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Abstract

This paper presents the double-layered nonlinear model predictive control method for a continuously stirred tank reactor and a pH neutralization process that are subject to input disturbances and output disturbances at the same time. The nonlinear systems can be described as a Hammerstein -Wiener model. Furthermore, two nonlinear parts of the Hammerstein -Wiener model should be transformed into linear combination of known input and unknown disturbances, respectively. By taking advantage of Kalman filter, disturbances and states can be estimated. The estimated disturbances and states can be considered to calculate steady-state target in steady-state target calculation layer. Moreover, the state feedback control law can be obtained in dynamic control layer. A simple proof for offset-free control is given in the proposed method. The simulation results show that the controlled variable can achieve the offset-free control. It can be seen that the proposed method has better disturbance rejection performance, strong robustness and practical value.

Keywords

Double-layered nonlinear model predictive control offset-free control disturbance rejection Kalman filter

Introduction

Model predictive control (MPC) is the most popular advanced control method in industrial control technology and academics, which can effectively overcome the disturbance and uncertainty and easily handle the constrain of controlled variables and manipulated variables.^1–5 It is a kind of model-based closed-loop optimization control strategy. The principle of the method is to predict the dynamic behavior of the system and to proceed rolling optimization and to achieve feedback correction of model error.^6–9 There are many researches on the MPC method for linear systems. However, all the practical systems are nonlinear in the physical world.^10–12 Therefore, in recent years, nonlinear model predictive control (NMPC) has been obtained more and more attention.^13–21

As for the NMPC, the accurate nonlinear prediction model is very important. At present, the nonlinear prediction models with block structure are becoming more and more popular, such as Hammerstein model, Wiener model or Hammerstein–Wiener model, which combined the dynamic linear model with the static nonlinear model. These models do not need to know the principle of nonlinear system, which can describe exactly the nonlinear system. Compared with Hammerstein model and Wiener model, the Hammerstein–Wiener model includes two static nonlinear parts and one dynamic linear part, so the model is closer to the nonlinear system than Hammerstein model or Wiener model. Moreover, it is also very effective for the asymptotic tracking and disturbance rejection of nonlinear systems. Nowadays, the NMPC based on the Hammerstein–Wiener model has achieved many research results.^22–26

However, disturbances appear in all stages of the control system.^27–29 In the control system, the disturbance sources can usually be divided into two categories: input disturbances and output disturbances. The input disturbances are considered to be acting on the input side of control system, which come from the nonlinear characteristics of the actuators, such as valve viscous, and come possibly from the change in operating conditions, such as the flow, level and composition fluctuation of upstream feed. The output disturbances are considered to be acting on the output side of control system, which come from the measurement of sensors. When the control system is subject to these disturbances, it may lead to model mismatch and steady-state deviation. Therefore, the offset-free control of nonlinear system should be worked as the important goal in the controller design. The offset-free character of the control system refers to the capability of controlled variable to asymptotically track the set point. In other words, the steady-state deviation of closed-loop system approaches to zero as soon as possible.³⁰

At present, an improved “steady-state target optimization + dynamic control” double-layered NMPC is proposed to reject disturbances and achieve offset-free control in control system.^31,32 It has three parts, such as prediction and estimation, steady-state target calculation and dynamic control. In other words, the NMPC, which contains steady-state target calculation and dynamic control, is called the double-layered NMPC. In the double-layered NMPC, the appropriate set point should be calculated in the steady-state target calculation layer with satisfying various constraints. Moreover, offset-free control can be obtained by the dynamic control layer very easily.

The double-layered NMPC with disturbance rejection has obtained a lot of research results. Rajamani et al.³³ investigated that the choice of disturbance models does not have an effect on the control performance of closed-loop system. Moreover, the appropriate covariances can be estimated by autocovariance technique. In Li et al.,³⁴ based on double-layered MPC, the target optimization and feasibility of constrained multiobjective multidegree of freedom were studied. In Horváth et al.,³⁵ the double-layered MPC was researched on the offset-free character with unknown disturbances. In Betti et al.,³⁶ for the infeasible reference problem of constrained systems subject to bounded disturbances, a double-layered robust MPC was proposed based on velocity form model. In Zou et al.,³⁷ for the non-square system, the double-layered MPC was proposed. The compatibility issue for the thin system and the optimal solution for the fat system can be tackled in the upper level steady-state optimization. The lower level predictive control algorithm is integrated with the control input target and guaranteed the uniqueness of the steady-state solution for fat system. Wang et al.³⁸ researched that the Hammerstein model was transformed into a linear model with one known input and unknown input, and a minimum-variance unbiased (MVU) filter was introduced to estimate both the state and the unknown input. Kim et al.³⁹ takes advantage of a disturbance observer (DOB) for the active/reactive powers of a three-phase alternating current (AC)/direct current (DC) converter. Furthermore, an offset-free MPC was studied. In Dang et al.,⁴⁰ in order to reject output disturbances and guarantee maximum power captured under the rated wind speed, a new double-layered MPC and new DOB were studied. In Wang et al.,⁴¹ for handling unmeasured disturbances with arbitrary statistics, a linear MPC with optimal filters was proposed. As a result, the offset-free control can be achieved by proposed method in the presence of asymptotically constant unmeasured disturbances. In Pan et al.,⁴² for a double-layered MPC, an on-line constraints softening strategy was given based on the infeasibility analysis of the dynamic controller. All in all, the above proposed approach can obtain offset-free control in the presence of unknown nonlinearity and unmeasured disturbances. However, the input disturbances or output disturbances were only considered in these papers, respectively. In the control system, most of the input disturbances and output disturbances appeared at the same time.

