A Novel Data-Driven Method for Simultaneous Performance

Article pubs.acs.org/IECR

A Novel Data-Driven Method for Simultaneous Performance Assessment and Retuning of PID Controllers Xinqing Gao,†,‡ Fan Yang,†,‡ Chao Shang,†,‡ and Dexian Huang*,†,‡ †

Department of Automation, Tsinghua University, Beijing 100084, China Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, China

‡

ABSTRACT: A novel data-driven method for simultaneous performance assessment and retuning PID controllers is presented in this study. The PID tuning problem is cast as a convex approximation problem of a reference model, which makes the resulting closed-loop step response resemble that of a predefined reference model. The proposed data-driven method only requires the current PID controller parameters and a set of closed-loop data with set point changes, while the process model is neither necessary to be known a priori nor to be identified directly. The controller effort limitation, which is closely related to variations of controller actions, is an important concern for controller design. To evaluate PID controller performance subject to control effort limitations, a trade-off curve displaying the integrated squared error against the total squared variation of controller actions is obtained, which can be calculated in a cost efficient way and serves as a useful tool for performance assessment and retuning of PID controllers. Illustrative case studies are presented to demonstrate the applicability of the proposed method.

1. INTRODUCTION Even though the importance of controllers in the process industry is explosively growing, a large portion of controllers are operating at unhealthy statuses due to various factors such as improper controller design, strong external disturbances or equipment problems.1−3 According to a recent investigation, more than 60% of industrial controllers suffer from certain types of malfunctions.4 Therefore, the technique controller performance assessment (CPA), which aims to detect poor control using routine closed-loop data and then deliver advanced alarm information, has become an attractive research area in the realm of process engineering. Traced back to the work of Harris,5 the most common approach to controller performance assessment relies on a comparison between the performance of monitored controllers and a prespecified benchmark, and a large deviation of the controller performance from the benchmark indicates poor control. The benchmarks are usually based on some kinds of optimal controllers that define the best achievable performance of monitored controllers, such as the MVC (minimum variance control) benchmarks,5 and FFC&FBC (feedforward and feedback control) benchmarks.6−8 From a practical perspective, pursuing theoretically optimal controller performance regardless of control effort limitations is not always advisible due to potential energy costs and safety risks.8 The controller effort is closely related to variations of controller actions, and performance benchmarks with control effort limitations considered are more reasonable references for monitored controllers. A straightforward and effective approach is to integrate the controller effort index and output performance index so as to better assess the control performance. An overall © 2017 American Chemical Society

performance IAE-TV index was defined as the product between the integrated absolute error (IAE) index and the total variation (TV) index, which can provide more reasonable monitoring information considering controller effort issues.9 Another approach is to resort to a trade-off curve displaying the optimal process output performance against the corresponding controller efforts. The trade-off curve for controller performance assessment is first proposed in the framework of linear quadratic Gaussian (LQG) benchmarks,10,11 where poor control can be detected if the current controller performance is far away from the trade-off curve. Nevertheless, the LQG benchmarks impose no structure limitations on monitored controllers. The vast majority of controllers are PID types,12 and industrial PID controllers commonly fail to achieve the performance defined by LQG benchmarks due to structural limitations. Consequently, it is necessary to establish accessible performance benchmarks for PID controllers. For general approaches, explicit process models are necessary,13 which are not always known a priori in industrial applications. A filter-based approach was proposed to derive the trade-off curve,14 where explicit process models are no longer needed. Nevertheless, this curve is derived by solving a nonlinear and nonconvex optimization problem, leading to a high computational burden that prohibits online implementation of control performance assessment. For retuning poorly behaved PID controllers, model-based methods have drawn considerable attention from researchers. Received: Revised: Accepted: Published: 2127

October 8, 2016 December 26, 2016 February 3, 2017 February 3, 2017 DOI: 10.1021/acs.iecr.6b03893 Ind. Eng. Chem. Res. 2017, 56, 2127−2139

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Industrial & Engineering Chemistry Research

behaved controllers, appropriate retuning parameters are derived with practical considerations about loop robustness. Compared to conventional model-based methods that require model reduction, the proposed data-driven method can achieve improved performance for processes that exhibit higher-order dynamics and cannot be well approached by lower-order linear models. The rest of this paper is organized as follows. In section 2, the data-driven PID design method is presented. In section 3, the performance assessment and retuning method based on the datadriven approach is depicted, and some implementation issues are discussed as well. In sections 4 and 5, both simulation and industrial case studies are presented to validate the effectiveness of the proposed method. Some concluding remarks are provided in the final section.

The majority of these methods generally involve the following two steps.15 First, low-order transfer function models, such as first- and second-order plus time delay models (FOPTD, SOPTD), are employed to approximate the process dynamics. Second, controller tuning parameters are derived on the basis of these simplified models, such as the internal model control (IMC) tuning16 and the approximated M-constrained integral gain optimization (AMIGO) tuning.17 A practical problem of these model-based methods is the modeling error arising from the model reduction. Classic model-based tuning strategies are based on the postulation that simplified FOPTD or SOPTD models suffice to describe the process dynamics of controlled processes. For processes exhibiting evident higher-order dynamics, however, modeling errors may be nonnegligible, which would result in a significant gap between the desired and resulting closed-loop performance. Beyond the model accuracy issue, another concern is that the process model for PID design is not always available in practice. To cope with this problem, a significant portion of PID autotuning methods would estimate process parameters related to PID design using experimental data with proper excitations under closed-loop condition.18,19 To address these issues, a novel data-driven method for simultaneous performance assessment and retuning of industrial PID controllers is presented. The proposed method requires the current PID parameters and process output data subject to step set point changes, and these requirements can usually be guaranteed without much difficulty in practice. The data-driven PID design methods have already gained considerable interest from academia recently, and approaches in this vein include the iterative feedback tuning (IFT),20 fictitious reference iterative tuning (FRIT),21,22 and virtual reference feedback tuning (VRFT).23,24 For the IFT method, iterative experiments and optimizations are involved, which may restrict the applicability in practice. This drawback is alleviated by the FRIT and VRFT methods, and recently the VRFT method has been extended to the continuous framework25 and closed-loop condition26,27 for improved applicability. In common with these existing datadriven methods, the proposed method utilizes a reference model to describe the tuning objective, and the process model is not necessarily known a priori. Nevertheless, what makes the proposed method different from these existing data-driven methods is that the expected PID parameters are derived by shaping the closed-loop step response to that of the reference model directly rather than minimizing the divergence between the experiment signal and the virtual signals in different forms generated by the reference model. For conventional data-driven approaches, only the PID tuning parameters are derived, and the closed-loop response of the retuned controller still remains unknown before the retuned PID controller is implemented. In contrast, the resulting closed-loop response can be estimated directly without resorting to practical implementations of retuned controllers by the proposed approach, and thus related key control performance indices, such as the integrated square error (ISE) and the total squared variations (TSV) of controller actions, can be calculated as well. Hence the proposed method is more suitable for controller performance assessment because the performance benchmark, which is actually the performance of well-behaved controllers, can be estimated based on the proposed method. To establish a reasonable benchmark with considerations on controller effort limitations, a trade-off curve displaying the ISE against the TSV of controller actions is obtained in a cost efficient way and serves as a useful tool for online controller performance assessment. To retune poorly

