C. Scherer, P. Gahinet, and M. Chilali, "Multiobjective output-feedback control via LMI optimization," IEEE Trans. Automat. Contr., vol. 42, no. 7, pp. 896--911, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....measures can be over # or #. Robust performance design questions concern the design of control laws that minimize the largest values of the performance measures over # or #. Design with multiple robust performance design constraints leads to the the so called multi objective design problem [5]. 2.1 Robust stability and performance analysis One approach towards answering question (P1) uses the notion of quadratic stability. A system is said to be quadratically stable if there exists a positive definite quadratic Lyapunov function V (#) # P # that decreases along every trajectory ....

....reformulation is known for the problem of even constant output feedback synthesis for even polytopic systems. It is worthy of note that a number of results are available for the LMI based synthesis of LTI controllers for LTI systems (i.e. a model with no uncertainties) a sampling is provided by [5, 14, 15, 16]. For robust control, gain scheduled controllers appear to hold promise: Designing gain scheduled output feedback controllers for polytopic systems using quadratic Lyapunov functions can be reduced to the solution of an optimization problem with a finite number of LMIs [10, 17] For LFR systems, ....

C. Scherer, P. Gahinet, and M. Chilali, "Multiobjective output-feedback control via LMI optimization, " IEEE Trans. Aut. Control, vol. 42, no. 7, pp. 896--911, July 1997.

....l 2 ( 0,N] IR ) # k =0 #z . 27) The control problem of discrete time nonlinear systems can be very di#cult due to the lack of geometric properties [14] We will show that for PWA systems this task turns out to be less impervious provided the use of some fundamental LMI techniques [16,6]. To begin with, we present some analysis results for the following closedloop system obtained by applying a feedback control law of type (16) to system (26) ij x k B ij x k D ij = A i B i K j , ij = C i D i K j and u k = K j x k . We observe again that the evolution of ....

Scherer, C. W., Gahinet, P., Chilali, M.: Multi-Objective Output-Feedback Control via LMI Optimization. IEEE Transactions on Automatic Control, 42(7), (1997), 896--911.

....some impressive designs. In recent years, it has been shown that when a state space description is available, then many of the infinite horizon costs and constraints can be represented as linear matrix inequalities (LMIs) and minimized exactly and efficiently as semidefinite programs (SDPs) see [4, 13] for a catalog of such constraints. So in this paper, we take the design procedure of [2] to the natural next level and formulate the multiobjective H2=H1 problem using LMIs for the objectives and constraints and solve it as an SDP. In this way, the errors due to cost, constraint and pulse ....

....case. The LMI formulation of the H2 and H1 costs [15, 4, 1, 7] introduces auxiliary Lyapunov matrices into the problem. As a result of product terms between these Lyapunov variables and the state space matrices of Q, the resulting constraints become nonlinear and hence, in general, nonconvex. In [13], convexity was recovered by a coordinate transformation of the controller variables, under the restriction that all the Lyapunov variables be equal. While this restriction makes the problem tractable in the LMI framework, it leads to conservatism in the overall design. There are as yet no results ....

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C. Scherer, P. Gahinet, M. Chilali,"Multiobjective OutputFeedback Control via LMI Optimization," IEEE TAC, vol.42, p.896, 1997.

....these inequalities affinely, the resulting search is convex and would yield the desired compensator, if one exists. The characterization presented here has several advantages. The first one is the possibility of adding other objectives, such as pole placements, to the search (as in Gahinet (1996) Chilali and Gahinet (1996), Chilali et al. 1996) and Scherer et al. 1996) Due to the similarity to the bounded real case, the details of such extensions are not presented here. The second benefit is the ability to remove some of assumptions on the 1 The corresponding author. Phone: 714)824 6433, Fax: 714) 824 8585, ....

....the resulting search is convex and would yield the desired compensator, if one exists. The characterization presented here has several advantages. The first one is the possibility of adding other objectives, such as pole placements, to the search (as in Gahinet (1996) Chilali and Gahinet (1996) Chilali et al. 1996) and Scherer et al. 1996) Due to the similarity to the bounded real case, the details of such extensions are not presented here. The second benefit is the ability to remove some of assumptions on the 1 The corresponding author. Phone: 714)824 6433, Fax: 714) 824 8585, fjabbari uci.edu 1 ....