In this article, the double-layered NMPC based on Hammerstein–Wiener model with disturbance rejection is proposed. The nonlinear system should be described as the Hammerstein–Wiener model and the Kalman filter is used to estimate the disturbances and states of the system. According to the estimated disturbances and states, the new steady-state target can be calculated in steady-state target optimization layer. Therefore, the state feedback control law is obtained by solving the algebraic Riccati equation. A simple proof for offset-free control is given in the proposed method. The simulation results show that the proposed method has better performance of disturbance rejection and strong robustness.

Notation $ℝ^{n}$ denotes the n-dimensional Euclidean space. $x (k | k)$ denotes the state of system at time $k$ . $A^{T}$ denotes the transpose of the matrix $A$ . $x (k + i | k)$ and $u (k + i | k)$ represent the predicted state and control action at step $k$ . $∥ x ∥^{2}$ denotes the Euclidean norm of the vector $x$ . $I$ denotes an identity matrix of known compatible dimensions.

Hammerstein–Wiener model

Consider the following discrete-time nonlinear system

\begin{array}{l} x (k + 1) = f (x (k), u (k), d (k)) \\ y (k) = g (x (k), p (k)) \end{array}

(1)

where $f, g$ are unknown nonlinear continuous and differentiable smooth functions, $x \in ℝ^{n_{x}}$ denotes the state vector, $u \in ℝ^{n_{u}}$ denotes the control input, $y \in ℝ^{n_{y}}$ denotes the system output, $d \in ℝ^{n_{d}}$ denotes the external input disturbances and $p \in ℝ^{n_{p}}$ denotes the external output disturbances.

The nonlinear system Equation (1) can be approximately described as the Hammerstein–Wiener model. The Hammerstein–Wiener model includes one linear part $G$ and two nonlinear parts $F$ and $H$ , as shown in Figure 1.

Figure 1.

Hammerstein–Wiener model.

The linear part $G$ can be described as follows

\begin{array}{l} x (k + 1) = A x (k) + B m (k) \\ n (k) = C x (k) \end{array}

(2)

The two nonlinear parts $F$ and $H$ can be expressed as

\begin{array}{l} m (k) = F [u (k)] \\ y (k) = H [n (k)] \end{array}

(3)

where $m (k)$ and $n (k)$ are intermediate variables without physical meaning. $F$ and $H$ are continuously differentiable and smooth functions. The following two conditions should be assumed.

Assumption 1

Consider the Hammerstein–Wiener model Equations (2) and (3); for all given $u$ and $y$ , there exists an unique solution $x^{*}$ such that $x^{*} = A x^{*} + B F [u (k)]$ and $y = H [n (k)]$ hold.

Assumption 2

Consider the Hammerstein–Wiener model Equations (2) and (3); for any given output $y$ , there exists an unique solution $x^{*}$ and $u^{*}$ such that $x^{*} = A x^{*} + B F (u^{*})$ and $y = H [n^{*}]$ hold.

In the vicinity of the working point $u_{0}$ , nonlinear part $F [.]$ can be Taylor expanded as follows

\begin{matrix} m (k) = F (u_{0}) + \frac{\partial F}{\partial u} |_{u_{0}} (u (k) - u_{0}) + ε (k) \\ = γ u (k) + θ (k) \end{matrix}

(4)

where $γ = (\partial F / \partial u) |_{u_{0}}$ , $ε (k)$ is the sum of all high-order infinitesimal except for the first-order differential term such that $θ (k) = F (u_{0}) - (\partial F / \partial u) |_{u_{0}} u_{0} + ε (k)$ . Therefore, Equation (4) can be expressed as

\begin{array}{l} m (k) = u (k) + d (k) \end{array}

(5)

where $d (k) = (γ - I) u (k) + θ (k)$ .

The purpose of the above transformation is that the nonlinear input can be expressed as the linear combination of known input and unknown disturbances. Similarly, in the vicinity of the working point $n_{0}$ , nonlinear part $H [.]$ can also be Taylor expanded as follows

\begin{matrix} y (k) = H (n_{0}) + \frac{\partial H}{\partial n} |_{n_{0}} (n (k) - n_{0}) + ζ (k) \\ = λ n (k) + ξ (k) \end{matrix}

(6)

where $λ = (\partial H / \partial n) |_{n_{0}}$ , $ζ (k)$ is the sum of all high-order infinitesimal except for the first-order differential term such that $ξ (k) = H (n_{0}) - (\partial H / \partial n) |_{n_{0}} n_{0} + ζ (k)$ . Therefore, Equation (6) can be expressed as

y (k) = n (k) + p (k)

(7)

where $p (k) = (λ - I) n (k) + ξ (k)$ . Equivalence between the Hammerstein–Wiener model and linear model with unknown input disturbance and output disturbance is shown in Figure 2.

Figure 2.

Equivalence between the Hammerstein–Wiener model and linear model with unknown input disturbance and output disturbance.

It should be mentioned that for all given pairs $(m (k), u (k)) and (y (k), n (k))$ , there exists only $d (k)$ and $p (k)$ such that Equations (5) and (7) hold. Therefore, the existence of these transformations can be illustrated if and only if assumptions 1 and 2 hold. Moreover, these transformations do not need to know prior knowledge of the nonlinear functions $F [u (k)]$ and $H [n (k)]$ and all the nonlinear dynamics and disturbances are included in $d (k)$ and $p (k)$ . Therefore, it can cope with the uncertainties and disturbances. The remaining problems are how to obtain the $d (k)$ and $p (k)$ .

