Assessment of PI control loop performance for input load disturbance

Article pubs.acs.org/IECR

Assessment of Proportional−Integral Control Loop Performance for Input Load Disturbance Rejection Zhenpeng Yu and Jiandong Wang College of Engineering, Peking University, Beijing, China 100871 ABSTRACT: This paper studies the performance assessment of proportional−integral (PI) control loops in terms of rejecting input load disturbances. The direct synthesis (DS-d) method for disturbance rejection is adopted to design PI controller parameters. The lower bound of the integrated absolute error (IAE) is established, from closed-loop responses subject to step or other general types of input load disturbance changes. A DS-d IAE-based index is formulated by using the lower bound of the IAE as a performance benchmark. A novel seminonparametric method is proposed to estimate the required parameters for calculating the performance index. Simulation and experimental examples are provided to validate the performance benchmark and to demonstrate the effectiveness of the proposed performance index.

1. INTRODUCTION Proportional−integral−derivative (PID) controllers undoubtedly play an important role in process industries. More than 95% of the industrial controllers are of the PID type; most of them are actually proportional−integral (PI) controllers.1 Thus, proper tuning of PI controllers is one of the key factors for industrial processes, in order to enhance product quality, reduce operation cost, and improve process safety. During the last two decades, performance assessment of control loops has been an active research topic and many tools have been developed, aiming at automatically delivering information to users, such as how well a control loop meets with the control target and how to improve the performance if necessary.2 Loosely speaking, there are two types of performance mainly concerned: set-point tracking performance and disturbance rejection performance. In terms of set-point tracking performance, the traditional performance indices include overshooting, rising time, and settling time, while some advanced indices for PID controllers appeared recently; see, for example, Yu et al.3 and references cited therein for a summary of the related techniques. In terms of the disturbance rejection performance, performance indices can be classified into two categories for stochastic disturbances and deterministic ones. For PID control loops, the celebrated minimum variance control (MVC) benchmark introduced by Harris4 was generalized to incorporate the structure limitations confined by PID controllers to assess the performance of suppressing stochastic disturbances.5−7 The MVC benchmark can also be used to assess the performance of rejecting deterministic load disturbances by modeling these disturbances as random processes (see Chapter 11 in ref 8). Besides the MVC-based index, there are some other performance indices for PID control loops to reject deterministic load disturbances. Hägglund9 proposed a so-called idle index to detect whether a control loop is sluggishly tuned; the idle index was later improved in refs 10 and 11. Visioli12 presented the area index to detect whether PI control loops are sluggish or oscillatory and to show a guideline to retune PI controllers if necessary. Salsbury13 assumed that the dynamics between load disturbance and error signal can be approximated by an © 2012 American Chemical Society

underdamped transfer function, and used a ratio index to reveal how damping a control loop is. The ratio index was further developed by Howard and Douglas14 using the autocorrelation function. This paper studies the performance assessment of PI control loops for rejecting a special deterministic disturbance, the input load disturbance. The main contribution of this paper is 2-fold. First, we establish the lower bound of the integrated absolute error (IAE) from closed-loop responses subject to input load disturbance changes, with PI controllers designed by the direct synthesis method for disturbance rejection (abbreviated as the DS-d method) proposed by Chen and Seborg.15 By use of the lower bound of the IAE as a performance benchmark, a DS-d IAE-based index is formulated to assess the performance of PI control loops in terms of rejecting input load disturbances. Second, a novel seminonparametric method is proposed to estimate the parameters required to calculate the performance index from closed-loop responses. The proposed DS-d IAE-based index has a major advantage over the MVC-based index that is applicable to the problem considered in this paper. That is, the proposed index is based on the DS-d method that is able to achieve a good balance among disturbance rejection, robustness, and control effort. By contrast, the MVC-based controller solely aims at minimizing the variance of the fluctuations in process output. As a result, the MVC is usually aggressive and sensitive to the mismatch between the model and actual process that may cause control loop instability.16,17 See example 3 in section 6 for a numerical illustration. Another control strategy widely adopted in industry is the internal model control (IMC) method. However, the IMC method is not effective in rejecting input load disturbance, especially for lag-dominated process (see, for example, the theoretical analysis in Chapter 6 of ref 1). By contrast, the DS-d Received: Revised: Accepted: Published: 11744

February 6, 2012 August 3, 2012 August 15, 2012 August 15, 2012 dx.doi.org/10.1021/ie300313g | Ind. Eng. Chem. Res. 2012, 51, 11744−11752

Industrial & Engineering Chemistry Research

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method has been shown to provide better input load disturbance rejection than the IMC method when the controllers are tuned to have the same degrees of robustness,15 and it has good performance in terms of robustness and control effort, especially for lag-dominated processes.18 Hence, the DSd method is selected to design PI controllers. The rest of the paper is organized as follows. Section 2 describes the problem to be solved. Section 3 establishes the lower bound of the IAE and proposes a DS-d IAE-based performance index. A seminonparametric method is presented in section 4 to estimate the required parameters in order to calculate the performance index. Sections 5 and 6 provide numerical and experimental examples to validate the performance benchmark and to show the effectiveness of the proposed performance index, respectively. Section 7 concludes the paper.

