Control via interconnection and damping assignment of linear time-invariant systems: a tutorial

Abstract

Interconnection and damping assignment is a controller design methodology that regulates the behaviour of dynamical systems assigning a desired port-Hamiltonian structure to the closed-loop. A key step for the application of the method is the solution of the so-called matching equation that, in the case of nonlinear systems, is a partial differential equation. It has recently been shown that for linear systems the problem boils down to the solution of a linear matrix inequality that, moreover, is feasible if and only if the system is stabilisable – making the method universally applicable. It has also been shown that if we narrow the class of assignable structures – e.g. to mechanical instead of the larger port-Hamiltonian – the problem is still translated to a linear matrix inequality, but now stabilisability is not sufficient to ensure its feasibility. It is additionally required that the uncontrolled modes are simple and lie on the jω axis, which is consistent with the considered scenario of mechanical systems without friction. The purpose of this article is to present these important results in a tutorial, self-contained form – invoking only basic linear algebra methods.

Keywords:

• passivity-based control
• interconnection and damping assignment
• linear matrix inequalities
• stabilisation

Acknowledgements

The authors would like to thank Arjan van der Schaft for many useful clarifications regarding his work. This work is partially supported by the National Creative Research Groups Science Foundation of China (NCRGSFC: 60721062); National Basic Research Program of China (973 Program 2007CB714006).

Notes

Notes

1. In Fujimoto and Sugie (2001) a coordinate transformation is added to IDA to yield the so-called canonical transformations method. A fundamental difference between these two methods is that, in the latter it is proposed – in the spirit of standard PBC – to fix the energy function, transforming the matching equation into a nonlinear algebraic equation in the unknown interconnection and damping structures.

2. Throughout this article, uniformity with respect to x of the properties of all functions is assumed.

3. All sign (semi)definite matrices are assumed to be symmetric, hence this fact is omitted in the sequel.

4. As shown in Ortega et al. (2002), selecting J ₂ linear in p, the dependence with respect to p of the PDE (8) can be removed. See Chang (2010a) for other choices of J ₂ that provide additional degrees of freedom to solve the kinetic energy PDE.

5. The following definitions are needed to map our notation to the one used in Prajna et al. 2002): A → (J − R)Q, F → (J _d  − R _d ) ≕ S _d , P → Q _d  ≕ X ⁻¹, K → F.

6. This is tantamount to saying that the system (13) has no uncontrollable pole at s = 0.

7. The following definitions are needed to map our notation to the one used in Zenkov (2002): .

8. This fact can also be proven using that fact that B(B ^⊤ B)⁻¹B^⊤ is an orthogonal projector.