustness (from Figure 3) Msk ) g(τok, τc)

(23)

Combining (22) and (23), controller’s normalized parameters may be expressed directly as a function of the closed-loop robustness as (κck, τik, βk) ) h(τok, Ms)

(24)

by changing the design parameter from the closed-loop relative speed τc to the control system robustness Ms. These new relations for each normalized dead-time are shown in Figures 4-6. The designer may resolve the performance-robustness tradeoff by selecting the lowest robustness allowed for the control system according to the expected variation of the controlled process parameters. This will give the designer a regulatory closed-loop control system behavior with the highest speed that can be obtained for the specified minimum robustness. Tuning Equations. Using the information given in Figures 4-6, a set of equations for the controller’s normalized parameters κc, τi, and the set-point weight factor β have been adjusted as a function of the robustness Ms for each one of the considered normalized model dead-times τo.

Figure 4. Controller’s normalized gain.

κc(τo, Ms) ) a0 + a1Ms + a2Ms2 + a3Ms3

(25)

τi(τo, Ms) ) b0 + b1Ms + b2Ms2 + b3Ms3

(26)

β(τo, Ms) ) c0 + c1Ms + c2Ms2 + c3Ms3

(27)

Tables 1-3 show the ai, bi, and ci constants for eqs 25-27 in terms of the corresponding τo. Due to its non-oscillatory


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Table 1. Controller Kc (25) Constants

Figure 5. Controller’s normalized integral time.

τo

a0

a1

a2

a3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

-21.82 -4.304 -8.375 -6.092 -5.779 -4.928 -8.351 -8.216 -8.707 -7.468 -7.033 -5.802 -5.497 -4.499 -4.979 -6.876 -5.893 -5.506 -5.610 -4.929

35.04 3.706 13.42 9.387 9.103 7.309 13.91 13.65 14.63 12.32 11.52 9.247 8.735 6.922 7.914 11.35 9.513 8.827 9.014 7.831

-15.93 1.498 -6.112 -3.973 -4.101 -2.940 -7.178 -7.073 -7.753 -6.376 -5.920 -4.555 -4.293 -3.221 -3.898 -5.961 -4.833 -4.437 -4.556 -3.879

2.733 -0.7238 1.034 0.6242 0.6829 0.4195 1.298 1.280 1.427 1.152 1.061 0.7881 0.7421 0.5301 0.6773 1.083 0.8538 0.7764 0.7999 0.6702

Table 2. Controller τi (26) Constants τo

b0

b1

b2

b3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

6.994 1.495 -1.529 -5.042 -6.467 -10.95 -16.98 -20.82 -23.21 -22.46 -22.21 -19.97 -19.49 -17.61 -17.84 -20.21 -18.92 -17.54 -17.22 -14.72

-10.28 0.1743 5.279 11.38 13.54 21.51 32.18 38.94 42.88 41.08 40.19 35.70 34.62 30.97 31.37 35.45 32.92 30.35 29.68 25.18

5.526 -0.7406 -3.444 -6.877 -7.899 -12.57 -18.78 -22.72 -24.85 -23.52 -22.76 -19.82 -19.06 -16.73 -16.96 -19.26 -17.64 -16.04 -15.58 -12.88

-1.011 0.2307 0.7081 1.357 1.517 2.431 3.634 4.404 4.788 4.489 4.300 3.679 3.514 3.034 3.083 3.512 3.175 2.853 2.756 2.221

Table 3. Controller β (27) Constants

Figure 6. Controller’s set-point weight factor.

