Nonlinear optimal control for wireless power transfer and EV charging

Purpose: Permanent magnet synchronous spherical motors can have wide use in robotics and industrial automation. They enable 3-DOF omnidirectional motion of their rotor. They are suitable for several applications as for instance actuation in robotics, traction in electric vehicles and use in several automation systems. Unlike conventional synchronous motors, permanent magnet synchronous spherical motors consist of a fixed inner shell which is the stator and a rotating outer shell which is the rotor. Their dynamic model is multivariable and strongly nonlinear. The treatment of the associated control problem is important. Design/methodology/approach: In this article the multivariable dynamic model of permanent magnet synchronous spherical motors is analyzed and a nonlinear optimal (H-infinity) control method is developed for it. Differential flatness properties are proven for the spherical motor's state-space model. Next, the motor's state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance around a time-varying operating point which is defined by the present value of the motors' state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the permanent magnet synchronous spherical motor a stabilizing H-infinity feedback controller is designed. To compute the controller's gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach implemented in successive loops. Findings: Due to the nonlinear and multivariable structure of the state-space model of spherical motors the solution of the associated nonlinear control problem is a non-trivial task. In this article a novel nonlinear optimal (H-infinity) control approach is proposed for the dynamic model of permanent magnet synchronous spherical motors. The method is based on approximate linearization of the motor's state-space model with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Furthermore, the article has introduced a different solution to the nonlinear control problem of the permanent magnet synchronous spherical motor which is based on flatness-based control implemented in successive loops. Research limitations/implications: The presented control approaches do not exhibit any limitations but on the contrary they have specific advantages. In comparison to global linearization-based control schemes (such as Lie-algebra-based control) they do not make use of complicated changes of state variables (diffeomorphims) and transformations of the system's state-space description. The computed control inputs are applied directly on the initial nonlinear state-space model of the PM spherical motor without the intervention of inverse transformations and thus without coming against the risk of singularities. The motion control problem of spherical motors is non-trivial because of the complicated nonlinear and multivariable dynamics of these electric machines. So far, there have been

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

A nonlinear optimal control approach for voltage source inverter-fed three-phase PMSMs

The control problem of the rotary double inverted pendulum (double Furuta pendulum) is nontrivial because of underactuation and strong nonlinearities in the associated state‐space model. The system has three degrees of freedom (one actuated and two unactuated joints) while receiving only one control input. In this article, a novel nonlinear optimal (H‐infinity) control approach is developed for the dynamic model of the rotary double inverted pendulum. First, the dynamic model of the double pendulum undergoes approximate linearization with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instance 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 inputs vector. At a next stage a stabilizing H‐infinity feedback controller is designed. To compute the controller's feedback gains an algebraic Riccati equation has to be solved at each time‐step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. To implement state estimation‐based control without the need to measure the entire state vector of the rotary double‐pendulum the H‐infinity Kalman filter is used as a robust state observer. The nonlinear optimal control method achieves fast and accurate tracking of setpoints by all state variables of the rotary double inverted pendulum under moderate variations of the control input.

This article aims to treat the nonlinear control problem for the complex dynamics of a wave energy unit (WEC) that consists of a Permanent Magnet Linear Synchronous Generator (PMLSG) and a Voltage Source Converter (VSC). The article has developed a globally stable nonlinear optimal control method for this wave power generation unit. The new method avoids complicated state-space model transformations and minimizes the energy dispersion by the control loop. A novel nonlinear optimal control method is proposed for the dynamic model of a wave energy conversion system, which includes a Permanent Magnet Linear Synchronous Generator (PMLSG) serially connected with an AC/DC three-phase voltage source converter (VSC). The dynamic model of this renewable energy system is formulated and differential flatness properties are proven about it. To apply the proposed nonlinear optimal control, the state-space model of the PMLSG-VSC wave energy conversion unit undergoes an approximate linearization process at each sampling instance. The linearization procedure relies on a first-order Taylor-series expansion and involves the computation of the system’s Jacobian matrices. It takes place at each sampling interval around a temporary operating point, which is defined by the present value of the wave energy conversion unit’s state vector and by the last sampled value of the control inputs vector. An H-infinity feedback controller is designed for the linearized model of the wave energy conversion unit. To compute the feedback gains of this controller, an algebraic Riccati equation is repetitively solved at each time step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis.