2. DATA-DRIVEN PID TUNING METHOD 2.1. Problem formulation. Consider the following SISO feedback loop shown in Figure 1. The process to be controlled

Figure 1. SISO feedback control loop.

is denoted by G(s), and the mathematical expression of the PID controller C(s) reads as ⎛ ⎞ 1 + Tds⎟ C(s) = K p⎜1 + Tsi ⎝ ⎠

(1)

where Kp, Ti, and Td represent the controller gain, integral time, and derivative time, respectively. The servo model T(s) from set point R(s) to the process output Y(s) can be derived as T (s ) =

Y (s ) G(s)C(s) = 1 + G(s)C(s) R (s )

(2)

The servo model T(s) is useful to quantify the tracking performance. In the stage of controller design, it is common to utilize a reference model Tr(s) defined in eq 3 to describe the desired closed-loop response:25,26 Y (s) = Tr(s)R(s)

(3)

Consequently, the objective of our method is to derive the retuned controller C̅ (s) that best shapes the resulting servo model T̅ (s) to the predefined reference model Tr(s). For the proposed method, closed-loop response data with set point changes are required to estimate the servo model T(s) of the initial PID controller C(s), and this prerequisite can usually be guaranteed without much difficulty in practice. 2.2. Data-Driven PID Controller Tuning Method Based on Convex Optimization. For simplicity of mathematical formulas, the form of PID controllers is reformulated as K C(s) = K p + i + Kds (4) s The retuned controller C̅ (s) is parametrized as C̅(s) = (K p + K pΔ) +

(K i + K iΔ) + (Kd + KdΔ)s s

(5)

Since the parameters (Kp, Ki, Kd) are already known a priori, the parameters (KΔp , KΔi , KΔd ) should be optimized to enable the 2128

DOI: 10.1021/acs.iecr.6b03893 Ind. Eng. Chem. Res. 2017, 56, 2127−2139

Article

Industrial & Engineering Chemistry Research resulting servo model T̅ (s) after C̅ (s) is complemented to approximate the reference model Tr(s) as closely as possible. The controller C̅ (s) is reformulated as C̅(s) = C(s)(1 + Δ(s))

effective approach to transfer function approximations. To guarantee a satisfactory approximation of the step responses, the length N·Ts of the transient should be longer than the settling time of the involved transfer function models in eq 12. A reasonable choice of the transient length is to enforce N·Ts to be equal to the sum of the settling time of Tr(s) and T(s). For optimization problem 12, it should be noted that both matrices Φ and Ω are constant. Thus, the Hessian matrix ∇2θ(J) of the optimization objective with respect to θ can be derived as

(6)

and the term Δ(s) can hence be derived as K pΔ +

Δ(s) =

Kp +

K iΔ s Ki s

+ KdΔs + Kds

(7)

∇2θ (J ) = ΦTΦ

The divergence between T(s) and T̅ (s) can be revealed by the ratio between the sensitivity functions:

It indicates that the unconstrained optimization problem 12 is convex since ∇2θ(J) is positive semidefinite.28 Hence the optimal solution θ* must meet the following condition:

1 − T (s ) S( s ) 1 + G(s)C(s) + G(s)C(s)Δ(s) = = 1 − T ̅ (s ) S ̅ (s ) 1 + G(s)C(s)

∇θ*(J ) = (ΦTΦ)θ* − ΦTΩ = 0

= 1 + T (s)Δ(s)

where is the gradient of J taken with respect to θ. Proposition 1. Assume that the current PID controller C(s) ≠ 0 and the servo model T(s) ≠ 0, then the matrix Φ def ined in (12) has f ull column rank. Proof: We denote by φi the i’th (i = 1,2,3) column vector of Φ. The vector φi is actually the unit step response sequence of the transfer function T(s)Δi(s). Let the vector ν be a linear combination of {φi}: v = α1φ1 + α2φ2 + α3φ3 (15)

where S(s) is the sensitivity function defined as 1 = 1 − T (s ) 1 + G(s)C(s)

(9)

Noting the formulation of Δ(s) in 7, we can rewrite 8 as S(s ) = 1 + T (s)[K pΔΔ1(s) + K iΔΔ2 (s) + KdΔΔ3(s)] S ̅ (s )

Obviously, ν is the unit step response sequence of the transfer function Pα(s):

(10)

where Δ1(s) = Δ 2 (s ) = Δ3(s) =

1 Kp +

Ki s

+ Kds

1/s Kp +

Ki s

Kp +

Ki s

+ Kds

Pα(s) = T (s)(α1Δ1(s) + α2Δ2 (s) + α3Δ3(s))

,

s (11)

(KΔp ,

KΔi ,

θ* = (ΦTΦ)−1ΦTΩ

KΔd )T

The PID parameter vector θ = can hence be derived by minimizing the divergence between S(s)/S̅(s) and S(s)/Sr(s), where Sr(s) = 1 − Tr(s), leading to the following optimization problem: θ

Ω = [Hr(Ts) − 1Hr(2·Ts) − 1⋯Hr(N ·Ts) − 1]T ⎤ ⎥ HΔ2(2· Ts) HΔ3 (2·Ts) ⎥ ⎥ ⎥ ⋮ ⋮ ⎥ 2 3 HΔ(N · Ts) HΔ(N ·Ts)⎥⎦ HΔ2(Ts)

(17)

To derive the retuned PID C̅ (s), the closed-loop servo model T(s) of the initial PID C(s) should be estimated, and the initial PID controller parameters (Kp, Ki, Kd) should be known a priori. In practice, these requirements are not difficult to guarantee. The controller parameters are usually available in practice for computer-based control. In addition, with closed-loop data with set point changes at hand, T(s) can be estimated based on the existing system identification methodologies with set point change signals and process outputs being the model input and output, respectively. Remark 1. To simulate the step responses, the transfer functions involved in 10 must be proper. For a proper transfer function P(s), the following condition must hold:

min J = || Ω − Φθ ||22

⎡ H1 (T ) ⎢ Δ s ⎢ H1 (2·T ) s Φ=⎢ Δ ⎢⋮ ⎢ 1 ⎢⎣ HΔ(N ·Ts)

