Stabilization of an experimental cart-pendulum system through approximate manifold decomposition

Ingram, David; Willson, S. S.; Müllhaupt, Ph.; Bonvin, Dominique

doi:10.3182/20110828-6-it-1002.01168

Cited by 4 publications

(9 citation statements)

References 29 publications

(29 reference statements)

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“…This is a consequence of Lemma 2 and Lemma 3. However, even if the distribution generated by the original system is not involutive, the approach can still be applied, as illustrated in [12] and [13].…”

Section: Discussionmentioning

confidence: 99%

“…This section presents an application of the algorithm to a FBL system. However, the algorithm is not restricted to FBL systems, and application of the quotient method to non-FBL system are illustrated in [12] and [13].…”

Section: Example Of a DC Motormentioning

confidence: 99%

“…The complete step can be summarized through the commutative diagram given in Figure 2. Iff z (x, z p ) depends linearly on z p , which is the case when the distribution ∆ = span{g, [f, g]} is involutive and a certain condition is satisfied (see Lemma 3 below), then (13) can be separated as follows:…”

Section: A Forward Stage For Achieving Cascade Formmentioning

confidence: 99%

“…That is, non-FBL systems will necessarily have an iteration where a linear relationship cannot be established. Hence, in this case, the algorithm needs to be modified as illustrated in [12] and [13]. Remark 2: This algorithm can transform the system into a chain of integrators by choosing γ(x) = L f φ i such that L g L f φ i = 0 at every iteration.…”

Section: Let Us Define Thementioning

confidence: 99%

“…Furthermore, with certain assumptions, the proposed approach can approximate non-FBL systems as FBL systems. For example, it has been possible to achieve both swingup and stabilization of the acrobot [12] and the inverted pendulum [13].…”

Section: Introductionmentioning

confidence: 99%

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A quotient method for designing nonlinear controllers

Willson

Müllhaupt

Bonvin

2013

European Journal of Control

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Example Of a DC Motormentioning

confidence: 99%

Section: A Forward Stage For Achieving Cascade Formmentioning

confidence: 99%

Section: Let Us Define Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A quotient method for designing nonlinear controllers

Willson

Müllhaupt

Bonvin

2013

European Journal of Control

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Numerical algorithm for feedback linearizable systems

Willson¹,

Müllhaupt²,

Bonvin³

2012

AIP Conference Proceedings

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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...

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Avoiding feedback-linearization singularity using a quotient method — The field-controlled DC motor case

Willson

Müllhaupt²,

Bonvin

2012

2012 American Control Conference (ACC)

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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.

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Stabilization of an experimental cart-pendulum system through approximate manifold decomposition

Cited by 4 publications

References 29 publications

A quotient method for designing nonlinear controllers

A quotient method for designing nonlinear controllers

Numerical algorithm for feedback linearizable systems

Avoiding feedback-linearization singularity using a quotient method — The field-controlled DC motor case

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Stabilization of an experimental cart-pendulum system through approximate manifold decomposition

Cited by 4 publications

References 29 publications

A quotient method for designing nonlinear controllers

A quotient method for designing nonlinear controllers

Numerical algorithm for feedback linearizable systems

Avoiding feedback-linearization singularity using a quotient method &#x2014; The field-controlled DC motor case

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Avoiding feedback-linearization singularity using a quotient method — The field-controlled DC motor case