Pole assignment for control of flexible link mechanisms

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Abstract

Although the dynamics of flexible link mechanisms and manipulators is nonlinear, motion and vibration control often relies on linear or piecewise-linear controllers based on linearized models in order to ensure real-time implementability. Keeping such an objective in mind, this paper proposes a general receptance-based method for pole assignment in flexible link mechanisms with a single rigid-body degree of freedom (dof) using a single control force (i.e. rank-one control). A chief advantage of the approach proposed is that it makes use of the second-order system model representation through the receptance matrix of the symmetric part of the asymmetric model. The asymmetric terms in the stiffness and damping matrices arise from the coupling between rigid-body motion and elastic motion. The proposed receptance-based formulation ensures numerical reliability and efficiency also for large dimensional and ill-conditioned system models originating from the simultaneous presence of high-frequency and weakly controllable oscillating modes, and of rigid-body motion low-frequency dynamics, which may also be unstable. The validation of the proposed technique is carried out by performing pole assignment through position and velocity feedback or acceleration and velocity feedback on a mechanism. Integral control is also introduced to improve the steady state system response. Numerical results indicate that the proposed method is more accurate and robust than two popular established methods.

Introduction

The need to perform hard real-time control at high sample rate in the motion and vibration control of flexible link multibody mechanisms often prevents the direct use of complete nonlinear models in controller implementations: indeed, the computational time required to solve nonlinear differential equations is generally too high, in particular when large dimensional and ill conditioned models involving both high-frequency and low-frequency dynamics are adopted to describe accurately the system behavior and to prevent spillover phenomenon. Nonetheless, it is widely recognized (see e.g. from [1] through [11]) that in the case of small deformations, the validity of linearized models about operating points is usually good enough to make their use successful in the synthesis of effective linear regulators. Therefore, although several nonlinear control methods have been proposed to date (e.g. [6], [9], [12]), linear control schemes are still quite popular, in particular in industry and in aerospace engineering ([13]), i.e. where controller implementability is a crucial requirement. Successful examples of linear or piecewise-linear controllers based on linearized models have been recently proposed in literature (see, for example, the simulation studies from [1], [2], [3], [4], [5], [6], or the experimental ones in [7], [8]). Linearized models have been also effectively employed in the synthesis of state observers (e.g. in [7], [10], [11]) which are often essential in the design of linear regulators.

Among the techniques commonly adopted for active vibration suppression in linear systems, a great deal of interest has been paid to pole assignment (pole placement). The selection of the desired real and imaginary parts of the system poles allows the straightforward definition of the system performances in terms of desired natural frequencies and damping. In flexible link mechanisms, this approach allows defining the bandwidth of the rigid-body motion controller too. However, the use of standard pole assignment techniques for the control of flexible link mechanisms may lead to unfeasible or wrong solutions. As a matter of fact, dynamic models, though linearized, are often still large dimensional and numerically ill conditioned because of the simultaneous presence of high frequency oscillating modes and rigid-body low-frequency dynamics. Additionally, rigid-body motion is often unstable, as a consequence of the presence of gravity force, and low controllability affects the highest frequency modes. This problem is exacerbated when mechanisms are driven by a single control force, i.e. when rigid-body motion has a single degree of freedom (dof) and no redundant actuators are devoted to vibration damping. Finally, accurate system modeling, capable of representing the mutual coupling between rigid-body and elastic motion, leads to asymmetric stiffness and damping matrices [7].

On the basis of an accurate linearized model, this paper proposes a general and effective technique for pole assignment in flexible link mechanisms with a single rigid-body dof and a single control force. This is the so-called rank-one control [14]. The technique takes advantage of the method proposed by Ouyang in [15] for the pole assignment in rank-one control of friction induced vibrations of second-order systems where asymmetric terms appear in the stiffness matrix [16]. Such a method is based on the receptances of the symmetric part of the asymmetric systems and represents a significant extension to the work developed by Ram and Mottershead [14], who first put forward receptance-based pole assignment to symmetric systems. The use of receptances has several advantages over those model-based methods in which a theoretical model of the system is required.

The receptance-based approach is exploited also in this paper to account for asymmetric damping and stiffness matrices in flexible link mechanisms. The method proposed here takes advantage of the receptance matrix of the symmetric part of the asymmetric system and makes direct use of the second-order system model representation without re-arranging the equations of motion into the first-order state space representation, as suggested in most of the methods that appeared in literature (see for example [17], [18] for state feedback, [19] for state derivative feedback). The first-order representation loses some relevant numerical properties of the second-order model matrices (e.g. bandedness, sparseness and definiteness [20]) having important implications on the efficiency of the numerical algorithms ([21]), imposes the mass matrix inversion and duplicates the dimension of the system. The last two properties often lead to increased numerical ill conditioning and unreliable controller gains.

