An optimal learning-based controller derived from Hamiltonian function combined with a cellular searching strategy for automotive coldstart emissions

Abstract

The main objective of this study is to propose a novel algorithmic framework for the implementation of a nonlinear controller for reducing the amount of tailpipe hydrocarbon emissions in automotive engines over the coldstart period. To this aim, the control problem for a given engine is formulated in the form of the standard Bolza problem, and then, the concepts of Euler–Lagrange equation and Hamiltonian function are taken into account to calculate the optimal states, co-states, and control input signals. An extreme learning machine is also linked to an experimentally validated nonlinear state-space representation of the engine during the coldstart to approximate the values of exhaust gas temperature and engine-out hydrocarbon emissions, which are two key variables for the considered control problem. To solve the resulting system of equations, a cellular variant of the particle swarm optimization technique is implemented and the existing nonlinear system of equations is solved heuristically. In addition, some constraints are exerted on the control signals to guarantee the smooth operation of the engine by applying the calculated controlling commands. Finally, the authenticity of the resulting optimal controller is validated against a classical Pontryagin’s minimum principle-based control system. Generally, the findings demonstrate the effectiveness of the proposed control methodology to reduce coldstart hydrocarbon emissions in automotive engines.

Appendix A

Details of difference-based formulations of the proposed controller

The following difference-based system of equations is obtained for the co-states:

$$ \,\left\{ \begin{array}{l} \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial x_{1}^{*} \left( k \right)}} = - \lambda_{1}^{*} \left( k \right) \hfill \\ \quad = - \,\,K_{{T_{exh} }} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{1} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{1} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{1} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{1} \left( k \right)}}{{\partial x_{1}^{*} \left( k \right)}}} \right)} - \left( {\frac{{k_{1} }}{{\tau_{1} }}\lambda_{1}^{*} \left( {k + 1} \right)} \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial x_{3}^{*} \left( k \right)}} = - \lambda_{2}^{*} \left( k \right) \hfill \\ \quad = - \,K_{{T_{exh} }} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{1} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{1} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{3} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{3} \left( k \right)}}{{\partial x_{3}^{*} \left( k \right)}}} \right)} - \left( {\frac{{k_{3} }}{{\tau_{3} }}\lambda_{2}^{*} \left( {k + 1} \right)} \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial x_{5}^{*} \left( k \right)}} = - \lambda_{3}^{*} \left( k \right) = \hfill \\ \quad = \;\;K_{HC} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{2} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{5} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{5} \left( k \right)}}{{\partial x_{5}^{*} \left( k \right)}}} \right)} - \left( {\frac{{k_{5} }}{{\tau_{5} }}\lambda_{3}^{*} \left( {k + 1} \right)} \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial x_{6}^{*} \left( k \right)}} = - \lambda_{4}^{*} \left( k \right) = \hfill \\ \quad = \;\;K_{HC} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{2} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{6} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{6} \left( k \right)}}{{\partial x_{6}^{*} \left( k \right)}}} \right)} - \left( {\frac{{k_{6} }}{{\tau_{6} }}\lambda_{4}^{*} \left( {k + 1} \right)} \right) \hfill \\ \end{array} \right. $$

(A.1)

Moreover, the following difference-based system of equations is obtained for the states:

$$ \,\left\{ \begin{array}{l} \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial \lambda_{1}^{*} \left( {k + 1} \right)}} = x_{1}^{*} \left( {k + 1} \right) = \delta t \cdot \frac{{u_{1} \left( k \right)}}{{\tau_{1} }} + \left( {1 - \frac{{\delta t \cdot k_{1} }}{{\tau_{1} }}} \right)x_{1} \left( k \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial \lambda_{2}^{*} \left( {k + 1} \right)}} = x_{3}^{*} \left( {k + 1} \right) = \delta t \cdot \frac{{16 - u_{2} \left( k \right)}}{{\tau_{3} }} + \left( {1 - \frac{{\delta t \cdot k_{3} }}{{\tau_{3} }}} \right)x_{3} \left( k \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial \lambda_{3}^{*} \left( {k + 1} \right)}} = x_{5}^{*} \left( {k + 1} \right) = \delta t \cdot \frac{{16 - u_{2} \left( k \right)}}{{\tau_{5} }} + \left( {1 - \frac{{\delta t \cdot k_{5} }}{{\tau_{5} }}} \right)x_{5} \left( k \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial \lambda_{4}^{*} \left( {k + 1} \right)}} = x_{6}^{*} \left( {k + 1} \right) = \delta t \cdot \frac{{\left| {u_{1} \left( k \right) - 55} \right| + \left( {u_{1} \left( k \right) - 55} \right)}}{{2\tau_{6} }} + \left( {1 - \frac{{\delta t \cdot k_{6} }}{{\tau_{6} }}} \right)x_{6} \left( k \right) \hfill \\ \end{array} \right. $$

(A.2)

