A Simultaneous Tuning Method for Cascade Control Systems Based

Article pubs.acs.org/IECR

A Simultaneous Tuning Method for Cascade Control Systems Based on Direct Use of Plant Data Jyh-Cheng Jeng* and Sinn-Jia Liao Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei 106, Taiwan ABSTRACT: This paper presents a novel method for tuning cascade control systems in which both primary and secondary controllers are tuned simultaneously by directly using plant data without resorting to process models. The required plant data are collected from a one-shot step test that can be conducted under either closed-loop or open-loop conditions. The goal of the proposed design is to obtain the parameters of two proportional-integral-derivative (PID) controllers such that the resulting inner and outer loops behave as closely to appropriately specified reference models as possible. The optimization problems related to the proposed design are derived. On the basis of the rationale behind cascade control, the secondary controller is designed to attenuate disturbance faster. The primary controller is designed to accurately account for the inner loop dynamics without requiring an additional test. Moreover, robustness consideration is included in the proposed tuning method, which enables the designer to explicitly address the trade-off between performance and robustness for inner and outer loops independently. Simulations show that the proposed method exhibits superior control performance compared with the existing (model-based) design methods, confirming the effectiveness of this novel method of PID controller design for cascade control systems.

1. INTRODUCTION Cascade control is one of the most successful structures for enhancing the performance of single-loop control, particularly when disturbances are associated with the manipulated variable.1 Thus, cascade control is applied extensively in chemical process industries. In the standard cascade-control approach, one feedback loop is nested inside another feedback loop using two controllers. The controller of the inner loop is called the secondary (or slave) controller, whereas the controller of the outer loop is the primary (or master) controller. The rationale behind this configuration is that the fast dynamics of the inner loop enable fast attenuation of disturbance and minimize the possible effects of disturbances before they affect the primary output, which is the controlled variable of interest. Although sophisticated schemes for cascade control have been proposed,2−4 the basic scheme still comprises two nested loops with two proportional-integral-derivative (PID) controllers. Because this scheme involves tuning two PID controllers, the design of cascade control systems is more complex than the design of single-loop control systems. The usual approach involves first tuning the secondary controller by setting the primary controller on a manual mode. The primary controller is then tuned by considering the action of the secondary controller on the inner loop. Such a tuning procedure is time-consuming because at least two runs of the plant test are typically required.5,6 However, the sequential tuning procedure has been improved so that only a single experiment is conducted for tuning the two controllers simultaneously.7−12 In most methods of simultaneous tuning, low-order models (usually first-order plus time delay models) of the primary and secondary processes are first identified, and then the two PID controllers are tuned using model-based tuning rules.8,11,12 Using these methods, the secondary controller can be tuned in a straightforward manner, and a © 2013 American Chemical Society

satisfactory PID controller can be designed because the secondary process inherently shows low-order dynamics. However, tuning the primary controller is more complex because the tuning requires an additional low-order apparent model (usually second-order plus time delay model), which represents the lumped dynamics of the primary process and the inner loop. Two difficulties may be encountered in identifying such an apparent model. First, because the primary process usually exhibits high-order dynamics, low-order modeling of the primary process introduces inevitable modeling errors. Second, the inner-loop dynamics is approximated using the inner-loop design target (e.g., a desired closed-loop transfer function of the inner loop)8,12−14 and then augmented with the primary process model to obtain a low-order apparent model. However, this approximation may be inaccurate because the implemented secondary controller cannot guarantee meeting the inner-loop design target. For example, in the secondary controller designed using the direct synthesis method, the approximation errors introduced by the Taylor or Pade approximation of time delay are involved in deriving the PID controllers from the synthesized controller, which causes the actual inner-loop dynamics to deviate from its design target. Consequently, the large modeling error associated with the apparent model diminishes both the effectiveness of model-based design in tuning the primary controller and the performance of cascade control. Jeng and Lee15 recently proposed a simultaneous tuning method for cascade control in which the identified apparent model can account for the actual inner-loop dynamics accurately by a frequency-domain identification technique. Received: Revised: Accepted: Published: 16820

June 23, 2013 October 17, 2013 October 21, 2013 October 21, 2013 dx.doi.org/10.1021/ie401978p | Ind. Eng. Chem. Res. 2013, 52, 16820−16831

Industrial & Engineering Chemistry Research

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attenuates disturbance. Figure 1 shows the configuration of a typical cascade control system, where G1 is the primary process

Nevertheless, the apparent model still contains modeling errors that result from model reduction. An attractive approach for alleviating the drawbacks of plantmodel mismatch is to design PID controllers directly from a set of process input and output data without resorting to process models. Because the measured plant data contain more direct and useful information on plant dynamics than mathematical models developed through system identification, direct approaches for controller design are expected to provide effective controllers that reflect the dynamics of a plant. Several mode-free or data-based methods for designing controllers have been developed, such as the iterative feedback tuning (IFT) method,16 the virtual reference feedback tuning (VRFT) method,17−19 and the fictitious reference iterative tuning (FRIT) method.20 Given a set of process input−output data, these methods determine the optimal controller parameters so that the corresponding closed-loop response is as close to a predefined (desired) reference trajectory as possible. The reference trajectory is typically defined on the basis of the servo response. However, using these methods, a suitable reference model cannot be easily determined because the process model is unknown. The use of an inappropriate reference model deteriorates the control performance and can lead to an unstable control system. To determine an appropriate reference model, extended data-based design methods have been developed,21,22 in which the parameters in the reference model are optimized with the PID controller parameters. These previous studies have addressed data-based design of controllers for unitary feedback systems, but the use of databased methods to design controllers for cascade control systems has not been reported. The use of data-based methods in designing controllers for cascade control systems is more attractive than the application of these methods to single-loop control systems, because the identification of models for cascade control systems is more complex and time-consuming than it is for single-loop systems. Thus, we seek to extend the data-based method of controller design to cascade control systems for the simultaneous tuning of two PID controllers by directly using plant data. The proposed method can be used to design two PID controllers in a cascade structure without depending on the availability of process models. This design of cascade control achieves satisfactory servo and regulatory control performance using the standard cascade control configuration (i.e., complicated control schemes are not used); this is accomplished by designing the inner loop for enhanced attenuation of disturbance and by considering the actual inner-loop dynamics in the primary controller design (without an additional experiment). Furthermore, robustness considerations of the inner and outer loops are explicitly included in the proposed procedure for controller tuning. The rest of this paper is organized as follows. Section 2 presents the structure of cascade control and the collection of plant data. Sections 3 and 4 present the PID controller designs for the secondary and primary controllers, respectively. Section 5 summarizes the controller tuning procedure. Section 6 provides several simulation examples demonstrating the effectiveness of the proposed method. Finally, section 7 offers concluding remarks.

