Ali Pakniyat

Dynamic modeling and controller design for a seamless two-speed transmission for electric vehicles

Transmission is one of the crucial elements of the driveline that affects vehicle fuel economy and comfort. It can transfer power in different combinations of torque and speed. This paper focuses on the modeling, simulation and control of a two-speed transmission for electric vehicles which has seamless gear shifting specification. The transmission incorporates two-stage planetary gear sets and two braking mechanisms to control the gear shifting. The dynamic model is developed by using the kinematic equations of the planetary gear trains and the Euler-Lagrange equations to derive the equations of motion. The mathematical model is validated by using the SimDriveLine library of MATLAB/Simulink®. The controller design employs optimal control methods to provide seamless shifting with minimum transition time. Then, by relaxing ideal constraints, a feasible controller is designed based on input-output and input-state feedback linearization. Simulation results demonstrate the ability of the proposed transmission to have smooth shifting without excessive oscillations in the output torque and speed.

Time Optimal Hybrid Minimum Principle and the Gear Changing Problem for Electric Vehicles

Abstract The statement of the Hybrid Minimum Principle is presented for time optimal control problems where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space. A key aspect of the analysis is the relationship between the Hamiltonians and the adjoint processes before and after the switching instants. As an example application, an electric vehicle equipped with a two-speed seamless transmission, that augments an additional degree of freedom during the transition period, is modelled within this framework. The state-dependant motor torque constraints are converted to state-independent control input constraints via a change of variable and the introduction of auxiliary discrete states. The problem of the minimum acceleration time required for reaching the speed of 100km/hr is formulated within the presented framework and the Time Optimal Hybrid Minimum Principle is employed in order to find the optimal control inputs and the optimal gear changing instants.

The gear selection problem for electric vehicles: An optimal control formulation

Optimal gear selection problem for energy minimization for two-gear electric vehicles is formulated within the framework of optimal control theory. Methods for restricting the number of switchings are analyzed and a computational example is presented.

The Hybrid Minimum Principle in the presence of switching costs

Hybrid optimal control problems are studied for systems where, in addition to running costs, switching between discrete states incurs costs. A key aspect of the analysis is the relationship between the Hamiltonian and the adjoint process before and after the switching instants. In this paper, the analysis is performed for systems for which autonomous and controlled state jumps are not permitted. First the results are established in the hybrid Mayer optimal control problem setup using the needle variation technique, and then the results for the hybrid Bolza optimal control problem are established via the calculus of variations methodology.

On the relation between the Minimum Principle and Dynamic Programming for Hybrid systems

Hybrid optimal control problems are studied for systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to running costs, switching between discrete states incurs costs. A key aspect of the analysis is the relationship between the Hamiltonian and the adjoint process in the Minimum Principle before and after the switching instants as well as the relationship between adjoint processes in the Minimum Principle and the gradient of the value function. In this paper we prove that under certain assumptions the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same dynamic equation and have the same boundary conditions and hence are identical to each other.

On the Minimum Principle and Dynamic Programming for Hybrid Systems

Abstract Hybrid optimal control problems are studied for systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to running costs, switching between discrete states incurs costs. A key aspect of the analysis is the relationship between the Hamiltonian and the adjoint process before and after the switching instants as well as the relationship between adjoint processes in the Minimum Principle and the value function in the Dynamic Programming. The results are illustrated through a simple, but yet very important, analytic example.

On the Relation Between the Minimum Principle and Dynamic Programming for Classical and Hybrid Control Systems

Hybrid optimal control problems are studied for a general class of hybrid systems, where autonomous and controlled state jumps are allowed at the switching instants, and in addition to terminal and running costs, switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework, and it is shown that under certain assumptions, the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results.

Observer-Based Backstepping Controller Design for Gear Shift Control of a Seamless Clutchless Two-Speed Transmission for Electric Vehicles

This paper proposes an observer-based backstepping controller design for gear shifting control of a seamless and clutchless two-speed transmission for electric vehicles. The state observer estimates the input and output torques of the transmission and the angular velocities of the gears, based on measuring the motor speed and the speed of the vehicle. Then, an observer-based backstepping controller is designed to provide seamless gear change while tracking the optimal trajectory corresponding to the minimum shifting time. Thereafter, the separation of the estimation and control is discussed. The driveline of an electric vehicle is modeled in MATLAB/Simulink by utilizing SimDriveLine library components to asses the performance of the designed observer and controller.

On the Relation between the Hybrid Minimum Principle and Hybrid Dynamic Programming: a Linear Quadratic Example

Abstract Hybrid optimal control problems are studied for systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to running costs, switching between discrete states incurs costs. Key aspects of the analysis are the relationship between the Hamiltonian and the adjoint process in the Hybrid Minimum Principle before and after the switching instants, the boundary conditions on the value function in Hybrid Dynamic Programming at these switching times, as well as the relationship between the adjoint process in the Hybrid Minimum Principle and the gradient process of the value function in Hybrid Dynamic Programming. The results are illustrated through an analytic example with linear dynamics and quadratic costs.

On the stochastic Minimum Principle for hybrid systems

A class of stochastic hybrid systems with both autonomous and controlled switchings and jumps is considered where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space. Optimal control problems associated with this class of stochastic hybrid systems are studied where in addition to running and terminal costs, switching between discrete states incurs costs. Necessary optimality conditions are established in the form of the Stochastic Hybrid Minimum Principle. A feature of special importance is the effect of hard constraints imposed by switching manifolds on diffusion-driven state trajectories which influence the boundary conditions for the stochastic Hamiltonian and adjoint processes.