Robust pole placement in LMI regions

Abstract—This paper discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality (LMI) regions, a class of convex regions of the complex plane that embraces most practically useful stability regions. The focus is on linear systems with static uncertainty on the state matrix. For this class of uncertain systems, the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary LMI regions. The resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities LMI’s and cover both unstructured and parameter uncertainty. These analysis results are then applied to the synthesis of dynamic outputfeedback controllers that robustly assign the closedloop poles in a prescribed LMI region. With some conservatism, this problem is again tractable via LMI optimization. In addition, robust pole placement can be combined with other control objectives, such as H2 or H1 performance, to capture realistic sets of design specifications. Physically motivated examples demonstrate the effectiveness of this robust pole clustering technique.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 2257 Robust Pole Placement in LMI Regions Mahmoud Chilali, Pascal Gahinet, and Pierre Apkarian, Associate Member, IEEE Abstract— This paper discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality (LMI) regions, a class of convex regions of the complex plane that embraces most practically useful stability regions The focus is on linear systems with static uncertainty on the state matrix For this class of uncertain systems, the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary LMI regions The resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities LMI’s and cover both unstructured and parameter uncertainty These analysis results are then applied to the synthesis of dynamic output-feedback controllers that robustly assign the closed-loop poles in a prescribed LMI region With some conservatism, this problem is again tractable via LMI optimization In addition, robust pole placement can be combined with other performance, to capture control objectives, such as or realistic sets of design specifications Physically motivated examples demonstrate the effectiveness of this robust pole clustering technique H H1 I INTRODUCTION S TABILITY is a minimum requirement for control systems In most practical situations, however, a good controller should also deliver sufficiently fast and well-damped time responses A customary way to guarantee satisfactory transients is to place the closed-loop poles in a suitable region of the complex plane We refer to this technique as regional pole placement, by contrast with pointwise pole placement, where the poles are assigned to specific locations in the complex plane For example, fast decay, good damping, and reasonable controller dynamics can be imposed by confining 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 or performance [20], [8], [28], [9], [32] as 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 pointwise pole placement [23], [22], [25] In comparison, few results are available on robust regional pole clustering These results include a Lyapunov approach to compute explicit robustness bounds for pole clustering in a disk [10] and extensions of the Manuscript received August 20, 1997; revised August 20, 1998 Recommended by Associate Editor, M Dahleh M Chilali was with INRIA Rocquencourt, 78153 Le Chesnay Cedex, France P Gahinet is with The MathWorks, Inc., Natick, MA 01760 USA (e-mail: pascal@mathworks.com) P Apkarian is with CERT-ONERA, 31055 Toulouse Cedex, France (e-mail: apkarian@cert.fr) Publisher Item Identifier S 0018-9286(99)09613-0 notion of quadratic stability to robust pole placement in a disk or a sector [3], [16], [15] This paper extends these results to more general clustering regions and to structured uncertainty The regions considered here are the linear matrix inequality (LMI) regions introduced in [9] This class of regions covers a large variety of useful clustering regions, including half-planes, disks, sectors, vertical/horizontal strips, and any intersection thereof The following analysis and synthesis problems are addressed: • robustness of pole clustering within a given LMI region in the face of unstructured or parameter uncertainty in the state matrix; • synthesis of output-feedback controllers that robustly assign the closed-loop poles in some arbitrary LMI region (assuming static and unstructured uncertainty on the plant matrices) With some conservatism, these problems are reduced to solving LMI’s Because LMI’s can be solved numerically using efficient optimization algorithms, such as those described in [29], [30], [6], and [35], or implemented in [14] and [2], our approach yields practical analysis and synthesis tools for robust regional pole placement See [7] for an overview of the applications of LMI techniques in control theory This paper is organized as follows Section II recalls the definition of LMI regions and key results on pole clustering in LMI regions Section III contains the main result, a generalization of the Bounded Real Lemma to arbitrary LMI regions This result gives a sufficient condition in terms of LMI’s for robust pole clustering within a given LMI region Section IV shows how some standard robustness analysis tests for parameter uncertainty can be generalized to LMI regions and illustrates the performance of the resulting robust pole clustering tests on a realistic example Section V applies the results in Section III 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 VI demonstrates the effectiveness of this approach on a physically motivated design example II BACKGROUND This section recalls the basics of LMI regions and some useful properties of Kronecker products A Notation and denote the sets of real and complex numbers, restands for the open left half-plane spectively The notation 0018–9286/99$10.00  1999 IEEE 2258 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 For a complex matrix and transpose of denotes the Hermitian is defined as For