Therefore, the system Equation (1) can be transformed into the following linear state-space model

\begin{matrix} x (k + 1) = A x (k) + B u (k) + B_{d} d (k) \\ y (k) = C x (k) + C_{p} p (k) \end{matrix}

(8)

where $d (k)$ can be considered as the unknown input disturbances and $p (k)$ can be taken as the unknown output disturbances.

Double-layered NMPC based on Hammerstein–Wiener model

The double-layered NMPC is mainly composed of steady-state target calculation and dynamic controller. In order to reject the disturbances, it should be added to state and disturbance estimation. Double-layered NMPC based on Hammerstein–Wiener model is shown in Figure 3.

Figure 3.

Double-layered nonlinear model predictive control based on Hammerstein–Wiener model.

Assume that the disturbances are constant disturbances

\begin{array}{l} d (k + 1) = d (k) \\ p (k + 1) = p (k) \end{array}

(9)

The disturbance models (Equation (9)) are expanded into Equation (8)

\begin{array}{l} [\begin{matrix} x (k + 1) \\ d (k + 1) \\ p (k + 1) \end{matrix}] = [\begin{matrix} A & B_{d} & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix}] [\begin{matrix} x (k) \\ d (k) \\ p (k) \end{matrix}] + [\begin{matrix} B \\ 0 \\ 0 \end{matrix}] u (k) \\ y (k) = [\begin{matrix} C & 0 & C_{p} \end{matrix}] [\begin{matrix} x (k) \\ d (k) \\ p (k) \end{matrix}] \end{array}

(10)

One can get

\begin{array}{l} \tilde{x} (k + 1) = \tilde{A} \tilde{x} (k) + \tilde{B} u (k) \\ y (k) = \tilde{C} \tilde{x} (k) \end{array}

(11)

where $\tilde{x} (k) = [\begin{matrix} x (k) \\ d (k) \\ p (k) \end{matrix}]$ , $\tilde{A} = [\begin{matrix} A & B_{d} & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix}]$ , $\tilde{B} = [\begin{matrix} B \\ 0 \\ 0 \end{matrix}]$ and $\tilde{C} = [\begin{matrix} C & 0 & C_{p} \end{matrix}]$ . $B_{d}$ and $C_{p}$ are input disturbance model and output disturbance model, respectively.

Dynamic control

The performance cost is described as follows

\begin{array}{l} J = \sum_{i = 1}^{H_{p}} {{[\tilde{C} \tilde{x} (k + i | k) - \tilde{C} x_{s}]}^{T} Q [\tilde{C} \tilde{x} (k + i | k) - \tilde{C} x_{s}]} \\ + \sum_{i = 1}^{H_{m}} {[u (k + i | k) - u_{s}]^{T} R [u (k + i | k) - u_{s}]} \end{array}

(12)

where $u$ should be calculated so that $J$ is minimized, $H_{p}$ is the prediction horizon, $H_{m}$ is the control horizon, $Q$ and $R$ are the positive symmetric weight matrices, $x_{s}$ is the steady-state target and $u_{s}$ is the manipulated input. The deviation variables are considered as

\begin{array}{l} ϕ (i) = \tilde{x} (k + i) - x_{s} \end{array}

(13)

\begin{array}{l} φ (i) = u (k + i) - u_{s} \end{array}

(14)

Therefore, the performance cost Equation (12) can be rewritten

\begin{array}{l} J = \sum_{i = 1}^{H_{p}} {{[\tilde{C} ϕ (i)]}^{T} Q [\tilde{C} ϕ (i)]} + \sum_{i = 1}^{H_{m}} {{[φ (i)]}^{T} R [φ (i)]} \end{array}

(15)

subject to

\begin{array}{l} ϕ (i + 1) = \tilde{A} ϕ (i) + \tilde{B} φ (i), i = 1, 2, \dots, H_{p} \end{array}

(16)

This is a classical linear quadratic regulator (LQR) problem. Its state feedback control law is designed as follows

\begin{array}{l} φ (i) = K ϕ (i) \end{array}

(17)

where $K$ can be solved by the following algebraic Riccati equation

\begin{matrix} P (i) = {[\tilde{A} (i) - \tilde{B} (i) K (i)]}^{T} P (i + 1) [\tilde{A} (i) - \tilde{B} (i) K (i)] \\ + K {(i)}^{T} R (i) K (i) + Q (i) \end{matrix}

(18)

\begin{array}{l} K (i) = {[\tilde{B} {(i)}^{T} P (i + 1) \tilde{B} (i)]}^{- 1} \tilde{B} {(i)}^{T} P (i + 1) \tilde{A} (i) \end{array}

(19)

Its terminal condition is

\lim_{i \to \infty} P (i) = 0

Then, it can be obtained that

\begin{array}{l} u (k) = K (\tilde{x} (k) - x_{s}) + u_{s} \end{array}

(20)

where $x_{s}$ and $u_{s}$ can be calculated by the following steady-state target calculation.