Figure 1. Configuration of a typical cascade control system.

and G2 is the secondary process. The primary controller Gc1 uses the primary process variable y1 (with set-point r1) to establish the set-point (r2) for the secondary controller Gc2. The secondary process variable y2 is transmitted to the secondary controller, which adjusts the manipulated variable u. Disturbance can enter at two possible points, d1 and d2. A cascade control scheme is effective because the disturbance d2 affecting the inner loop is promptly compensated before it affects the primary process variable y1. Assume that the processes G1 and G2 are unknown and only a set of process input−output data, u(t), y2(t), and y1(t), are available for tuning two PID controllers. The required plant data can be obtained by conducting a one-shot plant test under either closed-loop or open-loop conditions. When the control system has not been established and an initial set of controller parameters is unavailable, an open-loop test is conducted to tune the controller parameters. However, when the control system has been assembled and is in full operation, the closedloop test is desirable because the process can be conducted continuously throughout the entire test. In this case, the proposed tuning method plays a vital role in improving existing controllers that underperform (i.e., the poorly tuned controllers). Moreover, only the closed-loop test can be used for nonself-regulating processes. The step test is considered in this study because it is the simplest and most widely used test in process control applications. The step response data of the process variables, y1(t) and y2(t), and the manipulated variable, u(t), are collected until a new steady-state is reached.

3. DESIGN OF THE SECONDARY CONTROLLER Consider the inner loop structure shown in Figure 1, which consists of a process G2(s) and a PID controller Gc2(s). The PID controller Gc2(s) is given by ⎛ ⎞ 1 + τD2s⎟ Gc 2(s) = Kc 2⎜1 + τI 2s ⎝ ⎠

(1)

where Kc2, τI2, and τD2 denote the proportional gain, the integral time, and the derivative time of the secondary controller, respectively. In the proposed method for directly tuning the secondary PID controller, a model-reference problem is approximately solved as depicted in Figure 2, where the output Y2(s) is related to its set-point (reference signal) R2(s) through the secondary reference model, T2(s), by Y2(s) = T2(s)R 2(s)

2. CASCADE CONTROL SYSTEMS In a cascade control scheme, the introduction of an additional sensor creates a secondary (inner) loop that effectively

(2)

Figure 2. Reference model for inner loop. 16821

dx.doi.org/10.1021/ie401978p | Ind. Eng. Chem. Res. 2013, 52, 16820−16831


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information is not lost. Because the frequency responses of data within [0, ωmax,2] are of interest to controller design, the sampling rate fs (Hz) should be chosen to meet fs > 2(ωmax,2/ 2π) and equivalently fs > (ωmax,2/π), according to the sampling theorem. The frequency ωmax,2 denotes the upper bound of the frequency range for the minimization problem. Because ωmax,2 is closely related to the controller design, ωmax,2 can be specified as the inner-loop system’s critical frequency, ωc2, at which the phase angle of Gc2G2 (jω) equals −π. On the basis of the reference model T2(s), the critical frequency ωc2 can be calculated according to the following equation:

This secondary reference model describes the desired closedloop transfer function (complementary sensitivity function) of the inner control loop. The design goal of the proposed method is to obtain PID parameters of Gc2(s) such that the corresponding inner loop in Figure 1 behaves as closely to the prescribed reference model T2(s) as possible. To accomplish this, the available signal y2(t) is used and the reference signal in Figure 2 is obtained from eq 2 as

R 2(s) = T2(s)−1Y2(s)

(3)

where Y2(s) is the Laplace transform of y2(t), and R2(s) is the so-called virtual reference signal for the inner loop because it does not actually exist. As Y2(s) is considered to be the desired output of the inner loop when the reference signal of Y2(s) is specified by R2(s), the corresponding secondary controller’s output can be calculated by substituting eqs 1 and 3 as Û (s) = Gc 2(s)[R 2(s) − Y2(s)] ⎛ ⎞ 1 = Kc 2⎜1 + + τD2s⎟[T2(s)−1 − 1]Y2(s) τI 2s ⎝ ⎠

∠

P2

⎡ Re(Φ2)⎤ ⎥; Φ̃ 2 = ⎢ ⎢⎣ Im(Φ2)⎥⎦

(5)

Y2(s) = [1 − T2(s)]G2(s) D2(s)

G2(s) =

2

P2

(13)

K 2 −θ2s e τ2s + 1

(14)

According to eq 13, the output response to disturbance d2 could be sluggish if G2(s) is a lag-dominant process with τ2 ≫ θ2. To improve the rejection of disturbance by the inner loop, we propose that the reference model T2(s) could be specified as

(8)

where Φ2 = [U (jω0) U (jω1) ··· U (jωn)]T

T2(s) =

Φ̂ 2 = [Û (jω0) Û (jω1) ··· Û (jωn)]T

αs + 1 −θ2s e (λ 2s + 1)2

(15)

where θ2 is related to the apparent time delay of the secondary process, and λ2 is an adjustable parameter to address the tradeoff between the performance and robustness of the inner loop. The parameter α is determined so that 1 − T2(s) cancels the slow pole in G2(s), that is, lims→−1/τ2[1 − T2(s)] = 0, which leads to

Ψ2 = [ ψ2,0 ψ2,1 ··· ψ2, n ]

T

⎤T ⎡ −Ω 2(jωi) j Ω 2(jωi)jωi ⎥ ψ2, i = ⎢ Ω 2(jωi) ωi ⎦ ⎣

(12)

As it is typical in an industrial context, the secondary process can be modeled as first-order plus time delay (FOPTD) dynamics:

(7)

min J2 = || Φ2 − Φ̂ 2 || = || Φ2 − Ψ2P2 ||

⎡ Re(Ψ2)⎤ ⎥ Ψ̃2 = ⎢ ⎢⎣ Im(Ψ2)⎥⎦

where Re(A) and Im(A) denote the real matrix (or vector), and the elements are the real and imaginary parts of a complex matrix (or vector) A, respectively. The specification of the secondary reference model is crucial to the performance of the resulting inner loop. The reference model T2(s) describes the desired inner-loop behavior that is the target for tuning Gc2(s). Because the performance of cascade control in the presence of disturbance d2 is usually the principal concern,24 the inner loop must respond quickly to effectively reduce the effects of disturbance d2. The closed-loop transfer function between Y2(s) and D2(s) is given by

The aforementioned minimization problem is formulated in the frequency domain to minimize the difference between Û (jω) and U(jω) in a frequency range [0, ωmax,2]. Consequently, the PID parameters of Gc2(s) are obtained by solving 2

(11)

with

(4)

(6)

⎡ ⎤T Kc 2 τ K K P2 = ⎢ c 2 c 2 D2 ⎥ τI 2 ⎣ ⎦

(10)