Hermitian matrices, means is positive means is positive semidefinite definite and In symmetric block matrices, we use as an ellipsis for terms induced by symmetry, e.g., Finally, we use the shorthand • conic sector with apex at the origin and inner angle : Key facts about LMI regions include [9] the following • Intersections of LMI regions are LMI regions • Any convex region symmetric with respect to the real axis can be approximated by an LMI region to any desired accuracy is -stable, i.e., has all its eigenvalues • A real matrix in the LMI region , if and only if a symmetric matrix exists such that (2) This result can be seen as a generalization of the Lyapunov theorem because for the usual stability region , (2) reduces to B Kronecker Products The Kronecker product is an important tool for the subsequent analysis Recall the Kronecker product of two matrices and is a block matrix with generic block entry , that is, The following properties of the Kronecker product are easily established [17]: Pole clustering in LMI regions can be formulated as an LMI optimization problem, a convex semidefinite program that is easily tractable with recently available interior-point techniques Moreover, it is possible to combine such pole clustering specifications with other design objectives while preserving tractability [9], [32] III ROBUSTNESS OF POLE CLUSTERING IN LMI REGIONS The eigenvalues of are the pairwise products of the eigenvalues of and As a result, the Kronecker product of two positive-definite matrices is a positive-definite matrix Finally, the singular values of consist of all pairwise products of singular values and of The notions of robust and quadratic stability are useful tools for analyzing the stability of uncertain state-space models [7], [24] These notions are now generalized to pole clustering in arbitrary LMI regions, and a counterpart of the Bounded Real Lemma is derived for LMI regions Although our analysis is restricted to static (real or complex) uncertainty, its implications for more general classes of uncertainty (dynamic or time-varying) are briefly discussed at the end of the section A Robust and Quadratic -Stability Consider the uncertain linear system C LMI Regions An LMI region is any subset can be defined as of the complex plane that (3) (1) depends fractionally on the normwhere the state matrix bounded uncertainty matrix and are real matrices such that where matrix-valued function The is called the characteristic function of Below are a few examples of LMI regions: : ; • half-plane with radius : • disk centered at (4) The value corresponds to the nominal with and the parameter defines the level of state matrix uncertainty Although the uncertain model (3) is physically meaningful only for real uncertainty , we also consider the complex case because of its connection with dynamic uncertainty (see Section III-D) Let (5) CHILALI et al.: POLE PLACEMENT IN LMI REGIONS 2259 be is of , any LMI region, and suppose the nominal state matrix -stable, i.e., has all its eigenvalues in The question interest here is as follows: Given some uncertainty level the poles of remain in for all satisfying ? Definition 3.1 (Robust -Stability): The uncertain system lie (3)–(4) is robustly -stable if the eigenvalues of for all admissible uncertainties in Similar definitions can be found in [7] and [24] for the usual stability region and in [4] for uncertain polynomials Nonconservative assessment of robust -stability is difficult, for the except in special cases, e.g., complex unstructured open left half-plane Although conservative, the following notion of quadratic -stability proves more practical for analysis and synthesis purposes Definition 3.2 (Quadratic -Stability): Given any LMI region defined by (5), the uncertain system (3), (4) is said to be quadratically -stable if a real symmetric matrix exists such that We are now ready to state the main result, a sufficient LMI-based condition for quadratic -stability Theorem 3.3: The system (3) with uncertainty is quadratically -stable if matrices exist such that and (10) (11) with the notation (6) such that for all complex matrices is -stable if and only if Recall from Section II-C that exists such that Hence, quadratic -stability implies robust -stability, but the converse is generally false because quadratic -stability requires a single satisfies for all admissible ’s The assumption “ real” incurs no loss of generality and is motivated by the tractability of the synthesis problem discussed in Section V is the open left half-plane, it is well known that When is equivalent to quadratic starobust stability for complex bility for real or complex [24], which in turn is completely characterized by the Bounded Real Lemma: The uncertain system (3), (4) is quadratically stable if and only if exists such that Proof: See Appendix A The inequalities (10), (11) are LMI’s with unknown matrices and Hence, testing this sufficient condition numerically can be tackled efficiently with LMI solvers The matrix plays the role of Lyapunov matrix, and can be viewed as a scaling matrix that accounts for the block-diagonal structure in the relation (see proof in Appendix of A) The variable also accounts for the nonuniqueness of the Specifically, replacing by factorization is equivalent to replacing by The size of is typically small because for most useful LMI in (8) has rank less than three regions, the matrix It is insightful to explicitate the LMI (10) for well-known regions, such as the left half-plane and the disk and • The open left half-plane corresponds to (i.e., ) Taking , (10) reduces to (7) Using a bilinear shift [8], this result remains true for vertical half-planes and disks centered on the real axis Next, the Bounded Real Lemma condition for quadratic stability can actually be generalized to arbitrary LMI regions B Main Result with characteristic function Given an LMI region which coincides with the Bounded Real Lemma inequaland redefining ity (7) up to dividing by the scalar as with center and radius corre• The disk sponds