(16)

Therefore, a zero vector of ν reveals that Pα(s) = 0. Noting the formulations of Δi(s) in eq 11, Pα(s) equals to zero only if αi = 0 (i = 1,2,3). Therefore, the vectors {φi} are linearly independent, and the matrix Φ has full column rank. This completes the proof. On the basis of Proposition 1, ∇2θ(J) is strictly positive definite and nonsingular, and hence the optimal solution θ* can be uniquely determined by

,

+ Kds

(14)

∇2θ(J)

(8)

S(s ) =

(13)

HΔ3 (Ts)

(12)

lim P(s) < ∞

In this formulation, Hr(k·Ts) is the unit step response sample of S(s)/Sr(s) at the time instant k·Ts, HiΔ(k·Ts) (i = 1, 2, 3) are the unit step response samples of the terms T(s)Δi(s), and Ts is a prespecified sampling interval for simulation of these step responses. The problem 12 actually minimizes the difference between the step responses of the transfer functions S(s)/S(̅ s) and S(s)/Sr(s). Since step responses are sufficient to reveal both time-domain and frequency-domain characteristics of parametrized transfer function models, this approach is a simple yet

(18)

s →∞

Furthermore, if P(s) is strictly proper, then lim P(s) = 0

s →∞

(19)

It should be noted that the current servo model T(s) is strictly proper (this condition must be satisfied in industrial applications), and thus the term S(s) remains proper as well. As for the prespecified reference model Tr(s), it must be strictly 2129


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models can still be identified under closed-loop condition and then get utilized in PID retuning. Nevertheless, the proposed approach enjoys the following advantages compared to direct identification of process models. First, step set point changes are common in practice, and hence controller retuning can be achieved using such response data directly based on the optimization problem 22 without identification. If the set point experiences other types of changes, estimation of the servo model is essentially equivalent to open-loop identification, and the closed-loop identifiability issue can be avoided. Second, the data-driven nature of the method leads to improved applicability for processes with higher-order dynamics compared to classic model-based tuning methods. For model-based approaches such as the IMC tuning, model reduction is necessary to determine PID parameters appropriately. For processes with higher-order dynamics, the modeling error may be nonnegligible and tends to affect the controller tuning results. Alternatively, the data-driven approach imposes no assumptions on model order and can yield improved closed-loop performance. Third, the proposed approach is consistent with the objective of controller performance assessment, because the closed-loop step response itself should be estimated to evaluate tracking performance of the monitored controller. Therefore, the proposed data-driven method does not require extra prior information compared to performance assessment of tracking performance, and it is possible to design appropriate performance benchmarks by specifying reasonable reference models. 2.4. Implementation Issues. A very practical issue about the proposed method is the quality of the estimated servo model. According to the analysis in section 2.3, if the set point change signal is not step type, then the closed-loop servo model should be identified with the set point change and process output being the model input and output, respectively. In cases of low signal-to-noise ratio when variations of the process output are dominantly driven by the unmeasured disturbance or random noise rather than the set point change signal, an inaccurate servo model may be estimated and thus is not suitable for PID design. Hence, the quality of the estimated servo model, which is closely related to the signal-to-noise ratio, should be evaluated before PID design, and an unreasonable servo model should not be adopted. The model quality index QSNR is defined as follows:

proper (this issue will be discussed in detail in section 3.1), and it is easy to validate that the term S(s)/Sr(s) is also proper. Noting the formulation of Δi(s) in 11, the terms T(s)Δi(s) are proper as well. Therefore, all the transfer functions involved in 10 are proper, and the corresponding unit step responses can be simulated without difficulty. 2.3. Data-Driven Tuning Using Closed-Loop Step Response Data Directly. As the central part of the proposed method, the optimization problem 12 essentially approximates the step response of the involved transfer functions. It should be noted that the closed-loop operating data subject to step set point changes are common in industrial applications, based on which the unit step response of T(s) is directly available or can be estimated via one-dimensional signal smoothing approaches such as the second-order smoothing technique.28 Therefore, the optimal controller retuned parameters θ* can be derived using such data directly without resorting to identification of closedloop servo models. To accomplish this, we arrive at the following expression by multiplying both sides of eq 10 by Sr(s): S(s)Sr(s) = Sr(s) + T (s)Sr(s)[K pΔΔ1(s) + K iΔΔ2 (s) S ̅ (s ) + KdΔΔ3(s)]

(20)

If the retuned controller parameter θ* can guarantee that the S(̅ s) is exactly equivalent to Sr(s), eq 20 can be reformulated as S(s) = Sr(s) + Sr(s)T (s)[K pΔΔ1(s) + K iΔΔ2 (s) + KdΔΔ3(s)]

(21)

Nevertheless, it is rare that this condition can get fully satisfied, and thus θ* must be optimized to minimize the divergence between two sides of eq 21. On the basisof the same step response approximation strategy, the optimal parameters can be determined by solving the following optimization problem: min J = || Ω s − Φ sθ ||22 θ

Ω s = [Hrs(Ts) − HTs(Ts)Hrs(2·Ts) − HTs(2·Ts) ⋯Hrs(N ·Ts) − HTs(N ·Ts)]T ⎡ H1 (T ) HΔ2 , s(Ts) HΔ3 , s(Ts) ⎤ ⎢ Δ, s s ⎥ ⎢ H1 (2·T ) H2 (2· T ) H3 (2·T ) ⎥ Δ, s Δ, s s s s ⎥ Φ s = ⎢ Δ, s ⎢ ⎥ ⋮ ⋮ ⋮ ⎢ ⎥ ⎢⎣ HΔ1 , s(N ·Ts) HΔ2 , s(N · Ts) HΔ3 , s(N ·Ts)⎥⎦

N

Q SNR = 1 −

∑i = 1 [y(k·Ts) − y ̂(k·Ts)]2 N

∑i = 1 [y(k·Ts) − y ̅ ]2

(23)

where {y(k·Ts)} is the collected process output sequence, {ŷ(k·Ts)} is the simulated output of the estimated servo model, and y ̅ is the mean of {y(k·Ts)}. Actually, the sequence {ŷ(k·Ts)} can be viewed as the trend of the process output driven by the set point change signal. If QSNR is close to 1, it indicates that the model quality is satisfactory, and if QSNR is close to 0, the model quality is considered to be poor. For the proposed method, PID retuning can only be accomplished when the set point change response data are available. In practice, it is possible that set point changes do not always occur, yet this does not mean that our method is not suitable for practical applications. In fact, controller retuning is usually not necessary to be conducted in a real-time manner, and it is practical to attempt PID retuning when there exists set point response data, and meanwhile the signal-to-noise ratio is reasonable. Furthermore, in cases when there is an urgent demand for controller retuning, process operators can introduce proper changes to the set point such as step or ramp changes, and these