ω = ωc2

2 min J2 = || Φ̃ 2 − Ψ̃2P2 ||

where Ω 2(jω) = [T2(jω)−1 − 1]Y2(jω)

= −π

After algebraic calculations, eq 8 is recast as

When the secondary process is subject to the measured input signal u(t), it generates y2(t). Therefore, a secondary controller that shapes the inner closed-loop transfer function to the reference model generates u(t) or its Laplace transform U(s) when the error signal is given by R2(s) − Y2(s). Because u(t) is known, the controller design becomes a problem of minimizing the difference between Û (s) given in eq 4 and U(s) obtained from u(t). Substituting s = jω into eq 4 yields ⎡ ⎤ −Ω 2(jω) Û (jω) = ⎢ Ω 2(jω) j Ω 2(jω)jω ⎥P2 ⎣ ⎦ ω

T2(jω) 1 − T2(jω)

(9)

The frequency responses of U(jωi), Y2(jωi), and Y1(jωi) at various frequencies ωi(i = 0, 1, 2, ...) can be evaluated using fast Fourier transform (FFT) of the process input and output data collected from a step test.23 The sampling rate to collect process data must be large enough so that significant process

2 ⎛ ⎞ ⎛ λ ⎞ α = τ2⎜⎜1 − ⎜1 − 2 ⎟ e−θ2 / τ2⎟⎟ τ2 ⎠ ⎝ ⎝ ⎠

16822

(16)



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The sum of θ2 and τ2 can be obtained by evaluating the step response data as12 θ2 + τ2 =

∫0

θ2 + τ2 =

K1 h

∞⎛

y (t ) ⎞ ⎜1 − 2 ⎟ d t K 2h ⎠ ⎝

∫0

Therefore, the minimization problem given in eq 23 becomes the following problem: 2 min J2 = || Φ̃ 2 − Ψ̃2P*2 (θ2)||

(open‐loop data)

In the proposed design, the smallest J2(θ2) is identified in a specified range of θ2, such as J2 (θ2*), and its corresponding solution P2*(θ2*) is used to obtain the PID parameters of the secondary controller. Given a prescribed range of θ2, this onedimensional minimization problem can be readily solved using standard optimization algorithms such as golden-section search. Because θ2 is related to the apparent time delay of the secondary process, an appropriate choice of the prescribed range of θ2 should cover the apparent time delay of the secondary process, which is available from the step-response data. If the secondary process is not lag-dominant, the reference model given in eq 15 can be simplified by setting α = λ2 so that the reference model for the secondary controller design becomes

∞

[K 2u(t ) − y2 (t )] dt (closed‐loop data) (17)

where h is the magnitude of the step change, and K1 and K2 are the steady-state gains of the primary process and secondary processes, respectively. The steady-state gains are calculated by y y K1 = 1s ; K 2 = 2s us y2s (18) where y1s, y2s, and us are the final steady-state values of y1(t), y2(t), and u(t), respectively. Furthermore, the parameter λ2 can be selected to match a prescribed level of inner-loop robustness in terms of the inner-loop maximum sensitivity (MS2), which is defined as MS2 = max|S2(jω)| = max|1 − T2(jω)| ω

T2(s) =

(19)

ω

where S2 is the sensitivity function of inner loop. As the maximum sensitivity decreases, the dynamic response of the closed-loop system becomes more robust. The reference model can be normalized using τ2 by defining s ̅ = τ2s as α̅ s ̅ + 1 −θ2̅ s ̅ e (λ 2̅ s ̅ + 1)2

T2̅ ( s ̅ ) =

(20)

λ 2̅ =

2

−3.998θ2̅ + 4.279θ2̅ − 0.0397 2

θ2̅ − 3.242θ2̅ + 2.133θ2̅ + 0.139 2

p2 = p3 =

1410θ2̅ − 212θ2̅ − 4484 3

2

θ2̅ − 2228θ2̅ + 1962θ2̅ + 759.2 3 1.203θ2̅

−

2 1.463θ2̅

+ 0.789θ2̅ + 0.01257

θ2

P2

(22)

T ⎡ −Ω 2(jωi) ⎤ j⎥ ψ2, i = ⎢ Ω 2(jωi) ωi ⎦ ⎣

(28)

(29)

4. DESIGN OF THE PRIMARY CONTROLLER After the secondary controller is designed, the design of the primary controller should account for the inner loop dynamics so that Gc1(s) is designed on the basis of an apparent process G1a(s) that is seen by the primary controller as

(23)

For a given θ2, eq 11 is solved using the least-squares method as follows: T T P2*(θ2) = (Ψ̃2 Ψ̃2)−1Ψ̃2 Φ̃ 2

1.2 ≤ MS2 ≤ 2.0

T ⎡ Kc 2 ⎤ K P2 = ⎢ c 2 ⎥ τI 2 ⎦ ⎣

The design criterion given in eq 21 is suitable for 1.2 ≤ MS2 ≤2.0 and 0.05 ≤ θ̅2 ≤ 1. With this criterion, the value of λ2 can be determined conveniently and without trial and error. Consequently, when a desired value of MS2 is specified, the minimization problem given in eq 11 depends implicitly on only θ2 so that the optimal solution P2* becomes a function of θ2. Treating θ2 as a tuning parameter, the minimization problem for tuning Gc2(s) can be further formulated as 2 min min J2 = || Φ̃ 2 − Ψ̃2P2 ||

−0.7289MS2 + 1.555 ; MS2 − 1.006

When a desired value of MS2 is specified, the minimization problem given in eq 23 can be solved to obtain the PID parameters of the secondary controller; solving the equation is simplified because calculating the parameter τ2 is not required. In certain industrial applications, it is desirable for the secondary controller to be a PI controller. The proposed method can be applied to tune a PI controller by setting P2 in eq 7 and ψ2,i in eq 9 to

(21)

where 3

(26)

where λ̅2 = λ2/θ2. Because the value of MS2 depends only on λ̅2, the following robust design criterion provides the value of λ̅2 required to achieve a desired robustness level for MS2.15

where λ̅2 = λ2/τ2, θ̅2 = θ2/τ2, and α̅ = α/τ2 = 1− (1 − λ̅2) e−θ̅2. Equation 20 shows that the value of MS2 depends on λ̅2 and θ̅2. Therefore, the following correlated robust design criterion15 provides the required value of λ̅2, for a process parameter θ̅2, to achieve a desired level of robustness for MS2.

p1 =

1 e−θ2s λ 2s + 1

The above reference model can be normalized using θ2 by defining s ̅ = θ2s as 1 T2̅ ( s ̅ ) = e− s ̅ λ 2̅ s ̅ + 1 (27)

2

λ 2̅ = p1 MS2 p2 + p3

(25)

θ2

G1a(s) =

(24)

Gc 2(s)G2(s) G1(s) 1 + Gc 2(s)G2(s)

(30)