Steady-state target calculation

Steady-state target calculation is an important step of double-layered NMPC. Due to the external disturbances or new input information, the steady state of system may be changed, resulting in steady-state target deviation. Therefore, the new steady-state target $x_{s}$ and $u_{s}$ should be gained by considering the estimated disturbances. As for Equation (8), the following equation can be obtained in the steady state

\begin{array}{l} x_{s} = A x_{s} + B u_{s} + B_{d} \hat{d} (k) \\ y_{r} (k) = C x_{s} + C_{p} \hat{p} (k) \end{array}

(21)

$x_{s}$ and $u_{s}$ can be derived by solving the following quadratic programming (QP) problem

\begin{array}{l} \min_{x_{s}, u_{s}} {(u_{s} - u_{t})}^{T} R_{s} (u_{s} - u_{t}) \end{array}

(22)

subject to

\begin{matrix} [\begin{matrix} x_{s} \\ u_{s} \end{matrix}] = {[\begin{matrix} A & B \\ C & 0 \end{matrix}]}^{- 1} [\begin{matrix} - B_{d} \hat{d} (k) \\ y_{r} (k) - C_{p} \hat{p} (k) \end{matrix}] \\ [\begin{matrix} x_{\min} \\ u_{\min} \end{matrix}] \leq [\begin{matrix} x_{s} \\ u_{s} \end{matrix}] \leq [\begin{matrix} x_{\max} \\ u_{\max} \end{matrix}] \\ y_{\min} \leq y \leq y_{\max} \\ Δ u_{\min} \leq Δ u \leq Δ u_{\max} \end{matrix}

(23)

where $u_{t}$ is the expected value, which comes from the calculation of the external real-time optimization (RTO). $\hat{d} (k)$ and $\hat{p} (k)$ can be obtained by the estimation of state and disturbance.

The estimation of state and disturbance

The disturbances include system uncertainty, model mismatch and so on. In order to reject the disturbances, it is necessary to use the DOB to estimate system states and disturbances

\begin{array}{l} \hat{\tilde{x}} (k + 1) & = \tilde{A} \hat{\tilde{x}} (k) + \tilde{B} u (k) + L (y (k) - \hat{y} (k)) \end{array}

(24)

where $L$ is the DOB gain matrix.

To estimate the states and disturbances of the system, it is essential to ensure that the disturbances and states of augmented system are observable. Because these states are not necessarily asymptotically stable, the sufficient condition is when the augmented system is detectable.

Theorem 1

The augmented system $(\tilde{C}, \tilde{A})$ presented in Equation (8) is detectable if and only if $(C, A)$ detectable and

\begin{array}{l} r a n k [\begin{matrix} I - A & - B_{d} & 0 \\ C & 0 & C_{p} \end{matrix}] = n + s_{d} + s_{p} \end{array}

(25)

where $n$ is the dimension of system matrix, $s_{d}$ is the number of input disturbance states and $s_{p}$ is the number of output disturbance states. The proof of Theorem 1 is given in Muske and Badgwell.⁴³

In addition to the exact modeling of the system, it is important to accurately measure the system output. However, practically, the control system may be susceptible to external uncertainties, such as the aging of the system components. Therefore, in view of the complexity of the industrial environment, it is essential to introduce process noise $ω$ and measurement noise $ν$ into Equation (11)

\begin{array}{l} \tilde{x} (k + 1) = \tilde{A} \tilde{x} (k) + \tilde{B} u (k) + ω (k) \\ y (k) = \tilde{C} \tilde{x} (k) + ν (k) \end{array}

(26)

where $E (ω (k) ω {(j)}^{T}) = W (k) δ_{k j}$ and $E (ν (k) ν {(j)}^{T}) = V (k) δ_{k j}$ .

The accuracy of disturbances estimation is the key to realize the offset-free control of controlled variables. Kalman filter is an unbiased estimation method with simple design. Therefore, the Kalman filter is used to estimate the external disturbances in this paper

\begin{matrix} [\begin{matrix} \hat{x} (k + 1 | k + 1) \\ \hat{d} (k + 1 | k + 1) \\ \hat{p} (k + 1 | k + 1) \end{matrix}] = [\begin{matrix} \hat{x} (k + 1 | k) \\ \hat{d} (k + 1 | k) \\ \hat{p} (k + 1 | k) \end{matrix}] + [\begin{matrix} L_{1} (k + 1) \\ L_{2} (k + 1) \\ L_{3} (k + 1) \end{matrix}] \\ \times [y (k + 1) - \tilde{C} [\begin{matrix} \hat{x} (k + 1 | k) \\ \hat{d} (k + 1 | k) \\ \hat{p} (k + 1 | k) \end{matrix}]] \end{matrix}

(27)

\begin{array}{l} [\begin{matrix} \hat{x} (k + 1 | k) \\ \hat{d} (k + 1 | k) \\ \hat{p} (k + 1 | k) \end{matrix}] = \tilde{A} [\begin{matrix} \hat{x} (k | k) \\ \hat{d} (k | k) \\ \hat{p} (k | k) \end{matrix}] + \tilde{B} u (k) \end{array}

(28)

where $L (k + 1) = {[L_{1} (k + 1) L_{2} (k + 1) L_{3} (k + 1)]}^{T}$ , $L_{1} (k + 1)$ , $L_{2} (k + 1)$ and $L_{3} (k + 1)$ denote state filter gain, input disturbance filter gain and output disturbance filter gain, respectively. The recursive equation of filter gain can be shown

\begin{array}{l} L (k + 1) = S (k + 1 | k) {\tilde{C}}^{T} {(\tilde{C} S (k + 1 | k) {\tilde{C}}^{T} + V)}^{- 1} \end{array}

(29)

\begin{array}{l} S (k + 1 | k) = \tilde{A} S (k | k) {\tilde{A}}^{T} + {\tilde{B}}^{T} W \tilde{B} \end{array}

(30)

\begin{array}{l} S (k + 1 | k + 1) = [I - L (k + 1) \tilde{C}] S (k + 1 | k) \end{array}

(31)

According to the estimated disturbances, the steady-state target can be recalculated. And MPC can also realize to reject disturbances by using the feedforward compensation control strategy.