performance specification, the 2-DoF PI robust tuning method is referred to as NORT.38 Robustness Specific Simple Tuning Rules Tuning eqs 25-27 allows the design of the control system with any desired robustness level in the range Ms ∈ [1.2, 2.0] and to analyze the robustness-performance trade-off. However, in practice normally only a few discrete robustness levels are used. With this respect, as the Ms value is being established as a de facto standard measure of robustness, an Ms value of 2.0 is recognized as the minimum acceptable robustness level. This corresponds to the classical (Am g 2, φm g 30°) relative (gain and phase) stability margins specification. Then Ms ) 2.0 will be considered as the minimum degree of robustness, and in order to make the analysis simple, Ms ) 1.8 will be a low, Ms ) 1.6 a medium, and Ms ) 1.4 will be a high level of robustness. This broad classification allows a qualitative specification of the control system robustness.29 Simplified Tuning Rules. Using the information given in Figures 4-6, the controller’s parameters for the four different robustness levels Ms ) {1.4, 1.6, 1.8, 2.0} were obtained and shown in Figures 7-9. This information was used to obtain the new PI2 simplified Ms robust tuning (PI2Ms) equations

τo

c0

c1

c2

c3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

5.132 7.918 11.18 11.36 11.70 14.19 16.28 15.67 16.51 15.08 12.74 12.94 9.426 15.08 11.47 10.29 13.28 14.97 15.94 12.25

-7.699 -12.02 -17.64 -17.30 -17.44 -21.26 -25.04 -23.36 -24.55 -21.72 -17.49 -17.52 -11.27 -20.57 -14.44 -12.85 -17.55 -20.35 -21.97 -15.90

4.306 6.573 9.876 9.385 9.280 11.23 13.54 12.28 12.87 11.10 8.618 8.501 4.853 9.959 6.539 5.885 8.338 9.892 10.80 7.503

-0.8184 -1.216 -1.865 -1.724 -1.676 -2.007 -2.477 -2.187 -2.287 -1.928 -1.449 -1.406 -0.7054 -1.638 -1.009 -0.9308 -1.355 -1.642 -1.812 -1.218

κc(τo, Ms) ) a0′(Ms) + a1′(Ms)τao2′(Ms)

(28)

τi(τo, Ms) ) b0′(Ms) + b1′(Ms)τbo2′(Ms)

(29)

β(τo, Ms) ) c0′(Ms) + c1′(Ms)τco2′(Ms)

(30)

where constants {ai′, bi′, ci′} are shown in Tables 4-6. The parameters obtained using (28)-(30) are also shown in Figures 7-9 and labeled with “(eq.)”. Note that the highest robustness (Ms ) 1.4) may only be obtained for τo e 1.7.

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Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010 Table 4. Controller Kc Constants Ms

a0′

a1′

a2′

τ0 range

1.4 1.6 1.8 2.0

0.0674 0.1687 0.2118 0.3208

0.3775 0.4724 0.5633 0.5613

-0.9623 -0.9805 -0.9823 -1.0380

0.1-1.7 0.1-2.0 0.1-2.0 0.1-2.0

Table 5. Controller τi Constants Ms

b0′

b1′

b2′

τ0 range

1.4 1.6 1.8 2.0

1.440 8.672 -3.952 -2.105

-0.1744 -7.247 5.426 3.595

-0.659 -0.04929 0.08661 0.1476

0.1-1.7 0.1-2.0 0.1-2.0 0.1-2.0

Table 6. Controller β Constants

Figure 7. Normalized controller gain.

Ms

c0′

c1′

c2′

τ0 range

1.4 1.6 1.8 2.0

0.3803 0.2611 0.2296 0.2107

0.7794 0.5763 0.4711 0.4043

0.6851 0.4345 0.3588 0.3122

0.1-1.7 0.1-2.0 0.1-2.0 0.1-2.0

range. Then in such cases, the servo-control performance will be lower than the one that may be obtained using the recommended proportional set-point weight factor. The new tuning eqs 28-30 are more easy to use than the original ones (25)-(27). Examples Example 1. Consider the FOPDT model P1(s) )

Figure 8. Normalized controller integral time.

e-Ls , s+1

L ) {0.4, 0.8, 1.2, 1.6}

The PI2Ms controller’s parameters for different design robustness (Msd) are shown in Table 7 along with the achieved robustness (Msr). The control system response to a 20% set-point step change followed by a 10% load-disturbance step of such systems are shown in Figures 10 and 11. It was supposed that at its normal operation point all the system variables (r, u, and y) are at 70% of their normal operation range. As can be seen, PI2Ms eqs 28-30 produce a 2-DoF PI control system with the specified robustness (Msr ≈ Msd) for all the normalized dead-times used in the example. Figures 10 and 11 also show the existing trade-off between performance and robustness; if the control system robustness is increased, its response speed is reduced. They also show that the model deadTable 7. Example 1: PI2Ms Parameters and Robustness

Figure 9. Controller set-point weight factor.