The PID controller Gc1(s) is given by 16823


Industrial & Engineering Chemistry Research ⎞ ⎛ 1 + τD1s⎟ Gc1(s) = Kc1⎜1 + τI1s ⎠ ⎝

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⎡ ⎤T Kc1 Kc1τD1⎥ P1 = ⎢ Kc1 τI1 ⎣ ⎦

(31)

where Kc1, τI1, and τD1 denote the proportional gain, the integral time, and the derivative time of the primary controller, respectively. The primary controller’s output R2(s), which serves as the set-point of the inner loop, is used to generate the secondary controller’s output according to U (s) = Gc2(s)[R 2(s) − Y2(s)]

The minimization problem is formulated in the frequency domain to minimize the difference between R̂ 2(jω) and R2(jω) in a frequency range [0, ωmax,1]. Consequently, the PID parameters of Gc1(s) are obtained by solving min J1 = || Φ1 − Ψ1P1 ||2

where ⎡G (jω )−1U (jω ) + Y (jω )⎤ 0 2 0 ⎥ ⎢ c2 0 − 1 ⎢ G ( jω ) U ( jω ) + Y ( jω ) ⎥ 1 2 1 ⎥ Φ1 = ⎢ c 2 1 ⎢ ⎥ ⋮ ⎢ ⎥ ⎢⎣ Gc 2(jωn)−1U (jωn) + Y2(jωn) ⎥⎦

(33)

The rationale of design for the primary controller is similar to rationale of the secondary controller’s design. The primary reference model for the outer loop is as shown in Figure 3,

Ψ1 = [ ψ1,0 ψ1,1 ··· ψ1, n ]T ⎤T ⎡ −Ω1(jωi) j Ω1(jωi)jωi ⎥ ψ1, i = ⎢ Ω1(jωi) ωi ⎦ ⎣

Figure 3. Reference model for the outer loop.

(34)

∠

The design goal is to obtain PID parameters of Gc1(s) such that the corresponding overall cascade system (Figure 1) behaves as closely to the prescribed reference model T1(s) as possible. Using the available signal y1(t), the virtual reference signal for the outer loop in Figure 3 is obtained from eq 34 as follows: −1

T1(jω) 1 − T1(jω)

As Y1(s) is considered to be the desired output of the outer loop when the reference signal of Y1(s) is specified by R1(s), the corresponding primary controller’s output can be calculated by substituting eqs 31 and 35 as follows:

where ⎡ Re(Φ1)⎤ ⎥; Φ̃ 1 = ⎢ ⎢⎣ Im(Φ1)⎥⎦

(36)

T1(s) =

⎡ Re(Ψ1)⎤ ⎥ Ψ1̃ = ⎢ ⎢⎣ Im(Ψ1)⎥⎦

(44)

1 e−θ1as λ1s + 1

(45)

where θ1a is related to the apparent time delay of the process G1a(s), and λ1 is an adjustable parameter to address the tradeoff between the performance and robustness of the outer loop. The parameter λ1 can be selected to match a prescribed level of outer-loop robustness in terms of the outer-loop maximum sensitivity (MS1), which is defined as

(37)

where Ω1(jω) = [T1(jω)−1 − 1]Y1(jω)

(43)

The specification of the reference model T1(s) is crucial to the performance of the resulting outer loop. If an equation similar to eq 15 is specified for T1(s), a set-point filter for the outer loop must be incorporated to reduce the excessive overshoot in the set-point response of y1, which complicates the control system. To maintain the simplicity of the control system, we consider T1(s) specified in a typical form:

When the apparent process is subject to R2(s), it generates Y1(s). Therefore, a primary controller that shapes the outer closed-loop transfer function to the reference model generates R2(s) when the error signal is given by R1 (s) − Y1 (s). Because R2(s) can be calculated from eq 33, the controller design problem becomes a problem of minimizing the difference between R̂ 2 (s) given in eq 36 and R2(s) obtained from eq 33. Substituting s = jω into eq 36 yields ⎡ ⎤ −Ω1(jω) R̂ 2(jω) = ⎢ Ω1(jω) j Ω1(jω)jω ⎥P1 ⎣ ⎦ ω

(42)

2 min J1 = || Φ̃ 1 − Ψ1̃ P1 || P1

R̂ 2(s) = Gc1(s)[R1(s) − Y1(s)] ⎞ ⎛ 1 = Kc1⎜1 + + τD1s⎟(T1(s)−1 − 1)Y1(s) τI1s ⎠ ⎝

= −π ω = ωc1

Again, eq 40 is recast as

(35)

R1(s) = T1(s) Y1(s)

(41)

The upper bound of the frequency range for this minimization problem, ωmax,1, is specified as the outer-loop system’s critical frequency, ωc1, at which the phase angle of Gc1G1a (jω) equals −π. On the basis of the reference model T1(s), the critical frequency ωc1 can be calculated according to the following equation:

where T1(s) describes the desired closed-loop transfer function of the outer loop, and the output Y1(s) is related to its set-point (reference signal) R1(s) by Y1(s) = T1(s)R1(s)

(40)

P1

(32)

When the secondary controller is designed and the signals u(t) and y2(t) are available, the primary controller’s output, or the input to the apparent process, is obtained from eq 32 by R 2(s) = Gc 2(s)−1U (s) + Y2(s)

(39)

MS1 = max|S1(jω)| = max|1 − T1(jω)|

(38)

ω

16824

ω

(46)



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where S1 is the sensitivity function of the outer loop. Because the value of MS1 depends only on λ̅1, where λ̅1 = λ1/θ1a, the following robust design criterion, which is similar to eq 28, provides the value of λ̅1 required to achieve a desired level of robustness for MS1.15 λ1̅ =

−0.7289MS1 + 1.555 ; MS1 − 1.006

1.2 ≤ MS1 ≤ 2.0

θ2 + τ2 =

θ1a

K1 = (47)

For a given θ1a, eq 43 can be solved by the least-squares method as follows: T T P1*(θ1a) = (Ψ1̃ Ψ1̃ )−1Ψ1̃ Φ̃ 1

(49)

The minimization problem given in eq 48 becomes the following problem: 2 min J1 = || Φ̃ 1 − Ψ1̃ P1*(θ1a)||

(50)

θ1a

In the proposed design, the smallest J1(θ1a) is identified in a specified range of θ1a, such as J1(θ1a * ), and its corresponding solution P1*(θ1a * ) is used to obtain the PID parameters of the primary controller. Because θ1a is related to the apparent time delay of the apparent process G1a(s), an appropriate choice of the prescribed range of θ1a should cover the apparent time delay of the apparent process. The primary process can be an integrating process such as level-flow cascade control, which is common in typical industrial applications. When the primary process is an integrating process, the following asymptotic tracking constraint must be satisfied to enable the step-load disturbances d1 to be counteracted to eliminate the offset. lim

s→0

d [1 − T1(s)] = 0 ds

(51)

