Performance and Robustness Considerations for Tuning of

Article pubs.acs.org/IECR

Performance and Robustness Considerations for Tuning of Proportional Integral/Proportional Integral Derivative Controllers with Two Input Filters Víctor M. Alfaro*,† and Ramon Vilanova*,‡ †

Departamento de Automática, Escuela de Ingeniería Eléctrica, Universidad de Costa Rica, San José 11501 2060, Costa Rica Departament de Telecomunicació i Enginyeria de Sistemes, Escola d’Enginyeria, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

‡

ABSTRACT: The aim of this work is to present an extension of the closed-loop model-reference robust tuning (MoReRT) methodology to proportional integral and proportional integral derivative control algorithms enhanced with two input filters (setpoint and feedback signal filters) for control of stable, integrating, inverse response, and unstable controlled processes. The method is based on the use of target models that include two design parameters (closed-loop dominant poles relative speed and damping). The control system performance/robustness trade-off is analyzed to reduce the design parameters to only one directly related with the control system robustness measured with the maximum sensitivity (MS). The proposed design methodology takes into consideration the control system load-disturbance rejection, set-point tracking, control effort smoothness, measurement of high frequency noise attenuation, and robustness to changes on the controlled process dynamics. The incorporation of these two input filters allows taking into consideration two, rarely included, industrial application oriented features: high-frequency roll-off and lack of control effort abrupt changes. Comparisons and examples show that the proposed design strategy can be applied to diverse controlled process models to obtain robust control systems that produce smooth controller outputs without abrupt changes, and with performance comparable to or better than other tuning procedures.

■

INTRODUCTION Since its commercial introduction the proportional integral derivative (PID) control algorithm has, with no doubt, become the best option for industrial control applications,1 with most used as proportional integral (PI) controllers.2 Generally, PI control has been preferred rather than PID by arguing that the derivative mode amplifies measurement noise. This reticence to use the full capabilities of the PID control algorithm may be overcome if the measurement noise filter is taken as part of the controller design.3,4 An extension of the model-reference robust tuning5 unifying PI/PID controller design procedures for stable, integrating, and unstable controlled processes is presented in this work. It is based on the use of PI or PID controllers with two input filters, a feedback filter and a set-point filter, and on the use of a closedloop regulatory-control model-reference optimization procedure. The design takes into account the control system performance, control effort use, measurement of noise attenuation, and robustness. The proposed design procedure for PI and PID controllers with two input filters has its roots on the model-reference robust tuning (MoReRT) methodology earlier applied for robust tuning of two-degrees-of-freedom (2DoF) PI and PID controllers for overdamped,5,6 integrating,7,8 inverse response,9,10 and unstable11,12 controlled processes. The paper is organized as follows. First, the design framework is summarized including the control algorithm and input signals filters. Then, the general controller design methodology is outlined followed by its application to stable, integrating, and unstable controlled processes. Performance and robustness of © 2013 American Chemical Society

the resulting control systems with PI and PID controllers are analyzed.

■

PROBLEM FORMULATION Consider the general closed-loop control system depicted in Figure 1, where P is the controlled process model and Cr and Cy

Figure 1. Closed-loop control system.

are the controllers to design. In this system, r is the set-point, u the controller output signal, d the load disturbance, n the measurement noise, and y the process controlled variable. In general, the controlled process is represented by a linear model by means of a transfer function P(s) with parameters θp. The controller structure must allow for separating the servoReceived: Revised: Accepted: Published: 18287

April 22, 2013 November 25, 2013 November 25, 2013 November 25, 2013 dx.doi.org/10.1021/ie4012694 | Ind. Eng. Chem. Res. 2013, 52, 18287−18302

Industrial & Engineering Chemistry Research

Article

control design from the regulatory-control design; 2DoF are needed.13 Controller Equation. Considering the required features, the proportional integral derivative control algorithm of independent gains (ideal parallel implementation) is selected, whose output signal is given by14 u(s) = K p[r′(s) − y′(s)] +

Fr(s) =

(1)

where Kp is the controller proportional gain, Ki the integral gain, Kd the derivative gain, and γ the derivative mode selector (γ ∈ {0,1}). The set-point r and feedback y signals are filtered15 before they enter the controller. Then r′ and y′ in eq 1 are given by r′(s) = Fr(s) r(s),

y′(s) = Fy(s)[y(s) + n(s)]

Fy(s) =

(2)

Fy(s) =

(3)

(4)

where Cy with parameters θcy′ = {Kp, Ki, Kd} is the controller aspect that operates on the filtered controlled variable y′, the feedback controller; and Cr with parameters θ′cr = {Kp, Ki, γ = 0} is the controller aspect that operates on the filtered set-point r′, the set-point controller (see Figure 1). The closed-loop control system output y as a function of its inputs r, d, and n is y(s) = Myr (s) r(s) + Myd(s) d(s) + Myn(s) n(s)

(5)

where Myr (s) =

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

(6)

is the servo-control closed-loop transfer function, Myd(s) =

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

(7)

the regulatory-control closed-loop transfer function, and Myn(s) =

−P(s) Cy(s) Fy(s) 1 + P(s) Cy(s) Fy(s)

(8)

the measurement noise sensitivity function. The servo- and regulatory-control transfer functions eq 6 and eq 7 are related by Myr (s) = Cr(s) Fr(s) Myd(s)

1 1 = Dfy(s) Tf s + 1

(11)

1 1 = 2 2 Dfy(s) Tf /2s + Tf s + 1

(12)

to provide high-frequency roll-off (measurement noise attenuation) with either controller. The second-order filter damping ratio is selected as 1/√2.15 Input filters transfer functions gains are constrained to be equal, lims→0Fr(s) = lims→0Fy(s), to ensure that in steady state the controller integral action operates on the error signal. For simplicity we select both filters gains to have a value of 1. Considering the two input filters as part of the “controller” to be designed, their combination with the PI and PID are denoted as PI2IF and PID2IF, respectively. The selectable parameters of the set-point controller are θcr = {Kp, Ki, Tr, σ, γ = 0}, and those corresponding to the feedback controller are θcy = {Kp, Ki, Kd, Tf}. Then, parameters of the controller as a whole are θc ≐ θcr ∪ θcy = {Kp, Ki, Kd, Tf, Tr, σ, γ = 0}. The controller output instant change to a set-point step is zero, and then there is not an abrupt control effort change. At industrial process control applications, it is important to avoid a demand for abrupt and large control signals changes to the final control element. This is in order to prevent, for example, excess wear of a control valve or high instant changes in the input power demand of a variable-speed electrical pump, if it is not “softened” by the pump speed control. An abrupt opening of a control valve or a high acceleration of a pump may also originate upstream pressure problems. On the other hand the high-frequency gain of both, PI2IF and PID2IF, controllers tends to zero, providing the required highfrequency noise attenuation. This is thanks to the introduction of the feedback filter. However, in order to minimize the filter impact on the control system performance, its design has to be done as part of the controller tuning. The preceding controller characteristics are obtained using the proposed set-point and feedback filters. Feedback (input signal) filters are normally found in commercial PI/PID controllers, but sometimes they are only of first-order. In this case, highfrequency roll-off of measurement noise is obtained only with PI controllers. If the controller setup does not provide a set-point filter but it is of 2 degrees of freedom with a set-point weight, this can be used to reduce the controller output change in the case of a set-point step. If the controller is of only 1 degree of freedom, the use of a set-point ramp generator is recommended to avoid the abrupt changes in controller output. The lack of a set-point filter has no effect over the control system load-disturbance performance or over its robustness.