to (8) factorize the matrix as (9) have full column rank (such a factorization is where has rank , both easily obtained from the SVD of ) If and are matrices Because take has rank one, is again scalar and we can without loss of generality The LMI constraint 2260 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 (10) then reads as ————— ————- ——- ——- ——- ————— ————- ——- ——- ——- By a Schur complement with respect to the block (1, 1), this is equivalent to which is simply the discrete-time version of the Bounded This Real Lemma applied to the system has all its eigenvalues result stems from the fact that if and only if is stable in the in discrete-time sense, i.e., has all its eigenvalues in the unit disk C Intersections of LMI Regions In practical applications, LMI regions are often specified as the intersection of elementary regions, such as conic sectors, , disks, or vertical half-planes Given LMI regions the intersection has characteristic function If quadratic -stability is of interest, Theorem 3.3 should When be applied to the overall characteristic function robust -stability is the primary concern, however, it is more efficient and less conservative to test quadratic stability for independently Indeed, this process each elementary region , guarantees robust stability with respect to each region which in turn establishes robust -stability elementary More specifically, if is the intersection of LMI regions with characteristic functions a sufficient condition for robust bounded uncertainty -stability against norm- is the existence, for each region such that , of a pair of matrices (12) Fig Robustness analysis interconnection The LMI feasibility problems (12) should be because no coupling solved independently for each region exists between the constraints for each region By contrast, amounts to jointly applying Theorem 3.3 directly to and solving all LMI’s (12) with as variables This method is clearly more costly and more conservative because of the requirement that satisfy (12) for all regions a single D Comments on Quadratic -Stability Theorem 3.3 gives a sufficient condition for quadratic -stability in the face of complex and unstructured uncermust be real tainty As mentioned earlier, the uncertainty for the uncertain model (13) is the open left halfto be physically meaningful When plane and robust stability is of interest, the quadratic stability test is known to be conservative for real uncertainty It is therefore legitimate to question the value of Theorem 3.3 as a tool for assessing robust -stability While acknowledging conservatism for this particular uncertainty model, we now briefly review other benefits of quadratic -stability that strengthen its practical appeal Rewrite (13) as (14) Then, is simply the and let closed-loop matrix for the feedback loop of Fig 1, and robust -stability is therefore equivalent to requiring the closed-loop satisfying poles remain in for all is the open left half-plane, equivalence exists When between [24]: • quadratic stability; ; • robust stability against complex with where ; • • robust stability against stable dynamic uncertainty satisfying ; • feasibility of the Bounded Real Lemma LMI (7) Similar connections between quadratic -stability, robust -stability against dynamic uncertainty, and Theorem 3.3 can be established for general LMI regions Specifically, for analytic in (i.e., -stable), define the norm with CHILALI et al.: POLE PLACEMENT IN LMI REGIONS respect to 2261 as Proof: Multiplying (15) left and right by , respectively, we get for all Straightforward adaptations of the small gain and generalized Nyquist theorems [25] lead to the following results Theorem 3.4: The following properties are equivalent is robustly -stable for static complex uncertainty • satisfying • • The closed-loop system is robustly -stable against dynamic uncertainties that are -stable and satisfy has no poles on the boundary of and • If , the nominal poles of remain in More precisely, the number of poles of in is always equal to the number of nominal in poles (all in ) plus the number of poles of These results indicate quadratic -stability (and the related test in Theorem 3.3) also provides some robustness against dynamic uncertainty, which is desirable in practice Also, for general regions, -stability is difficult to handle numerically , the boundary of as it requires an exhaustive sweep of In this respect, quadratic -stability provides tractable, though possibly conservative, means for checking robust -stability E Time-Varying Uncertainties The proof of Theorem 3.3 remains valid when the uncertainty is time varying; i.e., Although the notion of “pole” disappears for linear timevarying systems, the generalized Bounded Real Lemma of Theorem 3.3 still provides the following guarantees: at all time ; • -stability of the matrix is con• exponential decay of the transients whenever with tained in some stable half-plane The second property is a consequence of the following lemma Lemma 3.5: Consider an LMI region with characteristic and suppose the dynamical function system Recalling ing by and , and divid, this process leads to which, by definition of , ensures This lemma shows, for quadratic -stable time-varying systems, stability and decay rate are essentially determined by It says time-invariant considerations, i.e., whether little more, however, about transient behaviors Can we also expect well-damped responses by choosing an adequate conic sector? Does a disk prevent fast dynamics? Such extensions to the time-varying case remain open for future research IV PARAMETER UNCERTAINTY This section discusses refinements of the previous robustness analysis results when the uncertainty is structured The main motivation is the assessment of robust -stability in the face of parameter uncertainties As is usual when dealing with structured uncertainty, the resulting tests are only sufficient conditions for robust pole clustering in a given LMI region Our analysis technique relies on the use of similar to the parametera parameter-dependent matrix