From Figure 9, it may also be noted that when a highly robust control system is required (Ms < 1.6) and the model normalized dead-time τo is high, a proportional set-point weight β > 1 is required. Up to this moment, the commercially available PI2 controllers allow selection of this weight factor only in the 0-1

τo

Msd

Kc

Ti

β

Msr

0.4 0.4 0.4 0.4 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 1.6 1.6 1.6 1.6

1.4 1.6 1.8 2.0 1.4 1.6 1.8 2.0 1.4 1.6 1.8 2.0 1.4 1.6 1.6 2.0

0.979 1.329 1.597 1.774 0.535 0.757 0.913 1.028 0.384 0.564 0.683 0.785 0.308 0.467 0.567 0.665

1.121 1.091 1.060 1.035 1.238 1.345 1.370 1.374 1.285 1.490 1.560 1.588 1.312 1.591 1.699 1.748

0.796 0.648 0.569 0.514 1 (1.049) 0.784 0.665 0.588 1 (1.263) 0.885 0.733 0.639 1 (1.456) 0.968 0.787 0.679

1.403 1.615 1.819 2.000 1.393 1.588 1.819 1.990 1.400 1.597 1.787 2.022 1.398 1.608 1.798 2.044


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Table 8. Example 1: PI2Ms Regulatory-Control and Servo-Control Performance τo

Msd

Jed/∆d

Jud/∆d

TVud/∆d

Jer/∆r

Jur/∆r

TVur/∆r

0.4 0.4 0.4 0.4 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 1.6 1.6 1.6 1.6

1.4 1.6 1.8 2.0 1.4 1.6 1.8 2.0 1.4 1.6 1.8 2.0 1.4 1.6 1.8 2.0

1.145 0.821 0.664 0.584 2.315 1.778 1.502 1.338 3.345 2.643 2.285 2.025 4.252 3.405 2.997 2.638

1.145 0.848 0.830 0.857 3.313 1.777 1.572 1.577 3.336 2.637 2.298 2.261 4.225 3.389 2.985 2.913

1.027 1.118 1.351 1.560 1.032 1.071 1.274 1.501 1.036 1.058 1.233 1.490 1.037 1.057 1.210 1.497

1.374 1.205 1.121 1.086 2.314 2.067 1.960 1.903 3.345 2.813 2.700 2.596 4.255 3.456 3.356 3.188

0.166 0.300 0.422 0.508 0.514 0.369 0.437 0.515 1.142 0.611 0.599 0.666 1.646 0.851 0.776 0.841

0.534 0.921 1.313 1.611 0.469 0.676 0.908 1.130 0.616 0.630 0.799 1.015 0.691 0.626 0.762 0.986

Table 9. Example 1: Servo-Control Performance with β > 1

Figure 10. Example 1: System responses (L ) 0.40).

τo

Msd

β

Jer/∆r

Jur/∆r

TVur/∆r

0.8 1.2 1.6

1.4 1.4 1.4

1.049 1.263 1.456

2.253 3.008 3.659

0.453 0.806 1.053

0.460 0.515 0.551

a defined trend in the other cases, and the control signal turns smoother (TVud and TVur decrease). Additionally, Table 9 shows the servo-control performance increment, the reduction in the control cost, and the control smoothness increment in those cases where the recommended proportional set-point weight (30) β > 1 is used not restraining it to the commercial controller limit of 1. Example 2. Consider the following fourth-order controlled process P2(s) )