In a compact form eq 3 is expressed as u(s) = Cr(s) Fr(s) r(s) − Cy(s) Fy(s)[y(s) + n(s)]

(10)

with time constant Tf, and of second-order for PID controllers, given by

Using eq 2 in eq 1 and selecting γ = 0, in order to avoid the controller output “derivative kick” when a set-point step change occurs, the following is obtained: ⎛ K⎞ u(s) = ⎜K p + i ⎟Fr(s) r(s) ⎝ s ⎠ ⎛ ⎞ K − ⎜K p + i + Kds⎟Fy(s)[y(s) + n(s)] ⎝ ⎠ s

(Trs + 1)2

where Tr is its time constant and σ an adjustable parameter. Filter eq 10 avoids having a step change in the controller output, a “proportional kick”, when a set-point step change is made.15 A simple first-order set-point filter can be obtained using σ = 1. For the feedback f ilter (“noise filter”) Fy(s) we use the structures suggested by Hägglund.15 It is selected of first-order for PI controllers, given by

Ki [r′(s) − y′(s)] s

+ Kds[γr′(s) − y′(s)]

σTrs + 1

(9)

Input Filters Transfer Functions. The set-point f ilter Fr(s) is selected strictly proper and given by the following transfer function: 18288

dx.doi.org/10.1021/ie4012694 | Ind. Eng. Chem. Res. 2013, 52, 18287−18302


Article

Table 1. Normalized Models and Controller/Filter Parameters controller normalized parameters process

K pe

(F)SOPDT

−Ls

(Ts + 1)(aTs + 1) K pe−Ls

IPDT

s

K pe−Ls

ISOPDT

s(Ts + 1) K pe−Ls

UFOPDT

Ts − 1

K p(− bTs + 1)

SOPRHPZ

kp

normalized model

(Ts + 1)(aTs + 1)

ki

kd

τf

τr

−τLs ̂

e (s ̂ + 1)(as ̂ + 1)

KKp

TKKi

KKd/T

Tf/T

Tr/T

e−s ̃ s̃

LKKp

L2KKi

KKd

Tf/L

Tr/L

e−τLs ̂ s (̂ s ̂ + 1)

TKKp

T2KKi

KKd

Tf/T

Tr/T

e−τLs ̂ ŝ − 1

KKp

TKKi

KKd/T

Tf/T

Tr/T

− bs ̂ + 1 (s ̂ + 1)(as ̂ + 1)

KKp

TKKi

KKd/T

Tf/T

Tr/T

τL ≐ L/T

■

CONTROL SYSTEM DESIGN FRAMEWORK Because the disturbance rejection is of main interest, the regulatory-control closed-loop transfer function eq 7, Myd(s), is analyzed first to design Cy(s)Fy(s) as a whole. Controller and Feedback Filter Design. Following the MoReRT design methodology,5 the regulatory-control closedloop transfer function for PI and PID controllers can be expressed by Myd(s) =

best match between the actual control system response and the target one. Set-Point Filter Design. Once the feedback controller Cy(s) and filter Fy(s) are designed, from eq 9 the servo-control closedloop transfer function can be obtained as Myr (s) =

(13)

where N+p (s) is the controlled process model nonminimum phase part and DM(s) is the denominator of all of the control system closed-loop transfer functions. The number of closed-loop poles depends on the controlled process model order and on the controller; PI or PID, and feedback filter combination. The feedback controller Cy(s) and filter Fy(s) design methodology made use of a regulatory-control closed-loop transfer function target for eq 13, Mtyd(s), that depends on the controlled process model nonminimum phase components, on the feedback controller parameters, and on the control system design parameters, θd. For the regulatory-control response model-reference optimization the cost functional to be optimized is defined as follows: Jd (θp , θcy , θd) ≐

∫0

∞

[ydt (θp , θcy , θd ,

° (s ) (Trs + 1)2 DM

(15)

Although usually of secondary interest for process control, in the event of a sporadic set-point step change, it is important to have a fast response with low overshoot and without an abrupt change or high variations of the controller output signal. The lack of any instant change at the controller output is already guaranteed by the above-mentioned selection of both controller and set-point filters. The optimality of the servo-control step response may be obtained by selecting the set-point filter parameters (σ, Tr) to optimize the integrated absolute error (IAE) given by the cost functional

(1/K i)sDfy(s) N p+(s) DM (s)

(σTrs + 1)(K °p /K i°s + 1)D°fy(s) N p+(s)

Jer ≐

∫0

∞

|er(t )| dt =

∫0

∞

|r(t ) − yr (t )| dt

(16)

the integrated absolute control effort (IAU) Jur ≐

2

t ) − yd (θp , θcy , t )] dt

∫0

∞

|ur (t ) − ur (∞)| dt

(17)

or a combination of these two indices. Another approach for the set-point filter design is by selecting its time constant as Tr = Kp°/Ki° in order to cancel one closed-loop zero. Then, making this substitution in eq 15, it reduces to

(14)

where θcy, θd, t) is the step response of the regulatorycontrol closed-loop target transfer function for eq 13 and yd(θp, θcy, t) is that of the regulatory-control system eq 7 with controlled process model P(s), feedback controller Cy(s), and feedback filter Fy(s) eq 11 or eq 12. At Jd eq 14 optimization, the design parameters θd, defined later, are selected in such a way that the control system robustness matches a target value (robust design) measured using the maximum sensitivity MS. The optimization of Jd is a model matching problem, instead of a performance optimization problem as in the traditional PI/PID controllers optimization procedures. The control system behavior is stated by the regulatory-control closed-loop transfer function target and the design parameters θd. The optimization procedure searches for the controller parameters that made the ytd(θp,

Myr (s) =

(σK °p /K i°s + 1)D°fy(s) N p+(s) ° (s ) (K °p /K i°s + 1)DM

(18)

In this case, the remaining filter parameter σ can be selected to optimize a servo-control cost functional such as eq 16 or eq 17. Even though for some situations the set-point filter may be unable to improve the set-point response characteristic, it must be used to avoid the controller output step change. Without a setpoint filter the instant controller output change is Δu° = KpΔr. In such cases a fast (low time constant Tr) first-order filter (σ = 1) can be used but the servo-control response is slowed a little bit. With this last step the complete set of controller and input filters parameters θc° is obtained. 18289



Article

Normalized Controlled Process Models and Controller Parameters. The design methodology outlined in the preceding text is applied to controlled processes represented by stable overdamped, and inverse response models, as well as by integrating, and unstable models. For control system performance analysis and controller tuning, it is convenient to work with dimensionless parameters to make it nondependent of the controlled process time scale and gain. Using the transformation ŝ ≐ Ts for overdamped, inverseresponse, unstable, and second-order integrating controlled processes and the transformation ŝ ≐ Ls for first-order integrating processes, their models can be normalized as shown in Table 1. This table also shows the corresponding controller filter combination normalized parameters. The independent variable transformations (time scaling)16 and normalized controlled processes models and controllers parameters are used to obtain controller dimensionless and consistent tuning equations. Regulatory Control Closed-Loop Transfer Function Targets. For the model-reference optimization procedure the desired control system output to a disturbance step change needs to be specified. For the normalized controlled process models in Table 1 the corresponding normalized targets Mtyd(s) for the regulatorycontrol closed-loop transfer function eq 13 are listed in Table 2.

selected the design criteria, the controller normalized parameters are of the form θ̂ cy = {f i(τL)}. If controller tuning equations are obtained for these models, the normalized deadtime τL variation range where they are valid for needs to be indicated. At the other hand, the control system analysis depends on the design parameters θd and two normalized model parameters for second-order plus dead-time (SOPDT) models, θ̂p = {a, τL}, and inverse response second-order plus dead time (SOPRHPZ) models, θ̂p = {a, b}. Optimization Procedure. The optimization problem stated by eq 14 starts by defining the regulatory control transfer function target corresponding to the normalized process to control as described in the previous sections. The process normalized parameters are θ̂p, the design parameters θd, and the normalized controller parameters to be found θ̂cy = {κp, κi, κd, τf}. The target robustness MtS is used as a sof t constraint. For a given process the resulting control system robustness depends on the design parameters; then, it is evaluated after each closed-loop model-reference response optimization and the design parameters adjusted in order to meet the robustness target. With the feedback controller and filter parameters at hand, θ̂°cy, the set-point filter parameters (τr, σ) are selected subsequently to improve the servo-control behavior under the selected cost functional. To select the (local or global) optimization algorithm to use the cost functional eq 14, convexity is analyzed. An example of the cost functional shape is shown in Figure 2 for a second-order

