Mohamed K. Helwa

Monotonic Reach Control on Polytopes

The technical note studies the problem of making the trajectories of an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope. The focus is on solvability by continuous piecewise affine feedback, and we formulate a variant of the problem in which trajectories exit in a monotonic sense. This allows to obtain necessary and sufficient conditions for solvability in certain geometric situations.

Epsilon controllability of nonlinear systems on polytopes

The paper introduces the notion of epsilon controllability of nonlinear systems on polytopes, and then uses it to approximately solve the mutual accessibility problems of nonlinear systems on polytopes. In particular, we first show that if one constructs a polytopic cover of a given polytope such that the affine system resulting from the linearization of the nonlinear system inside each polytopic region of the cover is in-block controllable, then starting from any initial state in the interior of the given polytope, one can steer the nonlinear system to an epsilon neighborhood of any final state in the interior of the polytope in finite time, where epsilon depends on the size of the polytopic regions of the cover. We then study a hierarchy of covers, representing the nonlinear system at different levels of accuracy, and provide a constructive algorithm for achieving approximate mutual accessibility of nonlinear systems on polytopes.

Flow functions, control flow functions, and the reach control problem

The paper studies the reach control problem (RCP) to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope. We introduce the notion of a flow function, which provides the analog of a Lyapunov function for the equilibrium stability problem. A flow function comprises a scalar function that decreases along closed-loop trajectories, and its existence is a necessary and sufficient condition for closed-loop trajectories to exit the polytope. It provides an analysis tool for determining if a specific instance of RCP is solved, without the need for calculating the state trajectories of the closed-loop system. Results include a variant of the LaSalle Principle tailored to RCP. An open problem is to identify suitable classes of flow functions. We explore functions of the form V ( x ) = max { V i ( x ) } , and we give evidence that these functions arise naturally when RCP is solved using continuous piecewise affine feedbacks. Next we introduce the notion of a control flow function. It is shown that the Artstein-Sontag theorem of control Lyapunov functions has direct analogies to RCP via control flow functions.

Constrained control of the synchromesh operating state in an electric vehicle's clutchless automated manual transmission

This paper considers the constrained control problem of the friction regimes in sliding lubricated surfaces with the purpose of speed synchronization, wear reduction and increasing the lifetime of the friction lining material. The case study here is the engagement process of the synchronizer cone clutch system. Such synchronizer performs the clutchless gear shifting in a 2-speed automated manual transmission (AMT) of an electric vehicle. In the present study, the frictional behavior of the cone clutch system is investigated by considering the involved lubricated friction regimes. By knowing the lubricated sliding friction regimes, the dynamic model of the system is derived according to the variable coefficient of friction. Moreover, the primary sources of the uncertainty and disturbance are recognized and considered in the dynamic model of the system. For the purpose of controlling the system, the control objectives and the constraints are defined, and a controller design method is proposed. The controller design method is based on solving a set of linear programming (LP) problems in the offline phase, which results in a piecewise affine (PWA) feedback law that can be easily applied on the system in the real-time closed-loop configuration. Finally, the performance of the proposed control approach is assessed by presenting the closed-loop control results for the ideal situation as well as the perturbed systems in the presence of the disturbance.

In-block controllability of affine systems on polytopes

The paper introduces the study of in-block controllability (IBC) of affine systems on polytopes. In particular, we study whether all the states in the interior of a given polytope are mutually accessible through its interior by applying uniformly bounded control inputs. We first provide easily checkable necessary conditions for IBC. Then, for an important class of polytopes, simplicial polytopes, we show that these conditions are also sufficient. Several illustrative examples are also given to clarify the main results.

Monotonic reach control on polytopes

The paper studies the problem of making an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope. The focus is on solvability by continuous piecewise affine feedback, and we formulate a variant of the problem in which trajectories exit in a monotonic sense. This allows to obtain necessary and sufficient conditions for solvability in certain geometric situations. Next, we show that, generically, solvability via arbitrary triangulations is equivalent to monotonic solvability. In contrast with existing simplex-based methods, this provides an avenue for reach control on polytopes that does not depend on the choice of triangulation of the polytope.

Hierarchical control of piecewise affine hybrid systems

This paper extends the hierarchical control structures in [4], [6], [17] to the important class of piecewise affine (PWA) hybrid systems. In particular, we extend the notions of dynamical consistency (DC), hybrid in-block controllable (HIBC) partitions and DC partition machines to PWA hybrid systems. Using these notions, we show that if an HIBC partition of the state space is constructed, then controllability of the PWA hybrid system can be easily checked from the associated DC partition machine. Then, we construct the lattice of HIBC partitions and use it to build a hierarchical control structure of the system in which a control objective at any level of abstraction can be realized by state dependent control sequences at the lower levels.

Generalized flow conditions for reach control on polytopes

The paper studies the reach control problem (RCP) to make an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope. We introduce the notion of generalized flow conditions, which give a necessary and sufficient condition for closed-loop trajectories to exit the polytope. In analogy with Lyapunov stability theory, the generalized flow condition comprises a functional that decreases along closed-loop trajectories. We provide a set of results to analyze whether an instance of RCP is solved, without resorting to exhaustive simulation of the closed-loop system. This includes a variant of the LaSalle principle tailored to RCP. An open problem is to identify suitable classes of functionals that give rise to a generalized flow condition. We explore functions of the form V (x) = max{Vi(x)}, and we give evidence that these functions arise naturally when RCP is solved using continuous piecewise affine feedbacks.

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 construction of in-block controllable covers of nonlinear systems on polytopes

This paper initiates the study of the problem of constructing in-block controllable covers of nonlinear systems on polytopes, which is useful in studying approximate mutual accessibility problems of nonlinear systems on polytopes. In particular, for a given nonlinear system and a given polytope, we study how to systematically construct polytopic covers of the given polytope such that the affine systems, resulting from the linearization of the nonlinear system inside the polytopic regions of the cover, are in-block controllable. By exploring the geometry of the problem, we first provide some constructive guidelines for building such covers. Then, for a special class of nonlinear systems, we provide a constructive algorithm. We also give illustrative examples of the main results.