In this case, the reference model T1(s) is chosen as T1(s) =

(2λ1 + θ1a)s + 1 (λ1s + 1)2

e−θ1as (52)

and the corresponding robust design criterion for the selection of λ1 is given by15 λ1̅ =

−0.4105MS1 + 2.044 ; MS1 − 1.012

1.2 ≤ MS1 ≤ 2.0

t

[K 2u(v) − y2 (v)] dv dt

(54)

y1s ∞

∫0 y2 (t ) dt

∞

;

K2 =

∫0 y2 (t ) dt ∞

∫0 u(t ) dt

(55)

5. CONTROLLER TUNING PROCEDURE The proposed method for tuning PID controllers for cascade control systems can be implemented as follows: Step 1. Collect the process data u(t), y2(t), and y1(t) from a plant step test and calculate their frequency responses U(jωi), Y2(jωi), and Y1(jωi) at various frequencies ωi (i = 0,1,2,...). Step 2. Set the prescribed searching ranges of θ2 and θ1a and the desired values of MS2 and MS1 (robustness specifications). The recommended values for MS are typically within the range 1.2 < MS < 2.0. Step 3. Solve the minimization problem given in eq 23 using golden-section search within the specified range of θ2. (i) If the secondary process is lag-dominant, calculate the sum of θ2 and τ2 using eq 17 or eq 54 for self-regulating or integrating primary processes, respectively. (Let M = θ2 + τ2). For each chosen θ2 and τ2 = M − θ2, calculate the corresponding λ2 and α using eq 21 and eq 16, respectively. Specify the reference model T2(s) according to eq 15, and set ωmax,2 = ωc2. Calculate P*2 (θ2) using eq 24. (ii) If the secondary process is not lag-dominant, for each chosen θ2, calculate the corresponding λ2 using eq 28. Specify the reference model T2(s) according to eq 26, and set ωmax,2 = ωc2. Calculate P2*(θ2) using eq 24. Step 4. Use the parameter P2* corresponding to the optimal value of θ2; that is, P2*(θ2), obtained from Step 3 to calculate the PID parameters of the secondary controller. Step 5. Solve the minimization problem given in eq 48 using golden-section search within the specified range of θ1a. (i) If the primary process is self-regulating, for each chosen θ1a, calculate the corresponding λ1 using eq 47. Specify the reference model T1(s) according to eq 45, and set ωmax,1 = ωc1. Calculate P*1 (θ1a) using eq 49. (ii) If the primary process is integrating, for each chosen θ1a, calculate the corresponding λ1 by eq 53. Specify the reference model T1(s) according to eq 52, and set ωmax,1 = ωc1. Calculate P*1 (θ1a) using eq 49. Step 6. Use the parameter P*1 corresponding to the optimal value of θ1a, that is, P1* (θ1a * ), obtained from Step 5 to calculate the PID parameters of the primary controller. Every control system should provide a certain degree of robustness to preserve the closed-loop dynamics despite possible variations in the processes. In the proposed approach, λ1 and λ2 are the tunable parameters to handle the trade-off between the speed of response and system robustness. By increasing the tunable parameters, the robustness margins are increased (MS decreased) at the expense of a faster response.

(48)

P1

∞

∫0 ∫0

where the steady-state gains are calculated by

With this design criterion, the value of λ1 can be determined conveniently and without trial and error. Thus, when a desired value of MS1 is specified, the minimization problem given in eq 43 depends implicitly on only θ1a so that the optimal solution P1* becomes a function of θ1a. Treating θ1a as a tuning parameter, the minimization problem for tuning Gc1(s) can be further formulated as 2 min min J1 = || Φ̃ 1 − Ψ1̃ P1 ||

K1 h

(53)

The method used for tuning the primary controller is similar to the method used for self-regulating processes. Only closed-loop plant data are available for tuning the controller because an open-loop test is not suitable for integrating processes. In addition, when the secondary controller is to be designed using the reference model given in eq 15, the sum of θ2 and τ2 is determined by evaluating the closed-loop step response data as follows:12 16825



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The rejection of disturbance d2 depends primarily on the secondary controller (or λ2), whereas set-point tracking and the rejection of disturbance d1 depend mainly on the primary controller (or λ1). As a result, λ1 and λ2 can be selected independently to achieve the desired trade-off for the outer and inner loops, respectively. However, in model-based design methods, the appropriate values of the tunable parameters are typically determined by a trial-and-error procedure, requiring prior information of process dynamics, to fulfill the robustness specifications. In contrast, the proposed design method explicitly includes the specifications for robustness based on MS1 and MS2 that automatically guide the selections of λ1 and λ2, respectively.

settings and using the open-loop data. The resulting PID controller parameters (not reported for the sake of brevity) are close to the previously noted parameter values, indicating that the proposed method is unaffected by the conditions used for collecting the plant data. Figure 4a shows the closed-loop response to a unit step setpoint change at t = 0 followed by step disturbance inputs d2 =

6. SIMULATION EXAMPLES Four simulation examples are provided to demonstrate the effectiveness of the proposed method for tuning PID controllers for cascade control systems. The processes for Examples 1−3 are presented in terms of transfer functions, whereas Example 4 is a jacketed continuous stirred tank reactor (CSTR) process described using nonlinear differential equations. 6.1. Example 1. Consider the process given by the following transfer functions: 1 ; (10s + 1)(4s + 1)(s + 1)2 1 G2(s) = e−0.1s 1.9s + 1

G1(s) =

Figure 4. Closed-loop responses for Example 1: (a) nominal plant; (b) perturbed plant. (56)

An initial cascade control system using the following two PID controllers proposed by Veronesi and Visioli12 is assumed for the application of the proposed method. ⎛ 1 ⎞⎟⎛ 0.38s + 1 ⎞ ⎜ ⎟; Gc1(s) = 1.13⎜1 + ⎝ 11.17s ⎠⎝ 0.038s + 1 ⎠ ⎛ 1 ⎞⎛ 1.9s + 1 ⎞ ⎟⎜ ⎟ Gc 2(s) = 0.104⎜1 + ⎝ 0.05s ⎠⎝ 0.04s + 1 ⎠

10 at t = 100 and d1 = 1 at t = 200, and also presents the results obtained with the controllers proposed in Veronesi and Visioli12 for comparison. The response achieved with the proposed tuning method is fast and smooth during the setpoint change and during the input of both disturbances, demonstrating the superior performance of the proposed method. Because a perfect secondary process model was used in Veronesi and Visioli,12 the proposed method exhibited a superior response to disturbance d2 because the secondary controller was designed for enhanced rejection of the disturbance. Moreover, the proposed method exhibited superior responses to set-point change and disturbance d1 because the primary controller’s design prevents the modeling error produced by the identification of an apparent process model. To illustrate the resistance to plant variations (robustness), the control systems are evaluated on a perturbed plant as follows.