Table 2. Regulatory Control Closed-Loop Transfer Function Targets model

θ̂p

(F)SOPDT

α, τL

IPDT

−

ISOPDT

τL

UFOPDT

τL

SOPRHPZ

α, b

Mtyd(s)

(1/κi)s ê −τLs ̂ 2 2

(τc s ̂ + 2ζτcs ̂ + 1)(aτcs ̂ + 1)

(1/κi)s ẽ −s ̃ 2 2

τc s ̃ + 2ζτcs ̃ + 1 (1/κi)s ê −τLs ̂ 2 2

(τc s ̂ + 2ζτcs ̂ + 1)(τcs ̂ + 1)

τf = 0.1τc τf = 0.1τc τf = 0.1τc

(1/κi)s ê −τLs ̂ 2 2

τc s ̂ + 2ζτcs ̂ + 1 (1/κi)s (̂ − bs ̂ + 1) (τc 2s 2̂ + 2ζτcs ̂ + 1)(aτcs ̂ + 1)

The regulatory-control transfer function targets include the normalized model parameters θ̂p and two design parameters, the closed-loop relative speed τc and the dominant poles damping ratio ζ, θd = {τc, ζ}. These two parameters change the closed-loop dominant poles location affecting the control system performance/robustness trade-off.14 From Table 2 it can be seen that the performance and robustness obtained with the selected regulatory-control targets depend on two to four normalized parameters. The simplest case corresponds to the integrated plus deadtime model (IPDT). For this particular case the analysis is independent of the controlled process model parameters. Only different sets of design parameters θd need to be considered. The analysis for first-order plus dead-time (FOPDT, a = 0) models, integrated second-order plus dead-time (ISOPDT) models, and unstable first-order plus dead-time (UFOPDT) models depends on only one model parameter, its normalized dead-time τL, and on the design parameters θd. Then, with

Figure 2. Cost functional Jd convexity.

plus dead-time model with parameters (K = 1, T = 1, a = 0.2, L = 0.4) and a PID2IF with parameters (Kp = 2.3215, Ki = 1.8066, Kd = 1.0883, Tf = 0.0693). Design specifications are ζ = 0.7 and MtS = 2.0. Controller parameters are changed from −10 to +10% of their nominal values. Controller parameters and the cost functional are normalized with their corresponding nominal values. Then, for both optimizations a direct search Nelder−Mead17 simplex-based algorithm is used.18 From the optimized 18290



Article

controller parameters sets, tuning equations were obtained using a curve fitting tool.19 In general, controller normalized parameters are θ̂c = {gi(θ̂p, θd; t MS)}, where the number of parameters on θ̂p are modeldependent, θd = {ζ, τc}, and MtS is the control system robustness target level. The closed-loop poles damping ratio ζ is selected based on the performance/control effort trade-off, and the control system speed τc is adjusted to meet the robustness target MtS. Then, controller normalized parameters can be expressed as θ̂c = {hi(θ̂p; MtS)}. For a robustness target level MtS controller tuning relations required for each controller normalized parameter depend on the number of normalized model parameters. It may be a constant for normalized models with no parameters, an equation on τL for normalized models with one parameter, and a “family” of equations for normalized models with two parameters. For this later case, it is impractical to use “tuning rules” and the direct use of the MoReRT controller design methodology to the controlled process model is recommended.

■

CONTROLLER PERFORMANCE/ROBUSTNESS ANALYSIS AND TUNING In order to reduce the number of design parameters, we analyze first the influence of the damping ratio ζ over the regulatory control system performance. The performance is compared for control systems providing the same robustness level; then, for a given damping ratio, during the regulatory control model matching process the closed-loop relative speed τc is selected in such a way that the robustness of the resulting closed-loop system reaches a specific target value MtS ∈ {2.0, 1.6}. For each design, the evaluated regulatory control response performance indices are as follows: the integrated absolute error (Jed), the controller output total variation (TVud), the maximum error (Emax), the time to reach the maximum error (tEmax), and the settling time (t5%Emax).20 Overdamped Stable Controlled Processes. For the stable (F)SOPDT controlled process models its normalized dead-time τL is selected in the range from 0.10 to 2.0 to include the time constant dominant as well as dead-time dominant processes. Because the time constants ratio a is in the range from 0 to 1.0, we analyze some extreme cases: first-order models (a = 0) with low (τL = 0.1) and high (τL = 2.0) normalized dead-time; dual pole models (a = 1.0) with low (τL = 0.1) and high (τL = 2.0) normalized dead-time; and also intermediate cases (a = 0.0, τL = 0.50), and (a = 0.75, τL = 0.50). This is made with PI and PID controllers. Figure 3 shows the robustness and normalized performance indices of the PI control system for the (a = 0.0, τL = 0.50) case and Figure 4 those corresponding to the PID system for the (a = 0.75, τL = 0.50) case. It is shown how their performance and robustness are influenced by the damping ratio selection. All of the performance indices have been normalized using their corresponding values for the nonoscillating target (ζ = 1). As can be seen from these figures both robustness level targets are perfectly accomplished, but it is not possible to have a highly underdamped system (low ζ) with high robustness at the same time. Therefore, allowing small oscillations at the control system output, it is possible to improve the regulatory control performance; a reduction on Jed and ts5%Emax values, but with a deterioration of the control effort smoothness (TVud), while

Figure 3. PI robustness and performance: a = 0.0; τL = 0.50.

Figure 4. PID robustness and performance: a = 0.75; τL = 0.50.

the effect over Emax and tEmax is negligible, particularly with the PID. From the information obtained, not all shown here for space reasons, it can be concluded that high values of the model time constants ratio a (models tends to be of dual pole) or of the normalized dead-time τL (models tends to be dead-time dominant) impose a constraint on the robustness of the highly underdamped responses. In addition it is seen that a good balance of the (Jed, ts5%Emax) versus TVud trade-off is obtained for damping ratios ζ in the range from 0.7 to 0.8. Then, for controller tuning we select a damping ratio ζ = 0.80 for the PI controller and a ζ = 0.70 for the PID. 18291



Article

Table 3. PI and PID Performance with (F)SOPDT Models a

0.0

0.0

1.0

1.0

0.0

0.75

τL

0.1

2.0

0.1

2.0

0.5

0.5

4.4950 1.4060 0.8342 5.6850 18.8000

0.8141 1.5183 0.5295 1.4400 3.0700

1.5884 1.6037 0.5592 2.5850 9.4350

3.0634 1.3035 0.7515 5.0400 9.4300

0.5476 1.4746 0.4357 1.1500 2.8750

0.7727 1.4419 0.3709 1.9700 4.6750

5.4158 1.0314 0.8602 5.9800 13.6600

1.1450 1.1064 0.5683 1.5850 4.3900

2.1151 1.1409 0.6185 2.8550 6.8000

4.2526 1.0641 0.8049 5.5000 11.6700

0.7641 1.1755 0.4631 1.2850 3.4350

1.0821 1.1116 0.4264 2.2000 5.3000

Jed TVud Emax tEmax ts5%Emax

0.0850 1.7488 0.1792 0.3750 1.0250

3.5152 1.3681 0.8912 4.3550 13.7500


0.0459 1.5676 0.1257 0.2800 0.7600

0.6313 1.7385 0.8715 4.0850 10.7850


0.1491 1.3994 0.2138 0.4550 1.4900

4.2589 1.0154 0.9014 4.5100 10.5400


0.0805 1.4099 0.1475 0.3350 1.0350

3.1115 1.1868 0.8742 4.1200 8.5700

PI Controller, MtS = 2.0 0.7246 1.9061 0.3236 1.7450 7.0900 PID Controller, MtS = 2.0 0.0577 1.8946 0.0615 0.7200 1.9300 PI Controller, MtS = 1.6 1.1547 1.3516 0.4069 2.0850 5.2300 PID Controller, MtS = 1.6 0.1238 1.5195 0.0954 0.9550 2.5500