(22)

HsT(k·Ts)

where is the unit step response sample of T(s) at the time instant of k·Ts, Hsr(k·Ts) is the unit step response sample of Tr(s), and HiΔ,s(k·Ts) (i = 1, 2, 3) are the unit step response samples of the terms T(s), Sr(s), Δi(s), respectively. Since the parametrized transfer function models Sr(s)Δi(s) are already known a priori, HiΔ,s(k·Ts) can hence be calculated by simulating the transfer functions Sr(s)Δi(s) with HiT(k·Ts) being the model input. In this way all step responses involved in problem 22 can be calculated directly using the closed-loop step response data HsT(k·Ts) without identification of the parametrized servo model T(s). For the proposed method, it must be emphasized that the closed-loop operating data are utilized to estimate the closedloop step response or the servo model from set points to process outputs rather than process models. One may argue that process 2130


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Therefore, the retuned parameters θ can be derived by solving the following optimization:

operations are usually allowable in practice. Hence the usability of the proposed method in industrial applications is ensured. Remark 2. If the initial PID is stable, the effectiveness of the proposed method can be guaranteed despite the divergence of the initial tuning that may generate set point response data with different excitation degrees. Closed-loop process output data from aggressive or sluggish controllers would yield close PID tuning by the proposed method provided that the same reference model is specified. Nevertheless, for unstable initial tuning, the servo model or the closed-loop response can hardly be estimated with sufficient accuracies from closed-loop operating data, and thus the applicability of the proposed method would be narrowed. Another concern is the practical forms of PID controllers. To restrain effects of measurement noise and guarantee the properness of the PID controllers, derivative control actions may be filtered by a first-order linear filter. In addition to the PID controller form formulated in eq 4, commonly applied implementation forms of PID controllers in DCS and PLC are formulated as follows, containing first-order filters to mitigate derivative control actions: 1 C(s) = K (K p + K i /s + Kds) d s + 1 (24) N C(s) =

1 + Kds Kd s N

+1

2 min J = || Ω − Φ̃ θ ||2 θ

Ω = [Hr(Ts) − 1Hr(2·Ts) − 1⋯Hr(N ·Ts) − 1]T ⎡ H̃ 1 (T ) HΔ̃ 2(Ts) HΔ̃ 3(Ts) ⎤ ⎢ Δ s ⎥ ⎢ H̃ 1 (2·T ) H̃ 2(2·T ) H̃ 3(2·T ) ⎥ s s s ⎥ Δ Δ Φ̃ = ⎢ Δ ⎢ ⎥ ⋮ ⋮ ⋮ ⎢ 1 ⎥ 2 3 ⎢⎣ HΔ̃ (N ·Ts) HΔ̃ (N ·Ts) HΔ̃ (N ·Ts)⎥⎦

where Hr(k·Ts) is the unit step response sample of S(s)/Sr(s) at the time instant k·Ts, and H̃ iΔ(k·Ts) are the unit step response samples of the terms T(s)Δ̃i(s) (i = 1, 2, 3), respectively.

3. PERFORMANCE ASSESSMENT AND RETUNING 3.1. Choice of the Reference Model. For the choice of the reference model Tr(s), it is very common in industrial applications to use a first-order plus time delay (FOPTD) model: Tr(s) =

(K p + K i / s ) (25)

Kds K C(s) = K p + i + s Kds /N + 1

(26)

Tr(s) =

⎞ (K i + K iΔ) 1 ⎛ ⎜⎜(K p + K pΔ) + C̅(s) = + (Kd + KdΔ)s⎟⎟ Tf s + 1 ⎝ s ⎠ (27)

(K + KdΔ)s (K i + K iΔ) + d s Tf s + 1

= C(s)(1 + K pΔΔ̃1(s) + K iΔΔ̃2(s) + KdΔΔ̃3(s))

(28)

where the terms Δ̃i(s) (i = 1, 2, 3) are formulated as Δ̃1(s) = Δ̃2(s) = Δ̃3(s) =

1 e−τs λ s + 1.6λs + 1 2 2

1 Kp +

Ki s

+

K ds Tf s + 1

TSV =

1/s Kp +

Ki s

+

Ki s

+

K ds Tf s + 1

1 Ts

(32)

∞

∑ uΔ(i·Ts)2 i=2

uΔ(i) = u(i·Ts) − u((i − 1) ·Ts)

K ds Tf s + 1

(33)

where u(t) is the controller output subject to step set point changes, and Ts is the sampling interval. The problem can thus be expressed as follows: what is the minimal ISE for step set point changes given the control effort constraint TSV ≤ α?

s /(Tf s + 1) Kp +

(31)

The delay τ of Tr(s) should be identical to the process delay, which can be estimated from closed-loop data. Compared to first-order reference models, this underdamped second-order reference model can bring faster responses. The damping ratio of this reference model is 0.8 and the overshoot is about 0.015, and thus a smooth response can still be guaranteed. 3.2. Assessment of Controller Performance Considering the Control Effort Trade-off Issue. Control effort limitations should be taken into consideration in controller design due to its close relationship to the loop stability and robustness. Hence in situations with limited controller efforts, it is reasonable to evaluate the possible improvement in controller performance under the same level of controller effort. The index total squared variation (TSV) of controller actions subject to unit step set point changes is utilized as an index to quantify the control effort and is defined as follows:

where the terms Δi(s) (i = 1, 2, 3) take the same formulation as those defined in eq 11. Therefore, the retuned controller parameters θ* can be derived in the same way as the PID controller forms defined in 4. For PID controllers formulated in eq 26, the retuned PID C̅ (s) can be formulated as C̅(s) = (K p + K pΔ) +

1 e−τs λs + 1

where λ is the user-specified closed-loop time constant with smaller values indicating faster responses. In fact, this objective is rather similar to that of the celebrated IMC tuning rules.29 However, for controlled processes with higher-order dynamics, such a reference model may result in conservative tuning, and the resulting closed-loop response has an excessively long settling time. This issue can be compensated by reference models with slightly underdamped dynamics,26 and hence the following reference model is adopted as an alternative:

Notice that the PID controllers formulated in eq 24 and eq 25 are essentially identical and can be mutually transformed. Denote by Tf = Kd/N the time constant of the filter for derivative actions. If Tf of the retuned controller is set to be equal to that of the PID controllers to be tuned, the remaining PID parameters θ = (KΔp , KΔi , KΔd )T can still be derived by the proposed data-driven method. For PID controllers described by eq 24, the retuned PID C̅ (s) can be formulated as

= C(s)(1 + K pΔΔ1(s) + K iΔΔ2(s) + KdΔΔ3(s))

(30)

(29) 2131


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derived without practical implementation of retuned PID controllers. Because the data-driven tuning method only involves convex optimization, the proposed trade-off curve can be derived in a cost efficient way. Remark 3. Common to the existing data-driven tuning methods, the tuning objective is described by a reference model. Nevertheless, what distinguishes from these conventional methods is that, the closed-loop servo model T̅ (s,λ) of the retuned controller C̅ (s,λ) can be directly estimated without resorting to practical implementation of C̅ (s,λ). Related key control performance indices, such as ISE and TSV, can be directly calculated as well. Therefore, an attractive feature about the proposed method is that PID performance assessment is enabled to be possible with the benchmark specified as the performance of the PID controllers aimed at approximating proper reference models. 3.3. PID Tuning with Loop Robustness Constraints. For a data-driven method, the PID tuning is converted to specifying a proper reference model. The user-specified parameter λ manages the trade-off between loop robustness and control performance. In general, a smaller λ would yield improved performance at the risk of decreasing the loop robustness. Since loop robustness is a very practical concern, λ can be determined based on the following constrained optimization problem:

Similar to the LQG benchmark, the best achievable PID performance considering control effort restrictions can be calculated based on the following optimization problem with the candidate controllers restricted to the PID type: min J(μ) = ISE + μTSV

(34)

PID

By varying μ, various optimal solutions of ISE and TSV subject to unit step set point changes can be obtained, thereby providing a trade-off curve between the optimal ISE and TSV. Any PID controllers can only operate in the region above the performance trade-off curve,14 as shown in Figure 2. Such a trade-off curve

min ISE(λ) λ

s.t.MS(S ̅ (s , λ)) ≤ γ Figure 2. An example of the performance assessment of PID controllers based on the trade-off curve.

The term ISE(λ) denotes the ISE of T̅ (s,λ) subject to the unit step set point change, γ is the user-specified robustness degree, and the following defined Ms is the maximum sensitivity peak with smaller values indicating improved robustness:

furnishes much useful information to assess control performance. For instance, a controller can be quantified to be satisfactorily tuned if the performance is close to this curve, and poor control can be detected if the performance of the monitored controller is far above the curve. Notice that problem 34 is substantially nonconvex. To derive the trade-off curve, this nonconvex optimization problem 34 must be optimized iteratively, resulting in a potentially high computational cost for online implementation of control performance assessment. A reasonable performance trade-off curve can be derived based on the proposed data-driven tuning method. The reference model Tr(s) with smaller λ yields improved control performance and increased control efforts, and hence the trade-off curve can be computed by sampling at different values of λ. Given a prespecified λ, the corresponding controller retuned parameters θ(λ) = (KΔp (λ),KΔi (λ),KΔd (λ))T can be determined directly based optimization problem 12 or 22, and the corresponding Δ(s,λ) can be derived based on 7. The resulting servo model T̅ (s,λ) can hence be derived accordingly: T ̅ (s , λ ) = 1 − S ̅ (s , λ ) = 1 −

S(s ) 1 + T (s)Δ(s , λ)

MS(S ̅ (s , λ)) = max|S ̅ (jω , λ)| ω

(38)

A reasonable choice of γ is 1.8, which can guarantee a phase margin about 32° and a gain margin about 2.25.29 The constraint in 37 is to derive proper PID tuning parameters with sufficient robustness degree. Since the servo model T̅ (s,λ) of the retuned controller can be directly derived based on eq 35, the index Ms of the retuned controller can also be estimated without implementation of the retuned controller. There is only a single optimization parameter, and thus problem 37 can be effectively solved using the golden-section search method. 3.4. Overall Steps of the PID Controller Performance Assessment and Retuning Method. The PID controller performance assessment and retuning method follows this protocol: Step 1: Monitor the set point signal and detect set point changes. Collect set point response data if set point change occurs. Step 2: Estimate the close-loop step response or the servo model using the collected response data and evaluate the model quality QSNR of the estimated step response or servo model via eq 23. If QSNR is low, for example QSNR ≤ 0.6, go to Step 3, otherwise return to Step 1. Step 3: With the current PID parameters available, derive the performance trade-off curve. The delay τ of the reference model in eq 32 is set to be identical to the delay of the estimated servo model or closed-loop step response. By varying λ of the reference model, solve the optimization problem 22 or 30 to derive the corresponding retuned controller parameters θ(λ) = (KpΔ(λ),KiΔ(λ),KdΔ(λ))T.

(35)

Thus, the control error e(t) of the retuned controller subject to unit step set point changes can be estimated directly based on T̅ (s,λ) without resorting to practical implementation of the retuned controller C̅ (s,λ). In addition, the corresponding controller output u(t) can be estimated as well:

u(t ) = C̅(s , λ)e(t )

(37)

(36)

where C̅ (s,λ) is the retuned PID controller formulated in eq 6. Hence the resulting ISE and TSV of C̅ (s,λ) subject to unit step set point changes can be directly estimated, and the trade-off curve is 2132


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Figure 3. Process outputs and the estimated closed-loop step response.

Then calculate the resulting servo model T̅ (s,λ) via eq 35 to estimate ISE(λ) and TSV(λ) subject to step set-point changes. With TSV(λ) against ISE(λ) for various λ values computed, the performance trade-off curve can be derived. Step 4: Assess control performance. Estimate the ISE and TSV subject to step set-point changes of the monitored PID and compare the performance to the trade-off curve. If the performance is close to the trade-off curve, then the controller tuning is reasonable, otherwise go to the controller retuning step. Step 5: Calculate the PID parameters with expected robustness degree through optimization 37 to retune the poorly behaved controller.