1 3

∏ (σ s + 1) n

n)0

Figure 11. Example 1: System responses (L ) 1.60).

time has an adverse effect over the closed-loop system performance. The performance of the obtained control systems was evaluated with the cost functional Je z

∫

∞

0

|r(t) - y(t)| dt

(31)

the system control cost with the functional Ju z

∫

∞

0

|u(∞) - u(t)| dt

(32)

and the control smoothness with its total variation ∞

TVu z

∑ |u(k + 1) - u(k)|

(33)

k)1

Cost functionals are computed for both set-point (r) and disturbance (d) step changes. The evaluated indexes (31)-(33) for the regulatory-control and servo-control are shown in Table 8. For a FOPDT process with a given dead-time, if the control system robustness Ms is increased (decreasing Ms) its performance decreases (Jed and Jer increase), its regulatory control cost Jud increases, its servo-control cost Jur decreases if the process normalized dead-time is very low but it does not have

with σ ) {0.25, 0.50, 1.0} taken from the benchmark proposal.39 Using a two-point identification procedure,40 FOPDT models were obtained whose parameters are show in Table 10. These parameters will be the ones used for tuning the PID controllers. Table 11 shows the 2-DoF PI controller parameters whereas in Table 12 the specified (Msd) and achieved robustness of the control system (Msrm with the model and Msrp with the process) are presented when the controller is tuned with kappa-tau28 and AMIGO29 rules and the corresponding parameters obtained with the proposed PI2Ms method. The control system’s and controller’s outputs with PI2Ms controllers are shown in Figure 12 for the case of σ ) 1.0. Besides allowing the selection of a design value for the control system robustness, Ms, in the 1.4-2.0 range, the closedloop system robustness obtained with the proposed PI2Ms method are closer to the desired values than the ones obtained with the kappa-tau and AMIGO tuning methods. Control systems performance, control cost, and smoothness are shown in Tables 13-15 for each case of σ. On both cases, regulatory and servo control, the following observations are to be highlighted. Table 10. Example 2: Process Model Parameters σ

Kp

T

L

0.25 0.50 1.0

1.0 1.0 1.0

1.049 1.247 2.343

0.298 0.691 1.860

5422


Table 11. Example 2: Controller Parameters σ ) 0.25

σ ) 0.5

σ ) 1.0

method

Msd

Kc

Ti

β

Kc

Ti

β

Kc

Ti

β

kappa-tau kappa-tau AMIGO

1.4 2.0 1.4

0.673 1.465 0.776

0.684 0.684 0.883

1.000 0.512 0.000

0.320 0.674 0.368

0.811 0.811 1.159

1.000 0.549 0.000

0.228 0.489 0.280

1.594 1.594 2.270

1.000 0.569 0.000

PI2Ms

1.4 1.6 1.8 2.0

1.335 1.791 2.151 2.393

1.091 1.008 0.958 0.924

0.709 0.595 0.530 0.484

0.734 1.011 1.218 1.357

1.475 1.510 1.501 1.484

0.900 0.707 0.611 0.547

0.539 0.761 0.919 1.034

2.898 1.145 3.202 3.209

1.046 0.782 0.663 0.587

Table 12. Example 2: Controller Robustness σ ) 0.25

σ ) 0.5

σ ) 1.0

method

Msd

Msrm

Msrp

Msrm

Msrp

Msrm

Msrp


1.4 2.0 1.4

1.27 1.59 1.24

1.26 1.46 1.21

1.29 1.66 1.21

1.30 1.67 1.21

1.29 1.71 1.23

1.31 1.77 1.24

PI2Ms

1.4 1.6 1.8 2.0

1.39 1.61 1.77 2.00

1.29 1.41 1.49 1.54

1.40 1.61 1.82 1.99

1.35 1.49 1.65 1.76

1.40 1.60 1.82 1.99

1.39 1.59 1.76 1.90

• If we concentrate on designs with an identical robustness specification (say same value of Msd), on both cases (Msd ) 1.4 and Msd ) 2.0) the PI2Ms performance is better than its

Figure 12. Example 2: PI2Ms Tuning (σ ) 1.0).

