Data-Driven Nonlinear Control Design Using Virtual-Reference

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Process Systems Engineering

Data-Driven Nonlinear Control Design Using Virtual Reference Feedback Tuning Based on Block-Oriented Modeling of Nonlinear Systems Jyh-Cheng Jeng, and Yi-Wei Lin Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00809 • Publication Date (Web): 07 May 2018 Downloaded from http://pubs.acs.org on May 7, 2018

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Industrial & Engineering Chemistry Research

Data-Driven Nonlinear Control Design Using Virtual Reference Feedback Tuning Based on Block-Oriented Modeling of Nonlinear Systems Jyh-Cheng Jeng,＊ Yi-Wei Lin Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei 106, Taiwan

Abstract Process nonlinearities impose difficulties for model identification and control system design. This paper presents a novel data-driven method for nonlinear control design based on the virtual reference feedback tuning (VRFT) framework and block-oriented modeling of nonlinear systems. Control design algorithms for Hammerstein, Wiener, and Hammerstein–Wiener systems were systematically developed. The proposed method can be applied to design a nonlinear controller for an unknown plant directly using one-shot input–output data generated by the plant. In the method, identifying a complete dynamic model of the nonlinear system is not necessary; only the static nonlinearity (or its inverse), represented by the B-spline series, requires estimation. Moreover, in the method, the nonlinearity estimation and control design processes are performed simultaneously without the need of nonlinear optimization or iterative procedures. The effectiveness of the proposed control design method is demonstrated herein through several simulation examples including two benchmark processes, namely a distillation column and a pH neutralization process.

＊

To whom correspondence should be addressed. Tel: 886-2-27712171 ext. 2540. E-mail: [email protected]. 1

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1. Introduction Nonlinearities are omnipresent and play a crucial role in most chemical and industrial process systems. Such dynamical systems can be effectively represented by nonlinear models, which can describe the global behavior of a system over wide ranges of operating conditions. One of the most frequently studied classes of nonlinear models is the block-oriented nonlinear model,1,2 which involves a cascade combination of a linear dynamic block and a nonlinear static (memoryless) block. Such a model is closely related to a linear model and can be easily adapted to linear control techniques. Two typical block-oriented model structures are Hammerstein and Wiener models. In the Hammerstein model structure, the static nonlinearity, f (•) , precedes the linear dynamic element, G (z). The order of connection is reversed in the Wiener model structure. .

A more general model structure is the Hammerstein–Wiener configuration, in which the linear dynamic element is placed between two nonlinear static functions, input nonlinearity f1 (•) and output nonlinearity

f 2 (•) . Figure 1 schematically illustrates the structures of these

block-oriented models, in which u and y are process input and output, respectively, whereas v and w are inaccessible intermediate variables. These model structures have been successfully used to describe nonlinear systems in a number of practical applications in the areas of chemical processes2–7 and biological processes.8,9 Over the past decades, a considerable amount of research has been conducted on modeling and identifying nonlinear systems using block-oriented representations for control system design. Typical control design approaches are often based on mathematical models that approximate the behavior of a dynamic process. Such model-based control design approaches require the identification of an empirical model for the controlled object. As the development of engineering technologies, industrial plants have been becoming more and more complex, and therefore identifying industrial processes is more challenging and demands considerable engineering effort. The identification process, however, usually relies on some prior assumptions such as model structure, order, and time delay, which are often unavailable or subject to uncertainties. Therefore, 2


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the complexity and modeling errors associated with such models increase the difficulty of the control design task and may lead to considerable degradation of control performance. Moreover, because model identification and control design are often treated as two separated pieces of work, the identified model, depending on the identification technique used, may not involve adequate control-relevant information for deriving an effective control design. Data-driven control (DDC) design methods are highly useful in various practical control applications, where obtaining a suitable model is a very difficult task. Because DDC design methods only need the data from a process, without requiring the intermediate step of model identification, these methods considerably relieve the effort of identifying a complicated process and also avoid the drawback of plant-model mismatch. Due to these appealing characteristics, DDC methods have drawn considerable attention as one of promising controller design methods in the control community in the past two decades. Several controller tuning methods have appeared such as iterative feedback tuning (IFT),10,11 virtual reference feedback tuning (VRFT),12,13 correlation based tuning,14 and fictitious reference iterative tuning.15 The IFT control scheme utilizes iterative algorithms so that several experiments have to be performed to determine the controller parameters. Consequently, extensions of the IFT method to reduce the number of experiments in each iteration16 and improve the convergence rate17 were explored. Huusom et al.18 investigated the use of IFT for optimization of the feedback gain and the state observer gain based on a state space system description. Precup et al.19 proposed a fuzzy control method combined with IFT. The data-driven controller tuning methods have been successfully applied to a variety of cases, as reported in a survey paper by Hou and Wang.20 Other DDC techniques include model-free adaptive control,21,22 model-free control,23,24 iterative learning control,25,26 and so on. The VRFT, originally introduced in Guardabassi and Saveresi,12 is a one-shot controller tuning method that enables the direct design of controllers using only a single set of process input–output data. When the controller is linearly parameterized, the VRFT performance 3



criterion is quadratic, allowing its minimization through a simple least-squares procedure to determine the controller parameters. The non-iterative feature with respect to both experimental and computational implementation is probably the major asset of the linear VRFT method. Hence, most existing results of the VRFT design are restricted to linear systems. Campi and Savaresi27 explored the extension of VRFT to nonlinear systems for a certain class of controller parameterizations; nevertheless, in the nonlinear setup, VRFT resorts to a multi-pass, iterative procedure. The nonlinear VRFT techniques have received attention more recently. Improved VRFT methods to cope with process nonlinearity have been developed by combining VRFT and machine learning techniques, such as reinforcement Q-learning,28 neural networks,29,30 and support vector machine.31 Adaptive VRFT methods in which the current process data were incorporated for designing proportional–integral–derivative (PID) controllers have been proposed for nonlinear process control.32–34 Bazanella and Neuhaus35 proposed the tuning of nonlinear controllers using the virtual reference paradigm for two classes of nonlinear plants: rational plants and Wiener plants. Unlike linear VRFT, the extended versions of VRFT to nonlinear systems are not one-shot methods or their solution cannot be determined by a single least-squares method; hence, a considerable advantage of the VRFT method is lost. Our previous work36 showed that VRFT can be extended to control design of nonlinear Hammerstein and Wiener systems in such a way as to retain the non-iterative feature of VRFT. This motivates the research presented in this paper to develop a data-driven control design method for nonlinear systems based on the one-shot (non-iterative) VRFT design framework and block-oriented modeling for the nonlinear system. In this study, specific control design algorithms for three types of block-oriented model structures (i.e., Hammerstein, Wiener, and Hammerstein–Wiener systems) are systematically derived. We adopt linearizing control schemes, which comprise the inverse of the static nonlinearity and a linear controller. The control scheme results in an equivalent linear control system that enables the application of the VRFT design method. Combining the B-spline parameterization of the nonlinearity with the VRFT framework 4


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enables putting the system in a linear regressor form; thus simple least-squares techniques can be used to determine the parameters of the static nonlinearity and the controller simultaneously, without identifying the linear dynamic subsystem. Distinctive features of the proposed nonlinear VRFT technique include (1) the identification of a complete block-oriented model of the nonlinear system is not necessary, but only the static nonlinearity, or its inverse, must be estimated as it is a part of the controller; (2) the nonlinearity estimation and the control design are performed simultaneously without the need of nonlinear optimization or iterative procedures. Compared to our previous work,36 the new contributions of this paper include (1) the control design algorithms for Hammerstein and Wiener systems are refined and presented in detail with complete derivations; (2) the control design method is extended to Hammerstein–Wiener systems; (3) the coordination of the VRFT reference model and the controller to ensure effective model matching is carried out and a two-stage procedure for designing low-order controllers (e.g., PID controllers) is proposed. We demonstrate the effectiveness of the proposed control design method through simulation examples of numerical systems and two benchmark processes (i.e., a distillation column and pH neutralization). The rest of this paper is organized as follows. Section 2 presents the original VRFT design method and the nonlinear approximation using B-splines as the preliminaries of the proposed method. Sections 3, 4, and 5 introduce the detailed control design algorithms for Hammerstein, Wiener, and Hammerstein–Wiener systems, respectively. Section 6 presents several simulation examples demonstrating the effectiveness of the proposed design method. Finally, Section 7 presents the concluding remarks.