dependent Lyapunov functions used in [19], [13], [11] for regular robust stability analysis Such approaches have proven to be significantly less conservative than quadratic stability for time-invariant parameter uncertainty The analysis below deals with the same basic uncertain is real and structured; i.e., model (3), but now assumes (16) where the ’s denote the (normalized) uncertain parameters the hypercube, in which We denote by ranges according to (16), and by the set of vertices of this hypercube; that is, To stress the dependence on the parameter vector uncertain state matrix is written as , the (17) is quadratically -stable, i.e., exists such that The relevant dimensionality parameters are defined by (15) for all time Then, the quadratic function satisfies, for all (18) For such parameter uncertainty, robust pole clustering in the LMI region (19) 2262 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 is equivalent to the existence of symmetric matrices parametrized by such that (20) To enforce tractability of (20), we restrict the search of to matrices with affine dependence on functions where Proof: See Appendix B This theorem provides a test for robust -stability that involves solving a finite set of LMI’s and is therefore tractable Applications to some aeronautics systems suggest it can be sharp In its most general form, this test can be computationally demanding for high-order systems with multiple uncertainties With additional conservatism, the computational cost can be reduced as follows • Use symmetric and skew-symmetric scalings in place of and the general and unconstrained scalings and and Specifically, impose consistent with the uncertainty make the structure of structure (23) Two robust -stability tests are derived next using such ’s The first test applies to general linear-fractional affine on whereas the second test is restricted dependence of These results are strongly to affine dependence related to the general integral quadratic constraint framework developed by Megretski and Rantzer in [27, pp 825–826] For simplicity, our results are stated for a single Lyapunov matrix regardless of the complexity of the LMI region For LMI regions that are intersections of elementary LMI regions , sharper tests can be obtained by using independent Lyapunov for each as indicated in Section III-C matrices A General Parameter Dependence Theorem 4.1: Given the parameter uncertainty specified in (19), a full-rank factorization by (16), the LMI region of , and the notation (18), the uncertain matrix in (17) is robustly -stable if the following matrices and (with exist: symmetric matrices ), scaling matrices , and (all such that in That is, require matrices of the form (23) • Set some of the matrices and for all to zero B Affine Parameter Dependence in (17), the uncertain state matrix depends When affinely on the parameters : (24) For such uncertain systems, the following robust -stability test is easily derived using the multiconvexity technique develis multiconvex oped in [13] Recall a function when it is convex with respect to each of its variables separately For differentiable functions, this property is equivalent to requiring the Hessian of has nonnegative diagonal entries Theorem 4.2: Given the parameter uncertainty specified by (16), the uncertain system with state matrix (24) is robustly -stable if symmetric matrices and scalars exist such that (25) (26) (27) ————- ————- ——- ————- ————- ——- and, for all vertices —- — —- hold at all vertices of and for , with (21) Proof: Condition (26) ensures, for any , the quadratic function of (22) is multiconvex in the ’s Using the same argument as in [13], (25) holds over the entire hypercube if it holds at the vertices of the uncertainty hypercube CHILALI et al.: POLE PLACEMENT IN LMI REGIONS 2263 TABLE I VARIABLE DESCRIPTION C Robust Analysis Application The analysis techniques developed in this section are applied to a realistic missile example (see [34] for additional details and insights) The purpose is to determine admissible uncertainty levels for which stability and adequate damping are preserved The dynamics of the controlled missile roll axis are described by Fig Closed-loop poles of A( ) + B ( )KC for some parameter values ( ;  ) 1; 0:5; 0; 0:5; f0 g where and the meaning of the different variables is given in Table I The output-feedback gain matrix is given and has been designed using eigenspace techniques The parameters and represent uncertainties whose effects on the missile dynamics are reflected in the matrices and Numerical values for these matrices can be found in Appendix C The objective is to estimate, in the parameter space , the largest square where closed-loop stability is maintained, and the largest square where closed-loop damping for the missile roll axis The is adequate, that is, uncertain parameters and enter affinely in the statespace matrices, so the techniques of Theorems 4.1 and 4.2 are both applicable The closed-loop pole locations for parameter values in the set are plotted in Fig Clearly, both stability and damping constraints are violated for some pairs in this uncertainty set The shaded area in Fig shows the region in the parameter where closed-loop stability is maintained This space area has been computed using an exhaustive search over a fine grid in the parameter space Based on the results of Theorem 4.1, we estimate the largest stability square using either fixed matrices As expected, a fixed or parameter-dependent leads to a conservative answer (dashed square in Fig 3) In contrast, using a parameter-dependent provides a sharp estimate of the largest allowable uncertainty (solid square) Similarly, Fig shows the uncertainty region where adeis maintained, and the dashed and solid quate damping lines delimit the estimated safe regions using Theorem 4.2 Fig Stability region estimates with fixed (dotted) and parameter-dependent (solid) X matrices Fig Adequate damping region estimates with fixed (dotted) and parameter-dependent (solid) X matrices with fixed or