4. SIMULATION CASE STUDIES In this section, various numerical examples covering different types of process dynamics are presented to illustrate the effectiveness of the proposed method. In all cases, there are unit step changes in the set-point with Gaussian white noise added to process outputs. To estimate the closed-loop step response, the method in the literature30 is adopted due to the robustness for measurement noise. The performance of the proposed PID is compared with that of commonly applied model-based tuning methods, including IMC-PID,16 SIMC-PID,15 and AMIGO-PID.17 The IMC-based PID is designed based on SOPTD models, and the AMIGO-PID is tuned based on FOPTD models. 4.1. SOPTD Process with a Large Input Delay. Consider the following SOPTD model with a large delay: G1(s) =

1 e−20s (20s + 1)(5s + 1)

Figure 4. Performance trade-off curve using different sets of closed-loop data.

curve based on these different sets of closed-loop data were derived and are plotted in Figure 4. It can be observed that there is little difference between these performance trade-off curves, which shows that the initial controller tuning has little influence on the proposed method. On the basis of the trade-off curve, the performance of these initial controllers can be assessed, as shown in Figure 4. It should be noted that there is an offset for C3(s), and the corresponding ISE against TSV is not plotted because the ISE is actually infinite. The performance of C1(s) is far from the trade-off curve and considered to be poor. For C2(s), the ISE is about 73 and can be significantly reduced by the proposed method. These results indicate that the controllers C1(s) ∼ C3(s) still have ample room for improvement of control performance. To retune these poorly behaved PID controllers, the optimization problem 37 was solved with the maximum sensitivity peak Ms restricted to be less than 1.8. The initial controllers were retuned to be C1*(s) ∼ C*3 (s), where C*i (s) (i = 1, 2, 3) values are derived using closedloop data generated by Ci(s), respectively. The parameters of these retuned PIDs are listed in Table 1. Owing to the inevitable modeling error of the estimated close-loop step responses, the actual Ms of the retuned controllers was slightly larger than 1.8, yet the resulting tuning is still reasonable because the actual Ms is close to the prespecified value. To quantify tracking and disturbance rejection performance, the ISE indices ISEsp and

(39)

As discussed in the aforementioned analysis, the proposed method is insensitive to the initial tuning. To illustrate this feature, three different sets of closed-loop data are generated for the controller performance assessment and retuning, as shown in Figure 3. The first set of closed-loop data is generated by an

(

1 15s

) 1 is generated by a sluggish controller C2(s) = 0.4(1 + 60s ), while aggressive controller C1(s) = 0.8 1 +

+ 1s , the second set

the third set is generated by a P controller C3(s) = 0.6. To attempt PID performance assessment and retuning, the closed-loop step responses were estimated and shown in Figure 3 as well. By varying λ of the reference model Tr(s), the performance trade-off 2133


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Industrial & Engineering Chemistry Research Table 1. Performance Indices of the PID after Retuning PID parameters design methods C1*(s) C*2 (s) C3*(s)

Kp 0.980 0.984 0.983

Ti 29.2 31.2 30.1

Table 2. PID Controllers Obtained by Various Design Methods for G1(s)

performance indices Td 7.72 7.97 7.64

ISEsp 27.13 27.01 27.15

ISEload 13.61 13.81 13.73

PID parameters

Ms 1.803 1.806 1.806

ISEload subject to the unit step set point change and load disturbance are computed and listed in Table 1. From these results, it can be concluded that the proposed method has improved the closed-loop performance significantly. Meanwhile, the parameters of C1*(s) ∼ C3*(s) are close, which validates that the proposed method is insensitive to the initial tuning. For comparison purposes, the proposed method is compared to IMC-PID, SIMC-PID, and AMIGO-PID. The SIMC-PID and IMC-PID are tuned to achieve the lowest ISE subject to the step set point change. The AMIGO tuning rule is based on the FOPTD model, and the following FOPTD model identified based on an open-loop step test is used: 1 G1f (s) = e−23.0s (40) 22.11s + 1

performance indices

design methods

Kp

Ti

Td

ISEsp

ISEload

Ms

proposed PID SIMC-PID IMC-PID AMIGO-PID

0.984 0.798 0.924 0.882

31.2 20 25 21.1

7.97 5 4 7.75

27.01 32.19 30.68 28.61

13.81 16.49 15.37 14.14

1.806 2.062 2.099 1.810

point changes can be reduced by the data-driven PID given the same control effort. 4.2. Higher-Order Processes. In addition to G1(s) studied above, the following higher-order processes are considered: 1 G2(s) = 2 e − 3s (s + 10s + 1)(s + 1)2 (41) G3(s) =

1 (s + 1)20

(42)

G4(s) =

−1.5s + 1 e−1.5s 3 (2s + 1)(s + 1)

(43)

Model reduction is necessary to derive model-based controllers. The simplified FOPTD and SOPTD models for these higher order processes are derived based on open-loop step tests and listed as follows. The process G2(s) can be approached as 1 G2f (s) = e−4.86s , 10.42s + 1 1 G2s(s) = e−3.59s (9.84s + 1)(1.51s + 1) (44)

For AMIGO-PID, there is one design parameter M that manages the balance between the closed-loop response and loop robustness,17 which is carefully selected to achieve a lower ISE while the closed-loop response is smooth. To compare the tracking and disturbance rejection performance, the left panel of Figure 5 plots the closed-loop response subject to a unit step set point change at t = 0 and a step load disturbance of magnitude −1 at t = 300, and Table 2 summarizes the tuning parameters and the corresponding performance indices of different PID design methods. It can be observed that the proposed PID can achieve the lowest ISE for both the step set point change and load disturbance, while the closed-loop response is still smooth with a slight overshoot and has the highest robustness degree. Furthermore, the trade-off curve in the right panel of Figure 5 indicates that the proposed PID can derive more reasonable tuning parameters. The performance of the model-based PID controllers lies above the trade-off curve. Since each point of the trade-off curve stands for the performance of the data-driven PID for a specified λ, it implies that the proposed method can achieve superior control performance because the ISE for set

For G3(s), however, the process dynamics is more complicated. In addition, there are repeated poles and thus dynamics would embody underdamped properties. By open-loop step tests, G3(s) can be approximated as 1 G3f (s) = e−15.0s , 5.99s + 1 1 G3s(s) = e−12.1s 2 (45) 20.66s + 7.386s + 1 For the SIMC tuning rule, it requires that all poles of simplified SOPTD models must be real, and hence the half-model

Figure 5. Performance assessment of different PID tuning methods for G1(s). 2134


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Industrial & Engineering Chemistry Research Table 3. PID Controllers Obtained by Various Design Methods for G2(s)−G4(s) PID parameters

performance indices

process

design methods

Kp

Ti

Td

ISEsp

ISEload

Ms

G2(s)

proposed PID SIMC-PID IMC-PID AMIGO-PID proposed PID SIMC-PID IMC-PID AMIGO-PID Proposed PID SIMC-PID IMC-PID AMIGO-PID