Lemma 1

Risfic et al.⁴⁴ consider the following discrete linear systems

\begin{matrix} \tilde{x} (k + 1) = \tilde{A} \tilde{x} (k) + \tilde{B} u (k) + Γ ω (k) \\ y (k) = \tilde{C} \tilde{x} (k) + ν (k) \end{matrix}

(32)

where $ω (k)$ and $ν (k)$ are the Gaussian white noise. If $(\tilde{A}, Γ)$ is completely stable and $(\tilde{A}, \tilde{C})$ is completely detectable, the Kalman filter is asymptotically stable.

The nonlinear system Equation (1) with the input and output disturbances can be transformed into Equation (8) based on the Hammerstein–Wiener model. Therefore, the offset-free control can be obtained by double-layered NMPC based on the augmented model. The result should be given in the following theorem.

Theorem 2

The system Equation (1) with the input disturbance $d (k)$ and the output disturbance $p (k)$ , under the action of the control rule Equation (20), can be tracked given the constant reference signal $y_{r}$ with offset-free in steady state

\begin{array}{l} \lim_{k \to \infty} y (k) = \lim_{k \to \infty} y_{r} (k) \end{array}

(33)

where $x_{s}$ and $u_{s}$ can be derived by Equation (22) and $d$ and $p$ can be obtained by equations (26)–(31). The asymptotic smooth tracking and disturbance rejection of system output can be achieved at the same time.

Proof

According to Equation (8)

\begin{array}{l} x (k + 1) = A x (k) + B u (k) + B_{d} d (k) \\ y (k) = C x (k) + C_{p} p (k) \end{array}

(34)

As for Equation (23)

\begin{array}{l} [\begin{matrix} x_{s} \\ u_{s} \end{matrix}] = {[\begin{matrix} A & B \\ C & 0 \end{matrix}]}^{- 1} [\begin{matrix} - B_{d} \hat{d} (k) \\ y_{r} (k) - C_{p} \hat{p} (k) \end{matrix}] \end{array}

(35)

Applying to the matrix inversion lemma⁴⁵

\begin{array}{l} {[\begin{matrix} A & B \\ C & D \end{matrix}]}^{- 1} = [\begin{matrix} A^{- 1} + A^{- 1} B {(D - C A^{- 1} B)}^{- 1} C A^{- 1} & - A^{- 1} B {(D - C A^{- 1} B)}^{- 1} \\ - {(D - C A^{- 1} B)}^{- 1} C A^{- 1} & {(D - C A^{- 1} B)}^{- 1} \end{matrix}] \end{array}

(36)

Then, Equation (35) can be changed that

\begin{matrix} [\begin{matrix} x_{s} \\ u_{s} \end{matrix}] = [\begin{matrix} A^{- 1} - A^{- 1} B {(C A^{- 1} B)}^{- 1} C A^{- 1} & A^{- 1} B {(C A^{- 1} B)}^{- 1} \\ {(C A^{- 1} B)}^{- 1} C A^{- 1} & - {(C A^{- 1} B)}^{- 1} \end{matrix}] \\ \times [\begin{matrix} - B_{d} \hat{d} (k) \\ y_{r} (k) - C_{p} \hat{p} (k) \end{matrix}] \end{matrix}

(37)

Assumption 3

The target optimization problem Equation (22) has a unique feasible solution. One can get

\begin{matrix} x_{s} = (A^{- 1} B {(C A^{- 1} B)}^{- 1} C A^{- 1} - A^{- 1}) B_{d} \hat{d} (k) \\ + A^{- 1} B {(C A^{- 1} B)}^{- 1} y_{r} (k) \\ - A^{- 1} B {(C A^{- 1} B)}^{- 1} C_{p} \hat{p} (k) \end{matrix}

(38)

\begin{matrix} u_{s} = - {(C A^{- 1} B)}^{- 1} C A^{- 1} B_{d} \hat{d} (k) - {(C A^{- 1} B)}^{- 1} y_{r} (k) \\ + {(C A^{- 1} B)}^{- 1} C_{p} \hat{p} (k) \end{matrix}

(39)

Similar to Equations (20) and (34), we have

\begin{array}{l} x (k) = {(A + B K)}^{- 1} [x (k + 1) - B (u_{s} - K x_{s}) - B_{d} d (k)] \end{array}

(40)

It is equivalent to

\begin{matrix} x (k) = {(A + B K)}^{- 1} x (k + 1) \\ + {(A + B K)}^{- 1} B [{(C A^{- 1} B)}^{- 1} + K A^{- 1} B {(C A^{- 1} B)}^{- 1}] y_{r} (k) \\ + {(A + B K)}^{- 1} B [(I + K A^{- 1} B) (C A^{- 1} B)^{- 1} C A^{- 1} - K A^{- 1}] B_{d} \hat{d} (k) \\ - {(A + B K)}^{- 1} B [(I + K A^{- 1} B) (C A^{- 1} B)^{- 1}] C_{p} \hat{p} (k) \\ - {(A + B K)}^{- 1} B_{d} d (k) \end{matrix}

(41)