parameter-dependent matrices The damping is captured by the conic LMI region constraint Again, the estimate based on parameter-dependent provides a sharp answer matrices V OUTPUT-FEEDBACK SYNTHESIS This last section shows how to use our main analysis result (Theorem 3.3) for synthesis purposes Specifically, we 2264 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 consider the problem of computing an output-feedback controller that robustly assigns the closed-loop poles in a prescribed LMI region For tractability reasons, the discussion is restricted to unstructured uncertainty The problem statement is as follows Consider the uncertain state-space model Theorem 5.1: A full-order output-feedback controller and a matrix exist such that (30) holds if and only if symmetric matrices and and matrices two and exists such that (31) and (32), shown at the bottom of the page, where (28) where and the static uncertainty Given the LMI region satisfies we are interested in designing a full-order dynamic controller (29) that robustly assigns the closed-loop poles in because this Without loss of generality, assume amounts to a mere change of variable in the controller matrices and considerably simplifies the formulas The closed-loop matrix is If these LMI’s are feasible, an th-order controller that rois bustly assigns the closed-loop poles in where the matrices are derived as follows and such that • Compute any square matrices where • Solve the following linear equations for : , and (33) From Theorem 3.3 with , a sufficient condition for is the existence quadratic (hence, robust) -stability of such that of (30) is a full-rank factorization of This where matrix inequality is not jointly convex in and the controller matrices It can be reduced, however, to a convex LMI problem by using the linearizing change of controller variables introduced in [26], [33], [9] This change leads to the following synthesis result Proof: The proof involves the changes of variable introduced in [26], [33], and [9] and is omitted for brevity Inequalities (31) and (32) are LMI’s in the variables and that can be solved numerically using LMI optimization software [14] Theorem 5.1 therefore provides a tractable (but somewhat conservative) approach to robust pole assignment in LMI regions is the intersection of several eleRemark 5.2: When as discussed in Section III-C, the mentary LMI regions synthesis LMI’s (31), (32) must be written for each region variables, and the resulting set of LMI’s using the same must be solved jointly Indeed, the synthesis problem is no (32) CHILALI et al.: POLE PLACEMENT IN LMI REGIONS 2265 longer convex when a different is used for each (this prevents using the linearizing change of variable) The extra conservatism introduced by this additional restriction is modest in most applications The linearized longitudinal dynamics of the missile are described by A Mixed Design Specifications From a practical viewpoint, enforcing quadratic -stability is rarely sufficient because most design problems are essentially multi-objective For instance, realistic designs are likely or (loop shaping) objectives in addition to to include robust pole assignment for transient tuning Fortunately, LMIbased synthesis can accomodate a rich variety of closed-loop specifications within a single LMI optimization problem, as shown in [32] This problem is achieved with some conservatism, but has proven effective in a number of applications The basic requirement is that a single closed-loop Lyapunov should account for all design function specifications As an immediate extension of the results in [26], [33], and [32], it is easy to mix quadratic -stability with other objecor performance, passivity constraints, tives, such as bounds on the impulse response, etc As an example, we can combine a quadratic -stability objective as captured by -norm bound on some input/output Theorem 5.1 with an channels of the closed-loop system For instance, if the nominal plant is described by the (nominal) closed-loop transfer function can be further constrained to from to by combining the LMI conditions (31) and (32) for robust pole assignment with the additional LMI constraint, shown in (34) at the bottom of the page where and denote the angle of attack, pitch rate, vertical accelerometer measurement, and fin deflection, and are auxiliary signals respectively The variables used to model variations of the aerodynamic coefficients for ranging between and 20 The parameter uncertainty has been normalized; that is, that meets We need to design a dynamic compensator the following specifications: • settling time of 0.2 s with minimal overshoot and zero in resteady-state error for the vertical acceleration sponse to a step command; • adequate high-frequency rolloff for noise attenuation and to withstand neglected dynamics and flexible modes; • maximum deflection of two (in normalized units) imposed ; on the control signal • time-domain specifications must be met over the uncer tainty range To attack this problem, we use the feedback structure denotes the reference accelerasketched in Fig Here, denote the weighted tracking error and tion signal, and control input, respectively An integrator is introduced in the acceleration channel to enforce zero steady-state error The full compensator is therefore given by To incorporate bounds on the size of unmodeled dynamics and penalize tracking error, we use the weighting filters (see [31]): VI DESIGN EXAMPLE This section illustrates the benefits of robust pole placement in LMI regions through a missile autopilot design example The problem setup comes from [31] and [21], where additional motivations and details can be found It has been slightly simplified to focus on aspects relevant to the technique proposed in this paper First, we perform a standard -optimal design in which gain between the inputs we minimize the closed-loop and the outputs This process is meant to enforce high-frequency rolloff as well as stability and performance for all admissible uncertainties The step controller