## A cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems (1997)

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Venue: | IEEE Transaction on Automatic Control |

Citations: | 35 - 0 self |

### BibTeX

@ARTICLE{Ghaoui97acone,

author = {Laurent El Ghaoui and Francois Oustry and Mustapha Aitrami},

title = {A cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems},

journal = {IEEE Transaction on Automatic Control},

year = {1997},

pages = {1171--1176}

}

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### Abstract

Abstract—This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, ny) independent inputs (respectively, outputs). The algorithm is based on a “cone complementarity ” formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(nu;ny), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura’s generic stabilizability result (m n0nu0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis. Index Terms — Complementarity problem, linear matrix inequality, reduced-order stabilization, static output feedback. I.

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Citation Context ...) is n. Problem (4) can be called a “cone complementarity” problem (CCP), as it is an extension linear complementarity problems (LCP’s) to the cone of positive semidefinite matrices; see, e.g., [26], =-=[16]-=-, and [27]. To solve such a problem, a linearization method (originally proposed by Franke and Wolfe and described in [16] for LCP’s) can be used. At a given point (X0;S0), a linear approximation of T... |

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Citation Context ...k+1 and go to Step 2). The first step of the algorithm and every Step 2) are simple LMI problems. There are many algorithms that are available for this, especially interior-point methods, e.g., [24], =-=[4]-=-, and [17]. In the sequel, we count each Step 2) as one outer iteration to make a distinction with the inner steps required to solve the LMI problems Pk. The following theorem shows that the algorithm... |

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Citation Context ...e ROF problem is still open. The complexity analysis of the problem is not quite complete yet. Anderson et al. have shown in [1] that the problem is decidable. A nice result of Blondel and Tsitsiklis =-=[3]-=- shows that the problem of finding an SOF controller with prespecified bounds on the controller matrix K is NP-hard. Of course, this does not prove that the SOF problem is NP-hard, since no a priori b... |

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Citation Context ...oblem (4) can be called a “cone complementarity” problem (CCP), as it is an extension linear complementarity problems (LCP’s) to the cone of positive semidefinite matrices; see, e.g., [26], [16], and =-=[27]-=-. To solve such a problem, a linearization method (originally proposed by Franke and Wolfe and described in [16] for LCP’s) can be used. At a given point (X0;S0), a linear approximation of TrXS takes ... |

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Citation Context ...mentation technique (see Section II-D). Despite its apparent simplicity, the ROF problem is still open. The complexity analysis of the problem is not quite complete yet. Anderson et al. have shown in =-=[1]-=- that the problem is decidable. A nice result of Blondel and Tsitsiklis [3] shows that the problem of finding an SOF controller with prespecified bounds on the controller matrix K is NP-hard. Of cours... |

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Citation Context ...ITY (BY A FULL-ORDER CONTROLLER) OF (A; B; C) IV. NUMERICAL EXPERIMENTS For every run, we have chosen (unless otherwise stated) M =10 5 ; =0:01; =2 . We have used the semidefinite programming code SP =-=[23]-=- and a MATLAB interface to SP, LMITOOL [9]. The SP parameters for absolute and relative convergence were both set to 10 010 . The random tests presented next are based on generic stabilizability resul... |

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Citation Context ...r value) of the matrix X, and X > 0(respectively, X 0) means X is symmetric and positive-definite (respectively, positive semidefinite). II. AN LMI-BASED ALGORITHM A. LMI-Based Formulation As seen in =-=[8]-=-, there exists an mth-order state-feedback stabilizing controller if and only if there exist matrices X; S; and scalar >0 such that B?(AX + XA T +2 X)B T ? 0 C T ?(A T S + SA +2 S)C? 0 K(X; S) = X I I... |

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Citation Context ...( ~ A; ~ B; ~ C) as in (6) and ~X = X R R T I Then, ~ X > 0and the LMI problem in K ( ~ A + ~ BK ~ C) ~ X + ~ X( ~ A + ~ BK ~ C) T +2 ~ X 0 (9) has a solution. Alternatively, the analytic formulas of =-=[13]-=- can be used to reconstruct K. : Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on October 31, 2008 at 18:59 from IEEE Xplore. Restrictions apply.IEEE TRANSACTIONS ON AUTOMATI... |

21 | Static output feedback – A survey
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Citation Context ...OF problem is NP-hard, since no a priori bounds on the controller matrix K are imposed. A number of numerical procedures have been proposed for solving the problem. A survey was done by Syrmos et al. =-=[22]-=-, and recent progress has been made for the related problem of pole placement; see [25], [19], and the references therein. Among the recent approaches that have been proposed, those based on linear ma... |

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Citation Context ...C) IV. NUMERICAL EXPERIMENTS For every run, we have chosen (unless otherwise stated) M =10 5 ; =0:01; =2 . We have used the semidefinite programming code SP [23] and a MATLAB interface to SP, LMITOOL =-=[9]-=-. The SP parameters for absolute and relative convergence were both set to 10 010 . The random tests presented next are based on generic stabilizability results given by Kimura [15]. In the sequel, we... |

14 |
A unified approach to linear control design
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Citation Context ... that have been proposed, those based on linear matrix inequalities (LMI’s) are promising since the same framework can be used for related problems such as robust control (see Section III). (The book =-=[21]-=- describes a large number of Manuscript received April 1, 1996; revised August 20, 1996. This work was supported by DRET under scholarship and under Grant #92017 BC 14. The authors are with the Labora... |

13 |
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Citation Context ... A number of numerical procedures have been proposed for solving the problem. A survey was done by Syrmos et al. [22], and recent progress has been made for the related problem of pole placement; see =-=[25]-=-, [19], and the references therein. Among the recent approaches that have been proposed, those based on linear matrix inequalities (LMI’s) are promising since the same framework can be used for relate... |

