## Robust control via sequential semidefinite programming (2002)

Venue: | SIAM J. CONTROL OPTIM |

Citations: | 23 - 8 self |

### BibTeX

@ARTICLE{Fares02robustcontrol,

author = {B. Fares and D. Noll and P. Apkarian},

title = {Robust control via sequential semidefinite programming},

journal = {SIAM J. CONTROL OPTIM},

year = {2002},

volume = {40},

number = {6}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper discusses nonlinear optimization techniques in robust control synthesis, with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality (LMI) constraints in tandem with nonlinear matrix equality constraints. The latter type of constraints renders the design numerically and algorithmically difficult. We solve the optimization problem via sequential semidefinite programming (SSDP), a technique which expands on sequential quadratic programming (SQP) known in nonlinear optimization. Global and fast local convergence properties of SSDP are similar to those of SQP, and SSDP is conveniently implemented with available semidefinite programming (SDP) solvers. Using two test examples, we compare SSDP to the augmented Lagrangian method, another classical scheme in nonlinear optimization, and to an approach using concave optimization.

### Citations

500 |
Constrained Optimization and Lagrange Multiplier Methods
- Bertsekas
- 1982
(Show Context)
Citation Context ...ian more convex than in the original form (D), and so the corrections are often very mild in practice and, according to the theory in polyhedral programming, are not even required asymptotically (cf. =-=[8, 10, 12]-=-. This observation is corroborated in our experiments with LMI constraints. We summarize the result of this section by presenting the following algorithmic approach to the robust gain-scheduling desig... |

473 | Primal-dual interior-point methods for semidefinite programming: convergence results, stability and numerical results
- Alizadeh, Haeberly, et al.
- 1998
(Show Context)
Citation Context ...˜ Φu has maximal rank, while I ⊗ Φu is invertible as soon as Φu has maximal rank. Remark. If either Φu or ˜ Φu is positive definite, we may symmetrize the equality constraint, as considered, e.g., in =-=[15]-=-. As mentioned before, this is typically not possible in the robust synthesis case but may help in different cases. The choice of (T ) is understood by inspecting the necessary optimality conditions, ... |

115 | Sequential quadratic programming
- Boggs, Tolle
- 1995
(Show Context)
Citation Context ...s x +∆x, but in practice a line search using an appropriate merit function is required. For appropriate choices avoiding the Maratos effect, we refer to the vast literature on the subject (see, e.g., =-=[10]-=-, [12]). In order to obtain the Lagrange multiplier updates, we have to inspect the necessary optimality conditions for ( ˜ T ). Let ˜ Λ + ≥ 0 be the Lagrange multiplier matrix variable in ( ˜ T ) ass... |

108 |
Gain Scheduling via Linear Fractional Transformations
- Packard
- 1994
(Show Context)
Citation Context ...orithmically, we keep this part rather cursory, as the individual steps of the method are essentially known. We rely on a recent excellent exposition of the material by Scherer [35] and related texts =-=[29, 1, 21]-=-. We have chosen this problem as our main motivating case study, as it seems to be among the most difficult and numerically demanding cases of the scheme (D). Section 3 aims at practical aspects. We o... |

64 | A linear matrix inequality approach to H∞ control
- Gahinet, Apkarian
- 1994
(Show Context)
Citation Context ...could readily transform this into an LMI plus a nonlinear matrix equality, KCX − W =0, to obtain the program (D). An alternative way to obtain the form (D) is to open the BMI via the projection lemma =-=[18]-=-. This leads to two LMIs, N T B T (AX + XAT )N B T < 0, N T C (YA+ A T Y )NC < 0, in tandem with X = Y −1 . Here N B T , NC are bases for the null spaces of B T , C. With the nonlinear equality constr... |

41 |
Constrained Global Optimization: Algorithms and Applications
- Pardalos, Rosen
- 1987
(Show Context)
Citation Context ...thod. In order to improve its performance, second-order information is at least partially included by approximating the concave second-order term of the objective fc(x) by a linear underestimate (see =-=[30]-=-). This modification improves convergence but still has the inconvenience of a high CPU cost. Altogether, concave methods cannot compete with the SSDP or augmented Lagrangian techniques, as they are v... |