(A3) The input disturbance d0(t) changes from a steady-state value to another one, while the set point r(t) is kept constant. (A4) The real-time measurements of r(t), u(t), y(t), and d0(t) are available with sampling period h, denoted as {r(nh), u(nh), y(nh), d0(nh)}Nn = 1, while the measurement of d(t) is not directly available. Here y(h) and y(Nh) are assumed to reach the initial and final steady states, respectively. The objective is to provide a quantitative index to assess the performance of the PI controller C(s) in rejecting a variation of the disturbance d0(t), and to recommend controller parameters to improve the performance index if necessary.

2. PROBLEM DESCRIPTION Consider a single-input single-output feedback control loop depicted in Figure 1. Here P(s) and C(s) are the process and

3. IAE BENCHMARK BASED ON DS-D METHOD This section reviews the DS-d method for designing PI controllers and derives the lower bound of the IAE in a closedloop response subject to step or other general types of load input disturbance changes. By taking the lower bound as a benchmark, a DS-d IAE-based index is proposed for performance assessment. 3.1. DS-d Method for PI Controller. This subsection briefly reviews the DS-d method proposed by Chen and Seborg15 for PI controllers. Applying the DS-d method to P(s) in eq 2, the parameters of the PI controller C(s) in eq 3 are as given in ref 15 (see eqs 35 and 36 therein):

Figure 1. A feedback control loop with input load disturbance.

Kc =

the PI controller, respectively; r(t), u(t), and y(t) are the set point, control signal, and process output, respectively; d(t) is the disturbance additive to the process input and is assumed to be generated by passing a disturbance source d0(t) through a static gain Kd, that is, d(t) = Kdd0(t). An important feature is that d0(t) acts as the input load disturbance. Many industrial processes have this feature: for example, a distillation column may have two inlets affecting the dynamics of the column in the same manner, and one is used for control while the other one plays a role as the disturbance. In the literature, there are also many studies on controller design with consideration of the input load disturbance.15,19−23 The following conditions are assumed to hold: (A1) The process P(s) is confined to be a linear time-invariant (LTI) process that is stable, without integrals and negative zeros, and can be described as P(s) =

Tc =

I

K e−θs τs + 1

τθ + 2ττc − τc 2 τ+θ

(4b)

Ks(τc + θ )2 e−θs Y (s ) P(s) = ≈ 1 + C(s)P(s) D(s) (τ + θ )(τcs + 1)2 (5)

where Y(s) and D(s) are the Laplace transforms of y(t) and d(t), respectively. The approximation in eq 5 results from e−θs ≈ 1 − θs for P(s) in the denominator 1 + C(s)P(s). Based on the desired closed-loop disturbance response in eq 5, the lower bound of the IAE is established in the next section. 3.2. IAE Lower Bound for Step Input Disturbance. Without loss of generality, consider the disturbance d(t) experiencing a step change from 0 to Ad, where Ad is a nonzero real number. From eq 5, we have

(1)

(2)

(A2) The PI controller is known a priori and takes a series formulation: ⎛ 1 ⎞ C(s) = Kc⎜1 + ⎟ Tcs ⎠ ⎝

(4a)

GCL(s) =

Such a process can be well approximated by a first-order plus dead time (FOPDT) model:24 P(s) ≈

K (τc + θ )2

where τc is a user-selected positive real number and can be regarded as the desired time constant of the closed-loop response to the input load disturbance d(t). With Kc from eq 4a and Tc from eq 4b, the closed-loop response from d(t) to y(t) is as given in ref 15 (eqs 33 and 37 therein):

K e−θ0s ∏i = 1 (τ0, is + 1)

τθ + 2ττc − τc 2

Y (s) = GCL(s)D(s) =

Ad K (τc + θ )2 e−θs (τ + θ )(τcs + 1)2

The inverse Laplace transform gives the time-domain expression of y(t):

(3) 11745

dx.doi.org/10.1021/ie300313g | Ind. Eng. Chem. Res. 2012, 51, 11744−11752


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⎧ 0 for 0 ≤ t < θ ⎪ 2 − ( t − θ )/ τ c y(t ) = ⎨ Ad K (τc + θ ) (t − θ )e for t ≥ θ ⎪ 2 (τ + θ )τc ⎩

Here d(ti) is the value of d(t) at the time instance ti; d(t0) and d(tI) are the initial and final steady-state values of d(t), respectively, that is, d(t) = d(t0) for t ≤ t0 and d(t) = d(tI) for t ≥ tI. The tracking error corresponding to di(t) is denoted as εi(t). As di(t) is a step signal, eq 8 is applicable and the lower bound of the IAE for di(t) is denoted as IAE0[di(t)]. From the superposition principle (the closed-loop system is LTI), it is ready to obtain the lower bound of the IAE for d(t) in eq 9 as

(6)

Assume that the set point signal r(t) is kept at a constant value equal to 0, that is, r(t) = 0, ∀t. If the constant value is nonzero, an initialization step is implemented to subtract r(t) and y(t) by the nonzero value. Then the tracking error is ε(t ) = r(t ) − y(t ) = −y(t )

IAE0[d(t )] =

∫0

≈

∫0

≤

∫0

(7)

We take y(t) in eq 6 as the desired closed-loop response to the step change of d(t). The corresponding IAE, denoted as IAE0[d(t)], is IAE0[d(t )] =

∫0

=

∫0

∞

+

| ε( t ) | d t

| ε( t ) | d t ∞

|ε(t0) + ε1(t ) + ... + εI (t )| dt ∞

|ε(t0)| dt +

∫t

∫t

∞

|ε1(t )| dt + ...