κp ≐ KK p = 0.2954 + 0.5065τL−0.9805

Table 3 shows the performance and control effort indices for the cases considered with PI (ζ = 0.8) and PID (ζ = 0.7) controllers for the two robustness levels. As it can be seen for the same controlled process model and robustness level, the PID controller provides more performance, lower Jed, Emax, and ts5%Emax than the PI but with more control effort variation, especially for MtS = 1.6. This is particularly notorious for the dual pole model with low dead-time (a = 1.0, τL = 0.1) where the Jed obtained with the PID is about 10 times lower than the one obtained with the PI with similar control effort variation. In general, for the same robustness level, the PID controller outperforms the PI in all aspects but the control effort variation. If the closed-loop poles damping ratio is selected following the above recommendation, the remaining design parameter τc may be used for further analysis: for example, for a given controlled process model, to study the effect of τc over the control system performance/control effort/robustness trade-off, or with selection of a robustness target level MtS, to adjust τc during the optimization process of cost functional eq 14 to obtain the controller parameters θcy for a range of model normalized deadtimes τL. This set of controller parameters may be used to develop a tuning rule. PID2IF Tuning for First-Order Plus Dead-Time Models. Because the overdamped normalized model with a = 0 has only one parameter (τL), controller normalized parameters for robust tuning can be obtained as functions of τL. For first-order plus dead-time models with 0.1 ≤ τL ≤ 2.0 the PID controller and feedback filter normalized parameters for an intermediate robustness MtS = 1.6 can be obtained with the following relations:

κi ≐ TKK i =

−0.04612 + 0.6407τL 0.005159 − 0.13885τL + τL 2

KKd = 0.2874 + 0.04719τL 0.9134 T Tf τf ≐ = −0.01613 + 0.1233τL 0.4922 T γ=0

κd ≐

(19)

The set-point filter parameters are given by the relations τr ≐

Tr −0.3141 + 18.66τL = T 2.347 + 18.77τL − 6.415τL 2 + τL 3 σ e = 1.042 + 0.3809τL 0.4615

(20)

if the integrated absolute error eq 16 is optimized, or by the following relations: τr ≐

Tr −0.3141 + 18.66τL = T 2.347 + 18.77τL − 6.415τL 2 + τL 3 σ u = −0.7676 + 1.828τL 0.2382

(21)

if is of interest to optimize the integrated absolute control effort eq 17. In a PI2FI/PID2IF robust tuning rule for second-order plus dead-time models (a > 0) the controller normalized parameters depend on the two normalized model parameters and the robustness target level, θ̂c = {hi(a, τL; MtS)}; therefore, obtaining tuning rules for these models is not a simple task, and several sets or rules are needed with consideration of different model time constant ratio a values and target robustness MtS levels, requiring 18292



Article

interpolations between the obtained controller parameters to arrive at the final tuning. Nevertheless, as has been pointed out elsewhere,5 except for models nearly of first-order, the more rich dynamics of the second-order models allow one to design control systems with the same robustness but more performance than the one obtained by tuning the controller using the first-order model parameters. For these more elaborated models direct application of the proposed model-reference optimization procedure is a more direct solution. Comparative Evaluation of PI2IF/PID2IF Controllers for Stable Overdamped Controlled Processes. For comparison purposes consider the fourth-order controlled process given by the benchmark21 transfer function: P(s) =

1 (s + 1)(0.5s + 1)(0.25s + 1)(0.125s + 1)

Table 5. PI2IF and PID2IF Performance PI2IF (FOPDT)

(22)

For controller tuning on the basis of the P(s) reaction curve using a three-point identification method22 the following firstand second-order (FOPDT, SOPDT) models are obtained P1(s) =

e−0.691s 1.247s + 1

(23)

a

PI2IF (SOPDT)

PID2IF (FOPDT)

PID2IF (SOPDT)

PID2IF (SOPDT)

Ms

1.60

1.60

1.60

1.60

2.00

K∞ Jed TVud Emax tEmax ts5%Emax Jera TVura Umaxa tra ts5%Δya Δu0 Jerb TVurb Umaxb trb ts5%Δy

0 1.590 1.172 0.561 2.290 5.485 2.352 1.499 1.229 2.080 5.740 0.0 2.562 1.211 1.084 2.650 4.315

0 1.488 1.207 0.538 2.220 5.340 2.266 1.519 1.229 2.050 5.115 0.0 2.484 1.263 1.099 2.530 4.160

0 1.095 1.223 0.437 2.000 6.700 2.215 1.728 1.338 1.750 5.890 0.0 2.366 1.320 1.148 2.295 3.745

0 0.574 1.260 0.302 1.610 3.825 1.813 1.923 1.455 1.520 4.750 0.0 2.200 1.174 1.083 2.455 4.045

0 0.371 1.577 0.240 1.415 3.160 1.569 2.596 1.769 1.265 4.080 0.0 2.119 1.272 1.126 2.440 4.025

Using σe. bUsing σu.

−0.277s

P2(s) =

e (0.876s + 1)(0.719s + 1)

Table 5 also allows one to compare the servo-control performance obtained by designing the set-point filter in order to improve the integrated absolute error (σe) with designing it to improve the integrating absolute control effort (σu). In the latter case TVur, Umax, and ts5%Δy decrease while Jer and tr increase. Figure 5 shows the PI2IF and PID2IF control systems responses to a 20% set-point step change followed by a 10% load-

(24)

The PI2IF and PID2IF controllers parameters using the FOPDT and SOPDT models for a robustness level MtS = 1.6 are listed in Table 4. The obtained servo- and regulatory control performance indices are listed in Table 5. These tables also include PID2IF controller parameters and performance for MtS = 2.0. Table 4. PI2IF and PID2IF Parameters PI2IF (FOPDT)

a

PI2IF (SOPDT)

PID2IF (FOPDT)

PID2IF (SOPDT)

PID2IF (SOPDT)

MtS

1.6

1.6

1.6

1.6

2.0

KP Ki Kd Tf Tra σeb σuc γ

0.802 0.629 0.0 0.117 1.275 1.581 1.146 0

0.916 0.672 0.0 0.093 1.363 1.542 1.201 0

1.197 1.034 0.392 0.096 1.158 1.343 0.836 0

2.345 1.841 0.958 0.067 1.274 1.212 0.647 0

3.374 2.873 1.387 0.054 1.174 1.155 0.446 0

Tr = KP/Ki. bUsing Jer, eq 16. cUsing Jur, eq 17.

As can be seen in Table 5, if we use a PI controller with the same robustness level, the performance obtained tuning the controller by using P1(s) or P2(s) is very similar with a small advantage for the second-order model. In the PID case the performance improvement obtained by tuning the controller with the SOPDT model P2(s) is evident. The PID2IF obtained from the SOPDT model outperforms the other controllers in all aspects but the control effort total variation. Similar conclusions are obtained using a 2DoF PI controller (PI2) with the same controlled process eq 22.5 More control system performance is obtained if the second-order model is used for controller tuning. Slightly better performance and control effort variation is obtained with the PI2 than with the PI2IF but with Δu0 = 0.75Δr (FOPDT) and 0.67Δr (SOPDT), and with K∞ = 0.98 (FOPDT) and 1.15 (SOPDT). For the PI2IF, Δu0 = 0 and K∞ → 0.