40 | Self-scheduled H∞ control of linear parameter-varying systems: a design example
- Apkarian, Gahinet, et al.
- 1995
(Show Context)
Citation Context ..., and the operator mapping w into z has L2gain bounded by γ for all admissible parameter trajectories Θ ∈ Θ. Proof. The argument is based on a solvability test for quadratic inequalities developed in =-=[34, 1, 2]-=-. The most recent reference is Lemma 10.2 in [35]. This result is used to eliminate the controller variable K from the solvability conditions (6) in Lemma 1. When applying the solvability test, due to... |

31 | A convex characterization of gain-scheduled H∞ controllers
- Apkarian, Gahinet
- 1995
(Show Context)
Citation Context ...orithmically, we keep this part rather cursory, as the individual steps of the method are essentially known. We rely on a recent excellent exposition of the material by Scherer [35] and related texts =-=[29, 1, 21]-=-. We have chosen this problem as our main motivating case study, as it seems to be among the most difficult and numerically demanding cases of the scheme (D). Section 3 aims at practical aspects. We o... |

30 |
Methods for Robust Gain Scheduling
- Helmersson
- 1995
(Show Context)
Citation Context ...gy discrepancy between outputs of the nominal and the reduced plant in response to arbitrary finite-energy input signals (see Figure 1). This approach admits a formulation of the form (D). See, e.g., =-=[21]-=- for more details. The nonlinear constraint B(x) =0 renders problem (D) highly complex and difficult to solve in practice (cf. [13]). Nonetheless, due to its importance, various1794 B. FARES, D. NOLL... |

28 |
Control system synthesis via bilinear matrix inequalities
- Safonov, Goh, et al.
- 1994
(Show Context)
Citation Context ...cable in the more complicated robust design problem we shall present in more detail in section 2. Example 4. Yet another important case is robust control design via generalized Popov multipliers (cf. =-=[33, 28]-=-), also known as km or µ synthesis. Here we encounter a BMI of the form (P + UKV) T S T + S(P + UKV) ≤ 0 to be solved for S and K for given P , U, V . By introducing a slack matrix variable G = SUKV +... |

25 | An Augmented Lagrangian Method for a Class of LMI-Constrained Problems in Robust Control Theory, Int
- Fares, Apkarian, et al.
(Show Context)
Citation Context ...which is known to be numerically difficult, produced unsatisfactory results. This occurs, in particular, when iterations get stalled, which is probably due to the lack of second-order information. In =-=[16]-=-, we therefore proposed a different approach to (D), again based on nonlinear optimization techniques. The augmented Lagrangian method from nonlinear optimization was successfully extended to program ... |

24 |
Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions
- Feron, Apkarian, et al.
- 1996
(Show Context)
Citation Context ...ngs in order to reduce the numerical burden in the design. Remark. Let us address the question of choosing the Lyapunov test function. Although parameter-dependent Lyapunov functions can be used (see =-=[7, 17]-=- for discussions), in the present paper, we shall restrict our attention to the more traditional Q T 2 Qm1798 B. FARES, D. NOLL, AND P. APKARIAN single quadratic Lyapunov approach based on a paramete... |

22 |
Local analysis of Newton-type methods for variational inequalitites and nonlinear programming
- Bonnans
- 1994
(Show Context)
Citation Context ...emerges from the classical SQP method. Local superlinear and quadratic convergence of SSDP is shown in section 5. While several convergence proofs for the SQP method are known in the literature, (cf. =-=[11, 12]-=-), they all seem to depend heavily on the polyhedrality of the classical-order cone, and no extension addressing the semidefinite cone seems available. The proof we present here is fairly general and ... |

21 | Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty
- Gahinet, Apkarian, et al.
- 1996
(Show Context)
Citation Context ...ent to the finite condition ) (14) ( Θui I S T u ) T ( Qu Su Ru )( Θui I ≥ 0 for every i =1,...,N. The proof is in fact a straightforward convexity argument based on Qu < 0 and may be found, e.g., in =-=[19, 35]-=-. This settles the question of finiteness for the uncertain part of (12) at the slight cost of conservatism introduced by assuming Qu < 0. Remark. We mention that in practice it is sufficient to let Θ... |

16 |
An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems
- Leibfritz, Mostafa
- 2002
(Show Context)
Citation Context ...the semidefinite programming (SDP) problem but does not present any numerical evidence as to the practicality of the approach. Theoretical and practical results are presented by Leibfritz and Mostafa =-=[25, 26]-=-, who consider static output feedback control and mixed H2/H∞-control. Our own numerical experiments [3] with interiorpoint methods for robust control design seem to indicate that those are generally ... |