Table 15. Example 2: Regulatory-Control and Servo-Control Performance (σ ) 1.0) method

Msd

Jed/∆d

Jud/∆d

TVud/∆d

Jer/∆r

Jur/∆r

TVur/∆r


1.4 2.0 1.4

6.98 4.38 8.10

6.98 5.09 8.10

1.00 1.47 1.00

6.98 5.40 10.37

2.98 2.34 6.37

0.77 1.34 1.00

PI2Ms

1.4 1.6 1.8 2.0

5.38 4.13 3.49 3.13

5.38 4.14 3.88 3.93

1.00 1.13 1.34 1.55

5.25 4.82 4.56 4.43

1.25 1.00 1.15 1.38

0.50 0.71 0.93 1.14

counterparts. Specially for a highly robust system (Msd ) 1.4). The PI2Ms performance improvement is around 20-50% depending of the case, giving also better control signal smoothness. • Another reading of the information supplied by these tables is also possible if we concentrate on the Msd ) 1.6 robustness level. This value for Msd is not considered on the AMIGO and kappa-tau tunings, but from the provided information, it is effectively verified that it constitutes a good candidate for a better robustess/performance tradeoff. Effectively, if we compare the PI2Ms Msd ) 1.6 with respect to the kappa-tau Msd ) 2.0, the achieved PI2Ms performance levels are along the same lines (or even better) while providing a higher robustness margin. • With reference to the behavior of the system respect to the σ value, from the data it is possible to see that if σ increases (i.e., the system becomes slower), the achieved performance decreases uniformly for all the tunings. On the other hand, the control signal smoothness, TV, remains along similar values for the three cases of σ, being the reason for this the imposed constraint for the system robustness. This is an indicator of the close link among robustness and control effort.


Msd

Jed/∆d

Jud/∆d

TVud/∆d

Jer/∆r

Jur/∆r

TVur/∆r


1.4 2.0 1.4

1.09 0.52 1.14

1.25 0.87 1.19

1.12 1.38 1.02

1.25 0.95 2.03

0.47 0.66 0.79

0.72 1.40 1.04

PI2Ms

1.4 1.6 1.8 2.0

0.82 0.56 0.45 0.39

0.82 0.69 0.66 0.65

1.01 1.16 1.30 1.41

1.13 0.97 0.90 0.86

0.24 0.42 0.54 0.60

0.56 0.92 1.24 1.48


Msd

Jed/∆d

Jud/∆d

TVud/∆d

Jer/∆r

Jur/∆r

TVur/∆r


1.4 2.0 1.4

2.60 1.56 3.15

2.65 2.01 3.15

1.03 1.45 1.00

2.65 2.16 4.31

0.87 1.05 2.44

0.74 1.39 1.00

PI2Ms

1.4 1.6 1.8 2.0

2.01 1.49 1.23 1.09

2.01 1.53 1.45 1.46

1.00 1.12 1.30 1.45

2.16 1.94 1.82 1.77

0.31 0.39 0.52 0.62

0.48 0.72 0.97 1.16

Figure 13. Example 2: Control system responses for kappa-tau (K-T), AMIGO, and PI2Ms tunings with Msd ) 1.4 (σ ) 0.50).

Ind. Eng. Chem. Res., Vol. 49, No. 11, 2010 d

Figure 13 compares the kappa-tau, AMIGO, and PI2Ms (Ms ) 1.4) tuned control systems, for the σ ) 0.50 case, showing the effectiveness of the proposed method. Conclusions