[Article contains additional citation context not shown here]

Chilali, M., Gahinet, P. and Scherer, C., 1996, "Multi-Objective Output-Feedback Control via LMI Optimization", Proceedings of International Federation of Automatic Control, Vol D, pp.249-254.

....a result of this, they share the same Youla parametrisation (Vidyasagar, 1985) Techniques which go part way towards solving these types of constrained optimisation problems for various combinations of norms j Deltaj a ; j Deltaj b already exist. The Linear Matrix Inequality (LMI) techniques in (Scherer et al. 1997) give sufficient conditions for the solvability in F (s) of jW e (I Gamma FG f )j b fl b , jW e FG d j a fl a for H1 , H 2 and mixed norms and hence can be used to bound fl a (fl b ) above. Alternatively (Boyd and Barratt, 1991) since all constraint and objective transfer functions share the ....

....5. CONCLUSION We present a general framework for designing linear fault detection filters for exactly modelled LTI systems by using a threshold criterion which requires H 2 , H1 or mixed methods to optimise. These problems are amenable to recently formulated LMI optimisation techniques such as (Scherer et al. 1997) or Youla optimisation as in (Boyd and Barratt, 1991) These methods approximately maximise the gain ratio: the filter response to the (expected or worst case) fault with respect to the response to the (expected or worst case) disturbances. Alternatively, as in the example, if the expected values ....

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Scherer, C., P. Gahinet and M. Chilali (1997). `Multiobjective output-feedback control via lmi optimization '. IEEE Transactions on Automatic Control 42, 896--911.

....########## ###### # # # # ### # ####### ###### ### ###### # ### # # 0: 7) For reasons of space, we used the abbreviation sy(M) M M # and we suppressed the blocks # de ned by symmetry as well as the function arguments in (7) Sketch of Proof. The H# synthesis inequalities for (5) are [3,4] # # # # # # # ## # # # # ####### ## # ### # #### ## # ## # # ### ###### # # ###### #### # ## ### ##### # # # # ### # ### ###### ### ##### # # ### ###### # ### # # # # # # in the variables X , Y , K, L, M , N . Let us partition X and Y according to the state of (4) ....

....set of a xed linear matrix inequality, verifying the existence of F ##and R, S, K, L, M , N with (6) 7) amounts to solving a standard LMI problem. After having found a solution, the corresponding dynamic controller component G is obtained by inverting the controller parameter transformation of [3,4] for the parameters K = S ## # ) # K(R ## # ) # , L = R ## # L, M = MS ## # and N . # As indicated in the proof, Theorem 1 can be viewed as a combination of what has been discussed in [3,4] for purely dynamic controller design and in [6] for designing purely static ....

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C. Scherer, P. Gahinet, and M. Chilali, \Multiobjective output-feedback control via LMI optimization, " IEEE Trans. Autom. Control, vol. 42, pp. 896{ 911, 1997.

....is to keep the time domain amplitude of a tracking error signal bounded in the presence of disturbances and norm bounded uncertainties. Since these two requirements are con icting, it is interesting to investigate what are the achievable trade o s between them. Using the mixed objectiveapproach [1], we construct Pareto optimal curves for this control problem, in correspondence to di erentchoices of norms to represent the speci cations. In this way we gain a considerable insight on what is achievable in this design. The resulting controllers are digitally implemented on the real system and ....

....the constraints the designer can achieve different trade o s without modifying the weighting functions. Unfortunately, so far there are no synthesis algorithms that allowtosolve the multi objective problems in its full generality. In this paper we use the mixed objective approach proposed in [3] [1] that reduces the problem to the solution of an LMI system at the price of introducing an arti cial dependence among the different objectives. Our main interest is to investigate to which extent mixed objectives techniques are useful in enlightening the intrinsic relation between performance and ....

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C. W. Scherer, P. Gahinet and M. Chilali. Multiobjective output-feedbackcontrol via LMI optimization. #### ###### ###### #######, 42:896-911, 1997.

....several possible formulations of the design problem of robust static state feedback controllers via LMI s with different optimization criteria. To assure closed loop stability, the LMI constraints associated to the robust control design always guarantee that the following Lyapunov equation holds [16, 17]: A cl W WA 0 cl 0 W 0 (8) A key idea behind the LMI approach for robust control design is the use of a single Lyapunov matrix W for guaranteeing several different constraints (H 2 and H norms in different system channels, for instance) This allows the explicit inclusion of a set of ....