Figure 5. PI2IF and PID2IF control systems responses.

disturbance step change. For the same robustness level (MtS = 1.6) the PID2IF (SOPDT) responses to a load disturbance have the lowest maximum error (Emax) and the shortest settling time (ts5%Emax). The Emax can be reduced even more if the desired control system robustness is relaxed and decreased. It also shows the advantages of designing the set-point filter by considering the 18293



Article

control effort; in all cases it is very smooth without any abrupt change. The PI2IF and PID2IF performance is compared with the following tuning rules that take into account the control system robustness: (1) Skogestad23 (SIMC) tuning for 1DoF PI and ideal PID controllers, (2) Sree et al.24 (SSC) tuning for 1DoF ideal PID controllers, and (3) Tavakoli et al.25 (TGF) for 2DoF PI (FOPDT) controllers. The SIMC and SSC tuning rules were developed using a nonproper ideal PID algorithm with derivative mode applied directly to the error signal. For evaluation a derivative filter is added to make realizable the controller, and the general 2DoF standard PID algorithm is used given by the following relation:2 ⎧ 1 u(s) = K p⎨βr(s) − y(s) + [r(s) − y(s)] Tsi ⎩ +

⎫ Tds [γr(s) − y(s)]⎬ αTds + 1 ⎭

(25)

For all PID controllers the derivative mode is applied only to the feedback signal (γ = 0). The controller parameters are listed in Table 6. SIMC PI, SSC PID, and TGF PI made use of the FOPDT model while SIMC

Figure 6. PI/PID control systems responses.

Table 6. PI and PID Controllers Parameters

Kp Ti Td β α γ

SIMC PI (FOPDT)

SIMC PID (SOPDT)

SSC PID (FOPDT)

TGF PI (FOPDT)

0.902 1.247

2.879 1.595 0.395 1 0.1 0

1.804 1.593 0.295 1 0.1 0

1.304 1.824

1

Although the PID2IF and SIMC PID regulatory-control performance and control effort (Jed, TVud) are very similar, the SIMC PID shows a large controller output instant step change (288%Δr) and an extreme control effort maximum (313%Δr). To reduce the SIMC PID control effort instant change to the PID2IF level, a set-point weight β = 0 is needed. In this latter case, its servo-control performance decreases 85% (Jer = 2.149) and its settling time increases 11% (ts5%Δy = 4.370). In addition to the preceding discussion, it is important to compare the controllers high-frequency gain (K∞). While for the PI2IF and PID2IF controllers it tends to zero, the SIMC PI and the TGF PI allow the measurement noise to pass through the controller proportional mode with small attenuation and a 30% amplification, respectively. On the other hand, the SSC PID controller amplifies the measurement noise about 20 times and the SIMC PID up to nearly 32 times. The noise filtering capabilities of the proposed PI and PID controller with a feedback input filter is shown in the controllers transfer functions Bode plot in Figure 7. It is also noted that the cutoff frequency ωcPI2IF < ωcPID2IF(FOPDT) < ωcPID2IF(SOPDT). From the high frequency noise attenuation point of view the PID2IF tuned with the FOPDT model did a better job than the one tuned with the SOPDT model. If a measurement noise filter is added to a PID already tuned with SIMC or SSC rules, the control system robustness and regulatory performance will be affected.4 While most of the tuning methods provide tuning relations for only one or two robustness levels, the proposed design methodology can be used to obtain control systems with the required robustness according to the expected controlled process dynamics variations. Integrating Controlled Processes. For the integrating processes, we have the integrating plus dead-time (IPDT) and the integrating second-order plus dead-time (ISOPDT) models. Integrating Plus Dead-Time Models. Similarly to that with the overdamped stable models for the integrated plus dead-time models, the influence of the closed-loop poles damping ratio over

0.708

PID of the SOPDT model. The obtained servo- and regulatorycontrol performance indices are listed in Table 7. Figure 6 shows the control systems responses to a 20% set-point step change followed by a 10% load-disturbance step change. Among these controllers the SIMC PID has the best overall performance, although its robustness (MS = 1.76) is lower than the method target level MtS = 1.56. A reason for this is that the SIMC design is based on zero-pole cancellation for an ideal PID that does not hold if a proper standard PID is used. Table 7. PI and PID Performance

Ms K∞ Jed TVud Emax tEmax ts5%Emax Jer TVur Umax tr ts5%Δy Δu0

SIMC PI (FOPDT)

SIMC PID (SOPDT)

SSC PID (FOPDT)

TGF PI (FOPDT)

1.59 0.902 1.383 1.176 0.514 2.175 5.245 1.590 0.913 1.350 1.595 4.085 0.902

1.76 31.667 0.554 1.281 0.258 1.510 5.170 1.161 3.100 3.130 0.800 3.940 2.879

2.27 19.844 0.883 1.079 0.354 1.795 5.970 1.251 1.398 2.030 1.080 3.525 1.804

1.97 1.304 1.299 1.327 0.456 1.990 8.015 1.832 1.143 1.354 1.770 5.905 0.923 18294



Article

Figure 7. Controllers magnitude and phase.

Figure 9. PID2IF robustness and performance, IPDT models.

the regulatory-control performance and control signal is analyzed. For a given damping ratio, during the optimization process the closed-loop relative speed τc is adjusted in such a way that the robustness of the resulting closed-loop system matches a specific target value MtS ∈ {2.0, 1.6}. Figure 8 and Figure 9 show the robustness and normalized performance indices of the IPDT control systems with PI and PID controllers, respectively. All of the performance indices have been normalized using their corresponding values for the nonoscillating target (ζ = 1).

As can be seen from these figures, for the same robustness level, if we allow the control system outputs to exhibit small oscillations, it is possible to improve the regulatory-control performance, a reduction on Jed and ts5%Emax values, but with a deterioration of the control effort smoothness (TVud), while the effect over Emax and tEmax is negligible, particularly with the PID. The improvement obtained by reducing the damping ratio is more evident for the PID. For controllers tuning by using IPDT models, we select a damping ratio ζ = 0.80 for the PI controller and a ζ = 0.70 for the PID. Note this selection is the same as for (F)SOPDT process models. Table 8 shows the performance and control effort indices for the IPDT models with PI (ζ = 0.8) and PID (ζ = 0.7) controllers for the two robustness levels. Table 8. PI and PID Performance with Integrating Plus DeadTime Models PI

PID

MtS

2.0

1.6

2.0

1.6


13.3756 1.9245 2.2536 4.2900 13.2503

29.4469 1.6531 3.0024 5.8450 20.5150

6.0845 1.6879 1.4365 2.9800 8.6300

12.3104 1.5913 1.7987 3.9400 12.5700

From this table it is seen that, for the same robustness level, the PID controller outperforms the PI in all aspects. PI2IF/PID2IF Tuning for Integrating Plus Dead-Time Models. Using the normalized IPDT model and regulatory-control closed-loop transfer function target, the cost functional eq 14 is optimized to obtain the PI and PID with feedback filter normalized parameters θ̃cy for both robustness levels. With the parameters θ̃°cy at hand the set-point filter parameters are obtained to improve the servo-control integrating absolute error.