In addition, the following equations are derived for the control signals:

$$ \,\left\{ \begin{aligned} \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial u_{1}^{*} \left( {k + 1} \right)}} = 0 = K_{HC} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{2} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{6} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{6} \left( k \right)}}{{\partial u_{1}^{*} \left( k \right)}}} \right)} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - K_{{T_{exh} }} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{1} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{1} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{1} \left( k \right)}}{{\partial u_{1}^{*} \left( k \right)}}} \right)} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{{\lambda_{1}^{*} \left( {k + 1} \right)}}{{\tau_{1} }} + \frac{{\lambda_{4}^{*} \left( {k + 1} \right)}}{{2\tau_{6} }}\left( {1 + \frac{1}{{\left| {u_{1}^{*} \left( k \right) - 55} \right|}}} \right) \hfill \\ \frac{{\partial {\mathbf{H}}\left( {{\mathbf{X}}^{*} \left( k \right),{\mathbf{U}}^{*} \left( k \right),\bar{\lambda }^{*} \left( {k + 1} \right)} \right)}}{{\partial u_{2}^{*} \left( {k + 1} \right)}} = 0 = K_{HC} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{2} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{5} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{5} \left( k \right)}}{{\partial u_{2}^{*} \left( k \right)}}} \right)} \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - K_{{T_{exh} }} \sum\limits_{j = 1}^{N} {\left( {\omega_{j}^{1} {\kern 1pt} \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{3} \left( k \right)}} \cdot \frac{{\partial {\kern 1pt} \Im_{3} \left( k \right)}}{{\partial u_{2}^{*} \left( k \right)}}} \right)} \hfill \\ \quad \quad \quad \quad \quad \quad + \frac{{\lambda_{2}^{*} \left( {k + 1} \right)}}{{\tau_{3} }} + \frac{{\lambda_{3}^{*} \left( {k + 1} \right)}}{{2\tau_{5} }} \hfill \\ \end{aligned} \right. $$

(A.3)

For Eqs. (A.1) to (A.3), the following differentiations should be employed:

$$ \begin{array}{l} \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{1} \left( k \right)}} = \frac{{\alpha_{j}^{1} \cdot e^{{ - \left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}} }}{{\left( {1 + e^{{ - \left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}} } \right)^{2} }} \hfill \\ \frac{{\partial {\kern 1pt} g\left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}}{{\partial {\kern 1pt} \Im_{3} \left( k \right)}} = \frac{{\alpha_{j}^{3} \cdot e^{{ - \left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}} }}{{\left( {1 + e^{{ - \left( {\alpha_{j}^{1} \Im_{1} \left( k \right) + \alpha_{j}^{2} \Im_{2} \left( k \right) + \alpha_{j}^{3} \Im_{3} \left( k \right) + b_{j}^{1} } \right)}} } \right)^{2} }} \hfill \\ \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{5} \left( k \right)}} = \frac{{\kappa_{j}^{2} \cdot e^{{ - \left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}} }}{{\left( {1 + e^{{ - \left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}} } \right)^{2} }} \hfill \\ \frac{{\partial {\kern 1pt} g\left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}}{{\partial {\kern 1pt} \Im_{6} \left( k \right)}} = \frac{{\kappa_{j}^{3} \cdot e^{{ - \left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}} }}{{\left( {1 + e^{{ - \left( {\kappa_{j}^{1} \Im_{4} \left( k \right) + \kappa_{j}^{2} \Im_{5} \left( k \right) + \kappa_{j}^{3} \Im_{6} \left( k \right) + b_{j}^{2} } \right)}} } \right)^{2} }} \hfill \\ \frac{{\partial {\kern 1pt} \Im_{1} \left( k \right)}}{{\partial x_{1}^{*} \left( k \right)}} = 1 - \frac{{\delta t \cdot k_{1} }}{{\tau_{1} }};\quad \frac{{\partial {\kern 1pt} \Im_{3} \left( k \right)}}{{\partial x_{3}^{*} \left( k \right)}} = 1 - \frac{{\delta t \cdot k_{3} }}{{\tau_{3} }};\quad \frac{{\partial {\kern 1pt} \Im_{5} \left( k \right)}}{{\partial x_{5}^{*} \left( k \right)}} = 1 - \frac{{\delta t \cdot k_{5} }}{{\tau_{5} }} \hfill \\ \frac{{\partial {\kern 1pt} \Im_{6} \left( k \right)}}{{\partial x_{6}^{*} \left( k \right)}} = 1 - \frac{{\delta t \cdot k_{6} }}{{\tau_{6} }};\quad \frac{{\partial {\kern 1pt} \Im_{1} \left( k \right)}}{{\partial u_{1}^{*} \left( k \right)}} = \frac{\delta t}{{\tau_{1} }};\quad \frac{{\partial {\kern 1pt} \Im_{6} \left( k \right)}}{{\partial u_{1}^{*} \left( k \right)}} = \frac{\delta t}{{2\tau_{6} }}\left( {1 + \frac{1}{{\left| {u_{1} \left( k \right) - 55} \right|}}} \right) \hfill \\ \frac{{\partial {\kern 1pt} \Im_{3} \left( k \right)}}{{\partial u_{2}^{*} \left( k \right)}} = \frac{ - \delta t}{{\tau_{3} }};\quad \frac{{\partial {\kern 1pt} \Im_{5} \left( k \right)}}{{\partial u_{2}^{*} \left( k \right)}} = \frac{ - \delta t}{{\tau_{5} }} \hfill \\ \end{array} $$