1

∞

|εI (t )| dt

I

∞

= IAE0[d1(t )] + ... + IAE0[dI (t )]

|−y(t )| dt

=

|Ad K |(τc + θ )2

=

|Ad K |(τc + θ )2

(τ + θ )τc 2 (τ + θ )τc

2

∫θ

∞

= [|d(t1) − d(t0)| + ... + |d(tI ) − d(tI − 1)|] |K |(τc + θ )2 τ+θ

(t − θ )e−(t − θ)/ τc dt

(10)

[−τc(t − θ )e−(t − θ)/ τc

The equality in eq 10 holds under two scenarios: (i) all step changes of di(t) have the same sign [in other words, the disturbance is mononically increasing or decreasing], or (ii) if there is a sign change between di(t) and di+1(t) [that is, the disturbance changes into the opposite direction], then di(t) has to keep invariant at the value d(ti) for a sufficiently long period of time for the closed loop to arrive at the steady state; that is, εi(t) ≡ 0 for t ∈ [ti+1, ∞). Under these scenarios, if the approximation error of d(t) in eq 9 is ignored, the lower bound of the IAE is

∞

− τc 2e−(t − θ)/ τc] t=θ 2

=

∞

|Ad K |(τc + θ ) τ+θ

(8)

This is the lower bound of the IAE for a DS-d-based PI controller in the closed-loop response subject to a step input load disturbance. 3.3. IAE Lower Bound for General Input Disturbance. Suppose that the load disturbance d(t) changes from one steady-state value to another, by following a general-type path

I−1

IAE0[d(t )] =

∑

|d(ti + 1) − d(ti)|

i=0

|K |(τc + θ )2 τ+θ

(11)

Apparently, eq 8 is a special case of eq 11 for a step disturbance. Thus, we have developed the lower bounds of IAEs in the closed-loop response subject to step or other general types of input disturbance changes. 3.4. DS-d IAE-Based Performance Index. Under assumption A4 in section 2, the actual IAE is ready to be obtained from the measurements {y(nh), r(nh)}Nn = 1 as N−1

IAEactual =

∑ |ε(nh + h) − ε(nh)|h (12)

n=1

where ε(t) is defined in eq 7. For the time being, assume that the disturbance gain Kd is known, so that the measurement of d(t) can be obtained from that of d0(t), that is, d(t) = Kdd0(t). If the parameters K, τ, and θ of P(s) in eq 2 are known, the lower bound of IAE for a user-selected parameter τc is calculated from eq 11 as

Figure 2. A general type of disturbance d(t) and its decomposition.

such as that shown in Figure 2. The disturbance can be approximated by a series of step signals: d(t ) ≈ d(t0) + d1(t ) + d 2(t ) + ... + dI (t )

(9)

N−1

where

IAE0 =

⎧ 0 for 0 ≤ t < ti ⎪ di(t ) = ⎨ ⎪ ⎩ d(ti) − d(ti − 1) for t ≥ ti

∑ |d(nh + h) − d(nh)| n=1

|K |(τc + θ )2 τ+θ

(13)

A dimensionless performance assessment index is defined on the basis of the actual IAE in eq 12 and IAE0 in eq 13: 11746


Industrial & Engineering Chemistry Research ηIAE =

min(IAE0, IAEactual) max(IAE0, IAEactual)

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4.1. Disturbance Gain Kd and Process Gain K. Under assumption A3 in section 2, the disturbance source d0(t) changes from one steady-state value to another. The amplitude change between the two steady-state values is denoted as Ad0. That is, under assumption A4 in section 2, Ad0 is obtained from the measurements {d0(nh)}Nn = 1:

(14)

Henceforth, ηIAE is referred to as the DS-d IAE-based performance index. The index can be used to evaluate the current control loop disturbance rejection performance. Remark 1. The performance index ηIAE in eq 14 does not just measure the distance between the actual IAE and the lower bound of IAE achievable from a DS-d-based controller. The DS-d-based controller is able to achieve a good balance among three conflicting factors, namely, disturbance rejection, robustness, and control effort.15,18 In this context, we take the same metrics Ms and TV used in ref 15 to quantify the robustness and control effort: Ms = max ω

1 P(jω)C(jω)

A d0 = d0(Nh) − d0(h)

The corresponding amplitude changes of d(t) and u(t), respectively denoted as Ad and Au, are the same but with opposite signs: Ad = −A u

Thus, the disturbance gain Kd can be determined as (15)

K̂ d = −

and N−1

TV =

Au A d0

(18)

The value of d(t) is estimated from that of d0(t):

∑ |u(nh + h) − u(nh)| n=1

(17)

(16)

d(̂ t ) = K̂ dd0(t )