(57)

12

Veronesi and Visioli used IMC-PID tuning based on the following identified process models: G1m(s) =

1 e−4.84s ; 11.17s + 1

G2m(s) =

1 e−0.1s 1.9s + 1 (58)

A unit step change at the set-point is applied to generate the closed-loop process data for controller design. The reference model given in eq 15 is used for the design of the secondary controller because the secondary process exhibits considerable lag-dominant dynamics. Choosing MS2 = 1.19, which is the value of MS2 resulting from the tuning of Veronesi and Visioli,12 the optimal solutions are θ*2 = 0.088 and P*2 = [4.718 4.560 0.101]. Consequently, the parameters of the secondary PID controller are given by Kc2 = 4.718, τI2 = 1.035, and τD2 = 0.0214. The resulting inner loop has MS2 = 1.20, which is close to the design value. To design the primary controller, with MS1 = 1.40, the optimal solutions are θ*1a = 2.34 and P*1 = [2.606 0.182 8.349]. The parameters of the primary PID controller are given by Kc1 = 2.606, τI1 = 14.36, and τD1 = 3.204. The resulting outer loop has MS1 = 1.36, which is also nearly equal to the design value. The PID controllers are also designed using the closed-loop data generated from distinct (poorly tuned) initial controller

2 , (8s + 1)(4s + 1)(s + 1)2 1.3 G2(s) = e−0.13s 1.33s + 1

G1(s) =

(59)

Figure 4b shows the responses of the perturbed systems. The proposed method provides a fast response and maintains excellent robustness. When the controller performance has degraded due to considerable process changes, the tuning procedure can be implemented again by conducting a (closed-loop) step test to retune the controller so that degraded control performance can be recovered. Assume that the process has changed to 16826


Industrial & Engineering Chemistry Research 0.7 e−1.5s , (15s + 1)(5s + 1)(2s + 1)2 1.3 G2(s) = e−0.2s 2.4s + 1

Article

various values of MS1 (1.2, 1.4, and 1.6) at a fixed value of MS2 = 1.2. The controller parameters for various design parameters are given in Table 1. The change of the inner-loop design parameter MS2 mainly affects the system performance for rejection of inner-loop disturbance d2; the set-point response and the disturbance response to d1 are almost unchanged. Conversely, the change of the outer-loop design parameter MS1 affects the system performance for set-point tracking and rejection of outer-loop disturbance d1, but has little influence over the disturbance response to d2. These results verify that the two design specifications MS1 and MS2 can be selected independently to achieve the desired performance/robustness trade-off for the outer and inner loops, respectively. Because the more common problem to face is the rejection of periodic (oscillating) disturbances, Marchetti et al.25 pointed out a very common problem encountered in industrial plants: the achievement of good performance on the primary process variable (that is acceptable attenuation on y1 amplitude) requires large oscillations in the inner loop (that is in r2 amplitude). There, a refinement of cascade control tuning to reduce oscillations in the inner loop without significantly deteriorating performance of the controlled (outer) variable was proposed. In their refinement, both proportional and integral actions for the primary controller are decreased, whereas, for the secondary controller, the proportional and integral actions are, respectively, unchanged and increased. The proposed tuning method can also achieve a similar effect of reducing oscillations in the inner loop. Because the secondary controller is designed to attenuate disturbance faster, the resulting parameter τI2 is smaller than that obtained from servobased design methods (such as IMC), corresponding to an increased integral action for the secondary controller. Furthermore, decreased proportional and integral actions (more conservative design) for the primary controller can be achieved by specifying a smaller MS1 (larger λ1). To verify the effectiveness of the proposed tuning in facing the periodic disturbance, the controllers are designed with MS1 = MS2 = 1.2 and the cascade control system in response to a sinusoidal inner-loop disturbance d2 (t) = sin(0.2t) is considered. Figure 7 shows the closed-loop responses of r2, y2, and y1, with also the results obtained with the controllers proposed in Veronesi and Visioli12 for comparison. The proposed controllers give smaller amplitude of y1 oscillations, while significant attenuation in the inner loop variables (r2 and y2) is achieved. This result extends the validity of the proposed tuning to a case of real interest in industrial applications. 6.2. Example 2. Consider a process that exhibits high-order and nonminimum phase dynamics given by the following transfer functions.8,13

G1(s) =

(60)

The closed-loop response after such a process change becomes unsatisfactory, as shown in Figure 5 (before retuning). After

Figure 5. Closed-loop responses after process change for Example 1.

conducting a step test on the perturbed system, the new PID parameters that adapt to this process change are suggested as Kc2 = 5.801, τI2 = 0.875, τD2 = 0.060, and Kc1 = 3.499, τI1 = 21.33, τD1 = 5.207. The system robustness after retuning is similar to that before retuning. With these new PID parameters, superior control performance is retrieved, as shown in Figure 5, which demonstrates the tuning method’s potential of adapting to plant variations. Consider the effect of design parameters, MS1 and MS2, on the response of the primary process variable. Figure 6a shows the responses resulting from the designs using various values of MS2 (1.2, 1.4, and 1.6) at a fixed value of MS1 = 1.2, whereas Figure 6b shows the responses resulting from the designs using

10( −5s + 1) e − 5s ; (30s + 1)3 (10s + 1)2 3 G2(s) = e − 3s 13.3s + 1

G1(s) =

(61)

A cascade control system with a PID/PI control mode using the parameters given in Mehta and Majhi8 (i.e., Kc1 = 0.046, τI1 = 68.81, τD1 = 3.057, and Kc2 = 0.703, τI2 = 13.108), which is a model-based design method, is adopted for applying the proposed tuning method. A unit step change at the set-point is applied to generate the closed-loop process data for controller design.

Figure 6. Effect of design parameters on the closed-loop response: (a) MS1 fixed; (b) MS2 fixed. 16827



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Table 1. PID Controller Parameters Resulting from Various Design Parameters for Example 1 outer loop

inner loop

MS1

θ*1a

Kc1

τI1

τD1

MS2

θ*2

Kc2

τI2

τD2

1.2 1.2 1.2 1.4 1.6

2.10 1.93 1.86 2.32 2.65

1.515 1.650 1.715 2.629 3.515

14.32 14.34 14.36 14.36 14.40

3.167 3.182 3.208 3.205 3.240

1.2 1.4 1.6 1.2 1.2

0.088 0.090 0.093 0.088 0.088

4.900 8.577 11.747 4.900 4.900

1.006 0.637 0.477 1.006 1.006

0.022 0.027 0.030 0.022 0.022

Figure 7. Closed-loop responses to a sinusoidal inner-loop disturbance for Example 1.

