- Research Article
- Open Access
New Dilated LMI Characterization for the Multiobjective Full-Order Dynamic Output Feedback Synthesis Problem
© J. Zrida and K. Dabboussi. 2010
Received: 23 April 2010
Accepted: 17 September 2010
Published: 21 September 2010
This paper introduces new dilated LMI conditions for continuous-time linear systems which not only characterize stability and performance specifications, but also, performance specifications. These new conditions offer, in addition to new analysis tools, synthesis procedures that have the advantages of keeping the controller parameters independent of the Lyapunov matrix and offering supplementary degrees of freedom. The impact of such advantages is great on the multiobjective full-order dynamic output feedback control problem as the obtained dilated LMI conditions always encompass the standard ones. It follows that much less conservatism is possible in comparison to the currently used standard LMI based synthesis procedures. A numerical simulation, based on an empirically abridged search procedure, is presented and shows the advantage of the proposed synthesis methods.
The impact of linear matrix inequalities on the systems community has been so great that it dramatically changed forever the usually utilized tools for analyzing and synthesizing control systems.The standard LMI conditions benefited greatly from breakthrough advances in convex optimization theory and offered new solutions to many analysis and synthesis problems [1–3]. When necessary and sufficient LMI conditions are not possible, as it is the case for the static output control [4, 5], the multi-objective control [6–8], or the robust control [9–12] problems, sufficient conditions were provided, but were known to be overly conservative. Some dilated versions of LMI conditions have first appeared in the literature in , wherein some robust dilated LMI conditions were proposed for some class of matrices. Since then, a flurry of results has been proposed in both the continuous-time [6, 7, 10, 14–17] and the discrete-time systems [5, 14, 18–20]. These conditions offer, though, no particular advantages for monoobjective and precisely known systems, but were found to greatly reduce conservatism in the multi-objective [6–8, 19] and the robust control problems [9, 10, 14–16, 18, 19]. In this respect, an interesting extension for the utilization of these dilated LMI conditions (as in, e.g., [21–23]) provided solutions to the problem of robust root-clustering analysis in some nonconnected regions with respect to polytopic and norm-bounded uncertainties. Generally, the main feature of these LMI conditions, in their dilated versions, consists in the introduction of an instrumental variable giving a suitable structure, from the synthesis viewpoint, in which the controller parameterization is completely independent from the Lyapunov matrix. A particular difficulty though with these proposed dilated versions in the continuous-time case is the absence of dilated conditions as it is visible in [6, 15].
This paper introduces new dilated LMIs conditions for the design of full-order dynamic output feedback controllers in continuous-time linear systems, which not only characterize stability and performance specifications, but also, performance specifications as well. Similarly to the existing dilated versions, these new dilated LMI conditions carry the same properties wherein an instrumental variable is introduced giving a suitable structure in which the controller parameterization is completely independent from the Lyapunov matrix. In addition, scalar parameters are also introduced, within these dilated LMI, to provide a supplementary degree of freedom whose impact is to further reduce, in a significant way, the conservatism in sufficient standard LMI conditions. It is also shown, in this paper, that the obtained dilated LMI conditions always encompass the standard ones. As a result, the conservatism which results whenever the standard LMI conditions are used is expected to considerably reduce in many cases. This paper focuses on the multi-objective full-order dynamic output feedback controller design in continuous-time linear systems for which the constraining necessity of using a single Lyapunov matrix to test all the objectives and all the channels, which constitutes a major source of conservatism, is no longer a necessity as a different Lyapunov matrix is separately searched for every objective and for every channel. It is shown, in this paper, that despite constraining the instrumental variable, the new dilated LMI conditions are, at worst, as good as the standard ones, and, generally, much less conservative than the standard LMI conditions. The proposed solution is quite interesting, despite an inevitable increase in the number of decision variables in the involved LMIs and a multivariable search procedure that could be abridged through empirical observations. A numerical simulation is presented and shows the advantage of the proposed synthesis method.
where the state vector , the perturbation vector , and the performance vector . All the matrices , , , and have appropriate dimensions. Let be the system transfer matrix from input to output . The following two lemmas are well known (see, e.g., [1, 3]) and provide necessary and sufficient conditions for System (2.1) to be asymptotically stable under an performance constraint and a performance constraint, respectively. These standard results will be extensively used in this paper.
3. Multiobjective Control Synthesis
The dynamic output feedback synthesis multi-objective problem consists of looking for a dynamic controller that stabilizes the closed loop system and, in the same time, achieves the desired and/or performance specifications for every single system channel. More specifically, the dynamic output feedback synthesis multi-objective problem aims at making System (3.1) possess the following propriety.
Theorem 3.1 (the standard sufficient conditions).