According to Equation (34)

\begin{matrix} y (k) = C x (k) + C_{p} p (k) \\ = C {(A + B K)}^{- 1} x (k + 1) \\ + C {(A + B K)}^{- 1} B [{(C A^{- 1} B)}^{- 1} + K A^{- 1} B {(C A^{- 1} B)}^{- 1}] y_{r} (k) \\ + C {(A + B K)}^{- 1} B \\ \times [(I + K A^{- 1} B) {(C A^{- 1} B)}^{- 1} C A^{- 1} - K A^{- 1}] B_{d} \hat{d} (k) \\ - C {(A + B K)}^{- 1} B [(I + K A^{- 1} B) {(C A^{- 1} B)}^{- 1}] C_{p} \hat{p} (k) \\ - C {(A + B K)}^{- 1} B_{d} d (k) + C_{p} p (k) \end{matrix}

(42)

From Equation (42), it can be seen that $y (k)$ and $x (k + 1)$ , $y_{r} (k)$ , $B_{d} \hat{d} (k)$ , $C_{p} \hat{p} (k)$ , $B_{d} d (k)$ and $C_{p} p (k)$ are related. Therefore, we have

1 As for $B_{d} \hat{d} (k)$

\begin{matrix} C {(A + B K)}^{- 1} B \\ [(I + K A^{- 1} B) {(C A^{- 1} B)}^{- 1} C A^{- 1} - K A^{- 1}] B_{d} \hat{d} (k) \\ = C {(A + B K)}^{- 1} B (I + K A^{- 1} B) {(C A^{- 1} B)}^{- 1} C A^{- 1} B_{d} \hat{d} (k) \\ - C {(A + B K)}^{- 1} B K A^{- 1} B_{d} \hat{d} (k) \\ = C {(A + B K)}^{- 1} B (B^{- 1} B + K A^{- 1} B) B^{- 1} B_{d} \hat{d} (k) \\ - C {(A + B K)}^{- 1} B K A^{- 1} B_{d} \hat{d} (k) \\ = C {(A + B K)}^{- 1} (I + B K A^{- 1}) B_{d} \hat{d} (k) \\ - C {(A + B K)}^{- 1} B K A^{- 1} B_{d} \hat{d} (k) \\ = C {(A + B K)}^{- 1} B_{d} \hat{d} (k) + C {(A + B K)}^{- 1} B K A^{- 1} B_{d} \hat{d} (k) \\ - C {(A + B K)}^{- 1} B K A^{- 1} B_{d} \hat{d} (k) \\ = C {(A + B K)}^{- 1} B_{d} \hat{d} (k) \end{matrix}

(43)

2 As for $C_{p} \hat{p} (k)$

\begin{array}{l} C {(A + B K)}^{- 1} B [(I + K A^{- 1} B) {(C A^{- 1} B)}^{- 1}] C_{p} \hat{p} (k) \\ = C {(A + B K)}^{- 1} [B B^{- 1} A C^{- 1} + B K A^{- 1} B B^{- 1} A C^{- 1}] C_{p} \hat{p} (k) \\ = C {(A + B K)}^{- 1} [A C^{- 1} + B K C^{- 1}] C_{p} \hat{p} (k) \\ = C {(A + B K)}^{- 1} [(A + B K) C^{- 1}] C_{p} \hat{p} (k) \\ = C_{p} \hat{p} (k) \end{array}

(44)

Defining $d (k) - \hat{d} (k) = e (k)$ and $p (k) - \hat{p} (k) = \bar{e} (k)$ , we get

\begin{matrix} y (k) = C {(A + B K)}^{- 1} x (k + 1) + C_{p} \bar{e} (k) \\ + C {(A + B K)}^{- 1} B_{d} e (k) \\ + C {(A + B K)}^{- 1} B [{(C A^{- 1} B)}^{- 1} + K A^{- 1} B {(C A^{- 1} B)}^{- 1}] y_{r} (k) \end{matrix}

(45)

It is assumed that the system is asymptotically stable, and then

\lim_{k \to \infty} x (k + 1) \to 0, \lim_{k \to \infty} e (k) \to 0, \lim_{k \to \infty} \bar{e} (k) \to 0

Therefore, it is obtained that

\begin{matrix} y (k) = C {(A + B K)}^{- 1} B \\ \times [{(C A^{- 1} B)}^{- 1} + K A^{- 1} B {(C A^{- 1} B)}^{- 1}] y_{r} (k) \\ = C {(A + B K)}^{- 1} B \\ \times [{(C A^{- 1} B)}^{- 1} + K A^{- 1} B B^{- 1} A C^{- 1}] y_{r} (k) \\ = C {(A + B K)}^{- 1} B [B^{- 1} A C^{- 1} + K C^{- 1}] y_{r} (k) \\ = C {(A + B K)}^{- 1} [B B^{- 1} A C^{- 1} + B K C^{- 1}] y_{r} (k) \\ = C {(A + B K)}^{- 1} [(A + B K) C^{- 1}] y_{r} (k) \\ = y_{r} (k) \end{matrix}

(46)

The proof is complete.

Numerical examples

Example 1

The continuously stirred tank reactor (CSTR) is used as the simulation example to verify the effectiveness of the proposed method. Figure 4 depicts CSTR schematic. Consider a CSTR in which the first-order irreversible exothermic reaction $\vec{A} \to \vec{B}$ occurs. The control target is to retain the desired concentration as close as possible to its steady-state value.^46–49

Figure 4.

Continuously stirred tank reactor (CSTR) schematic.

