1. Introduction
Modeling and control of systems governed by partial differential equations (PDEs) have attracted the interest of many researchers. See, for example, [
1] and the references cited therein. These systems are known as distributed parameter systems (DPSs). Among the controller synthesis methods for DPSs, there are direct design strategies [
2], post-centralized strategies [
3], and prior-centralized strategies. For these three strategies, the prior-centralized strategy is easier to be understood, so it has been widely used to deal with DPSs. With the prior-centralized approach, a DPS is first approximated by a system of ordinary differential equations, referred to as a lumped parameter system (LPS) [
4]. However, the obtained LPS tends to have a high dimension, and hence it is essential to get a low-order representation of the LPS and use it for the design of the controller. There are many effective control schemes in the literature for the design of the controller.
For a system described by hyperbolic PDEs, the eigenspectrums of its differential operator contain almost all the energy. As a consequence, the order of the model is higher than that of the parabolic system [
4]. The classical finite difference method (FDM) has long been recognized as a popular numerical method for DPSs among the control community. For example, several FDM numerical models have been developed to deal with various control problems, including optimal control problems governed by first-order hyperbolic systems. A coupled system involving PDE and ODE is transformed into a zero-order neutral system with distributed delays [
5]. An optimal boundary control problem, where its underlying dynamical system is governed by linear hyperbolic partial differential equations with a turnpike structure, is being applied to solve a control problem in gas pipeline operation [
6]. A boundary input-to-state stability of a class of coupled systems involving hyperbolic PDE and ODE in the presence of uncertainties and external disturbances is investigated and the results obtained is applied to an interconnected continuous stirred tank reactor coupled with a plug-flow reactor [
7]. Furthermore, the weighted residual methods (WRMs), such as the Galerkin method, collocation method, and finite element method (FEM), have also attracted the attention of many researchers among the control community. WRM has been widely used in various industrial applications due to its high computational efficiency and flexibility in handling complex geometries and boundaries [
8]. For control problems governed by various DPSs, the weighted residual method based on a wavelet basis is found to be effective, showing great potential [
9]. The collocation method involves finding a functional basis for the solution space of the equation, then making the residual zero at
n points within the domain [
10]. Generally, wavelet-collocation method (WCM), created by choosing a wavelet and some grid structure, is most appropriate for solving nonlinear PDEs with general boundary conditions [
11]. The computational model of hyperbolic wave equations is proposed by using Haar wavelet operation matrix [
12]. A non-Fourier heat conduction problem is solved through the use of Haar wavelet basis implicitly in combination of finite difference and pseudospectral methods, thereby avoiding numerical oscillations near wavefront jumping discontinuities [
13]. Although the Haar wavelet function has an explicit expression, it is non-differentiable. Thus, an integral smoothing calculation is introduced. Furthermore, since the Haar wavelet has only one vanishing moment, it does not perform well for the approximation of a smooth function. The Pennes bioheat transfer equation is solved by the WCM and a multiresolution analysis is conducted using Daubechies wavelets [
14]. The Daubechies wavelet is tightly supported, smooth, and has
n-order vanishing moments, but its differential operator is difficult to obtain due to the lack of analytical expression. By combining the operation matrix of integration for Legendre and Chebyshev wavelets with the collocation method, an approximate solution of a one-dimensional hyperbolic PDEs with given initial conditions is derived [
15]. By the operation matrix of fractional integration combined with the Galerkin method and the collocation method, a coupled system of Burgers’ equations with the time-fractional derivative of Gegenbauer wavelets is obtained [
16]. The operation matrix of Hermite wavelet is derived, and on this basis a direct numerical method is developed for solving optimal control problems [
17]. By combining WRM with Shannon wavelet basis, a numerical method is proposed to calculate the stationary solution of Fokker-Planck-Kolmogorov equation [
18]. Shannon wavelet is continuous, smooth, and infinitely differentiable, but it is not tightly supported. For the quasi-Shannon wavelet [
19], it is also tightly supported.
Model predictive control (MPC) is a successful method for the control of industrial processes modeled as LPSs or DPSs with constraints and time delays. For the control of hyperbolic systems, such as long duct heating systems [
20], and convection-dominated chemical reactor [
21], the need to handle constraints and time delays is unavoidable. The key step in the design of an MPC algorithm for DPSs is to build a prediction model through model reduction. For DPS, a linear MPC scheme using Cayley-Tustin transformation is proposed in [
22]. The FDM has also become a popular method for constructing prediction models. MPC using FDM’s prediction model has been successfully applied to several industrial processes, such as convection-dominated transport-reaction processes [
21], cocurrent fluid flow problems [
22], non-isothermal plug-flow reactor [
23,
24], wind tunnel propagation delay problems [
25], and open irrigation canals [
26]. The characteristic line method (CLM) is a spatio-temporal discretization method. It has been applied to the study of control problems, where the
concentration is to be controlled in a circulating fluidized bed [
27]. It has also been applied to uncertain hyperbolic systems [
28], and a counterflow plug-flow reactor [
29]. On the other hand, model predictive control based on the weighted residual method is being developed for hyperbolic systems. The control problem of selective catalytic reduction in diesel-powered vehicles is solved by CLM combined with spectral decomposition [
30]. Wavelet-based numerical methods have great potential in MPC application design, but there is little related literature. A prediction model based on Haar wavelet basis has been applied to the design of the MPC controller of discrete DPS [
31]. However, in order to improve the computational efficiency, it is necessary to develop a more suitable wavelet-based method.