12 | Oustry,Nonsmooth algorithms to solve semidefinite programs
- Lemaréchal, F
- 2000
(Show Context)
Citation Context ...en be required. Remark. A special type of LMI solver which replaces the SDP by an eigenvalue optimization and uses the bundle method from nonsmooth optimization was presented by Lemaréchal and Oustry =-=[27]-=- and was reported to work well for certain large-sizeROBUST CONTROL VIA SSDP 1819 LMI problems. On the other hand, for large-size problems where most SDP solvers are at ill, the direct approach via i... |

11 |
Concave Programming in Control Theory
- Apkarian, Tuan
- 1999
(Show Context)
Citation Context ... [22]). As a result, controllers obtained via such methods are highly questionable and bear the risk of unnecessary conservatism. A new optimization approach to robust control design was initiated in =-=[5]-=-, where the authors showed that reduced-order H∞ control could be cast as a concave minimization problem. It was observed, however, that in a number of cases local concave minimization, which is known... |

11 |
Algorithms for Analysis and Design of Robust Controllers
- David
- 1994
(Show Context)
Citation Context ...e 1). This approach admits a formulation of the form (D). See, e.g., [21] for more details. The nonlinear constraint B(x) =0 renders problem (D) highly complex and difficult to solve in practice (cf. =-=[13]-=-). Nonetheless, due to its importance, various1794 B. FARES, D. NOLL, AND P. APKARIAN heuristics and ad hoc methods have been developed over recent years to obtain suboptimal solutions to (D). Method... |

11 |
A full block S-procedure with applications
- Scherer
- 1997
(Show Context)
Citation Context ..., and the operator mapping w into z has L2gain bounded by γ for all admissible parameter trajectories Θ ∈ Θ. Proof. The argument is based on a solvability test for quadratic inequalities developed in =-=[34, 1, 2]-=-. The most recent reference is Lemma 10.2 in [35]. This result is used to eliminate the controller variable K from the solvability conditions (6) in Lemma 1. When applying the solvability test, due to... |

9 |
Trust region methods for solving the optimal output feedback design problem
- Leibfritz, Mostafa
- 2003
(Show Context)
Citation Context ...the semidefinite programming (SDP) problem but does not present any numerical evidence as to the practicality of the approach. Theoretical and practical results are presented by Leibfritz and Mostafa =-=[25, 26]-=-, who consider static output feedback control and mixed H2/H∞-control. Our own numerical experiments [3] with interiorpoint methods for robust control design seem to indicate that those are generally ... |

8 |
µ-Analysis and Synthesis Toolbox” Users' Guide for Use with
- Balas, Doyle, et al.
- 1993
(Show Context)
Citation Context ...rnatively and iteratively fix parts of the coordinates of the decision vector, x, trying to optimize the remaining indices. The D-K (scaling-controller) iteration procedure is an example of this type =-=[6, 37]-=-, whose popularity may be attributed to the fact that it is conceptually simple and easily implemented as long as the intermediate steps are convex LMI programs. The latter may often be guaranteed thr... |

8 | Coordinate optimization for bi-convex matrix inequalitites
- Helton, Merino
- 1997
(Show Context)
Citation Context ...ecision variables held fixed at each step. However, a major drawback of coordinate descent schemes is that they almost always fail to converge, even for starting points close to a local solution (see =-=[22]-=-). As a result, controllers obtained via such methods are highly questionable and bear the risk of unnecessary conservatism. A new optimization approach to robust control design was initiated in [5], ... |

7 | Measurement-scheduled control for the RTAC problem
- Dussy, Ghaoui
- 1998
(Show Context)
Citation Context ...2 14 2.145 e−005 15 1.217 e−006 128 where ϑ and ˙ ϑ denote the angular position and velocity of the pendulum and Z, Z˙ denote the position and velocity of the cart. We normalize these equations as in =-=[14]-=-: ¨ζ + ε ¨ ϑ cos ϑ = ε ˙ ϑ 2 sin ϑ − ζ +w , ε¨ ζ cos ϑ + ¨ ϑ = u, where [ζ ˙ ζϑ ˙ ϑ] T is the new state vector. We assume θm =cos ϑ is measured, and we express the remaining nonlinear term in the left... |