2.04 2.31 2.31 1.69 0.573 0.321 0.445 0.501 0.643 0.401 0.590 0.611

12.1 9.84 11.35 6.87 10.90 6.05 7.39 9.04 4.55 2.56 3.66 3.56

1.91 1.51 1.31 1.83 4.09 1.48 2.80 3.97 1.48 1.00 0.97 1.51

5.23 5.54 5.60 6.08 16.42 21.35 19.02 17.15 6.27 7.82 6.91 6.47

1.39 1.09 1.19 1.41 12.91

1.804 2.073 2.106 1.700 1.808 2.090 2.086 1.747 1.811 2.108 2.073 1.882

G3(s)

G4(s)

17.77

15.31 13.68 4.53 6.00 5.04 4.62

Figure 6. Performance assessment of different PID tuning methods (G2(s)−G4(s)).

reduction rule proposed in the literature15 is adopted to derive the SIMC-PID: G3si(s) =

1 e−11.6s (6.05s + 1)(1.48s + 1)

1 e−5.1s , 2.85s + 1 1 G4s(s) = e−4.1s 3.55s 2 + 3.65s + 1

G4f (s) =

(47)

(46)

The process G4(s) also exhibits slightly underdamped dynamics, and the SIMC-PID is designed based on G4si(s) that is obtained by the same half-model reduction rule:15

The process G4(s) has a negative zero. On the basis of step tests, G4(s) can be approximated by the following simplified models: 2135


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Industrial & Engineering Chemistry Research G4si(s) =

1 e−4.2s (2.56s + 1)(s + 1)

For model-based controllers, an accurate process model is required to derive reasonable controller parameters. In these higher-order cases, however, the modeling error arising from model reduction is still non-negligible. This issue restricts the performance of the model-based controllers, and the response speed should be restricted to avoid loop oscillatory behaviors. As a consequence, the performance lies above the trade-off curve, which means that the closed-loop response speed can be reasonably accelerated by the data-driven PID. This is confirmed in Figure 6, and it can be seen that the closed-loop response of the data-driven PID is more rapid and smooth in comparison with that of the model-based ones. The data-driven PID imposes no limitations on model order and directly derives the controllers that can best approach the smooth reference model with rapid responses, and this is the main reason why the closed-loop performance is superior. 4.3. Nonlinear Process. Consider a continuous polymerization reaction process that takes place in a jacketed CSTR.31 The process dynamics is described by the following differential equations:

(48)

The proposed PID is tuned based on eq 37 with Ms restricted not to exceed 1.8. The IMC-PID and SIMC-PID are tuned to achieve the lowest ISE, while the AMIGO-PID is tuned to achieve an ISE that is as low as possible while the closed-loop response is still smooth. The closed-loop performance indices of different PID controllers are listed in Table 3. In addition, the closed-loop responses subject to the step set-point change and load disturbance together with the performance trade-off curve are shown in Figure 6. It can be observed that the proposed PID can achieve improved performance for both set-point tracking and load disturbance rejection for almost all cases, except for G2(s) where the ISEsp is larger than that of the IMC-based PIDs. Nevertheless, it should be noted the Ms of the IMC-based PIDs is much larger than that of the proposed method. If the Ms of the proposed PID is specified to be 2.1 such that the robustness degree is similar to the IMC-based PIDs, the resulting PID

(

is C(s) = 2.35 1 +

1 12.21s

F(Cmin − Cm) dC1 dCm , = −(kp + k f )CmP0 + m V dt dt FI C Iin − FCI = −kICI + V FD0 dD1 dD0 , = (0.5k Tc + k Td)P02 + k fmCmP0 − V dt dt FD1 = Mm(kp + k fm)CmP0 − V

)

+ 2.24s with ISE sp and ISE load

reduced to 4.90 and 1.11, respectively. For this tuning, the load disturbance rejection performance is comparable to the IMC-based PIDs, while the tracking performance is better. Even though the FOPTD and SOPTD models that can best approach the process dynamics are adopted to design modelbased controller in this study, the closed-loop performance is still inferior to that of the data-driven PID according to the results. Table 4. PID Controllers Obtained by Various Design Methods for the CSTR Process

⎛ 2fk C ⎞0.5 I I ⎟ P0 = ⎜⎜ ⎟ , k + ⎝ Tc k Td ⎠

performance indices

PID parameters design methods

Kp

Ti

Td

ISEsp

ISEload

proposed PID SIMC-PID IMC-PID AMIGO-PID

−1.210 × 10−5 −1.220 × 10−5 −1.290 × 10−5 −4.345 × 10−6

0.141 0.122 0.149 0.0761

0.060 0.027 0.022 0.017

0.0214 0.0286 0.0294 0.0709

0.0116 0.0112 0.0119 0.0660

y=

D1 , u = FI D0

(49)

The control objective is to regulate the product numberaverage molecular weight (y = D1/D0) by manipulating the flow rate of the initiator (u = FI). The model parameters and the steady conditions are the same as those adopted in the literature.31 The model-based controllers are tuned based on the

Figure 7. Performance assessment of different PID tuning methods on the CSTR process. 2136


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yields a much smaller overshoot and the closed-loop response is smoother than those of model-based PIDs. This example demonstrates that the data-driven method is promising for nonlinear processes.

5. INDUSTRIAL CASE STUDY In this section, the proposed method is validated on an asphalt furnace temperature controller in a refinery plant in north China. The brief sketch of the asphalt furnace temperature control system is shown in Figure 8. The fuel and air are mixed and burnt in the combustion chamber, and the asphaltene solution is preheated in the furnace tube by the heat generated in the chamber. The outlet temperature is the key controlled variable and regulated by a PID controller, and the control performance would be assessed by the proposed method. The initial PID controller is ⎛ ⎞ 1 C(s) = 0.55⎜1 + + 20s⎟ ⎝ ⎠ 360s

and the closed-loop response with set point changes were collected for PID control performance assessment and retuning, as shown in Figure 9. Using the estimated closed-loop responses, the trade-off curve was derived and is plotted in Figure 10.

Figure 8. Sketch of the asphalt furnace with temperature control.