Thus, ηIAE in eq 14 measures the level of improvement in terms of the overall control-loop performance with a balance among IAE, Ms, and TV. It is possible that IAEactual may be smaller than IAE0 in eq 13; in this case, it does not mean that the current performance is desirable, with no space for improvement. If example 3, which appears later in section 6, is taken as an example, the IAE of the MVC-based controller is smaller than IAE0, but the MVC-based controller is associated with poor performance in terms of Ms and TV; thus, the control loop performance can be further improved, not in terms of IAE but in terms of a good balance among IAE, Ms, and TV. The level of improvement is measured by the performance index ηIAE. Remark 2. The index ηIAE in eq 14 is in the range of [0, 1] with the ideal value equal to 1. If ηIAE → 0, the current control loop performance is far away from what can be achieved by using a PI controller tuned by the DS-d method. If ηIAE → 1, then a further step of performance assessment is required before a conclusion can be drawn. The mapping from {y(nh)}Nn = 1 to IAEactual in eq 12 is clearly not bijective; as a result, there might exist several controllers having the same value of IAEactual but being associated with different amounts of control effort and robustness. This problem is not unique to ηIAE in eq 14 but exists for other performance indices as well, for example, the MVC-based index. If ηIAE is close to 1, we recommend an additional step of estimating the robustness Ms in eq 15. If the estimate of Ms meets with its specification, for example, M̂ s ≤ 1.6, then the current control loop performance is satisfactory; otherwise, there is some space for improvement. The required process information to estimate Ms will be available in the procedure of estimating ηIAE that is presented in the next section.

(19)

Next, we will estimate the process gain K. For the feedback control loop in Figure 1, we have U (s) = −[U (s) + D(s)]P(s)C(s)

(20)

where U(s) is the Laplace transform of u(t). Using P(s) from eq 2 and C(s) from eqs 3 and 20, we have U (s ) lim − = lim s→0 s→0 [U (s) + D(s)]/s

(

KKc s +

1 Tc

)e

τs + 1

−θs

=

KKc Tc

From the final-value theorem, the process gain K can be estimated as K̂ = −

A uTc Kc ∫

∞

0

[u(t ) + d(̂ t )] dt

On the basis of the measurements {u(nh), d0(nh)}nN= 1 (assumption A4 in section 2), K̂ is calculated as K̂ = −

A uTc N Kc ∑n = 1

[u(nh) + d(̂ nh)]h

(21)

4.2. Process Time Constant τ and Time Delay θ. In this subsection, we first determine the range of the sum of τ and θ, and we estimate the two parameters afterward by solving a onedimensional optimization problem. Inspired by Veronesi and Visioli,26 we define the sum of time constants and the dead time of process P(s) as I

T0 = θ0 +

4. ESTIMATION OF PERFORMANCE INDEX Calculating ηIAE in eq 14 requires process information: the parameters K, τ, and θ of P(s) in eq 2 and the disturbance gain Kd, as well as a user-selected parameter τc. All of these parameters may not be available at hand or experience significant changes for various operating points. This section proposes a seminonparametric method to estimate the required parameters and the index ηIAE in eq 14.

∑ τ0,i ≈ θ + τ i=1

(22)

Thus, T0 directly confines the range of θ and τ. We devise a new signal n(t): n(t ) = y(t ) − K̂ [d(̂ t ) + u(t )]

(23)

whose Laplace transform is 11747



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N (s) = Y (s) − K̂ [D̂ (s) + U (s)] =

Y ̂ (s | θ , τ ) =

⎡ D̂ (s)P(s) D̂ (s)C(s)P(s) ⎤ − K̂ ⎢D̂ (s) − ⎥ 1 + C(s)P(s) 1 + C(s)P(s) ⎦ ⎣

⎛ || y (̂ t |θ , τ ) − y(t )||2 ⎞ ⎟ F(θ , τ ) = 100⎜⎜1 − || y(t ) − E{y(t )}||2 ⎟⎠ ⎝ (24)

N (s) = D̂ (s)

−

I ∏i = 1

(θ ̂, τ )̂ = arg max F(θ , τ )

q

∫ ∫0 q →∞ 0 = lim s s→0

subject to the constraint in eq 22

I Tcs ∏i = 1 (τ0, is + 1) + KcK̂ (Tcs + 1)e−θ0s

θ + τ = T0̂

where T̂ 0 is determined in eq 26. The one-dimensional optimization problem in eq 30 can be solved easily, for example, via a grid search method. Remark 3. In section 2, the process is assumed to be well approximated by the FOPDT model in eq 2. A comparison between the actual output y(t) and the simulated output ŷ(t): = ŷ(t|θ̂, τ̂) in eq 28 based on θ̂ and τ̂ in eq 30 also plays a role in evaluating whether the approximation is acceptable and the calculated performance index is reliable. If ŷ(t) can capture the main characteristics of y(t), then we are confident that the model quality is satisfactory and the calculated performance index is reliable. Otherwise, if ŷ(t) severely deviates from y(t), we would be aware of the fact that the above-mentioned assumption may be violated, so that the calculated performance index may be unreliable. 4.3. Selection of τc. The appearance of τc makes IAE0 in eq 13 a user-specified benchmark. In other words, users can select τc as the desired time constant of the closed-loop response to the input load disturbance and evaluate the performance of the current control loop against the desired one. In the case that users do not have prior knowledge of a proper value of τc, it is recommended to select τc to achieve an acceptable robustness of the control loop. To ensure positive controller parameters, the range of τc is confined as given in ref 15 (eq 38 therein):

p

n(t ) dt dp

N (s ) s2

= lim sD̂ (s) s→0

̂ c[e−θ0s − ∏I (τ0, is + 1)] KT i=1 I

s[Tcs ∏i = 1 (τ0, is + 1) + KcK̂ (Tcs + 1)e−θ0s] (25)

