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Spatiotemporal LS-SVM inverse control for nonlinear distributed parameter systems with application to heating process

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Abstract

In actual industrial systems, numerous processes are characterized as distributed parameter systems (DPSs) that exhibit strong nonlinearity, complex boundary conditions, and unknown dynamics. Achieving accurate control of such processes poses a significant challenge. In light of this, a data-driven spatiotemporal least squares support vector machine (LS-SVM) inverse control method has been developed specifically for complex DPSs. First, a spatiotemporal LS-SVM model is proposed to capture the dynamics of DPSs by leveraging available data. Subsequently, by employing Taylor expansion on this spatiotemporal model, a state model is derived to explicitly establish the relationship between the control input and output variables. Building upon this foundation, an explicit control input is obtained through inversion and spatial fuzzy strategy, allowing for effective tracking of spatiotemporal dynamics. This control approach takes into account the influence of each input on all spatial points, thereby ensuring a favorable control effect for DPSs. Theoretical analysis and stability proofs affirm the stability of the proposed control approach for nonlinear DPSs. Furthermore, the efficacy of this controller is demonstrated through two actual experiments. First, the proposed approach exhibits superior control performance, as evidenced by a nearly threefold improvement in tracking accuracy on several sensors compared to the fuzzy controller. Second, the proposed controller maintains a smaller error during the tracking process, with deviations bounded within 1 °C even in the presence of disturbances.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

DPS:

Distributed parameter system

LS-SVM:

Least square support vector machine

PDE:

Partial differential equations

MIMO:

Multi-input multi-output

RKHS:

Reproducing kernel Hilbert space

RBF:

Radial basis function

RE:

Relative error

ODE:

Ordinary differential equation

LPS:

Lumped parameter system

SBFs:

Spatial basis functions

MPC:

Model predictive control

UUB:

Uniformly ultimately bounded

MAPE:

Mean absolute percentage error

\(x_{i}\) :

The ith spatial point \((i = 1,2, \cdots ,N)\)

\(\varphi (x)\) :

A nonlinear mapping function to reflect the spatial distribution from low dimensions to high dimensions

\(N\) :

Spatial sensor number

\(y(x, \, t)\) :

System output at position x

\(\gamma\) :

Regularization factor

\({{\varvec{\upalpha}}}(t_{k} )\) :

Lagrange multiplier coefficient

\({\mathbf{z}}(t_{k} )\) :

The time-series variable, which is a vector related with Lagrange multiplier coefficients and control input.

\(\eta_{\tau }\)(\(\tau = 1,2, \cdots ,L\)):

The Lagrange multiplier for \({{\varvec{\upalpha}}}(t_{\tau } )\)

\(k_{\alpha } (z(t_{i} ),z(t_{j} ))\) :

Kernel function for \({{\varvec{\upalpha}}}(t_{k} )\)

\(g({\mathbf{u}}(t_{k} ))\) :

A function with respect to \({\mathbf{u}}(t_{k} )\) that is represented by the LS-SVM model

\(r(x,t_{k} )\) :

The target temperature for controlling

\(G(x_{p} )\) :

Membership function for pth rule of the fuzzy model

\(e_{r} (x, \, t_{k} )\) :

The tracking error at \(t_{k}\)

\(\Lambda\) :

Diagonal matrix that denotes PDEs \((\partial y/\partial x,\partial^{2} y/\partial x^{2} , \cdots )\) of the system output for spatial location point x

\(m\) :

The number of system input

\(K(x,x_{i} )\) :

Kernel functions with respect to spatial point \(x_{i}\) and satisfying \(K(x,x_{i} ) = \varphi (x)\varphi (x_{i} )\)

\(w( \cdot )\) :

Weight coefficient in the LS-SVM

\(b(t_{k} )\) :

Bias term for the LS-SVM model

\(\hat{y}(x, \, t)\) :

Estimation of system output at position x

\(u_{j} (t)\) :

The jth control input \((j = 1,2, \cdots ,m)\)

\({{\varvec{\Omega}}} = \left[ {K(x, \, x_{1} ), \cdots ,K(x, \, x_{N} )} \right]\) :

A vector of kernel functions

\({\mathbf{E}}(x)\) :

A vector with respect to spatial point x

\(t_{k}\) :

The kth sampling time

\(\theta_{{\text{s}}}\) :

The bias term for the modeling of \({{\varvec{\upalpha}}}(t_{k} )\)

\(R_{k} [\Delta {\mathbf{u}}(t_{k} )]\) :

The higher-order infinitesimal term with

\(\Delta {\mathbf{u}}(t_{k} )\;\rho_{0}\) :

A positive number

\(\overline{\upsilon }\) :

A small positive number

\(r(x, \, t_{k} )\) :

The reference signal

\(R_{{k{ + }1}}\) :

Abbreviation of the high-order term \(R_{k} \left[ {\Delta u\left( {t_{k} } \right)} \right]\)

\(h_{p}\) :

The weight coefficient of the control input at \(x_{p} (p = 1,2, \cdots ,N)\) with values between (0, 1)

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Funding

This work was partially supported by National Natural Science Foundation of China (52075556), and the Key R&D Program of Hunan Province (2021SK2016).

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Authors and Affiliations

  1. State Key Laboratory of High Performance Complex Manufacturing, School of Mechanical and Electrical Engineering, Central South University, Changsha, 410012, China

    Bowen Xu & Xinjiang Lu

  2. School of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Bowen Xu

Authors
  1. Bowen Xu
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  2. Xinjiang Lu
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Correspondence to Xinjiang Lu.

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Xinjiang Lu is the corresponding author, and all the authors declare that they have no conflict of interests that could have appeared to influence the work reported in connection with the submitted work.

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Xu, B., Lu, X. Spatiotemporal LS-SVM inverse control for nonlinear distributed parameter systems with application to heating process. Nonlinear Dyn 111, 17229–17246 (2023). https://doi.org/10.1007/s11071-023-08771-6

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