The material and energy balance equations are

\begin{array}{l} \frac{d V}{d t} = F_{0} - F \\ \frac{d C_{A}}{d t} = \frac{F_{0} (C_{A 0} - C_{A})}{V} - k_{0} \exp (\frac{- E}{R T}) C_{A} \\ \frac{d T}{d t} = \frac{F_{0} (T_{0} - T)}{V} - \frac{Δ H}{ρ C_{\tilde{p}}} \exp (\frac{- E}{R T}) C_{A} - \frac{U A (T_{c} - T)}{ρ C_{\tilde{p}} V} \end{array}

(47)

The controlled variable is the molar concentration $C_{A}$ and the manipulated variable is the coolant liquid temperature $T_{c}$ . It is assumed that the inlet flow rate acts as unmeasured disturbance; In this paper, the control target is to regulate $C_{A}$ by manipulating $T_{c}$ . The nominal parameters are $F_{0} = 100 L / \min$ , $C_{A 0} = 0.5 mol / L$ , $T_{0} = 350 K$ , $ρ = 1000 g / L$ , $C_{\tilde{p}} = 0.239 J / g K$ , $Δ H = - 5 \times 10^{4} J / mol$ , $E / R = 8750 K$ , $k_{0} = 7.2 \times 10^{10} \min^{- 1}$ and UA = 5 × 104 J/min K. The steady-state operating conditions are $C_{A}^{e q} = 0.5 mol / L$ , $V^{e q} = 1001 L$ , $T^{e q} = 350 K$ and $T_{c}^{e q} = 300 K$ .

The open-loop response of CSTR is shown in Figure 5. It is clear from the picture that the process has strong nonlinear and presenting oscillation. The identification result of system is shown in Figure 6. The error of between Hammerstein–Wiener model output and system model output is shown in Figure 7.

Figure 5.

Open-loop responses for (a) +5K and (b) −5K step changes in T_c.

Figure 6.

Identification results of CSTR.

Figure 7.

The error of between Hammerstein–Wiener model output and system model output.

Let the state vector $x = {[\begin{matrix} C_{A} - C_{A}^{e q} & T - T^{e q} & V - V^{e q} \end{matrix}]}^{T}$ and the input vector $u = T_{c} - T_{c}^{e q}$ . Sampling time is taken into account $T_{s} = 0.1 s$ . The corresponding discrete time linear model is shown that

\begin{array}{l} x (k + 1) = A x (k) + B u (k) + B_{d} d (k) \\ y (k) = C x (k) + C_{p} p (k) \end{array}

(48)

The differential equation of CSTR is used to generate process data through MATLAB. Furthermore, the system matrices can be obtained by using the subspace identification method of system identification toolbox in MATLAB, where

\begin{matrix} A = [\begin{matrix} 0.9146 & 0 & 0.0405 \\ 0.1665 & 0.14 & 0.0058 \\ 0 & 0 & 0.1353 \end{matrix}], B = [\begin{matrix} 0.0544 \\ 0.0053 \\ 0.8647 \end{matrix}] \\ C = [\begin{matrix} 1.7993 & 13.2160 & 0 \end{matrix}] \end{matrix}

The disturbance models are chosen as $B_{p} = B, C_{p} = I$ . Initial values are $x_{0} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T}$ . The weight matrices are selected as $Q = d i a g {0.5, 0.5, 0.5}$ , $R = 1$ .

The simulation results are shown in Figures 8 –10. Figure 8 shows the output response of closed-loop system. Figure 9 shows the control input of closed-loop system. Figure 10 shows that the closed-loop system subject to the input and output disturbances.

Figure 8.

The output response of closed-loop system.

Figure 9.

The control input of closed-loop system.

Figure 10.

Input and output disturbances.

From Figure 8, we can see that the output of the closed-loop system can achieve smooth tracking set point by proposed method. When the system is subject to input disturbance (80s) and output disturbance (170s), respectively, the closed-loop system can actively reject these disturbances. Therefore, the offset-free control of system can be obtained by the proposed method. Although the method of Wang et al.³⁸ can also reject these disturbances, the system output appears to have steady-state error and the frequent fluctuations. In other words, the example shows that the method that consider only the input disturbance cannot deal with the control system with having input disturbances and output disturbances at the same time.

Example 2

pH neutralization process is very common in the industrial production, such as chemical industry, water treatment and power, which has the strong nonlinear, time varying, time delay and uncertainties. Therefore, pH neutralization process is considered as one of the most difficult control problems in control and engineering,^50,51 as shown in Figure 11.

Figure 11.

pH neutralization process.

The differential equations of pH neutralization process can be obtained⁵²

\begin{array}{l} \frac{d h}{d t} = \frac{1}{S} (q_{1} + q_{2} + q_{3} - C_{ν} h^{0.5}) \\ \frac{d W_{a 4}}{d t} = \frac{1}{S h} [(W_{a 1} - W_{a 4}) q_{1} + (W_{a 2} - W_{a 4}) q_{2} + (W_{a 3} - W_{a 4}) q_{3}] \\ \frac{d W_{b 4}}{d t} = \frac{1}{S h} [(W_{b 1} - W_{b 4}) q_{1} + (W_{b 2} - W_{b 4}) q_{2} + (W_{b 3} - W_{b 4}) q_{3}] \\ W_{a 4} + 10^{pH - 14} + W_{b 4} \frac{(1 + 2 \times 10^{pH - p K_{2}})}{1 + 10^{p K_{1} - pH} + 10^{pH - p K_{2}}} - 10^{- pH} = 0 \end{array}

