Elsevier

Systems & Control Letters

Volume 62, Issue 9, September 2013, Pages 770-780
Systems & Control Letters

Review
Robust energy shaping control of mechanical systems

https://doi.org/10.1016/j.sysconle.2013.05.011Get rights and content

Abstract

The problem of robustness improvement, vis à vis external disturbances, of energy shaping controllers for mechanical systems is addressed in this paper. First, it is shown that, if the inertia matrix is constant, constant disturbances (both, matched and unmatched) can be rejected simply adding a suitable integral action—interestingly, not at the passive output. For systems with non-constant inertia matrix, additional damping and gyroscopic forces terms must be added to reject matched disturbances and, moreover, enforce the property of integral input-to-state stability with respect to matched disturbances. The stronger property of input-to-state stability, this time with respect to matched and unmatched disturbances, is ensured with further addition of nonlinear damping. Finally, it is shown that including a partial change of coordinates, the controller can be significantly simplified, preserving input-to-state stability with respect to matched disturbances.

Introduction

Passivity-based controllers (PBC), which achieve stabilization shaping the energy function of the system, are widely popular for mechanical systems. It is well-known that PBC is robust with respect to parametric uncertainty and passive unmodeled dynamics (like friction), in the sense that stability–with respect to a shifted equilibrium–is preserved. However, very little is known about their robustness in the face of external disturbances, due to measurement or system noise. The main objective of this paper is to address this practically important issue for fully actuated fully damped mechanical systems whose energy function has an isolated minimum at the desired equilibrium, but are subject to external, matched and unmatched, disturbances.

As witnessed by the ubiquity of PI controllers, one of the most popular and natural approaches to robustify a controller design is to add an integral action on the signal to be regulated. If this signal turns out to be a passive output, stability is preserved in spite of the addition of the integral action. In this paper it is first shown that applying this procedure to mechanical systems, where the passive output is velocities, generates, even in the absence of disturbances, a set of equilibria and an invariant foliation in the extended state space, rendering asymptotic stability (practically) impossible.

Surprisingly enough, if the inertia matrix is constant the robustification problem has a very simple solution. Indeed, it is shown in the paper that adding a PI controller around the potential energy forces ensures the rejection of matched and unmatched constant disturbances. To quantify the robustness for time-varying disturbance we adopt the, by now standard, formalism of input-to-state stability (ISS), and the weaker property of integral ISS (IISS). (See  [1] for a recent survey on ISS and IISS properties.) More precisely, several controllers, with increasing complexity, that ensure these properties are proposed for mechanical systems. Finally, it is shown that including the partial change of coordinates proposed in  [2], we obtain a very simple controller that ensures ISS with respect to matched disturbances.

The remaining of the paper is organized as follows. In Section  2, we present the problem formulation. Section  3 contains the derivations for constant inertia systems, while the general case is treated in Section  4. The controllers in both cases are derived following the same procedure, but the ones obtained for constant inertia matrix are much simpler. Hence, the decision to split the material in this form is done to enhance readability. The new results using change of coordinates in momenta are presented in Section  5. Simulations of the controllers of Section  4, using a prismatic robot, are given in Section  6. Finally, we present some conclusions in Section  7.1

Notation. For xRn,SRn×n,S=S>0 we denote the Euclidean norm |x|2xx, and the weighted-norm xS2xSx. Given a function f:RnR we define the operators f(fx),2f(2fx2),xif(fxi), where xiRp is an element of the vector x. For a mapping g:RnRm, its Jacobian matrix is defined as g[(g1)(gm)], where gi:RnR is the i-th element of g.

Section snippets

Problem formulation

Throughout the paper we consider n-degrees of freedom, fully-actuated mechanical system described in port-Hamiltonian (pH) form by [q̇ṗ]=[0InInKp]H(q,p)+[0In]u+[d1d2] with Hamiltonian function H(q,p)=12pM1(q)p+V(q)q,pRn are generalized positions and momenta, respectively, and are assumed measurable, uRn is the control input, d1 and d2Rn are the matched and unmatched disturbances—possibly time-varying, but bounded and unmeasurable. The mass matrix M(q)=M(q)>0, and satisfies m1InM1(q)

Constant inertia matrix

In this section, the particular case of constant inertia matrix is considered. For this case, the problem of rejection of constant disturbances has a surprisingly simple solution: adding a PI control around the potential energy forces. However, to enforce the important property of ISS, damping must be added to all the coordinates, which is achieved incorporating suitable gyroscopic forces.

Non-constant inertia matrix

The derivation of the controller for non-constant inertia matrix M follows the same procedure used above. However, the expressions of the control laws become more complicated because of the need to differentiate M.

Throughout the section the following well-known identity is used q[pM1(q)p]=q[q̇M(q)q̇].

A simplified controller for matched disturbances

As discussed in Remark 8 the controllers for non-constant inertia matrix are highly complex. To overcome this practical shortcoming we follow  [2] and propose to change the generalized momentum coordinates to “remove” the inertia matrix from the energy function.2 Unfortunately, this modification achieves the desired objective only if there are no unmatched disturbances, i.e. if d1=0, an assumption that is made

Case study: prismatic robot

In this section, we use the two DoF prismatic robot4 example of  [9] to illustrate in simulations our results. Similarly to  [9], the initial condition vector is [q10,q20,p10,p20,z310,z320]=[0,0.1,0.1,0.1,0.1,0.2] and the desired equilibrium is the origin. The bounded disturbance vector is taken as d2=αtanh(q̇), with α=3,10. The parameters of the model are the same as in  [9], and are repeated here for ease of reference. The mass matrix is M=

Conclusions

In this paper, we have presented a control design that improves the robustness of energy shaping controllers for mechanical systems with external disturbances. Robustness is achieved with a dynamics state feedback that adds integral actions, as well as gyroscopic and damping forces. It should be underscored that no controllers carry out cancellation of nonlinearities, instead they inject the required forces to achieve the robustification objective.

The solution for mechanical systems with

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