Figure 8. PI2IF robustness and performance, IPDT models. 18295



Article

For robust tuning of PI2IF and PID2IF controllers for IPDT models the normalized parameters are determined by the following simple relations:

κp ≐ TKK p =

κi ≐ L2KK i = b ,

κi ≐ T 2KK i =

κp ≐ LKK p = a , τf ≐

Tf L

= d,

τr ≐

Tr = e, L

κd ≐ KKd = c ,

σ = f,

σ=

Table 9. PI2IF and PID2IF Tuning Constants for IPDT Models PID2IF

MtS

2.0

1.6

2.0

1.6

a b c d e f

0.4579 0.0751

0.2991 0.0340

0.2662 6.0972 1.1847

0.4320 8.7972 1.1945

0.6955 0.1825 0.3909 0.1876 3.8110 1.0786

0.4738 0.0895 0.2902 0.2863 5.2939 1.1140

(28)

For the PID2IF case it is found that for models with low normalized dead-time τL and low target robustness (MtS = 2.0) it is possible to improve the control system performance by reducing the closed-loop poles damping ratio, but as the normalized dead-time increases, it is not possible to obtain control systems with high robustness (MtS = 1.6) for damping ratios ζ < 1. Only ζ = 1.0 can guarantee one will obtain MtS = 1.6 for normalized dead-times in the range from 0.1 to 2.0. Using the proposed design procedure, similar relations can also be obtained for PID2IF controllers for ISOPDT models. Comparative Evaluation of PI2IF/PID2IF Controllers for Integrating Controlled Process. For comparison purposes, consider first the distillation column bottom level control using the steam flow rate where the process model is given by the transfer function:26,27

0.5163 + 0.5095τL 0.5788 + 2.099τL + τL 2 0.09323 0.5438 + 2.302τL + τL 2

Tf = 0.1407 + 0.1687τL 0.8976 T T τr ≐ r = 5.603 + 5.829τL T 1.393 + 2.379τL + 1.215τL 2 0.9866 + 1.991τL + τL 2

6.847 + 13.13τL + τL 2

Figure 10. PI2IF robustness, ISOPDT models.

τf ≐

σ=

8.773 + 15.53τL + 1.197τL 2

Set-point filter parameters in eq 27 and eq 28 are obtained by selecting τr = κp/κi and σ is such that the integrated absolute error eq 16 is optimized. Figure 10 shows the robustness obtained with eq 27 and eq 28.

Integrating Second-Order Plus Dead-Time Models. To analyze the influence of the closed-loop poles damping ratio over the PI2IF and PID2IF controllers regulatory-control performance with ISOPDT models, models with τL ∈ {0.1, 1.0, 2.0} are considered for two robustness levels MtS ∈ {2.0, 1.6}. It is found that for PI2IF controllers in all analyzed cases a reduction in the closed-loop poles damping ratio ζ worsens the control system performance (Jed, TVud, Emax, and tEmax increase). Then, for the PI2IF controller, tuning is made with ζ = 1.0. PI2IF Tuning for Integrating Second-Order Plus Dead-Time Models. Normalized integrating second-order plus dead-time models has only one parameter, τL. Then, controller parameters can be obtained as functions of τL and the robustness target level, MtS. Normalized parameters for robust tuning of PI2IF controllers for ISOPDT models with 0.1 ≤ τL ≤ 2.0 can be obtained with the following relations for MtS = 2.0

κi ≐ T 2KK i =

0.04682 0.7068 + 2.192τL + τL 2

τf ≐

Table 9 lists the a, b, c, d, e, and f constants for eq 26 for the two robustness levels.

κp ≐ TKK p =

0.5108 + 1.672τL + τL 2

Tf = 0.2163 + 0.2416τL 0.9322 T T τr ≐ r = 7.528 + 7.663τL T

γ=0 (26)

PI2IF

0.2461 + 0.3372τL

(27)

P3(s) =

and for MtS = 1.6 with 18296

0.2e−7.4s s

(29)



Article

Table 10. Example IPDT: Robustness, Performance, and Control Effort PI2

PI2IF

PID2IF

MtS

2.0

1.6

2.0

1.6

2.0

1.6

Ms K∞ Jed TVud Emax Jer TVur Δu0 Umax

1.998 0.382 93.019 1.919 2.676 19.639 0.550 0.182 0.262

1.599 0.280 164.959 1.639 3.168 25.568 0.386 0.145 0.190

1.999 0 145.773 1.936 3.356 38.843 0.263 0.0 0.130

1.600 0 323.029 1.649 4.457 58.359 0.161 0.0 0.079

2.001 0 66.557 1.689 2.126 30.178 0.398 0.0 0.188

1.600 0 134.588 1.589 2.663 41.433 0.251 0.0 0.118

Table 10 shows the robustness, performance, and control effort variation obtained using PI2IF, PID2IF controllers, as well as the 2DoF PI controller (PI2).8 From this table it can be seen that for both robustness levels PID2IF has the highest regulatory-control performance, lowest Jed, smoothest control effort, lowest TVud, and lowest maximum error, Emax. By design, the PI2IF and PID2IF controllers do not generate an abrupt change to a set-point change and their control effort is very smooth. These features made their servo-control responses slower than the responses obtained with a controller without set-point filter, such as those with the PI2. Figure 11 shows the control systems responses to a 20% setpoint change followed by a −5% disturbance step change with PID2IF and PI2 controllers for both robustness levels.

Table 11. PID2IF Parameters and Performance MtS

2.0

1.60

Kp Ki Kd Tf Tr σ γ Jed TVud Emax Jer TVur Umax

2.0201 0.6099 1.3896 0.0977 3.3122 1.1973 0 1.643 1.610 0.472 3.050 1.072 0.525

1.3172 0.3042 0.9138 0.1308 4.3300 1.2097 0 3.287 1.488 0.701 3.913 0.705 0.346

performance is lost (Jed, Emax, and Jer increase), but the control effort is smoother (TVud, TVur, and Umax decrease). Figure 12 shows the control system responses to a 20% setpoint change followed by a −10% disturbance step change. Unstable Controlled Processes. As it is expected, the dynamic characteristics of unstable controlled processes impose

Figure 11. Example IPDT control systems responses.

As a second example consider the integrating third-order process28 and its ISOPDT approximation5 given by the transfer functions P4(s) =

0.833e−0.2s 0.833e−0.353s ≈ s(0.1s + 1)(0.833s + 1) s(0.780s + 1)

(30)

Table 11 lists the PID2IF controller parameters for ∈ {2.0, 1.6} and the regulatory- and servo-control performances. If control system robustness needs to be increased, some MtS

Figure 12. Example IPDT control systems responses. 18297



Article

Figure 13. PI2IF and PID2IF controllers maximum attainable robustness with unstable processes.

Figure 14. PID2IF controllers robustness and performance.

severe limitations to the achievable control system robustness with PI/PID controllers. Therefore, the first step to follow is a verification of the maximum robustness attainable level with the proposed PI2IF and PID2IF controllers. This is shown in Figure 13. The first thing that can be noticed from this figure is that the attainable robustness levels are low in comparison with the best

known picture for stable processes. We have to constrain ourselves to higher values of MS. As an example for τL = 0.3 MSmin = 3.88 (ζ = 1.0) with PI2IF and MSmin = 2.06 (ζ = 0.6) with PID2IF. For τL = 0.7 it is only MSmin = 4.78 (PID2IF, ζ = 0.5). For a given process the maximum robustness with a PI2IF controller is obtained for the critically damped closed-loop poles 18298