are depicted in responses of the resulting (pure) (34) 2266 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 Fig Synthesis structure Fig LMI region Fig for (nominal) and (perturbed) Although this first design could be deemed acceptable, it suffers from up to 30% overshoot in the perturbed transient responses To improve transient behavior, we add a robust pole clustering constraint to achieve better damping across the uncertainty range Specifically, we require robust pole clustering in the LMI region represented in Fig This region is defined as the intersection of the following: • disk with center zero and radius 1500 (to prevent fast dynamics); and angle • shifted conic sector with apex at Its characteristic function is and the corresponding and Fig Pure H1 design—nominal and perturbed (1 = 61) step responses TABLE II CONTROLLER ZERO-POLE-GAIN DESCRIPTION matrices are This particular region is chosen to achieve differential damping at low and high frequency (the damping constraint takes effect ) Because the constraint already enforces for closed-loop stability, it is inconsequential that this LMI region intersects the right half-plane The resulting synthesis problem is multi-objective because norm subject to it involves minimizing the closed-loop robust pole clustering in the selected region In the LMI framework, this problem is attacked by minimizing the closedloop gain subject to the LMI constraints of Theorem 5.1 for robust pole clustering and the LMI constraint (34) for the ” (see Section V-A for details) constraint “closed-loop gain This LMI optimization problem was solved with [14] and produced the compensator with zero-pole-gain description in Table II The corresponding step responses are shown in Fig The transients are smoother control for both nominal than those obtained with pure and perturbed plants More importantly, thanks to the disk constraint, this result is achieved with significantly slower CHILALI et al.: POLE PLACEMENT IN LMI REGIONS 2267 APPENDIX A Proof of Theorem 3.3: Assume without loss of is always invertible because generality First, observe (10) implies which, from and the properties of eigenvalues and hence of Kronecker products, secures for all admissible By definition, quadratic -stability holds if and only if exists such that, for all satisfying Fig Final design—nominal and perturbed (1 = 61) step responses Equivalently, the inequality (35) should hold for any nonzero vector and admissible , this result amounts to requiring For fixed whenever (36) Fig Rolloff in final design controller dynamics Indeed, the fastest mode in the pure controller is , whereas it is only in the multi-objective controller These improvements are secured performance because without tangible degradation of the both designs have nearly optimal performance Finally, Fig shows the final controller has adequate rolloff properties is the unique solution where Observing of the equation (37) An equivalent and simpler characterization of is Now, any ensures, for together with matrix VII CONCLUSION Tractable analysis and synthesis techniques have been derived for robust pole placement in LMI regions For analysis with unstructured uncertainties, the Bounded Real Lemma characterization of quadratic stability has been generalized to pole clustering in arbitrary LMI regions For parameter uncertainty, two robust -stability tests have been derived that rely on scaling and multiconvexity techniques Although both tests provide only sufficient conditions, they have proven sharp in a number of applications Finally, we have proposed a tractable LMI-based approach to the synthesis of outputfeedback controllers that robustly assign the closed-loop poles in a prescribed LMI region Combination of robust pole assignment with other closed-loop design specifications has also been discussed Consequently, a sufficient condition for (36) to hold is that whenever (38) or, equivalently, from the expression (37) for whenever 2268 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 Using a standard -procedure argument [36], [12], this condition is in turn equivalent (up to rescaling ) to the single LMI constraint inequalities (40) are collectively equivalent to (39) Finally, a Schur complement with respect to the block (3,3) of (10) shows the equivalence between (39) and (10) Remark A.1: Theorem 3.3 is nonconservative when has rank one Indeed, is then characterized and coincides with the set of by the relation for some vectors such that Because the gap between these two sets is the only source of conservatism in the proof, the LMI constraints (10), (11) then become necessary and sufficient for quadratic -stability against complex unstructured uncertainty No scaling is needed incurs no conservatism in this case; i.e., setting whenever (42) Now, it is easily shown that APPENDIX B Proof of Theorem 4.1: First, observe for valued function , the matrix- is concave in As a result, (22) holds for all if it holds at the vertices of To establish well posedness, i.e, invertibility of for all , suppose is singular for some , and such that consider • Premultiplying and postmultiplying (21) by the full-rank and its transpose shows the matrix matrix ————– —- — — Using this simpler characterization, we can show any satisfies – —- – —- ———- ———– is negative definite and • Premultiplying and postmultiplying (22) by its transpose shows the same matrix is positive definite, a contradiction The remainder of the proof parallels that of Theorem 3.3 -stability for the uncertain system (17) is guaranteed if satisfies, for all and Using the expressions of direct calculations show, for fixed with the notation —- – ——— —– — —- ———- ——– —– (40) (simply premultiply and postmultiply (22) by transpose) Consequently (41) (43) holds and , , the -dependent and a sufficient condition for (40) is (42) with (43) and its replaced CHILALI et al.: POLE PLACEMENT IN LMI REGIONS by , the latter being equivalent to (21) by a standard procedure argument [36], [12] To complete the proof, we need to show (41) holds for all