7 |
Real/Complex Multivariable Stability Margin Computation via Generalized Popov Multiplier { LMI Approach
- Ly, Safonov, et al.
- 1994
(Show Context)
Citation Context ...cable in the more complicated robust design problem we shall present in more detail in section 2. Example 4. Yet another important case is robust control design via generalized Popov multipliers (cf. =-=[33, 28]-=-), also known as km or µ synthesis. Here we encounter a BMI of the form (P + UKV) T S T + S(P + UKV) ≤ 0 to be solved for S and K for given P , U, V . By introducing a slack matrix variable G = SUKV +... |

5 |
Robust mixed control and linear parameter-varying control with full block scalings
- Scherer
- 2000
(Show Context)
Citation Context ... let alone attacked algorithmically, we keep this part rather cursory, as the individual steps of the method are essentially known. We rely on a recent excellent exposition of the material by Scherer =-=[35]-=- and related texts [29, 1, 21]. We have chosen this problem as our main motivating case study, as it seems to be among the most difficult and numerically demanding cases of the scheme (D). Section 3 a... |

4 |
H.D.: A prototype primal-dual LMI interior point algorithm for non-convex robust control problems
- Apkarian, Noll, et al.
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Citation Context ... of the approach. Theoretical and practical results are presented by Leibfritz and Mostafa [25, 26], who consider static output feedback control and mixed H2/H∞-control. Our own numerical experiments =-=[3]-=- with interiorpoint methods for robust control design seem to indicate that those are generally less robust and that the different parameters may be difficult to tune. We emphasize that the method pro... |

4 |
Parameter-dependent control of an under-actuated mechanical system
- Becker
- 1995
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Citation Context ...ngs in order to reduce the numerical burden in the design. Remark. Let us address the question of choosing the Lyapunov test function. Although parameter-dependent Lyapunov functions can be used (see =-=[7, 17]-=- for discussions), in the present paper, we shall restrict our attention to the more traditional Q T 2 Qm1798 B. FARES, D. NOLL, AND P. APKARIAN single quadratic Lyapunov approach based on a paramete... |

4 | Modifying the inertia of matrices arising in optimization
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Citation Context ...s is presently too difficult numerically to become a functional scheme. We therefore have to use the well-known convexification methods used in nonlinear optimization over many years, and we refer to =-=[9, 23]-=- for several such strategies. In our numerical experiments, we tested Powell’s idea of doing a Cholesky factorization, and adding correction terms as soon as negative square roots appear, and a direct... |

4 | A QQP-Minimization Method for Semidefinite and Smooth Nonconvex Programs
- Jarre
- 1999
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Citation Context ...h special emphasis on SSDP since it performed best. The reader might be missing an approach via interior-point techniques—perhaps more in the spirit of the age. In fact, in a different context, Jarre =-=[24]-=- proposes such a method based on the log-barrier function known from the interior-point approach to the semidefinite programming (SDP) problem but does not present any numerical evidence as to the pra... |

4 |
Generalized equations and their solutions. II: Applications to nonlinear programming
- Robinson
- 1982
(Show Context)
Citation Context ..., (3) gI(¯x) ∈ K0 ,λI ∈ K, 〈gI(x), ¯ λI〉 =0. Observe that the existence of ¯ λ is guaranteed under a weak regularity assumption like, for instance, Robinson’s constraint qualification hypothesis (cf. =-=[32]-=-). The Lagrangian associated with (P )is (22) L(x; λ) =f(x)+〈g(x),λ〉 = f(x)+〈gE(x),λE〉 + 〈gI(x),λI〉. We consider Newton’s method for solving the Kuhn–Tucker system (KT), which generates a sequence (x ... |

2 |
H.D.: Robust control via concave minimization—local and global algorithms
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- 2000
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Citation Context ..., a well-implemented primal SDP solver like [20] often outperformed existing primal-dual software, even though the latter is preferred by theory. Let us finally recall an approach to (D) discussed in =-=[4]-=-. Primarily, this scheme is suited for the feasibility problem (find x such that B(x) =0,A(x) ≤ 0) but may be modified to apply to (D). Consider (D) with a nonlinear equality constraint of the form B(... |

1 |
Robust autopilot design using µ-synthesis
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1 | Software for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure, User’s Guide - Wu, Boyd - 1996 |