By use of the concept of equivalent infinitesimal and eq 17, eq 25 becomes q

∫ ∫0 q →∞ 0 lim

p

n(t ) dt dp

̂ c lim sD̂ (s) = KT s→0

I

1 − θ0s − ∏i = 1 (τ0, is + 1) I

s[Tcs ∏i (τ0, is + 1) + KcK̂ (Tcs + 1)(1 − θ0s)] I

̂ c lim sD̂ (s) = KT s→0

=−

−s(θ0 + ∑i = 1 τ0, i) KcKŝ

TcT0 lim d(̂ t ) Kc t →∞

0 < τc < τ ̂ +

A TT = u c 0 Kc

KcA s A uTc

(26)

where As is the steady-state value of the output of a double integrator 1/s2 driven by n(t). Based on T̂ 0, if the time delay θ is available, the process time constant τ is easily determined as τ̂ = T̂ 0 − θ owing to eq 22; otherwise, θ and τ can be estimated as follows. By use of the estimated parameter K̂ in eq 21, the FOPDT model is K̂ e−θs P(̂ s|θ , τ ) = τs + 1

τ 2̂ + τθ̂ ̂

(31)

where τ̂ and θ̂ are given in eq 30. In general, it is desired to have a smaller value of τc in order to reject the input load disturbance more quickly; however, the robustness of the control loop is monotonically getting worse with the decrement of τc. Hence, τc can be determined as the minimal value to achieve the acceptable upper bound of robustness as follows. First, choose a positive integer value L, for example, L = 100, and separate the confined range of τc in eq 31 into equally spaced grids:

Thus, T0 can be estimated as

T0̂ =

(30)

θ ,τ

(τ0, is + 1)]

According to the final-value theorem, we have lim

(29)

Then, θ and τ are estimated as

By use of the complete forms of P(s) from eq 1 and C(s) from eq 3, N(s) in eq 24 can be written as −θ0s

(28)

Define a fitness between ŷ(t|θ, τ) and the actual output y(t):

P(s) − K̂ = D̂ (s) 1 + C(s)P(s)

̂ cs[e KT

P(̂ s|θ , τ ) D̂ (s) 1 + C(s)P(̂ s|θ , τ )

τc, l =

l (τ ̂ + L

τ 2̂ + τθ̂ ̂ )

for l = 1, 2, ..., L

Following the DS-d method in eqs 4a and 4b, we define a controller Cl(s) as ⎛ 1 ⎞ ⎟⎟ Cl(s) = Kc, l ⎜⎜1 + Tc, ls ⎠ ⎝

(27)

̂ With the knowledge of d(t) in eq 19 and the current PI controller setting, the simulated output ŷ(t|θ, τ) for P̂ (s|θ, τ) can be obtained as

where 11748


Industrial & Engineering Chemistry Research Kc, l =

τθ̂ ̂ + 2ττ̂ c, l − τc, l 2 K̂ (τc, l + θ )̂ 2

Tc, l =

τθ̂ ̂ + 2ττ̂ c, l − τc, l 2 τ̂ + θ̂

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4a and Tc from eq 4b with the estimated process parameters K̂ , τ̂, and θ̂ and the selected value of τc. (Step 10) If η̂IAE is close to 1, estimate the robustness Ms in eq 15 based on the estimated process parameters K̂ , τ̂, and θ̂. If M̂ s ≤ Ms,0, then the current control loop has a satisfactory performance; otherwise, retune the PI controller as described in step 9.

Second, the robustness is measured by the maximum of the sensitivity function: Ms, l = max ω

5. SIMULATION EXAMPLES In this section, three simulation examples are presented. The first example validates the established IAE benchmark, the second one illustrates the effectiveness of the proposed DS-d IAE-based performance index, and the last example compares the MVC-based index with the proposed one. 5.1. Example 1. The process in Figure 1 is 2 P(s) = e − 5s (33) 10s + 1

1 ̂ P(jω)Cl(jω)

where P̂ (jω) and Cl(jω) are the frequency responses of P̂ (s) and Cl(s), respectively. Here the estimated process model P̂ (s) is obtained from K̂ in eq 21 and τ̂ and θ̂ in eq 30: P(̂ s) =

̂ K̂ e−θs τŝ + 1

and the PI controller C(s) follows the DS-d method. The sampling period is h = 0.1 s. We set the disturbance gain Kd = 1 and let the disturbance d0(t) experience some changes (shown in Figure 3a):

Next, a proper value of τc is determined as τc = min(τc, l) l

such that Ms, l ≤ Ms,0

(32)