Suppose for some and nonzero Then, the left-hand side in (40) evaluates to zero when setting , a contradiction Hence, cannot be singular over , which together with guarantees (41) APPENDIX C 2269 [12] A L Fradkov and V A Yakubovich, “The S -procedure and duality relations in nonconvex problems of quadratic programming,” Vestnik Leningrad Univ Math., vol 6, pp 101–109, 1979 (In Russian, 1973) [13] P Gahinet, P Apkarian, and M Chilali, “Parameter-dependent Lyapunov functions for real parametric uncertainty,” IEEE Trans Automat Contr., vol 41, pp 436–442, Mar 1996 [14] P Gahinet, A Nemirovski, A J Laub, and M Chilali, The LMI Control Toolbox Natick, MA: The MathWorks, 1995 [15] G Garcia, J Daafouz, and J Bernussou, “Output feedback disk pole assignment for systems with positive real uncertainty,” IEEE Trans Automat Contr., vol 41, no 9, pp 1385–1391, 1996 [16] G Garcia and J Bernussou, “Pole assignment for uncertain systems in a specified disk by state feedback,” IEEE Trans Automat Contr., vol 40, pp 184–190, 1995 [17] A Graham, Kronecker Product and Matrix Calculus with Applications Chichester, U.K.: Ellis Horwood, 1981 [18] S Gutman and E I Jury, “A general theory for matrix root clustering in subregions of the complex plan,” IEEE Trans Automat Contr., vol AC-26, pp 853–863, 1981 [19] W M Haddad and D S Bernstein, “Parameter-dependent Lyapunov functions, constant real parameter uncertainty, and the Popov criterion in robust analysis and synthesis,” in Proc CDC, Brighton, U.K., Dec 1991, pp 2274–2279, pp 2632–2633 , “Controller design with regional pole constraints,” IEEE Trans [20] Automat Contr., vol 37, pp 54–61, 1992 [21] Y Huang and W Lu, “An LFT approach to autopilot design for missiles,” in Proc Amer Contr Conf., June 1996, pp 2990–2994 [22] Y T Juang, “A fundamental multivariable robustness theorem for robust eigenvalue assignment,” IEEE Trans Automat Contr., vol 25, pp 588–591, 1988 [23] J Kautsky, N K Nichols, and P Van Dooren, “Robust pole assignment in linear state feedback,” Int J Contr., vol 41, pp 1129–1155, 1985 [24] P P Khargonekar, I R Petersen, and K Zhou, “Robust stabilization of uncertain linear systems: Quadratic stabilizability and control theory,” IEEE Trans Automat Contr., vol 35, pp 356–361, 1990 [25] J M Martin, “State space measures of robustness of pole locations for structured and unstructured perturbations,” Syst Contr Lett., vol 16, pp 423–433, 1991 [26] I Masubuchi, A Ohara, and N Suda, “LMI-based controller synthesis: A unified formulation and solution,” Int J Robust Nonlin Contr., submitted for publication [27] A Megretski and A Rantzer, “System analysis via integral quadratic constraints,” IEEE Trans Automat Contr., vol 42, pp 819–830, 1997 [28] A S Milfont, S A A Filho, and J C Geromel, “ robust control design with regional pole placement for discrete time linear systems,” in Proc 3rd Eur Contr Conf., ECC’95, pp 3388–3393 [29] A Nemirovskii and P Gahinet, “The projective method for solving linear matrix inequalities,” Math Programming Series B, vol 77, pp 163–190, 1997 [30] Y Nesterov and A Nemirovskii, “Interior point polynomial algorithms in convex programming: Theory and applications,” SIAM Studies Appl Math., vol 13, 1994 [31] R T Reichert, “Robust autopilot design using -synthesis,” in Proc Amer Contr Conf., May 1990, pp 2368–2373 [32] C Scherer, P Gahinet, and M Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans Automat Contr., vol 42, pp 896–911, 1997 [33] C W Scherer, “Trends in control: A European perspective,” in Volume of the Special Contributions to the ECC’95, A Isidori, Ed New York: Springer-Verlag, 1995, pp 173–216 [34] K M Sobel, W Yu, J E Piou, J R Cloutier, and R Wilson, “Robust eigenstructure assignment with structured state-space uncertainty,” Int J Syst Sci., vol 23, pp 765–788, 1992 [35] L Vandenberghe and S Boyd, “Primal-dual potential reduction method for problems involving matrix inequalities,” Math Programming Series B, vol 69, pp 205–236, 1995 [36] V A Yakubovich, “The S -procedure in nonlinear control theory,” Vestnik Leningrad Univ Math., vol 4, pp 73–93, 1977 (In Russian, 1971) H1 H REFERENCES [1] J Ackermann, Robust Control: Systems with Uncertain Physical Parameters London, U.K.: Springer-Verlag, 1993 [2] F Alizadeh, J.-P Haeberly, M V Nayakkankuppam, M L Overton, and S Schmieta, SDPpack Version 0.9 Beta for Matlab 5.0—Semidefinite-Quadratic-Linearly Constrained Programs, 1997 [3] D Arzelier, J Bernussou, and G Garcia, “Pole assignment of linear uncertain systems in a sector via a Lyapunov-type approach,” IEEE Trans Automat Contr., vol 38, pp 1128–1131, 1993 [4] B R Barmish, New Tools for Robustness of Linear Systems New York: Macmillan, 1994 [5] S P Bhattacharyya, Robust Stabilization Against Structured Perturbations Lecture Notes Contr Inf Sci., vol 99 New York: SpringerVerlag, 1987 [6] S Boyd and L El Ghaoui, “Method of centers for minimizing generalized eigenvalues,” Lin Alg Applicat., pp 63–111, 1993 [7] S Boyd, L El Ghaoui, E Feron, and V Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, vol 15 Philadelphia: SIAM, 1994 [8] R Y Chiang and M G Safonov, “ synthesis using a bilinear pole shifting transform,” AIAA J Guid Contr Dyn., vol 15, pp 1111–1117, Sept 1992 [9] M Chilali and P Gahinet, “ design with pole placement constraints: An LMI approach,” IEEE Trans Automat Contr., pp 358–367, 1996 [10] J H Chou, S J Ho, and I R Horng, “Pole assignment robustness in a specified disc.,” Syst Contr Lett., vol 16, pp 41–44, 1991 [11] E Feron, P Apkarian, and P Gahinet, “Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions,” IEEE Trans Automat Contr., vol 41, pp 1041–1046, July 1996 H1 H1 Mahmoud Chilali, photograph and biography not available at the time of publication 2270 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 44, NO 12, DECEMBER 1999 Pascal Gahinet received the degree from the Ecole Polytechnique in 1984 and from the E.N.S.T.A., Paris, France, in 1986, and the M.S degree in electrical and computer engineering and the Ph.D degree in system theory from the University