C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control, 42(7):896--911, 1997.

....for a stabilizer amounts to nd a X( 0 that solves the LMI A( X( B # ( Y ( X( A( # Y ( # B # ( # (5) over , and then computing F ( Y ( X( ## . A completely analogous procedure applies to the state feedback # # problem. We state the following result (see e.g. [12] for the nite dimensional versions) Proposition 1 (State Feedback) Given 0, there exists a feedback gain F ( that internally stabilizes the closed loop system (4) and satis es #Gzw # # ## if and only if for each # [# ; d , there exist X( X( # , Z( Z( # and Y ( ....

.... d # ## ; # d ##[#(#) ## # ## (8) where ) C # ( X( D ## ( Y ( A suitable feedback is given by F ( Y ( X( ## : Another result we will need is the following proposition on output feedback synthesis, which is a straightforward derivation from the nite dimensional version (see [12], for the nite dimensional version) Proposition 2 (Output Feedback) There exists a controller that renders #Gzw ### if and only if for each # [# ; d there exist X( Y ( K( L( M( andZ( that satisfy # #( # #( #( #( # # I # 0 # # ( #( T #( ....

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C. Scherer, P. Gahinet, and M. Chilali. \ Multiobjective output-feedback control via lmioptimization. " #### ############ ## ######### #### ####, 42:896-911, 1997.

....min Phi Tr (X Gamma Y Gamma1 Gamma V V T ) s.t. 18) 19) and (10) Psi ; 22) By similar arguments, various performance indexes can be handled such as H 2 norm performance, passivity constraints, and general quadratic constraints and their combinations. The reader is referred to [22] for a thorough discussion on these constraints. Again, in the case of full order control (i.e. k = n) the dynamic stabilization problem reduces to the feasibility of LMIs (18) 19) and (11) an easy convex problem. These results readily extend to reduced order control with guaranteed ....

....(i.e. k = n) the dynamic stabilization problem reduces to the feasibility of LMIs (18) 19) and (11) an easy convex problem. These results readily extend to reduced order control with guaranteed performance problems including H1 , H 2 syntheses and problems involving LMI region constraints. See [22, 16] for a catalog of possible extensions. Concave representations are also easily obtained for robust control problems involving many dioeerent classes of scalings or multipliers [3] 4 Reduction of LFT representations Many robust control techniques make use of an LFT representation of the system. ....

C. Scherer, P. Gahinet, and M. Chilali, Multi-Objective Output-Feedback Control via LMI Optimization, IEEE Trans. Aut. Control, 42 (1997), pp. 896 911.

.... in a highly efficient manner using interior point methods [21] Several generic SDP solvers are available, including the Matlab based SeDuMi package [32] Semidefinite programming techniques have been applied to efficiently solve a number of other engineering problems, including many in control [33, 34] and a few in signal processing [24, 35, 36] More generally, interior point methods have brought convex programmes in particular, linear and semidefinite programmes into the realm of (adaptive) signal processing [23, 24, 35 40] By solving Formulation 1 we obtain a feasible autocorrelation, ....

C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE Trans. Automat. Control, 42(7):896--911, Jul. 1997.

....result is the solution to the problem of performance oriented controller synthesis for T S models [11] In this result, the controller synthesis is again formulated as an LMI problem. Multiple design objectives can also be achieved by finding a feasible solution to an augmented LMI problem as in [16]. Throughout the paper, the notation M 0 will mean that M is positive definite symmetric matrix, and the notation L(A; P ) will denote the mapping from R n Thetan Theta R n Thetan to R n Thetan defined such that (A; P ) PA A T P T . The same holds for L(A T ; Q T ) AQ Q T ....

C. Scherer, P. Gahinet and M. Chilali, " Multiobjective Output-Feedback Control via LMI Optimization," IEEE Trans. on Automatic Control, vol. 42, no. 7, pp. 896-911, 1997.

....formulation of the controller synthesis problems as LMIs become much more straightforward. The second contribution in this paper is the performance oriented controller design for LPV systems. It is known that many performance specification can be written in the form of LMIs via Lyapunov methods [5]. Thus, by combining the novel controller structures and these LMIs, the performance specified controller synthesis can formulated as an LMI problem in a similar way as stabilizing controller design. Therefore, the synthesis framework in this paper can handle both the quadratically stabilizing ....