In this paper, the WCM is used to construct the reduced-order model of first-order hyperbolic system. The interval quasi-Shannon wavelet scaling function is used as a basis function to suppress the boundary error. Then, a nonlinear MPC scheme is developed based on the above prediction model. Nonlinear MPC schemes are also developed based on the models obtained by FDM and the Haar wavelet transformation proposed in [
31]. They are applied to solve the disturbance rejection problem of long duct heating systems. Their computational efficiency has been compared.
The rest of the paper is organized as follows. In
Section 2, the first-order hyperbolic system and its control objective are introduced. Then, the interval quasi-Shannon wavelet is introduced, and a nonlinear model predictive control scheme is developed based on the wavelet-collocation method for the first-order hyperbolic system in
Section 3. In
Section 4, a simulation study involving a long duct heating system is performed to verify the effectiveness of the algorithm proposed in
Section 3. Some concluding remarks are given in
Section 5.
2. Problem Formulation
We consider the following class of distributed parameter systems described by first-order hyperbolic evolution equations:
subject to initial condition
and boundary condition
, where
is the spatial variable,
is the set of real numbers,
is the time variable,
is the controlled variable,
is an infinite-dimensional Hilbert space containing integrable spatial differentiable functions defined in the interval
,
is differentiable with respect to
x and
t,
,
,
is a smooth and bounded manipulatable variable,
are sufficiently smooth nonlinear functions of
x and
t,
is a smooth and bounded function of time
t,
,
, where
is
-norm.
The optimal control problem may now be stated as follows:
where
is the objective function, and
,
,
and
are, respectively, the reference trajectory, the prediction output and the control increment at position
x and time
t.
and
represent, respectively, the upper bound and the lower bound of the control variables, while
and
represent, respectively, the upper bound and the lower bound of the control increment.
is the weighting coefficient.
To proceed further, let us consider an actual situation, which is referred to as Problem (P), as stated below.
Consider the situation where there are sensors, which are uniformly located, and l actuators, and let , where H is the standard Heaviside function.
The discrete forms of the system and the cost function will be constructed using the wavelet-collocation method to be described in
Section 3.
4. Case Study
In this section, we consider the temperature control problem of a long duct heating system based on flow control [
20], as shown in
Figure 5.
This is a one-dimensional incompressible convection heat transfer process. The fluid flows from one end of the long duct to the other end. We assume that the axial direction of the long duct is the cylindrical surface of the same diameter and that the ambient temperature of the entire system, the inlet temperature and the thermal conductivity of the fluid in the long duct are constant. Furthermore, both the fluid flow and the convection heat transfer are fully developed flows, and the diffusion phenomenon is neglected. Under the above assumptions, the temperature distribution in the duct depends only on the flow velocity. This problem is divided into three categories, corresponding to the different relationships between the flow velocity and the delay time : (i). The flow velocity is high, but the delay time is short. In this case, the influence of the flow velocity on the temperature is fast; (ii). The flow velocity is slow, but the delay time is large. In this case, the influence of the flow velocity on the temperature is slow; and (iii). The flow velocity is particularly slow. In this case, the change of the temperature in response to the flow velocity is negligible. In practice, the first two treatments are similar, and the last case is rare. In actual industrial systems, the flow velocity is often affected by unknown disturbances. When the flow velocity is affected, the axial temperature of the long tube will be affected. The purpose of the controller is to quickly restore the axial temperature of the long duct back to the equilibrium state by adjusting the opening of the regulating valve.
The dimensionless form of the dynamical model can be expressed as:
where
is the axis temperature of the long duct,
is the fluid velocity,
and
are, respectively, the partial derivative with respect to
t and
The numerical solution is to be obtained by using FDM as described in [
32]. The spatial region is evenly divided by uniform grid points
, and
. Similarly, let the time step be fixed as
,
. Then, the discretized temperature value is
at the space location
and the time
. The brackets are used to distinguish between time points and indices. Let Krone number be
. Then, the discretized version of system (21) can be expressed as:
The temperature value of the long duct is calculated iteratively by (22). The sensors are evenly located in a sufficiently small interval
to measure the real temperature value, whereas the time interval
is chosen such that Krone stability condition is satisfied, that is
. Thus, to ensure that the stability condition is satisfied, we choose
, where
is the maximum flow velocity of the fluid in the long duct. The mesh is chosen to be more refine than that in [
20] with the same flow velocity. Thus, the numerical solution obtained is more accurate. The length of the spatial interval
is chosen to be
, which is small enough to minimize the inaccuracy to be an integer power of
such that it is consistent with the binary wavelet. Meanwhile, the discrete time step
is chosen to be
such that the inaccuracy is minimized and the CFL condition is satisfied. The flow velocity
is selected randomly between
to
. All the experiments work on IBM PC-Intel Core I5-2500K using
. The actual temperature in the long duct after 10-time units is depicted in
Figure 6.
