Abstract
Although assignment and stabilization problems discussed in the SISO case can be formulated and solved for MIMO systems, this extension requires discussion of deeper algebraic-geometric concepts making the controller design less direct: machine computation and coding is ultimately required. In most practical situations, on the other hand, assigning precise values to the closed-loop eigenvalues is not strictly necessary, being sufficient to prescribe their membership to certain subsets of the complex plane (for example, the unit circle in the discrete-time case). For this reason in the MIMO case we prefer to reformulate the problem with the use of modern optimization tools [2] that are computationally very efficient and allow to address design aspects like constraints on input and output variables that would be impossible to deal with by assignment-based methods.
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- 1.
In case of continuous-time systems the same applies with the proviso that f is Lipschitz continuous at 0 and ℒ is continuously differentiable.
- 2.
In the continuous-time case methods similar to those discussed in the present and next sections can be employed but are not discussed for brevity.
- 3.
In recent releases this has been changed to conform to standard notation.
- 4.
For a thorough discussion of the correspondence between eigenvalue location in the complex plane and behavior of the trajectories, the reader is referred to the discussion of natural modes in basic linear system theory.
- 5.
Note that c cannot be chosen independently of r if we want to ensure that C(c, r) belongs to the asymptotic stability region of the complex plane.
- 6.
Actually G(z) > 0 implies that the set of the feasible z is open and a minimum for f(z) may not exist. However a highest lower bound inff(z) always exists.
- 7.
Without regional pole assignment, the minimization of | | u | | tends to be ill-conditioned as it displaces closed-loop eigenvalues towards the boundary of the stability region.
- 8.
See appendix.
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Caravani, P. (2013). MIMO, x Observed, w = 0. In: Modern Linear Control Design. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6943-8_4
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DOI: https://doi.org/10.1007/978-1-4614-6943-8_4
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