⎧0 ⎪ ⎪ 0.1(t − 10) d0(t ) = ⎨ ⎪ e(t − 20)/10 ⎪ ⎩e

where Ms,0 is the upper bound of the acceptable robustness. As the typical range of Ms,0 is [1.2, 2.0] (see section 4.8 in ref 1), the value of Ms,0 here is recommended to be no larger than 1.6 to make gain and phase margins greater than 2.67 and 36.4°, respectively. Thus, with K̂ in eq 21, τ̂ and θ̂ in eq 30, and τc in eq 32, it is possible to calculate the lower bound IAE0 in eq 13 and the performance index ηIAE in eq 14. 4.4. Steps of Performance Assessment. Under assumptions A1−A4 in section 2, the steps of the performance assessment are summarized as follows: (Step 1) Detrend the measurements of r(t), u(t), y(t), and d0(t) by their own initial steady-state values. (Step 2) Calculate the amplitude change Ad0 of d0(t) and the amplitude change Au of u(t). (Step 3) Obtain K̂ d in eq 18, d̂(t) in eq 19, and K̂ in eq 21. (Step 4) Generate n(t) in eq 23 and calculate T̂ 0 in eq 26. (Step 5) Estimate τ and θ in eq 30. If the simulated output ŷ(t) can capture the main characteristics of y(t), proceed to step 6; otherwise, select another set of data and go back to step 1. (Step 6a) If a proper value of τc is available, go to step 7; otherwise, continue to step 6b. (Step 6b) Determine τc in eq 32 to achieve the upper bound of the acceptable robustness, for example, Ms,0 = 1.6. (Step 7) Calculate IAEactual from eq 12 and estimate IAE0 from eq 13 by use of d̂(t), K̂ , τ̂, θ̂, and τc: N

̂ 0= IAE

∑ n=1

10 ≤ t < 20 20 ≤ t < 30 30 ≤ t < 500

(34)

Figure 3. Signals in example 2: (a) d0(t) in eq 34; (b) y(t), () for initial controller parameters Kc = 0.3 and Tc = 40 and (− · −) after retuning with Kc = 0.4827 and Tc = 8.9965; (c) u(t), () for initial controller parameters and (− · −) after retuning.

|K̂ |(τc + θ )̂ 2 d(̂ nh + h) − d(̂ nh) τ̂ + θ̂

To validate IAE0 in eq 13, we vary the value of τc in eq 4b as an integral multiple of the time delay θ = 5, that is, τc = λθ. Table 1 Table 1. Actual IAE and IAE0 for Different Values of λ

(Step 8) Estimate ηIAE from eq 14 as η̂IAE =

0 ≤ t < 10

̂ 0, IAEactual) min(IAE ̂ 0, IAEactual) max(IAE

(Step 9) If η̂IAE is not satisfactory, for example, η̂IAE < 0.8, then retune PI controller parameters by using Kc from eq 11749

λ

Kc

Tc

IAE0 (eq 13)

IAEactual (eq 12)

1 2 3 4

0.6250 0.3333 0.1563 0.04

8.3333 10 8.3333 3.3333

36.2438 81.5485 144.9750 226.5235

36.3658 81.5484 144.9744 226.5188



Article

compares the actual IAE from eq 12 and IAE0 from eq 13. The two values are very close to each other for different values of λ. The minor difference between them is due to the approximation in eq 5. 5.2. Example 2. The configuration is the same as that in example 1 with different PI controller parameters. Initially, C(s) in eq 3 takes the parameters Kc = 0.3 and Tc = 40. The corresponding process output y(t) and control signal u(t) are shown as solid lines in Figure 3 panels b and c, respectively. By following the steps summarized in section 4.4, we reach the results listed in the first row in Table 2. In particular, η̂IAE = Table 2. Comparison between Initial Control-Loop Performance and That after Retuning for Example 2 initial after retuning

Kc

Tc

IAEactual

IÂ E0

η̂IAE

0.30 0.4832

40 8.9965

359.5317 50.6139

50.4833 51.7002

0.1408 0.9924

Figure 4. Signals in example 3 without process variation: (a) y(t) and (b) u(t) for (---) MVC-based controller and () DS-d-based controller

0.1408 indicates that the current performance is far from being satisfactory. In the process of calculating η̂IAE, the estimated disturbance gain is K̂ d = 0.9993 and the estimated process model is obtained: P(̂ s) =

1.9961 e−4.926s 9.5622s + 1

For the MVC-based controller, the IAE is smaller than the counterpart from the DS-based controller. However, the control effort measured by TV and the robustness measured by Ms from the MVC-based controller are worse. In particular, Ms = 12.4213 is far away from the typical range [1.2, 2.0] of Ms (section 4.8 in ref 1). This large value of Ms implies that the MVC-based controller is very sensitive to process variations. For instance, if the steady-state gain of P(s) in eq 33 is changed from 2 to 2.3: 2.3 P(s) = e − 5s 10s + 1

(35)