....time invariant controller matrices A c (p) B c (p) C c (p) and D c (p) which will stabilize the system (3) at the fixed value of p. The parameters of the unknown controller does not enter linearly into equation (6) so this equation does not represent an LMI condition. However, in the paper [5] the authors present a transformation procedure which results in a modified set of inequalities which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will first partition the constant matrices P and P Gamma1 into components. P ....

C. Scherer, P. Gahinet and M. Chilali, " Multiobjective Output-Feedback Control via LMI Optimization," IEEE Trans. on Automatic Control, vol.42, no. 7, pp. 896-911, 1997.

....parameterization, which is one of the three parameterizations presented in [2] The controller synthesis is formulated as an LMI problem. If multiple design objectives are required, we only need to group the LMI conditions together and find a feasible solution to this augmented LMI problem [1]. In this paper, a selective collection of performance objectives is considered, which include the general quadratic constraint, H 2 gain performance, constraint on system output or input. Derivation of most results will be omitted due to lack of space. In this paper, the notation M 0 stands ....

....same holds for L(A T ; Q T ) AQ T QA T . P GammaT is the same as (P Gamma1 ) T . 2 From Design Specifications to LMIs In this section, we consider the LMI formulations corresponding to different design specifications. Many results in this section have already been presented in [1] [5] However, conditions become only sufficient in this paper since we are dealing with nonlinear and or parameter varying systems. We consider the class of systems G : A cl (p) B cl (p) C cl (p) D cl (p) which can be described the equations x cl (t) A cl (p)x cl (t) B cl (p)w(t) z(t) ....

[Article contains additional citation context not shown here]

C. Scherer, P. Gahinet and M. Chilali, " Multiobjective Output-Feedback Control via LMI Optimization," IEEE Trans. on Automatic Control, vol.42, no. 7, pp. 896-911, 1997.

....of a set of linear, time invariant controller matrices A c (p) B c (p) C c (p) and D c (p) which will stabilize the system (2) at the fixed value of p. The unknown controller does not enter linearly into equation (5) so this equation does not represent LMI conditions. However, in the paper [4] the authors present a transformation procedure which results in a modified set of inequalities in which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will first partition the constant matrices P cl and P Gamma1 cl into ....

C. Scherer, P. Gahinet and M. Chilali, " Multiobjective Output-Feedback Control via LMI Optimization, " IEEE Trans. on Automatic Control, vol.42, no. 7, pp. 896-911, 1997.

....generalized H 2 performance, output and input constraints. The controller synthesis procedures are formulated as LMI problems. In the case of meeting multiple design objectives, we only need to group these LMI conditions together and find a feasible solution to the augmented LMI problem [10]. In this paper, the notation M 0 stands for a positive definite symmetric matrix. And L(A; P ) A T P T PA is defined as a mapping from R n Thetan Theta R n Thetan to R n Thetan . The same holds for L(A T ; Q) AQ T QA T . P GammaT is the same as (P Gamma1 ) T . ....

....of a set of linear, time invariant controller matrices A c (p) B c (p) C c (p) and D c (p) which will stabilize the system (4) at the fixed value of p. The unknown controller does not enter linearly into equation (7) so this equation does not represent an LMI condition. However, in the paper [10] the authors present a transformation procedure which results in a modified set of inequalities which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will first partition the constant matrices P and P Gamma1 into components. P = ....

[Article contains additional citation context not shown here]

C. Scherer, P. Gahinet and M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Trans. on Automatic Control 42(7):896-911 (1997).

.... decay, good damping, and reasonable controller dynamics can be imposed by con ning the poles in the intersection of a shifted half plane, a sector, and a disk [18, 1, 4, 5] Regional pole assignment has also been considered in conjunction with other design objectives such as H1 or H 2 performance [20, 8, 28, 9, 32]. Because real systems always involve some amount of uncertainty, it is natural to worry about the robustness of pole clustering, i.e. whether the poles remain in the prescribed region when the nominal model is perturbed. Such robustness issues have been thoroughly studied in the context of ....