Figure 9. Estimated closed-loop response of the PID before maintenance. Figure 10. Trade-off curve and the performance of the monitored controller.

following models identified via step tests: −5.58 × 105 −0.0438s e , 0.141s + 1 −5.49 × 105 M 2 (s ) = e−0.039s (0.122s + 1)(0.027s + 1)

It can be observed that the performance of C(s) is far from the trade-off curve. Meanwhile, the large overshoot and long settling time implies that the controller tuning was overly tight. On the basis of the proposed data-driven approach, the controller

M1(s) =

(50)

(

was suggested to be retuned as C*(s) = 0.35 1 +

The tuning parameters for the model-based PID are adjusted by trial and error to achieve a low ISE and a smooth response. Table 4 summarizes the tuning parameters and related performance indices of different methods, and the corresponding closed-loop response subject to the unit step set point change and load disturbance are compared in Figure 7. Clearly, the proposed method can achieve greater tracking performance than the model-based ones because the performance of model-based controllers is farther from the trade-off curve than the proposed method, while the load disturbance rejection performance is comparable to that of the IMC-based PIDs and better than that of the AMIGO-PID. In addition, the data-based controller

1 780s

)

+ 20s .

The closed-loop response of C*(s) is plotted Figure 11, and the control performance is compared with the trade-off curve in Figure 10 as well. Since the performance of C*(s) is close to the trade-off curve and the closed-loop response is rapid and smooth, it can be concluded that the control performance after controller maintenance was satisfactory.

6. CONCLUSIONS In this article, a novel data-driven method is proposed for simultaneous performance assessment and retuning of PID controllers. The central idea is to enable the step response 2137


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(4) Desborough, L.; Miller, R. Increasing customer value of industrial control performance monitoring-Honeywell’s experience. AIChE Symp. Ser. 1998, 169−189. (5) Harris, T. J. Assessment of control loop performance. Can. J. Chem. Eng. 1989, 67, 856−861. (6) Huang, B.; Shah, S. L.; Miller, R. Feedforward plus feedback controller performance assessment of MIMO systems. IEEE Trans. Control Syst. Technol. 2000, 8, 580−587. (7) Desborough, L.; Harris, T. Performance assessment measures for univariate feedforward/feedback control. Can. J. Chem. Eng. 1993, 71, 605−616. (8) Huang, B.; Shah, S. L. Performance assessment of control loops: theory and applications; Springer: Berlin, 1999. (9) Yu, Z.; Wang, J.; Huang, B.; Li, J.; Bi, Z. Performance assessment of industrial linear controllers in univariate control loops for both set point tracking and load disturbance rejection. Ind. Eng. Chem. Res. 2014, 53, 11050−11060. (10) Kadali, R.; Huang, B. Controller performance analysis with LQG benchmark obtained under closed loop conditions. ISA Trans. 2002, 41, 521−537. (11) Pour, N. D.; Huang, B.; Shah, S. L. Performance assessment of advanced supervisory−regulatory control systems with subspace LQG benchmark. Automatica 2010, 46, 1363−1368. (12) Åström, K. J.; Hägglund, T. Advanced PID control; ISA-The Instrumentation; Systems, and Automation Society: Research Triangle Park, NC, 2005. (13) Assessment of achievable PI control performance for linear processes with dead time. Proc. 1998 Am. Control Conf. 1998, 1548− 1552. (14) 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. (15) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291−309. (16) Morari, M.; Zafiriou, E. Robust process control; Prentice Hall: Englewood Cliffs, NJ, 1989. (17) Åström, K.; Hägglund, T. Revisiting the Ziegler−Nichols step response method for PID control. J. Process Control 2004, 14, 635−650. (18) Veronesi, M.; Visioli, A. Simultaneous closed-loop automatic tuning method for cascade controllers. IET Control Theory and Appl. 2011, 5, 263−270. (19) Veronesi, M.; Visioli, A. Performance assessment and retuning of PID controllers. Ind. Eng. Chem. Res. 2009, 48, 2616−2623. (20) Halmarsson, H.; Gevers, M.; Gunnarsson, S.; Lequin, O. Iterative feedback tuning: theory and applications. IEEE Control Syst. Magazine 1998, 18, 26−41. (21) Soma, S.; Kaneko, O.; Fujii, T. A new approach to parameter tuning of controllers by using one-shot experimental data-a proposal of fictitious reference iterative tuning. Transactions of the Institute of Systems, Control and Information Engineers 2004, 17, 528−536. (22) Masuda, S.; Kano, M.; Yasuda, Y. A fictitious reference iterative tuning method with simultaneous delay parameter tuning of the reference model, Int. Conf. Network., Sens. Control 2009, 422−427. (23) Campi, M. C.; Lecchini, A.; Savaresi, S. M. Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automatica 2002, 38, 1337−1346. (24) Kansha, Y.; Hashimoto, Y.; Chiu, M. S. New results on VRFT design of PID controller. Chem. Eng. Res. Des. 2008, 86, 925−931. (25) Yang, X.; Xu, B.; Chiu, M. S. PID controller design directly from plant data. Ind. Eng. Chem. Res. 2010, 50, 1352−1359. (26) Jeng, J. C.; Fu, E. P. Closed-Loop tuning of set-point-weighted proportional-integral-derivative controllers for stable, integrating, and unstable Processes: a unified data-based method. Ind. Eng. Chem. Res. 2015, 54, 1041−1058. (27) Jeng, J. C.; Tseng, W. L.; Chiu, M. S. A one-step tuning method for PID controllers with robustness specification using plant step-response data. Chem. Eng. Res. Des. 2014, 92, 545−558. (28) Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press, 2004.

Figure 11. Closed-loop response of the PID after maintenance.

of the tuned control loop to approach that of a prespecified reference model. The PID tuning problem is converted to a reference model approximation problem, which can be solved using elegant convex optimization techniques, and the controller design procedure can be accomplished in a cost efficient way. A performance trade-off curve can be obtained and serves as a useful tool for assessment of controller performance. In comparison with existing model-based methods, the proposed method imposes no limitations on the model order and can achieve improved control performance, especially for the processes with complicated dynamics. Furthermore, the implementation cost is reduced since the closed-loop identification is no longer necessary, and therefore this datadriven method enjoys satisfactory applicability in practical applications.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Dexian Huang: 0000-0001-7743-0023 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported in part by the National Natural Science Foundation of China (61433001 and 61673236), the National Basic Research Program of China (2012CB720505), the seventh framework programme of the European Union (P7-PEOPLE2013-IRSES-612230), and Tsinghua University Initiative Scientific Research Program.

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REFERENCES

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Industrial & Engineering Chemistry Research (29) Rivera, D. E.; Morari, M.; Skogestad, S. Internal model control: PID controller design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252− 265. (30) Wang, Q. G.; Guo, X.; Zhang, Y. Direct identification of continuous time delay systems from step responses. J. Process Control 2001, 11, 531−542. (31) Doyle, F. J.; Ogunnaike, B. A.; Pearson, R. K. Nonlinear modelbased control using second-order Volterra models. Automatica 1995, 31, 697−714.

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A Novel Data-Driven Method for Simultaneous Performance

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