2. Preliminaries This section briefly describes the idea of VRFT and nonlinear approximation using basis functions of B-splines (basic splines). These two techniques play critical roles in the proposed control design. 5



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2.1 VRFT method VRFT is a data-driven control approach in which the tuning of a discrete-time feedback controller approximately solves a model-reference control problem and is transformed into a controller parameter identification problem. Consider a feedback control system consisting of a process G(z) and a controller C(z; θ) with a controller parameter vector θ, as illustrated in Figure 2. Here, rk, ek, uk, and yk are the reference signal, error signal, process input, and process output at the kth sampling instant, respectively. Suppose that a model of the process G(z) is unavailable but a set of N process input–output data points {uk, yk}k=1~N can be obtained from an open-loop experiment. By specifying a reference model M(z) that represents the desired closed-loop dynamics, the goal of

the control design is to obtain the controller parameter θ so that the

behavior of the corresponding feedback control system in Figure 2 is sufficiently close to that of the reference model. The desired reference model and the measured output signal yk can be used to compute a virtual reference signal r%k as follows: r%k = M −1 ( z ) yk

(1)

where z is the shift operator (i.e., z −1 xk = xk −1 ). The reference signal r%k is called “virtual” because it does not exist in reality and is not in place when data set {uk, yk} is collected. The virtual signal represents the reference signal that must be applied in a closed-loop to obtain yk as the closed-loop response. If the control strategy, as shown in Figure 3, is applied, the virtual controller output u%k is calculated as follows:

(

)

u%k = C ( z; θ ) e%k = C ( z; θ )( r%k − yk ) = C ( z; θ ) M −1 ( z ) − 1 yk

(2)

A controller that shapes the closed-loop behavior to the reference model would generate uk when its input is given by e%k = r%k − yk . The idea is to search for C(z; θ) whose output u%k (Eq.(2)) sufficiently matches the process input signal uk. Hence, the controller tuning task becomes a standard identification problem, and the controller parameter θ can then be determined by 6


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solving the following minimization problem: N

min J ( θ ) = min ∑ ( uk − u%k ) θ

θ

2

(3)

k =1

More detailed discussions on VRFT are provided in Campi et al.13 If the controller C(z; θ) is linearly parameterized with respect to its parameters (e.g., a PID controller), the minimization problem in Eq.(3) can be solved by the simple least-squares technique. 2.2 B-spline functions Polynomials are the preferred functions for approximating a smooth function locally. However, when a function is to be approximated on a greater interval, the degree of the approximating polynomial might have to be unacceptably large. The alternative is to subdivide the interval of approximation into sufficiently small intervals so that in each such interval, a polynomial of a relatively low degree can provide a suitable approximation. The process can be executed in such a manner that the polynomial pieces combine smoothly and the resulting composite function has several continuous derivatives. Any such smooth piecewise polynomial function is called a spline.37 To represent a polynomial spline, the B-splines on a bounded interval [ a, b ] are introduced. Let m and n be positive integers and x = { xi } be a non-decreasing sequence of real numbers defined as follows: x : a = x1 = L = xm < x1+ m < L < xn + m = L = xn + 2 m −1 = b

(4)

For each i = 1,L , n + m − 1 , Bx ,m;i ( x) is called the ith B-spline of order m with knot sequence x given in Eq.(4). In particular, Bx ,m;i ( x) is a piecewise polynomial of degree m − 1 , with breaks xi ,L , xi + m , is nonnegative, and is zero outside the interval normalized such that

∑

p i =1

[ xi , xi + m ] .

The B-splines are

Bx,m;i ( x) = 1 for all x ∈ [ a, b ] , where p is the number of B-splines

and p = n + m − 1 . The B-splines Bx,m;i ( x) , i = 1,L , m − 1 and i = n + 1,L , n + m − 1 , are the 7



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“boundary B-splines” for boundaries a and b, respectively, and Bx ,m;i ( x) , i = m,L , n are the

“interior B-splines”. The B-spline Bx ,m;i ( x) can be computed from Bx ,m −1;i ( x) by using the recurrence relation:38

Bx ,m;i ( x) =

x − xi x −x Bx,m −1;i ( x) + i + m Bx ,m −1;i +1 ( x) xi + m−1 − xi xi + m − xi +1

(5)

with an initial condition  1 if xi ≤ x < xi +1 Bx,1;i ( x) = χ[ xi , xi+1 ) ( x) ≡  ,  0 otherwise

i = 1, 2,L , n

(6)

Here, a simple example is used to illustrate the B-splines. Figure 4 presents the third-order B-splines on the interval [ −2, 2] with an equally spaced knot sequence given as follows:

x = {−2, −2, −2, −1, 0,1, 2, 2, 2}

(7)

The number of B-splines is p = 4 + 3 − 1 = 6 . The collection of p B-splines Bx ,m;i ( x) , i = 1,L , n + m − 1 , with knot sequence x given by Eq.(4) is a basis of the spline space S x,m on the interval [ a, b ] . Therefore, the B-spline series describes a spline S ( x) ∈ Sx ,m as a linear combination given by p

S ( x) = ∑ ai Bx ,m;i ( x)

(8)

i =1

where ai represents the B-spline coefficients. The spline given by Eq.(8) can be used for representing a nonlinear function and also presents a mathematical model of the nonlinear function. In this study, a quadratic spline ( m = 3 ) is used to represent the nonlinear function for the block-oriented modeling process. Notably, the nonlinear function f crosses the origin (i.e., f (0) = 0 ) because it is defined on the basis of deviations from the operating point. When 0 is selected as one of the knots (i.e., xs + 2 = 0 for some s), the condition f (0) = 0 can be readily embedded into the B-spline series expansion by imposing the following constraint on the B-spline coefficients: 8


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f (0) = as Bx ,3; s (0) + as +1 Bx ,3; s +1 (0) = 0

(9)

as = κ as +1

(10)

or equivalently

where κ is a known constant given by the following: (0) B κ = − x,3;s +1 Bx ,3;s (0)

(11)

The use of B-spline series for representing a function has several advantages. B-splines are a local basis (with compact support) that has more flexibility in function representation, compared with the use of a global basis such as trigonometric function or Laguerre function.39,40 Moreover, the distance between knots is relative to the resolution of the B-splines. B-splines with a finer knot sequence have enhanced capability for representing a function that has sharp changes. A suitable choice of the knot sequence depends on the “shape” of the function to be represented, which enables the incorporation of a priori information on the process nonlinearity in the control design.

3. Control design based on Hammerstein modeling The control structure for a Hammerstein system is schematically shown in Figure 5, where C(z) and f −1 (•) represent the controller and the inverse of the nonlinearity, respectively. If the VRFT design framework is directly applied to this control system, the control design task requires the minimization of the difference between uk and u%k where u%k = f −1 ( C ( z ) e%k ) . Because u%k is not linear in the parameters, the minimization problem cannot be formulated as a linear regression to calculate the controller parameters using the least-squares technique. Therefore, a different approach is proposed to overcome this difficulty. The linearizing control scheme (Figure 5) results in an equivalent linear control system illustrated in Figure 6(a). We assume that the static nonlinear function f (u ) can be represented by the B-spline series as follows:

9



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p

vk = f (uk ) = ∑ ai Bu ,i (uk )

(12)

i =1

where Bu ,i (•) represents the third-order B-splines with a knot sequence u defined in the operating range of process input u and ai represents unknown coefficients. Based on Eq.(10),

as = κ u as +1 , where κ u = − Bu , s +1 (0) Bu , s (0) , for some (known) s, is required to assure f (0) = 0 . Consequently, the inaccessible intermediate variable vk can be expressed as follows: s −1

vk = ∑ ai Bu ,i (uk ) + as +1 κ u Bu , s (uk ) + Bu , s +1 (uk )  + i =1

p

∑ aB i

u ,i

(u k )

