Abstract-This paper describes a two-sweep control design method to stabilize the acrobot, an input-affine under-actuated system, at the upper equilibrium point. In the forward sweep, the system is successively reduced, one dimension at a time, until a two-dimensional system is obtained. At each step of the reduction process, a quotient is taken along one-dimensional integral manifolds of the input vector field. This decomposes the current manifold into classes of equivalence that constitute a quotient manifold of reduced dimension. The input to a given step becomes the representative of the previous-step equivalence class, and a new input vector field can be defined on the tangent of the quotient manifold. The representatives remain undefined throughout the forward sweep. During the backward sweep, the controller is designed recursively, starting with the twodimensional system. At each step of the recursion, a well-chosen representative of the equivalence class ahead of the current level of recursion is chosen, so as to guarantee stability of the current step. Therefore, this stabilizes the global system once the backward sweep is complete. Although stability can only be guaranteed locally around the upper equilibrium point, the domain of attraction can be enlarged to include the lowerequilibrium point, thereby allowing a swing-up implementation. As a result, the controller does not require switching, which is illustrated in simulation. The controller has four tuning parameters, which helps shape the closed-loop behavior.

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Abstract-An algorithmic method is proposed to design stabilizing control laws for a class of nonlinear systems that comprises single-input feedback-linearizable systems and a particular set of single-input non feedback-linearizable systems. The method proceeds iteratively and consists of two stages; it converts the system into cascade form and reduces the dimension at every step by creating quotient manifold in the forward stage, while it constructs the feedback law iteratively in the backward stage. The paper shows that the construction of these quotient manifolds is well defined for feedbacklinearizable system and, furthermore, it can also be applied to a class of non feedback-linearizable systems.

Feedback linearization requires a unique feedback law and a unique diffeomorphism to bring a system to Brunovský normal form. Unfortunately, singularities might arise both in the feedback law and in the diffeomorphism. This paper demonstrates the ability of a quotient method to avoid or mitigate the singularities that typically arise with feedback linearization. The quotient method does it by relaxing the conditions on diffeomorphism, which can be achieved since there is an additional degree of freedom at each step of the iterative procedure. This freedom in choosing quotients and the resulting advantage are demonstrated for a field-controlled DC motor. Using a Lyapunov function, the domain of attraction of the control law obtained with the quotient method is proved to be larger than the domain of attraction of a control law obtained using feedback linearization.

Abstract.A numerical algorithm that achieves asymptotic stability for feedback linearizable systems is presented. The nonlinear systems can be represented in various forms that include differential equations, simulated physical models or lookup tables. The proposed algorithm is based on a quotient method and proceeds iteratively. At each step, the dynamic system is desensitized with respect to the current input vector field. Control is obtained by tracking a desired value along the input vector field at each step. The numerical algorithm uses the direction on the tangent manifold at a given point and its variation around that point. This enables the algorithm to produce control values simply using a simulator of the nonlinear system. Keywords: Numerical algorithms, feedback-linearization, quotient, control law design, nonlinear dynamical systems PACS: 05.10.-a Feedback linearization is an effective method to handle nonlinear systems. However, it requires exact knowledge of the system and, furthermore, there exist conditions for a nonlinear system to be feedback linearizable (FL). Moreover, for feedback linearization, an output function of relative degree n must be known [1,2], where n is the dimension of the system. Computing such an output requires a systematic procedure, such as the one proposed in [3,4], which proceeds by successively generating quotients that are desensitized with respect to the input vector field. This method has been extended to produce a control design technique applicable to FL input-affine single-input systems of the formFor non-FL systems, the method requires approximations [6,7]. However, for some non-FL systems, the system of equations turns out to be so complicated that, even after approximations, the method is not applicable due to the lack of closed-form solutions to some of the equations. This difficulty has motivated the use of numerical algorithms to compute the control input. This paper presents a numerical algorithm that computes the input for a given state of the system. The algorithm requires the time derivatives of the states at particular state and input values. The time derivatives can be obtained from the system's differential equations (through traditional modeling) or through advanced computer-generated physical modeling. Computer-generated physical modeling is an interactive technique that connects electro-mechanical components to simulate dynamical systems [8]. Simscape™ is such a simulation tool. With it, a designer can simulate complicated electro-mechanical systems without having full knowledge of the underlying differential equations. The inability of these tools to provide the governing differential equations make them unsuited for developing nonlinear control laws. Using the proposed numerical algorithm, it is possible to harness the simulation capabilities of such tools. Moreover, if measured data are available for a system, it is also possible to compute the derivatives via numerical differentiation and use that for the proposed numerical algorithm NUMERICAL...