Figure 8. Closed-loop responses for Example 2.

The reference model given in eq 15 is used for the design of the secondary controller. Choosing MS2 = 2, the optimal solutions are θ*2 = 0.508 and P*2 = [1.970 2.017 0.315]. Consequently, the parameters of the secondary PID controller are given by Kc2 = 1.970, τI2 = 0.976, and τD2 = 0.160. The resulting inner loop has MS2 = 1.98, which is close to the design value. Next, with MS1 = 1.5 and the reference model given in eq 52, the optimal solutions are θ1a * = 6.36 and P1* = [0.0895 0.0017 0.7894]. The parameters of the primary PID controller are given by Kc1 = 0.0865, τI1 = 53.13, and τD1 = 8.822. The resulting outer loop has MS1 = 1.49, which is nearly equal to the design value. Figure 9 shows the closed-loop response to a unit step setpoint change at t = 0 followed by step disturbance inputs d2 = 0.5 at t = 500 and d1 = 0.1 at t = 1000, and also presents the results obtained with the initial controllers for comparison. The improvement is evident. In the automatic tuning method used by Veronesi and Visioli,12 the primary controller is suggested as a PD controller when the primary process is integrating. However, an offset results after a step load disturbances input from d1 occurs. For comparison, the reference model given in eq 45 is also used to design the primary controller as a PD controller, which results in Kc1 = 0.0767 and τD1 = 10.91. The resulting closed-loop responses to a unit step set-point change at t = 0 followed by a step disturbance input d2 = 1 at t = 100 are compared in Figure 10, where it can be seen that the PD/ PID control mode designed using the proposed method achieves better control performance. This example confirms the effectiveness of the proposed tuning method for integrating processes. 6.4. Example 4. A common application of cascade control is the jacketed CSTR system in which a coolant flows through the reactor jacket to regulate the reactor temperature. Figure 11

The reference model given in eq 15 is used for the design of the secondary controller because the secondary process exhibits lag-dominant dynamics. The choice of MS2 = 1.505, which is the value of MS2 resulting from the tuning of Lee et al.,13 gives the optimal solutions θ*2 = 2.71 and P*2 = [0.0882 0.088 0.782]. Consequently, the parameters of the secondary PID controller are given by Kc2 = 0.882, τI2 = 10.07, and τD2 = 0.887. The resulting inner loop has MS2 = 1.513, which is similar to the design value. Next, with MS1 = 1.647, which is the value of MS1 resulting from the tuning of Lee et al.,13 the optimal solutions are θ1a * = 53.86 and P1* = [0.1042 0.0013 3.194]. The parameters of the primary PID controller are given by Kcl = 0.1042, τI1 = 82.48, and τD1 = 30.66. The resulting outer loop has MS1 = 1.636, which is close to the design value. Figure 8 shows the closed-loop response to a unit step setpoint change at t = 0 followed by step disturbance inputs d2 = 0.5 at t = 600 and d1 = 0.1 at t = 1300, and also presents the results obtained with the controllers proposed in Lee et al.13 and Mehta and Majhi8 for comparison. These results illustrate that the proposed method for designing PID controllers outperforms previous methods. 6.3. Example 3. As a third example, consider the following cascade control system for an integrating process12 1 1 −0.5s G1(s) = e−3s , G2(s) = e s(10s + 1)(2s + 1) s+1 (62)

where the initial settings for the controllers are given by Kc1 = 0.1, τI1 = 50, τD1 = 2 (in series form) and Kc2 = 1, τI2 = 2. Process data are collected by introducing a unit step change at the set-point. 16828



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secondary controller TC2 before the reactor temperature (i.e., the primary process variable) is affected substantially. A dimensionless, nonlinear dynamic model for this CSTR process is26 ⎛ ⎞ dx1 x2 = q(x1f − x1) − ϕx1 exp⎜ ⎟ dτ ⎝ 1 + (x 2 / γ ) ⎠ dx 2 = q(x 2f − x 2) − δ(x 2 − x3) + βϕx1 dτ ⎞ ⎛ x2 × exp⎜ ⎟ ⎝ 1 + (x 2 / γ ) ⎠ dx 3 = δ1(qc(x3f − x3) − δδ2(x 2 − x3)) dτ

(63)

where τ is the dimensionless time; x1, x2, x3, and qc are the dimensionless concentration, reactor temperature, jacket temperature, and coolant flow rate, respectively. Refer to Russo and Bequette26 for the detailed definitions of the variables and parameters in eq 63. The values of the parameters used for the simulation are given in Table 2. The time delays in

Figure 9. Closed-loop responses for Example 3.

Table 2. Parameter Values for the Jacketed CSTR Process (Example 4) parameter

value

parameter

value

ϕ β δ γ q

0.11 7 0.5 20 1

δ1 δ2 x1f x2f x3f

10 1 1 0 −1

measurements for the inner and outer loops are both assumed to be 1. A (stable) steady-state operating point for this process is given as qc = 1.5, x1 = 0.8478, x2 = 0.5021, and x3 = −0.6245. Assume that the controllers of the cascade control system are initially tuned roughly (Kc1 = 0.7, τI1 = 1.5, τD1 = 0.5, and Kc2 = −0.5, τI2 = 0.3). The proposed method is applied for retuning the controllers by introducing a step change of 10% in the setpoint of the dimensionless reactor temperature x2. The secondary PI controller is designed based on the reference model given in eq 26 because the secondary process exhibits negligible lag-dominant dynamics. Choosing MS2 = 1.6, the optimal solution is θ*2 = 1.14 so that the parameters of the secondary PI controller are given by Kc2 = −1.108 and τI2 = 0.572. Next, with MS1 = 1.6, the optimal solution is θ1a * = 1.38 so that the parameters of the primary PID controller are given by Kc1 = 1.259, τI1 = 2.288, and τD1 = 0.750. Figure 12 shows the closed-loop response to a step change of 10% in the set-point at τ = 0 followed by a 5% step disturbance in the dimensionless inlet coolant temperature at τ = 40 and a step disturbance of magnitude 0.02 in the dimensionless feed temperature at τ = 70. Clearly, the retuned cascade control system using the proposed method achieves greater control performance than the initial system. The set-point response is smooth and fast, and the responses to disturbances from the inlet coolant temperature and feed temperature are considerably improved. These results illustrate that the proposed design method is effective for a real nonlinear process. In practice, process measurements are inevitably corrupted by measurement noise. The sensitivity of the proposed tuning method to the effects of measurement noise is investigated.

Figure 10. Closed-loop responses using the PD/PID control mode for Example 3.