Tuning rules have been developed for two-degree-of-freedom (2-DoF) PI controllers for first-order-plus-dead-time (FOPDT) models that consider the closed-loop performance-robustness trade-off. The proposed PI2Ms method allows the designer to design closed-loop control systems with a specified minimum, low, medium, or high robustness level measured using the maximum sensitivity Ms. The performance-robustness trade-off is tackled by selecting the lowest closed-loop robustness allowed, considering the expected variation of the controlled process dynamics, resulting in the faster nonoscillatory response to a load-disturbance step. The used examples show the advantages of the application of a tuned two-degree-of-freedom controller taking into account the control system robustness requirements. Future research efforts are conducted on the extension of the proposed approach to second-order-plus-dead-time (SOPDT) models and 2-DoF PID controllers and trying to get full autotuning formulas in order to get all the controller’s parameters. Acknowledgment This work has received financial support from the Spanish CICYT program under grant DPI2007-63356. Also, the financial support from the University of Costa Rica and from the MICIT and CONICIT of the Government of the Republic of Costa Rica is greatly appreciated. Literature Cited (1) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. ASME Trans. 1942, 64, 759–768. (2) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules, 2nd ed.; Imperial College Press: London, UK, 2006. (3) Lo´pez, A. M.; Miller, J. A.; Smith, C. L.; Murrill, P. W. Tuning Controllers with Error-Integral Criteria. Instrument. Technol. 1967, 14, 57– 62. (4) Rovira, A.; Murrill, P. W.; Smith, C. L. Tuning Controllers for Setpoint Changes. Instrument. Control Syst. 1969, 42, 67–69. (5) Åstro¨m, K. J; Ha¨gglund, T. Automatic Tuning of Simple Regulators with Specification on Phase and Amplitude Margins. Automatica 1984, 20 (5), 645–651. (6) Ho, W.-K.; Hang, C.-C.; Cao, L. S. Tuning PID Controllers Based on Gain and Phase Margin Specifications. Automatica 1995, 31 (3), 497– 502. (7) Voda, A. A.; Landau, L. D. A Method for the Auto-calibration of PID Controllers. Automatica 1995, 31 (1), 41–53. (8) Ho, W. K.; Lim, K. L.; Hang, C. C.; Ni, L. Y. Getting more Phase Margin and Performance out of PID controllers. Automatica 1999, 35, 1579– 1585. (9) Vilanova, R. IMC based Robust PID design: Tuning guidelines and automatic tuning. J. Process Control 2008, 18, 61–70. (10) Ingimundarson, A.; Ha¨gglund, T.; Åstro¨m, K. Criteria for Desing of PID Controllers; Technical Report, ESAII, Universitat Politecnica de Catalunya, (n.d.), 2004. (11) Ge, M.; Chiu, M.; Wang, Q. Robust PID Controller design via LMI approach. J. Process Control 2002, 12, 3–13. (12) Toscano, R. A Simple PI/PID controller design method via numerical optimization approach. J. Process Control 2005, 15, 81–88. (13) Silva, G.; Datta, A.; Battacharayya, S. New Results on the synthesis of PID controllers. IEEE Trans. Automat. Control 2002, 47, 241–252. (14) Ho, M.; Lin, C. PID controller design for Robust Performance. IEEE Trans. Automat. Control 2003, 48, 1404–1409.