Article

target (ζ = 1). If the closed-loop poles damping ratio ζ is reduced, the control system robustness decreases. Comparing the PID2IF robustness with the one for the PI2IF, we can see that for the same process PID2IF gives more robustness, as indicated by the down-pointing solid arrow, and that the range of processes that can be tuned with the same robustness is wider, as indicated by the right-pointing solid arrow. Then, we dismiss the use of the PI2IF controller for controlling unstable processes. For the PID2IF controller it can be seen that by reducing the closed-loop poles target damping ratio, it is possible to increase the control system robustness for a given process and to extend the range of processes that can be tuned with the same robustness, specially for normalized dead times τL in the higher side, as indicating by the dashed−dotted arrows. Before going directly to a maximum robustness design procedure for unstable processes, it is necessary to evaluate the influence of the damping ratio ζ regarding the performance/ control effort/robustness trade-off. This is shown in Figure 14 for processes with τL ∈ {0.4, 0.8}. If we use these two cases as a general indication of the closeloop poles damping ratio influence over the system robustness and performance, it is seen that, for unstable first-order controlled processes with dead time, if ζ is reduced to a certain limit, indicated by the down-pointing arrows, robustness increases to a limit M°Smin and almost all the other indices also improve, as indicated by the up-pointing arrows. Then, for the UFOPDT case a lower damping ratio ζ (maximum robustness) design strategy for PID2IF controllers is recommended. PID2IF Tuning for Unstable First-Order Plus Dead-Time Models. For the particular case of the unstable controlled processes, it is found that for the servo-control response it is better not to force the set-point filter time constant to Tr = Kp/Ki. For UFOPDT models both set-point filter parameters, (σ, Tr) are used to optimize the servo-control integrated absolute error cost functional Jer. Tuning of the normalized parameters for maximum robustness for UFOPDT models with normalized dead time 0.1 ≤ τL ≤ 1.0 obeys the following relations:

MS =

Comparative Evaluation of PID2IF Controllers for Unstable Controlled Processes. Consider the unstable first-order plus dead-time model given by the transfer function:29 P(s) =

e−0.2s s−1

(33)

The characteristics of the proposed PID2IF controller are compared with Wang and Cai28 (WC), Wang and Xu (WX),30 and Sree et al.24 (SSC) tuning methods. The PID2IF parameters are shown in Table 12, and the resulting robustness as well as regulatory- and servo-control performance is given in Table 13. Table 12. PID2IF Parameters MtS

min

2.0

3.0

Kp Ki Kd Tf Tr σ γ

2.025 0.696 0.311 0.110 2.907 0.963 0

2.974 1.807 0.380 0.060 1.621 1 0

4.625 5.609 0.520 0.030 0.819 1 0

Table 13. PID2IF Performance MtS

min

2.0

3.0

MS K∞ Jed TVud Emax Jer TVur Umax Δu0

1.83 0 1.790 3.055 0.725 1.849 1.823 0.272 0.0

2.00 0 0.612 2.317 0.434 1.172 2.317 0.573 0.0

3.00 0 0.223 2.805 0.311 0.715 4.113 1.417 0.0

The maximum attainable robustness with the PID2IF controller for this process (τL = 0.2) estimated with eq 32 is MS = 1.819 and MS = 1.83 is obtained. If robustness is reduced to the usual minimum for a stable process, MS = 2.0, all of the regulatory performance indices improve. They can be improved even more, with the exception of TVud, if the control system robustness is decreased more. For the same unstable process eq 33 the PID2IF controller outperforms the 2DoF PI controller in all indices except the control effort total variation to a set-point change.11 The WC, WX, and SSC tunings were developed for Ideal PID controllers (without derivative filter and with derivative mode applied directly to the error signal). To make the controller proper, and to avoid an impulse change at the controller output to a set-point step, a change derivative filter is added and the derivative mode is applied only to the feedback signal. A 2DoF standard PID controller with α = 0.1 and γ = 0 is used. The controller parameters are shown in Table 14, and the resulting robustness and regulatory- and servo-control performance, in Table 15. The WC tuning method was developed with a design robustness corresponding to a gain margin Am = 3 and a phase

1 0.3866 + 0.5494τL − 0.0634τL 2 + 0.0002184τL 3

κi ≐ TKK i =

0.7656 − 0.2024τL − 0.3848τL 2 + 0.0003126τL 3 (32)

κp ≐ KK p =

1 + 1.4561τL

1 1.5422 − 4.9568τL + 22.1484τL 2

KKd = 0.07318 + 0.6722τL 0.646 T Tf τf ≐ = 0.08649 + 0.1452τL1.132 T T τr ≐ r = 3.957 − 15.65τL + 55.41τL 2 − 26.58τL 3 T

κd ≐

σ = 1.156 − 1.684τL + 3.562τL 2 − 2.411τL 3 γ=0 (31)

The attainable robustness with tuning eq 31 can be estimated using the following relation: 18299



Article

Table 14. Comparative Analysis: Controller Parameters

Kp

Ti Td β α γ

WC

WX (MtS = 2.0)

SSC

3.683 1.138 0.032 0.393 0.1 0

4.163 0.966 0.037 0.463 0.1 0

4.903 1.108 0.104 1.0 0.1 0

Table 15. Comparative Analysis: Robustness Performance Ms K∞ Jed TVud Emax Jer TVur Umax Δu0

WC

WX (MtS = 2.0)

SSC

245 40.513 0.309 2.731 0.395 0.531 4.451 2.095 1.428

2.75 45.793 0.234 2.448 0.356 0.520 6.361 2.789 1.927

3.17 53.933 0.227 3.064 0.275 0.811 15.271 5.788 4.903

margin ϕ = 30° (an intermediate robustness level), but a system with low robustness (MS = 2.45) is obtained. In the WX method using a robustness design level MtS = 2.0 a system with a MS = 2.75 is obtained. Nyquist diagrams of PID2IF controllers for MSmin and MtS = 2.0, and of WC and WX PID2 controllers are shown in Figure 15.

Figure 16. UFOPDT models comparative analysis: control systems responses.

the servo-control response delayed. The other important characteristic of the proposed controllers is their frequency response roll-off. It is normal that in process control applications the feedback signal is corrupted with high-frequency noise. If this measurement noise is not properly filtered, it will generate high control signal variations resulting in a deterioration of the final control element. If a measurement noise filter is added to a standard PID controller af ter its tuning, the filter dynamics will affect the control system robustness and performance. Then, it is essential that both of these characteristics be part of the controller design from the beginning. Inverse Response Controlled Processes. It is expected that the SOPRHPZ model right-plane zero relative position b and the selected closed-loop poles damping ratio ζ affect the attainable robustness level of the resulting control system. In order to analyze the effect of the damping ratio ζ and the target robustness MtS, a sample choice for the inverse response model parameters (a = 0.1, b = 0.1), (a = 1.0, b = 1.0), and (a = 1.0, b = 2.0) with a PID2IF controller is used. It is found that the minimum robustness (MtS = 2.0) can be obtained in all cases even if the damping ratio is decreased to 0.7. The intermediate robustness level (MtS = 1.6) can be obtained for ζ < 1.0 only if b is very low. Also, the performance Je and t5%Emax can be improved by decreasing ζ but only if b is very low although the control signal total variation TVud increases. If b increases, the obtainable robustness decreases if the damping ratio is reduced. The target robustness can be guaranteed for right-plane zero relative positions from 0.1 to the maximum value listed in Table 16. The PI2IF controller allows one to obtain robust control systems for a wider range of inverse response models. For comparison of the proposed controllers consider the frequently used second-order plus right-plane zero model given by the following transfer function:31

Figure 15. UFOPDT models comparative analysis: Nyquist diagrams.