(13)

i =s + 2

The linear controller with integral action is given by the following: C ( z) =

c0 + c1 z −1 + L + cr z − r 1 − z −1

(14)

where ci represents the controller parameters and r represents the order of the controller. Because the steady-state gain of a block-oriented nonlinear system can be arbitrarily distributed in nonlinear and linear blocks due to the product, the gains of the linear controller and inverse of the nonlinearity are actually not unique. To obtain a unique result, additional constraints must be imposed on the parameters. For example, one of either the controller parameters or the B-spline coefficients can be arbitrarily set to a nonzero value without loss of generality. Here, we assume that c0 = 1 . The problem of controller design is to determine the controller parameters cj (j = 1,…, r) and the unknown coefficients ai (i = 1,…, p) from an N-point data set {uk, yk}k=1~N of observed input–output measurements. The VRFT design framework is applied to the equivalent linear control system presented in Figure 6(a), and the corresponding reference model is depicted in Figure 6(b). Based on the virtual reference signal r%k = M −1 ( z ) yk , the virtual controller output v%k is calculated as follows:

 1 + c1 z −1 + L + cr z − r  −1 v%k = C ( z ) ( r%k − yk ) =   M ( z ) − 1 yk −1 − 1 z  

(

)

(15)

Therefore, the controller parameters cj and the unknown coefficients ai can be determined by solving the following minimization problem: 10


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N

min J H0 (c j , ai ) = min ∑ ( vk − v%k ) c j , ai

c j , ai

2

(16)

k =1

In the proposed control design, the following second-order transfer function is adopted as the reference model:

M (s) =

1

(τ s + 1)

2

e − s ( d ∆t )

(17)

where ∆t denotes the sampling interval, d ∆t denotes the time delay, and τ denotes the time constant related to the speed of the closed-loop response. The discrete-time reference model M(z) corresponding to Eq.(17) is given by the following:

(α + β z ) z M ( z) = −1

− d −1

(18)

1 − 2λ z −1 + λ 2 z −2

where α = 1 − λ + λ ln λ , β = λ 2 − λ − λ ln λ , and λ = exp ( − ∆t τ ) is a user-specified tuning parameter. Substituting Eq.(18) into Eq.(15) yields

 1 + c1 z −1 + L + cr z − r v%k =  1 − z −1 

−1 2 −2 −1 − d −1    1 − 2λ z + λ z − ( α + β z ) z   yk   (α + β z −1 ) z − d −1  

(19)

which can be further rewritten as follows:

Q ( z ) yk = − ( c1 z −1 + L + cr z − r ) Q ( z ) yk + (α + β z −1 ) v%k

(20)

where  z − ( λ + λ ln λ )  2 Q( z ) =  z − ( 2λ − 1) z + β  d +1 d −1 l 2 d  z − ( 2λ − 1) z + (1 − λ ) ∑ l =1 z + β

for d = 0 for d = 1

(21)

for d ≥ 2

The time-domain expression of Eq.(20) is given by the following:

φk ,0 = − ( c1φk ,1 + L + crφk ,r ) + α v%k + β v%k −1

(22)

where φk , j (j = 0, 1,…, r) are defined as follows:

φk , j

 yk +1− j − ( λ + λ ln λ ) yk − j  yk + 2 − j − ( 2λ − 1) yk +1− j + β yk − j =  d −1 2  yk + d +1− j − ( 2λ − 1) yk + d − j + (1 − λ ) ∑ l =1 yk + l − j + β yk − j 11


for d = 0 for d = 1 for d ≥ 2

(23)


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To minimize the difference between vk and v%k (Eq.(16)), we replace v%k and v%k −1 in Eq.(22) with vk and vk−1, respectively, and then substitute Eq.(13) to define a new variable:

µ k = − ( c1φk ,1 + L + crφk ,r ) + α vk + β vk −1 = ψ k θ

(24)

ψ k =  −φk ,1 L −φk , r η k ,1 L η k , s −1 (κ uηk , s + ηk , s +1 ) η k , s + 2 L η k , p  η k ,i = α Bu ,i (uk ) + β Bu ,i (uk −1 ), i = 1, 2,L , p

(25)

where

θ = c1 L cr

a1 L as −1

as +1

as + 2 L a p 

T

(26)

Therefore, the minimization problem given in Eq.(16) becomes the problem of minimizing the difference between φk ,0 and µk , and the parameter vector θ is determined by solving the following minimization problem: min J H ( θ ) = min θ

θ

N − d −1

∑ (φ

− ψ k θ ) = min φ − Ψθ 2

k ,0

k =1

θ

2 2

(27)

where

φ = φ1,0 φ2,0 L φN − d −1,0  Ψ =  ψ1T

ψ 2T

T

L ψ N − d −1T 

T

(28)

The solution can be calculated using the least-squares technique as follows: −1 θˆ = ( ΨT Ψ ) ΨT φ

(29)

The controller parameters cj (j = 1, …, r) and estimates of the coefficients ai (i = 1, …, s–1, s+1, …, p) can now be obtained by partitioning the estimate θˆ , according to the definition of θ in Eq.(26). The coefficient as is then computed by as = κ u as +1 . With the estimated nonlinear function, its inverse f −1 (•) can be readily obtained (e.g., represented by the B-spline series) for control implementation.

4. Control design based on Wiener modeling The linearizing control scheme for a Wiener system is as shown in Figure 7, which results in an equivalent linear control system illustrated in Figure 8(a), where rv = f −1 (r ) denotes the 12


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reference signal for the intermediate variable v. We assume that the inverse function of the static nonlinearity f −1 ( y ) can be represented by the B-spline series as follows: p

vk = f −1 ( yk ) = ∑ ai By ,i ( yk )

(30)

i =1

where By ,i (•) represents the third-order B-splines with a knot sequence y defined in the operating range of process output y and ai represents unknown coefficients. To assure f −1 (0) = 0 , we require that as = κ y as +1 , where κ y = − By , s +1 (0) By , s (0) , for some (known) s. Hence, the inaccessible intermediate variable vk can be expressed by the following: s −1

vk = ∑ ai By ,i ( yk ) + as +1 κ y By , s ( yk ) + By , s +1 ( yk )  + i =1

p

∑ aB i

y ,i

( yk )

(31)

i= s +2

The structure of the linear controller is as given in Eq.(14). The problem of control design is to determine the controller parameters cj (j = 0, 1, …, r) and the unknown coefficients ai (i = 1, …, p) from an N-point data set {uk, yk}k=1~N of observed input–output measurements. The VRFT design framework is applied to the equivalent linear control system presented in Figure 8(a), and the corresponding reference model is depicted in Figure 8(b). In the case of the Wiener system, notably, the reference model M(z) does not represent the desired closed-loop behavior of the overall control system. Based on the virtual reference signal r%v ,k = M −1 ( z ) vk , the virtual controller output u%k is calculated as follows:

 c + c z −1 + L + cr z − r  −1 u%k = C ( z ) ( r%v ,k − vk ) =  0 1  M ( z ) − 1 vk −1 1− z  

(

)

(32)

Therefore, the controller parameters cj and the unknown coefficients ai can be determined by solving the following minimization problem N

min JW0 (c j , ai ) = min ∑ ( uk − u%k ) c j , ai

c j , ai

2

(33)

k =1

As in the case of the Hammerstein system, the second-order dynamics given in Eq.(18) is adopted as the reference model. Hence, Eq.(32) is written as follows:

13



 c + c z −1 + L + cr z − r u%k =  0 1 1 − z −1 

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−1 2 −2 −1 − d −1    1 − 2λ z + λ z − ( α + β z ) z   vk  −1 − d −1  (α + β z ) z  

(34)

which can be further rearranged as follows:

(α + β z ) u% = ( c −1

0

k

+ c1 z −1 + L + cr z − r ) Q( z )vk

(35)

where Q(z) is given in Eq.(21). Substituting Eq.(31) into Eq.(35) for the unknown intermediate variable vk yields the following time-domain expression:

η%k = ψ k θ

(36)