5423

(15) Pedret, C.; Vilanova, R.; Moreno, R.; Serra, I. A refinement procedure for PID controller tuning. Comput. Chem. Eng. 2002, 26, 903– 908. (16) Arrieta, O.; Vilanova, R. Servo/Regulation tradeoff tuning of PID controllers with a robustness consideration. CDC07, 46th IEEE Conference on Decision and Control, New Orleans, LA, December 12-14, 2007. (17) Araki, M. On Two-Degree-of-Freedom PID Control System; Technical Report, SICE Research Commitee on Modeling and Control Design of Real Systems, 1984. (18) Araki, M. PID Control Systems with Reference Feedforward (PIDFF Control System) Proceedings of the 23rd SICE Anual Conference, 1984; pp 31-32. (19) Araki, M. Two-Degree-of-Freedom Control System - I. Syst. Control 1985, 29, 649–656. (20) Taguchi, H.; Araki, M. Two-Degree-of-Freedom PID Controllers Their Functions and Optimal Tuning. IFAC Digital Control: Past, Present and Future of PID Control, Terrassa, Spain, April 5-7, 2000; pp 91-96. (21) Taguchi, H.; Araki, M. Survey of Researches on Two-Degree-ofFreedom PID Controllers. The 4th Asian Control Conference, Singapore, September 25-27, 2002; pp 1880-1885. (22) Åstro¨m, K.; Hang, C. C.; Persson, P.; Ho, W. K. Towards Intelligent PID Control. Automatica 1992, 28 (1), 1–9. (23) Hang, C.; Cao, L. Improvement of Transient Response by Means of Variable Set Point Weighting. IEEE Trans. Ind. Electr. 1996, 4, 477– 484. (24) Åstro¨m, K.; Panagopoulos, H.; Ha¨gglund, T. Design of PI controllers based on non-convex optimization. Automatica 1998, 34, 585–601. (25) Gorez, R. New Design relations for 2-DOF PID-like control systems. Automatica 2003, 39, 901–908. (26) Åstro¨m, K.; Ha¨gglund, T. Revisiting the Ziegler-Nichols step respose method for PID control. J. Process Control 2004, 14, 635–650. (27) Alfaro, V. M.; Vilanova, R.; Arrieta, O. Analytical Robust Tuning of PI Controllers for First-Order-Plus-Dead-Time Processes. 13th IEEE International Conference on Emerging Technologies and Factory Automation, Hamburg, Germany, September 15-18, 2008. (28) Åstro¨m, K. J.; Ha¨gglund, T. PID Controllers: Theory, Design and Tuning; Instrument Society of America, Research Triangle Park, NC, 1995. (29) Åstro¨m, K.; Ha¨gglund, T. AdVanced PID Control; ISA - The Instrumentation, Systems, and Automation Society, 2006. (30) Leva, A.; Bascetta, L. On the Design of the Feedforward Compensator in Two-Degree-of-Freedom Controllers. Mechatronics 2006, 16 (9), 533–546. (31) Leva, A.; Bascetta, L. Set point tracking optimization by causal nonparametric modelling. Automatica 2007, 11, 1984–1991. (32) Bascetta, L.; Leva, A. FIR based causal design of 2-DOF controllers for optimal set point tracking. J. Process Control 2008, 18, 465–478. (33) Grimble, M. J. Robust Industrial Control. Optimal design Approach for Polynomial Systems; Prentice-Hall International, 1994. (34) Limebeer, D.; Kasenalli, E.; Perkins, J. On the design of Robust Two Degree of Freedom Controllers. Automatica 1993, 29, 157–168. (35) Vilanova, R.; Serra, I.; Pedret, C.; Moreno, R. Optimal reference processing in 2-DOF control. IET-Control Theory Appl. 2007, 1, 1322– 1328. (36) Visioli, A. Practical PID Control; Springer Verlag Advances in Industrial Control Series, 2006. (37) Alfaro, V.; Vilanova, R.; Arrieta, O. Considerations on Set-Point Weighting choice for 2-DoF PID Controllers. IFAC International Symposium on AdVanced Control of Chemical Processes (ADCHEM2009), Istanbul, Turkey, July 12-15, 2009. (38) Alfaro, V.; Vilanova, R.; Arrieta, O. NORT: a Non-Oscillatory Robust Tuning Approach for 2-DoF PI Controllers. IEEE International Conference on Control Applications (CCA2009), St. Petersburg, Russia, July 8-10, 2009. (39) Åstro¨m, K.; Ha¨gglund, T. Benchmark Systems for PID Control. IFAC Digital Control: Past, Present and Future of PID Control, Terrassa, Spain, April 5-7, 2000. (40) Alfaro, V. M. Low-Order Models Identification from Process Reaction Curve. Ciencia y Tecnologıá (Costa Rica) 2006, 24 (2), 197– 216; in Spanish.

ReceiVed for reView October 16, 2009 ReVised manuscript receiVed March 11, 2010 Accepted April 8, 2010 IE901617Y

Maximum Sensitivity Based Robust Tuning for Two-Degree-of

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