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New Dilated LMI Characterization for the Multiobjective Full-Order Dynamic Output Feedback Synthesis Problem
Journal of Inequalities and Applications volume 2010, Article number: 608374 (2010)
Abstract
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.
1. Introduction
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 [13], 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.
2. Background
Consider the linear time-invariant continuous-time and input-free system
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.
Lemma 2.1.
System (2.1) with 
is asymptotically stable and 
if and only if there exist symmetric matrices 
and 
such that
Lemma 2.2.
System (2.1) is asymptotically stable and 
if and only if there exists a symmetric matrix
in 
such that
3. Multiobjective Control Synthesis
Consider the continuous-time time-invariant linear system with input
where the state vector 
,the perturbation vector 
, the input command vector 
, the performance vector 
, and the controlled output vector 
, and all the matrices 
, 
, 
, 
, 
, 
, 
, and 
have the appropriate dimensions. In the dynamic output feedback case, the control law is given by the state equations
As this controller is supposed to be of a full order
, 
, 
, and 
. The closed-loop system is then described by the augmented state equations
where
The closed loop system transfer matrix from input 
to output 
then becomes
It is supposed that this system is of a multichannel type meaning that the perturbation vector 
is partitioned into a given number (say 
) of block components,
and the performance vector 
is partitioned into a given number (say 
) of block components,
It is supposed that some performance specifications are defined with respect to a particular channel 
(a path relating input component 
to output component 
) or a combination of channels. It is also supposed that, for a given control law strategy, these performance specifications can always be expressed in terms of an 
and/or a 
transfer matrix norm of the corresponding channel, namely, 
, where the matrices 
and 
are set to select the desired input/output channel from the system closed-loop transfer matrix 
. In fact, 
is a 
-block row matrix of dimension 
such that only the 
th block is nonzero and is the identity matrix in 
. Similarly, 
is an 
-block column vector of dimension 
such that only the 
th block is nonzero and is the identity matrix in 
. The problem of the multi-objective controller synthesis is to construct a controller that stabilizes the closed loop system and, simultaneously, achieves all the prescribed specifications. It is easy to see that, for each channel 
, the closed loop transfer matrix is given by
On the channel basis, the closed-loop system is then described by
where
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.
Propriety P
System (3.1) is stabilizable by a dynamic output feedback law (3.2) such that, for every channel 
, either or both of the following two conditions hold:
with 
;
.
Theorem 3.1 (the standard sufficient conditions).
If there exist symmetric matrices 
and 
, general matrices 
, 
, and 
and, for every channel ij, there exists a symmetric matrix 
such that either or both of the following two conditions are satisfied:
[StdH2]
[StdH∞]
then, Propriety 
holds, and furthermore, a set of the controller parameters defined in (3.2) is given by
where the nonsingular matrices 
and 
are obtained via the equation
Proof.
If either or both of conditions [StdH2] and [StdH∞] are satisfied, let 
and let 
be a nonsingular transformation matrix, with 
and 
selected from (3.14) (among infinitely many possibilities) via the singular value decomposition of 
. In view of (3.13) and (3.14), the following useful identities are easily derived:
As either or both of conditions [StdH2] and [StdH∞] are satisfied, by the congruence lemma applied to each LMI and in view of the identities listed just above, either or both of the following conditions are also satisfied, respectively,
-
(i)
(3.16)
-
(ii)
(3.17)
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 3.1 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:
(i)[DilH2]
(ii)[DilH∞]
Then, Propriety P holds, and furthermore, a set of the controller parameters defined in (3.2) is given by
where the nonsingular matrices 
and 
are obtained via the equation
Proof.
If either or both of conditions [DilH2] and [DilH∞] are satisfied, let 
and let 
be a nonsingular transformation matrix with 
and 
selected from (3.22) (among infinitely many possibilities) via the singular value decomposition of 
. In view of (3.21) and (3.22), the following useful identities are easily derived:
On the other hand, let us introduce
As either or both of conditions [DilH2] and [DilH∞] are satisfied, by the congruence Lemma applied to each LMI and in view of the identities listed just above, either or both of the following conditions are also satisfied, respectively.
-
(i)
(3.25)
-
(ii)
(3.26)
To summarize, we have proven that if either or both conditions [DilH2] and [DilH∞] are satisfied, then either or both of the following conditions are also satisfied:
-
(i)
(3.27)
-
(ii)
(3.28)
The third LMI of the first item condition is equivalent to
which, according to the elimination lemma [3], leads to
The two previous LMIs are equivalent to 
and 
, that is, for any 
, 
.
Similarly, the LMI of the second item condition is equivalent to
According to the Elimination lemma, this leads to
The previous two matrix inequalities are equivalent to
Via the Schur lemma, the latter inequality is equivalent to 
and
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 [19] and the continuous-time case [6, 7]. The continuous-time results were, however, strictly concerned with the multi-channel 
synthesis problem and only in [7] 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.
Theorem 4.1.
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 
.
Proof.
If the standard LMI conditions of Theorem 3.1 are satisfied for a given controller and achieve, for every channel, the upper bounds 
and 
, then there exist symmetric matrices 
and 
such that
and/or
Let us prove that these standard LMI conditions imply that the dilated inequality conditions of Theorem 3.2 are also satisfied with the same controller. When expressed in terms of the system closed-loop parameters, the right-hand sides of the dilated LMI conditions of Theorem 3.2 take the following form:
and/or
Let, in these matrices, 
, 
, 
, 
, 
and 
, these right-hand sides become
and/or
Let us prove, for these four matrices above, that the second matrix is positive definite while the third and/or the fourth matrices are both negative definite. Clearly, the standard conditions imply that
By virtue of the Schur complement lemma, the third matrix and/or the fourth matrix will be negative definite if and only if 
,
and/or
As, from the standard 
and 
conditions,
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
Consider the LTI unstable third-order plant
The system is supposed to be consisting of two channels. Channel 1 connects the perturbation vector 
to the performance component 
, while Channel 2 connects the perturbation vector 
to the performance component 
. The objective here is to find a stabilizing full-order (i.e., a third order) dynamic output feedback controller which achieves simultaneously and optimally the performance specifications 
and 
, relatively to Channel 2 and Channel 1, respectively. Optimality is here defined as the minimization of 
, giving equal importance to the two channels. The use of the dilated LMI conditions can be carried out through a search procedure in the plane 
. Figure 1 is a three-dimensional plot which depicts the waveform of 
in that plane. This figure clearly shows that optimality is achieved close to the direction where 
. In this example, it is found that the minimum value of 
which guarantees recovery is 
. The abridged search procedure along the line 
produced a near optimal global performance of 
and 
when 
. Clearly, in this example,improvement is being made in the region below 
where recovery is not necessarily there. Table 1 lists the simulation results obtained with the sufficient standard LMI conditions of Theorem 3.1 and with the sufficient dilated LMI conditions of Theorem 3.2.
representing the LMI produced full-order dynamic output feedback controller.
3D-plot of the waveform 
in the plane 
.
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).
6. Conclusion
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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Zrida, J., Dabboussi, K. New Dilated LMI Characterization for the Multiobjective Full-Order Dynamic Output Feedback Synthesis Problem. J Inequal Appl 2010, 608374 (2010). https://doi.org/10.1155/2010/608374
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DOI: https://doi.org/10.1155/2010/608374