η%k = α u%k + β u%k −1

(37)

where

(κ φ

ψ k = φ k ,1 L φ k , s −1

y

k ,s

+ φ k , s +1 ) φ k , s + 2 L φ k , p 

φ k ,i = φk ,i ,0 φk ,i ,1 L φk ,i ,r 

φk , i , j

 for d = 0 By ,i ( yk +1− j ) − ( λ + λ ln λ ) By ,i ( yk − j )  By ,i ( yk + 2− j ) − ( 2λ − 1) By ,i ( yk +1− j ) + β By ,i ( yk − j ) = for d = 1  d −1 2  By ,i ( yk + d +1− j ) − ( 2λ − 1) By ,i ( yk + d − j ) + (1 − λ ) ∑ l =1 By ,i ( yk + l − j ) + β By ,i ( yk − j ) for d ≥ 2 (38)

θ = ρ1 L ρ s −1 ρ s +1 ρ s + 2 L ρ p  ρi = [ c0 ai c1ai L cr ai ]

T

(39)

By defining η k = α uk + β uk −1 , we find that the minimization problem given in Eq.(33) is equivalent to minimizing the difference between η k and η%k , and the parameter vector θ is determined by solving the following minimization problem: min JW ( θ ) = min θ

N − d −1

θ

∑ (η

− ψ k θ ) = min η − Ψθ

2

2

k

θ

k =1

2

(40)

where

η = [η1 η2 L η N − d −1 ]

T

Ψ =  ψ1T

ψ 2T

L ψ N − d −1T 

T

(41)

The solution can be determined using the least-squares technique as follows: −1 θˆ = ( ΨT Ψ ) ΨT η

14


(42)

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The problem now is how to obtain the controller parameters cj and estimates of the unknown coefficients ai from the estimate θˆ . Define the matrix

Θ = ρ1T

L ρ s −1T

ρ s +1T

ρ s + 2T

L ρ pT  = c ⋅ aT

(43)

where

c = [ c0

c1 L cr ]

T

a =  a1 L as −1

as + 2 L a p 

as +1

(44)

T

ˆ of the matrix Θ can then be obtained from the estimate θˆ . The problem An estimate Θ

ˆ . The closest, in the evolves into obtaining parameter vectors c and a from the estimate Θ 2-norm sense, estimates of c and a are those that solve the following minimization problem:

( cˆ, aˆ ) = arg min c, a

ˆ − c ⋅ aT Θ

2

(45)

2

The solution to this minimization problem is provided by the singular value decomposition (SVD)

ˆ .41 Let the economy-size SVD of Θ ˆ be given by the following: of the matrix Θ

ˆ = UΣV T Θ = [u1 u 2

σ 1 0 L 0  0 σ L M  2  [v L u r +1 ]  1 M M O 0     0 L 0 σ r +1 

v 2 L v r +1 ]

T

(46)

Then, the estimates cˆ and aˆ in Eq.(45) are given as follows:41 cˆ = u1 aˆ = σ 1 v1

(47)

ˆ is a unitary matrix, the algorithm intrinsically delivers estimates Because U in the SVD of Θ that satisfy the uniqueness condition cˆ 2 = 1 .

5. Control design based on Hammerstein–Wiener modeling The control scheme for a Hammerstein–Wiener system is as illustrated in Figure 9, which results in an equivalent linear control system illustrated in Figure 10(a), where rw = f 2−1 (r ) denotes the reference signal for the intermediate variable w. The aforementioned algorithms for 15



Page 16 of 49

Hammerstein and Wiener systems can be combined to design the controller for a Hammerstein–Wiener system. We assume that both the input nonlinear function and the inverse of the output nonlinear function can be represented by the B-spline series as follows: p1

vk = f1 (uk ) = ∑ a1,i Bu ,i (uk ) i =1

(48)

p2

wk = f 2 −1 ( yk ) = ∑ a2,i By ,i ( yk ) i =1

where a1,i and a2,i are unknown coefficients. For the Hammerstein–Wiener system, two constraints must be imposed on the parameters to obtain a unique representation. As the first constraint, it is assumed, without loss of generality, that a1,1 = 1. In addition, to assure f1 (0) = 0 and f 2−1 (0) = 0 , it is required that a1, s1 = κ u a1, s1 +1 and a2, s2 = κ y a2, s2 +1 , respectively, for some (known) s1 and s2, where

κu = − κy = −

Bu , s1 +1 (0) Bu , s1 (0)

(49)

By , s2 +1 (0) By , s2 (0)

Consequently, the inaccessible intermediate variables vk and wk can, respectively, be expressed by the following: s1 −1

vk = Bu ,1 (uk ) + ∑ a1,i Bu ,i (uk ) + a1, s1 +1 κ u Bu , s1 (uk ) + Bu , s1 +1 (uk )  + i=2

s2 −1

wk = ∑ a2,i By ,i ( yk ) + a2, s2 +1 κ y By , s2 ( yk ) + By , s2 +1 ( yk )  + i =1

p1

∑a

1,i

Bu ,i (uk )

i = s1 + 2

p2

∑

(50)

a2,i By ,i ( yk )

i = s2 + 2

The structure of the linear controller is as given in Eq.(14). The problem of control design is to determine the controller parameters cj (j = 0, 1, …, r) and the unknown coefficients a1,i (i = 2, …, p1) and a2,i (i = 1, …, p2) from an N-point data set {uk, yk}k=1~N of observed input–output measurements. The VRFT design framework is applied to the equivalent linear control system shown in Figure 10(a), and the corresponding reference model is depicted in Figure 10(b). As in the case 16


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of the Wiener system, here, the reference model M(z) does not represent the desired closed-loop behavior of the overall control system. Based on the virtual reference signal r%w,k = M −1 ( z ) wk , the virtual controller output v%k is calculated as follows:

 c + c z −1 + L + cr z − r  −1 %vk = C ( z ) ( r%w,k − wk ) =  0 1  M ( z ) − 1 wk −1 1− z  

(

)

(51)

Therefore, the controller parameters cj and the unknown coefficients a1,i and a2,i can be determined by solving the following minimization problem 0 min J HW (c j , a1,i , a2,i ) = min

c j , a1,i , a2,i

c j , a1,i , a2,i

N

∑ (v

k

− v%k )

2

(52)

k =1

Substituting the reference model in Eq.(18) into Eq.(51) yields

 c + c z −1 + L + cr z − r v%k =  0 1 1 − z −1 

−1 2 −2 −1 − d −1    1 − 2λ z + λ z − ( α + β z ) z  wk   (α + β z −1 ) z − d −1  

(53)

which can be further rearranged as follows:

(α + β z ) v% = ( c −1

k

0

+ c1 z −1 + L + cr z − r ) Q ( z ) wk

(54)

where Q(z) is given in Eq.(21). The time-domain expression of Eq.(54) is given by the following:

α v%k + β v%k −1 = c0ϕk ,0 + c1ϕ k ,1 + L + crϕk ,r

(55)

where ϕ k , j (j = 0, 1, …, r) is defined as follows:

ϕk , j

 wk +1− j − ( λ + λ ln λ ) wk − j  wk + 2 − j − ( 2λ − 1) wk +1− j + β wk − j =  d −1 2  wk + d +1− j − ( 2λ − 1) wk + d − j + (1 − λ ) ∑ l =1 wk + l − j + β wk − j

for d = 0 for d = 1

(56)

for d ≥ 2

To minimize the difference between vk and v%k (Eq.(52)), we define a new variable

 s1 −1

µk = −α ∑ a1,i Bu,i (uk ) + a1, s +1 κ u Bu , s (uk ) + Bu , s +1 (uk )  + 1

1

1

p1

∑a

1,i

 Bu ,i (uk )  

i = s1 + 2  i=2 s1 −1 p1     − β  ∑ a1,i Bu ,i (uk −1 ) + a1, s1 +1 κ u Bu ,s1 (uk −1 ) + Bu , s1 +1 (uk −1 )  + ∑ a1,i Bu ,i (uk −1 )  i = s1 + 2  i=2  + c0ϕ k ,0 + c1ϕ k ,1 + L + crϕk ,r

and then substitute wk given in Eq.(50) to yield 17


(57)