12 | Output feedback pole placement with dynamic compensators
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Citation Context ...ber of numerical procedures have been proposed for solving the problem. A survey was done by Syrmos et al. [22], and recent progress has been made for the related problem of pole placement; see [25], =-=[19]-=-, and the references therein. Among the recent approaches that have been proposed, those based on linear matrix inequalities (LMI’s) are promising since the same framework can be used for related prob... |

11 |
Algorithms for analysis and design of robust controllers
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Citation Context ...MI-based methods include the D-K iteration method mentioned in [20], the alternating projections method [12], the projection method of [18], the min–max algorithm [10], the potential reduction method =-=[6]-=-, and the XY -centering algorithm [14]. This paper describes yet another LMI-based method for the problem. One of the strong points of the proposed algorithm is that it is guaranteed to find a control... |

9 |
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Citation Context ...97)04279-7. (2) such problems.) The LMI-based methods include the D-K iteration method mentioned in [20], the alternating projections method [12], the projection method of [18], the min–max algorithm =-=[10]-=-, the potential reduction method [6], and the XY -centering algorithm [14]. This paper describes yet another LMI-based method for the problem. One of the strong points of the proposed algorithm is tha... |

7 |
Robust Stabilization Against Structured Perturbations
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Citation Context ...ura condition. Our numerical results match the fact that every system is generically stabilizable with a first-order controller. C. Robust Output Feedback Consider a more “realistic” model taken from =-=[2]-=-. The plant matrices are given by A(p1;p2)= 0:0366 :0271 :0188 0:4555 :0482 01:01 :0024 04:0208 :1002 p1 0:707 p2 0 0 1 0 :4422 :1761 Bu(p3)= p3 07:5922 05:52 4:49 0 0 C = [0 1 0 0]: The three uncerta... |

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4 |
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Citation Context ... “cone complementarity” formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(nu;ny), matching a generic stabilizability result of Davison and Chatterjee =-=[7]-=-. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura’s generic stabilizability result (m n0nu0ny+1). A similar alg... |

4 |
H1 robust control by static output feedback
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Citation Context ...Item Identifier S 0018-9286(97)04279-7. (2) such problems.) The LMI-based methods include the D-K iteration method mentioned in [20], the alternating projections method [12], the projection method of =-=[18]-=-, the min–max algorithm [10], the potential reduction method [6], and the XY -centering algorithm [14]. This paper describes yet another LMI-based method for the problem. One of the strong points of t... |

3 |
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Citation Context ... elghaoui@ensta.fr). Publisher Item Identifier S 0018-9286(97)04279-7. (2) such problems.) The LMI-based methods include the D-K iteration method mentioned in [20], the alternating projections method =-=[12]-=-, the projection method of [18], the min–max algorithm [10], the potential reduction method [6], and the XY -centering algorithm [14]. This paper describes yet another LMI-based method for the problem... |

3 |
An alternative to the D-K iteration
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Citation Context ...es Avancées, 75739 Paris, France (e-mail: elghaoui@ensta.fr). Publisher Item Identifier S 0018-9286(97)04279-7. (2) such problems.) The LMI-based methods include the D-K iteration method mentioned in =-=[20]-=-, the alternating projections method [12], the projection method of [18], the min–max algorithm [10], the potential reduction method [6], and the XY -centering algorithm [14]. This paper describes yet... |

2 | The X-Y centering algorithm for the dual LMI problem: a new approach to fixed order control design - Iwasaki, Skelton - 1995 |

1 |
Ait Rami. An LMI-based linearization algorithm for static outputfeedback and related problems
- Ghaoui, Oustry, et al.
- 1995
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Citation Context ...in Algorithm 1 by V = X 01=2 (X 1=2 YX 1=2 ) 1=2 X 01=2 W =V 01 =Y 01=2 (Y 1=2 XY 1=2 ) 1=2 Y 01=2 : In our experiments, we have found that this modified algorithm behaves similarly. For details, see =-=[11]-=-. C. System Augmentation As stated in the introduction, the ROF problem can be addressed as an SOF problem for an augmented system. Define ~A A 0 = ; ~ 0 B B = 0 0m Im 0 ; C ~ 0 Im = : (6) C 0 It can ... |

1 |
Pole placement by gain output feedback
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- 1975
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Citation Context ...face to SP, LMITOOL [9]. The SP parameters for absolute and relative convergence were both set to 10 010 . The random tests presented next are based on generic stabilizability results given by Kimura =-=[15]-=-. In the sequel, we say that (A; B; C) satisfies the mth-order Kimura property if m>n0nu0ny. This property guarantees the existence of an output-feedback stabilizing controller of order m. A. SOF In T... |

1 |
fully polynomial-time algorithm for computing a stationary point for the general linear complementarity problem
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Citation Context ...lem (4) is n. Problem (4) can be called a “cone complementarity” problem (CCP), as it is an extension linear complementarity problems (LCP’s) to the cone of positive semidefinite matrices; see, e.g., =-=[26]-=-, [16], and [27]. To solve such a problem, a linearization method (originally proposed by Franke and Wolfe and described in [16] for LCP’s) can be used. At a given point (X0;S0), a linear approximatio... |

1 |
XY -centering algorithm for the dual LMI problem: A new approach to fixed order control desgin
- “The
- 1995
(Show Context)
Citation Context ...ation method mentioned in [20], the alternating projections method [12], the projection method of [18], the min–max algorithm [10], the potential reduction method [6], and the XY -centering algorithm =-=[14]-=-. This paper describes yet another LMI-based method for the problem. One of the strong points of the proposed algorithm is that it is guaranteed to find a controller with order less than or equal to n... |