Heat pumps can be used for thermal management of electric vehicles (EVs) in a low-cost and environmentally friendly manner. Heat pumps can exploit the heat that is diffused by the vehicle’s batteries so as to perform temperature control of the passengers cabin. A heat-pump-based thermal management system of EVs has the dual objective of (a) stabilizing temperature in the passenger cabin at the desirable levels so as to ensure the passengers’ comfort and (b) stabilizing temperature at the batteries so as to avoid degradation of the batteries’ power storage capacity and of their regenerative charging capability. So far, nonlinear model predictive control (NMPC) methods have been mainly applied for the nonlinear optimal control problem of heat pumps in EVs. In this article a novel nonlinear optimal control method is proposed for solving the nonlinear optimal control problem of heat pumps. This method is computationally simple, has clear implementation stages, and is of proven global stability. The dynamic model of the heat pump undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. Linearization takes place at each sampling instance, around a temporary operating point that is defined by the present value of the heat pump’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the heat pump an H-infinity stabilizing controller is designed. The feedback gains of the controller are computed through the solution of an algebraic Riccati equation, at each time step of the control method. The global stability properties of the control scheme are proven through Lyapunov analysis. Moreover, to enable state estimation-based control, the H-infinity Kalman filter is used as a robust observer. The method achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

The paper proposes a nonlinear optimal control method for guidance and three-dimensional trajectory tracking of missiles. It is proven that the kinematic–dynamic model of the missile is differentially flat, which confirms the controllability of this system. Next, to apply the nonlinear optimal control scheme the dynamic model of the missile undergoes approximate linearization with the use of first-order Taylor-series expansion and through the computation of the associated Jacobian matrices. The linearization takes place at each sampling instance around a time-varying operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the missile an H-infinity optimal feedback controller is designed. To compute the controller’s stabilizing feedback gains an algebraic Riccati equation has to be solved repetitively at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control scheme are proven through Lyapunov analysis. This control method exhibits specific advantages: (i) unlike global linearization-based control schemes it avoids complicated changes of state variables and state-space model transformations, (ii) unlike Nonlinear model-predictive control approaches its convergence to the optimum does not depend on [Formula: see text] initialization and empirical selection of the controller’s parameters, (iii) unlike multi-model feedback control schemes with linearization around multiple operating points, it avoids the solution of an exponentially large number or Riccati equations or the solution of LMIs of large dimensionality.

In this article, the problem of nonlinear optimal (H-infinity) control for a translational oscillator with rotating actuator (TORA) system is treated. The dynamic model of a translational oscillator with rotational actuator is generated through Euler-Lagrange analysis. This model undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the system, a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains, an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control without the need to measure the entire state vector of the TORA the H-infinity Kalman Filter is used as a robust state estimator.

The paper proposes a nonlinear optimal control method for treating the control and stabilization problem of supply chain networks and inventories under time-delays. The state-space model of the supply chain network is considered to have as state variables the customer’s demand and the inventories of the manufacturer, of the retailers and of the distributors. The control inputs of the model are the manufacturer’s production and the retailer’s ordering quantities. The model is subject to time-delays. It is proven that the dynamic model of the supply chain is differentially flat and a nonlinear optimal control method is applied to it. To implement this control method, approximate linearization is performed with the use of Taylor-series expansion while an algebraic Riccati equation has to be solved at each sampling instance. The proposed control method avoids complicated changes of state variables and state-space model transformations while the control inputs it computes are applied directly on the initial nonlinear model of the controlled system. It achieves optimality because of enabling convergence of the state variables of the supply chain network to the targeted setpoints under minimal variations of the control inputs. The article’s nonlinear optimal control method is novel when compared to past approaches for treating the optimal control problem in nonlinear dynamical systems. Unlike Nonlinear Model Predictive Control (NMPC), the proposed nonlinear optimal control scheme ensures convergence to optimum without dependence on initialization and empirical selection of the controller’s parameters.