Page 18 of 49

µk = ψ k θ

(58)

where

(

)

ψ k =  ηk φ k ,1 L φ k , s2 −1 κ y φ k , s2 + φ k ,s2 +1 φ k , s2 + 2 L φ k , p2  ηk =  −ηk ,2 L −ηk , s1 −1 − κ uηk , s1 + ηk , s1 +1 −η k , s1 + 2 L −ηk , p1  η k ,i = α Bu ,i (uk ) + β Bu ,i (uk −1 )

(

)

φ k ,i = φk ,i ,0 φk ,i ,1 L φk ,i ,r   By ,i ( yk +1− j ) − ( λ + λ ln λ ) By ,i ( yk − j ) for d = 0  φk , i , j =  By ,i ( yk + 2− j ) − ( 2λ − 1) By ,i ( yk +1− j ) + β By ,i ( yk − j ) for d = 1  d −1 2  By ,i ( yk + d +1− j ) − ( 2λ − 1) By ,i ( yk + d − j ) + (1 − λ ) ∑ l =1 By ,i ( yk + l − j ) + β By ,i ( yk − j ) for d ≥ 2 (59) θ = a1 ρ1 L ρ s −1 ρ s +1 ρ s + 2 L ρ p  a1 =  a1,2 L a1,s1 −1 ρi =  c0 a2,i

T

a1, s1 +1 a1, s1 + 2 L a1, p1 

(60)

c1a2,i L cr a2,i 

Therefore, the minimization problem given in Eq.(52) evolves into minimizing the difference between η k ,1 and µ k , and the parameter vector θ is determined by solving the following minimization problem: min J HW ( θ ) = min θ

N − d −1

θ

∑ (η k =1

− ψ k θ ) = min η − Ψθ 2

k ,1

θ

2 2

(61)

where η = η1,1 η2,1 L η N − d −1,1  Ψ =  ψ1T

ψ 2T

T

L ψ N − d −1T 

T

(62)

The solution can be calculated using the least-squares technique as follows: −1 θˆ = ( ΨT Ψ ) ΨT η

(63)

The estimates of the coefficients a1,i (i = 2, …, s1–1, s1+1, …, p1) can be directly obtained from the first p1 − 2 elements of the estimate θˆ , according to the definition of θ in Eq.(60). The controller parameters cj (j = 0, 1, …, r) and estimates of coefficients a2,i (i = 1, …, s2–1, s2+1, …, p2) can be obtained by the aforementioned SVD-based algorithm using the remaining portion of 18


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θˆ . The algorithm delivers estimates that satisfy cˆ 2 = 1 as the second uniqueness condition. The coefficient a1,s1 and a2,s2 are then calculated by a1, s1 = κ u a1, s1 +1 and a2, s2 = κ y a2, s2 +1 . With the estimated nonlinear function f1 (•) , its inverse f1−1 (•) can be readily obtained for control implementation.

Remark 1. The closed-loop system behavior depends on the dynamics of the controlled process and the controller. Because the control design is aimed at solving a model-reference problem, the dynamics of the reference model, process, and controller should match so that the design target (reference model) can be effectively achieved and the static nonlinearity can be accurately estimated. Specifically, the time delay parameter d in the reference model should account for the non-minimum-phase dynamics of the process and the controller order r should be selected to allow effective model matching. Because the process is unidentified, appropriate values of d and r cannot be determined a priori. We propose that the control design procedures can be repeated for several values of d and r, and then the resulting error criteria (JH, JW, or JHW) can be evaluated to identify the appropriate values of d and r, as illustrated in the following simulation examples. Remark 2. Low-order controllers (e.g., PID controllers) are usually preferred for control applications in chemical process industries. A PID controller can be directly obtained from the proposed design algorithms by setting r = 2 . However, such a method could lead to an ineffective control design because the estimation of the static nonlinearity is affected when the PID controller cannot provide a satisfactory model matching. Here, a two-stage procedure is suggested for designing the PID controller. In the first stage, the design algorithm is implemented using a controller with an adequate order so that the static nonlinearity can be accurately estimated. In the second stage, the estimated nonlinearity is used to reconstruct the intermediate variable (Eq.(13), (31), or (50)). Consequently, the input–output data of the linear subsystem become available, and the PID controller can be readily designed using the original linear VRFT 19



Page 20 of 49

method.

Remark 3. When the process output is corrupted with additive measurement noise, the proposed design algorithms using the simple least-squares method may yield biased parameter estimates because of the correlation between the regressors and the residuals. In the noisy environment, the use of the instrumental variable (IV) technique13,39 is recommended to solve this problem. Nevertheless, simulation results have revealed that the presented design algorithm for Hammerstein systems can still provide an effective control design when the output is corrupted with moderate noise. Notably, the robustness of the control design against noise has been improved by choosing the uniqueness condition as c0 = 1 . However, the design algorithms for Wiener and Hammerstein–Wiener systems tend to be rather sensitive to noise, and an effective control design can only be obtained when the noise level is relatively low.

6. Simulation examples To illustrate the effectiveness of the proposed control design methods, this section presents simulation examples for three numerical systems and two physical processes drawn from the literature.

6.1 Example 1: Hammerstein system Consider a nonlinear process described by the following Hammerstein system: 1 − 0.5 1 + e −2 u y ( s) 1 G ( s) = = v( s ) ( s + 1) 4 v = f (u ) =

(64)

where the nonlinearity can often be observed in control systems involving the behavior of valve saturation. The proposed control design method for Hammerstein systems was applied to design the control system. For this purpose, the process was excited with uniformly random steps distributed in the range of [−3, 3]. To simulate a more realistic condition involving measurement noise, the process output was corrupted by Gaussian white noise with zero mean and a standard 20


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deviation of 0.05. The noise-to-signal ratio (NSR) was approximately 15%. A set of 500 data points was collected (N = 500) with a sampling interval of 1, as illustrated in Figure 11. The tuning parameter was chosen as λ = 0.607 ( τ = 2 ), and the nonlinear function f (u ) was represented using third-order B-splines with a knot sequence given by the following:

u = {−3, − 3, − 3, − 1.5, 0, 1.5, 3, 3, 3}

(65)

To determine the appropriate time delay parameter d and the controller order r, the minimization problem Eq.(27) was repeatedly solved for several values of d and r. For each chosen d and r, the corresponding matrices φ and Ψ are constructed (Eqs.(28), (23), and (25)) and the controller parameter vector θˆ is calculated using Eq.(29) so that the error criterion JH can be evaluated. The resulting JH is presented in Figure 12, indicating that d = 1 and r = 4 constituted a suitable choice for control design because further increasing the controller order yields a negligible improvement of the error criterion. The obtained fourth-order controller is expressed as follows:

C ( z) =

1 − 0.334 z −1 − 0.390 z −2 − 0.084 z −3 + 0.176 z −4 1 − z −1

(66)

whereas the estimated static nonlinearity is plotted in Figure 13 in which a scaled version of the estimated nonlinearity is also depicted for comparison with the actual one. We can see that the estimated static nonlinearity accurately described the actual nonlinearity. Based on the estimated nonlinearity, the intermediate variable vk was reconstructed, and then a PID controller was designed from the data set {vk, yk}, which yielded c0 = 0.971 , c1 = −0.387 , and c2 = −0.255 . For comparison, a conventional PID controller was also designed using the original (linear) VRFT method, without considering the process nonlinearity, based on the same process data {uk, yk} and reference model. The resulting PID controller parameters were c0 = 1.307 , c1 = −0.016 , and c2 = −0.677 . Figure 14 shows the closed-loop responses of the proposed nonlinear control systems and the linear PID control system to successive set-point changes. The nonlinear control system using the fourth-order controller demonstrated similar dynamics and tracked almost perfectly the 21



Page 22 of 49

behavior of the desired reference model for all the set-point changes; this clearly demonstrates the effectiveness of the proposed control design. The nonlinear control system using the PID controller from the proposed two-stage design method slightly deviated from the reference model, but it still exhibited consistent and satisfactory responses for the set-point changes. However, the behavior of the linear PID control system considerably deviated from the reference model, and it displayed oscillatory or sluggish responses for the set-point changes due to process nonlinearity. 6.2 Example 2: Wiener system Consider a nonlinear process described by the following Wiener system:

G ( s) =

v( s) 1.5 e− s = u ( s ) ( 2 s + 1)( s + 4 )

(67)

y = f (v) = 0.5v + v + v 3

2

The proposed control design method for Wiener systems was applied to design the control system. Accordingly, the process was excited with uniformly random steps distributed in the range of [−3, 2]. Gaussian white noise with zero mean and a standard deviation of 0.01 was added to the process output as measurement noise. A set of 500 data points was collected with a sampling interval of 0.5, as shown in Figure 15. The tuning parameter was chosen as λ = 0.852 ( τ = 1.25 ). Because the output data were distributed in the range of [−0.52, 0.92], the inverse of the nonlinear function f −1 ( y ) was represented using the third-order B-splines with a knot sequence given by the following:

y = {−0.52, − 0.52, − 0.52, − 0.35, − 0.17, 0, 0.18, 0.37, 0.55, 0.73, 0.92, 0.92, 0.92} (68) The minimization problem Eq.(40) was repeatedly solved for several values of d and r. For each chosen d and r, the corresponding matrices η and Ψ are constructed (Eqs.(41) and (38)) and the parameter vector θˆ is calculated using Eq.(42). Then, the controller parameters and the B-spline coefficients of f −1 ( y ) are calculated using Eq.(47), and the error criterion JW can be evaluated. The resulting error criterion JW is presented in Figure 16. Notably, the calculation of JW is based on the separated parameters obtained from Eq.(47). Figure 16 reveals that d = 2 and r = 2 (PID controller) constituted a suitable choice for control design. The corresponding 22


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parameters of the PID controller were c0 = −0.857 , c1 = 0.053 , and c2 = 0.513 , whereas the estimated inverse of the static nonlinearity is plotted in Figure 17, in which a scaled version of the estimated nonlinearity is also depicted for comparison with the actual one. The process nonlinearity was observed to be satisfactorily estimated. For comparison, a conventional PID controller was also designed using the linear VRFT method based on the same process data and reference model. The resulting controller parameters were c0 = 1.028 , c1 = −0.053 , and c2 = −0.572 . Figure 18 illustrates the closed-loop responses of the proposed nonlinear control system and the linear PID control system to successive set-point changes. The proposed nonlinear control system provided smooth responses for all set-point changes and obviously outperformed the PID control system from the linear VRFT method. 6.3 Example 3: Hammerstein−Wiener system Consider a nonlinear process described by the following Hammerstein−Wiener system:

v = f1 (u ) = (1 − e −0.75u ) u G(s) =

1 w( s ) = v( s) ( s + 1)3

(69)

y = f 2 ( w) = 2 (1 − e −0.693 w ) The proposed design control method for Hammerstein−Wiener systems was applied to design the control system. For this purpose, the process was excited with uniformly random steps distributed in the range of [−1, 2]. The process output was corrupted with Gaussian white noise with zero mean and a standard deviation of 0.002. A set of 600 data points was collected with a sampling interval of 0.5, as shown in Figure 19. The tuning parameter was chosen as λ = 0.717 ( τ = 1.5 ), and f1 (u ) and f 2−1 ( y ) were represented using the third-order B-splines with knot sequences respectively given by the following:

u = {−1, − 1, − 1, − 0.5, 0, 0.5, 1, 1.5, 2, 2, 2} y = {−1.51, − 1.51, − 1.51, − 1.13, − 0.75, − 0.38, 0, 0.40, 0.80, 1.2, 1.2, 1.2} 23


(70)


Page 24 of 49

To determine the appropriate time delay parameter d and controller order r, the minimization problem Eq.(61) was repeatedly solved for several values of d and r. For each chosen d and r, the corresponding matrices η and Ψ are constructed (Eqs.(62) and (59)) and the parameter vector θˆ is calculated using Eq.(63). Then, the controller parameters and the B-spline coefficients of f 2−1 ( y ) are obtained using the SVD-based algorithm similar to Eq.(47),

and the error criterion JHW can be evaluated. The resulting JHW is presented in Figure 20, indicating that d = 1 and r = 3 constituted a suitable choice for control design. The obtained third-order controller is expressed as follows: C(z) =

−0.605 + 0.125 z −1 + 0.701z −2 − 0.356 z −3 1 − z −1

(71)

whereas the estimated input and output nonlinearities are plotted in Figure 21, in which the scaled versions of the estimated nonlinearities are also depicted for comparison with the actual ones. Both the estimated nonlinearities accurately described the actual nonlinearities. Based on the estimated nonlinearities, the intermediate variables vk and wk were reconstructed, and then a PID controller was designed from the data set {vk, wk}, which yielded c0 = −0.772 , c1 = 0.797 , and c2 = −0.151 . For comparison, a conventional PID controller was also designed using the linear VRFT method based on the same process data and reference model. The resulting PID controller parameters were c0 = 0.882 , c1 = −0.973 , and c2 = 0.239 . Figure 22 shows the closed-loop responses of the proposed nonlinear control systems and the linear PID control system to successive set-point changes. The two nonlinear control systems exhibited similarly satisfactory control performance, whereas the nonlinear PID control system provided a smoother controller output compared with the nonlinear control system using a third-order controller. However, the linear PID control system demonstrated sluggish output responses for most of the applied set-point changes. 6.4 Example 4: Distillation column In this example, the proposed control design method for Hammerstein systems was applied 24


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to design a nonlinear control system based on simulation data of a binary distillation column. The considered 30-tray (32-stage) binary distillation column, studied by Horton et al.,42 was described by a set of coupled nonlinear differential equations and assumed to have a constant molar overflow as well as a constant relative volatility of 1.6. The feed stream was introduced at the middle of the column on stage 17 and had a feed composition of 0.5. In the control structure considered in this study, the reflux ratio (R) was considered as the manipulated variable to control the distillate composition (xD). The nominal operating conditions of the column were R0 = 3 mol/min and xD ,0 = 0.935 . A time delay of 2 min was assumed in the composition

measurements. For control design, the model of the column was excited with uniformly random steps around the nominal value of the reflux ratio. Changes in the reflux ratio were produced every 20 min, with a maximum amplitude of ±20% of the nominal value. To simulate a more realistic condition, the process output xD was corrupted with Gaussian white noise with NSR = 15%. A set of 600 data points was collected from the simulation with a sampling interval of 2 min, as presented in Figure 23, for implementing the proposed control design. The tuning parameter was chosen as λ = 0.670 ( τ = 5 ), and the nonlinear function f (u ) , where u = R − R0 , was represented using the third-order B-splines with a knot sequence given by the following:

u = {−0.6, − 0.6, − 0.6, − 0.3, 0, 0.6, 0.6, 0.6}

(72)

The minimization problem Eq.(27) was repeatedly solved for several values of d and r to identify the appropriate time delay parameter and controller order. The resulting error criterion JH is shown in Figure 24. To reach a compromise between the accuracy and the complexity of the

controller, d = 1 and r = 3 constituted a suitable choice for control design. The obtained third-order controller is expressed as follows:

C ( z) =

1 − 0.166 z −1 − 0.428 z −2 − 0.287 z −3 1 − z −1

(73)

whereas the estimated static nonlinearity is plotted in Figure 25. The estimated nonlinearity was 25