Figure 11. Cascade control diagram for a jacketed CSTR process.

shows the cascade control system for a jacketed CSTR in which an exothermic, irreversible first-order reaction (A → B) takes place. In this control scheme, the disturbance in the inlet coolant temperature can be effectively rejected by the 16829



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to disturbance because the cascade control is devised mainly to rapidly attenuate disturbance; (2) without performing an additional experiment, the inner-loop dynamics is accurately considered in the design of the primary controller; (3) the robustness consideration is explicitly included in the procedure of controller tuning to fulfill the robustness specifications of inner and outer loops. The effectiveness of the proposed method is apparent from the satisfactory control performance in both set-point tracking and disturbance rejection, as shown in the simulation examples. Therefore, the proposed design approach is an effective alternative to the existing design methods for cascade control systems. This kind of data-based tuning method is even more attractive for multivariable systems, because the model identification becomes more and more difficult as the system complexity increases. Future research will focus on extensions of current work to multiloop or multivariable control systems.

■

Figure 12. Closed-loop responses for Example 4 (a jacketed CSTR process).

AUTHOR INFORMATION

Corresponding Author

*Tel: 886-2-27712171 ext. 2540. E-mail: [email protected].

Figure 13 shows the closed-loop plant data under Gaussian white noises with a standard deviation of 0.005, which are now

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the National Science Council (NSC) of Taiwan for supporting this research under Grant NSC 1012221-E-027-110.

■

REFERENCES

(1) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley & Sons: New York, 2004. (2) Alfaro, V. M.; Vilanova, R.; Arrieta, O. Robust tuning of twodegree-of-freedom (2-DoF) PI/PID based cascade control systems. J. Process Control 2009, 19, 1658−1670. (3) Kaya, I.; Tan, N.; Atherton, D. P. Improved cascade control structure for enhanced performance. J. Process Control 2007, 17, 3−16. (4) Liu, T.; Gu, D.; Zhang, W. Decoupling two-degree-of-freedom control strategy for cascade control systems. J. Process Control 2005, 15, 159−167. (5) Hang, C. C.; Loh, A. P.; Vasnani, V. U. Relay feedback autotuning of cascade controllers. IEEE Trans. Control Syst. Technol. 1994, 2, 42−45. (6) Vivek, S.; Chidambaram, M. Cascade controller tuning by relay auto tune method. J. Indian Inst. Sci. 2004, 84, 89−97. (7) Leva, A.; Donida, F. Autotuning in cascaded systems based on a single relay experiment. J. Process Control 2009, 19, 896−905. (8) Mehta, U.; Majhi, S. On-line identification of cascade control systems based on half limit cycle data. Int. Soc. Automation Trans. 2011, 50, 473−478. (9) Song, S.; Cai, W.; Wang, Y. G. Auto-tuning of cascade control systems. Int. Soc. Automation Trans. 2003, 42, 62−72. (10) Tan, K. K.; Lee, T. H.; Ferdous, R. Simultaneous online automatic tuning of cascade control for open loop stable processes. Int. Soc. Automation Trans. 2000, 39, 233−242. (11) Visioli, A.; Piazzi, A. An automatic tuning method for cascade control systems. Proc. IEEE Int. Conf. Control Appl., Munich, Germany, 2006; pp 2968−2973. (12) Veronesi, M.; Visioli, A. Simultaneous closed-loop automatic tuning method for cascade controllers. IET Control Theory Appl. 2011, 5, 263−270. (13) Lee, Y.; Park, S.; Lee, M. PID controller tuning to obtain desired closed loop responses for cascade control systems. Ind. Eng. Chem. Res. 1998, 37, 1859−1865. (14) Lee, Y.; Oh, S.; Park, S. Enhanced control with a general cascade control structure. Ind. Eng. Chem. Res. 2002, 41, 2679−2688.

Figure 13. Noisy plant data used in controller design for Example 4.

used in the proposed tuning method. By using the same specified value of MS2 and MS1, the resulting controller parameters are Kc2 = −1.046, τI2 = 0.535, and Kc1 = 1.228, τI1 = 2.231, τD1 = 0.765. These controller parameters are similar to those obtained under noise-free conditions, which illustrates the proposed tuning method’s resistance to measurement noises.

7. CONCLUSIONS A systematic procedure of simultaneously tuning the two PID controllers in a cascade control system directly based on the process data collected from a one-shot plant (closed-loop) step test was proposed in this study. The proposed method can be used to design PID controllers without depending on the availability of process models. This is in sharp contrast to the model-based PID design methods that require accurate process models and demand considerable engineering efforts for identifying processes. The distinctive features of the presented tuning approach for cascade control include the following: (1) The secondary controller is designed for enhanced performance 16830



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(15) Jeng, J. C.; Lee, M. W. Simultaneous automatic tuning of cascade control systems from closed-loop step response data. J. Process Control 2012, 22, 1020−1033. (16) Hjalmarsson, H.; Gevers, M.; Gunnarsson, S.; Lequin, O. Iterative feedback tuning: Theory and applications. IEEE Control Syst. Mag. 1998, 18, 26−41. (17) Guardabassi, G. O.; Savaresi, S. M. Virtual reference direct design method: an off-line approach to data-based control system design. IEEE Trans. Autom. Control 2000, 45, 954−959. (18) Campi, M. C.; Lecchini, A.; Savaresi, S. M. Virtual reference feedback tuning: a direct approach for the design of feedback controllers. Automatica 2002, 38, 1337−1346. (19) Kansha, Y.; Hashimoto, Y.; Chiu, M. S. New results on VRFT design of PID controller. Chem. Eng. Res. Des. 2008, 86, 925−931. (20) 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. Trans. Inst. Syst., Control Inf. Eng. 2004, 17, 528−536. (21) Kano, M.; Tasaka, K.; Owaga, M.; Ootakara, S.; Takinami, A.; Takahashi, S.; Yoshii, S. Practical direct PID/I-PD controller tuning and its application to chemical processes. Proceedings of 2010 IEEE International Conference on Control Applocations; IEEE: Yokohama, Japan, 2010; pp 2426−2431. (22) Yang, X.; Xu, B.; Chiu, M. S. PID controller design directly from plant data. Ind. Eng. Chem. Res. 2011, 50, 1352−1359. (23) Wang, Q. G.; Zhang, Y.; Guo, X. Robust closed-loop identification with application to auto-tuning. J. Process Control 2001, 11, 519−530. (24) Shinskey, F. G. Process Control Systems, 4th ed.; McGraw−Hill: New York, 1996. (25) Marchetti, E.; Esposito, A.; Scali, C. Refinement of cascade tuning to improve control performance without requiring any additional knowledge on the process. Ind. Eng. Chem. Res. 2013, 52, 6193−6200. (26) Russo, L. P.; Bequette, B. W. State−space versus input/output representations for cascade control of unstable systems. Ind. Eng. Chem. Res. 1997, 36, 2271−2278.

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A Simultaneous Tuning Method for Cascade Control Systems Based

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