Dual-arm robotic manipulators are used in industry and for assisting humans since they enable dexterous handling of objects and more agile and secure execution of pick-and-place, grasping or assembling tasks. In this paper, a nonlinear optimal control approach is proposed for the dynamic model of a dual robotic arm. In the considered application, the dual-arm robotic system has to transfer an object under synchronized motion of its two end-effectors so as to achieve precise positioning and to compensate for contact forces. The dynamic model of this robotic system is formulated while it is proven that the state-space description of the robot’s dynamics is differentially flat. Next, to solve the associated nonlinear optimal control problem, the dynamic model of the dual-arm robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the dual-arm robot, a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

Control of the milling process of mining products (ore milling) is a non-trivial problem due to being related with a strongly nonlinear and multivariable state-space model. To provide an efficient solution to this problem, in this article a nonlinear optimal (H-infinity) control method is developed. In the considered nonlinear optimal control method, the dynamic model of the mining products' mill undergoes first approximate linearization with the use of Taylor series expansion and with the computation of the associated Jacobian matrices. The linearization point (temporary equilibrium) is recomputed at each time step of the control method and comprises the present value of the system's state vector and the last value of the control inputs' vector that was exerted on it. For the linearized description of the mill's functioning the optimal control problem is solved by applying an H-infinity controller. The feedback gain is computed again at each iteration of the control algorithm through the solution of an algebraic Riccati equation. The stability of the control scheme is confirmed through Lyapunov analysis. First, it is shown that the control method satisfies the H-infinity tracking performance, and this signifies elevated robustness against model uncertainty and external perturbations. Next, under moderate conditions, it is proven that the control loop is globally asymptotically stable.

The multivariable tumor-growth dynamic model has been widely used to describe the inhibition of tumor-cells proliferation under the simultaneous infusion of multiple chemotherapeutic drugs. In this article, a nonlinear optimal (H-infinity) control method is developed for the multi-variable tumor-growth model. First, differential flatness properties are proven for the associated state-space description. Next, the state-space description undergoes approximate linearization with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization process takes place at each sampling instant around a time-varying operating point which is defined by the present value of the system’s state vector and by the last sampled value of the control inputs vector. For the approximately linearized model of the system a stabilizing H-infinity feedback controller is designed. To compute the controller’s gains an algebraic Riccati equation has to be repetitively solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. Finally, the performance of the nonlinear optimal control method is compared against a flatness-based control approach.

ABSTRACTGantry cranes of the H‐type with dual electric‐motor actuation are widely used in industry. In this article, the control problem of an H‐type gantry crane which is driven by a pair of linear permanent magnet synchronous motors (dual PMLSMs) is considered. The integrated system that comprises the H‐type gantry crane and its two LPMSMs is proven to be differentially flat. The control problem for this robotic system is solved with the use of a nonlinear optimal control method. To apply the nonlinear optimal control method, the dynamic model of the H‐type gantry crane with dual LPMSM undergoes approximate linearization at each sampling instant with the use of first‐order Taylor series expansion and through the computation of the associated Jacobian matrices. The linearization point is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time‐step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. The proposed control scheme achieves stabilization of the H‐type gantry crane with dual LPMSMs without the need of diffeomorphisms and complicated state‐space model transformations.

Nonlinear optimal control problems usually require solutions using iterative procedures and, hence, they fall naturally in the realm of 2-D systems where the two dimensions are response time horizon and iteration index, respectively. The paper uses this observation to employ 2-D systems theory, in the form of unit memory repetitive process techniques, to investigate optimality, local stability, and global convergence behavior of an algorithm, based on integrated-system optimization and parameter estimation, for solving continuous nonlinear dynamic optimal control problems. It is shown that 2-D systems theory can be usefully applied to analyze the properties of iterative procedures for solving these problems.