Page 26 of 49

used to reconstruct the intermediate variable vk for designing a PID controller from the data set {vk, yk}, where y = xD − xD ,0 . The resulting PID controller parameters were c0 = 1.219 , c1 = −0.409 , and c2 = −0.674 . For comparison, a conventional PID controller was also designed

using the linear VRFT method, without considering the process nonlinearity, based on the same process data and reference model, which yielded c0 = 1.068 , c1 = 0.102 , and c2 = −0.829 . Figure 26 shows the closed-loop responses of the proposed nonlinear control systems and the linear PID control system to successive set-point changes. The two nonlinear control systems exhibited similar dynamics and effectively tracked the behavior of the reference model for all the set-point changes, indicating that the proposed nonlinear controllers could effectively compensate the process nonlinearity. However, the behavior of the linear PID control system remarkably differed from that of the reference model because the response of linear control displayed an oscillatory behavior and longer settling time for the set-point changes. Clearly, the linear VRFT design is inadequate to provide a satisfactory control performance level for the distillation column, thus validating the usefulness of the proposed nonlinear VRFT design. 6.5 Example 5: pH neutralization process The pH neutralization process is crucial in various chemical processes such as wastewater treatment, biochemical, and polymerization processes. In this example, both the proposed control design methods for Wiener and Hammerstein−Wiener systems were applied for designing a nonlinear control system based on simulation data of a pH neutralization process. The considered pH process, studied by Palancar et al.,43 involved the neutralization of acetic acid (AcH), propionic acid (PrH), and sodium hydroxide (NaOH) in a single tank. The process can be described by three nonlinear differential equations and a nonlinear algebraic equation for the pH: dCAcH = qa CAcH,i − ( qa + qb )CAcH dt dCPrH V = qa CPrH,i − ( qa + qb )CPrH dt dCNaOH V = qb CNaOH,i − ( qa + qb )C NaOH dt

V

26


(74)

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CAcH CPrH + + 10 pH −14 − CNaOH − 10− pH = 0 pK AcH − pH pK PrH − pH 1 + 10 1 + 10

(75)

where qa and qb are the acid and base flow rates, respectively, V is the tank volume, and C is the concentration. Without buffering, the process exhibited a high degree of nonlinearity due to the implicit output equation (Eq.(75)), known as the titration curve, and hence was difficult to control. The objective was to control the pH of the effluent solution by manipulating the base flow rate qb. The system parameters were V = 1 L , pK AcH = 4.75 , pK PrH = 4.87 , CAcH,i = 1 M , CPrH,i = 1 M , CNaOH,i = 2 M , and qa = 0.0142 L/s . The nominal operating conditions were qb ,0 = 0.0142 L/s and pH 0 = 9.407 . For control design, uniformly random steps were introduced to the base flow rate, with changes produced every 1 s and a maximum amplitude of ±50% of the nominal value. The process output was corrupted with Gaussian white noise with zero mean and a standard deviation of 0.003. A set of 600 data points was collected from the simulation with a sampling interval of 0.1 s, as illustrated in Figure 27, for implementing the proposed control design. The tuning parameters were chosen as λ = 0.717 ( τ = 0.3 ), d = 0 , and r = 2 . By modeling the pH process as a Wiener system, we represented the inverse of nonlinear function f −1 ( y ) , where y = pH − pH 0 , using the third-order B-splines with a knot sequence given by the following:

y = {−3.24, − 3.24, − 3.24, − 2.43, − 1.62, − 0.81, 0, 0.79, 1.58, 2.37, 3.16, 3.16, 3.16} (76) The resulting controller parameters were c0 = −0.642 , c1 = −0.122 , and c2 = 0.757 , whereas the estimated inverse of static nonlinearity is plotted in Figure 28. The estimated nonlinearity was observed to resemble the behavior of a typical titration curve. By modeling the pH process as a Hammerstein−Wiener system, we represented the nonlinear function

f1 (u ) , where

u = qb − qb ,0 , using the third-order B-splines with a knot sequence given by the following:

u = {−0.007, − 0.007, − 0.007, 0, 0.0062, 0.0062, 0.0062} 27


(77)


and f 2−1 ( y ) was represented using the same B-splines used in the design based on Wiener modeling. The resulting controller parameters were c0 = −0.639 , c1 = −0.121 , and c2 = 0.760 , whereas the estimated input nonlinearity and inverse of the output nonlinearity are plotted in Figure 29. For comparison, a conventional PID controller was also designed using the linear VRFT method based on the same process data and reference model, which resulted in c0 = 2.48 ×10−4 , c1 = 3.05 × 10−5 , and c2 = −2.43 ×10−4 . Figure 30 shows the closed-loop responses of the aforementioned three control systems to successive set-point changes. Both the nonlinear control systems exhibited more favorable and consistent responses than the linear control system for the set-point changes because the proposed nonlinear control design enables effective compensation of the process nonlinearity. However, the response derived from the linear PID controller exhibited a considerable overshoot and consumed a long time to settle down due to the nonlinear behavior of the pH process. When the two nonlinear control systems were compared, the control system designed based on the Hammerstein−Wiener modeling achieved a higher control performance level because it provided a more rapid tracking response. The result can be expected because, compared with Wiener modeling, Hammerstein−Wiener modeling provides more degrees of freedom for modeling and controlling the nonlinear process.

7. Conclusion This paper presents a novel VRFT-based method for the control design of nonlinear systems modeled by the Hammerstein, Wiener, and Hammerstein−Wiener structures. In this method, the controller parameters and system nonlinearity are simultaneously obtained by directly using a single set of plant data. The proposed algorithms combine the VRFT framework with B-spline-based nonlinear approximation to enable the system to be established in a linear regressor form; thus, the simple least-squares technique can be used to design the controller, which avoids implementation problems due to computational limitations. The proposed 28


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nonlinear VRFT technique retains the advantage of one-shot and easy implementation of the linear VRFT. Moreover, in the proposed method, the used reference model and the order of controller are coordinately selected to ensure an effective VRFT design for the nonlinear control, and a two-stage design procedure is suggested accordingly when a low-order (i.e., PID) controller is preferred. The superiority of the proposed nonlinear control design over the linear VRFT control design is demonstrated herein through three numerical systems and two benchmark processes. Further research will focus on the extension of the nonlinear control design method to multivariable systems.

Supporting Information The programs and simulation schemes for the proposed nonlinear VRFT control design in the Matlab/Simulink environment corresponding to the presented simulation examples are provided. The Spline Toolbox of Matlab is required for executing the programs.

Acknowledgement The authors thank the Ministry of Science and Technology of Taiwan for supporting this research under grant 104-2221-E-027-105.

29



References (1) Haber, R.; Unbehauen, H. Structure identification of nonlinear dynamic systems – A survey on input/output approaches. Automatica 1990, 26, 651–677. (2) Pearson, R.; Pottmann, M. Gray-box identification of block-oriented nonlinear models. J. Process Control 2000, 10, 301–315. (3) Bloemen, H.; Chou, C.; van den Boon, T. ; Verdult, V.; Verhaegen, M.; Backx, T. Wiener model identification and predictive control for dual composition control of a distillation column. J. Process Control 2001, 11, 601–620. (4) Eskinat, E.; Johnson, S.; Luyben, W. Use of Hammerstein models in identification of nonlinear systems. AIChE J. 1991, 37, 255–268. (5) Hwang, S. H.; Hsieh, C. Y.; Chen, H. T.; Huang, Y. C. Use of discrete Laguerre expansions for noniterative identification of nonlinear Wiener models. Ind. Eng. Chem. Res. 2011, 50, 1427–1438. (6) Nagammai, S.; Sivakumaran, N.; Radhakrishnan, T. Control system design for a neutralization process using block oriented models. Instrumentation Science and Technology 2006, 34, 653–667. (7) Norquary, S.; Palazoglu, A.; Romagnoli, J. Model predictive control based on Wiener models. Chem. Eng. Sci. 1998, 53, 75–84. (8) Hunter, I.; Korenberg, M. The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biol. Cybern. 1986, 55, 135–144. (9) Kotz, K.; Cinar A.; Mei, Y.; Roggendorf, A.; Littlejohn, E.; Quinn, L.; Rollins, D. Multiple-input subject-specific modeling of plasma glucose concentration for feedforward control. Ind. Eng. Chem. Res. 2014, 53, 18216–18225. (10) Hjalmarsson, H.; Gevers, M.; Gunnarsson, S.; Lequin, O. Iterative feedback tuning: theory and applications. IEEE Control Syst. Mag. 1998, 18, 26–41. (11) Hjalmarsson, H. Iterative feedback tuning – an overview. Int. J. Adapt. Control Signal Process. 2002, 16, 373–395. (12) 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. 30