The fitness in eq 29 between y(t) and the simulated output ŷ(t) based on P̂(s) is 98.8%, saying that the quality of P̂(s) is very good. By choosing the maximum sensitivity Ms,0 = 1.6, the desired closed-loop time constant τc is determined as τc = 1.36θ̂ = 6.6994. On the basis of P̂ (s) in eq 35 and τc = 6.6994, the controller parameters in eqs 4a and 4b are retuned to Kc = 0.4832 and Tc = 8.9965. The closed-loop responses y(t) and u(t) after retuning are shown as dashed lines in Figure 3b,c, from which the performance is assessed again under the same value of τc = 6.6994 to give the results in the second row in Table 2. As expected, η̂IAE = 0.9924 after retuning is very close to 1. 5.3. Example 3. As stated in section 1, the MVC-based index can be used to assess the performance of rejecting load disturbances for PI controllers. This example compares the MVC-based index with the proposed DS-d IAE-based index. Calculating the MVC index for PI controllers requires the almost same information as the proposed index in this paper, namely, the disturbance gain Kd and the process information of P(s) in eq 2 in the form of impulse response coefficients. For a fair comparison, P(s) and Kd are assumed to be known, and the configuration used in this example is the same as that in example 1. For the MVC-based index, the minimum achievable output variance is 0.0668 and the corresponding settings of the PI controller are Kc = 1.3426 and Tc = 8.4298, which are obtained by generalizing the method proposed by Ko and Edgar5 for the problem considered in this paper. The DS-d method gives τc = 1.38θ = 6.9 so that Ms in eq 15 reaches the recommended upper bound of Ms, Ms,0 = 1.6, and yields the PI controller parameters Kc = 0.4957 and Tc = 9.3593. The corresponding output variance is 0.3209. Figure 4 presents the process outputs and control signals for the MVC-based controller (dashed lines) and the DS-d-based one (solid lines). Table 3 lists the actual IAE from eq 12, IAE0 from eq 13, and the DS-d IAEbased index ηIAE from eq 14, as well as Ms from eq 15 and TV from eq 16, for the two controllers.

then the MVC-based controller leads to an unstable closedloop control. The corresponding process output y(t) and control signal u(t) are shown as dashed lines in Figure 5. By contrast, the DS-based controller is robust to such a process variation: y(t) and u(t) in Figure 5 (solid lines) are similar to the counterparts in Figure 4, and the performance metrics IAEactual = 51.336 and TV = 3.1720 are close to the corresponding values in Table 3.

6. EXPERIMENTAL EXAMPLES In this section, two experiments are carried out at a laboratory at the Peking University to validate the effectiveness of the proposed performance index. The experimental setup is schematically shown in Figure 6, and the control loop is in the same configuration as that in Figure 1. In the experiments, the process is a water tank system, whose cross-sectional area is about 320 cm2. The water level of the tank is selected as the process output y(t). The set point r(t) takes a constant value 30. The discrete-time counterpart of a PI controller C(s) is implemented with the sampling period 0.5 s at a DCS platform of Siemens PCS7. Two control devices are used to inject water into the tank: the controller output u(t) is sent to a frequency convertor to change the inlet flow rate, while the opening position of an electrical control valve is used as the disturbance source d0(t). In the first experiment, the PI controller in eq 3 takes the parameters Kc = 0.2 and Tc = 40. The measured position of electrical control valve d0(t), process output y(t), and control 11750



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Table 3. Performance Metrics of PI Controllers Based on the MVC and DS-d Methods DS-d MVC

Kc

Tc

IAEactual

Ms

TV

IAE0

ηIAE

0.4957 1.3426

9.3593 8.4298

51.3248 33.0518

1.6 12.4213

2.9849 10.8149

51.3248 51.3248

1 0.644

Figure 7. Signals in the first experiment: (a) disturbance source d0(t); (b) () actual output y(t) and (---) simulated output ŷ(t) based on the estimated process model in eq 36; (c) control signal u(t).

Figure 5. Signals in example 3 with process variation: (a) y(t) and (b) u(t) for (---) MVC-based controller and () DS-d-based controller.

Table 4. Comparison of Control-Loop Performance for the Two Experiments expt

Kc

Tc

IAEactual

IÂ E0

η̂IAE

first second

0.3 1.3276

40 50.3499

2887.3 460.7582

425.3 415.4579

0.1473 0.9017

27.099, eqs 4a and 4b give the recommended controller parameters Kc = 1.3276 and Tc = 50.3499. The second experiment is implemented with the aboverecommended controller parameters Kc = 1.3276 and Tc = 50.3499. The corresponding signals d0(t), y(t), and u(t) are shown in Figure 8. The steps summarized in section 4.4 are implemented with the same value of τc = 27.099 to give the Figure 6. Schematic diagram of the experimental setup.

signal u(t) are shown in Figure 7 panels a−c, respectively. The steps summarized in section 4.4 are implemented to yield the results listed in the first row in Table 4. In particular, the estimated disturbance gain is K̂ d = 0.5906 and the estimated process model is 5.7830 P(̂ s) = e−2.7099s (36) 132.7851s + 1 The fitness in eq 29 between y(t) and the simulated output ŷ(t) (thick dashed line in Figure 7b) is 91.28%, so that the estimated model is reliable. The desired time constant τc of the closedloop response is selected as τc = 10θ̂ = 27.099 and the resulting maximum sensitivity is M̂ s = 1.16. Note that this sensitivity value is chosen to be less than 1.6 in order to accommodate more process modeling errors. In Table 4, the performance index for the current closed-loop response is η̂IAE = 0.1473, which indicates that there is a large space for improvement. On the basis of the estimated FOPDT model in eq 36 and τc =

Figure 8. Signals in the second experiment: (a) disturbance source d0(t); (b) () actual output y(t) and (---) simulated output ŷ(t) based on the estimated process model in eq 37; (c) control signal u(t). 11751



