Purpose – Voltage source inverterfed permanent magnet synchronous motors (VSIPMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSIPMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSIPMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSIfed threephase PMSMs. Design/methodology/approach – The present article proposes a nonlinear optimal control approach for VSIPMSMs. The nonlinear dynamic model of VSIPMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is deﬁned by the present value of the voltage source inverterfed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverterfed PMSM, an Hinﬁnity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each timestep of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimationbased control for this system, the Hinﬁnity Kalman ﬁlter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSIfed PMSM under moderate variations of the control inputs. Findings – The proposed Hinﬁnity controller provides the solution to the optimal control problem for the VSI PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min– max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each timestep of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI PMSM loop has the Hinﬁnity tracking performance property, which signiﬁes robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSIPMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSIPMSM control loop can be minimized. Practical implications – The proposed nonlinear optimal control method for the VSIPMSM system exhibits several advantages: Comparing to global linearizationbased control methods, such as Lie algebrabased control or differential ﬂatness theorybased control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSIPMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with slidingmode control, there is no need to deﬁne in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local modelbased control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort. Social implications – Voltage source inverterfed permanent magnet synchronous motors (VSIPMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSIfd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSIfed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective. Originality/value – Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is deﬁned by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (Hinﬁnity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of statedependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.
A nonlinear optimal control approach for voltage source inverterfed threephase PMSMs
Marignetti, Fabrizio;
20230101
Abstract
Purpose – Voltage source inverterfed permanent magnet synchronous motors (VSIPMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSIPMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSIPMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSIfed threephase PMSMs. Design/methodology/approach – The present article proposes a nonlinear optimal control approach for VSIPMSMs. The nonlinear dynamic model of VSIPMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is deﬁned by the present value of the voltage source inverterfed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverterfed PMSM, an Hinﬁnity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each timestep of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimationbased control for this system, the Hinﬁnity Kalman ﬁlter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSIfed PMSM under moderate variations of the control inputs. Findings – The proposed Hinﬁnity controller provides the solution to the optimal control problem for the VSI PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min– max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each timestep of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI PMSM loop has the Hinﬁnity tracking performance property, which signiﬁes robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSIPMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSIPMSM control loop can be minimized. Practical implications – The proposed nonlinear optimal control method for the VSIPMSM system exhibits several advantages: Comparing to global linearizationbased control methods, such as Lie algebrabased control or differential ﬂatness theorybased control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSIPMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with slidingmode control, there is no need to deﬁne in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local modelbased control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort. Social implications – Voltage source inverterfed permanent magnet synchronous motors (VSIPMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSIfd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSIfed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective. Originality/value – Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is deﬁned by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (Hinﬁnity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of statedependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.File  Dimensione  Formato  

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