Computationally Efficient Nonlinear Model Predictive Control Using the L₁ Cost-Function

• Main Authors: Maciej Ławryńczuk, Robert Nebeluk
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Sensors
• Subjects: process control; model predictive control; L₁ cost function; optimisation
• Online Access: https://www.mdpi.com/1424-8220/21/17/5835

Model Predictive Control (MPC) algorithms typically use the classical L<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula> cost function, which minimises squared differences of predicted control errors. Such an approach has good numerical properties, but the L<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>1</mn></msub></semantics></math></inline-formula> norm that measures absolute values of the control errors gives better control quality. If a nonlinear model is used for prediction, the L<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>1</mn></msub></semantics></math></inline-formula> norm leads to a difficult, nonlinear, possibly non-differentiable cost function. A computationally efficient alternative is discussed in this work. The solution used consists of two concepts: (a) a neural approximator is used in place of the non-differentiable absolute value function; (b) an advanced trajectory linearisation is performed on-line. As a result, an easy-to-solve quadratic optimisation task is obtained in place of the nonlinear one. Advantages of the presented solution are discussed for a simulated neutralisation benchmark. It is shown that the obtained trajectories are very similar, practically the same, as those possible in the reference scheme with nonlinear optimisation. Furthermore, the L<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>1</mn></msub></semantics></math></inline-formula> norm even gives better performance than the classical L<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mn>2</mn></msub></semantics></math></inline-formula> one in terms of the classical control performance indicator that measures squared control errors.