4.1. The Performance of the Prediction Model
In this subsection, we will test the performance of the prediction model using the method proposed in
Section 3.3. Assume that there are
temperature sensors evenly located along the long duct.
. The
P-step prediction of the temperature value of (21) given by (17) is
The choice of the parameters
of the interval quasi-Shannon interpolating wavelet will significantly affect the performance. In
Table 1, we show the
-error and the mean square error (MSE) under two different sample times
for three different meshes
with
. Based on the results obtained in
Section 3.2, the quasi-wavelet parameters
are chosen. Let the
error function be defined as follows:
and let the MSE function be given by:
From
Table 1, we can see that the sample time
has a significant impact on the prediction performance. The single-step prediction performance meets the control requirements for both the cases of
and
. On the other hand, the multi-step prediction performance fails to meet the control requirements for the case of
. As
j is increased from 2 to 3, the
error is deduced significantly. However, as
j continues to increase, the decrease of the error is no longer significantly. The optimal value of the parameter
r is around
. The
error and MSE are the smallest for
and 3. However, the parameter
r becomes insensitive to prediction performance when
.
The WCM, which is developed based on Haar wavelet basis in [
31] and the FDM developed in [
32], are selected for comparison with the method proposed in this paper. The sample time
is chosen small enough such that all errors are mainly originated from the spatial discretization. Without loss of generality, we choose
. In
Table 2, a comparison of performance between the FDM with
, the WCM using Haar wavelet basis with
, and the proposed method with
is given. The parameters of the interval quasi-Shannon interpolating wavelet are selected as
for
;
for
; and
for
. The predictive temperature and the error of the model, which are obtained by the FDM with
, the WCM using Haar wavelet basis with
, and the proposed method with
, are depicted in
Figure 7.
In
Table 2, the prediction errors obtained by FDM and WCM using Haar wavelet basis decrease slowly when
M increases. On the other hand, the prediction errors obtained by the proposed method decrease exponentially when
is increased to
. Please note that with the increase of the integer
j, the truncation error of the proposed method using the regularized Shannon’s kernel decays exponentially, but the Euler forward difference method has only the first-order accuracy [
35]. Therefore, we can see that the prediction performance of the model using the proposed method is better than those obtained by using the other two methods. The model based on the WCM using Haar wavelet basis with
is divergent, and the prediction performance is the worst among the three methods. From
Figure 7, the error caused by using the proposed method is smaller than those induced by using the FDM and the WCM using Haar wavelet basis. The errors of the models caused by using the proposed method and the FDM are smoother than that obtained by the WCM using Haar wavelet basis. This is because Haar wavelet basis is discontinuous.
4.2. The Performance of the Proposed MPC Scheme
Based on the above prediction model, a nonlinear predictive controller is designed to deal with the temperature control problem. The controller is required to adjust the flow velocity to drive the temperature back to the desired initial steady state. Assume that the quadratic criterion to be minimized is given by
where
is the objective function, and
and
are, respectively, the reference temperature value and the correction prediction temperature value defined at time
and position
.
is the increment of the flow velocity at time
,
and
represent, respectively, the upper bound and the lower bound of the flow velocity,
and
represent, respectively, the upper bound and lower bound of the increment of the flow velocity.
,
,
, and
are defined in (18).
In this section, using the proposed method, the FDM, and the WCM using Haar wavelet basis, three sets of comparison experiments are performed. The purpose is to estimate the temperature at various spatial points along the long duct when the flow velocity
is suddenly increased by
from 1 to 1.1 at the entrance. We suppose that there are 17 temperature sensors when the proposed method and the FDM are used, while there are 16 controlled temperature sensors when the WCM using Haar wavelet basis is used. Assume that
,
,
,
. The predictive horizon
, and the control horizon
, the weighting factors
in the quadratic criterion expressed by (26) are chosen. The temperature response at the terminal position and the flow velocity
are depicted in
Figure 8, and the response curves of the temperature values at other positions
are depicted in
Figure 9.
From
Figure 8 to
Figure 9, we see that the flow velocity
changes smoothly, and the temperature values at positions
are driven back to the steady state position for all the three methods. The results show that all the controllers obtained can solve the temperature control problem at any measured position along the long duct. However, for the proposed method, the temperature is driven back to the steady state position at a faster rate with smaller overshoots.