Theorem 3.1 provides sufficient conditions for the existence of a single multi-objective dynamic output controller in terms of LMI conditions in which common Lyapunov matrices are enforced for convexity. This is known to produce, in general, overly conservative results. The following theorem attempts at reducing the effect of this limitation.
Theorem 3.2 (the dilated sufficient conditions).
If there exist general matrices , , , , , and and for every channel ij, for some scalars and , there exist symmetric matrices , , , , , general matrices and such that either or both of the following two conditions are satisfied:
Clearly, as , there always exists a sufficiently large which satisfies this LMI. According to Lemmas 2.1 and 2.2, these are precisely the sufficient standard LMI conditions, expressed on a channel basis, for Propriety to hold.
Theorem 3.2 also provides sufficient conditions for the existence of a single multi-objective dynamic output controller in terms of LMI conditions in which the constraint of a common Lyapunov matrix is no longer needed. Convexity is rather insured by constraining the instrumental variables to be common. This is known to produce, in general, less conservative results than those obtained with the standard conditions of Theorem 3.1. Reducing further this conservatism is also possible through the positive scalar parameters and . A simple multidimensional search procedure can be carried out in the space of these parameters in order to obtain the values of these parameters for which LMI (3.19) and/or LMI (3.20) are feasible and produce the best multi-objective dynamic output controller with optimal performance levels. This multidimensional search procedure can, however, be overly expensive if the number of channel gets larger. A solution to this rather annoying limitation will be proposed in the next section. Yet, the important results of Theorem 3.2 constitute a significant contribution to the multi-objective control problem.
Next, the important question on whether or not the standard conditions could possibly be recovered by the dilated conditions will be addressed in the following section.
4. Recovery Condition
In the following theorem, it will be shown that our proposed dilated LMI conditions for the design of multiobjective full-order dynamic output feedback controllers do indeed encompass the standard conditions. This situation will be of great importance, as it will guarantee that conservatism will be almost always reduced. Similar results do exist in the literature in both the discrete-time  and the continuous-time case [6, 7]. The continuous-time results were, however, strictly concerned with the multi-channel synthesis problem and only in  that the recovery of the standard approach is proven. In view of this, the following theorem extends the discrete-time results to the continuous-time case. This point constitutes the major contribution of this paper.
For, the multi-objective dynamic output feedback synthesis problem, if the standard LMI conditions of Theorem 3.1 are satisfied and achieve, with a given controller, the upper bounds and , then the dilated inequality conditions of Theorem 3.2 are also satisfied with the same controller and with the upper bounds and .
there always exists an which achieves, simultaneously, these two conditions. As a result, the dilated inequality conditions of Theorem 3.2 are also satisfied. This proves that the dilated LMI multi-objective conditions always encompass the standard ones. Clearly, this means that the dilated-based approach yields upper bounds that are always and .
Theorem 4.1 has proven that the dilated LMI conditions of Theorem 3.2 do indeed encompass the standard ones of Theorem 3.1. The multidimensional search procedure carried out in the space of the scalars being exhaustive, up to a given discretization step that could be made as small as needed, does indeed cover every region, and in particular, the region where the standard conditions are recovered and which is defined by , where is greater than a minimum value defined by the two LMIs just in the proof above. In practice, the value of can be easily computed through a simple one dimensional line search procedure over these two LMIs.
On the other hand, at the light of the results of Theorem 3.2, a controller which achieves the best global performance level can be obtained through the minimization of the global objective function .Under this setting, it appears that optimality is always achieved very close to where all the and all the coincide. This purely empirical rule,observed with many examples we have tried, fits nicely to where the recovery of the standard conditions can be proved. In order to achieve optimality, it is then reasonable to abridge the costly multi-dimensional search procedure to a much cheaper one-dimensional search in the line for all channels. In this way, this proposed simple line search procedure not only provides a near optimal solution, but achieves the recovery condition which guarantees that this solution is, at least, as good as the one provided by the standard conditions.
5. An Example
The advantage of using the dilated rather than the standard LMI conditions is quite visible with this example. Indeed, around a 30% improvement on and a 25% improvement on performance levels were possible. However, this improvement comes at the expense of almost tripling the number of decision variables involved in the proposed dilated LMI conditions (see Table 1).
This paper has presented new dilated LMI conditions for the design of multiobjective full-order dynamic output controllers in continuous-time systems that are able to cope not only with stability analysis and performance specifications, but also, with performance specifications as well. The paper developed new controller synthesis procedures which offer no particular advantage for precisely known monoobjective systems, but significantly reduce conservatism in the multi-objective control problem, as the main property of these new dilated LMI conditions, besides the fact thattheyallow a complete independence between the standard Lyapunov matrix and the controller parametersis that they alwaysencompass the standard ones.A numerical simulation is presented which supports these claims. The extension of these results to other control issues such as the robust controller, model predictive controller, and filter design problems is rather straightforward and yet very useful.
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