....5 applies the results in Section 3 to the synthesis of output feedback controllers that robustly assign the closed loop poles in a given LMI region. This section also shows how to combine robust pole clustering with other synthesis objectives using the multi objective design framework developed in [26, 33, 32]. Finally, Section 6 demonstrates the e ectiveness of this approach on a physically motivated design example. 2 Background This section recalls the basics on LMI regions and some useful properties of Kronecker products. 2.1 Notation R and C denote the sets of real and complex numbers, ....

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C. Scherer, P. Gahinet, and M. Chilali. Multiobjective Output-Feedback Control via LMI Optimization. IEEE Trans. Aut. Contr., pages 896911, 42 (1997).

....or that it achieves certain desired loop shapes. All this is discussed in detail for the corresponding single objective control problems in the literature [1] Remark. The extension of our results to constraints with fixed bounds is straightforward and left out for notational simplicity. In [2, 3] it has been discussed how to compute an upper bound on fl that is achievable with controllers of the same McMillan degree as P . Moreover, 4, 5, 6] provide a detailed discussion of how to improve this upper bound by computing a non increasing sequence of upper bounds fl u of fl through ....

....computed upper bound is 60:23 and the best computed lower bound is 58:87; hence the optimal value has been determined with a relative accuracy of 2:3 . 5 Conclusions In the literature, several techniques have been proposed how to approach multi objective control problems for various specifications [14, 15, 5, 16, 17, 3, 18, 19, 4]. Based on the Youla parameterization, these problems are typically translated to infinite dimensional optimization problems. Lower and upper bounds on the optimal values are then determined by truncating both the primal and the dual problem and by proving convergence of the resulting scheme. If ....

C. Scherer, P. Gahinet, and M. Chilali, "Multi-objective output-feedback control via LMI optimization," IEEE Trans. Autom. Control, vol. 42, pp. 896--911, 1997.

....if A is stable. Stabilizing controllers exists iff (A; B) is stabilizable and (A; C) is detectable (with respect to the open unit disk) this is assumed from now on. A typical multi objective control problem imposes different specification on different channels of the closed loop system [1, 2, 3, 4, 5, 6, 7, 8, 9]. In this paper we concentrate on those requirements that can be formulated in terms of the solvability of a linear matrix inequality (LMI) For a pretty comprehensive list of possible choices we refer to [10, 6, 11] Only to simplify the exposition we confine ourselves to the discussion of a ....

.... specification on different channels of the closed loop system [1, 2, 3, 4, 5, 6, 7, 8, 9] In this paper we concentrate on those requirements that can be formulated in terms of the solvability of a linear matrix inequality (LMI) For a pretty comprehensive list of possible choices we refer to [10, 6, 11]. Only to simplify the exposition we confine ourselves to the discussion of a paradigm example that has received considerable attention in the literature [12, 13, 3, 4] the so called multi objective H 2 =H1 control problem. In this problem the goal is to keep bounds on the H 2 norm of, say, the ....

[Article contains additional citation context not shown here]

C. Scherer, P. Gahinet, and M. Chilali, "Multi-objective output-feedback control via LMI optimization," IEEE Trans. Autom. Control, vol. 42, pp. 896--911, 1997.

No context found.

C. Scherer, P. Gahinet, and M. Chilali, "Multiobjective output-feedback control via LMI optimization," IEEE Trans. Automat. Contr., vol. 42, no. 7, pp. 896--911, 1997.

No context found.

C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via LMI optimization. IEEE Trans. Auto. Contr., 42(7):896--911, 1997.

No context found.

C. Scherer, P. Gahinet, M. Chilali,"Multiobjective output-feedback control via LMI optimization", IEEE Transactions on Automatic Control, vol.42, no.7, pp.896-911, 1997

No context found.

C. W. Scherer, P. Gahinet, and M. Chilali. Multi-objective output-feedback control via LMI optimization. IEEE Trans. Aut. Control, 1995 (submitted).

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C.W. Scherer, P. Gahinet and M. Chilali. Multiobjective output-feedbackcontrol via LMI optimization. IEEE Trans. Automat. Contr., AC-42(7):896-- 911, 1997.

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C. Scherer, P. Gahinet and M. Chilali, " Multiobjective Output-Feedback Control via LMI Optimization," IEEE Trans. on Automatic Control, vol.42, no. 7, pp. 896-911, 1997.

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