The second-order formulation has been used by a number of researchers. In [22] Schulz and Inman developed an eigenstructure assignment technique for vibration suppression and isolation through active control. In [23] Datta et al. studied eigenstructure assignment for quadratic pencil by multi-input state feedback control. Ram et al. implemented in [24] multi-input partial pole assignment as a series of single-input control actions. In [25] Singh and Ram put forward both passive and active-control methods for vibration absorption. Robust pole assignment has also been developed by using second-order formulation. For instance, in [26], [27] the eigenvalue assignment with minimum sensitivity in descriptor second-order dynamical systems through position and velocity feedback is investigated. In [28] robust pole placement on second-order linear systems affected by norm-bounded uncertainty is solved through a Linear Matrix Inequality (LMI) approach. All the aforementioned works were directed towards conventional vibration problems represented by symmetric systems and receptances were not used. In contrast, very few researchers have studied asymmetric second-order systems. A general approach for pole assignment in second-order systems, with no assumptions on the system symmetry, was proposed in [29] and then refined in [21]. The method, in both formulations, requires the selection of the eigenvectors to be assigned among those spanned by a null-space. When dealing with single-input systems, this calculation is useless since rank-one control gives no chance of selecting the closed-loop system eigenvectors [30], and additionally can introduce numerical errors or increase the computational time in large-dimensional systems. In [21] another generalized method for multi-input and non-symmetric linear second-order systems is proposed by Chu. The method relies on the computation of the left eigenvectors, whose extraction from measured noisy FRFs in complex systems is usually an ill-conditioned procedure [31], even though it may be improved by regularization. It should be pointed out that the method presented in [21], [26] is model-based.

The work proposed in this paper tries to overcome these limitations by using receptances to achieve pole assignment with rank-one control. The technique developed allows performing pole assignment either through state feedback (active stiffness and active damping), or through state derivative feedback (active mass and active damping), or finally through suitable combinations of position, velocity and acceleration feedback. Integral control is also accounted for, since it is commonly required in the motion control of mechanisms in order to improve steady state response. All these features are significant strengths and peculiarities of the proposed method and justify its use in the rank-one control of flexible link mechanisms.

The paper is organized as follows. Section 2 describes an effective model of flexible link mechanism with one rigid-body motion dof. Only the most relevant equations are reported here: the interested reader should refer to [7] and the references therein for a more detailed description of the model adopted. The pole assignment technique is thoroughly described in Section 3, and then extended in Section 4 to include integral control. Several test cases, including state feedback, state derivative feedback and integral control, are proposed in Section 5 in order to validate the technique. Concluding remarks and future developments of the work are provided in Section 6.

Section snippets

Model of a flexible link mechanism

In order to extend the technique introduced in [15] to pole assignment of flexible link mechanisms, the system equations of motion should be written in terms of second-order linear-time-invariant Ordinary Differential Equations (ODE):Mẍ(t)+Cẋ(t)+Kx(t)=d(t)+bvA(t)where M, C, KRN×N are the system mass, damping and stiffness matrices, x(t) ∈RN×1 is the N-dimensional displacement vector (with respect to the equilibrium configuration), d(t) ∈RN×1 represents the effect of the external nodal

Problem formulation

In the following it is supposed that the system is controllable, i.e. rank([λi2M+λiC+Kb])=N for all eigenvalues λi (i=1,…,2N) [21] and therefore all the system poles may be assigned arbitrarily to prescribed positions in the complex plane through position and velocity feedback.

It will also be supposed that system full state (or state derivative) is available for feedback. In practice, state estimators are often required to overcome the presence of unmeasured states. This is neither a drawback

Pole assignment in the presence of integral control

Effective control of flexible link mechanisms often requires integral actions in the controller so as to reduce steady state error and compensate external disturbances. Unfortunately, integral control can decrease the effectiveness of vibration control and lead to actuator saturation. In order to trade off between the aforementioned problems, integral control should be applied only to the ERLS tracking error or to the absolute motion of the actual system. For this purpose, the control action

System description

The theory developed above is now applied to pole assignment in a four-bar planar linkage with all the links flexible, except for the ground link (the frame, named link 0). The mechanism is almost identical to the one adopted in some previous works (e.g. [7]) for experimental investigations. The system has one rigid-body dof and the ERLS generalized coordinate q is the (rigid-body) rotation of the crank (link 1). The links are assumed to be straight and slender steel bars and are modeled

Conclusions

The motivation for this paper is proposing a general technique for pole assignment in the rank-one control of flexible link mechanisms, by means of suitable combinations of position, velocity and acceleration feedback. Integral control has also been addressed. The proposed technique is based on the theory proposed for the pole assignment of friction-induced vibrations, and takes advantage of the receptance matrix of the symmetric part of the asymmetric system. The formulation adopted makes

Conflict of interest

None.

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