(49)

where $q_{1}$ , $q_{2}$ , $q_{3}$ and $q_{4}$ are acid solution stream, buffer solution stream, alkaline solution stream and effluent solution stream, respectively; $W_{a 1}$ , $W_{a 2}$ , $W_{a 3}$ and $W_{a 4}$ are the reaction invariants of the acid solution, buffer solution, alkaline solution and effluent solution, respectively; $h$ denotes the liquid level height in vessel; $S$ denotes the cross-sectional area of cylindrical container and $C_{ν}$ denotes the coefficient of the valve. The controlled variable is the effluent pH value; the manipulated variable is the flow rate of alkaline solution; the flow rate of the buffer solution is treated as the disturbance and the nominal parameters are $S = 207 {cm}^{2}$ , $C_{ν} = 8.75 mL / (cm s)$ , $q_{1} = 16.6 mL / s$ , $q_{2} = 0.55 mL / s$ , $p K_{1} = 6.35$ , $p K_{2} = 10.25$ , $W_{a 1} = 3 \times 10^{- 3} mol / L$ , W_a2 = $- 3 \times 10^{- 2} mol / L$ , $W_{a 3} = - 3.05 \times 10^{- 3} mol / L$ , W_b1 = 0mol/L, $W_{b 2} = 3 \times 10^{- 2} mol / L$ , $W_{b 3} = 5 \times 10^{- 5} mol / L$ and $pH = 7.0$ . The steady-state operating conditions are $h^{e q} = 14 cm$ , $W_{a 4}^{e q} = 4.32 \times 10^{- 4} mol / L$ , $W_{b 4}^{e q} = 5.28 \times 10^{- 4}$ and $q_{3}^{e q} = 15.6 mL / s$ .

The nonlinear characteristics of the system can be shown in Figure 12. It can be seen that when the pH value is away from the neutral point (pH = 7), the sensitivity of pH value is very low for the flow rate of neutralization solution. However, when the pH value is close to the neutral point (pH = 7), the sensitivity of pH value is very high, so that the pH value can subject to strong change even if the flow rate of neutralization solution is slow. The identification result of pH neutralization process is shown in Figure 13. The error between Hammerstein–Wiener model output and system model output is shown in Figure 14.

Figure 12.

pH neutralization process titration curve.

Figure 13.

The identification results of pH neutralization process.

Figure 14.

The error of between Hammerstein–Wiener model output and system model output.

Based on the deviation variables, the state vector $x = {[\begin{matrix} h - h^{e q} & W_{a 4} - W_{a 4}^{e q} & W_{b 4} - W_{b 4}^{e q} \end{matrix}]}^{T}$ and the input vector $u = q_{3} - q_{3}^{e q}$ . Sampling time is taken into account $T_{s} = 0.1 s$ . Then, the corresponding discrete time linear model is obtained that

\begin{matrix} x (k + 1) = A x (k) + B u (k) + B_{d} d (k) \\ y (k) = C x (k) + C_{p} p (k) \end{matrix}

(50)

Similar to the CSTR example, the differential equations of pH neutralization process can be used to generate process data. Furthermore, by taking advantage of the system identification toolbox in MATLAB, the system matrices can be identified very easily in the nominal parameters. The system matrices can be obtained as follows

\begin{matrix} A = [\begin{matrix} 0.3026 & 0.3 & 0.021 \\ 1.3659 & - 0.2358 & - 0.12 \\ 0.5 & - 0.2 & 1.25 \end{matrix}], B = [\begin{matrix} 0.16 \\ 0.56 \\ 0.58 \end{matrix}] \\ C = [\begin{matrix} 2.56 & 12.6 & 1.2 \end{matrix}] \end{matrix}

The disturbance models are still chosen as $B_{p} = B, C_{p} = I$ . Initial values are $x_{0} = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T}$ . The weight matrices are selected as $Q = d i a g {0.3, 0.3, 0.3}$ , $R = 1$ .

The simulation results are shown in Figures 15 –17. It is well known that the neutral point (pH = 7) is the control goal of pH neutralization process. When the closed-loop system is subject to the input disturbance (80s) and the output disturbance (300s) at the same time, the system has instantaneous fluctuation and it can immediately reach the neutral point. It can be illustrated that the proposed method possesses the strong robustness. However, the method of Wang et al.³⁸ cannot stabilize in the neutral point, and the system presents the frequent fluctuations near the neutral point. It can lead to the poor quality of pH neutralization process, so that the environment should be subject to very serious pollution. Therefore, it can be proved that the proposed method is very effective and practical.

Figure 15.

The output response of closed-loop system.

Figure 16.

The control input of closed-loop system.

Figure 17.

Input and output disturbances.

Conclusion

For the nonlinear systems that have simultaneously input and output disturbances, a double-layered NMPC based on the Hammerstein–Wiener model has been proposed. The proposed method takes advantage of the Hammerstein–Wiener model to describe the nonlinear systems, which has been transformed into combination of one linear part and two nonlinear parts, respectively. Disturbances and states have been estimated by applying to the Kalman filter. Moreover, the estimated disturbances were used for calculating steady-state targets in steady-state target calculation layer. Finally, the state feedback control law of MPC can be obtained in dynamic control layer. A simple proof for offset-free control is given in the proposed method. The proposed method has been applied to the industrial CSTR and pH neutralization process. The simulation results show that it has better disturbance rejection performance, and the controlled variables have been achieved the offset-free control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

This work was supported by the National Natural Science Foundation of China under Grants 61673199 and the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (ICT1800400).

ORCID iD

Hongbin Cai

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Double-layered nonlinear model predictive control based on Hammerstein–Wiener model with disturbance rejection

Abstract

Keywords

Introduction

Hammerstein–Wiener model

Assumption 1

Assumption 2

Double-layered NMPC based on Hammerstein–Wiener model

Dynamic control

Steady-state target calculation

The estimation of state and disturbance

Theorem 1

Lemma 1

Theorem 2

Proof

Assumption 3

Numerical examples

Example 1

Example 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References