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Continuous-time analysis, eigenstructure assignment, and \(H_2\) synthesis with enhanced linear matrix inequalities (LMI) characterizations. (English) Zbl 1003.93016

Since the mid 1990s, optimization over linear matrix inequalities (LMI), also called semidefinite programming in the mathematical programming terminology, has become a key ingredient for control systems analysis and design. Indeed, a wide range of control problems can be formulated in a unified fashion as LMI problems. Unlike purely algebraic approaches based on Riccati equations and the like, the LMI approach is flexible enough to allow for multiple closed-loop performance and robustness specifications, ubiquitous in modern control. Moreover, several efficient and reliable LMI solvers (mostly based on interior-point methods and Nesterov and Nemirovski’s theory of self-concordant barrier functions) are now available.
Most LMI control methods are based on Lyapunov’s rationale of seeking a positive quadratic form of the system state whose derivate is negative along the system trajectories. Analysis or design problems can then be formulated as non-linear matrix inequality problems involving products of (given) system open-loop matrices, (sought) controller matrices and (sought) Lyapunov matrices. In some special cases (state-feedback design or output-feedback design when the controller order is equal to the open-loop system order) an adequate change of variable can be used to reformulate the (generally non-convex and hard to solve) non-linear matrix equality problem as a (convex and tractable) LMI problem. Performance and robustness specifications are then ensured with a unique quadratic Lyapunov function valid over the whole system operation range and shared by each performance channel. Seeking a unique quadratic Lyapunov function is at the core of the whole approach, because it ensures convexity of the matrix inequality problem. On the other hand, the uniqueness restriction can prove overly conservative, i.e. there may exist a controller ensuring all the specifications, but the LMI approach is unable to find it just because the corresponding Lyapunov function necessarily varies with the system operating range or the performance channel.
In order to overcome this drawback, researchers are constantly trying to replace the unique Lyapunov function with multiple, or varying, or parametrized Lyapunov functions. So far most of the achievements in this domain have been robustness analysis results for systems affected by parametric uncertainty, producing parameter-dependent Lyapunov functions at the price of solving an LMI of increased dimension. Following the seminal paper [M. C. de Oliveira, J. Bernussou and J. C. Geromel, Syst. Control Lett. 37, 261-265 (1999; Zbl 0948.93058)], several researchers realized that some of these analysis results can also be extended to design. The underlying idea is in decoupling the matrix product of the system open-loop matrices, controller matrices and Lyapunov matrices, with the help of additional slack variables appearing somewhat artificially in the LMI. Due to the particular matrix structure of the problem, it was however very soon realized that the trick can be applied only to discrete-time systems so as to outperform systematically the original unique Lyapunov function LMI design framework.
The paper under review proposes a partial extension of these results to continuous-time systems. Using some standard matrix tricks, several equivalent continuous-time LMI conditions are proposed to ensure stability or performance. It is then shown that the degrees of freedom resulting from the additional slack variables can be exploited to solve some specific control problems with the help of multiple Lyapunov matrices. The extension is partial because, as mentioned by the authors themselves just after Remark 4.1, for robustness requirements it is not proved that the proposed continuous-time LMI conditions are systematically less conservative than the original unique Lyapunov function LMI conditions (contrary to their discrete-time counterparts). However, the particular structure inherent to the proposed LMI conditions generally provides more flexibility in the design procedure, as shown in section V in the case of eigenstructure assignment or in section VI in the case of multichannel H2 output-feedback design.

MSC:

93B50 Synthesis problems
93D09 Robust stability
93B60 Eigenvalue problems
90C22 Semidefinite programming
15A39 Linear inequalities of matrices
93B51 Design techniques (robust design, computer-aided design, etc.)

Citations:

Zbl 0948.93058
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