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(10) Peter, K.; Alexander, H. Detection of sluggish control loops: experiences and improvements. Control Eng. Pract. 2005, 13, 1029− 1025. (11) Hägglund, T. Industrial implementation of on-line performance monitoring tools. Control Eng. Pract. 2005, 13, 1383−1390. (12) Visioli, A. Method for proportional-integral controller tuning assessment. Ind. Eng. Chem. Res. 2006, 45, 2741−2747. (13) Salsbury, T. I. A practical method for assessing the performance of control loops subject to random load changes. J. Process Control 2005, 15, 393−405. (14) Howard, R.; Douglas, C. A novel pattern-based approach for diagnostic controller performance monitoring. Control Eng. Pract. 2010, 18, 279−288. (15) Chen, D.; Seborg, E. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807−4822. (16) Qin, S. J. Control performance monitoring − a review and assessment. Comput. Chem. Eng. 1998, 23, 173−186. (17) Uduehi, D.; Ordys, A. W.; Grimble, M. J.; Majecki, P.; Xia, H. Controller benchmarking proceduresdata-driven methods. In Process Control Performance Assessment: From Theory to Implementation; Oredys, A. W., Uduehi, D., Johnson, M. A., Eds.; Springer: London, 2007; pp 81−126. (18) Michael, W. F.; Julien, R. H.; Brian, R. C. A comparision of PID controller tuning methods. Can. J. Chem. Eng. 2005, 83, 712−722. (19) Matausek, M. R.; Micic, A. D. A modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Trans. Automatic Control 1996, 41, 1199−1203. (20) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291−309. (21) Liu, T.; Gao, F. New insight into internal model control filter design for load disturbance rejection. IET Control Theory Appl. 2010, 4, 448−460. (22) Alberto, L. Autotuning process controller with improved load disturbance rejection. J. Process Control 2005, 15, 223−234. (23) Skogestad, S. Tuning for smooth PID control with acceptable disturbance rejection. Ind. Eng. Chem. Res. 2006, 45, 7817−7822. (24) Seborg, D.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, 2nd ed.; John Wiley & Sons: New York, 2004. (25) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64−759. (26) Veronesi, M.; Visioli, A. Performance assessment and retuning of PID controllers. Ind. Eng. Chem. Res. 2009, 48, 2616−2623.

performance assessment results listed in the second row in Table 4. The estimated model in the second experiment is P(̂ s) =

5.6750 e−2.7831s 136.3702s + 1

(37)

and the fitness in eq 29 between y(t) and ŷ(t) (thick dashed line in Figure 8b) is 75.1%. Note that this fitness is smaller than the counterpart in the first experiment, due to the larger measurements. Even so, the main characteristics of y(t) are well captured by ŷ(t) so that the estimated model is still reliable. In Table 4, the actual IAE in the second experiment is significantly reduced, and as expected, the performance index η̂IAE = 0.9017 is close to 1.

7. CONCLUSION This paper studied the problem of assessing the performance of a PI controller in rejecting input load disturbances. By adopting the DS-d method, the lower bound of the IAE was established in eq 13 from closed-loop responses subject to step or other types of input disturbance changes. Taking the lower bound as the benchmark, the DS-d IAE-based performance index in eq 14 was proposed to assess the performance of rejecting input load disturbances. A novel seminonparametric method was proposed to estimate the parameters required by the calculation of the performance index, with the detailed steps summarized in section 4.4. Simulation and experimental examples were provided to verify the performance benchmark and to illustrate the effectiveness of the proposed performance index.

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AUTHOR INFORMATION

Corresponding Author

* E-mail: [email protected]. Telephone: +86 (10) 62753856. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was partially supported by the National Natural Science Foundation of China under Grants 61074105 and 61061130559. We thank the anonymous reviewers for their constructive comments and helpful suggestions.

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REFERENCES

(1) Aström, K. J.; Hägglund, T. Advanced PID Control; International Society of Automation: Research Triangle Park, NC, 2005. (2) Jelali, M. An overview of control performance assessment technology and industrial applications. Control Eng. Pract. 2006, 14, 441−466. (3) Yu, Z.; Wang, J.; Huang, B.; Bi, Z. Performance assessment and retuning of PID controllers. J. Process Control 2011, 21, 1164−1171. (4) Harris, T. J. Assessment of closed loop performance. Can. J. Chem. Eng. 1989, 67, 856−861. (5) Ko, B. S.; Edgar, T. F. PID control performance assessment: The single-loop case. AIChE J. 2004, 50 (6), 1211−1218. (6) Jain, M.; Lakshminarayanan, S. A filter-based approach for performance assessment and enhancement of SISO control systems. Ind. Eng. Chem. Res. 2005, 44, 8260−8276. (7) Sendjaja, A. Y.; Kariwala, V. Achievable PID performance using sums of squares programming. J. Process Control 2009, 19, 1061−1065. (8) Huang, B.; Shah, S. L. Performance Assessment of Control Loops: Theory and Applications; Springer-Verlag: Berlin, 1999. (9) Hägglund, T. Automatic detection of sluggish control loops. Control Eng. Pract. 1999, 7, 1505−1511. 11752


Assessment of PI control loop performance for input load disturbance

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