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(13) 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. (14) Karimi, A.; Miskovic, L.; Bonvin, D. Iterative correlation-based controller tuning. Int. J. Adapt. Control Signal Process. 2004, 18, 645–664. (15) 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 Institute of Systems, Control and Information Engineers 2004, 17, 528–536. (16) Sjoberg, J.; Gutman, P. O.; Agarwal, M.; Bax, M. Nonlinear controller tuning based on a sequence of identifications of linearized time-varying models, Control Engineering Practice

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(39) Chan, K.; Bao, J.; Whiten, W. Identification of MIMO Hammerstein systems using cardinal spline functions. J. Process Control 2006, 16, 659–670. (40) Jeng, J. C. Closed-loop identification of multivariable systems, using B-spline series expansions for step responses. Ind. Eng. Chem. Res. 2012, 51, 2376–2387. (41) Gómez, J. C.; Baeyens, E. Identification of block-oriented nonlinear systems using orthonormal bases. J. Process Control 2004, 14, 685–697. (42) Horton, R. R.; Bequette, B. W.; Edgar, T. F. Improvements in dynamic compartmental modeling for distillation. Comput. Chem. Eng. 1991, 15, 197–201. (43) Palancar, M. C.; Aragon, J. M.; Torrecilla, J. S. pH-control systems based on artificial neural networks. Ind. Eng. Chem. Res. 1998, 37, 2729–2740.

33



Figure 1. Schematic diagram of block-oriented nonlinear models: (a) Hammerstein (b) Wiener and (c) Hammerstein–Wiener models.

Figure 2. Feedback control system.

34


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Page 35 of 49

e%k

r%k

u%k

Figure 3. Schematic diagram of VRFT.

1

i=1

i=6

0.8

i=3

i=4

i=2

Bx,3; i(x)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


i=5

0.6

0.4

0.2

0 -2

-1

0

1

x Figure 4. The third-order B-splines on the interval [–2, 2].

35


2


Figure 5. Linearizing control scheme for nonlinear processes based on Hammerstein modeling.

r%k

Figure 6. (a) Equivalent linear control system for Hammerstein systems and (b) its reference model.

36


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Figure 7. Linearizing control scheme for nonlinear processes based on Wiener modeling.

r%v,k

Figure 8. (a) Equivalent linear control system for Wiener systems and (b) its reference model.

37



Figure 9. Linearizing control scheme for nonlinear processes based on Hammerstein–Wiener modeling.

r%w,k

Figure 10. (a) Equivalent linear control system for Hammerstein–Wiener systems and (b) its reference model.

38


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Page 39 of 49

3 2

u

1 0 -1 -2 -3

0

100

200

300

400

500

300

400

500

Sample (k) 0.6 0.4 0.2

y

0 -0.2 -0.4 -0.6

0

100

200

Sample (k)

Figure 11. Process input-output data used in controller design for Example 1.

3.2 d=0 d=1 d=2

3

2.8

2.6

JH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.4

2.2

2

1.8

1.6

1

2

3

4

5

Controller orde (r)

Figure 12. Error criterion JH with different values of d and r for Example 1. 39



1.5 Actual nonlinear function Estimated nonlinear function Estimated nonlinear function (scaled) 1

v

0.5

0

-0.5

-1 -3

-2

-1

0

1

2

3

u

Figure 13. Comparison of estimated and actual process nonlinearity for Example 1.

0.5

y

0

-0.5

0

20

40

60

Time 2 1

u

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 49

80

100

120

Set-point Reference model response Nonlinear control (4th-order controller) Nonlinear control (PID controller) Linear PID control

0 -1 -2

0

20

40

60

80

100

Time

Figure 14. Closed-loop responses for Example 1. 40


120

Page 41 of 49

2 1

u

0 -1 -2 -3

0

100

200

300

400

500

300

400

500

Sample (k) 1

y

0.5

0

-0.5 0

100

200

Sample (k)


2.5

2 d=1 d=2 d=3 1.5

JW

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

0.5

0

1

1.5

2

2.5

3

3.5

4

4.5

5

Controller orde (r)

Figure 16. Error criterion JW with different values of d and r for Example 2. 41



1.5 Actual inverse nonlinear function Estimated inverse nonlinear function Estimated inverse nonlinear function (scaled)

1

0.5

v

0

-0.5

-1

-1.5 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y

Figure 17. Comparison of estimated and actual process nonlinearity for Example 2.

1 Set-point Nonlinear control (PID controller) Linear PID control

y

0.5

0

-0.5

0

10

20

30

40

50

60

70

80

50

60

70

80

Time

1 0

u

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-1 -2 0

10

20

30

40

Time



Page 43 of 49

2

u

1

0

-1

0

100

200

300

400

500

600

400

500

600

Sample (k ) 1

y

0.5 0 -0.5 -1 -1.5 0

100

200

300

Sample (k )


0.08 d=0 d=1 d=2

0.07

0.06

0.05

JHW

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.04

0.03

0.02

0.01

0

1

1.5

2

2.5

3

3.5

4

4.5

5

Controller orde (r)

Figure 20. Error criterion JHW with different values of d and r for Example 3. 43



1.5 1 0.5

v

0 -0.5

Actual nonlinear function Estimated nonlinear function Estimated nonlinear function (scaled)

-1 -1.5 -1

-0.5

0

0.5

1

1.5

2

u 1.5 Actual inverse nonlinear function Estimated inverse nonlinear function Estimated inverse nonlinear function (scaled)

1

w

0.5 0 -0.5 -1 -2

-1.5

-1

-0.5

0

0.5

1

1.5

y

Figure 21. Comparison of estimated and actual process nonlinearities for Example 3 (up: input nonlinearity; down: output nonlinearity).

1

y

0.5

0

-0.5

0

10

20

30

40

50

60

70

80

90

60

70

80

90

Time Set-point Nonlinear control (3rd-order controller) Nonlinear control (PID controller) Linear PID control

1.5 1

u

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 49

0.5 0 -0.5 0

10

20

30

40

50

Time



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R (mol/min)

3.6 3.3 3 2.7 2.4

0

100

200

300

400

500

600

400

500

600

Sample (k) 1

xD

0.9

0.8

0.7

0

100

200

300

Sample (k)


0.17

d=0 d=1 d=2

0.16 0.15 0.14 0.13

JH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.12 0.11 0.1 0.09 0.08

1

2

3

4

5

Controller orde (r)

Figure 24. Error criterion JH with different values of d and r for Example 4. 45



0.1

0.05

0

v

-0.05

-0.1

-0.15

-0.2

-0.25 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

R -R0 (mol/min)

Figure 25. Estimated process nonlinearity for Example 4.

0.98

xD

0.96 0.94 0.92 0.9 0.88

0

100

200

300

400

500

600

700

800

500

600

700

800

Time (min) Set-point Reference model response Nonlinear control (3rd-order controller) Nonlinear control (PID controller) Linear PID control

3.4

R (mol/min)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 49

3.2 3 2.8 0

100

200

300

400

Time (min)



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0.022

qb (L/s)

0.018 0.014 0.01 0.006

0

100

200

300

400

500

600

400

500

600

Sample (k) 14

pH

12 10 8 6

0

100

200

300

Sample (k)


0.025 0.02 0.015 0.01 0.005

v

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 -0.005 -0.01 -0.015 -0.02 -4

-3

-2

-1

0

1

2

3

4

pH - pH0

Figure 28. Estimated process nonlinearity based on Wiener modeling for Example 5. 47



1

v

0.5

0

-0.5 -8

-6

-4

-2

0

2

4

6

qb - qb,0

8 -3

x 10

2

w

1 0 -1 -2 -4

-3

-2

-1

0

1

2

3

4

pH - pH0

Figure 29. Estimated process nonlinearities based on Hammerstein-Wiener modeling for Example 5 (up: input nonlinearity; down: output nonlinearity).

11

pH

10 Set-point Nonlinear control (H-W) Nonlinear control (W) Linear control

9 8 0

5

10

15

20

25

15

20

25

Time (s) 0.016

qb (L/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.015

0.014

0.013

0

5

10

Time (s)



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