The control system responses with WC and WX PID2 and PID2IF to a 20% set-point change followed by a 10% disturbance step change are shown in Figure 16. For comparison of the tuning rules, in addition to the quantitative indices and the responses shape, the process control oriented characteristics of the proposed PI2IF and PID2IF controllers must be brought to the front. With these controllers there is no abrupt change at the controller output when a step change is made at the set-point. To mimic this characteristic with a 2DoF PID controller, its proportional set-point weight β must be made zero. With this, the second degree of freedom is lost and

P5(s) = 18300

3( −2s + 1) (2s + 1)(s + 1)

(34)



Article

control effort variation than the PID2IF but with Δu0 = 0.17Δr and K∞ = 0.21.10

Table 16. Nonminimum Phase Zero Position PI2IF

■

PID2IF

MtS

2.0

1.6

2.0

1.6

bmax

6.5

2.8

4.8

2.1

CONCLUSION This work presents a unified design procedure for proportional integrating and proportional integrating derivative control algorithms of independent gains with two input filters, feedback and set-point signal filters (PI2IF, PID2IF), to control stable, integrating, and unstable controlled processes. The feedback controller and measurement signal filter design is based on a regulatory-control closed-loop transfer function target with two dominant poles that depends on the model nonminimum phase part and on the design parameters θd = {τc, ζ}. The closed-loop poles damping ratio is selected in order to improve the control system performance (Jed) without much deterioration on the control effort total variation (TVud). The closed-loop target poles relative speed τc is selected to match a target robustness level measured with the maximum sensitivity (MtS). The set-point filter parameters are obtained by optimizing a selected servo-control performance cost functional. The proposed model-reference robust tuning (MoReRT) methodology provides a controller design strategy suitable for obtaining robust control systems for a wide variety of controlled process models. The proposed PI2IF and PID2IF controllers have two important characteristics: they do not introduce a controller output signal abrupt change to a step set-point change, and their high frequency gain tends to zero providing measurement noise attenuation. These two features are of importance in process control applications.

The PI2IF and PID2IF controller parameters are listed in Table 17. This table also lists the performance and control effort usage with these controllers. Table 17. PI2IF and PID2IF Parameters and Performance PI2IF

PID2IF

MtS

2.0

1.6

2.0

1.6

Kp Ki Kd Tf Tr σ γ Jed TVud Emax Jer TVur Umax

0.1844 0.05953 0.0 0.2302 3.0976 1.7739 0 20.235 1.653 2.931 6.377 0.438 0.378

0.1229 0.04685 0.0 0.2852 2.6233 2.1592 0 24.558 1.345 2.783 6.989 0.382 0.358

0.2166 0.07158 0.1710 0.2186 3.0260 1.6366 0 17.689 2.200 2.397 6.512 0.395 0.364

0.1710 0.05900 0.1241 0.2518 2.8983 1.8506 0 20.412 1.775 2.463 6.890 0.391 0.362

Control system responses to a 10% set-point change followed by a 5% disturbance step change are shown in Figure 17.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has received financial support from the Spanish CICYT program under Grant DPI2010-15230 and from the University of Costa Rica.

■

REFERENCES

(1) Åström, K. J.; Hägglund, T. Control Eng. Pract. 2001, 9, 1163− 1175. (2) Åström, K.; Hägglund, T. Advanced PID Control; ISAThe Instrumentation, Systems, and Automation Society: Research Triangle Park, NC, USA, 2006. (3) Luyben, W. L. Ind. Eng. Chem. Res. 2001, 40, 3605−3611. (4) Isaksson, A. J.; Graebe, S. F. IEE Proc.: Control Theory Appl. 2002, 149 (1), 41−45. (5) Alfaro, V. M.; Vilanova, R. J. Process Control 2012, 22, 359−374. (6) Alfaro, V. M.; Vilanova, R. Model reference based robust tuning of five-parameter 2DoF PID controllers for first-order plus dead-time models. 12th European Control Conference (ECC13), Zurich, Switzerland, Jul 17−19, 2013. (7) Alfaro, V. M.; Vilanova, R. Ind. Eng. Chem. Res. 2012, 51, 13182− 13194. (8) Alfaro, V. M.; Vilanova, R. Model Reference Robust Tuning of 2DoF PI Controllers for Integrating Controlled Processes. IEEE 20th Mediterranean Conference on Control and Automation (MED’12), Barcelona, Spain, Jul. 3−6, 2012.

Figure 17. SOPRHPZ models comparative analysis: control systems responses.

As the table and figure show, all of the responses are very similar but for the same robustness level better regulatorycontrol performance (Jed, Emax) is obtained with the PID2IF controller. For the inverse-response process eq 34 of the example the PI2 controller provides a little better performance with similar 18301



Article

(9) Alfaro, V. M.; Vilanova, R. Two-Degree-of-Freedom Proportional Integral Control of Inverse Response Second-Order Processes. 16th International Conference on Systems Theory, Control and Computing (ICSCC 2012), Sinaia, Romania, Oct. 12−14, 2012. (10) Alfaro, V. M.; Vilanova, R. J. Process Control 2013, 23 (4), 453− 462. (11) Alfaro, V. M.; Vilanova, R. Robustness-Based Tuning of TwoDegree-of-Freedom Proportional Integral Control for Unstable Processes. 16th International Conference on Systems Theory, Control and Computing (ICSCC 2012), Sinaia, Romania, Oct. 12−14, 2012. (12) Alfaro, V. M.; Vilanova, R. Robust Tuning of 2DoF PID Controllers with Filter for Unstable First-Order Plus Dead-Time Processes. 18th IEEE International Conference on Emerging Technologies & Factory Automation (ETFA 2013), Cagliary, Italy, Sep. 10−13, 2013. (13) Horowitz, I. M. Synthesis of Feedback Systems; Academic Press: New York, 1963. (14) Åström, K. J.; Hägglund, T. PID Controllers: Theory, Design and Tuning; Instrument Society of America, : Research Triangle Park, NC, USA, 1995. (15) Hägglund, T. Control Eng. Pract. 2013, 21 (8), 994−1006. (16) Atherton, D. P.; Boz, A. F. Trans. Inst. Meas. Control (London) 2009, 31 (5), 425−433. (17) Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; Wright, P. E. SIAM J. Optim. 1998, 9, 112−147. (18) Optimization Toolbox User’s Guide (R2012a); The MathWorks, Inc.: Natick, MA, USA, 2012. (19) Curve Fitting Toolbox User’s Guide (R2012a); The MathWorks, Inc.: Natick, MA, USA, 2012. (20) Alfaro, V. M.; Vilanova, R. Performance Analysis of ModelReference Robust Tuned 2DoF PID Controllers for Over Damped Processes. IEEE 20th Mediterranean Conference on Control and Automation (MED 2012), Barcelona, Spain, Jul. 3-6, 2012. (21) Åström, K. J.; Hägglund, T. Benchmark Systems for PID Control. IFAC Digital Control: Past, Present and Future of PID Control (PID’00), Terrasa, Spain, Apr. 5−7,2000. (22) Alfaro, V. M. Cienc. Tecnol. (San Jose, Costa Rica) 2006, 24 (2), 197−216 (in Spanish). Available at http://www.revistas.ucr.ac.cr/index. php/cienciaytecnologia/article/view/2647/2598. (23) Skogestad, S. J. Process Control 2003, 13, 291−309. (24) Sree, R. P.; Srinivas, M. N.; Chidambaram, M. Comput. Chem. Eng. 2004, 28, 2201−2218. (25) Tavakoli, S.; Griffin, I.; Fleming, P. J. Robust PI Controller for Load Disturbance Rejection and Setpoint Regulation. IEEE Conference on Control Applications, Toronto, Canada, Aug. 28−31, 2005; pp 1015− 1020. (26) Chien, I.-L.; Fruehauf, P. S. Chem. Eng. Progress 1990, 86, 33−41. (27) Chen, D.; Seborg, D. Ind. Eng. Chem. Res. 2002, 41, 4807−4822. (28) Wang, Y.-G.; Cai, W.-J. Ind. Eng. Chem. Res. 2002, 41, 2910−2914. (29) Ho, W. K.; Xu, W. IEE Proc.: Control Theory Appl. 1998, 145 (5), 392−396. (30) Wang, Y.-G.; Xu, X.-M. PID Tuning for unstable processes with sensitivity specification. 2009 Chinese Control and Decision Conference (CCDC2009), Guilin, China, Jun. 17−19, 2009. (31) Zhang, W.; Xu, X.; Sun, Y. Ind. Eng. Chem. Res. 2000, 39, 2056− 2061.

18302


Performance and Robustness Considerations for Tuning of

Recommend Documents