of California, Santa Barbara, in 1987 and 1989, respectively He was with the French National Research Institute in Computer and Control Sciences (I.N.R.I.A.) from 1990 to 1995, taught control at E.N.S.T.A from 1991 to 1994, and served as a Consultant for THOMSON-CSF and A´erospatiale during this period He has been with The MathWorks, Inc., since April 1996 and is currently Senior Project Leader for the control and advanced control product developments His research interests include robust control theory, linear matrix inequalities, numerical linear algebra, and numerical software for control He is coauthor of the LMI Control Toolbox and the Control System Toolbox for use with MATLAB Pierre Apkarian (A’94) received the degree in engineering from the Ecole Sup´erieure d’Informatique, Electronique, Automatique, Paris, in 1985, the M.S degree and “Diplˆome d’Etudes Appronfondies” in mathematics from the University of Paris VII, in 1985 and 1986, and the Ph.D degree in control engineering from the “Ecole Nationale Sup´erieure de I’A´eronautique et de Espace” (ENSAE) in 1988 He has been a Research Scientist at ONERACERT and Associate Professor at ENSAE since 1988 His research interests include robust and gainscheduling control theory, linear matrix inequality techniques, mathematical programming, and applications in aeronautics Dr Apkarian is a member of SIAM [...]... generalized to pole clustering in arbitrary LMI regions For parameter uncertainty, two robust -stability tests have been derived that rely on scaling and multiconvexity techniques Although both tests provide only sufficient conditions, they have proven sharp in a number of applications Finally, we have proposed a tractable LMI- based approach to the synthesis of outputfeedback controllers that robustly assign... method for solving linear matrix inequalities,” Math Programming Series B, vol 77, pp 163–190, 1997 [30] Y Nesterov and A Nemirovskii, “Interior point polynomial algorithms in convex programming: Theory and applications,” SIAM Studies Appl Math., vol 13, 1994 [31] R T Reichert, Robust autopilot design using -synthesis,” in Proc Amer Contr Conf., May 1990, pp 2368–2373 [32] C Scherer, P Gahinet, and M... Notes Contr Inf Sci., vol 99 New York: SpringerVerlag, 1987 [6] S Boyd and L El Ghaoui, “Method of centers for minimizing generalized eigenvalues,” Lin Alg Applicat., pp 63–111, 1993 [7] S Boyd, L El Ghaoui, E Feron, and V Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, vol 15 Philadelphia: SIAM, 1994 [8] R Y Chiang and M G Safonov, “ synthesis using a bilinear pole shifting transform,”... Matlab 5.0—Semidefinite-Quadratic-Linearly Constrained Programs, 1997 [3] D Arzelier, J Bernussou, and G Garcia, Pole assignment of linear uncertain systems in a sector via a Lyapunov-type approach,” IEEE Trans Automat Contr., vol 38, pp 1128–1131, 1993 [4] B R Barmish, New Tools for Robustness of Linear Systems New York: Macmillan, 1994 [5] S P Bhattacharyya, Robust Stabilization Against Structured... potential reduction method for problems involving matrix inequalities,” Math Programming Series B, vol 69, pp 205–236, 1995 [36] V A Yakubovich, “The S -procedure in nonlinear control theory,” Vestnik Leningrad Univ Math., vol 4, pp 73–93, 1977 (In Russian, 1971) H1 H REFERENCES [1] J Ackermann, Robust Control: Systems with Uncertain Physical Parameters London, U.K.: Springer-Verlag, 1993 [2] F Alizadeh,... Apkarian (A’94) received the degree in engineering from the Ecole Sup´erieure d’Informatique, Electronique, Automatique, Paris, in 1985, the M.S degree and “Diplˆome d’Etudes Appronfondies” in mathematics from the University of Paris VII, in 1985 and 1986, and the Ph.D degree in control engineering from the “Ecole Nationale Sup´erieure de I’A´eronautique et de Espace” (ENSAE) in 1988 He has been a Research...CHILALI et al.: POLE PLACEMENT IN LMI REGIONS 2267 APPENDIX A Proof of Theorem 3.3: Assume without loss of is always invertible because generality First, observe (10) implies which, from and the properties of eigenvalues and hence of Kronecker products, secures for all admissible By definition, quadratic -stability holds if and only if exists such that, for all satisfying Fig 8 Final design—nominal and perturbed... source of conservatism in the proof, the LMI constraints (10), (11) then become necessary and sufficient for quadratic -stability against complex unstructured uncertainty No scaling is needed incurs no conservatism in this case; i.e., setting whenever (42) Now, it is easily shown that APPENDIX B Proof of Theorem 4.1: First, observe for valued function , the matrix- is concave in As a result, (22) holds... root clustering in subregions of the complex plan,” IEEE Trans Automat Contr., vol AC-26, pp 853–863, 1981 [19] W M Haddad and D S Bernstein, “Parameter-dependent Lyapunov functions, constant real parameter uncertainty, and the Popov criterion in robust analysis and synthesis,” in Proc CDC, Brighton, U.K., Dec 1991, pp 2274–2279, pp 2632–2633 , “Controller design with regional pole constraints,” IEEE... pp 1111–1117, Sept 1992 [9] M Chilali and P Gahinet, “ design with pole placement constraints: An LMI approach,” IEEE Trans Automat Contr., pp 358–367, 1996 [10] J H Chou, S J Ho, and I R Horng, Pole assignment robustness in a specified disc.,” Syst Contr Lett., vol 16, pp 41–44, 1991 [11] E Feron, P Apkarian, and P Gahinet, “Analysis and synthesis of robust control systems via parameter-dependent ... robust pole clustering constraint to achieve better damping across the uncertainty range Specifically, we require robust pole clustering in the LMI region represented in Fig This region is defined... complexity of the LMI region For LMI regions that are intersections of elementary LMI regions , sharper tests can be obtained by using independent Lyapunov for each as indicated in Section III-C... • conic sector with apex at the origin and inner angle : Key facts about LMI regions include [9] the following • Intersections of LMI regions are LMI regions • Any convex region symmetric with